■^pfvi. 


1 


KA.GM'.niant  IKOl 


y       ^    ^NEW   AMERICAN 

PRACTICAL  NAVIGATOR: 


EPITOME   OF  NAVIGATION; 

CONTAINING    ALL    THE 

TABLES 

NECESSARY  TO  BE  USED  WITH  THE  NAUTICAL  ALMANAC 

IN 

DETERMINING    THE    LATITUDE,    AND    THE    LONGITUDE 
BY    LUNAR   OBSERVATIONS, 

AND 

KEEPING  A   COMPLETE   RECKONING  AT   SEA; 

ILLUSTRATED    BY 

PROPER  RULES   AND   EXAMPLES: 

THE    WHOLE    EXEMPLIFIED    IN    A    JOURNAL, 

KEPT  FROM  BOSTON  TO  MADEIRA, 
IN   WHICH 

ALL  THE  RULES  QF  NAVIGATION  ARE  INTRODUCED  i 

ALSO, 

THE     DEMONSTRATION     OF     THE     USUAL     RULES     OF     TRIGONOMETRY  ;     TROBLiftlS     IB 

MENSURATION,  SURVEYING,  AND    GAUGING  ,    DICTIONARY   OF    SEA    TERMS  ; 

AND    THE    MANNER    OF    PERFORMING   THE    MOST     USEFUL 

EVOLUTIONS    AT    SEA  : 

WITH 


AN    APPENDIX, 


CONTAINING 

METHODS     OF     CALCULATING    ECLIPSES    OF   THE     SUN   AND    MOON,  AND     OCCULTATIONS    OF   THE 

FIXED    STARS ;   RULES    FOR    FINDING    THE    LONGITUDE   OF   A    PLACE    BY   OBSERVATIONS 

OF     ECLIPSES,    OCCULTATIONS,    AND     TRANSITS    OF    THE     MOON'S    LIMB     OVER 

THE     MERIDIAN  ;    ALSO   A    NEW    METHOD     FOR     FINDING    THE 

LATITUDE    BY   TWO   ALTITUDES. 


BY  NATHANIEL  BOWDITCH,  LL.  D. 

Fellow  of  the  Royal  Societies  of  London,  Edinburgh,  and  Dublin;  of  the  Astronomical  Society  in  London:  "j 

(At  American  Philosophical  Society,  h  eld  at  Philadelphia ;  of  the  American  Academy  of  Arts  and 

Sciences;  of  the  Connecticut  Academy  of  Arts  and  Sciences ;  of  the  Literary  and 

Philosophical  Society  of  JWm   York ;  Corresponding  Member  of  the 

Royal  Societies  of  Berlin,  Palermo,  &-c., — and,  since 

his  decease,  continued  by  his  son, 

J.  INGERSOLL  BOWDITCH. 


THIRTIETH     NEW    STEREOTYPE    EDITION. 


NEW-YORK: 
PUBLISHED  BY  E.  &  G.  W.  BLUNT,  PROPRIETORS, 

No.  179  WATER-STREET,  CORNER  OF  BITRLING  SLIP. 

STEREOTYPED  AT  THE 
BOSTON  TYPE  AND  STEREOTYPE  FOUNDRY. 

1861. 


X       .    «NEW   AMERICAN 

PRACTICAI.  NAVIGATOR: 

BEING    AN 

EPITOME   OF  NAVIGATION; 

CONTAINING    ALL    THE 

TABLES 

NECESSARY  TO  BE  USED  WITH  THE  NAUTICAL  ALMANAC 

IN 

DETERMINING    THE    LATITUDE,    AND    THE    LONGITUDE 
BY    LUNAR   OBSERVATIONS, 

AND 

KEEPING  A   COMPLETE   RECKONING  AT   SEA; 

ILLUSTRATED    BY 

PROPER  RULES   AND   EXAMPLES: 

THE    WHOLE    EXEMPLIFIED    IN    A    JOURNAL, 

KEPT  FROM  BOSTON  TO  MADEIRA, 
IN    WHICH 

ALL  THE  RULES  OF  NAVIGATION  ARE  INTRODUCED  i 

ALSO, 

THE     DEMONSTRATION     OF     THE     USUAL     RULES     OF     TRIGONOMETRT ;    FROBLiMS     IS 

MENSURATION,  SURVEYING,  AND    GAUGING  ,    DICTIONARY    OF    SEA    TERMS  *. 

AND    THE    MANNER    OF    PERFORMING    THE    MOST     USEFUL 

EVOLUTIONS    AT    SEA  : 

WITH 

AN    APPENDIX, 

CONTAINING 

METHODS    OF     CALCULATING    ECLIPSES   OF   THE     SUN    AND    MOON,  AND     OCCULTATIONS   OF   THE 

FIXED    stars;   RULES    FOR    FINDING    THE    LONGITUDE   OF   A    PLACE    BY    OBSERVATIONS 

OF     ECLIPSES,    OCCULTATIONS,   AND     TRANSITS   OF    THE     MOON'S    LIMB     OVER 

THE     MERIDIAN;    ALSO   A    NEW    METHOD     FOR     FINDING   THE 

LATITUDE   BY   TWO   ALTITUDES. 


BY  NATHANIEL  BOWDITCH,  LL.  D. 

Fellow  »f  the  Royal  Societies  of  London,  Edinburgh,  and  Dublin;  of  the  Astronomical  Society  in  London;  of 

the  American  Philosophical  Society,  held  at  Philadelphia ;  of  the  Jimerican  Academy  of  Arts  and 

Sciences;  of  the  Connecticut  Academy  of  Arts  and  Sciences;  of  the  Literary  and 

Philosophical  Society  of  J^Tew   York;  Corresponding  Member  of  the 

Royal  Societies  of  Berlin,  Palermo,  &-c., — and,  since 

his  decease,  continued  by  his  son, 

J.  INGERSOLL  BOWDITCH. 


THIRTIETH     NEW    STEREOTYPE    EDITION. 


NEW-YORK: 
PUBLISHED  BY  E.  &  G.  W.  BLUNT,  PROPRIETORS, 

No.  170  WATER-STREET,  CORNER  OF  BURLING  SLIP. 


STEREOTYPED  AT  THE 
BOSTON  TYPE  AND  STEREOTYPE  FOUNDRY. 

1861. 


NOTICE   TO   THE  30th   EDITION. 


Some  corre'ctrbn's  in  tlie  Latitude  and  Longitude  of  points  on  the  coast  of  Cuba 
iia^^e  Ijee.n  made. 
'  TLe  Pole 'Star 'table,  o:  page  206,  has  been  altered  to  correspond  nearly  to  the 
year  1860. 

Table  LV.  has  been  corrected  from  the  "  Tide  Tables  for  the  English  and  Irish 
Ports  for  1860."     From  Mr.  Portales  I  have  received  valuable  aid. 

The  article  on  "  Tides,"  on  pages  120,  &c.,  was  prepared  by  Dr.  Bache. 


Captain  Josiah  Snow,  of  the  ship  Asterion,  informs  me,  that  the  doubtful  shoal 
in  Macassar  Straits,  laid  down  in  4°  50'  S.  and  116°  50'  E.,  is  about  three  miles 
broad  in  the  shoalest  part,  and  bears  N.  W.  by  W.  from  the  North  Seras  seven 
miles ;  in  some  places  there  is  not  over  six  feet  of  water.  By  good  observations  he 
places  it  in  116°  58'  E. 

On  Sahul  Bank,  off  Timor,  in  lat.  10°  50'  S.  and  127°  40'  E.,  he  passed  a  shoal 
or  reef  a  quarter  of  a  mile  long  East  and  West ;  some  of  the  rocks  were  even  with 
the  water's  edge.  Captain  Snow,  in  the  lat.  10°  54^  S.  and  127°  05'  E.,  passed  a 
line  of  breakers  two  miles  long,  running  East  and  West,  and  from  appearances 
thought  there  must  be  many  shoal  spots  in  the '  neighborhood.  He  thinks  nearly 
all  the  islands  East  of  Mindoro  are  laid  down  erroneous,  and  should  not  be  depended 
upon — especially  Semerara,  the  largest,  which  should  be  12  miles  further  north. 


From  Mr.  Daniel  P.  Upton,  I  learn  that,  in  1859,  the  Dutch  bark  "Hoop  von 
Capello"  struck  on  a  sharp  coral  rock,  about  20  feet  long,  in  the  Straits  of  Augier, 
having  16  feet  of  water  on  it.     The  compass  bearings  are — 

The  Cap N.  W.  by  W.  4°  W. 

Point  Tanjong  Lemuing S.  -j  W. 

4  Point  Lio-ht S.  W. 


The  "  Caimsmore"  Rock  is  a  small,  dangerous,  and  very  abrupt  rock,  30  or  40 
feet  in  diameter,  on  which  the  ship  Caimsmore  was  totally  lost  26th  June,  1858. 
Lat.  30°  42'  10"  N.     Long.  122°  34'  40"  E. 

The  geographical  position  of  some  of  the  Shoals,  Rocks,  &c.,  in  the  "  Coral  Sea," 
have  been  corrected  from  the  Surveys  of  1859,  and  new  positions  added  on  p.  451. 

I860.  J.  INGERSOLL  BOWDITOH. 


Entered  according  to  Act  of  Congress,  la  the  year  of  our  Lord  ISSY,  by  E.  &  G.  W.  Blunt, 
in  the  Clerk's  Office  of  the  District  Court  for  the  Southern  District  of  K"ew  York. 


For  new  Nautical  Publications,  &c.,  of  E.  &.  G.  ^Y.  Blunt, 
see  adv'ertibiment  at  the  end. 


Printed  by     Joseph     Russell,  79  John  bx. 
IN  MEMORrAM 


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tPage  1. 


CORRECTIONS  AND  ADDITIONS 


TO    GEOGRAPHICAL   POSITIONS    IN   TABLE    LIV.    OF    BOWDITCH  S    NAVIGATOR, 
EDITION    OF    1851. 

Kindly  furnished  by  Dr.  Bache,  Superintendent  of  the  U.  S.  Coast  Survey,  by  authority  of  the  Treasury  Department, 


J^AME  OF  PLACES. 


Cape  Elizabeth  West  Light 
Cape  Elizabeth  East  Light  , 

Wood  Island  Light 

Agamenticus  Hill 


Plum  Island  East  Light 

Plum  Island  West  Light 

Beverley  Spire 

Ipswich  East  Light 

Ipswich  West  Light 

Squara  Light 

Straits  mouth  Island  Light 

Thatcher's  Island  South  Light 
Thatcher's  Island  North  Light 

Ten  pound  Island  Light 

Eastern  Point  Light 

Eaker's  Island  Light  

Salem,  Tall  Spire 

Marblehead,  Black  Top  Church 

Nahant  Hotel 

BOSTON,  State-House 

Cambridge  Observatory  Dome 

Bunker  Hill  Monument 

Scituate  Light 

Boston  Light 

Long  Island  Light 

Plymouth  Light 

Race  Point  Light  

Cape  Cod  Light 

Long  Point  Light 

WeUfleet  Light 

Billingsgate  Point  Light 

Nausett  Centre  Light.. 

Nausett  South  Liglit 

Chatham  South  Light 

ilouomoy  Light , 

New  Bedford  Light 

Cape  Pogue  Light 

Great  Point  Light 

Brant  Point  Beacon 

Saukaty  Head  Light 

Nantucket  Harbor  Light 

Nantucket  Old  South  Shoal  .... 
Nantucket  Old  South  Shoal  .... 

Davis'  New  South  Shoal 

Fishing  Rip,  5^  fath 

Barnstable  Light 

Point  Gammon  Light 

Edgartown  Light 


43  33.8 
43  33.9 
43  27.4 
43  i3.4 


42 
42 
42 
42 
42 
42 
42 
42 
42 
42 
42 
42 
42 
42 
42 
42 
42 
42 
42 
42 
42 
42 
42 
42 
42 


48.4 
48.4 
43.0 
4i.i 
4i.i 
39.7 
39.7 
38.2 
38.3 
36.1 
34.8 

32.2 
3l  .2 

3o.4 

25.  I 

21  .5 

22.9 

22  .6 
12.3 

19.6 

19.8 

00.2 
o3.7 
02.4 
02 .0 
55.8 
5i.6 
5i.6 
5i.i 
4o.2 
33.5 
35.5 

25.2 

23.4 

17.4 

17.0 

16.4 

o5.5 

o4.2 

58. o 

o3 

07.5 

43.3 

36.5 

23.4 


Longitude, 


D.   M. 

70  1 1 . 8 
70  II .7 
70  19.4 
70  41.2 


70  48.7  / 
70  48.8  ] 
70  52.4 
70  45.6 
70  45.8 
70  40.6 
70  35.0 
70  34.2  ) 
70  34.2  J 
70  39.6 
70  39.5 
70  46.8 
70  53.6 
70  5o.5 

70  54 -O 

71  o3.5 
71  07-4 
71  o3.3 
70  43.6 
70  53. I 
70  57.1. 
70  35.7 
70  i4.3 
70  o3.3 
70  09.8 
70  01 .7 
70  03.9 
69  56.7 
69  56.6 
69  56.6 

69  59.3 

70  53.7 
70  26.7 
70  02.4 

70   o5.2  i, 

69  57.6 

70  o4.4 
69  5o.o 
69  5i .4 
69  5 1 . 5 
69  26 

69  29 

70  16.5 
70  1 5. 6 
70  29.8 


REMARKS. 


Trig.  Point  of  C.  S. 


Newbury  port  Lights. 
Spire  with  Tm-rets. 


Cape  Ann, 


Boston  Bay. 
Gurnet  South  Light. 

Highland  Light. 


Clark's  Point  Light. 
Nantucket. 


Eastern  Spot. 
Western  Spot. 
Nantucket. 


Q95^RQi 


Pa«e  2.] 


TABLE  LIY. 
Corrections  and  Additions. 


^r^ME  OF  PLACES. 


West  Chop  Light 

Nobska  Light  

Bird  Island  Light 

Tarpauhn  Cove  Light 

Ned's  Point  Light 

Pahner's  Ishmd  Light 

New  Bedford,  Baptist  Spire 

Round  Hill  Light 

Cuttyhunk  Light 

Gay  Head  Light 

No  Man's  Land 


Newport  Spire 

Goat  Island  Light 

Nayat  Light 

Warwick  Light 

Wickford  Light 

Providence,  Baptist  Church 

Dutch  Island  Light 

Beavertail  Light 

Point  Judith  Light 

Watchill  Light  

Block  Island  Lie;ht 


Stonington  Light 

Mystic  Light 

Say  brook  Light 

Little  Gull  Island  Light. 

New  London  Light 

Falkner's  Island  Liglit  . . 

New  Haven  Light 

Stratford  Point  Light.... 

Black  Rock  Light , 

Sheffield  Island  Light  .., 
Captain's  Island  Light.. 


Latitude. 


D. 

4i 
4i 
4i 
4i 
4i 
4i 
4i 
4i 
4i 
4i 
4i 


M. 
28.9 
30.9 

4o.i 
28.1 
39.0 
37.6 
38.2 
32.3 
24.8 
20.9 

l5.2 


Plum  Island  Light 

Montauk  Light  

Cedar  Island  Light , 

Oldfield  Point  Light 

Eaton's  Point  Light 

Sands  Point  Light 

NEW  YORK,  City  Hall.. 

Robin's  Reef  Light 

Navy  Yard,  Flagstaff 

Castle  Garden,  Flagstaff 

Fire  Island  Light 

Prince's  Bay  Light 


4i 
4i 
4i 
4o 
4o 
4o 
4o 
4o 
40 
40 
4o 
40 


19.6 
19.0 
16.2 
12.3 
19.0 
12.7 
14.9 
09.1 
08.5 
02.9 
58.9 


10.4 
04.2 
02.4 
58.6 
57.2 
5i  .9 
42.7 
39.4 
42.0 
42.0 
37.9 
3o.4 


Longitude. 


M. 

35.8 
39.0 
42.7 
45.1 
47-4 
54.2 
55.3 
55.0 
56.7 
49-8 
48.5 


4i 

29.2 

71 

18.5 

4i 

29.6 

71 

19.3 

4i 

43.5 

71 

20.0 

4i 

4o.o 

71 

22.4 

4i 

34.2 

71 

26.0 

4i 

49.6 

71 

24.2 

4i 

29.8 

71 

23.9 

4i 

26.9 

71 

23.6 

4i 

21.6 

71 

28.6 

4i 

18.2 

71 

5i  .2 

4i 

i3.4 

71 

34.2 

71  54.0 

71  59.0 

72  20.3 
72  06.1 
72  o5.i 
72  38.9 

72  53.9 

73  05.9 
73  12.7 
73  24.8 
73  37.1 


72  12.4 

71  5i.i 

72  i5.3 

73  06.8 
73  23.4 

73  43.5 

74  00. 1 
74  o3.6 

73  58.5 

74  00.5 

73  12.8 

74  12.5 


REMARKS. 


Trig.  Point  of  C.  S. 


Rhode  Island  Light 


Brooklyn. 
New  York. 


TABLE  LIV. 
Corrections  and  Additions. 


[Page  3. 


J^^ME  OF  PLACES. 


Longitude, 


REMARKS. 


Sandy  Hook  Light 

Navesink  Light 

Ocean  House  Flagstaff 

Barnegat  Light 

Tucker's  Island  Light.. 

Cohansey  Light 

Egg  Island  Light 

CajDe  May  Light 


PHILADELPHIA,  State-House 


"Wilmington  Light 

Bombay  Hook  Light . 

Mispilion  Light 

Breakwater  Light 

Cape  Henlopen  Light 


Susquehanna  Light 

Turkey  Point  Light 

Baltimore,  Washington  Monument 

Poole's  Island  Light 

North  Point  Lower  Light 

North  Point  Upper  Light 

Bodkin  Light 

Annapolis,  State-House 

Sharp's  Island  Light 

Clay  Island  Light 

Point  Lookout  Light 


National  Observatory 

WASHINGTON  City,  Dome  of  Cap. 


Fog  Point  Light 

Assateague  Light 

Smith's  Point  Light 

I  Watts'  Island  Light 

!  New  Point  Comfort  Light, 

Old  Point  Comfort  Light . . 

Smith's  Island  Light ,. 

Cape  Charles 

1  Cape  Henry  Light 


D.      M. 

4o  27.7 
4o  23.7 
4o  22.8 
39  46.0 
39  3o.3 
39  20.3 
39  10.5 
38  55.8 


39  56.9 


38  53.6 
38  53.3 


D.      M. 

73  59.8 
73  58.8 

73  58.2 

74  06.0 

74  16.8 

75  21 .3 
75  08.0 
74  57.3 


75  08.7 


39  43.3 

75  30.9 

39  21.8 

75  3o.3 

38  56.6 

75  18.5 

38  47-9 

75  06.1 

38  46.6 

75  04.7 

39  32.4 

76  04.8 

39  26,9 

76  00.2 

39  17.8 

76  36.6 

39  17.4 

76  i5.7 

39  II .6 

76  26.2 

39  1 1. 8 

76  27.3 

39  08.0 

76  25.1 

38  58.7 

76  29.1 

38  37.7 

76  22.6 

38  i3.9 

75  58.1 

38  02.3 

76  19.0 

77  02.8 
77  00.2 


38  02.1 

76  02.2 

37  54.6 

75  21 . I 

37  53.2 

76  i4.o 

37  46.9 

75  53.3 

37  18.0 

76  16.4 

37  00.0 

76  18. I 

37  07.8 

75  52.2 

37  07.3 

75  57.9 

36  55.5 

76  00.2 

Southern  Light. 

Little  Egg  Harbor  Light. 


Washingtoa. 


Mouth  of  Potomaa 


Trig.  Point  of  C.  S. 


*■)*. 


Page  4.] 


TABLE   LIV. 
Corrections  and  Additions, 


J^AME  OF  PLACES. 


KOI  Devil  Hill 

Bodies'  Island  Light 

Ne-R'  Inlet,  South  Point... 

Cape  Hatteras  Light 

Ocracoke  Light 

Fort  Pinkney 

Charleston  Light 

SAVAKNAH,  Exchange. 

Cape  Florida  Light 

Key  West  Light 

Sand  Key  Light , 

MobUe,  Barton's  Academy 

Choctaw  Point  Light 

Grant's  Light , 

Mobile  Point  Light , 

Sand  Island  Light 

Biloxi  Light 

Pass  Christian  Light , 

Round  Island  Light , 

Cat  Island  Light 

Ship  Island  West  , 

Chandeleur  Light 

Galveston,  Entrance 

Galveston,  Cathedral 

Point  Lobos 

South  Farallon 

Point  Pinos  

Point  Conception 

Point  Loma 


Latitude. 


D.      M. 

36  01 . 1 
35  47-3 
35  4i.i 
35  i5.2 
35  o6.5 


32  46.4 
32  4i -9 


32  04.9 


3o  23.8 
3o  18.9 
3o  17.5 
3o  13.9 
3o  12.9 


3o  o3.4 


29  20.5 
29  18.3 


3?  47.0 
37  4i.6 
36  38. o 
34  26.9 
32  4o.2 


Longitude. 


25  39.9 

80  o5.o 

24  33.0 

81  47.3 

24  27.2 

81  51.9 

D.      M. 

75  39.7 
75  3i.6 
75  28.5 
75  30.9 
75  58.9 


79  ^^-^ 
79  52.5 


81  o5.2 


3o 

4i 

4 

88 

01 

9 

do 

4o 

2 

88 

01 

I 

3o 

17 

6 

88 

07 

5 

3o 

i3 

8 

88 

00 

5 

3o 

II 

3 

88 

02 

0 

88  53.1 

89  14.0 

88  34.1 

89  08.7 
88  57.0 


5i.8 


94  45.0 
94  47.0 


122  32.0 

122  59.2 

* 

120  25.7 


REMARKS. 


Trig.  Point  of  C.  S. 
Trig.  Point  of  C.  S. 


Charleston. 


Trig.  Point  of  C.  S. 


San  Francisco  Bay. 

Monterey. 

San  Diego  Bay. 


[Page  5. 

Stations  on  tJie  Pacific  Coast ;  determined  astronomically  hy  the  U.  S.  Coast  Survey. 


STATIONS. 


Latitude. 


Longitude. 


San  Diego 

San  Nicolas 

San  Catalina 

San  Pedro 

Prisoner's  Harbor 

Santa  Barbara 

Point  Conception 

San  Luis  Obispo 

San  Simeon 

Point  Pinos 

Santa  Crnz  

Presidio  Hill 

Piinta  de  los  Reyes 

Bodega  Bay 

Havens'  Anchorage 

Mendocino  City 

Shelter  Cove 

Bucksport 

Trinidad  Bay 

Crescent  City 

Telegraph  Hill 

Cuyler's  Harbor 

San  Clemcnte 

Ewing  Harbor 

Uraquah  River 

Cape  Hancock 

Point  Hudson 

False  Dungeness 

Scarborough  Harbor 

Lunimie  Island 

Astor  Point 

Heard's  Islands,  a  new  discovery, .  -j 


32 

33 
33 
32 
2i 
34 
M 
35 
35 
36 
36 
37 
3? 
38 
38 

39 
4o 
4o 
4i 
4i 
3? 


4i  Sy. 

l4  12- 

26  34- 
43  19. 

01  lO^ 
24  24' 

26  56. 
10  37. 
38  24' 
37  59. 
57  26. 
47  36. 
59  34. 
18  20. 

47  57. 
18  06. 
01  i3. 
46  37. 
o3  20. 
AA  44' 

48  06. 


96  N. 

71 

84 

59 

20 

71 
3o 
48 
43 
86 
93 
i5 
20 

37 
87 
16 
67 
09 
o4 
10 
43 


44  21-73 
4i  45-3i 
16  34-85 

07  03-02 

07  52 -03 
21  48-78 
44  01-74 
II  27-61 


17  i3 
19  25 

18  28 

18  16 

19  40 

19  4o 

20  25 

20  43 

21  10 

21  54 

22  00 
22  26 

22  57 

23  02 

23  34 

23  47 

24  o3 

24  10 

24  08 

24  II 

22  23 

20  20 

18  34 

24  28 

24  09 

24  02 

22  44 

23  27 

24  37 

22  4o 

23  49 


25-00  TV 

00-00 

45-00 

o3-oo 

00-00 

18-00 

39-00 

3i  -00 

22-00 

25-00 

10-00 
i5-oo 
4o-io 
28-80 
00-70 
25-65 
02-85 
43 -80 
07.95 
13-95 
19.42 
27-00 

00 -DO 

47 -40 
57-00 
00-81 
33-00 
21.00 

12-00 

37.35 
31.65 


53  o3 
53  00 


73  3o  E. 
72  3o 


Positions  0/  points  in  the  North  Pacific  Ocean.,  prepared  ly  Lieut.  Bent.,  dy  direction 
of  Commodore  Perry.,  commander  of  the  late  expedition  to  Japan. 


NAMES  OF  PLACES. 


Latitude. 


Longitude. 


Formosa. — S.  E.  point 

Islands,  <tc. — Vele  Rete  Rocks 

Agenhue 

Lew  Chew,  Napha 

"      N.  Pt.  Cape  Hope... 

Borodino,  Northern  Island 

"         Southern  Island 

Disappoinlment  or  Rosario  Island 

Bonin  Islands,  Port  Lloyd 

Ponafidin  or  St.  Peters 

Lot's  Wife  (high  rock) 

Eedfield  Rocks,  Northern 

"  "        Southern 

Broughton  Rocks 

Rock  Island 

Japan  Islands. — NirnoN,  Cape  Idzou 

Simoda  (Centre  Island) 

Cape  Sagami 

Webster  Island,  Yedo  Bay 

Treaty  Building,  Yoku-haina. . . . 

Cape  Susaki 

Siriji  Saki,  Northern  point 

Teso Cape  Blunt,  Sangar  Straits 

Hakodadi,  Kamida  Creek 


21  56  00  N. 

120  56  00  E 

21  42  00 

120  49  00 

26  32  00 

127  12  00 

26  12  00 

127  43  00 

26  48  00 

128  16  00 

25  52  00 

i3i  I 3  00 

25  48  00 

i3i  12  00 

27  I 4  00 

i4o  57  00 

27  o5  00 

142  1 5  00 

3o  35  00 

i4o  20  00 

29  47  00 

i4o  22  3o 

33  57  3i 

i38  49  i3 

33  56  i3 

i38  48  3i 

33  43  00 

139  17  00 

34  34  20 

i38  57  10 

3/i  36  o3 

i38  5o  35 

34  39  49 

i38  57  3o 

35  06  3o 

139  42  45 

35  18  3o 

139  4o  34 

35  27  i5 

139  4o  23 

34  55  00 

139  47  00 

4r  23  00 

i4i  3o  00 

4 t  44  00 

i4i  o3  00 

4i  49  00 

i4o  47  45 

GULF  STREAM. 


Under  the  direction  of  Dr.  Bache,  the  Superintendent  of  the  Coast  Survey, 
the  exploration  of  the  Gulf  Stream,  extending  from  about  42°  N.  latitude  to 
about  2SJ°,  and  from  about  C5j°  to  80j°  W.  longitude,  has  been  made— and 
from  his  notes  on  the  same,  we  have  extracted  the  following: — 

The  ocean  within  the  region  of  the  Gulf  Stream  is  divided  into  several 
bands  of  higher  and  lower  temperature,  of  which  the  axis  of  the  Gulf  Stream 
is  the  hottest,  the  temperature  falling  rapidly  inshore  and  more  slowly  outside. 

Thus,  on  a  line  perpendicular  to  the  axis  of  the  stream,  drawn  from  Sandy 
Hook,  the  temperature  at  the  depth  of  15  fathoms  and  100  miles,  was  63°  ;  at 
150  miles,  67°;  at  240  miles,  63i°;  280  miles,  80 J°. 

The  late  Lieut.  G.  M.  Bache  discovered  a  band  of  water  so  much  colder 
than  the  rest  that  he  called  it  the  "  Cold  Wall''  S^^  cold  water  appearing  to 
confine  the  hot  water  as  by  a  wall  on  the  inshore  side.  Its  distance  from  San- 
dy Hook  is  from  230  to  280  miles;  its  distance  from  Cape  May  is  between  132 
and  178  miles:  the  thermometer  at  15  fathoms  on  the  Sandy  Hook  section 
rising  from  62^°  to  80-J°,  or  18°  in  50  miles;  on  the  Cape  May  section  rising 
from  62°  to  83j°,  or  21^°  in  46  miles :  at  Charleston,  at  the  depth  of  20  fath- 
oms, rising  from  67  J°  to  79°  in  15  miles,  and  at  St.  Simons  from  70°  to  76°  in 
12  miles,  being  at  the  rate  of  4^  tenths  to  9  tenths  of  a  degree  to  a  mile.'  Be- 
sides this  remarkable  cold  band  there  are  two  outside  ones,  suflBciently  well 
defined,  though  the  differences  of  temperature  are  less  marked,  the  existence 
of  which  should  be  known  to  the  navigator,  that  he  be  not  perplexed  in  cross- 
ing the  stream  and  finding  warm  water,  to  meet  with  cold,  then  warm  and  then 
cold  again.  The  positions  of  these  bands  may  be  somewhat  changed,  when 
more  thoroughly  considered.  Inside  of  the  "Cold  Wall"  there  is  a  warm  laoid 
and  then  the  cold  water  of  the  shore.  The  axis  of  the  stream  takes,  in  gen- 
eral, the  curve  of  the  coast,  below  rather  than  above  the  water,  being  turned 
to  the  eastward  by  the  shoals  off  the  southern  coast  of  New  England.    The 


GULF    STREAM.  7 

axis  of  the  cold  hand,  the  minimum  of  temperature  which  forms  the  "Cold 
Wall,"  follows  the  shore  and  shoals  in  its  bendings  more  closely  than  the  axis 
of  the  Gulf  Stream;  and  is  traced  with  considei*able  probability  to  longitude 
66°.  The  warm  water  of  the  Gulf  Stream  rests  on  a  cold  current,  flowing  to- 
wards Cape  Florida,  the  coldest  water  keeping  near  the  Atlantic  coast,  below 
the  surface  if  not  at  it.  By  observations  at  several  points  along  the  coast  in 
400  fathoms,  between  Sandy  Hook  and  Cape  Florida,  the  surface  temperature 
exceeding  80°,  the  thermometer  indicated  46^°  to  ^b°',  off  Ilatteras,  in  1000 
fathoms,  40°. 

The  warm  water  of  the  Gulf  Stream  is  of  very  diflfcrent  depths  at  different 
points  of  its  course,  and  in  different  parts  of  any  one  of  the  sections  across  it. 
From  the  deepest  portion  in  the  cross  sections  the  warmer  water  flows  off  to- 
wards the  shore,  and  outwards,  overlying  the  cold.  This  thins  out  as  it  ap- 
proaches the  shore,  the  cold  water  which  lies  at  the  bottom  coming  up  in  the 
northern  sections,  but  the  warm  water  prevailing  to  the  very  shore  and  at  con- 
siderable depths  in  the  southern.  When  the  cold  water  is  forced  up  by  a  bank 
or  shoal,  or  when  it  comes  to  the  surface  from  the  thinning  out  of  the  warm, 
there  is  of  course  a  considerable  change  of  temperature.  This  cold  water  from 
the  north  prevails  on  the  inside  of  the  cold  axis,  at  moderate  depths,  as  far 
south  as  Hatteras,  and  probably  to  the  south  of  it.  Acting  Master  Jones  found 
It,  50  miles  S.  E.  of  Charleston  light,  running  to  the  S.  W.,  the  surface  water 
being  75°,  and  at  20  fathoms  68°,  the  axis  of  the  Gulf  Stream  being  82°,  mo- 
derately warm  water  extending  to  the  bottom. 

The  direction  of  the  axis  of  the  stream  indicates  the  set  of  the  current  in 
that  band.  To  the  right  and  left  of  it,  the  current  is  outward  and  onward,  and 
to  the  left  as  far  as  the  Cold  Wall  is  inward  and  onward.  Inside  of  the  Cold 
Wall,  north  of  Cape  Hatteras,  and  probably  south  of  it,  the  current  is  south- 
erly, along  the  coast. 

The  velocity  of  the  current  in  the  axis  of  the  stream,  on  the  Cape  Canaveral 
section,  is  about  3  miles  per  hour;  on  the  Cape  Fear  section,  about  2  miles 
per  hour,  and  on  the  Sandy  Hook  section,  about  1  mile  per  hour. 

In  the  Charleston  section,  and  to  the  south,  the  bands  of  cold  and  warm 
water,  with  scarcely  an  exception,  are  inodxiced  by  the  sliaj^e  of  the  hottom. 
The  elevated  portions  of  the  bottom,  forcing  up  the  cold  water  into  the  warm, 
cause  cold  streaks,  and  the  division  into  cold  and  warm  bands. 

The  variations  in  temperature  in  different  years  and  at  different  seasons  is 
considerable,  the  more  southerly  sections  in  the  same  season  giving  usually 


8  GULP   STREAM. 

the  highest  temperature.  But  in  July,  1846,  on  the  axis  of  the  Gulf  Stream, 
the  temperature  was  higher  at  Sandy  Hook  than  in  June,  1853,  at  Canaveral, 
by  1  J°,  and  higher  than  at  Charleston  by  5j°. 

The  low  temperatures  observed,  show  that  the  Gulf  Stream  is  comparatively 
a  superficial  current  on  the  surface  of  an  ocean  of  cold  water.  The  tempera- 
tures have  been  observed  from  the  surface  to  the  depth  of  500  fathoms — in  a 
few  instances  as  low  as  13  to  1500  fathoms. 

Navigators  are  advised  to  make  their  observations  at  the  depth  of  20  fath- 
oms. Saxton's  metallic  thermometer  is  highly  recommended.  A  common 
Six's  self-registering  thermometer,  or  a  common  thermometer  enveloped  in 
cotton  or  other  bad  conducting  material,  allowed  to  remain  below  the  surface 
long  enough  to  take  the  temperature,  will  answer. 

Mr.  George  W.  Blunt  in  his  "Atlantic  Memoir"  remarks — 

That  in  summer  the  temperature  of  the  Gulf  water,  south  of  Hatteras,  is 
about  the  same  as  the  water  on  soundings.  In  the  months  of  July  and  August, 
1845,  the  temperature  of  the  water,  from  the  Mississippi  to  Cape  Hatteras, 
both  in  and  out  of  the  stream,  even  to  the  very  mouth  of  the  Atlantic  rivers, 
was  84°  to  82°. 

The  current  on  the  western  edge  of  the  Gulf  Stream,  from  Sandy  Hook  to 
Cape  Hatteras,  sets  south,  a  little  westerly,  about  20  miles  in  24  hours. 

The  current  on  the  eastern  edge  of  the  Gulf  Stream,  nearly  down  to  Mata- 
nilla  Reef,  sets  to  the  south  and  west,  almost  opposite  to  the  flow  of  the  Gulf, 
.it  an  average  of  20  miles  in  24  hours. 

For  further  information  respecting  the  Gulf  Stream,  the  navigator  is  referred 
to  Dr.  Bache's  "Notes"  on  the  same,  in  Blunt's  Coast  Pilot,  and  the  "Memoir 
on  the  dangers  and  ice  in  the  North  Atlantic  Ocean,"  by  G.  W.  Blurt. 


PREFACE. 


In  the  Preface  to  the  first  edition  of  this  work,  it  was  observed,  that  the 
object  of  the  publication  was  to  collect  into  one  volume  all  the  rules,  ex- 
amples, and  tables,  necessary  for  forming  a  complete  system  of  practical 
navigation.  To  do  this,  those  authors  were  consulted  whose  writings  afforded 
the  best  materials  for  the  purpose ;  and  such  additions  and  improvements 
were  introduced  as  were  suggested  \y  a  close  attention  to  the  subject ;  and 
the  accuracy  of  the  tables  accompanying  the  work  was  ensured  by  actually 
going  through  all  the  calculations  necessary  to  a  complete  examination  of 
them,  making  the  last  figure  exact  to  the  nearest  unit.  In  performing  this, 
above  eight  thousand  errors  were  discovered  and  corrected  in  Moore's  Practi- 
cal Navigator,  and  above  two  thousand  in  the  second  edition  of  Maskelyne's 
Requisite  Tables.  Almost  all  the  errors  in  Maskelyne's  collection  were  in 
the  last  decimal  place,  and  in  most  cases  would  but  little  affect  the  result  of 
any  nautical  calculation ;  but  when  it  is  considered  that  most  of  those  tables 
are  useful  in  other  calculations,  where  great  accuracy  is  required,  it  v/ill  not 
be  deemed  an  unnecessary  improvement  to  have  corrected  so  great  a  number 
of  small  errors. 

Several  articles  were  added  in  the  second  edition,  particularly  the  description 
and  use  of  the  circular  instrument  of  reflection,  methods  of  surveying  harbors, 
new  tables,  &lc.  In  the  third,  and  subsequent  editions,  several  improvements 
were  made,  and  an  Appendix  was  given,  containing  methods  of  projecting  and 
calculating  eclipses  of  the  moon  and  sun,  and  occultations  of  the  fixed  stars  or 
alanets  by  the  moon ;  rules  for  deducing  the  longitude  of  a  place  from  obser- 
vations of  eclipses  of  the  sun  or  occultations;  a  new  and  short  method  of 
calculating  the  altitude  and  longitude  of  the  nonagesimal  degree  of  the  ecliptic  ; 
solutions  of  several  useful  problems  of  nautical  astronomy,  and  an  improvement 
of  Napier's  rules  for  the  solution  of  spheric  triangles.  Several  new  tables 
were  added.  The  table  of  latitudes  and  longitudes  was  much  increased  and 
corrected, 

A  new  article  was  given  in  the  sixth  and  seventh  editions,  on  the  method  of 
finding  the  latitudes  by  two  altitudes  of  the  same  or  of  different  objects,  being 


iv  PREFACE. 

an  improvement  of  Mr.  Ivory's  solution.  The  method  we  have  given  is  direct 
and  simple,  embracing  all  the  cases  of  the  problem ;  a  point  which  is  not 
sufficiently  attended  to  in  some  works  of  celebrity.  This  article  is  an  impor 
tant  addition  to  the  work,  and  it  is  recommended  to  the  consideration  of 
navigators. 

The  tables,  published  separately  in  the  Appendix  of  the  first  edition,  are 
introduced  into  the  body  of  this  work,  and  are  extended  so  as  to  render  the 
use  of  them  more  simple.  The  first  method  of  working  a  lunar  observation, 
published  in  that  Appendix,  which  has  one  great  advantage  over  all  other 
approximate  methods,  in  the  manner  of  applying  the  corrections,  (all  of  them 
being  additive,)  is  here  explained  and  illustrated  by  several  examples.  The 
second  is  an  improvement  of  Lyons's  method,  which  had  been  known  for  many 
years,  but  had  not  been  generally  used,  because  the  tables  were  not  sufficient- 
ly extended.  This  difficulty  is  now  obviated,  by  means  of  Tables  XLVII. 
XLVIIL,  which  have  been  compared  with  Thompson's  tables,  and  many  of 
them  recomputed  by  the  aid  of  Shephard's  tables.  The  third  method  was 
given  by  the  author  of  this  work,  in  1795.  The  fourth  method  is  an  improve- 
ment of  Witchell's  process,  in  which,  without  altering  materially  the  calcula- 
tion, the  number  of  cases  is  considerably  reduced. 

Any  person  who  wishes  to  examine  the  tables,  may  do  it  by  the  methods 
used  for  that  purpose,  which  will  here  be  explained,  with  some  additional 
remarks  : 

Tables  I.  and  II.  were  calculated  by  the  natural  sines  taken  from  the  fourth 
edition  of  Sherwin's  logarithms,  which  were  previously  examined,  by  differ- 
ences ;  when  the  proof-sheets  of  the  first  edition  were  examined,  the  numbers 
were  again  calculated  by  the  natural  sines  in  the  second  edition  of  Button's 
logarithms ;  and  if  any  difference  was  found,  the  numbers  were  calculated  a 
third  time  by  Taylor's  logarithms. 

Table  III.  contains  the  meridional  parts  for  every  degree  and  minute  of  the 
quadrant,  calculated  by  the  following  rule,  viz. 

M  =:r  T  X  0.000791 57044G8, 
in  which  T  is  the  log.  tangent  less  radius  of  half  the  latitude,  increased  by  45°, 
taken  to  seven  places  of  figures,  reckoned  as  integers;  and  M  is  the  meridional 
parts  of  that  latitude  in  miles. 

Table  IV.  contains  the  declination  of  the  sun,  which  was  compared  with  the 
Nautical  Almanacs  for  the  years  *1 833,  1834,  1835,  and  1836,  and  marked  to 
the  nearest  minute. 

Table  IV.  A.  The  equation  of  time,  for  the  years  *1833,  1834,  1835,  and 
1836. 

Table  V.  contains  the  correction  of  the  sun's  declination,  as  published  by 
Dr.  Maskelyne.     The  correction  taken  from  this  table  will  rarely  differ  more 

than  sixteen  or  seventeen  seconds  from  the  truth. 

*  Altered  to  correspond  to  the  yeiirs  1848,  1840,  1850,  and  1351. 


PREFACE.  \ 

Table  VI.  contains  the  mean  of  the  sun's  right  ascension,  taken  from  the 
Nautical  Almanacs  for  the  years  1833,  1S34,  1835,  and  1830. 

Table  VI.  A.  contains  the  correction  for  the  daily  variation  of  the  equation 
of  time 

Table  VII.  contains  the  amplitudes  of  the  sun  for  various  latitudes  and 
declinations,  calculated  by  Taylor's  logarithms,  by  this  rule  : 

Log.  scc.lat.-|-log.  sine  declination — lO.OOOOOOOrrlog.  sine  amplitude. 

Table  VI II.  contains  the  right  ascensions  and  declinations  of  one  hundred 
and  eighty  stars  of  the  first,  second,  and  third  magnitudes,  with  their  annual 
variations,  adapted  to  the  beginning  of  the  year  1830.  This  table  was  abridged 
from  that  published  by  the  astronomer  royal  at  Greenwich,  (Mr.  Pond,)  in  the 
year  1833. 

Table  IX.  contains  the  time  of  the  sun's  rising  and  setting,  calculated  by 
Taylor's  logarithms,  by  this  rule  : 

Log.  COS.  hour  =  log.  tang,  declin.-j-  log.  tang.  latitude  — 10.0000000. 

Table  X.  contains  the  distances  at  which  any  object  is  visible  at  sea.  calcu- 
lated by  the  rule  given  in  §  195  of  Vince's  Astronomy,  in  which  the  terrestrial 
refraction  is  noticed.  This  circumstance  was  neglected  by  Robertson. 
Moore,  and  others,  and  of  course  their  tables  are  erroneous.  The  rule  given 
by  Mr.  Vince,  expressed. in  logarithms,  is  this: 

0.12155 -f- half  log.  of  height  in  feet  =  log.  of  dist.  in  statute  miles. 
In    reducing   the   rule    to    logarithms,    the   radius   of  the   earth    was   called 
20911790  feet,   which  agrees  nearly  with  the  mean  value  given  in  De  La 
Lande's  Astronomy. 

Table  XI.  is  a  common  table  of  proportional  parts,  the  construction  of 
which  does  not  need  any  explanation. 

Table  XII.  contains  the  refraction  of  the  heavenly  bodies,  calculated  by 
Dr.  Bradley's  rule,  supposing  the  refraction  to  be  as  the  tangent  of  the  apparent 
zenith  distance  of  the  object,  decreased  by  three  times  the  refraction,  the 
horizontal  refraction  being  supposed  equal  to  33'.  The  rule,  expressed  in 
logarithms,  is  this  : 

Log.  tang.  (app.  zen.  dist. — 3.  refraction) — 8. 2438534= log.  of  ref  in  sec. 
The   numbers  calculated  by  this  rule  agree   nearly  with  those  published  in 
Table  1  of  Maskelyne's  Requisite  Tables. 

Table  XIII.  contains  the  dip  of  the  horizon  for  various  heights,  calculated 
by  the  rule  in  ^  197  of  Vince's  Astronomy,  in  which  the  terrestrial  refraction 
is  allowed  for.  All  the  numbers  of  this  table  differ  a  little  from  those  published 
by  Dr.  Maskelyne,  who  had  made  a  different  allowance  for  that  refraction. 
The  rule  given  by  Mr.  Vince,  expressed  in  logarithms,  is, 

1.7712711  -f-  half  the  log.  of  the  height  in  feet  =  log.  dip  in  seconds. 

Table  XIV.  contains  the  sun's  parallax  in  altitude,  calculated  by  multiplying 


vi  PREFACE. 

the  natural  sine  of  the  apparent  zenith  distance  by  the  sun's  horizontal  parallax 
8f '.     The  numbers  in  this  table  agree  with  those  published  by  Dr.  Maskelyne. 

Table  XV.  contains  the 

Augmentation  of  the  moon's  semi-diameter  =  15".626  X  sine  J)'  s  altitude. 
This  table  agrees  nearly  with  that  published  by  Maskelyne. 

Table  XVI.  contains  the  dip  for  various  distances  and  heights,  calculated  by 
this  rule, 

D  =  -d4-  0.56514  X  -' 

in  which  D  represents  the  dip  in  miles  or  minutes,  d  the  distance  of  the  land 
in  sea  miles,  and  h  the  height  of  the  eye  of  the  observer  in  feet. 

Tables  XVII.,  XVIII.,  and  XIX. ,  were  first  calculated  by  the  author  of  this 
work,  and  published  in  the  Appendix  to  the  first  edition.  The  correction  in 
the  first  of  these  tables  is  equal  to  the  difference  between  the  star's  refraction 
and  CO'.  The  correction  of  Table  XVIII.  is  equal  to  the  diiference  between 
60'  and  the  correction  of  the  sun's  altitude  for  parallax  and  refraction.  The 
correction  of  Table  XIX.  is  equal  to  the  difference  between  59'  42"  and  the 
correction  of  the  moon's  altitude  for  parallax  and  refraction.  The  logarithms 
in  each  of  these  tables  may  be  found  by  adding  together  the  constant  log. 
9.6990,  the  log.  cosine  of  the  apparent  altitude  of  the  object,  the  proportional 
logarithm  of  the  correction  of  the  altitude  of  the  object  for  parallax  and 
refraction,  and  rejecting  20  from  the  index.  The  methods  of  performing  these 
calculations  are  so  obvious,  that  it  is  unnecessary  to  enter  into  any  further 
explanation.  Most  of  the  numbers  in  these  tables  were  calculated  three 
different  times. 

Table  XX.  Corrections  in  seconds,  additive.  This  was  computed  by 
means  of  Shephard's  tables. 

Table  XXL,  for  turning  time  into  degrees,  is  the  same  as  in  other  works  of 
this  kind. 

Table  XXII.  contains  the  proportional  logarithms  for  three  hours.     The 

numbers  of  this  table  may  be  found  by  subtracting  the  logarithm  of  the  time  in 

seconds  from  the  log.  of  10800",  or,  which  is  the  same  thing,  by  the  following 

rule : 

Prop.  log.  T  =  4.0334738  —  log.  of  T  in  seconds, 

neglecting  the  three  right-hand  figures  of  the  remainder. 

Table  XXIII.  was  first  constructed  by  Mr.  Douwes  of  Amsterdam,  about  the 
year  1740,  for  which  he  received  £50  of  the  commissioners  of  longitude  in 
England.  This  table  was  published  in  the  first  and  second  editions  of  the 
Requisite  Tables  ;  in  the  former  of  which  it  was  carried  as  far  as  6  hours  ;  in 
the  latter,  the  table  of  log.  rising  was  extended  to  9  hours ;  in  the  present 
edition  of  this  work,  it  is  extended  to  12  hours.     The  numbers  in  this  table  are 


PREFACE.  VH 

easily  deduced  from  the  log.  sines,  log.  cosecants,  and  log.  versed  sines  of  the 
hour  to  which  they  correspond.  Thus  if  the  time,  opposite  to  any  number  in 
these  tables,  turned  into  degrees,  is  H,  we  shall  have 

Log.  ^  elapsed  time  of  H  =  log.  cosecant  H  —  10.0000000. 
Log.  middle  time  =  log.  sine  H  —  4.C9S9700. 

.  .  I  —  log.  versed  sine  H  — 5.0000000. 

Log.  nsmg  H  |  ^^  x  log.  sine  ^  H-  14.G989700. 
By  means  of  these  formulas,  the  numbers  of  Table  XXIIL  were  calculated  by 
Sherwin's,  Hutton's,  and  Taylor's  logarithms,  and  above  a  thousand  errors 
were  discovered  in  the  second  edition  of  the  Requisite  Tables,  most  of  which 
were  in  the  additional  three  hours  (from  six  to  nine  hours)  not  published  in 
the  first  edition.  About  two  thirds  of  these  additional  numbers  differ  from 
their  true  values  by  one  or  two  units. 

Table  XXIV.  was  compared  with  Sherwin's  and  Hutton's  tables,  and  a  few 
errors  corrected. 

Table  XXV.  contains  the  log.  sines,  log.  tangents,  &c.  corresponding  to 
points  and  quarter  points  of  the  compass.  This  was  compared  with  Sherwin's, 
Hutton's,  and  Taylor's  logarithms. 

Table  XXVI.,  containing  the  common  logarithms  of  numbers,  was  compared 
with  Sherwin's,  Hutton's,  and  Taylor's  logarithms. 

Table  XXVII.  contains  the  common  log.  sines,  tangents,  secants,  &c.  This 
was  compared  with  Sherwin's,  Hutton's,  and  Taylor's  tables.  Two  additional 
columns  are  given  in  this  table,  which  are  very  convenient  in  finding  the  time 
from  an  altitude  of  the  sun;  also,  three  columns  of  proportional  parts  for 
seconds  of  space ;  and  a  small  table  at  the  bottom  of  each  page,  for  finding  the 
proportional  parts  for  seconds  of  time.  The  degrees  are  marked  to  180°, 
which  saves  the  trouble  of  subtracting  the  given  angle  from  180°  when  il 
exceeds  90°. 

Table  XXVIII.  was  calculated  by  proportioning  the  daily  variation  of  the 
time  of  the  moon's  passing  the  meridian. 

Table  XXIX.  contains  the  correction  of  the  moon's  altitude  for  parallax 
and  refraction,  corresponding  to  the  parallax  57'  30". 

Tables  XXX.  and  XXXI.  are  tables  of  proportional  parts,  taken  from  the 
Requisite  Tables,  with  a  few  corrections. 

Table  XXXII.  contains  the  variation  of  the  altitude  of  any  heavenly  body, 
for  one  minute  of  time  from  noon,  for  various  degrees  of  latitude  and  declina- 
tion. The  following  method  was  used  in  constructing  the  table  :  A  and  B 
were  calculated  for  each  degree  of  declination  by  these  formulas; 

Log.  A  =  log.  l".96349  4-2  log.  cos.  declination  — 20.00000, 
Log.  B  =  log.  A  -\-  log.  tang,  declination  —  1 0.00000 ; 
find  then  the  correction  of  the  table  corresponding  to  the  zenith  distance 


viii  PREFACE. 

Z  (  =  lat. '^  dec.)  was  found  by  this  formula:  A  X  cotang.  Z±B.  To 
facilitate  the  computation  of  these  numbers,  a  table  of  the  products  of  A  by 
the  whole  numbers  from  1  to  9  was  calculated. 

Table  XXXIII.  contains  the  squares  of  the  minutes  and  parts  of  a  minute 
of  time  corresponding  to  every  second  from  0"  to  12"  59*.  This  requires 
no  explanation. 

Table  XXXIV.  contains  the  error  of  an  observed  angle  arising  from  a 
deviation  of  1'  in  the  parallelism  of  the  surfaces  of  the  central  mirror,  those 
surfaces  being  supposed  to  be  perpendicular  to  the  plane  of  the  instrument. 
The  correction  in  the  fifth  column  of  this  table  corresponding  to  any  angle  A 
in  the  first  column,  may  be  found  nearly  by  Hutton's  logarithms,  as  follows : 
To  the  constant  logarithm  0.07345  add  the  log.  secant  of  4^  A ;  find  this  in  the 
column  of  log.  tangents,  and  take  out  the  corresponding  natural  secant  B ;  then 
the  correction  will  be  2'  (B  —  1.55.)  The  numbers  in  the  second  column  are 
nearly  equal  to  those  in  the  fifth  corresponding,  o  the  angle  A  -j-  20°,  decreased 
by  1".G8.  The  numbers  in  the  third  column  are  equal  to  the  difference  be- 
tween 1".68,  and  the  numbers  in  the  fifth  corresponding  to  A  20  20°.  The 
numbers  in  the  fourth  column  are  equal  to  the  half-difierence  of  the  numbers 
on  the  same  horizontal  line  in  columns  second  and  third,  when  it  exceeds  40°, 
otherwise  their  half-sum. 

Table  XXXV.  contains  the  correction  to  be  applied  to  an  observation  taken 
in  a  direction  inclined  to  a  plane  of  the  instrument.  The  following  rule  was 
used  in  calculating  this  table :     Find  an  arc  A  such  that 

Log.  sine  A  =  log.  sine  ^  observed  angle  -|- log.  cosine  of  error  of  inclination. 
Then  the  difference  between  2  A  and  the  observed  angle  will  be  the  tabular 
correction. 

Table  XXXVI.  contains  the  variation  of  the  mean  refraction  (given  in 
Table  XII.)  for  various  temperatures  and  densities  of  the  air.  The  correction 
given  in  this  table  is  nearly  the  same  as  that  deduced  from  Dr.  Bradley's  rule, 
which  is  as  follows :  As  the  mean  height  of  the  barometer,  29.6  inches,  is  to 
the  true  height,  so  is  the  mean  refraction  to  the  corrected  refraction ;  and  as 
350,  increased  by  the  height  of  Fahrenheit's  thermometer,  is  to  400,  so  is  the 
corrected  to  the  true  refraction. 

Table  XXXVII.  contains  the  latitudes  and  longitudes  of  the  fixed  stais  of 
the  1st,  2d,  and  3d  magnitudes.  The  nine  stars  from  v/hich  the  distances  are 
marked  in  the  Nautical  Almanac,  are  given  from  the  table  published  in  the 
Nautical  Almanac  for  1820,  allowing  for  10  years'  annual  variation,  to  reduce 
them  to  1830.  The  rest  were  deduced  from  the  table  published  in  the  second 
edition  of  Dr.  Mackay's  treatise  on  longitude,  supposing  the  annual  precession 
50".35,  and  the  secular  equation  as  in  his  table. 

Table  XXXVIII.  was  calculated  by  this  rule  :     Suppose  I-  to  be  the  lati- 


PREFACE,  ix 

tude,  R  the  reduction  of  latitude ;  then  log.  cotang  (L  —  R)  =  0  0029001  -|- 
log.  cotang.  L.  The  reduction  of  parallax  corresponding  to  53',  57',  and  61'. 
was  found  by  the  formulas  respectively, 

5//.3_5'/.3  COS.  2L;  5".7  — 5".7  cos.  2L;  6".l  — 6".l  cos.2L. 
Table  XXXIX.  was  calculated  by  the  rule  in  Vol.  I.,  page  334,  of  Vince's 
Astronomy,  supposing  S  to  be  the  place  of  the  sun,  P  that  of  the  planet,  and 
T  that  of  the  earth : 

Aberration  =  — 20".  cos.  STP--20"  \y/~  cos.  SPT, 

making  use  of  the  distances,  &lc.  given  by  La  Place  in  Vol  III.  of  his  Meca- 
nique  Celeste.  A  small  alteration  was  made  in  the  rule  in  calculating  the 
aberration  of  Mercury. 

Table  XL.  was  calculated  by  —  17".9  sine  long.  3)'s  node. 
Table  XLI.  was  calculated  by — 20".  cos.  argument. 
Table  XLII.  Part  I.    =:  —  19".  173  cos.  arg. 
PartIL  =0".827cos.  arg. 
Part  III.  =  — 3".9814  cos.  arg. 
Table  XLIIL  Part  I.     =  — 8".33  cos.  arg. 
Part  n.  ^— 1".22  cos.  arg. 
Part  III.  =  — 16".382  sine  arg. 
Table  XLIV.  Part  I.     —  8".  1845  sine  arg. 

Part  II    —  ("g-  '"  seconds)^ 

960" 
Part  III.=  960"  X  sine  J)  's  par.  in  lat.  X  tang.  J>  's  true  lat, 
—  960"  X  versed  sine  par.  in  lat. 
If  we  suppose  the  sum  of  these  three  parts  to  be  S  seconds,  and  the  moon's 
horizontal  semi-diameter  to  be  D  minutes. 

Part  IV.  corresponding  to  S  and  D,  will  be  S  X 

^  *=  256 

Table  XLV.     The  arguments  at  the  side  being  B  and  12 — B  hours,  and 

the   second   difference  at  the   top  A,   the  correction   of  this   table   will  be 

288 

Table  XLVI.  gives  the  variation  of  the  altitude  of  any  heavenly  body, 
arising  from  a  change  of  100  seconds  in  the  declination. 

Table  XLVII.  contains  the  proportional  logarithms  as  in  Table  XXII., 
increasing  the  argument  at  the  bottom  of  the  table  by  5°,  and  inverting  the 
order  of  the  numbers. 

Table  XL VIII.  contains  the  third  correction  of  a  lunar  obseivation  in 
Lyons's  improved  method.  These  numbers  may  be  easily  computed  from 
Shephard's  tables,  using  the  moon's  parallax  57'  30",  which  is  nearly  its 
mean  value. 


X  PREFACE. 

Table  XLIX.  For  computing  the  parallax  in  altitude  of  a  planet,  supposing 
its  horizontal  parallax  to  be  35". 

Table  L.  Proportional  parts,  to  reduce  the  numbers  of  Table  XLIX.  to  the 
values  corresponding  to  the  actual  horizontal  parallax  of  a  planet. 

Table  LI.     To  change  mean  solar  time  into  sideral  time. 

Table  LII.     To  change  sideral  time  into  mean  solar  time. 

Table  LIII.  Variation  of  the  compass  in  different  parts  of  the  world, 
deduced  from  Barlow's  chart. 

Table  LIV.  contains  the  latitudes  and  longitudes  of  the  most  remarkable 
ports,  harbors,  &.c.  in  the  world,  from  the  latest  and  best  authorities. 

Table  LV.  contains  the  times  of  high  water  on  the  full  and  change  of 
the  moon,  with  the  vertical  rise  of  the  tide,  at  many  ports,  harbors,  &c.  in  the 
world.  This  table,  (like  the  preceding,)  depending  wholly  on  observations,  is 
therefore  liable  to  be  erroneous,  though  great  pains  have  been  taken  to  make 
it  as  correct  as  possible,  using  for  this  purpose  the  observations  collected  by 
Dr.  Whewell. 

Table  LVI.  Extracts  from  the  Nautical  Almanac  for  the  year  1836,  cor 
responding  to  the  examples  which  are  given  in  this  work. 

The  tables  have  all  been  newly  cast  from  a  clear  and  beautiful  type,  and 
above  ninety  pages  have  been  added  to  the  collection.  Various  improvements 
have  been  made  in  the  body  of  the  work,  which  is  now  for  the  first  time 
completely  stereotyped.  Among  the  additions  made  to  the  work,  may  be 
mentioned  the  description  of  a  portable  transit  instrument,  with  its  uses  in 
regulating  a  chronometer,  and  in  finding  the  longitude  by  observations  of  the 
moon's  transits  over  the  meridian  of  the  place  of  observation ;  methodi  for 
making  allowance  for  any  observed  change  in  the  rate  of  a  chronometer ;  new 
methods  and  improvements  in  the  computation  of  lunar  observations,  &c. 

In  preparing  this  edition,  I  have  been  very  much  assisted  by  my  son, 
J.  Ingersoll  Bowditch,  who  computed  most  of  the  new  tables,  and  care- 
fully examined  those  which  were  taken  from  other  works.  By  associating 
him  with  me,  many  improvements  have  been  made  which  otherwise  would 
not  have  been  introduced. 

N.  BOWDITCEI 

Boston,  October  1,  1837. 


CONTENTS. 


Page 

Signs  and  abbreviations  used  in  this  work xvi 

Decimal  arithmetic 1 

Geometry 4 

Demonstration  of  the  most  useful  propositions  of  geometry 7 

Demonstration  of  tlieorems  in  plane  trigonometry 13 

Geometrical  problems 15 

Construction  of  the  plane  scale 18 

Description  of  Gunter's  scale 20 

Description  and  use  of  the  sliding  rule 23 

Description  and  use  of  the  sector 25 

To  find  the  logarithm  of  any  number,  and  the  contrary 28 

Multiplication  by  logarithms 30 

Division  by  logarithms 31 

Involution  by  logarithms 31 

Evolution  by  logarithms 32 

The  rule  of  three  by  logarithms 32 

To  calculate  compound  interest  by  logarithms 32 

To  find  the  log.  sine,  tangent,  »&c.  corresponding  to  any  number  of  degrees  and  minutes  33 

To  find  the  degrees,  minutes,  and  seconds,  corresponding  to  any  log.  sine,  cosine,  «Sbc.  35 

To  find  the  arithmetical  complement  of  any  logarithm 35 

Plane  trigonometry 36 

Table  of  solutions  of  the  various  cases  of  trigonometry 37 

Right-angled  plane  trigonometry , 38 

Questions  to  exercise  the  learner  in  right-angled  plane  trigonometry 41 

Oblique  trigonometry 41 

A  short  introduction  to  astronomy  and  geography 45 

Explanations  of  the  terms  used  in  astronomy  and  geography 47 

Examples  in  geography 51 

Plane  sailing 52 

A  table  of  the  angles  which  every  point  of  the  compass  makes  with  the  meridian 53 

A  table  of  solutions  of  the  several  cases  of  plane  sailing 53 

Questions  to  exercise  tlie  learner  in  plane  sailing 58 

Traverse  sailing - 59 

Parallel  sailing 63 

Theorems  for  solving  the  several  cases  of  parallel  sailing 63 

A  table  showing  how  many  miles  of  meridian  distance  correspond  to  a  degree  of  longi- 
tude at  every  degree  of  latitude 64 

Questions  to  exercise  the  learner  in  parallel  sailing 65 

Middle  latitude  sailing 66 

Theorems  in  middle  latitude  sailing 67 

Table  of  solutions  of  the  several  cases  of  middle  latitude  sailing 68 

Table  to  correct  the  middle  latitude 76 

Questions  to  exercise  the  learner  in  middle  latitude  sailing 77 

Mercator's  sailing 78 

To  find  the  meridional  parts  corresponding  to  any  degree  and  minute 78 


Xii  CONTENTS. 

Page 

Table  of  solutions  of  the  various  cases  of  Mercator's  sailing 79 

'"■o  work  a  compound  course  by  middle  latitude  or  Mercator's  sailing 86 

Construction  and  use  of  Mercator's  chart 87 

Problems  useful  in  navigation  and  surveying 89 

To  find  the  difference  between  the  true  and  apparent  directions  of  the  wind 97 

To  determine  the  height  of  a  mountain  by  barometers 97 

Mensuration 99  " 

Gauging ■ 103 

Surveying 106 

To  find  the  contents  of  a  field  by  the  table  of  difference  of  latitude  and  departure 107—- 

To  survey  a  coast  in  sailing  along  shore 109 

To  survey  a  harbor  by  observations  on  shore Ill 

Methods  of  surveying  a  small  bank  or  shoal  "where  great  accuracy  is  required 112 

To  reduce  soundings,  taken  at  any  time  of  the  tide,  to  low  water 115-1- 

To  reduce  a  drauglit  to  a  smaller  scale 115 

Of  winds 117 

Directions  for  sailing  from  America  to  India 118 

Tides 120 

To  find  the  time  of  high  water  by  a  Nautical  Almanac 121 

To  find  the  time  of  high  water  by  the  tables  C  and  D 122 

Tables  for  calculating  tlie  time  of  high  water 123 

Currents 124 

Gulf  stream 124 

Of  the  log-line  and  half-minute  glass 126 

Description  and  use  of  a  quadrant  of  reflection  1 28 

To  adjust  a  quadrant 129 

To  take  an  altitude  by  a  fore  observation 130 

To  take  the  sun's  altitude  by  a  back  observation 131 

Advice  to  seamen  in  the  choice  of  a  quadrant 1 31 

Description  and  use  of  a  sextant  of  reflection 133 

To  adjust  a  sextant 134 

To  measure  the  angular  distance  of  the  sun  from  the  moon 135 

To  measure  the  angular  distance  of  the  moon  from  a  sVai     ,   136 

Verification  of  the  mirrors  and  colored  glasses 136 

Description  and  uses  of  the  circle  of  reflection 137 

Adjustments  of  a  circle  of  reflection 133 

To  observe  the  meridian  altitude  of  an  object  by  a  circle 140 

To  measure  the  angular  distance  of  the  sun  from  the  moon  by  a  circle 141 

To  measure  the  angular  distance  of  the  moon  from  a  star  by  a  circle  .    142 

Verification  of  the  mirrors  and  colored  glasses 143 

Description  and  use  of  a  portable  transit  instrument 145 

Adjustments  of  a  transit  instrument 146 

To  observe  the  transit  of  any  heavenly  body  over  the  meridian 150 

Tables  for  correcting  the  adjustments  of  a  transit  instrument 151 

On  parallax,  refraction,  and  dip  of  the  horizon • 153 

To  find  the  distance  of  the  land  in  order  to  calculate  the  dip 155 

To  find  the  sun's  declination 156 

Variation  of  the  compass 158 

To  observe  an  amplitude  or  azimuth  by  a  compass 158 

To  calculate  the  true  amplitude 159 

To  calculate  the  true  azimuth 160 

Questions  to  exercise  the  learner  in  calculating  an  azimuth 160 

Having  tlie  true  and  magnetic  amplitude  or  azimuth,  to  find  the  variation 161 

To  calculate  the  variation  by  azimuths,  observed  at  equal  altitudes,  before  and  after 

passing  the  meridian 161 

Variation  observed 163 

On  the  dip  of  the  magnetic  needle 164 

To  find  the  latitude  by  a  meridian  altitude  of  the  sun  or  a  fixed  star 166 

To  find  the  time  of  the  moon's  passing  the  meridian 170 

To  find  the  moon's  declination 170 


CONTENTS.  xiii 

Page 

To  find  the  latitude  by  the  moon's  meridian  altitude 171 

To  find  the  latitude  by  the  meridian  altitude  of  a  planet 174 

(  of  the  sun 176 

of  a  star 176 

of  a  planet 177 

To  find  the  latitude  by  double  altitudes  \  of  the  moon 177 

of  two  different  objects,  taken  within  a  few 

minutes  of  each  other,  by  one  observer  ....   178 
^  of  two  different  objects,  taken  at  different  times  178 

To  estimate  the  effects  of  small  errors  in  the  observations 179 

First  method  of  calculating  double  altitudes 180 

Second  method 185 

Third  method 189 

Questions  to  exercise  the  learner  in  working  double  altitudes 193 

Fourth  method,  when  the  declinations  are  different 193 

Fifth  method,  to  fifld  the  latitude  from  altitudes  and  distances  used  in  taking  a  lunar 

observation 197 

To  find  the  latitude  by  one  altitude  of  the  sun,  having  your  watch  previously  regulated  200 
To  find  the  latitude  by  the  mean  of  several  altitudes  of  the  sun,  taken  near  noon  by  a 

sextant  or  circle 202 

To  find  the  latitude  on  shore  by  means  of  an  artificial  horizon 204 

'  To  find  the  latitude  by  the  polar  star 206 

To  find  the  time  at  sea,  and  regulate  a  watch 208 

Examples  to  exercise  the  learner  in  finding  the  mean  time 210 

Second  method  of  finding  the  mean  time  at  sea 210 

Third  method  of  finding  the  mean  time  at  sea 211 

To  find  the  time  at  sea  by  the  moon's  altitude 213 

To  find  the  time  at  sea  by  a  planet's  altitude 215 

To  find  the  apparent  time  by  an  altitude  of  a  fixed  star 217 

To  regulate  a  chronometer  by  equal  altitudes  of  the  sun 219 

To  regulate  a  chronometer  by  means  of  a  transit  instrument 221 

To  find  the  longitude  at  sea  by  lunar  observations 225 

Method  of  finding  the  stars  used  in  lunar  observations 226 

General  remarks  on  the  taking  of  a  lunar  observation 228 

To  work  a  lunar  observation 229 

Examples  of  lunar  observations 232 

Second  method  of  working  a  lunar  observation 239 

Third  method  of  working  a  lunar  observation 242 

Fourth  method,  or  Witchell's  improved  method  of  finding  the  true  distance 243 

Table  of  corrections  for  second  differences 245 

Method  of  taking  a  lunar  observation  when  you  have  only  one  observer 246 

To  calculate  the  sun's  altitude  at  any  time 247 

To  calculate  the  moon's  altitude 248 

To  calculate  a  planet's  altitude 249 

To  calculate  a  star's  altitude 250 

Method  of  combining  several  lunar  observations,  and  determining  the  error  of  the  chro- 

nometer 251 

To  find  the  longitude  by  the  eclipses  of  Jupiter's  satellites 252 

To  find  the  longitude  by  an  eclipse  of  the  moon 253 

To  find  the  longitude  by  a  time-keeper  or  chronometer 253 

do  do  do  do 289 

To  allow  for  the  change  of  rate  in  a  chronometer 257 

Precautions  in  using  a  chronometer 259 

On  a  variation  chart 259 

Method  of  keeping  a  reckoning  at  sea 260 

To  find  the  lee-way,  and  allow  for  it 261 

To  correct  the  dead  reckoning 263 

Rules  for  working  a  day's  work 264 

Examples  for  working  a  day's  work .• 266 

Journal  from  Boston  to  Madeira 270 


ZIT 


CONTENTS. 


ARRANGEMENT    OF    THE    TABLES. 


Table.  Page 

I  Difference  of  latitude  and  departure  for  points 1 

II.  Difference  for  degrees 17 

III.  Meridional  parts 62 

IV.  Sun's  declination 63 

IV.  A.  Equation  of  time 68 

V.  For  reducing  the  sun's  declination 72 

VI.  Sun's  right  ascension 77 

VI.  A.  Correction  for  the  daily  variation  of  the  equation  of  time 77 

VII.  Amplitudes 78 

VIII.  Right  ascensions  and  declinations  of  the  fixed  stars * 80 

IX.  Sun's  rising  and  setting 84 

X.  For  finding  the  distance  of  terrestrial  objects  at  sea 8t 

X.  A.  Parallax  in  altitude  of  a  planet 86 

XI.  Proportional  parts , . .  , .     87 

XII.  Refraction  of  the  heavenly  bodies 88 

XIII.  Dip  of  the  horizon 88 

XIV.  Sun's  parallax  in  altitude 88 

XV.  Augmentation  of  the  moon's  semi-diameter 88 

XVI.  Dip  for  different  heights  and  distances 88 

XVII.  To  find  the  correction  and  logarithm  of  a  lunar  observation  when  a  star 

or  either  of  the  planets  Venus,  Mars,  Jupiter  or  Saturn  is  observed 89 

XVIII.  To  find    the  correction  and  logarithm  of  a  lunar  observation  when  the 

sun  is  used 97 

XIX.  To  find  the  correction  and  logarithm  of  a  lunar  observation  depending  on 

the  moon's  altitude 98 

XX.  For  finding  the  third  correction  of  a  lunar  observation 130 

— — •     XXI.  For  turning  degrees  and  minutes  into  time,  and  the  contrary 131 

XXII.  Proportional  logarithms 132 

XXIII.  For  finding  the  latitude  by  two  altitudes  of  the  sun 148 

XXIV.  Natural  sines  and  cosines 160 

XXV."f  Log.  sines,  tangents,  &c.  to  points  and  quarter  points 169  — f"       . 

XXVI.  Logarithms  of  numbers . /P.,.'.  .•■. . .-. .'  ^.^^4^ 169  flt^^JU 

XXVII.  Logarithmic  sines,  tangents,  and  secants 185 ' 

XXVIII.  To  find  the  time  of  the  moon's  passing  the  meridian 230 

XXIX  Correction  of  the  moon's  altitude  for  parallax  and  refraction 230 

XXX.  To  find  the  variation  ot  the  moon's  decimation,  &c 231 

XXXI.  To  find  the  sun's  right  ascension 23? 

XXXII.  Variation  of  the  sun's  altitude  in  one  minute  from  noon 23S 

XXXIII.  To  reduce  the  numbers  of  Table  XXXII.  to  other  given  intervals  from 

noon 243 

XXXIV.  Errors  arising  from  a  deviation  of  one  minute  in  the  parallelism  of  the 

surfaces  of  the  central  mirror    244 

XXXV.  Errors  arising  from  a  deviation  of  the  telescope  from  a  plane  parallel  to  the 

plane  of  the  instrument 2^14 

XXXVI.  Corref  .ion  of  the  mean  refraction  for  various  heights  of  the  thermometer 

and  barometer 244 

XXXVII.  Longitudes  and  latitudes  of  the  fixed  stars 245 

XXXVIII.  Reductions  of  latitude  and  horizontal  parallax 246 

XXXIX.  Aberration  of  tlie  planets  in  longitude 246 

XL.  Equation  of  the  equinoxes  in  longitude 246 

XLI.  Aberration  of  the  fixed  stars  in  latitude  and  longitude 246 

XLII.  Aberration  of  the  fixed  stars  in  right  ascension  and  declination 247 

XLIII.  Nutation  in  right  ascension  and  declination 248 

XLIV.  Augmentation  of  the  moon's  semi-diameter,  found  by  the  nonagesimal  . . .  249 

XLV  Equation  of  second  differences » 250 


-r 


CONTFaNTS.  XV 

Table.  Page. 

XLVI.  Table,  showing  the  variation  of  the  altitude  of  an  object  arising  from  a 

change  of  100  seconds  in  its  declination 251 

XLVIl.  Logarithms  in  Lyons's  improved  method 253 

XLVIII.  Third  correction  in  Lyons's  improved  method , 275 

XLIX.  Correction  for  a  planet  whose  horizontal  parallax  is  35" 326 

L.  Reduction  to  any  other  parallax 328 

LL  To  change  solar  time  into  sideral  time 329 

LIL  To  change  sideral  time  into  solar  time 329 

LIII.  Variation  of  the  compass,  by  Barlow 330 

LI V.  Latitudes  and  longitudes 332 

LV.  Tide  table 379 

LVI.  Extracts  from  the  Nautical  Almanac 383 

Catalogue  of  the  Tables,  with  examples  of  the  uses  of  those  not  explained  in  other 
parts  of  the  work 385 

APPENDIX. 

Addition  and  subtraction,  using  the  signs  as  in  algebra 395 

Problem  L     To  find  the  longitude,  latitude,  &c.  of  the  moon 395 

Problem  11.     To  find  the  horary  motion  of  the  moon 398 

Problem  IIL     To  find  the  ecliptic  conjunction  or  opposition  of  the  moon  and  sun,  or 

a  star 400 

Problem  IV.     To  find  the  altitude  and  longitude  of  the  nonagesimal 402 

Table  to  facilitate  the  calculation 403 

Abridged  rule  for  calculating  the  altitude  and  longitude  of  the  nonagesimal 403 

Problem  V.     To  calculate  the  moon's  parallax  in  latitude  and  longitude 404 

Problem  VI.     To  calculate  the  longitude  of  a  place  from  the  observed  beginning  and 

end  of  a  solar  eclipse 407 

Problem  VII.     To  calculate  the  longitude  of  a  place  from  the  observed  beginning  and 

end  of  an  occultation 410 

Problem  VIII.     To  find  the  longitude  of  a  place  from  the  beginning  or  end  of  a  solar 

eclipse  . . . . , 413 

Problem  IX.     To  find  the   longitude  of  a  place   from  the  beginning  or  end  of  an 

occultation 414 

Problem  X.     To  project  an  eclipse  of  the  moon 415 

Problem  XI.     To  calculate  an  eclipse  of  the  sun    417 

Problem  XII.     To  project  an  occultation  of  a  fixed  star 421 

Problem  XIII.     To  calculate  the  beginning  or  end  of  an  eclipse  or  occultation 425"' 

Problem  XIV.     To   find   the   apparent  time   at    Greenwich   from  the   moon's  longi- 
tude   426 

Problem  XV.     To  find  the  longitude  of  a  place  by  measuring  the  distance  of  the  moon 

from  a  fixed  star  not  marked  in  the  Nautical  Almanac 427 

Problem  XVI.     To  find  the  longitude  of  a  place  by  the  moon's   passage  over  the 

meridian 429 

Pnoblem  XVII.     Given  the  latitude  of  the  moon,  and  longitude  of  the  moon  and  sun, 

to  find  their  angular  distance 433 

Problem  XVIII.     Given  the  longitudes  and  latitudes  of  the  moon  and  a  star,  to  find 

their  angular  distance 434 

Problem  XIX.     Given  the  right  ascension  and  declination,  to  find  the  longitude  and 

latitude 435 

Problem  XX.     Given   the   longitude   and   latitude,  to  find   the   right  ascension  and 

declination 436 

Spheric  trigonometry 436 

Improvement  of  Napier's  rules  for  the  circular  parts 436 

Theorems  in  spherics 439 

Redfield's  theory  of  storms,  &c 440 

Problem  XXI.     To  find  the  longitude  of  a  place  from  the  beginning  or  end  of  a  solar 

eclipse 443 

Problem  XXII.     To  find  the  longitude  of  a  place  from  the  beginning  or  end  of  an  occul- 
tation   446 


INDEX  TO  TABLE  LIV. 

LATITUDES  AND  LONGITUDES. 


PAGE 

ADRIATIC,  coast  of 350 

AFRICA,  North  coast 351 

West  coast 354 

South  coast 856 

East  coast 356 

Red  Sea 356 

ALBANIA,  coast  of 350 

ALGIERS,  coast  of 352 

AMERICA — Eastern  coast. 

Greenland 343 

• Hudson's    and    Davis's    Bay   and 

Straits 343 

Labrador 342 

•  Newfoundland 342 

Gulf  of  St.  Lawrence 341 

Canada 342 

Nova  Scotia 841 

■ United  States 332 

Mexico  836 

Honduras  337 

Mosquitoes 83*7 

Panama 83'7 

Darien 337 

Cartagena 337 

. Maracaybo 337 

— Caracas 337 

Cumana 337 

Surinam  838 

. Maranham  838 

Brazil 338 

River  Plata 339 

. Patagonia 839 

Terra  del  Fuego 839 

AMERICA—  Western  coast. 

• Patagonia 340 

Chili 340 

Peru 840 

Quito 840 

■ Panama 840 

Mexico  841 

California 341 

Oregon  341 

British  and  Russian  possessions....  341 

ANAMBAS  ISLANDS 365 

ANDAMAN         «  362 

APO  BANK 369 

ARABIA,  coast  of 357 

ASCENSION— Island 355 


PA.  01 

ASIA: 

Red  Sea 356 

Arabian  coast 357 

Gulf  of  Persia..: 857 

Malabar 357 

Ceylon 358 

Coromandel 358 

Bengal 858 

Pegu 359 

Malay 359 

Siam 359 

Cochin  China 359 

Hainan — Island 860 

Cliina,  coast  of,  to  Canton 860 

"      from  Canton  to  Kamtskatka, 

with  adjacent  islands 370 

AZOF,  sea  of 851 

BANCA— Island 364 

BANDA  SEA 370 

BASHEE  ISLANDS 369 

BENGAL,  coast  of 358 

BENIN,  coast  of 355 

BIAFRA 855 

BLACK  SEA 351 

BORNEO— Island 368 

BOTHNIA 348 

BRAZIL,  coast  of 338 

CALIFORNIA,  coast  of 341,451 

CANARY  ISLANDS  354 

CANDIA 353 

CAPE  BRETON— Island 841 

CAPE  VERDE  ISLANDS 354 

CARACAS,  coast  of 337 

CARIBBEAN  SEA 333 

CAROLINE  ISLANDS 374 

CARTAGENA,  coast  of '.  837 

CELEBES— Island 368 

CERAM— Island 870 

CEYLON— Island 853 

CHAGOS  ARCHIPELAGO 361 

CHILI,  coast  of 340 

CHINA,  Southern  coast 360 

Eastern  coast 370 

CHINA  SEA,  islands  in 365 

Southeast  part 366 

COCHIN  CHINA,  coast  of 359 


INDEX  TO  TABLE  LIV. 


xvu 


COMORO  ISLAJSnOS 361 

CONGO,  coast  of 355 

COROMANDEL  COAST 358 

CORSICA— Island 352 

CRIMEA,  coast  of 351 

CUMANA,  coast  of 337 

CYPRUS— Island 353 

DARIEN,  coast  of 337 

DENMARK,  coast  of 347 

Islands  348 

EGYPT,  Mediterranean 351 

Red  Sea 356 

ENGLAND,  South  coast 343 

East  coast 344 

West  coast 346 

EOLIAN  ISLANDS 352 

FALKLAND  ISLANDS 355 

FEEJEE  GROUP  451 

FERRO  ISLANDS 345 

FRANCE,  coast  of 

North  coast .T 344,347 

West  coast 349 

South  coast 350 

FRIENDLY  ISLANDS 377 

GALAPAGOS  ISLANDS 375 

GERMANY,  Baltic 348 

North  Sea 347 

GEORGIA— Island 355 

GOOD  HOPE,  Cape  of 355 

GREECE,  Western  coast 350 

Southern  coast 351 

Eastern  coast 351 

GRECIAN  ARCHIPELAGO 353 

GREENLAND,  coast  of 343 

GUINEA,  coast  of 355 

GULF  OF  VENICE,  coast 350 

•  "  Islands 353 

GULF  OF  PERSIA 357 

GULF  OF  BOTHNIA 348 

Finland 348 

Guinea  355 

Mexico 336 

Persia 357 

Slam 359 

St.  Lawrence 841 

Tartaiy 371 

Tonquin 860 

Venice 350 

HAINAN— Island  360 

HOLLAND,  coast 347 

HONDURAS,  coa^t 337 

ICELAND 343 

lONLAN  ISLANDS 353 

IRELAND,  East  coast 346 


IRELAND,  North  coast 346 

South  coast 347 

West  coast 345 


ISLANDS,  Anambas 355 

Andaman 352 

Apo  Bank ; 359 

Ascension 355 

Asia,  eastern  coast  from  Canton  ...  370 

Baltic 348 

Banca 364 

Banda  Sea 370 

Bashee 369 

between  Batavia  and  New  Guinea, 

south  of  the  Celebes 367 

between  Cape  Verde,  Cape  of  Good 

Hope,  and  Cape  Horn 355 

Borneo,  Celebes,  Luconia,  with  those 

adjacent  as   far   east  as  New 
Guinea  368 

Borneo 368 

Canary 354 

Caudia 353 

Cape  Breton 341 

Cape  Verde 354 

Caroline  874 

Celebes 368 

Ceram 370 

Ceylon  , 358 

Chagos  Arcliipelago 361 

China,  Eastern  coast 370 

China  Seas 365 

"  and  adjacent 862 

"  S.  E.  part 366 

Comoro 361 

Corsica 35?. 

Cyprus 853 

Dangerous  Archipelago 378 

Denmark,  coast  of 348 

Eolian 352 

Falkland 355 

Feejee 451 

Ferro 345 

Friendly S77 

•  Galapagos  375 

Georgia 355 

Grecian  Archipelago 353 

Gulf  of  St.  Lawrence 341 

Gulf  of  Venice 353 

Hainan 360 

Indian   Ocean,  between    Caj)e    of 

Good  Hope  and  Sumatra,  iLc...  S60 

Iceland 363 

Isle  of  Wight 344 

Isle  of  Man 347 

Ivica 352 

Ionian 353 

Italian 352 

Japan 871 

Java 863 

Java  Sea 367 


INDEX  TO  TABLE  LIV. 


ISLAISTDS,  Kirigsmill  Group 

Laccadive  Archipelago 

Luconia 

Madagascar 

Madeira 

Magdalen 

Mahe  Bank 

Majorca 

Maldivia  Archipelago 

Marquesas 

•  Mediterranean , 

Mindanao 

Mindoro 

Minorca 

ly  I'lecas 

]Natnnas  

■  Navigator 

New  Caledonia 

Newfoundland 

New  Hebrides  .  

New  Holland,  adjacent  to 

New  Pliillipines 

New  South  Shetland 

New  South  Wales 

' New  Zealand 

Nicobar 

Orkney  

Pacific  Ocean,  north  313. 

"  "      south 

Panay 

Paracels 

Paternosters 

Paumotu  Group 

Fratas  Shoals 

■ Prince  Edward 

Radack 

PtaUck 

Sandwich  Land 

Sandwich  Islands 315 

Sardinia  

Shetland 

—  Sicily 

-= "     Islands  adjacent 

Society 

Solomons 

Sooloo  Sea 369, 

St.  Helena 

Spitzbcrgen  

Straits  of  Banca 

"        BiUington 

"         Gaspar 

Straits  east  of  Java 

Straits  of  Malacca 

"        Macassar 

• "         Sincapore 

"         Sunda 

"        Torres 


Staten ... 
Sumatra 


west  of.. 


PAGE 

450 
362 
369 
361 
354 
342 
361 
352 
361 
318 
362 
369 
369 
852 
810 
365 
450 
316 
342 
316 
312 
314 
318 
312 
311 
362 
345 

,450 
315 
369 
865 
861 
450 
866 
342 
314 
314 
856 

,450 
352 
345 
852 
853 
811 
316 
451 
356 
343 
864 
864 
364 
361 
365 
868 
364 
364 
315 
839 
362 
363 


ISLANDS,  Tambelan 365 

Terra  del  Fuego 339 

Timor 368 

Van  Dieman's  Land 312 

Wasliington 318 

Western 354 

West  India 333 

XuUa  Besseys 369 

Zealand 341 

ISLE  OF  MAN 341 

ITALIAN  ISLANDS 352 

ITALY 350 

IVICA 352 

JAPAN  ISLANDS 311 

JAVA 363 

Sea,  islands  in 361 

Straits  east  of  Java 361 

JUTLAND,  coast  of. 341 

KINGSMILL  GROUP 450 

LABRADOR,  coa#  of. 342 

LACCADIVE  ARCHIPELAGO 362 

LAPLAND,  coast  of 349 

LOANGO,  coast  of 355 

LUCONIA— Island 369 

MACASSAR  STRAITS 868 

MADEIRA  ISLANDS 854 

MADAGASCAR—Island 361 

MAGDALEN  ISLANDS  342 

MAHE  BANK 361 

MAJORCA— Island 352 

MALABAR,  coast  of 851 

MALAY,  coast  of 359 

MALDIVIA  ARCHIPELAGO 361 

MARMORA,  sea  of 351 

MARANHAM,  coast  of 388 

MARACAYBO,  coast  of 331 

MARQUESAS  ISLANDS 318 

MEDITERRANEAN  SEA : 

East  coast 351 

South  coast 351 

North  coast 849 

Islands  in 352 


MEXICO,  South  and  East  coast 836 

West  coast 341 


MINORCA— Island  852 

MINDANO— Island 369 

MINDORO— Island  369 

MOLUCCAS— Islands 3l0 

MOREA,  coast  of 851 

MOROCCO,  North  coast 852 

West  coast 354 


MOSQUITOES,  coast  of 831 

NAPLES,  coast  of 350 

NATUNAS  ISLANDS 365 

NAVIGATOR  ISLANDS 460 


INDEX  TO  TABLE  LIV. 


NEW  CALEDONIA— Island 376 

NEW"  HEBRIDES— Island 376 

NEW  HOLLAND  372 

NEW  PHILLIPINE  ISLANDS 374 

NEW  SOUTH  SHETLAND— Island 378 

NEW  SOUTH  WALES,  coast  of. 372 

NEW  ZEALAND— Island  377 

NICOBAR  ISLANDS 362 

NORTHWEST  COAST  OF  AMERICA  ...  341 
NORWAY,  Cattegat.... 347 

West  coast 349 

North  coast 349 

NOVA  SCOTIA,  coast  of 341 

OREGON,  coast  of 341 

ORKNEY  ISLANDS  345 

PACIFIC  OCEAN,  N.,  Islands  in 373,  450 

PACIFIC  OCEAN,  S.,  Islands  in S75,  450 

PANAY— Island  369 

PANAMA,  East  coast 337 

AYest  coast 340 

PARACELS  ISLANDS 365 

PAUMOTU  GROUP 450 

PATAGONIA,  E.  coast  of 339 

W.  coastof 340 

PATERNOSTERS— Islands 367 

PEGU,  coastof. 359 

PERU,  coastof 340 

PICO  ISLANDS 354 

PORTUGAL,  coastof 349 

PRATAS  SHOALS 366 

PRINCE  EDWARD'S  ISLAND  342 

QUITO,  coastof 340 

RADACK  ISLANDS 374 

RALICK  ISLANDS 374 

RED  SEA,  coastof 356 

RIVER  PLATA,  coastof 339 

RUSSIA,  Black  Sea 351 

Baltic  Sea  and  Islands 348 

Gulfs  of  Finland  and  Bothnia 348 

N.  Eastern  coast 371 

Sea  of  Azof. 351 

SACHALIN— Island 371 

SANDWICH  LAND., 356 

SANDWICH  ISLANDS 375,  450 

SARDINIA— Island  352 

SCOTLAND,  East  coast 345 

N.  and  West  coasts 345 

SENEGAMBIA,  coastof 354 

SHETLAND  ISLANDS 345 

SIAM,  coastof. 359 


PAOB 

SICILY— Island 352 

SICILIAN  ISLANDS 353 

SOCIETY  ISLANDS  377 

SOLOMONS  ISLANDS 376 

SOOLOO  SEA 369,  451 

SPAIN,  North  coast 349 

South  coast 349 

SPITZBERGEN— Island 343 

ST.  ANTHONY— Island 354 

ST.  HELENA— Island 355 

STATEN  ISLAND 339 

SUMATRA ^62 

Islands  west  of 363 

SURINAM,  coastof 338 

SWEDEN,  Cattegat  and  Sound 347 

Baltic 348 

TAMBELAN  ISLANDS 365 

TARTARY,  coastof 371 

TENERIFFE— Island  354 

TERRA  DEL  FUEGO,  coastof 339 

TIMOR— Island  368 

TORRES  STRAITS,  Islands  in 375 

TRIPOLI,  coastof 351 

TUNIS,  coastof 351 

TURKEY 351 

Adriatic  or  West  coast 350 

UNITED  STATES  OF  AMERICA : 

Eastern  coast 332 

Western  coast 341 

VAN  DIEMAN'S  LAND  372 

WASHINGTON  ISLANDS .-...  378 

WEST  INDIES : 

Bahama  Bank,  Great 335 

"  "      Little 336 

Bermuda  336 

Caycos  Islands 335 

Cuba,  North  side  335 

Cuba,  South  side 334 

Jamaica  334 

Passage  Islands 335 

Porto  Rico 334 

Salt  Key 336 

St.  Domingo 334 

Vu-gm  Islands 334 

Windward  Islands 333 

WESTERN  ISLANDS 354 

WHITE  SEA 349 

XULLA  BESSEYS— Island 369 

ZEALAND— Island , 341 


SIGNS    AND    ABBREVIATIONS 

USED    IN    THIS    WORK. 


|-  IS  the  sign  of  addition,  and  denotes  that  whatever  number  or  quantity  follows  the  sigu, 
must  be  added  to  those  that  go  before  it;  thus,  9-|-8  signifies  that  8  is  to  be  added  to 
9 ;  or  A  +  B  implies  that  the  quantities  represented  by  A  and  B  are  to  be  added  to- 
gether. The  sign  -j-  is  called  the  positive  sign. 
-  the  sign  of  subtraction,  and  denotes  that  the  number  following  it  must  be  subtracted 
from  those  going  before  it;  thus,  7  —  5  signifies  that  5  must  be  subtracted  from  7. 
The  sign  —  is  called  the  negative  sign. 

X  is  the  sign  of  multiplication,  and  shows  that  the  numbers  placed  before  and  after  it 
are  to  be  multiplied  together  ;  thus,  7x9  signifies  7  multiplied  by  9,  which  makes  63; 
and  7x8x2  signifies  the  continued  product  of  7  by  8  and, by  2,  which  makes  112. 
Multiplication  is  also  denoted  by  placing  a  point  between  the  quantities  to  be  multi- 
plied together ;  thus,  A  .  B  signifies  that  A  is  to  be  multiplied  by  B. 

-f-  is  the  sign  of  division,  and  signifies  that  the  number  that  stands  before  it  is  to  be  divided 
by  the  number  following  it ;  as,  72  -r-  12  shows  that  72  is  to  be  divided  by  12 
Division  may  also  be  denoted  by  placing  two  points  between  the   numbers;  thus, 

72 
72  :  12  represents  72  divided  by  12 ;    or  by  placing  the  numbers  thus,  — >   which 

signifies  72  divided  by  12. 
( )     or  -.     Either  of  these  marks  is  used  for  connecting  numbers   together ;  thus, 

3  -(-  4  X  6,  or  (3  -|-  4)  X  ^j  signifies  that  the  sum  of  3  and  4  is  to  be  multiplied  by  6 
t=    is  the  sign  of  equality,  and  shows  that  the  numbers  or  quantities  placed  before  it  are 

equal  to  those  following  it;  thus,  8  X  12  =  96  ;  or,  8  multiplied  by  12  are  equal  to  96; 

and  7  4-  2  X  4  =  36. 
•: :  are  the  signs  of  proportion,  and  are  used  thus ;  7  :  14  : :  10  :  20,  that  is,  as  7  is  to  14,  so 

is  10  to  20 ;  or,  A  :  B  :  :  C  :  D,  that  is,  as  A  is  to  B,  so  is  C  to  D. 
**    signifies  degrees  ;  thus,  45°  represents  45  degrees. 

signifies  minutes  ;  thus,  24',  or  24  minutes. 
"    signifies  seconds;  thus,  44",  or  44  seconds. 

'"    signifies  thirds,  or  sixtieth  parts  of  seconds ;  thus,  44'",  or  44  thirds. 
In  noting  any  time,  d  is  the  mark  for  days,  h  for  hours,  m  for  minutes,  &c 
S.  signifies  sine.     N.  S.  signifies  natural  sine. 
Sec.  signifies  secant. 
Tan.  signifies  tangent. 
Cosine,  Cotangent,  or  Cosecant  of  an  arc,  signifies  the  sine,  tangent,  or  secant  of  tlia 

complement  of  that  arc  respectively. 
<^     signifies  angle. 

/\     signifies  triangle.     /\'s,  triangles, 
□     signifies  a  square. 

0  or  @,  the  sun.  Q)  or  "^  ,  the  moon.  *  a  star.  L.  L.  lower  limb.  U.  L.  upper  limb. 
N.  L.  nearest  limb.  S.  D.  semi- diameter.  P.  L.  proportional  logarithm,  N.  A.  Kautical 
Almanac.    Z.  D.  zenith  distance.     D.  R.  dead  reckoning. 


DIRECTIONS  FOR  THE  BINDER. 


Plate    I.  to  front  the  title-page. 

II.  to  front  page     17. 

III.  to  front  page    48. 

IV.  to  front  page    60. 
V.  to  front  page    52. 

VI.  to  front  page    64. 

VII.  to  front  page  112. 


Plate  VIII.  to  front  page  116. 

IX.  to  front  page  136. 

X.  to  front  page  144. 

XI.  to  front  page  150. 

XII.  to  front  page  156. 

XIII.  to  front  page  426,  Appendix 


DECIMAL    ARITHMETIC. 


Mant  persons  wlio  have  ac(iuircJ  coiisiderublp  skill  in  common  arithmetic,  are 
unacquauited  with  the  method  of  calcuhitiiiji  by  decimals,  which  is  of  great  use  in 
Navigation ;  for  which  reason  it  was  thought  proper  to  prefix  the  foUowuig  brief 
explanation. 

Fractions,  or  Vulgar  Fractions,  are  expressions  for  any  assignable  part  of  a  unit ; 
they  are  usually  denoted  by  two  numbers,  placed  the  one  above  the  other,  with  a  line 
between  them  ;  thus  |  denotes  the  fraction  one  fourth,  or  one  part  out  of  four  of  some 
whole  quantity,  considered  as  divisible  into  four  equal  ])arts.  The  lower  number,  4, 
is  called  the  denominator  of  the  fraction,  showing  into  how  many  parts  the  whole  or 
integer  is  divided  ;  and  the  upper  numbei',  1,  is  called  the  numerator,  and  shows  how 
many  of  those  equal  parts  are  contained  in  the  fraction.  And  it  is  evident  that  if  the 
numerator  and  denominator  bo  varied  in  the  same  ratio,  the  value  of  the  fraction  will 
remain  unaltered ;  thus,  if  the  numerator  and  denominator  of  the  fraction,  ^,  be 
multiplied  by  2,  3,  or  4,  &c.,  the  fractions  arising  will  be  -|,  -^^,  3^,  &c.,  which  are 
evidently  equal  to  ^. 

A  Decimal  Fraction  is  a  fraction  whose  denominator  is  always  a  unit  with  some 
number  of  ciphers  annexed,  and  the  numerator  any  number  whatever ;  as,  ^2^, 
tB^ttj  T^aTT)  &c.  And  as  the  denominator  of  a  decimal  is  always  one  of  the 
numbers  10,  100,  1000,  &c.,  the  inconvenience  of  writing  the  denominator  may  be 
avoided,  hf  placing  a  point  between  the  integral  and  the  fractional  part  of  the 
number ;  thus,  -fjy  is  \vi-itten  .3 ;  and  xV\  is  written  .14 ;  the  viired  number  3^^^^ 
consisting  of  a  whole  number  and  a  fractional  one,  is  written  3.14. 

In  setting  down  a  decimal  fraction,  the  numerator  must  consist  of  as  many  places  as 
tliere  are  ciphers  in  the  denominator ;  and  if  it  lias  not  so  many  figures,  the  defect 
must  be  supplied  by  placing  ciphers  before  it  ;  thus,  ^'^^j  =z  .16,  li^'iJ  =  '016, 
fTjxyiyu  =^  .0016,  &c.  And  as  ciphers  on  the  right  hand  side  of  integers  increase  their 
value  in  a  tenfold  proportion,  as,  2,  20,  200,  &c.,  so,  when  set  on  the  left  hand  of 
decimal  fractions,  they  decrease  their  value  in  a  tenfold  jiroportion,  as,  .2,  .02,  .002,  &c.; 
but  ciphers  set  on  the  right  hand  of  these  fractions  make  no  alteration  in  their  value, 
neither  of  increase  or  decrease  ;  thus,  .2  is  the  same  as  .20  or  .200.  The  common 
arithmetical  operations  are  performed  the  same  way  in  decimals  as  they  are  in 
integers ;  regard  being  had  only  to  the  particular  notation,  to  distinguish  the  uitegral 
from  the  fractional  part  of  a  sum. 

ADDITION   OF   DECIMALS. 

Addition  of  decimals  is  performed  exactly  like  that  of  whole  numbers,  placing  the 
numbers  of  the  same  denomination  under  each  other,  in  which  case  the  decimal 
separatmg  points  will  range  straight  in  one  column. 


EXAMPLEfc 

Miles. 

Feet. 

Inches. 

26.7 

1.26 

272.3267 

32.15 

2.31 

.0134 

143.206 

1.785 

2.1576 

.003 

2.0 

31.4 

Sura        202.059  7.355  305.8077 

1 


2  DECIMAL   ARITHMETIC. 

SUBTRACTION   OF   DECIMALS. 

Subtraction  of  decimals  is  performed  in  the  same  manner  as  in  whole  numbere,  by 
observing  to  set  the  figures  of  the  same  denomination  and  the  separating  points  dii-ectly 
under  each  ct*ier. 

EXAMPLES. 

From  3L267  36.75  L254  1364.2 

Take     2.63  .026  .316  25.163 


Difftrence  28.637 


36.724 


.938 


1339.037 


MULTIPLICATION   OF   DECIMALS. 

Multiply  the  numbers  together  the  same  as  if  they  were  whole  numbers,  and  ponit 
ofi'as  many  decimals  from  the  right  hand  as  tliere  are  decimals  in  both  factors  together; 
and  when  it  happens  that  there  are  not  so  many  figures  in  the  product  as  there  must 
be  decimals,  then  prefix  as  many  ciphers  to  the  lefl;  hand  as  will  supply  the  defect 

EXAMPLE  IV. 
Multiply  .17  by  .06. 
.17 


EXAMPLE   I. 

Multiply  3.25  by  4.5. 

3.25 

4.5 


1.625 
13.00 

Answer    14.625 

In  one  of  the  factors  is  one  decimal,  and 
In  the  other  two ;  their  sum,  3,  is  the 
uuml)er  of  decimals  of  the  product. 

EXAMPLE  II. 

Multiply  0.5  by  0.7. 

0.5 

0.7 

Answer    0.35 


EXAMPLE  III. 
Multiply  3.25  by  .05. 
3.25 
■05 

Answer    .1625 


.06 


Answer    .0102 


In  each  of  the  factors  are  two  decimals 
the  product  ought  therefore  to  contain  4 
and,  there  being  only  three  figures  in  the 
product,  a  cipher  must  be  prefixed. 

EXAMPLE  V. 

Multiply  .18  by  24. 

.18 

_24 

72 
36 


Answer    4.32 

EXAMPLE  VI. 

Multiply  36.1  by  2.5. 

36.1 

2.5 

18.05 
72.2 


Answer    90.25 


DIVISION   OF   DECIMALS. 

Division  of  decimals  is  performed  in  the  same  manner  as  in  whole  numbei-s  ;  only 
observing  that  the  number  of  decimals  in  the  quotient  must  be  equal  to  the  excess  of 
the  number  of  decimals  of  the  dividend  above  those  of  the  divisor.  When  the  divisor 
contains  more  decimals  than  the  dividend,  ciphers  must  be  affixed  to  the  right  hand 
of  the  latter  to  make  the  number  equal  or  exceed  that  of  the  divisor. 

EXAMPLE   II. 
Divide  3.1  by  .0062. 
Previous  to    the    division,    I    affix    a 
number  of  ciphers  to  the  right  hand  of 
3.1,  which  does  not  alter  its  value. 
.0062)3.100000(500.00 
310 


EXAMPLE  I. 

Divide  14.625  by  3.25. 

3.25  )  14.625  { 4.5 

1300 


1625 
1625 


In  this  example,  there  are  two  decimals 
in  the  divisor,  and  three  in  the  dividend  ; 
hence  there  is  one  decimal  in  the  quotient. 


00000 
Therefore  the  answer  is  500  00  or  500. 


DECIMAL  ARITHMETIC. 


EXAMPLE   in. 

Divide  0.35  by  0.7. 

.7  )  .35  ( .5 

.35 

EXAMPLE  IV. 
Divide  9.6  by  .06. 
.06 )  9.60 

160  Answer. 

Here,  by  affixing  a  cipher  to  9.6,  it 
becomes  9.60,  and  has  then  two  decimals 
in  it,  which  is  the  same  number  as  is  in 
the  divisor;  therefore  the  quotient  is  an 
integral  number. 


EXAMPLE  V. 
Divide  17.256  by  1.16. 
L13)  17.25600  (14.875 
116_ 

565 
464 

1016 
928 

880 
812 

680 
580 

100 


REDUCTION  OF   DECIMALS. 

If  you  wish  to  reduce  a  vulgar  fraction  to  a  decimal,  you  may  add  any  number  of 
ciphers  to  tlie  numerator,  and  divide  it  by  the  denominator ;  the  quotient  will  be  the 
decimal  fraction  ;  the  decimal  point  must  be  so  placed  that  there  may  be  as  many  figures 
to  the  right  hand  of  it  as  you  added  ciphers  to  the  numerator ;  if  tiiere  ai"e  not  as  many 
figures  m  the  quotient,  you  must  place  ciphers  to  the  left  hand  to  make  up  the  number 

EXAMPLE   I. 


Reduce  i^  to  a  decimal. 
5)1.0 

.2  Answer. 

EXAMPLE  II. 
Reduce  f  to  a  decimal. 
8 )  3.000 

.375  Answer. 


EXAMPLE  III. 
Reduce  3  inches  to  the  decimal  of  a 
foot. 

Since  12  inches  =z  1  foot,  this  fraction 
is  ^% 

12 )  3.00 


.25  Answer. 


EXAMPLE  IV. 
Reduce  3^  mches  to  the  decimal  of  a  foot 
3J-  =  J ;  this  divided  by  12  is  /j. 

24  )  7.000  ( .291  Answer,  nearly. 


40 
24 

16 


EXAMPLE  V. 

Reduce  1  foot  and  6  inches  to  the 
decimal  of  a  yard. 

Here  1  foot  6  inches  ■=z  18  inches. 

And  1  yard rr: 36  inches;  therefore  this 
fraction  is  ^|. 

36  )  18.0  ( .5  Answer. 
180 


If  you  have  any  decimal  fraction,  it  is  easy  to  find  its  value  in  the  lower 
denominations  of  the  same  quantity ;  thus,  if  the  fraction  was  the  decimal  of  a  yard, 
by  multiplying  it  by  3  we  have  its  value  in  feet  and  parts ;  if  we  multiply  this  by 
12,  the  product  is  its  value  in  inches  and  pai-ts ;  and  in  the  same  manner  the  valuea 
may  be  obtained  in  other  cases. 

EXAMPLE  VII. 


Answer,  3  yards,  0  feet,  9  inches. 


EXAMPLE   VI. 

Required  the  value  of  7.231  days 

ired  the  value  of  3.25  yards. 
3.25 

7.231 
24 

3 
.75 

924 

4G2 

12 

5.544 

60 

32.640 
60 

38.400 


Answer,  7  days,  5  hours,  32  minute% 
38  seconds,  and  4  tenths  of  a  second. 


GEOMICTRY. 


Geometry  is  the  science  which  treats  of  the  description,  properties,  and  relation3 
of  magnitudes  in  general,  of  which  there  are  three  kinds  or  species  ;  viz.  a  Ihie,  which 
has  only  length  without  either  breadtli  or  thickness ;  a  suptrficies,  comprehended  by 
length  and  breadth ;  and  a  solid^  wliich  has  length,  breadth,  and  thickness. 

I. 

A  roiJv'T,  considered  mathematically,  has  no  leng*Ji,  breadth,  or  thickness. 

II. 

A  STRAIGHT  LINE,  Or  RIGHT  LINE,  is  tlic  shortcst  distance  between  the  two  pointa 

wiiich  limit  its  length,  as  AC.  4 c 

III. 

A  PLANE  SUPERFICIES  is  that  ui  which  any  two  pomts  being  taken,  the  straight  line 
between  them  lies  wholly  in  that  surface. 

IV. 

Parallel  lines  are  such  as  are  m  the  same  plane,  and  which,      A '■ — 3 

extended  infinitely,  do  never  meet,  as  AB,  DC.  j) c 

V. 

A  CIRCLE  is  a  plane  figure,  bounded  by  a  uniform  cune  line ;  it  is  commonly 
described  with  a  pair  of  compasses  ;  one  point  of  which  is  fixed,  whilst  the  other  13 
turned  round  to  the  place  where  the  motion  first  began  ;  the  fixed  point  is  called  tlie 
CENTRE,  and  the  line  described  by  the  other  point  is  called  the  circumference. 

VI. 

The  radius  of  a  circle,  or  semi-diameter,  is  a  right  line  drawn 
fi-om  the  centre  to  the  circumference,  as  AC  ;  or  it  is  that  line 
which  is  taken  between  the  points  of  the  compasses  to  describe  the 
cii  cle. 

A  diameter  of  a  circle  is  a  right  line  drawn  through  the  centre, 
and  terminated  at  both  ends  by  the  chcumference,  as  ACB  ;  and  is 
the  double  of  the  radius,  AC.  A  diameter  divides  the  circle  and  its 
circumference  into  two  equal  parts. 

VII. 

An  ARC  of  a  circle  is  any  part  or  portion  of  the  circumference,  as  DFE. 

VIII. 

The  CHORD  of  an  arc  is  a  straight  line  joining  the  ends  of  the  arc;  it  divides  the 
circle  into  two  unequal  parts,  called  segments,  and  is  a  chord  to  them  both  ;  as  DE  is 
the  chord  of  the  arcs  DFE  and  DGE. 

IX. 

A  semicircle,  or  half  circle,  is  a  figure  contained  under  a  diameter  and  the  arc 
terminated  by  that  diameter,  as  AGB  or  AFB.  Any  iwirtof  a  circle  contained  between 
two  radii  and  an  arc,  is  called  a  sector. 

X. 

A  QUADRANT  is  half  a  semichcle,  or  one  fourth  part  of  a  whole  circle,  as  the  Dguve 
CAG. 

Note.  All  circles,  whether  gi-eat  or  small,  are  supposed  to  have  their  circumference 
divided  into  3G0  equal  parts,  called  degrees;  and  each  degree  into  60  equal  parts,  called 
minutes ;  mid  each  minute  into  GO  equal  parts,  called  seconds  ;  and  so  on  into  tliirds, 


GEOMETRY. 


fouiths,*  &c. ;  and  an  arc  is  said  to  be  of  as  many  degrees  as  it  contains  parts  of  the 
3G0,  into  Yvliich  tlie  circumference  is  divided. 

XI. 

An  ANGLE  is  the  inchuation  of  two  luies  which  meet,  but  not  in 
the  same  direction. 

An  angle  is  usually  expressed  by  the  letter  placed  at  the  angular 
point,  as  the  angle  A.  But  when  two  or  more  angles  are  at  the 
same  point,  it  is  tlien  necessary  to  express  each  by  three  letters, 
and  the  letter  at  the  angular  point  is  placed  between  the  otlier 
two.  Thus  the  angle  formed  by  the  lines  AB,  AC,  is  called  the 
angle  BAG,  or  CAB ;  and  that  formed  by  AB,  AD,  is  called  the 
angle  BAD,  or  DAB. 

An  angle  is  measured  by  the  arc  of  a  circle  comprehended  between  the  two  le^s  that  form 
the  angle ;  the  centre  of  the  circle  being  the  angular  point,  and  the  whole  circumference 
considered  as  equal  to  360°. 

Thus  the  angle  A  is  measured  by  the  arc  BC  described  round 
the  point  A  as  a  centre,  and  the  angle  is  said  to  be  of  as  many 
degrees  as  the  arc  is ;  that  is,  if  the  arc  BC  is  30°,  then  the  angle 
BAC  is  said  to  be  an  angle  of  30  degi-ees. 

XIT. 

If  a  right  line,  AB,  fall  upon  another,  DC,  so  as  to  incline  neither  to  the  one  side  nor 
the  other,  but  makes  the  angles  ABC,  ABD,  equal  to  each  other, 
then  the  line  AB  is  said  to  be  perpendicular  to  the  line  DC,  and 
each  of  these  angles  is  called  a  right  angle,  being  each  equal  to  a 
quadrant,  or  90°;  because  the  sum  of  the  two  angles,  ABC,  ABD, 
is  measured  by  the  semicircle  DAC,  described  on  the  diameter 
DBC,  and  centre  B. 

XIII. 

An  ACUTE  ANGLE  is  Icss  than  a  right  angle,  as  ABC. 


A 


B       I> 


XIV. 

An  OBTUSE  ANGLE  is  greater  than  a  right  angle,  as  GEH. 

The  least  number  of  right  lines  that  can  include  a  space  are 
three,  which  form  a  figure  called  a  triangle,  consisting  of  six  parts, 
viz.  three  sides  and  three  angles ;  it  is  distinguished  into  three 
sorts,  viz.  a  nght-angled  triangle,  an  ohtuse-angled  tnangle,  and 
an  acute-angled  triangle. 

XV. 

A  RIGHT-ANGLED  TRIANGLE  lias  ouc  of  its  anglcs  right ;  the  side 
opposite  the  right  angle  is  called  the  hypotenuse ;  and  the  other 
two  sides  are  called  legs;  that  which  stands  upright  is  called  the 
perpendicular,  and  the  other  the  base;  thus  BC  is  the  hypotenuse, 
AC  the  perpendicular,  and  AB  the  base  ;  the  angles  opposite  the 
two  lesrs  are  both  acute. 


A 


H    M 


A 


XVI. 

An  ACUTE-ANGLED  TRIANGLE  Iias  cacli  of  its  angles  acute,  as 
DEG. 

XVII. 

An  OBTUSE-ANGLED  TRIANGLE  has  ouc  of  its  auglcs  obtusc,  or 
greater  than  a  right  angle,  as  BAF ;  the  other  two  angles  are 
acute. 

N'ote.  All  triangles  that  are  not  right-angled,  whether  they  arc  acute  or  obtuse,  are 
in  general  called  oblique-angled  triangles,  without  any  other  distuiction. 

*  A  new  division  of  the  circumference  of  tiie  circle  has  lately  been  adopted  by  several  eminent  French 
mathematicians,  in  which  the  quadrant  is  divided  into  100°,  each  degree  into  100',  each  minute  into 
100",  &c.,  and  tables  of  logarithms  have  been  published  conformable  thereto.  The  general  adoption 
of  this  division  would  tend  greatly  to  facilitate  most  of  the  calculations  of  navigation  and  astronomy. 


6 


GEOMETRY. 


H 


XVIII. 

A  QUADRILATERAL  figure  is  one  bounded  by  four  sides,  as 
ACDB.  If  the  opposite  sides  are  parallel,  they  ai-e  called  paral- 
lelograms. Thus,  if  AC  be  parallel  to  BD,  and  AB  pai-allel  to 
CD,  the  figure  ACDB  is  a  parallelogram.  A  parallelogram  having 
all  its  sides  equal,  and  its  angles  right,  is  called  a  square,  as  B. 
When  the  angles  are  right,  and  the  opposite  sides  only  equal,  it  is 
«alled  a  rectangle,  as  A 

XIX. 

The  sine  of  an  arc  is  a  line  drawn  from  one  end  of  the  arc  pei-pendicular  to  a 
diameter  drawn  through  the  other  end  of  the  same  arc ;  thus  RS  is  the  sine  of 
the  arc  AS,  RS  being  a  line  drawn  from  one  end,  S,  of 
that  arc,  pei-pendicular  to  DA,  which  is  the  diameter 
passing  through  the  other  end,  A,  of  the  arc. 

XX. 

The  COSINE  of  an  arc  is  the  sine  of  the  complement  of 
that  arc,  or  of  what  that  arc  wants  of  a  quadrant ;  thus, 
AH  being  a  quadrant,  the  arc  SH  is  the  complement  of 
the  arc  AS ;  SZ  is  the  sine  of  the  arc  SH,  or  the  cosme  of 
the  arc  AS. 

XXI. 

The  VERSED  SINE  of  an  arc  is  that  part  of  the  diameter 
contained  between  the  sine  and  the  arc  ;  thus  RA  is  the 
versed  sine  of  the  arc  AS  and  DCR  is  the  versed  sine  of 
tlie  arc  DHS. 

XXII. 

The  tangent  of  an  arc  is  a  right  line  drawn  perpendicular  to  the  diameter,  passing 
through  one  end  of  the  arc,  and  terminated  by  a  line  di-awn  from  the  centi'e  through 
the  other  end  of  the  arc  ;  thus  AT  is  the  tangent  of  the  arc  AS. 

XXIII. 

The  cotangent  of  an  arc  is  the  tangent  of  the  complement  of  that  arc  to  a 
quadrant ;  tlius  HG  is  the  tangent  of  the  arc  HS,  or  the  cotangent  of  the  ai'c  AS. 

XXIV. 

The  SECANT  of  an  arc  is  a  right  line  drawn  from  the  centre  through  one  end  of  the 
arc  to  meet  tlie  tangent  drawn  from  the  other  end ;  thus  CT  is  the  secant  of  the 
arc  AS. 

XXV. 

The  COSECANT  of  an  arc  is  the  secant  of  the  complement  of  that  arc  to  a  quadrant ; 
thus  CG  is  the  secant  of  the  arc  SH,  or  cosecant  of  the  arc  AS. 

XXVI. 

What  an  ra-c  wants  of  a  semicircle  is  called  the  supplement  of  the  arc ;  thus  the 
arc  DHS  is  the  supplement  of  the  arc  AS.  Tlie  sine,  tangent,  or  secant  of  an  arc,  is 
the  same  as  the  sine,  tangent,  or  secant  of  its  supplement ;  thus  the  sine  of  80°  =z  sine 
of  100°,  and  the  sine  of  70°  =  sine  of  110°,  &c. 

XXVII. 

If  one  line,  Alj,fall  any  way  upon  another,  CD,  the  sum  of  the  two  angles,  ABD,  ABC, 
is  always  equal  to  two  right  angles. 

For,  on  the  point  B  as  a  centre,  describe  the  circular  arc  CAD, 
cutting  the  line  CD  in  C  and  D ;  then  (by  Art.  G),  this  arc  is  equal 
to  a  seinichcle,  but  it  is  also  equal  to  the  sum  of  the  arcs  CA  and 
AD,  the  measures  of  the  two  angles  ABC,  ABD ;  therefore  the 
sum  of  the  two  angles  is  equal  to  a  semicircle,  or  two  right  angles.  Hence  it  is 
evident  that  all  tlie  angles  which  can  be  made  from  a  point  in  any  line,  towards  on« 
Bide  of  the  line,  are  equal  to  two  right  angles,  and  that  all  the  angles  which  can  b' 
made  about  a  point,  are  equal  U  four  right  angles. 


J) 


GEOMETRY.  7 

XXVIII. 

If  a  line,  AC,  cross  another,  BD,  in  the  point  E,  the  opposite  angles  will  he  equal,  viz. 
BEA  =r  CED,  and  BEC  =  AED. 

Upon  the  point  E  as  a  centre,  describe  the  circle  ABCD  ;  then 
it  is  evident  that  ABC  is  a  semicircle,  as  also  BCD  (by  AH.  G) ; 
therefore  the  arc  ABC  =:  arc  BCD  ;  taking  from  both  the  common 
arc  BC,  there  remains  arc  AB  =  arc  CD  ;  that  is,  the  angle  BEA 
is  equal  to  the  angle  CED.  After  the  same  manner  we  may  prove 
that  the  angle  BEC  is  equal  to  the  angle  AED. 

XXIX. 

If  a  line,  GH,  cross  two  parallel  lines,  AB,  CD,  it  makes  the  external  opposite  angles 
equal  to  each  other ;  viz.  GEB  =  CFH,  and  AEG  =  HFD. 

For  since  AB  and  CD  are  parallel  to  each  other,  they  may  be 
considered  as  one  hroad  line,  and  GH  crossing  it ;  then  the  vertical 
or  opposite  angles,  GEB,  CFH,  are  equal  (by  Art.  28),  as  also 
AEG  =  HFD. 


XXX.  —j^-^ 

If  a  line,  Gil,  cross  tivo  parallel  lines,  AB,  CD  (see  the  figure),      ^     -^ 

the  alternate  angles,  A^F  and  EFD,  or  CFE  and  FEB, are  equal. 
For  GEB  =:  AEF  [Art.  28),  as  also  CFH  =:  EFD  (by  the  same 

Art),  but  GEB  =  CFH  by  the  last ;  therefore  AEF  is  equal  to  EFD;  in  the  same 

way  may  we  prove  FEB=:  CFE. 

XXXI. 

If  a  line,  GH,  cross  two  parallel  lines,  AB,  CD  (see  the  preceding  figure),  the  external 
angle,  GEB,  is  equal  to  the  internal  opposite  one,  EFD,  or  AEG  equal  to  CFE. 

For  the  angle  AEF  is  equal  to  the  angle  EFD  by  the  last,  and  AEF  =  GEB  (by 
Art.  28) ;  therefore  GEB  =  EFD ;  in  the  same  way  we  may  prove  AEG  rr  CFE. 

XXXII. 

If  a  line,  GH,  cross  tivo  parallel  lines,  AB,  CD  (see  the  preceduig  figure),  the  sum  of 
the  two  internal  angles,  BEF  and  DFE,  or  AEF  and  CFE,  is  equal  to  two  right 
angles. 

For  since  the  angle  GEB  is  equal  to  the  angle  EFD  (by  Ad.  31),  to  both  add  the 
angle  BEF,  and  we  have  GEB  +  BEF  =  BEF  +  EFD  ;  but  GEB  -\-  BEF  =  two 
right  angles  [Art.  27).  Hence,  BEF  -|-  EFD  =:  two  righl  angles ;  and  in  the  same 
manner  we  may  prove  AEF  -f-  CFE  =  two  right  angles. 

XXXIII. 

In  any  triangle,  ABC,  one  of  its  legs,  as  BC,  being  produced  towards  D,  the  externa, 
angle,  ACD,  is  equal  to  the  sum  of  the  internal  and  opposite  angles,  ABC,  BAC. 

To  prove  this,  through  C  draw  CE  parallel  to  AB ;  then,  since 
CE  is  parallel  to  AB,  and  the  lines  AC,  BD  cross  them,  the  angle 
ECD:r=ABC  (by  ^7-/.  31),  and  ACE  =  BAC  (by  ^r^  30);  adding 
these  together  we  have   ECD  4-  ACE  =  ABC  +  BAC  ;   but  B 
ECD  +  ACE  z=  ACD  ;  therefore  ACD  =  ABC  +  BAC. 

XXXIV. 

Hence  it  may  be  proved  that  if  any  two  lines,  AB  and  CD,  be  crossed  by  a  third  line, 
EF,  arid  the  alternate  angles,  AEF  and  EFD,  be  equal,  the  lines  AB  aiia  CD  will  be 
parallel. 

For,  if  they  are  not  parallel,  they  must  meet  each  other  on  one 
side  of  the  line  EF  (su])pose  at  G),and  so  form  the  triangle  EGF,  / 

one  of  whose  sides,  GE,  being  produced  to  A,  the  exterior  angle,  r/ n 

AEF,  must  (by  the  preceding  article)  be  equal  to  the  sum  of  the  

two  angles  EFG  and  EGF  ;  but  by  sui)position  it  is  equal  to  the 
angle  EFG  alone;  therefore  the  angle  AEF  must  be  equal  to 
the  sum  of  the  two  angles  EFG  and  EGF,  and  at  the  same  time 
equal  to  EFG  alone,  which  is  absurd;  therefore  tlie  ImesAB,CD, 
camiot  meet,  and  must  be  parallel. 


GEOMETRY. 


XXXV. 


In  any  right-lined  triangle,  ABC,  the  sum  of  the  three  angles  is  equal  to  ticu  right 
angles. 

To  prove  this,  you  must  produce  BC  (in  the  fig.  ^4/-i. 83)  towards  D;  then  (by^rf.33), 
the  external  angle  ACD  =  ABC  -|-  BAG  ;  to  both  add  the  angle  ACB,  and  we  have 
ACD  -[- ACB  =  ABC  +  B AC  +  ACB ;  but  ACD  -[-  ACB  =  two  right  angles  (by  Jlrt.  27). 
Hence,  ABC-f-BAC-f- ACB  =  two  right  angles;  therefore  the  sum  of  the  tlu'ce  angles 
of  any  plaui  triangle,  ACB,  is  equal  to  two  right  angles. 

XXXVI. 

Hence  in  any  plain  triangle,  if  one  of  its  angles  he  known,  the  sum  of  the  other  two  will 
he  also  knoivn. 

For  by  the  last  article  the  sum  of  all  three  angles  is  equal  to  two  right  angles,  or 
180°;  hence,  by  subtractmg  the  given  angle  from  180°,  the  remainder  will  be  the  sum 
of  the  other  two. 

In  any  right-angled  triangle,  the  two  acute  angles  taken  together  are  just  equal  to  a  right 
angle  ;  for,  all  three  angles  being  equal  to  two  right  angles,  and  one  angle  bemg  right 
by  supposition,  the  sum  of  the  other  two  must  be  equal  to  a  right  angle  ;  consequently, 
any  one  of  the  acute  angles  being  given,  the  other  one  may  be  foimd  by  subtracting 
the  given  one  from  90  degrees. 

XXXVII. 

If  in  any  two  tnangles,  ABC,  DEF,  two  legs  of  the  one,  AB,  AC,  he  equal  to  two  legs 
of  the  other,  DE,  DF,  each  to  each  respectively,  that  is,  AB  =  DE,  and  AC  z=z  DF,  and 
the  angles  BAC,  EDF,  included  between  the  equal  legs  he  equal ;  then  the  remaining  leg 
of  the  one  will  he  equal  to  the  remaining  leg  of  the  other,  and  the  angles  opposite  to  the 
equal  legs  will  be  equcd ;  that  is,  BC  rz:  EF,  ABC  =  DEF,  and  ACB  =  DFE. 

For  if  the  triangle  ABC  be  supposed  to  be  lifted  up  and 

Eut  upon  the  triangle  DEF,  with  the  point  A  on  the  point  -A  ^ 

►,  and  the  hue  AB  upon  DE,  it  is  plain,  since  AB=z:DE, 
that  the  pouit  B  will  fall  upon  E  ;  and  since  the  angles  BAC, 
EDF  are  equal,  the  Ime  AC  will  fall  upon  DF ;  and  these  __ 
lines  being  of  equal  length,  the  pomt  C  will  fall  upon  F ; 
consequently  the  line  BC  will  fall  exactly  upon  the  line  EF,  and  the  triangle  ABC  will 
in  all  respects  be  exactly  equal  to  the  triangle  DEF,  and  the  angle  ABC  wiU  be  equal 
to  the  angle  DEF,  also  the  angle  ACB  will  be  equal  to  the  angle  DFE. 

XXXVIII. 

After  the  same  manner  it  may  be  proved  that  if  in  any  two  triangles,  ABC,  DEF  (see 
the  preceding  figure),  tiuo  angles,  ABC  and  ACB,  of  the  one  he  equal  to  two  angles, 
DEF,  DFE,  of  the  other,  and  the  included  side,  BC,  be  equal  to  EF,  the  remaining  sides 
and  included  angles  tvill  also  be  equal  to  each  other  respectively;  that  is,  ABnrDE, 
AC  =  DF,  and  the  angle  BAC  =  the  angle  EDF. 

For  if  the  triangle  ABC  be  supposed  to  be  lifted  up  and  laid  upon  the  triangle  DEF, 
the  point  B  being  upon  the  point  E,  and  the  line  BC  upon  the  line  EF,  then,  since 
BCinEF,  the  point  C  will  fall  upon  the  point  F ;  and,  as  the  angle  ACB  =  the  angle 
DFE,  the  line  CA  will  fall  upon  the  line  FD  ;  by  the  same  way  of  reasoning,  the  line 
BA  will  fall  upon  the  line  ED;  therefore  the  pouU  of  intersection.  A,  of  the  two  lines, 
BA,  CA,  will  fall  upon  D,  the  point  of  "intersection  of  the  lines  ED,  FD ;  consequently 
AB  =  DE,  AC  =  DF,  and  the  angle  BAC  =  the  angle  EDF. 

XXXIX. 

If  two  sides  of  n  triangle  are  equal,  the  angles  opposite  these  sides  tvill  also  be  equal; 
that  is,  if  AB=z  AC,  the  angles  ABC,  ACB,  tvill  also  be  eqtial. 

For,  draw  the  line  AD,  bisecting  the  angle  BAC,  and  meeting 
the  line  BC  in  D,  dividing  the  triangle  BAC  into  two  triangles, 
ABD,  ACD,  in  which  the  side  AB  =  AC,  the  side  AD  is  common  to 
both  triangles,  and  the  angle  BAD  =:  the  angle  DAC  ;  consequently 
(by  Jlrt.  37),  tlie  angle  ABD  must  be  equal  to  tlie  angle  ACD. 

The  converse  of  this  proposition  is  also  true;  that  is,  if  two  angles  of  a  triangle  are 
equal,  the  opposite  sides  are  also  equal.  This  is  demonstrated  nearly  in  tlie  same 
manner,  by  means  of  Art.  38. 


GEOMETRY. 


XL. 

Any  angle  at  the  circumference  of  a  cirde  is  equal  to  half  the  angle  at  the  centre, 
standing  upon  the  same  arc. 

Thus  the  angle  CAD  is  half  the  angle  BCD,  standing  upon 
the  same  arc,  BD,  of  the  cu'cle  BEDA  whose  centi-e  is  C.  To 
demoustfate  this,  draAV  through  A  and  the  centre  C,  the  right  line 
ACE;  then  (by  Art.  33)  the  angle  CAD  +  angle  CDAr=  angle 
ECD  ;  but  AC  :=  CD  (being  two  radii  of  the  same  circle) ;  therefore 
(by  ^/-^  39),  the  angle  CAD  =  tlie  angle  CD  A,  and  the  sum  of  these 
two  angles  is  the  double  of  either  of  them ;  tliat  is,  CAD  -\-  CDA  — 
tivice  CAD ;  therefore  ECD  =  twice  CAD  ;  in  the  same  manner  it  maybe  proved  that 
BCE  =  twice  BAC.  and  by  adding  these  together,  we  have  ECD  4- BCE  =:  twice 
CAD  +  twice  BA  C  ;'  that  is,  BCD  =  twice  BAD,  or  Bx'VD  equal  to  half  of  BCD.  The 
demonstration  is  sbnilar  Avhen  B,  D,  fall  on  the  same  side  of  E. 

XLI. 

An  angle  at  the  circumference  is  measured  by  half  the  arc  it  subtends. 

For  an  angle  at  the  centre,  standing  on  the  same  arc,  is  measm-ed 
by  the  whole  arc  (by  Art.  11);  but  since  an  angle  at  the  centre  is 
double  that  at  the  circumference  [Art.  40),  it  is  evident  that  an 
angle  at  the  cuxumfei-ence  must  be  measured  by  half  the  arc  it 
stands  upon.  Hence  all  angles,  ACB,  ADB,  AEB,  &c.,  at  the 
circumference  of  a  circle  standing  on  the  same  chord,  AB,  are  equal  to 
each  other ;  for  they  ai-e  all  measm-ed  by  the  same  arc,  viz.  half  the 
arc  AB. 

XLII. 

An  angle  in  a  segment  greater  than  a  semicircle  is  less  than  a  light 
angle. 

Thus,  if  ABC  be  a  segment  gi-eater  than  a  semicii-cle,  the  arc  AC 
on  which  it  stands  must  be  less  than  a  semicircle,  and  the  half  of 
it  less  than  a  quadrant  or  a  right  angle ;  but  the  angle  ABC  in  the 
segment  is  measured  by  the  half  of  the  ai*c  AC ;  therefore  it  is  less 
tlian  a  right  angle. 

An  angle  in  a  semicircle  is  a  right  angle. 

For  since  DEE  is  a  semicircle,  the  arc  DKF  must  also  be  a 
eemicircle ;  but  the  angle  DEF  is  measured  by  half  the  arc  DKF, 
that  is,  by  half  a  semicu'cle  or  by  a  quadi'ant ;  thcrefoi'e  the  angle 
DEF  is  a  right  one. 

An  angle  in  a  segment  less  than  a  semicircle  is  greater  than  a  rigid 
angle. 

Thus,  if  GHI  be  a  segment  less  than  a  semi-circle,  the  arc  GKI 
on  which  it  stands  must  be  gi-eater  than  a  semicircle,  and  its  half 
greater  than  a  quadrant  or  right  angle ;  tlierefore  the  angle  GHI, 
which  is  measured  by  half  the  arc  GKI  is  gi-eater  than  a  riglit 
angle. 

XLIII. 

If  from  the  centre,  C,  of  the  circle  ABE  there  be  let  fall  the  fcrpcndicidar  CD  on  tht 
chord  AB,  it  ivill  bisect  the  chord  in  the  point  D. 

Draw  the  radii  CA,  CB;  ihen  (by  Art.  39)  the  angle  CBA  =  the 
angle  CAB,  and  as  the  angles  at  D  are  right,  the  angle  ACD  must 
be  equal  to  the  angle  BCD  (by  Art.  3G).  Hence  in  the  triangles 
ACD,  BCD,  we  have  the  angle  ACD  equal  to  the  angle  BCD, 
CA  =  CB.  and  CD  common  to  both  triangles,  consequently  (by 
Art.  37)  AD  =  DB  ;  that  is,  AB  is  bisected  at  D. 

XLIV. 

If  from  the  centre,  C,  of  the  circle  ABE  thei-e  be  draivn  a  perpendicular,  CD,  to  thf 
chord  AB,  and  it  be  continued  to  meet  the  circle  in  F,  it  loill  bisect  the  arc  AFB  in  F 
(See  the  preceding  figure.) 

For  in  the  last  article  it  was  proved  that  the  angle  ACD  =r  the  angle  BCD  ;  hence 
(by  Art.  11)  the  arc  AFz=the  arc  FB. 
2 


10  GEOMETRY. 

XLV. 

Any  line  bisecting  a  chord  at  right  angles  is  a  diameter. 

For  since  (by  JlrtA'3)  a  line  dl•a^vn  from  the  centre  pei-pendicular  to  a  chord,  bisects 
that  chord  at  riglit  angles,  therefore  conversely  a  line  bisecting  a  chord  at  right  aiiglea 
ipust  pass  through  the  centre,  and  consequently  be  a  diameter. 

XLVI. 

TTic  sine  of  any  arc  is  equal  to  half  the  chord  of  twice  that  arc. 

For  (in  the  last  scheme)  AD  is  the  sine  of  the  arc  AF,  and  AF  is  equal  to  half  the 
aj'C  AFB,  and  AD  half  the  chord  AB ;  hence  the  proposition  is  manifest. 

XLVII. 

If  two  equal  and  parallel  lines,  AB,  CD,  be  joined  by  two  others,  AC,  BD,  these  ivill  6t 
also  equal  ami  parallel. 

To  demonstrate  this,  joui   the   two   oj)posJte  angles  A  and   D 
with  the  line  AD  ;  then  it  is  evident,  that  the  line  AD  divides  the 
quadrilateral  ACDB  uito  two  triangles,  ABD,  ACD,  in  which  AB 
is  equal  to  CD,  by  sup])osition,  and  AD  is  common  to  both  triangles  ; 
and  since  AB  is  i)arallel  to  CD,  the  angle  BAD  is  equal  to  the  angle 
ADC  (by  Art.  30) ;  therefore,  in  the  two  triangles,  the  sides  AB,  AD,  and  the  angle 
BAD,  are  equal  respectively  to  the  sides  CD,  AD,  and  the  angle  ADC  ;   hence  (by 
Art.  37)  BD  is  equal  to  AC,  and  the  angle  DAC  equal  to  the  angle  ADB ;  therefore 
(by  Art.  34)  the  lines  BD,  AC,  must  be  parallel. 

Co7\  Hence  it  follows,  that  the  quadrilateral  ABDC  is  a  parallelogram,  since  the 
opposite  sides  are  parallel.  It  is  also  evident  that,  in  any  parallelogram,  the  line  joining 
the  opposite  angles  (called  the  diagorud),  as  AD,  divides  the  figure  into  two  equal  parts, 
since  it  has  been  proved  that  the  triangles  x\BD,  ACD,  are  equal  to  each  other. 

XLVIII. 

It  follows  also  from  the  preceding  article,  that  a  triangle,  ACD  (see  the  preceding 
figure),  on  the  same  base,  and  betiveenthe  same  parallels  loith  a  parallelogram,  ABDC,  is 
the  half  of  that  parallelogram. 

XLIX. 

From  the  same  article  it  also  follows,  that  the  opposite  sides  of  a  parallelogram  are 
equal ;  for  it  has  been  proved,  tiiat,  ABDC  being  a  parallelogi-am,  AB  is  equal  to  CD, 
and  AC  equal  to  BD. 

L. 

All  parallelograms  on  the  same  or  equal  bases,  and  between  the  same  parallels,  are  equal 
to  each  other ;  that  is,  if  BD  and  GH  be  equal,  and  the  lines  BH,  AF,  be  parallel,  the 
parallelograms  ABDC,  BDFE,  and  EFHG,  ivill  be  equal  to  each  other. 

For  AC  is  equal  to  EF,  each  being  equal  to  BD  (by  Art.  49) ; 
to  both  add  CE,  and  we  liave  AE,  equal  to  CF ;  therefore  in 
the  two  triangles  ABE,  CDF,  AB  is  equal  to  CD,  AE  is  equal 
to  CF,  and  the  angle  BAE  is  equal  to  DCF  {hy  Art.  31); 
therefore  the  two  trianglesABE,  CDF,  are  equal  (liy  Art.  37), 
and  taking  the  triangle  CKE  from  both,  the  figure  ABKC  is 
equal  to  the  figure  KDFE,  to  both  which  add  tlie  triangle  KBD,  and  we  have  the 
parallelogi-am  ABDC,  equal  to  the  parallelogram  BDFE.  In  the  same  way  it  may  be 
proved  that  the  parallelogram  EFHG  is  equal  to  the  parallelogram  BDFE  ;  therefore 
the  three  parallelograms  ABDC,  BDFE,  and  EFHG,  arc  equal  to  each  other. 

Cor.  Hence  it  follows,  that  triangles  on  the  same  base,  and  between  the  same  parallels, 
are  eqiud,  since  they  are  the  half  of  the  })arallelograms  on  the  same  base  and  between 
the  same  parallels  (by  Art.  48). 

LI. 

In  any  right-angled  triangle,  the  square  of  the  hypotenuse  is  equal  to  the  su77i  of  the 
squares  of  the  two  sides.  Thus,  if  BAG  be  a  right-angled  tiiangle,  the  square  of  the 
hypotenuse  BC,  viz.  BCMH,  is  equal  to  the  sum  of  the  squares  made  on  the  two  sides,  AB 
and  AC,  viz.  to  ABDE  and  ACGF. 

To  demonstrate  this,  through  the  point  A  draw  AKL  per|)cndicular  to  the  hypotenuse 
BC.    Join  AH,  AM,  DC,  and  BG ;  then  it  is  evident,  that  DB  is  equal  to  BA  (by  Art.  18' 


GEOMETRY. 


II 


and  BH  equal  to  BC ;  therefore  iii  the  triangles  DBC,  ABH,  the  two  legs,  DB,  BC,  of 
the  one,  are  equal  to  the  two  legs,  AB,  BH,  of  the  other; 
and  the  moluded  angles,  DBC  and  ABH,  m-e  also  equal  ; 
(for  DBA  is  equ;d  to  CBH,  being  both  right ;  to  each  add 
ABC,  and  we  have  DBC,  equal  to  ABH);  therefore  the 
triangles  DBC,  ABH,  are  equal  (by  Jlrt.S7) ;  but  the  trian- 
gle DBC  is  half  of  the  square  ABDE  (by  ArtA8),  and  the 
ti'iangle  ABH  is  half  the  parallelogram  BKLH  (by  the  same 
article) ;  consequently  the  square  ABDE  is  equal  to  the 
parallelogi-ani  BKLH.  In  the  same  way  it  may  be  proved 
that  the  square  ACGF  is  equal  to  the  parallelogram 
KCML.  Therefore  the  sum  of  the  squares  ABDE  and 
ACGF  is  equal  to  the  sum  of  the  parallelograms  BKLH 
and  KCML;  but  the  sum  of  these  parallelograms  is  equal 

to  the  square  BCJ^HI  ;  therefore  the  sum  of  the  squares  on  AB  and  AC  is  equal  to  the 
squai-e  on  BC. 

Coj:  Hence,  in  any  right-angled  triangle,  if  we  have  the  hypotenuse  and  one  of  the 
legs,  we  may  easily  find  the  other  leg,  by  taking  the  square  of  tlie  given  leg  from  the 
square  of  the  hypotenuse;  the  square  root  of  the  remainder  will  be  the  sought  leg. 
Thus,  if  the  hypotenuse  was  13,  and  one  leg  was  5,  the  other  leg  would  be  12,  for  the 
square  of  5  is  25,  and  the  square  of  13  is  169;  subtracting  25  from  IG'J  leaves  144,  the 
square  root  of  which  is  12.  If  both  legs  are  given,  the  liy])otenuse  may  also  be  found 
by  extracting  the  square  root  of  die  sum  of  the  squares  of  tlie  legs;  thus,  if  one  leg  was 
6,  and  the  other  8,  the  square  of  the  first  is  36,  the  stiuare  of  the  second  is  04  ;  adding 
36  and  64  together  gives  100,  whose  square  root  is  10,  which  is  the  sought  hyi)Otenuse. 

LIL 

Four  quantities  are  said  to  be  proportional,  when  the  magnitude  of  the  Jirst  compared 
tinth  the  second  is  the  same  as  the  magnitudt  of  the  third  compared  with  the  fourth. 

Thus  4,  8,  12,  and  24,  are  proportional,  because  4  is  half  of  8,  and  12  is  half  of  24 ; 
and  if  we  take  equi-multi))les,  A  X  «>  •'i  X  b,  of  the  quantities  a  and  6,  and  other 
equi-multiples,  B  y,  a,  B  Xb,  of  the  same  quantities  a  and  b,  the  four  quantities, 
A  X.  a,  A  S<,  b,  B  y^  a,  B  X  b,  will  be  proportional ;  for  Ay,  a  com])ared  with  Ayb'is 
of  the  same  magnitude  as  a  compared  with  6,  and  By  a  compared  with  B  yb  is  also 
of  the  same  magnitude  as  a  compared  with  b. 


LIIL 

In  any  triangle,  AGg,  if  a  line,  Ee,  be  draivn  parallel  to  either  of  the  sides,  as  Gg,  the 
side  AG  ivill  be  to  AE  05  Ag  to  Ae,  or  as  Gg  to  Ee. 

To  demonstrate  this,  upon  the  line  AG  take 
the  line  AB  so  that  a  certain  multii)le  of  it  may 
be  equal  to  AE,  and  another  multiple  of  it  may 
be  equal  to  AG;  this  may  be  always  done 
accurately  when  AE  and  AG  are  connuensura- 
ble;  if  they  are  not  accurately  commensurable, 
the  quantity  AB  may  be  taken  so  small  that 
certain  multiples  of  it  may  differ  from  AE  and 
AG  respectively  by  quantities  less  dian  any 
assignable.  On  the  line  AG,  take  BC,  CD, 
DE,  EF,  FG,  &c.,   each   equal   to  AB;   and 

through  these  points  draw  the  lines  Bb,  Cc,  &c.,  parallel  to  Gg,  cutting  the  line  Ag  in 
the  points  b,  c,  d,  e,  &c. ;  draw  also  the  lines  BM,  CL,  DK,  &.C.,  parallel  to  Ag,  cutting 
the  former  parallels  in  the  points  N,  O,  P,  &c.,  and  the  line  Gg  in  the  points  M,  L,  K,  &c. 
Thon  the  triangles  ABb,  BCN,  CDO,  &c.,  are  similar  and  equal  to  each  otlier  ;  for  the 
lint*  Bb,  CN,  are  parallel;  dierefore  the  angle  ABb —  BCN  (by  Art.  31),  and  by  the 
san-e  article  the  angle  BAb  is  equal  to  CBN  (becaiLse  BN  is  parallel  to  Ab),  and 
by  i;onstruction  ABz=BC;  therefore  (by  Art.  38)  die  triangles  ABb  and  BCN  are 
eqivi.  to  each  other;  and  in  the  same  manner  we  may  prove  tliat  the  others,  CDO, 
DEP,  EFQ,  &c.,  are  equal  to  ABb.  Therefore  Al)=:  BN=:CO  =  DP,  &,c.,  and 
Bb  ^  CN  =  DO  =  EP,  &c. ;  but  (by  Art.  49)  BN  ==  be,  CO  =  cd,  DP  =  <le  ;  therefore 
Ab —:  be  =  cd  =  de,  &c. ;  and  since  (by  construction)  ABr::BC  =  CD,  &c.,  any  line 
AE  fs  the  same  multijile  of  AB  as  the  corresponding  line  Ae  is  of  Ab;  and  AG  is  the 
same  multiple  of  AB  as  Ag  is  of  Ab ;    therefore  the  lines   AG,  AE,  Ag,  Ae,  aro 


12 


GEOMETRY. 


propoi-tional  {hy  Art.  52] ;  that  is,  AG  is  to  AE  as  Ag  is  to  Ae ;  and  in  a  similai-  raannei 
we  may  prove  that  AG  is  to  AE  as  Gg  is  to  Ee. 


LIV. 

If  any  two  tiiangks,  ABC,  abc,  are  similar,  or  lave  all  the  angles  of  the  one  equal  to 
all  the  angles  of  the  other,  each  to  each  respectively,  thai  ts,CAB  =  cab,  ACB  =  acb, 
ABC  =  abc  ;  the  legs  opposite  to  the  cl  TtZ  angles  ivHl  be  propoHional,  viz.  AB :  ab : :  AC  :  ac ; 
AB  :  ab  : :  BC :  be  ;  and  AC  :  ac  : :  t\J  :  be. 

To  prove  this,  set  ofF,  upon  a  side,  AB,  of  the  largest 
triangle,  AE  =z  ab,  and  through  E  draw  ED  parallel  to  BC, 
to  meet  AC  in  D  ;  then  since  DE,  BC,  are  parallel,  the  angle 
AED  is  equal  to  ABC  (by.^r^  31),  and  this  (by  supposition) 
is  equal  to  the  angle  abc ;  also  the  angle  DAE  is  (by  sup- 
position) equal  to  cab ;  therefore  in  the  triangles  ADE,  abc, 
the  two  angles,  DAE,  AED,  of  the  one,  are  equal  to  the  two 
angles,  cab,  abc,  of  the  other,  each  to  each  respectively,  and 
the  included  side  AE  is  (by  construction)  equal  to  the  included  side  ab ;  therefore 
(by  Jlii.  38)  AD  is  eciual  to  ac,  and  DE  equal  to  be  ;  but  since,  in  the  triangle  ABC,  there 
is  drawn  DE  parallel  to  BC,  one  of  its  sides,  to  meet  the  other  two  sides  in  the  points 
DE,  therefore  (by  the  preceding  article)  AB  :  AE  : :  AC  :  AD,  and  AB  :  AE  : .  BC  :  DE, 
and  AC  :  AD  : :  BC  :  DE ;  if,  in  these  three  proportions,  for  DE  we  put  its  equal 
be,  for  AE  put  ab,  and  for  AD  put  ac,  they  will  become  AB  :  ab : :  AC  :  &c,  and 
AB  :  ab  : :  BC  :  be,  and  AC  :  ac  : :  BC  :  be. 


LV. 

T7ie  chord,  sine,  tangent,  4'c.,  of  any  arc  in  one  circle,  is  to  the  chord,  sine,  tangeyi,  S^c 
of  the  same  arc  in  another,  as  the  radius  of  the  one  is  to  the  7'adius  of  the  other. 

Let  x\BD,  abd,  be  two  circles  ;  BD,  bd,  two  arcs  of 
these  circles,  equal  to  one  another,  or  consisting  of  the 
same  number  of  degrees ;  also  FD,  fd,  the  tangents ; 
BD,  bd,  the  chords ;  BE,  be,  the  sines,  &c.,  of  these 
two  arcs  BD,  bd  ;  and  CD,  cd,  the  radii  of  the  ciixles ; 
then  CD  :  cd  : :  FD  :  fcl,  and  CD  :  cd  : :  BD  :  bd,  and 
CD  :  cd  : :  BE  :  be,  &c.  For  since  the  arcs  BD,  bd, 
are  equal,  the  angles  BCD,  bed,  are  also  equal,  and 
FD,  fd,  being  tangents  to  the  points  D  and  d,  the 
angles  CDF,  cdf,  are  each  equal  to  a  right  angle  (by 
Art.  22) ;  therefore,  since  in  the  two  triangles  CDF, 
cdf,  the  two  angles  FCD,  CDF,  of  the  one,  are  equal 
to  the  two  angles,  fed,  cdf,  of  the  other,  each  to 
each,  the  rcmainuig  angle,  CFD,  is  also  equal  to  the 
remaining  angle,  cfd  (by  Art.  36) ;  consequently  the 
triangles  CFD,  cfil,  are  similar.  The  triangles  BCD,  bed,  are  also  similar,  for  the 
angle  CBD  is  equal  to  the  angle  CDB,  being  each  subtended  by  the  radius ;  therefore 
(by  Art.  3G),  eacli  of  these  angles  is  equal  to  half  the  supplement  of  the  angle  BCD ; 
and  in  the  same  manner  the  angle  cbd  or  cdb  is  equal  to  half  the  supi)lement  of  the 
angle  bed ;  and  since  the  angle  BCD  is  equal  to  bed,  the  angles  of  these  two  triangles 
must  be  equal ;  consequently  they  are  similar.  The  triangles  BCE,  bee,  are  also  similar, 
because  BE  is  parallel  to  FD,  and  be  parallel  to  fd.  Hence  we  obtain  (by  Art.  54)  the 
followmg  analogies.     CD  :  cd  : :  FD  :  fd  ;  CD  :  cd  : :  BD  :  bd  ;  CB  :  cb  :  :BE  :  l^e,  &c 


LVl. 

Let  ABD  be  a  quadrant  of  a  circle,  described  by  the  radius  CD,  BD 
any  arc  of  it,  BvV  its  complement,  BG  or  CF  the  sine,  CG  or  BF  the 
cosme,  DE  tlie  tangent,  AH  the  cotangent,  CE  the  secant,  and  CII  the 
cosecant  of  that  arc  BD.  Then,  since  the  triangles  CDE,  CGB,  are 
similar,  we  shall  have  (by  Art.  54)  DE  :  CE  : :  BG  :  CB ;  that  is,  the 
tangent  of  an  arc  is  to  secant  of  the  same  as  tlie  sine  of  it  is  to 
radius.  Also,  CE  :  CD  ; :  CB  :  CG ;  that  is,  the  secant  is  to  radius  as 
tlie  radius  to  the  cosine  of  the  arc.  Also,  CF  :  CA  ::  CB  :  ClI ;  that  is,  tlie  sine  is  to 
radius  as  radius  to  the  cosecant  of  the  arc;  and  since  the  triangle  CAII  is  similar  to 
the  triangle  CDE,  -we  have  AH  :  CA  : :  CD  :  DE  ;  that  is,  the  cotangent  is  to  the  radius 
as  die  radius  to  the  tangent  of  the  arc 


GEOMETRY. 


13 


LVII. 

In  all  circles,  the  snie  of  90°,  the  tangent  of  45°,  and  the  chord  of  60°,  are  each  equal  to 
the  radius. 

For,  in  the  circle  DFAEB,  let  the  arc  BE  be  45°,  the  arc  BA  60°, 
and  BF  90°.  Draw  through  the  centre,  C,  the  diameter  DCB,  and 
perpendicular  thereto  the  tangent  BG,  meeting  CE  produced  in  G ; 
di"aw  the  chord  BA,  and  jom  CF,  CA.  Then,  since  the  arc  BF  is 
90°,  DF  must  be  90° ;  whence  (by  Art.  12  and  19)  the  radius  CF 
is  equal  to  the  sine  of  the  arc  BF,  or  sine  of  90°.  Again,  in  the 
ti-langle  CBG,  since  the  angle  CBG  is  90°,  and  BCG  is  45°,  by 
supposition,  the  angle  CGB  is  also  45°  (by  Art.  36) ;  therefore  (liy  A)i.  39)  BG  is  equal 
to  CB  ;  that  is,  the  tangent  of  45°  is  equal  to  the  I'adius.  Again,  the  angle  ACB  is  60° 
(being  measured  by  the  arc  BA),  and  the  angle  CBA  is  also  60°  (being  measured  by 
half  the  arc  AD  =:  120°,  by  Art.  40) ;  therefore  (by  Art.  39)  CA  =  AB ;  that  is,  the  chord 
of  60°  is  equal  to  the  i-adius. 


The  four  following  propositions  contain  the  demonstration  of  the  iidcs  by  which  all 
the  calcidations  of  ti'igonometry  may  be  made ;  they  are  inserted  here  in  order  to 
prevent  any  embarrassment  of  the  young  calculator,  from  the  uitroduction  of  t'<e 
demonstrations  among  the  precepts  for  calculation. 

LVIII. 

In  any  plane  triangle,  the  sides  are  propoHional  to  the  sines  of  the  opposite  angles 

Let  ABC  be  the  triangle  ;  produce  the  shorter  side,  AB,  to 
F,  making  AF  equal  to  BC  ;  from  B  and  F  let  fall  the 
perpendiculars  BD,  FE,  upon  AC  (produced  if  necessary) ; 
then  FE  is  the  sine  of  the  angle  A,  and  BD  is  the  sine  of  the 
angle  C,  the  radius  bemg  BC,  equal  to  AF  ;  now,  the  triangles 

ABD,  AFE,  having  the  angle  A  common  to  both,  and  the 
angle  D  equal  to  the  angle  E  (being  each  equal  to  a  right 
angle),  are  similar  ;  hence  [by  Art.  5i),  as  AF  (or  its  equal  BC) 
is  to  AB,  so  is  FE  to  BD  ;  that  is,  BC  is  to  AB  as  the  sine  of 
the  angle  A  is  to  the  sine  of  the  angle  C. 

LIX. 

In  any  triangle  [supposing  any  side  to  he  the  base,  and  calling  the  other  tioo  the  sidcS) 
the  sum  of  the  sides  is  to  their  difference  as  the  tangent  of  half  the  sum  of  the  angles  at  the 
ba^e  is  to  the  tangent  of  half  the  difference  of  the  same  angles. 

Thus,  in  the  triangle  ABC,  if  we  call  AB  the  base,  it  will  be.  As 
the  sum  of  AC  and  CB  is  to  then-  difference,  so  is  the  tangent  of 
half  the  sum  of  the  angles  ABC,  BxlC,  to  the  tangent  of  half  then- 
difference. 

Dem.  With  the  longest  leg,  CB,  as  radius,  describe  a  cb-cle 
about  the  centre,  C,  meeting  the  shorter  side,  AC  (produced  on  each 
side),  in  the  points  D  and  E  ;  join  EB,  DB ;  draw  AH  perpendicular 
to  DB,  and  AF  perpendicular  to  EB ;  then  (by  Art.  42)  the  angle 
EBD,  being  in  a  semicircle,  is  a  right  angle ;  and  the  triangles  AHD,  AFE,  are  similar, 
and  AF  is  equal  to  HB.  Moreover,  since  CB  is  equal  to  CD  or  CE,  AD  is  the  sum 
and  AE  is  the  difference  of  the  legs  AC,  CB  ;  likewise  (by  Art.  .33)  the  angle  BCD  is 
equal  to  the  sum  of  die  angles  BAC,  ABC,  and  thei-efore  (by  Art.  40),  the  angle  DEB, 
or  its  equal  DAII,  is  equal  to  half  the  sum  of  the  angles  at  the  base  ABC,  BAC, 
Again  [by  Art.  33)  the  angle  BAC  is  equal  to  the  sum  of  the  angles  CEB  (or  CBE)  and 

ABE,  and  therefore  is  equal  to  the  sum  of  the  angle  ABC,  and  twice  the  angle  ABE  ; 
hence  the  angle  ABE,  or  its  equal  BAH,  is  equal  to  half  the  difference  of  die  angles 
at  the  base.  But  in  the  right-angled  triangles  AHD,  AHB,  making  AH  radius,  the 
legs  DH,  HB,  are  the  tangents  of  the  angles  DAH,  BAH,  or  the  tangents  of  half  the 
sum  and  half  the  difference  of  the  angles  at  the  base  ;  but  l)y  reason  of  the  similar 
triangles  AHD,  AFE,  we  have  AD  :  AE  ::  DH  :  AF  or  HB ;  that  is,  AD,  the  sum  of 
the  legs,  AC  and  CB,  is  to  AE,  their  difference,  as  DH,  the  tangent  of  half  the  sum 
of  the  angles  at  die  Ijase  (the  radius  being  AH),  is  to  HB,  the  tangent  of  half  the 
difference  of  the  same  angles  (to  the  same  radius);  and  therefore  [by  Art.  55)  as  the 
tabular  tangent  of  half  the  sum  of  the  angles  at  the  biise  is  to  the  tabular  tangent  of 
half  the  difference  of  the  same  angles. 


14  GEOMETRY. 

LX. 

In  any  plane  triangle,  ABC,  if  the  line  CD  be  di-awn 
perpendicular  to  the  base,  AB,  dividing  it  into  two 
segments,  AD,  DB,  and  the  base,  AB,  be  bisected  in  the 
point  H,  Ave  shall  have, 

As  the  base,  AB,  is  to  the  sum  of  the  sides,  AC,  BC,  so 
is  the  difference  of  tJie  sides  to  twice  the  distance,  DH,  of 
the  perpendicular  from  the  middle  of  the  base. 

Dem.  With  the  greater  side,  CB,  as  radius,  describe  about  the  centre,  C,  the  circle 
BFGE,  meeting  the  other  side  produced  in  the  points  E  and  F,  and  the  base  AB 
produced  in  G ;  join  GF  and  BE.  Then  AE  is  the  sum,  and  AF  the  difference,  of 
the  sides  AC,  CB ;  and  since  CD  is  perpendicular  to  GB,  the  Ime  GB  is  bisected  in  D 
(by  Art.  43),  and  as  AB  is  bisected  in  H,  the  line  xlG  is  equal  to  twice  DH.  Now,  in 
the  triangles  BAE,  GAF,  the  angles  ABE,  GFA,  are  equal  (by  Art.  41),  and  the  angle 
BAE  is  equal  to  GAF  (by  Art.  28) ;  therefore  the  remaming  angles  AEB,  AGF, 
ai"e  equal,  and  the  triangles  BAE,  GAF,  are  similar ;  consoquendy  (by  Art.  54) 
AB  :  AE  : :  AF  :  AG,  or  twice  HD,  which  is  the  proposition  to  be  demonstrated. 
Having  thus  obtained  HD,  we  may  find  the  segments  AD,  DB,  by  adding  HD  to  the 
half  base  HA  or  HB,  and  by  taking  theii*  difference. 

LXI. 

In  any  plane  triangle,  the  square  of  radius  is  to  the  square  of  the  cosine  of  half  of  either 
of  the  angles,  as  the  rectangle  contained  by  the  two  sides  including  that  angle,  is  to  the 
rectangle  contained  by  the  half  sum  of  the  sides,  and  that  half  sum  decreased  by  the  side 
opposite  to  that  angle. 

Thus,  in  the  triangle  CBE,  the  square  of  radius  is  to  the  square 
of  the  cosine  of  half  the  angle  C,  as  the  I'ectangle  CB  X  CE  is  ^ 

(CB  +  CE  +  BE)      (CB  +  CE  — BE)    „  ^^  ^ 

to  ^ ^ ! '  X .     For  continue  EC  to 

2  2  -^  o    HD 

A,  making  CA  =  CB ;  di-aw  BD  perpendicular  to  CE  ;  bisect  CE 
in  H,  and  join  AB.     Then  (supposing  CB  to  be  greater  than  EB)  we  have  (by  Art.  60) 

r;g2 BE2 

CE  :  CB  +  BE  : :  CB  —  BE  :  ■ i=2  X  HD  ;  by  adding  half  this  to  half  the 

CE 

base  rz:  CH,  we  have  the  segment  CD  =  J^ ;  to  this  adding  CA  or 

2XCE 

^„  ,  .P,        CB2  — BE^  +  CE2  +  2CEXCB        (CB  +  CE)3  —  BE2 

CB,   we   have   AD  =  • i ! — rr  i ! '- ■=. 

2  X  CE  2  X  CE 

(CB  +  CE  +  BE)X(CB  +  CE-BE)  AD  =  AC  +  CD  =  CB  +  CD; 

2XCE  ^  ^  ^ 

hence    AD2  —  CB^-f  2CB  X  CD  -}-  CD^  ;     also,    ^YT-  —  CB2  _  CD^  ;     hence 
AB'-  —  AD2  -|-  BD3  =  2  X  CB^  -f  2CB  X  CD~2CB  X  (CB-t-CD)  =  2CB  X  AD  ; 

hence  AB^  :  AD^  ::  2CB  :  AD  =  (CB  +  CE  +  BE)  X  (CB  +  CE -BE)     ^^^^  ^^ 

2XCE 
being  radius,  x\D  is  the  cosine  of  the  angle  A,  whi'(;h  is  equal  to  half  the  angle  C  (by 
^rf.40);  therefore  the  square  of  radius  is  to  the  square  of  the  cosine  of  half  the  angle  C, 

m  the  rectangle  CE  X  CB  is  to  the  rectangle  (CB+CE  +  BE)       (CB  +  CE— BE\ 

»  2  2 

The  other  cases  of  this  proposition  may  be  demonstrated  in  the  same  manner 


GEOMETRICAL   PROBLEMS. 


15 


GEOMETRICAL    PROBLEMS. 


PROBLEM  I. 

To  draw  a  right  line,  CD,  parallel  to  a  given  i-ight  line,  AB,  at  any  given  distance,  as  at 

the  point  D. 

With  a  pair  of  compasses  take  the  nearest  distance  between  the      — .^—^ 

point  D  and  the  given  right  Une,  AB ;   with  that  distance  set  one       •'    ^    '"•     -^ 

foot  of  tiie  compasses  any  where  on  the  hue  AB,  as  at  A,  and  ch'aw      . "■■ ^^ 

the  arc  C  on  the  same  side  of  the  Une  AB  as  the  jjoint  D ;  from  the  -^  -^ 

point  D  (h-aw  a  Mne  so  as  just  to  touch  the  arc  C,  and  it  is  done ;  for  the  hne  CD  will 
be  parallel  to  the  line  AB,  and  at  the  distance  of  the  pouit  given,  D,  as  was  requu'ed. 


A    i 


F\ 


\ 


/K 


K\     B 


JO 


PROBLEM   II. 

To  bisect  or  divide  a  given  line,  AB,  into  two  equal  parts. 
Take  any  distance  in  your  compasses  greater  than  half  the  line 
AB  ;  then,  with  one  foot  in  B,  describe  the  arc  CFD  ;  with  the  same 
distance,  and  one  foot  m  A,  describe  tlie  arc  CGD,  cutting  the 
former  aic  in  C  and  D ;  di'aw  the  line  CD,  and  it  will  bisect  AB  in 
the  point  E. 

PROBLEM  III. 

To  erect  a  perpendicular,  BA,  on  the  end  of  a  given  right  line,  DB 
Take  any  extent  in  your  compasses,  and  with  one  foot  in  B  fix 
the  otiier  in  any  point,  C,  without  the  given  line  ;  then,  with  one 
point  of  the  compasses  in  C,  descriljc,  with  tiie  other,  the  cu'cle 
ABD ;  through  D  and  C  draw  the  diameter  DCA,  meeting  the 
circle  in  A ;  join  B  and  A,  and  it  is  done ;  for  BA  will  be  the 
required  line  (by  Art.  42,  Georaetiy). 

Or  thus ; 

Take  any  convenient  distance,  as  BH,  in  your  compasses,  and, 
with  one  foot  in  B,  descrilie  the  arc  HFG,  ui)on  which  set  off  the 
same  distance  as  a  chord  from  H  to  F,  and  from  F  to  G,  upon  F 
and  G,  des<;ribe  two  arcs  intersecting  each  other  in  X ;  draw  a  line 
from  B  to  A,  and  it  is  done ;  for  BA  will  be  the  peqiendicular 
required.  ^'  ^ 

PROBLEM  IV. 

Fro^n  a  given  point,  as  C,  to  let  fall  a  perpendicidar,  CO,  on  a  given  right  line,  AB. 

Take   any   extent    in   your   compasses    greater  than  the  least  C 

distance  between  C  and  the  given  line  AB;  with  one  foot  in  C, 
describe  an  arc  to  cut  the  given  line,  AB,  in  F  and  G;  with  one  foot 
in  G,  {lcocril)e  an  arc,  and  with  the  same  distance,  and  one  foot  in 
F,  describe  another  arc  cutting  the  former  in  D  ;  from  C  to  D  draw 
the  line  COD,  cutting  AB  in  O ;  then  CO  will  be  the  pei-pendicular 
requii-ed. 

PROBLEM  V. 

Pi-om  a  given  point,  C,  to  let  fall  a  perpendicidar,  CB,  on  a  given  line,  AB,  when  the 
perpendicular  is  to  fall  so  near  the  end  of  the  given  line  that  it  cannot  be  done  as 
above. 

Upon  any  point.  A,  of  the  line  AB  as  a  centre,  and  with  the 
distance  AC,  describe  an  arc,  E ;  choose  any  other  point  in  the 
line  AB,  as  D,  and  with  the  distance  DC  describe  anodier  arc 
intersecting  the  former  in  E  ;  join  CE  cutting  A  B  in  B,  and  it  is 
done ;  for  CB  will  be  the  perpendicular  required. 


.F 


A- 


^B 


D 


16 


GEOMETRICAL   PROBLEMS. 


D 


H 


G 


-A 


PROBLEM  VL 

To  make  an  angle  that  shall  contain  any  pi  oposed  number  of  degrees,  from  a  given  poini 

in  a  given  line. 

Case  I.  When  the  given  angle  is  riglit,  or  contains  90°,  let  CA  be  the  given  line 
and  C  the  given  point. 

On  C  erect  a  pei'pendicular,  CD,  and  it  is  done ;  for  the  angle 
DC  A  is  an  angle  of  90°.  Or  thus ;  on  the  point  C,  as  a  centre,  with 
the  chord  of  (50°*,  describe  an  arc,  GH,  and  set  off  thereon,  from 
G  to  H,  the  distance  of  the  chord  of  90°,  and  from  C  through  H 
di-aw  CHD,  which  will  form  the  angle  DC  A  of  90°  requu-ed. 

Case  2.  When  the  angle  is  acute,  as,  for  example,  36°  30',  let 
CB  be  the  given  line,  and  C  the  pouit  at  which  the  angle  is  to  be 
made. 

With  the  chord  of  60°  in  your  compasses,  and  one  foot  on  C,  as 
a  centre,  draw  the  arc  FB,  on  which  set  off,  from  B  to  F,  the 
given  angle,  36^°,  taken  from  the  line  of  chords;  through  F  and 
the  centre  C,  draw  the  right  line  AC,  and  it  is  done ;  for  the  angle  ACB  will  be  an 
angle  of  36°  30',  as  was  required. 

Case  3.  When  the  given  angle  is  obtuse,  as,  for  examjjle,  127°,  let  CB  be  the  given 
line,  and  C  the  angular  point. 

Take  the  chord  of  60°  in  yoin*  compasses,  and  with  one  foot  on 
C  as  a  centre,  describe  an  arc,  BGHE,  upon  which  set  off  the  chord  \e  H 

of  60°  (which  you  already  have  in  your  compasses)  from  B  to  G, 
and  from  G  to  H  ;  then  set  oif  from  G  to  E  the  excess  of  tlie  given  ^,     _g 

angle  above  60°,  which  is  67°,  taken  from  the  luie  of  chords ;  or 
you  may  set  off,  from  H  to  E,  the  excess  of  the  given  angle  above  120,  which  is  7° ; 
draw  the  Ihie  CE,  and  it  is  done ;  for  the  angle  ECB  will  be  an  angle  of  127° 

Were  it  required  to  measure  a  given  angle,  the  process  would  have  been  nearly  the 
sf,me,  by  sweeping  an  arc,  as  BE,  and  mcasurmg  it  on  the  line  of  chords,  as  is  evident. 


PROBLEM   VII. 

To  bisect  a  given  arc  of  a  circle,  AB,  ivliose  centre  is  C 
Take  in  your  compasses  any  extent  gi'eater  tnan  the  half  of  AB, 
and,  with  one  foot  in  A,  describe  an  arc ;  with  the  same  extent, 
and  one  foot  in  B,  describe  another  arc,  cutting  the  former  in  D ; 
join  CD,  and  it  is  done ;  for  this  line  will  bisect  the  arc  AB  in  the 
point  E.  It  is  also  evident  that  the  line  CD  bisects  the  angle  BCA, 
or  divides  it  into  two  equal  parts. 


,i:¥P 


PROBLEM  VIIL 

To  fnd  the  centre  of  a  given  circle. 
With  any  radius,  and  one  foot  in  the  circumference,  as  at  A,  describe  an  ai'C  ol 
a  circle,  as  CBD,  cutting  the  given  circle  in  B ;  with  the  same 
extent,  and  one  foot  in  B,  describe  another  arc,  CAD,  cutting  the 
former  in  C  and  D ;  through  C  and  D  draw  the  line  CD,  which 
will  pass  through  the  centre  of  the  circle;  in  like  manner  may 
another  right  line  be  draAvn,  as  EIIG,  which  shall  cross  the  first 
right  line  at  the  centre  required.  This  construction  depends  upon 
Jlrt.  43  of  Geometry. 

PROBLEM   IX. 

To  draw  a  circle  through  any  three  given  points  not  situated  in  a  right  line. 
Let  A,  B,  and  D,  be  the  given  points.  Take  in  your  compasses 
any  distance  greater  than  half  AB,  and,  with  one  foot  in  A,  describe 
an  arc,  EF ;  with  the  same  extent,  and  one  foot  in  B,  describe 
another  arc  cutting  the  former  in  the  ])oints  E,  F,  through  which 
draw  the  indefinite  right  line  EFC  ;  then  take  in  your  compasses 
any  extent  greater  than  half  BD,  and,  with  one  loot  in  B,  describe 
an  arc,  GII ;  with  the  same  extent,  and  one  foot  in  D,  describe 


*  For  a  descrlptioji  of  the  line  of  chords,  see  pag'e  IS. 


PlaieYL 


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1B61 


GEOMETRICAL    PROBLEMS. 


17 


another  ai-c  cutting  the  former  in  tlie  points  G,  II,  through  which  ch-iw  tlio  right  Hne 
GHC,  cutting  the  former  right  line  EFC  in  the  point  C  ;  upon  the  ])oiiit  C  as  a  centre 
with  an  extent  equal  to  CA,  CB,  or  CD,  as  radius,  describe  the  sought  circle. 


PROBLEM  X. 

7'o  divide  a  circle  into  2,  4,  8,  IC,  or  32  equal  parts. 
Draw  a  diaineter  through  the  centre,  dividuig  the  circle  into  two 
equal  parts ;  bisect  this  diameter  by  another,  drawn  perpendicular 
tliereto,  and  the  circle  will  be  div'ided  into  four  eijual  i)arts  or 
quadrants;  bisect  each  of  these  quadrants  again  by  right  lines 
di'awn  through  the  centre,  and  the  cu'cle  will  be  divided  into  eight 
equal  j)arts ;  and  so  you  may  continue  the  bisections  any  number 
of  times.  This  problem  is  useful  in  constructing  the  mariner's 
compass. 

PROBLEM  XL 

To  divide  a  given  line  into  any  nwnber  of  equal  parts 
Let  it  be  required  to  divide  the  line  AB  into  five  ecjual  })arts. 
From  the  pohit  A  draw  any  line,  AD,  making  an  angle  with  the 
lin«  AB ;  then  through  the  pohit  B  draw  a  line",  BC,  parallel  to  AD  ; 
and  from  A,  with  any  small  ojjening  in  your  compasses,  set  off  a 
number  of  equal  parts  on  the  line  AD,  less  by  one  than  the  proposed 
number  (which  number  of  equal  parts  in  this  examjtle  is  4);  then 
from  B,  set  off  the  same  number  of  the  same  parts  on  the  line  BC; 
then  join  4  and  1,  3  and  2,  2  and  -3.  I  and  4,  and  these  lines  will 
cut  the  given  line  as  requii-ed. 


J  ,  f- 


18 


CONSTRUCTION    OF    THE    PLANE    SCALE 


Isi.  With  the  nuliu5  you  intend  for  your  scale,  describe  a  semicircle,  ADB  (Plate  II 
fig.  1),  and  from  the  centre,  C,  draw  CD  perpendicular  to  AB,  which  will  divide  the 
semicu-cle  into  two  quadrants,  AD,  BD ;  continue  CD  towards  S,  draw  BT  perpen- 
dicular to  CB,  and  join  BD  and  AD. 

2dly.  Divide  the  quadrant  BD  into  9  equal  parts ;  then  will  each  of  these  be  10 
degrees;  subdivide  each  of  these  parts  into  single  degrees,  and,  if  your  radius  will 
admit  of  it,  into  minutes  or  some  aliquot  parts  of  a  degree  greater  than  mmutes. 

3dly.  Set  one  foot  of  the  comi)asscs  in  B,  and  transfer  each  of  the  divisions  of  the 
quadrant  BD  to  the  right  line  BD,  tlum  will  BD  be  a  line  of  chords. 

4thly.  From  the  points  10,20,  30,  &c.,  in  the  quadrant  BD,  draw  right  lines  parallel 
to  CD,  to  cut  the  radius  CB,  and  they  will  divide  that  luie  into  a  line  of  sines  which 
must  be  ninnbered  from  C  towards  B. 

5thly.  If  the  same  line  of  sines  be  numbered  from  B  towards  C,  it  will  become  a  line 
of  versed  sines,  which  may  be  continued  to  180°,  if  the  same  divisions  be  transferred 
on  the  same  line  on  the  other  side  of  the  centi'e  C. 

Gthly.  From  the  centre  C,  through  the  several  divisions  of  the  quadrant  BD,  draw 
right  lines  till  they  cut  the  tangent  BT ;   so  will  the  line  BT  become  a  line  of  tangents. 

7thly.  Setting  one  foot  of  the  compasses  in  C,  extend  the  other  to  the  several 
divisions,  10,  20,  30,  &c.,  in  the  tangent  line,  BT,  and  transfer  these  extents  severally  to 
the  right  line,  CS  ;  then  will  that  hue  be  a  line  of  secants. 

8thly.  Right  lines  drawn  from  A  to  the  several  divisions,  10,  20,  30,  &c.,  in  the 
quadrant  BD,  will  divide  the  radius  CD  into  a  line  of  semi-tangents. 

Dthly.  Divide  the  quadrant  AD  into  eight  e(]ual  parts,  and  from  A,  as  a  centre, 
tj-ansfer  these  divisions  severally  into  the  line  AD  ;  then  will  AD  be  a  line  of  rhumbs, 
each  division  answering  to  11°  15'  upon  the  line  of  chords.  The  use  of  this  line  is  for 
protracting  and  measuring  angles,  according  to  the  connnon  division  of  the  marmer's 
compass.  If  the  radius  AC  be  divided  into  100  or  1000,  &c.,  equal  parts,  and  tlie 
lengths  of  the  several  sines,  tangents,  and  secanis,  corresj)onding  to  the  several  arcs  of 
the  quadrant,  be  measured  thereby,  and  these  numbers  be  set  down  in  a  table,*  each 
in  its  proper  column,  you  will  by  these  means  have  a  collection  of  numbers  by  which 
the  several  cases  in  trigonometry  may  be  solved.  Right  lines,  graduated  as  above, 
beuig  f)laced  severally  upon  a  ruler,  form  the  instrument  called  the  Plane  Scale  'see 
Plate  II.  fig.  2),  by  wliich  the  lines  and  angles  of  all  triangles  may  be  measured.  All 
right  lines,  as  the  sides  of  plane  triangles,  &c.,  wlicn  they  are  considered  sunply  as 
such,  without  having  any  relation  to  a  circle,  are  measured  by  scales  of  equal  parts, 
one  of  which  is  subdivided  e(iually  into  10,  and  this  serves  as  a  common  division  to 
all  the  rest.  In  most  scales,  an  inch  is  tak(Mi  for  a  common  measure,  and  what  an  inch 
is  di\  ided  into  is  generally  set  at  the  end  of  the  scale.  By  any  conunon  scale  of  equal 
parts,  divided  in  this  manner,  any  number  less  than  100  may  be  readily  taken  ;  but  if 
the  number  should  consist  of  three  places  of  figures,  the  value  of  the  third  figure 
cannot  be  exactly  ascertained,  and  in  this  case  it  is  better  to  use  a  diagonal  scale,  by 
which  any  number  consisting  of  three  places  of  figures,  may  be  exactly  found.  'The 
figiu-e  of  this  scale  is  given  in  Plate  11.  fig.  3;.  its  construction  is  as  follows: — 

Having  jjrepared  a  ruler  of  convenient  breadth  (or  your  scale,  draw  near  the  edges 
thereof  two  right  lines,  af,  eg,  parallel  to  each  other ;  divide  one  of  these  lines,  as  of 
into  equal  parts,  according  to  the  size  of  your  scale;}  and,  through  each  of  these 
divisions  draw  right  lines  perpendicular  to  q/",  to  meet  eg*;  then  divide  the  breadth  into 
10  equal  ])arts,  and  through  each  of  these  divisions  draw  right  lines  parallel  to  af  and 
eg;  divide  the  lines  ab,  cd,  into  10  etjual  parts,  and  from  the  pomt  a  to  the  first  division 

*  In  Tabic  XXIV.  are  given  the  sine  and  cosine  to  every  iniinite  of  the  quadraJit,  to  five  places  of 
decimals. 

t  The  length  of  one  of  these  equal  parts  at  the  end  of  the  scale  to  which  this  dcscriiJtion  refers  is  ah 
or  cd ;  the  length  of  one  of  liie  equal  parts  of  the  scale  of  the  other  end  being  ihe  half  of  cib. 


COiNSTRUCTION   OF   THE   PLANE   SCALE.  19 

in  tlie  line  cd,  draw  a  diagonal  line ;  then,  parallel  to  that  line,  draw  diagonal  linos 
through  all  the  other  divisions,  and  the  scale  is  complete.  Then,  if  any  number, 
consisting  of  three  places  of  figures,  as  256,  be  required  from  the  larger  scale,  gd,  you 
must  place  one  foot  of  the  compasses  on  the  figure  2  on  the  line  gd,  then  the  extent 
from  2  to  the  point  d  will  represent  200.  The  second  figure  being  5,  count  five  of  the 
smaller  divisions  from  d  towards  c,  and  the  extent  from  2  to  that  point  will  be  250. 
Move  both  points  of  the  compasses  downwards  till  they  are  on  the  sixth  parallel  line 
below  gd,  and  open  them  a  little  till  the  one  pohit  rests  on  the  vertical  line  dra^vn 
througli  2,  and  the  other  on  the  diagonal  line  drawn  through  5 ;  the  extent  then  in  the 
compasses  will  represent  256.  In  the  same  way  the  quantities  25.6,  2.56,  0.256,  &c., 
are  measured. 

Besides  the  lines  already  mentioned,  there  is  another  on  the  Plane  Scale,  marked 
ML,  which  is  joined  to  a  line  of  chords,  and  shows  how  many  miles  of  easting  or 
westing  correspond  to  a  degree  of  longitude  in  every  latitude.*  These  several  lines  are 
generally  put  on  one  side  of  a  ruler  two  feet  long ;  and  on  the  other  side  is  laid  do\vn 
a  scale  of  the  logai-ithms  of  the  sines,  tangents,  and  numbers,  which  is  commonly 
called  Gunter's  Scale ;  and,  as  it  is  of  general  use,  it  requires  a  particular  descri[)tion. 

*  As  it  would  confuse  the  adjoined  figure  to  describe  on  it  the  line  of  longitudes,  it  is  neglected,  but 
tlie  crnslruction  is  as  follows  ;  divide  the  line  CB  into  GO  equal  parts  (if  it  can  be  done),  and  through 
each  point  draw  lines  parallel  to  CD,  to  intersect  the  arc  BD ;  about  15,  as  a  centre,  transfer  the  severiil 
points  of  intersection  to  the  line  of  chords,  BD,  and  then  number  it  from  D  towards  B,  from  0  to  GO, 
aiid  it  will  be  the  line  of  longitudes,  corresponding  to  the  degrees  on  the  line  of  cliords. 


20 


GL'NTER'S    SCALE. 


On  Gunter's  Scale  are  eight  lines,  viz. 

1st.  Sine  rhumbs,  marked  (SR),  correspontling  to  the  logarithms*  of  the  natural 
smes  of  every  point  of  the  maruier's  compass,  numbered  from  the  left  hand  towards 
the  right,  with  1,  2,  3,  4,  5,  6,  7,  to  8,  where  is  a  brass  pm.  This  line  is  also  divided, 
where  it  can  be  done,  into  halves  and  quarters. 

2dly.  Tangent  rhumbs,  marked  (TR),  correspond  to  the  logarithms  of  the  tangents  of 
eveiy  point  of  the  compass,  and  are  numbered  1,  2,  3,  to  4,  at  the  right  hand,  where 
there  is  a  })in,  and  thence  towards  the  left  hand  with  5,  0,  7  ;  it  is  also  divided,  where 
it  can  be  done,  mto  lialvcs  and  quarters. 

3d]y.  The  line  of  numbers,  marked  (Num.),  corresponds  to  the  logarithms  of  numbers, 
and  is  marked  thus:  near  tlie  left  hand  it  begins  at  1,  and  towards  the  right  hand  are 
2,  3,  4,  5,  G,  7,  8,  9 ;  and  1  in  the  middle,  at  which  is  a  brass  pin ;  then  2,  3,  4,  5,  6, 
7,  8,  9,  and  10,  at  the  end,  where  there  is  another  pin.  The  values  of  these  numbers 
and  their  intermediate  divisions  depend  on  the  estimated  values  of  the  extreme  numbers 
1  and  10 ;  and  as  this  line  is  of  great  imy)ortance,  a  particular  description  of  it  Avill  be 
given.  The  first  1  may  be  counted  for  1,  10,  100,  or  1000,  &c.,  and  then  the  next  2 
will  be  2,  20,  200,  or  2000,  &c.,  respectively.     Again,  tlie  first  1  may  be  reckoned 

1  tenth,  1  hundredth,  or  1  thousandth  part,  «fcc. ;  then  the  next  will  be  2  tenth,  or 

2  hundredth,  or  2  thousandth  parts,  &c. ;  so  that  if  the  first  1  be  esteemed  1,  the 
middle  1  will  be  10 ;  2  to  its  right,  20 ;  3,  30 ;  4,  40 ;  and  10  at  the  end,  100.  Again,  if 
the  first  1  is  10,  the  next  2  is  20,  3  is  30,  and  so  on,  making  the  middle  1,  100;  the 
next  2  is  200,  3  is  300,  4  is  400,  and  10  at  the  end  is  1000.  In  like  manner,  if  the  fii-st 
I  be  esteemed  1  tenth  part,  the  next  2  will  be  2  tenth  parts,  and  the  niiddle  1  will  be  1 ; 
the  next  2,  2 ;  and  10  at  the  end  will  be  10.  Again,  if  the  first  1  be  counted  1  hundredth 
pait ;  the  next,  2  hundredth  parts ;  the  middle  1  will  be  10  hundredth  parts,  or  1  tenth 
part;  and  the  next  2,  2  tenth  parts;  and  10  at  the  end  will  be  but  one  whole  number 
or  imeger. 

As  the  figures  are  increased  or  diminished  in  their  value,  so  in  like  manner  must  all 
the  intermediate  strokes  or  subdivisions  be  increased  or  diminished;  that  is,  if  the  first 
I  at  the  left  hand  be  counted  1,  then  2  (next  followmg  it)  will  be  2,  and  each  subdivision 
between  them  will  be  1  tenth  part;  and  so  all  the  way  to  the  middle  1,  Avhicli  -will  be 
10  ;  the  next  2,  20  ;  and  the  longer  strokes  between  1  and  2  are  to  be  counted  from  1 
thus,  11,  12  (where  is  a  brass  pm);  then  13,  14,  15,  sometimes  a  longer  stroke  than  the 
rest ;  then  IG,  17,  18,  19,  20,  at  the  figure  2 ;  and  hi  the  same  manner  the  short  strokes 
between  the  figures  2  and  3,  3  and  4,  4  and  5,  &c.,  are  to  be  reckoned  as  units,  Agam, 
if  1  at  the  left  hand  be  10,  the  figures  between  it  and  the  middle  1  will  be  conunon 
tens,  and  the  subdivisions  between  each  figure  will  be  units ;  from  the  middle  1  to  10 
ut  the  end,  each  figure  will  be  so  many  hunch-eds ;  and  between  these  figures  each 
longer  division  will  be  10.  From  this  description  it  will  be  easy  to  find  the  divisions 
representing  any  given  lunnber,  thus:  Supj)ose  the  j)oint  representing  the  number  12 
were  required;  take  the  division  at  tlie  figure  1  in  tlie  middle,  for  the  firet  figure  of 
12 ;  then  for  the  second  figure  count  two  tenths,  or  longer  strokes  to  the  right  hand, 
and  this  will  be  the  point  representing  12,  where  the  brass  pin  is. 

Again,  suj)pose  the  number  22  were  required;  the  first  figure  2  is  to  be  found  on 
the  scale,  and  for  the  second  figure  2,  count  2  tenths  onwards,  and  that  is  the  point 
representuig  22. 

Again,  sup])ose  1728  were  required;  for  the  first  figure  1, 1  talce  the  middle  1,  for 
tlie  second  figure  7,  count  onwards  as  before,  and  that  will  be  1700.  And,  as  the 
remaming  figures  are  28,  or  nearly  30,  I  note  the  jjouit  which  is  nearly  fV  of  the 
distance  between  the  marks  7  and  8,  and  this  will  be  the  point  representing  1728. 

*  The  description  and  use  of  logarilhms  arc  given  in  page  23,  et  seq.  The  log.  sines,  tangents,  &c., 
are  marked  on  these  scales  by  means  of  a  line  of  equal  parts,  corresponding  to  the  size  of  the  scale. 


UKrfCRlPTIOiN    AND    USE   OF    GUTTERS    SCALE.  21 

If  the  point  representing  435  was  i-eqinred,  from  the  4  in  tlie  second  inten-al  count 
towards  5  on  the  right,  three  of  the  larger  divisions  and  one  of  the  smaller  (this  smaller 
division  being  midway  between  the  marks  3  and  4),  and  that  will  be  the  division 
expressing  435.     In  a  similar  manner  other  numbers  may  be  found. 

All  fractions  found  in  this  line  nuist  be  decimals;  and  if  they  are  not,  they  must  be 
reduced  into  decimals,  which  is  easily  done  by  extending  the  compasses  from  the 
denominator  to  the  numerator ;  that  extent  laid  the  same  way,  from  1  m  the  middle  or 
right  hand,  will  reach  to  the  decimal  required. 

Example.  Requu-ed  the  decunal  fraction  equal  to  ^.  Extend  from  4  to  3 ;  that 
extent  will  reach  from  1  on  the  middle  to  .75  towards  the  left  hand.  The  like  may  be 
observed  of  any  other  vidg-ar  fraction. 

Multii)lication  is  performed  on  this  Ihie  by  extendhig  from  1  to  the  multiplier;  that 
extent  will  reach  from  the  multiplicand  to  the  product. 

Suppose,  for  example,  it  were  requu-ed  to  find  the  product  of  16  multiplied  by  4 ; 
extend  from  1  to  4  ;  that  extent  will  reach  from  10  to  G4,  the  product  required. 

Division  being  the  reverse  of  multiplication,  therefore  extend  from  the  divisor  to 
unity ;  that  extent  will  reach  from  the  dividend  to  the  quotient. 

Sa))pose  C4  to  bo  divided  by  4 ;  extend  from  4  to  1 ;  that  extent  will  reach  from  64 
to  IG,  the  quotient. 

Questions  in  the  Rule  of  Three  are  solved  by  this  luie  as  follows :  Extend  from  the 
first  terir  to  the  second  ;  that  extent  will  reach  fi'om  the  thu'd  term  *  to  the  fourth. 
And  it  ought  to  be  particularly  noted,  that  if  you  extend  to  the  left,  from  the  first 
number  to  the  second,  you  nuist  also  extend  to  the  left,  from  the  third  number  to  the 
fourth  ;  and  the  contrary. 

ExASiPLE.  If  the  diameter  of  a  circle  be  7  inches,  and  the  circumference  22,  what 
is  the  circumference  of  another  circle,  tJie  diameter  of  which  is  14  inches  .''  Extend 
ti"om  7  to  22  ;  that  extent  -will  reach  from  14  to  44,  the  same  way. 

The  superficial  content  of  any  parallelogram  is  foimd  by  extending  from  1  to  the 
breadth  ;  that  extent  will  reach  from  the  length  to  the  superficial  content. 

Example.  Siqipose  a  plank  or  board  to  be  15  uiches  broad  and  27  feet  long,  the 
content  of  which  is  required.  Extend  from  1  to  1  foot  3  inclies  (or  1.25) ;  that  extent 
will  reach  from  27  feet  to  33.75  feet,  the  superficial  content.  Or  extend  from  12  inches 
to  1.5,  &c. 

The  solid  content  of  any  l)ale,  box,  chest,  &c.,  is  found  by  extending  from  1  to  the 
breadth  ;  that  extent  will  reach  from  the  depth  to  a  fourth  number,  and  the  extent  from 
1  to  that  fourth  number  will  reach  from  the  length  to  the  solid  content. 

Example  I.  What  is  the  content  of  a  square  j)illar,  whose  lenglh  is  21  feet  9 
inches,  and  breadth  1  foot  3  inches  ?  The  extent  from  1  to  1.25  will  reach  from  1.25 
to  1.56,  the  content  of  one  foot  in  length ;  again,  the  extent  from  1  to  1.56,  will  reach 
from  the  length  21.75  to  33.9,  or  34,  the  solid  content  m  feet. 

Example  II.  Suppose  a  squai-e  piece  of  timber,  1.25  feet  broad,  .56  deep,  and  36 
long,  be  given  to  find  the  content.  Extend  from  1  to  1.25;  that  extent  will  reach  from 
.56  to  .7 ;  then  extend  from  1  to  .7 ;  that  extent  will  reach  from  36  to  25.2,  the  solid 
content.  In  like  manner  may  the  contents  of  bales,  &-c.,  be  found,  which,  being  divided 
by  40,  will  give  the  number  of  tons. 

4thly.  Tiie  line  of  sines,  marked  (Sin.),  coiTcsponduig  to  the  log.  sines  of  the  degi-ees 
of  the  quadrant,  liegins  at  the  lefi;  hand,  and  is  numbered  to  the  right,  1,  2,  3,  4,  5,  &c., 
to  10  ;  then  20,  30,  40,  &:c.,  ending  at  90  degrees,  where  is  a  brass  centre-pin,  as  there 
is  at  the  right  end  of  all  the  luies. 

5thly.  The  line  of  versed  shies,  marked  (V.  S.),  corresponding  to  the  log.  versed  sines 
of  the  degrees  of  the  quadrant,  begins  at  the  right  hand  against  90°  on  tlie  sines,  and 
from  thence  is  niuiibei-ed  towards  the  left  hand,  10,  20,  30,  40,  &c.,  ending  at  the  lefi 
hand  at  about  169° ;  each  of  the  subdivisions,  from  10  to  30,  is  in  general  two  degrees ; 
from  thence  to  90  is  suigle  degrees ;  from  thence  to  the  end,  each  degree  is  divided  mto 
15  minutes. 

Gthly.  The  line  of  tangents,  marked  (Tang.),  corresponding  to  the  log.  tangents  of  the 
degrees  of  the  quadrant,  begins  at  the  left  hand,  and  is  numbered  towards  the  right, 
1,  2,  3,  &c.  to  10,  and  so  on,  20,  30,  40,  and  45,  where  is  a  brass  pin  under  90°  on  the 
shies  ;  from  thence  it  is  numbered  backwards,  50,  60,  70,  80,  &c.  to  89,  ending  at  the 
lefi;  hand  where  it  began  at  1  degree.  The  subdivisions  arc  nearly  similar  to  those  of 
the  sines.  When  you  have  any  extent  in  your  comjiasses,  to  be  set  off"  from  any 
number  less  than  4.5°  on  the  line  of  tangents,  towards  the  right,  and  it  is  found  to  reach 

*  Or  ynii  may  cxteiul  from  the  first  to  tlie  third  ;  for  lliat  extent  will  roach  from  the  second  to  the 
fourth.  This  inelliod  must  be  adopted  vviicn  usiiisj  the  lines  of  sines,  tangents,  &c.,  if  the  first  and  third 
terms  are  of  the  same  name,  and  different  from  the  second  and  fourth. 


22  DESCRIPTION   AND    USE   OF   GUNTERS   SCALh. 

beyond  the  mark  of  45°,  you  must  see  how  far  it  extends  beyond  that  mark,  and  set  it 
off  from  45°  towards  the  left,  and  see  what  degi-ee  it  falls  upon,  which  will  be  the 
number  sought,  which  must  exceed  45° ;  if,  on  the  contrary,  you  are  to  set  off  such  a 
distance  to  the  right  from  a  number  greater  than  45°,  you  must  proceed  as  before,  only 
remembering,  that  the  answer  must  be  less  than  45°,  and  you  must  always  consider 
the  degi'ees  above  45°,  as  if  they  were  marked  on  the  continuation  of  tlie  line  to  the 
right  hand  of  45°. 

7thly.  The  line  of  the  meridional  parts,  marked  (Mer.),  begins  at  the  right  hand,  and 
is  numbered,  10,  20,  30,  &c.,  to  the  left  hand,  where  it  ends  at  87  degrees.  This  line, 
with  the  line  of  equal  parts,  marked  (E.  P.),  under  it,  are  used  together,  and  only  in 
Mercator's  Sailing.  The  upper  line  contains  the  degrees  of  the  meridian,  or  latitude 
in  a  Mercator's  chart,  corresponding  to  the  degrees  of  longitude  on  the  lower  line. 

The  use  of  this  Scale  in  solving  the  usual  problems  of  Trigonometry,  Plane  Sailing, 
Middle  Latitude  Sailing,  and  Meixator's  Sailing,  will  be  given  in  the  course  of  this 
work  ;  but  it  will  be  unnecessary  to  enter  uito  an  explanation  of  its  use  in  calculating 
the  common  pi-oblems  of  Nautical  Asti'onomy  as  it  is  much  more  accui'ate  to  perform 
those  calculations  by  logarithms. 


23 


ON    THE    SLIDING    RULE. 


The  Sliding  Rule  consists  o^  di  fixed  part  and  a  slider,  and  is  of  the  same  dimensiona. 
and  has  the  same  Imes  marked  on  it  as  on  a  common  Gunter's  Scale  or  Plane  Scale, 
which  may  be  used,  with  a  pair  of  compasses,  in  the  same  manner  as  tliose  scales ; 
and  as  a  description  of  those  lines  has  already  been  given,  it  will  be  uimecessary  to 
repeat  it  here,  it  being  sufficient  to  observe,  that  there  are  two  lines  of  numbers,  a  line 
of  log.  sines,  and  a  line  of  log.  tangents,  on  the  slider,  and  that  it  may  be  shifled  so  as 
to  fix  any  face  of  it  on  cither  side  of  the  fixed  part  of  the  scale,  accordmg  to  the  nature  . 
of  the  question  to  be  solved. 

In  solving  aiay  problem  in  Ai'ithmetic,  Trigonometry,  Plane  Sailing,  &c.,  let  the 
proposition  be  so  stated  that  the  first  and  third  terms  may  be  alike,  and  of  course  the 
second  and  fourth  terms  alike  ;  then  biing  the  first  term  of  the  analogy  on  the  fixed  part, 
against  the  second  term  on  the  slider,  and  against  the  third  term  on  the  fixed  part  ivill  be 
found  the  fourth  term  07i  the  slider  ;*  or,  if  necessary,  the  first  and  third  terms  may  be 
found  on  the  slider,  and  the  second  and  fourth  on  the  fixed  part.  Multiplication  and 
division  are  performed  by  this  rule,  in  considering  unity  as  one  of  the  terms  of  the 
analogy. 

Thus,  to  perform  multiplication ;  set  1  on  the  line  of  numbere  of  the  fixed  pai-t, 
against  one  of  the  factors  on  the  line  of  numbers  of  the  slider ;  then  agamst  the  other 
factor,  on  the  fixed  part,  will  be  found  the  product  on  the  slider. 

Example,  To  find  the  product  of  4  by  12 ;  draw  out  the  slider  till  1  on  the  fixed 
part  comcides  with  4  on  the  slider ;  then  opposite  12  on  the  fixed  part  will  be  found  48 
on  the  slider. 

To  perform  division ;  set  the  divisor  on  the  line  of  numbers  of  the  fixed  part  against 
1  on  the  slider ;  then  against  the  dividend  on  the  fixed  pait  will  be  found  the  quotient 
on  the  slider. 

Example.  To  divide  48  by  4 ;  set  4  on  the  fixed  part  against  1  on  the  slider ;  then 
against  48  on  the  fixed  pait  will  be  found  12  on  the  slider. 

EXAMPL*ES  IN  THE  RULE  OF  THREE. 

If  a  ship  sail  25  miles  m  4  houre,  how  many  miles  will  she  saU  in  12  houre  at  the 
same  rate  ? 

Bring  4  on  the  line  of  numbei-s  of  the  fixed  part  against  25  on  the  line  of  numbers 
of  the  slider ;  then  against  12  on  the  fixed  part  will  be  found  75  on  the  slider,  which  is 
the  answer  required. 

Example.     If  3  pounds  of  sugar  cost  21  cents,  Avhat  will  27  jiounds  cost? 

Bring  3  on  the  line  of  numbers  of  the  fixed  part,  against  21  on  tlie  line  of  numbers 
of  the  slider ;  then  against  27  on  the  fixed  i)art  will  be  found  189  on  the  slider. 

EXAMPLE   IN   TRIGONOMETRY. 

In  the  oblique-an<rled  triangle  ABC,  let  there  be  given  ABz=5G, 
AC=:G4,  angle  ABC  =  4G°30',  to  find  the  other  angles  and  the 
Bid 0  BC. 

In  this  case  we  have  (by  Art.  58,  Geometiy)  the  following 
canons : — 

AC  (64) :  sine  angle  B  (46°  3(y) : :  AB  (56) :  sine  angle  C  ;  and  sine  angle  B  :  AC  : :  sine 
angle  A  :  BC  Therefore,  to  work  the  first  projwrtion  by  the  sliding  rule,  we  must 
bi-ing  64  on  the  line  of  numbers  of  the  fixed  part  against  46°  30'  on  the  line  of  sines  of 
tlie  slider ;  then  against  56  on  the  former  will  be  39°  24'  on  the  latter,  which  will  be 

*  If  tlie  first  and  second  terms  are  alike,  instead  of  the  first  and  third,  you  must  bring  the  first  term 
on  the  slider  against  the  third  on  the  fixed  part,  and  against  the  second  term  on  tne  slider  will  be  found 
the  fourth  term  on  the  fi.xed  part ;  or,  if  necessarj-,  the  first  and  second  terms  may  be  found  on  tho 
fixed  part,  and  the  third  and  fourth  on  the  slider. 


24  OJN    THE   SLIDING  RULE. 

the  angle  C.  The  sum  of  the  angles  B  and  C,  being  subti'acted  from  180°,  leaves  the 
angle  A  =:  94°  C.  Then,  by  the  second  canon,  bnng  the  angle  B^46°30',  on  the 
Ime  of  sines  of  the  slider  against  AC  =  64,  on  the  line  of  numbers  of  the  fixed  part; 
then  against  the  angle  A  =  94°  6'  (or  its  supplement,  85°  54')  on  the  slider  will  be  found 
the  side  BC  =  88  on  the  fixed  pait. 

In  a  similar  manner  may  the  other  propositions  in  Trigonometry  be  solved. 

From  what  has  been  said,  it  will  be  easy  to  work  all  the  problems  in  Plane,  IMiddle 
Latitude,  and  Mercator's  Sailing,  as  iu  the  three  followuig  examples,  which  the  learner 
may  pass  over  until  he  can  solve  the  same  problems  by  Uie  scale.  If  any  one  wishes 
to  laiow  the  use  of  the  Sliding  Rule  in  problems  of  Spherical  Trigonometiy,  he  may 
consult  the  treatises  written  expressly  on  that  subject;  but  it  may  be  observed,  that  in 
such  calculations  the  Sliding  Rule  is  rather  an  oljject  of  curiosity  than  of  real  use,  as  it 
is  much  more  accurate  to  make  use  of  logarithms. 

Example  I.  Given  tlie  course  sailed  1  pomt,  and  the  distance  85  mUes ;  required 
the  difference  of  latitude  and  departure. 

By  Case  I.  of  Plane  Sailing,  we  have  these  canons : — 

Radius  (8  points)*  Distance  (85) : :  Sine  Co.  Course  (7  points)  :  Difference  of  Latitude  ; 
and  Radius  (8  points)  :  Distance  (85)  : :  Sine  Course  (1  pomt)  :  Departure. 

Hence  we  must  bring  the  radius,  8  points,  on  the  fixed  part  of  the  sine  rhumbs 
agamst  85  on  the  line  of  niunbers  on  the  slider ;  then  agauist  7  points  on  the  sine 
rhumbs  will  be  found  the  difference  of  latitude,  83^,  on  the  slider,  and  against  1  point 
will  be  found  the  departure,  16i  miles. 

If  the  com-s8  is  given  in  degrees,  you  must  use  the  Ime  marked  (Sin.) 

ExAJiPLE  II.  Given  the  difference  of  latitude,  40  miles,  and  departure,  30  miles ; 
requu-ed  the  course  and  distance. 

By  Case  VI.  of  Plane  Sailing,  we  have  tins  canon  for  the  course : — 

Difference  of  Latitude  (40)  :  Radius  (45°)  : :  Departure  (30)  :  Tangent  Course. 

Hence  we  must  bring  40  on  the  line  of  numbers  of  the  slider  against  45°  on  the  line 
of  tangents  on  the  fixeci  pait;  then  agauist  30  on  the  slider  will  be  found  the  course, 
37°,  nearly. 

Again,  the  canon  for  the  distance  gives 

Sine  Course  (87°)  :  Departure  (30)  : :  Radius  (90°)  :  Distance. 

Hence  we  must  braig  37°  on  the  line  of  sines  of  the  fixed  jwrt  against  30  on  the  line 
of  numbers  on  the  slider  ;  then  against  90°  on  the  line  of  sines  of  the  fixed  part  will  be 
found  the  distance,  50,  on  the  slider. 

Example  IIJ  Given  tlie  middle  latitude,  40°,  and  tlie  departure,  30  mUes ;  requu'ed 
the  diffei'ente  of  longitude. 

By  Case  VI.  of  31iddle  Latitude  Sailing,  we  have  this  canon : — 

Sme  Comp.  Middle  l^atitude  (50°)  :  Dei)arture  (30) ::  Radius  (90°) :  Difference  Long. 

Hence  by  bruiging  50°,  on  the  ILiie  of  siues  of  the  fixed  part,  against  30  on  the  line 
of  numbers  on  the  slider,  then  against  90°  on  the  fixej  part  we  sliall  find  39  on  the 
slider,  wliicli  wiLl  be  the  difference  of  longitude  required. 


25 


DESCRIPTION  AND   USE  OF  THE  SECTOR. 


This  instrument  consists  of  two  rules  or  legs,  movable  round  an  axis  or  joint,  as  a 
centre,  having  several  scales  drawn  on  the  faces,  some  single,  others  double  ;  the  single 
scales  are  like  those  upon  a  common  Gunter's  Scale  ;  the  double  scales  are  those  which 
proceed  from  the  centre,  each  beuig  laid  twice  on  the  same  foce  of  the  instrument,  viz. 
once  on  each  leg.  From  these  scales,  dimensions  or  distances  are  to  be  taken,  when 
the  legs  of  the  instrument  are  set  in  an  angular  position. 

The  single  scales  being  used  exactly  like  those  on  the  common  Gunter's  Scale,  it  is 
unnecessary  to  notice  them  particiUarly ;  we  shall  therefore  only  mention  a  few  of  the 
uses  of  tlie  double  scales,  the  number  of  which  is  seven,  viz.  the  scale  of  Lines,  marked 
Lin.  or  L. ;  the  scale  of  Choi-ds,  marked  Cho.  or  C. ;  the  scale  of  Sines,  marked  Sin. 
or  S. ;  the  scale  of  Tangents  to  45°,  and  another  scale  of  Tangents,  from  45°  to  about 
70°,  both  of  which  are  mai'ked  Tan.  or  T. ;  the  scale  of  Secants,  marked  Sec.  or  S. ; 
and  the  scale  of  Polygons,  marked  Pol. 

The  scales  of  lines,  chords,  sines,  and  tangents  under  45°,  are  all  of  the  same  i*adius, 
beginning  at  the  centre  of  the  instnuncnt,  and  terminating  near  the  other  extremity  of 
each  leg,  viz.  the  Imes  at  the  division  10,  the  chords  at  G0°,  the  sines  at  90°,  and  the 
tangents  at  45° ;  the  remainder  of  the  tangents,  or  those  above  45°,  are  on  other  scales, 
beginning  at  a  quarter  of  the  length  of  the  former,  counted  from  the  centi-e,  where  they 
are  marked  with  45°,  and  extend  to  about  76  degrees.  The  secants  also  begin  at  the 
same  distance  from  the  centre,  Avhere  they  are  mai'ked  with  0,  and  are  from  thence 
continued  to  75°.  The  scales  of  polygons  are  set  near  the  inner  edge  of  the  legs,  and 
where  these  scales  begin,  they  ai"e  marked  with  4,  and  fi'om  thence  are  numbered 
backward  or  towards  the  centre,  to  12. 

In  describing  the  use  of  the  Sector,  the  terms  lateral  distance  and  transverse  distance 
often  occm-.  By  the  former  is  meant  the  distance  taken  with  the  compasses  on  one 
of  the  scales  only,  beginning  at  the  centre  of  the  sector ;  and  by  the  latter,  the  distance 
taken  between  any  two  corresponding  divisions  of  the  scales  of  the  same  name,  the 
le^  of  the  sector  beuig  m  an  angular  position. 

The  use  of  the  Sector  depends  upon  the  proportionality  of  the  con-esponding  sides 
of  similar  triangles  (demonstrated  in  ^rt.  53,  Geometiy).  For  if, 
in  the  triangle  ABC,  we  take  ABr=AC,  and  AD=r  AE,  and  draw 
DE,  BC,  it  is  evident  that  DE  and  BC  will  be  parallel ;  therefore, 
by  the  above-mentioned  proposition,  AB  :  BC  : :  AD  :  DE ;  so 
that,  whatever  part  AD  is  of  AB,  the  same  part  DE  will  be  of  BC ; 
hence,  if  DE  be  the  chord,  sine,  or  tangent,  of  any  arc  to  the  radius 
A.D,  BC  will  be  the  same  to  the  radius  AB. 

Use  of  the  line  of  Lines. 

The  line  of  lines  is  useful  to  divide  a  given  line  into  any  number  of  equal  parts,  oi 
in  any  proportion,  or  to  find  third  and  fourth  proportionals,  or  mean  proportionals,  or 
to  increase  a  given  line  in  any  proportion. 

ExAjiPLE  I.  To  divide  a  given  line  into  any  number  of  equal  pai-ts,  as  suppose  9; 
make  the  length  of  the  given  line  a  transverse  distance  to  9  and  9,  the  number  of  parts 
[)roposed ;  then  will  the  transverse  distance  of  1  and  1  be  one  of  the  parts,  or  the  ninth 
])art  of  the  whole  ;  and  the  transverse  distance  of  2  and  2  will  be  two  of  the  equal  paits, 
or  f-  of  the  whole  line,  &c. 

ExAiMPLE  IL  If  a  ship  sails  52  miles  m  8  hours,  how  much  would  she  sail  in 
3  hours  at  the  same  rate  ? 

Take  52  in  your  compasses  as  a  transverse  distance,  and  set  it  off  from  8  to  8 ;  then 
the  transverse  distance,  3  and  3,  being  measured  laterally,  will  be  found  equal  to  19  and 
.1  half,  which  is  the  number  of  miles  required. 
4 


26  DESCRIPTION  AND  USE  OF  THE  SECTOR. 

Example  III.  Having  a  chart  constructed  upon  a  scale  of  6  miles  to  an  inch,  it  is 
requu-ed  to  open  the  Sectoi-,  so  that  a  corresponding  scale  may  be  taken  from  the  line 
of  lines. 

Make  the  transverse  distance,  6  and  G,  equal  to  ]  inch,  and  this  position  of  the  sector 
will  produce  the  given  scale. 

Example  IV.  It  is  requured  to  reduce  a  scale  of  6  mches  to  a  degi-ee  to  another 
of  3  inches  to  a  degree. 

Make  the  ti-ansvei-se  distance,  6  and  6,  equal  to  the  lateral  distance,  3  and  3 ;  then  set 
off  any  distance  from  the  chart  laterally,  and  the  coiTesponduig  transverse  distance 
will  be  the  reduced  distance  requu'ed.  jj 

Example  V.     One  side  of  any  triangle  being   given,  of  any 
length,  to  measure  the  other  two  sides  on  the  same  scale. 

Suppose  the  side  AB  of  the  triangle  ABC  measures  50,  what 
are  the  measures  of  the  other  two  sides  ?  -^ 

Take  AB  in  your  comjiasses,  and  apply  it  transversely  to  50  and 
50 ;  to  this  opening  ot  the  Sector  apply  the  distance  AC,  in  your  compasses,  to  the 
same  number  on  both  sides  of  the  rule  transvei-sely  ;  and  where  the  two  pouits  fall  will 
be  the  measure  un  the  line  of  lines  of  the  distance  requned ;  the  distance  AC  will  fall 
against  63,  63,  and  BC  against  45,  45,  on  the  line  of  lines. 

Usa  of  the  line  of  Chords  on  the  Sector. 

The  line  of  chords  upon  the  Sector  is  veiy  useful  for  protracting  any  angle,  when 
the  paper  is  so  small  that  an  arc  cannot  be  dra%vn  upon  it  with  the  radius  of  a  common 
line  of  chords. 

Supjjose  it  was  required  to  set  off  an  arc  of  30°  from  the  point  C  of  the  small  circle 
ABC,  whose  centre  is  D. 

Take  the  radius,  DC,  in  your  compasses,  and  set  it  off  transversely 
ftorn  60°  to  60°  on  the  chords ;  then  take  the  transveree  extent  from 
30°  to  30°  on  the  chords,  and  place  one  foot  of  the  compasses  in  C  ; 
the  other  will  reach  to  E,  and  CE  will  be  the  arc  required.  And 
by  the  converse  operation,  any  angle  or  arc  may  be  measured,  viz. 
with  any  radius  describe  an  arc  about  the  angular  point ;  set  that 
radius  transversely  from  60°  to  60° ;  then  take  the  distance  of  the 
arc,  intercepted  between  the  two  legs,  and  apply  it  transversely  to  the  chords,  which 
will  show  the  degrees  of  the  given  angle. 

N'ole.  When  the  angle  to  be  protracted  exceeds  60°,  you  must  lay  off  60°,  and  then 
the  remaining  part ;  or  if  it  be  above  120°,  lay  off  60°  twice,  and  then  the  remamhig 
part.     And  in  a  similar  manner  any  arc  above  60°  may  be  measured. 

Uses  of  the  lines  of  Sines,  Tangents,  and  Secants. 

By  the  several  lines  disposed  on  the  Sector,  we  have  scales  of  several  radii;  so  that, 

1st.  Having  a  length  or  radius  given,  not  exceeding  the  length  of  the  Sector  when 
opened,  we  can  find  the  chord,  sine,  &c.  of  an  arc  to  that  radius ;  thus,  suppose  the 
chord,  sine,  or  tangent  of  20  degrees  to  a  radius  of  2  inches  be  required.  Make  2 
niches  the  transverse  opening  to  60°  and  60°  on  the  chords;  then  will  the  same  extent 
reach  from  45°  to  45°  on  the  tangents,  and  from  90°  to  90°  on  the  sines ;  so  that,  to 
whatever  radius  the  lines  of  chords  is  set,  to  the  same  are  all  the  others  set  also.  In 
this  disposition,  therefore,  if  the  transverse  distance  between  20°  and  20°  on  the  chords 
be  taken  with  the  compass,  it  will  give  the  chord  of  20  degi'ees ;  and  if  the  transverse 
of  20°  and  20°  be  in  like  manner  taken  on  the  sines,  it  will  be  the  sine  of  20  degrees; 
and  lastly,  if  the  transvei-se  distance  of  20°  and  20°  be  taken  on  the  tangents,  it  will  be 
the  tangent  of  20  degrees  to  the  same  radius  of  two  inches. 

2dly.  If  the  chord  or  tangent  of  70°  were  required.  For  the  chord  you  must  first  set 
off  the  chord  of  60°  (or  the  radius)  upon  the  arc,  and  then  set  o-ff  the  chord  of  10°.  To 
find  the  tangent  of  70  degrees,  to  the  same  radius,  the  scale  of  upper  tangents  must  be 
used,  the  under  one  only  reaching  to  45° ;  making  therefore  2  inclies  the  transverse 
distance  to  45°  and  45°  at  the  beginning  of  that  scale,  the  extent  between  70°  and  70° 
on  the  same  will  be  the  tangent  of  70  degrees  to  2  inches  radius. 

3dly.  To  find  the  secant  of  any  arc  ;  make  the  given  radius  the  transverse  distance 
between  0  and  0  on  the  secants;  then  will  the  transvei-se  distance  of  20°  and  20°,  or 
70°  and  70°,  give  the  secant  of  20°  or  70°  respectively. 


DESCRIPTION   AND   USE   OF  THE   SECTOR.  27 

4thly.  ll  the  radius  and  any  line  representing  a  sine,  tangent,  or  secant,  be  given,  the 
degrees  corresponding  to  that  line  may  be  found  by  setting  the  Sector  to  the  given 
radius,  according  as  a  sine,  tangent,  or  secant,  is  concerned  ;  then,  taking  the  given  line 
between  the  coni])asses,  and  aj)j)Iying  the  two  feet  transversely  to  the  proper  scale,  and 
sliding  the  feet  alon^  till  they  both  rest  on  like  divisions  on  both  legs,  then  the  divisions 
will  show  the  degrees  and  pai'ts  corresponduig  to  the  given  Ime. 

Use  of  the  line  of  Polygons. 

The  use  of  this  line  is  to  inscribe  a  regular  polygon  in  a  cu'cle.  For  example,  let  it 
be  required  to  mscribe  an  octagon  or  polygon  of  eight  equal  sides,  in  a  circle.  Open 
the  Sector  till  the  transverse  distance  G  and  6  be  equal  to  the  radius  of  the  cu'cle ;  then 
will  tlie  transverse  distance  of  8  and  8  be  the  side  of  the  mscribed  polygon. 

Use  of  the  Sector  in  Trigonometry. 

All  proportions  in  Trigonometiy  are  easily  worked  by  the  double  lines  on  the  Sector; 
observing  that  the  sides  of  triangles  are  taken  upon  the  line  of  lines,  and  the  angles  are 
taken  upon  tiie  sines,  tangents,  or  secants,  according  to  the  nature  of  the  proportion. 
Thus,  it|  in  the  triangle  ABC,  we  have  given  AB  rr:  56,  AC  =  64,  and  the  angle 
ABC  =  46° 30', to  find  the  rest;  in  this  case  we  have  (byw4/-<. 58,  Geometry)  the  follow- 
ing proportions ;  As  AC  (64) :  sine  angle  B  (46°  30') : :  AB  (56) :  sine  angle  C  ;  and  as 
sine  B  :  AC  : :  sine  A  :  BC.  Therefore,  to  work  these  proportions 
by  the  Sector,  take  the  lateral  distance,  64  z=:  AC,  from  the  line  of 
lines,  and  open  the  Sector  to  make  this  a  transverse  distance  of 
46°  30'  =r  angle  B  on  the  sines ;  then  take  the  lateral  distance 
56  rr  AB  on  the  lines,  and  apply  it  transversely  on  the  sines,  which 
will  give  39°  24'  =:  angle  C.  Hence  the  sum  of  the  angles  B  and 
C  is  85°  54',  which  taken  from  180°,  leaves  the  angle  A  =  94°  6'.  Then,  to  work  this 
second  proportion,  the  Sector  being  set  at  the  same  opening  as  before,  take  the 
transvei-se  distance  of  94°  6'=  the  angle  A  on  the  sines,  or,  which  is  the  same  thing, 
the  transverse  distance  of  its  supplement,  85°  54' ;  then  this,  applied  laterally  to  the 
Imes,  gives  the  sought  side,  BC  nr  88.  In  the  same  manner  we  might  solve  any 
probiem  m  Trigonometry,  where  the  tangents  and  secants  occur,  by  only  measuring 
the  transverse  distances  on  the  tangents  or  secants,  uistead  of  measiu'ing  them  on  the 
sines,  as  in  the  preceding  example.  All  the  problems  that  occm-  in  Nautical  Astronomy 
may  be  solved  by  the  sector ;  but  as  the  calcidation  by  logarithms  is  much  more 
accurate,  it  will  be  useless  to  enter  into  a  further  detail  on  this  subject 


2d 


LOGARITHMS. 


In  order  to  abbreviate  tbe  tedious  operations  of  multiplication  and  division  with  large 
numbers,  a  series  of  numbers,  called  Logarithms,  was  invented  by  Lord  Napier,  Baron 
of  Marchi4iston  in  Scotland,  and  published  in  Edinburgh  in  1G14 ;  by  means  of  which 
the  operation  of  multii)lication  may  be  performed  by  addition,  and  division  by  subtrac- 
tion j^nunibers  may  be  involved  to  any  power  by  simple  multii)lication,  and  the  root 
of  any~p9wer  extracted  by  sunple  division."^ 

In  Table  XXVL  are  given  the  logarithms  of  all  numbers  from  1  to  9999 ;  to  each 
one  must  be  prefixed  an  index,  with  a  period  or  dot  to  separate  it  from  the  other  part, 
as  in  decimal  fractions ;  the  numbers  from  1  to  100  are  published  in  that  table  with 
then-  indices ;  but  from  100  to  9999  the  index  is  left  out  for  the  sake  of  brevity ;  but  it 
may  be  supplied  by  this  general  rule,  viz.  The  index  of  the  logarithm  of  any  integer  or 
mixed  number  is  always  one  less  than  the  number  of  integral  places  in  the  natural  number. 
Thus  the  mdex  of  the  logarithm  of  any  number  (integral  or  mixed),  between  10  and 
100,  is  1  •,_from  100  to  1000,  it  is  2 ;  from  1000  to  10000  is  3,  &c. ;  the  method  of  finding 
the  logaritln^^s  from  this  table  will  be  evident  from  the  followmg  examples. 

To  find  the  logarithm  of  any  number  less  than  100. 

Rule.    Enter  the  first  page  of  the  table,  and  opposite  the  given  number  will  be 
found  the  logarithm  with  its  index  prefixed. 
Thus  opposite  71  is  1.85126,  which  is  its  logarithm. 

To  find  the  logarithm  of  any  number  between  100  and  1000. 

Rule.  Find  the  given  number  m  the  left-hand  column  of  the  table  of  logarithms, 
and  immediately  under  0  in  the  next  column  is  a  number,  to  which  must  be  prefixed 
the  number  2  as  an  index  (because  the  number  consists  of  three  places  of  figiu'es),  and 
you  will  have  the  sought  logarithm. 

Thus,  if  the  logarithm  of  149  was  required ;  this  number  being  found  in  the  left- 
hand  column,  agauist  it,  in  the  colunm  marked  0  at  the  toj)  (or  bottom),  is  found  17319, 
to  which  prefixing  the  index  2,  we  have  the  logarithm  of  149  =i  2.17319. 

To  find  the  logarithm  of  any  niwibcr  between  1000  and  10000. 

Rule.  Fuid  the  three  left-hand  figures  of  the  given  number,  in  the  left-hand  column 
of  the  table  of  logaridmis,  opposite  to  which,  in  the  column  that  is  marked  at  the  top 
(or  bottom)  with  the  fourth  figure,  is  to  be  found  the  sought  logarithm ;  to  which  must 
be  prefixed  the  mdex  3,  because  the  number  contains  four  places  of  figures. 

Thus,  if  the  logarithm  of  1495  was  required ;  opposite  to  149,  and  in  the  column 
marked  5  at  the  tn[i  (or  bottom),  is  174G4,  to  which  prefix  the  index  3,  and  we  have  the 
sought  logarithm,  3.17404. 

To  find  the  logarithm  of  any  number  above  10000. 

Rule.  Find  the  three  first  figiu'cs  of  the  given  number  in  the  left-hand  column  of 
ilie  table,  and  tli<3  fourth  figure  at  the  top  or  bottom,  and  take  out  the  corresponding 
number  as  m  the  preceding  rule  ;  take  also  the  ditlerence  between  this  logarithm  and 
the  next  greater,  and  multiply  it  by  the  given  number  exclusive  of  the  four  first  figures ; 
rross  off"  at  the  right  hand  of  the  product  as  many  figures  as  you  had  figures  of  the 
jiven  number  to  multiply  by;  then  add  the  remaining  left-hand  figures  of  this  product 
lO  the  logarithm  taken  from  tlie  t<*ible,  and  to  the  sum  prefix  an  index  equal  to  one  less 


LOGARITHMS.  29 

tlian  the  iiumber  of  integral  figures  in  the  given  number,  and  you  will  have  the  sought 
iogarithm.  To  facilitate  the  calculation  of  these  proportional  [)arts,  several  small  tables 
are  ])Iaccd  in  the  margin,  which  give  the  correction  coiTCsponduig  to  the  difference  D, 
and  to  the  f/lh  figure  of  the  proposed  number.  The  use  of  these  tables  will  be  seen  ill 
the  following  examples. 

Thus,  if  the  logarithm  of  14957  was  required;  opposite  to  149,  and  under  5,  is  17464; 
the  difference  between  this  and  the  next  greater  number,  17493,  is  29,  the  difference  D ; 
this  multiplied  by  7  (the  last  figm-e  of  the  given  number)  gives  203;  crossing  off"  the 
right-hand  figin-e  leaA'es  20.3  or  20  to  be  added  to  17404,  which  makes  17484 ;  to  this 
prefixing  the  index  4,  we  have  the  sought  logiu'ithm,  4.17484.  Tliis  correction,  20, 
may  also  be  found  by  inspection  in  the  small  table  hi  the  margin,  marked  at  the  top 
with  D  rr29,  and  opposite  to  ihe  Jiflh  figure  of  the  number,  namely  7,  at  the  side ;  tlie 
corresponding  number  is  the  correction,  20. 

Again,  if  the  logarithm  of  1495738  was  required ;  the  logarithm  coiresponding  to 
149  at  the  left,  and  5  at  the  top,  is,  as  in  the  last  example,  174G4 ;  the  difference  between 
this  and  the  next  gi-eater  is  29;  multiplyhig  this  by  738  (which  is  equal  to  the  given 
ninnber,  excluding  the  four  first  figures)  gives  21402 ;  crossing  off'  the  three  right-hand 
figures  of  this  product  (because  the  number  738  consists  of  three  figures),  we  have  the 
coiTection  21  to  be  added  to  174G4 ;  and  the  index  to  be  prefixed  is  G,  because  the 
given  number  consists  of  7  j)laces  of  figures;  therefore  the  sought  logarithm  is  6.17485. 
riiis  correction,  21,  may  be  found  as  above,  by  means  of  the  marginal  table,  marked  at 
the  top  with  D^29,  and  at  the  side  7.38  or  7^  nearly,  to  which  corresponds  21,  as 
before. 

To  jind  the  logarithm  of  any  mixed  decimal  number. 

Rule.  Find  the  logarithm  of  the  number,  as  if  it  was  an  integer,  by  the  last  rule,  to 
which  prefix  the  index  of  the  integral  part  of  the  given  number. 

Thus,  if  the  logarithm  of  the  mixed  decimal  149.5738  was  required;  find  the 
logarithm  of  1495738,  without  noticing  the  decimal  point ;  this,  in  tlie  last  example, 
was  found  to  be  17485  ;  to  this  we  nuist  prefix  the  index  2,  corresponding  to  the  integral 
jiait  149 ;  the  logarithm  sought  will  therefore  be  2.17485. 

To  Jind  the  logarithm,  of  any  decimal  fraction  less  than  unity. 

The  uidex  of  the  logarithm  of  any  number  less  than  unity  is  negative ;  but  to  avoid 
the  mixture  of  positive  and  negative  quantities,  it  is  common  to  borrow  10  or  100  in 
the  uidex,  which  must  afterwards  be  neglected  in  summing  them  with  other  mdices ; 
thus,  instead  of  ^vi-iting  the  index — 1,  it  is  usually  written  -\-9,  or-|-99;  but  in 
general  it  is  sufficient  to  borroAV  10  ui  the  index ;  and  it  is  toliat  ive  shall  do  in  the  rest 
of  this  ivork.  In  this  way  we  may  find  the  logarithm  of  any  decunal  fraction  by  the 
following  rule. 

Rule.  Find  the  logarithm  of  a  fraction  as  if  it  was  a  whole  number;  see  how  many 
ciphers  precede  the  first  figure  of  the  decimal  fraction,  subtract  that  number  from  9, 
and  the  remainder  will  be  the  index  of  the  given  fraction. 

Thus  the  logarithm  of  0,0391  is  8.59218 ;  the  logarithm  of  0.25  is  9.39794 ;  the 
logarithm  of  0.0000025  is  4.39794,  &c. 

To  find  the  logarithm  of  a  vidgar  fraction. 

Rule.  Subtract  the  logaridim  of  the  denominator  from  the  logarithm  of  the 
numerator  (borrowing  10  in  the  index  when  the  denominator  is  the  greatest) ;  the 
cemauider  will  be  the  logarithm  of  the  fraction  sought. 


EXAMPLE   I. 

Required  the  logarithm  of  |. 

From  log.  of  3 0.47712 

Take  log.  of  8 0.90309 

Remainder,  log.  of  |  or  .375 9.57403 


EXAMPLE   II. 
Required  the  logarithm  of  3^,  or  ■^-. 

From  log.  of  13 1.11394 

Take  log.  of  4 0.60206 

Remahider,  log.  of  3i  or  3.25.. .  0.51188 


To  find  the  number  corresponding  to  any  logarithm. 

Rule.   In  the  column  marked  0  at  the  top  (and  bottom)  of  the  table,  seek  for  the  next 
loss  logarithm,  neglecting  the  uidex ;  note  the  number  against  it,  and  cany  your  eye 


30 


LOGARITHMS. 


along  that  line  until  you  find  the  nearest  less  logarithm  to  the  given  one,  ann  you  will 
have  the  fourth  figure  of  the  given  number  at  the  top,  which  is  to  be  placed  to  tlie 
right  of. the  three  other  figures;  if  you  wish  for  greater  accuracy,  you  nnist  take  the 
difference,  D,  between  this  tabular  logarithm  and  the  next  gi'eater,  also  the  difference, 
d,  between  that  least  tabular  logarithm  and  the  given  one ;  to  the  latter  difference,  d, 
annex  two  or  more  ciphers  at  the  right  hand,  and  divide  it  by  the  fomier  difference,  D, 
and  place  die  quotient*  to  the  right  hand  of  the  four  figures  already  found,  and  you 
will  have  tlie  number  sought,  expressed  in  a  mixed  decimal,  the  integral  part  of  which 
will  consist  of  a  number  of  figiu'es  (at  the  left  hand)  equal  to  the  mdex  of  the  logarithm 
increased  by  unity ,f 

Thus,  if  the  number  coiresponding  to  the  logarithm  1.52634  was  required,  we  find 
52634  in  the  column  marked  0  at  the  top  or  bottom,  and  opposite  to  it  is  336 ;  now, 
the  index  beuig  1,  the  sought  number  must  consist  of  two  integral  places;  therefore  it 
is  33.6. 

If  the  given  logarithm  was  2.32838,  we  find  that  32838  stands  in  the  column 
mai-ked  0  at  the  top  or  Iwttom,  directly  opposite  to  213,  which  is  the  number  sought, 
because,  tlie  index  beuig  2,  the  number  must  consist  of  three  places  of  figures. 

If  the  number  corresponding  to  the  logarithm  2.57345  was  I'equired,  we  must  look 
in  the  cohunn  0;  and  we  find  hi  it,  against  the  number  374,  the  logarithm  57287 ;  and, 
guiding  the  eye  along  that  line,  we  find  the  given  logarithm,  57345,  in  the  column 
marked  5  ;  therefore  the  mixed  number  sought  is  3745 ;  and,  since  the  index  is  2,  tlie 
integi-al  part  must  consist  of  3  places  ;  therefore  the  number  sought  is  374.5.  If  the 
index  be  1,  the  number  will  be  37.45;  and  if  the  index  be  0,  the  number  will  be  3.745. 
If  the  index  be  8,  coiTesponding  to  a  number  less  than  unity,  the  answer  will  be 
0.03745,  &c. 

Again,  if  the  number  corresponding  to  the  logarithn  5.57811  was  required,  look  in 
the  column  0,  aJid  find  in  it,  against  378,  and  under  5,  the  logarithm  57807,  the  difference 
between  this  a;id  the  next  greater  logarithm,  57818,  being  1 1,  and  the  difference  between 
57807  and  the  given  number,  57811,  being  4  ;  to  this  4  aflix  two  ciphers,  which  make 
400,  and  divide  it  by  11  ;  the  quotient  is  36  nearly  ;  this  number,  being  connected  with 
the  former  four  figures,  makes  378536,  which  is  the  number  required,  shice,  the  index 
being  5,  the  number  must  consist  of  six  places  of  figures. 

To  ghow,  at  one  view,  the  mdices  coiresponding  to  mixed  and  decimal  numbers,  wc 
have  given  the  followhig  table. 


Mixed  niunber. 


Logarithms. 


405)43.0 Log.  4.61218 

4094.3 Log.  3.61218 

409.4:3 Log.  2.61218 

40.943 Log.  1.61218 

4.0943 Log.  0.61218 


Decimal  number. 


Lofrarilkms. 


0.4C943 Log.  9.61218 

0.04094:3 Log.  8.61218 

0.0040943 Log.  7.61218 

0.00040943 Log,  6.61218 

0.000040943 Lo-'.  5.61218 


MULTIPLICATION   BY   LOGARITHMS. 

flcLE.    Add  the  logarithms  of  the  two  numbera  to  be  multiplied,  and  the  sum  will 
be  the  logarithm  of  their  product 


EXAMPLE   I. 
Multiply  25  by  35. 

25 Lof 

35 Loi 


1.39794 
1.54407 


Product,  875 Log.  2.94201 


EXAMPLE  II. 
31ultiply  22.4  by  1.8. 

22.4 Log,  1.35025 

1.8 Log.  0.25527 


Product,  40.32 Log.  1.60552 


*  This  quotient  must  consist  of  as  many  places  of  figures  as  there  were  ciphers  annexed,  conformable 
to  the  rules  of  the  division  of  decimals.  Thus,  if  the  divisor  was  40,  and  the  number  to  iviiich  two 
ciphers  were  annexed  was  2,  making  2.00,  the  quotient  must  not  be  estimated  as  6,  but  as  05,  and  then 
two  figures  must  be  placed  to  the  rigiit  of  the  four  figures  before  found. 

t  If  the  index  corresponds  to  a  fraction  less  than  unity,  you  must  place  as  many  ciphers  to  the  left  of 
that  number  as  are  equal  to  the  index  subtracted  from  9,  the  decimal  point  being  placed  to  the  left  of 
these  ciphers ;  in  this  manner  you  will  obtain  the  sought  number. 

We  may  find  the  fifth  figure  of  the  required  number  by  means  of  the  marginal  tables,  by  entering  the 
table  corresponding  at  the  top  to  the  proposed  value  of  1),  and  in  the  rigln-hand  column  with  di  the 
corresponding  number  is  the  fifth  figure  of  the  required  natural  number. 


LOGARITHMS. 


31 


EXAMPLE  in. 
Multiply  3.2C  by  0.0025. 

3.26 Log.  0.51322 

0.0025 Log.  7.39794 

Product,  0.00815 Log.  7.91116 


EXAMPLE  IV. 
Multiply  0.25  by  0.003. 

0.25 Log.  9.39794 

0.003 Log.  7.47712 

Product,  0.00075 Log.  6.87506 


In  the  last  example,  the  sum  of  the  two  indices  is  16 ;  but  since  10  was  borrowed  in 
each  number,  we  have  neglected  10  m  the  sum ;  and  the  remainder,  6,  being  less  than 
the  other  10,  is  evidently  tlie  index  ol"  the  logaritlim  of  a  fraction  less  than  unity. 


DIVISION   BY   LOGARITHMS. 

Rule.    From  the  logarithm  of  the  dividend  subtract  the  logaritlun  of  the  divisor ; 
the  remainder  Avill  be  the  logarithm  of  the  quotient. 


EXAMPLE   I. 
Divide  875  by  25. 

875 Log.  2.94201 

25 Log.  1.39794 


Quotient,  35 Log.  1.54407 

EXAMPLE  II. 
Divide  40.32  by  22.4. 


1.60552 
1.35025 

Quotient,   1.8 Log.  0.25527 


40.32 
22.4  . 


•Log. 
•  Log. 


EXAMPLE   III. 
Divide  0.00815  by  0.0025. 

0.00815 Log.  7.91116 

0.0025 Log.  7.39794 

Quotient,  3.26 Log.  0.51322 

EXAMPLE   IV. 
Divide  0.00075  by  0.025. 

0.00075 Log.  6.87506 

0.025 Loff.  8.39794 


Quotient,  0.03 Log.  8.47712 


In  Example  III.  both  the  divisor  and  dividend  are  fractions  less  than  unity,  and  the 
divisor  is  the  least ;  consequently  the  quotient  is  gi-eater  than  unity.  In  Example  IV. 
both  fractions  are  less  than  unity  ;  and,  since  the  divisor  is  the  gi-eatest,  its  logaritlim  is 
gi"eater  than  that  of  the  dividend  ;  for  this  reason  it  is  necessaiy  to  borrow  10  in  the 
indax  before  making  the  subtraction  ;  hence  tlie  quotient  is  less  than  unity. 


INVOLUTION   BY   LOGARITHMS. 

Rule.  Multiply  the  logarithm  of  the  number  given,  by  the  index  of  the  power  to 
which  the  quantity  is  to  be  raised  ;  the  product  will  be  the  logarithm  of  the  power 
souglit.  But  in  raising  the  powers  of  any  decimal  fraction,  it  must  be  observed,  that 
the  first  significant  figure  of  the  power  must  be  put  as  many  places  below  the  place 
of  units  as  the  index  of  its  logaritlim  wants  of  10  multiplied  by  the  uidex  of  the  power. 


EXAMPLE   1. 
Required  the  square  of  18. 

18 Log.  1.25527 

2 


Ajjswer,  324 Lo^.  2.51054 

EXAMPLE   II. 
Required  the  cube  of  13. 

13 Log.  1.11394 

3 

Ans\vcr,  2197 Log,  3.34182 


EXAMPLE   III. 
Required  the  squai-e  of  6.4. 

6.4 Log.  0.80618 


Answer,  40.96 Log.  1.61236 

EXAMPLE  IV. 
Requu'ed  the  cube  of  0.25. 

0.25 Log.    9.39794 

3 

Ans^ver,  0.015623 Log.  28.19382 


In  the  last  example,  the  index  28  wants  2  of  30  (the  product  of  10  by  the  power  3) ; 
therefore  the  fii-st  sip|nLfic3jit  figure  of  the  answer,  viz.  1,  is  placed  two  figures  distant 
f""opq  the  pl°ce  of  unit''. 


32 


LOGARITHMS 


EVOLUTION   BY   LOGARITHMS. 


Rule.  Divide  the  logaritiim  of  the  number  hy  the  mdex  of  the  power ;  the  quotieni 
will  be  the  logarithm  of  the  root  sought.  Bat  if  the  power  whose  root  is  to  be 
extracted  is  a  decimal  fraction  less  than  unity,  prefix  to  the  index  of  its  logarithm  a 
figiu'C  less  by  one  than  the  index  of  the  power,*  and  divide  the  whole  by  the  mdex  of 
the  power  ;  the  quotient  will  be  tlie  logarithm  of  the  root  sought. 


EXAMPLE   I. 

Wliat  is  the  squai-e  root  of  324  ? 
324 Log.  2)2.51055 

Answer,  18 Log.       1.25527 

EXAMPLE  II. 
Required  the  cube  root  of  2197. 

2197 Log.  3)3..34]83 

Answer,    13 Log.        1.11394 


EXAMPLE  III. 
Requii-ed  the  square  root  of  40.90. 

40.96 Log.  2 )  1.61236 

Answer,  6.4 Log.       0.80618 

EXAMPLE  IV. 
Requu-ed  the  cube  root  of  0.015625. 

0.015625 ..Log.    8.19382 

Prefix  2  to  the  mdex 3 )  28.19382 

Answer,  0.25 Log.    9.39794 


TO  WORK   THE   RULE   OF  THREE  BY   LOGARITHMS. 


When  three  numbers  are  given  to  find  a  fourth  proportional,  in  arithmetic,  we  make 
a  statement,  and  say.  As  the  first  number  is  to  the  second,  so  is  the  thu'd  to  the  fourth ; 
and  by  multi})lying  the  second  and  thu'd  together,  and  dividuig  the  product  by  the 
first,  we  obtain  the  fourth  number  sought.  To  obtain  the  same  result  by  logarithms, 
we  must  add  the  logarithms  of  the  second  and  third  nunibers  together,  and  from  the  sum 
subtract  the  logarithm  of  the  first  number;  the  remainder  will  he  the  logarithm  of  the  sought 
fourth  number. 


EXAMPLE   I. 

If  6  yards  of  cloth  cost  5  dollars,  what 
will  20  yards  cost  ? 
As  6 Log.  0.77815 

Is  to  5 Log.  0.69807 

So  is  20 Log.  1.30103 


Sum  of  2(1  and  3d 2.00000 

Subtract  the  first 0.77815 

To  16.67 


.Log.  1.22185 


The  answer,  therefore,  is  16  doUare  and 
tVu^j  or  16  dollars  and  67  cents. 


EXAMPLE  II. 

If  a  ship  sails  20  miles  in  7  hours,  how 
much  will  she  sail  in  21  hours  at  the 
same  I'ate  ? 

As  7 Log.  0.84510 

Is  to  20 Log.  1.30103 

So  is  21 Log.  1.32222 

Sum  of  2d  and  3d 2.02325 

Subtract  the  first 0.84510 


To  60 Log.  1.77815 


The  ans^ver  is  60  miles. 


TO  CALCULATE  COMPOUND  INTEREST  BY  LOGARITHMS. 

To  100  dollai-s  add  its  interest  for  one  year;  find  the  logarithm  of  this  sinn,  and 
reject  2  in  the  mdex  ;  then  multiply  it  l)y  the  number  of  jears  and  parts  of  a  year  for 
which  the  interest  is  to  be  calculatetl ;  to  tlie  ])roduct  add  the  logarithm  of  the  sum 
put  at  interest ;  the  sum  of  these  two  logai'ithms  will  be  the  logarithm  of  the  amount 
of  the  given  sum  for  the  given  time. 


*  In  this  rule  it  is  supposed  that  10  is  borrowed  in  finding  tlie  index  to  the  decimal  according  to 
the  nile,  page  29. 


LOGARlTHiMS. 


33 


EXAMPLE. 

Requu-ed  the  amoimt  of  the  principal  and  interest  of  355  dollars,  let  at  6  per  cenL 
compound  interest,  for  7  years. 

Adding  6  to  100  gives  106  ;  whose  logarithm,  rejecting 

2  in  the  index,  is 0.02531 

Multiphed  by 7 

Product 0.17717 

Pi-incipal,  355  dollars Log.  2.55023 

Sum  gives  the  logarithm  of  533.83 Log.  2.72740 

Therefore  the  amount  of  principal  and  interest  is  533  dollai-s  and  83  cents. 


To  find  the  logarithm  of  the  sine,  tangent,  or  secant,  corresponding  to  any 
number  of  degrees  and  minutes,  hy  Table  XXVII. 

The  given  number  of  degrees  must  be  found  at  the  bottom  of  the"  page  when 
between  45°  and  135°,  otlicrwise  at  the  top  ;  tlie  muuites  being  found  in  the  column 
marked  31,  wliich  stands  on  the  side  of  the  page  on  which  the  degi-ees  are  marked ; 
thus,  if  the  degrees  are  less  than4^5,  the  inijuites  are  to  be  found  in  the  left-hand  column,  &c. ; 
and  it  must  be  noted  that  if  the  degrees  are  found  at  the  top,  the  names  of  hour,  sine,  cosine, 
tangent,  &c.,  must  also  be  found  at  the  top ;  and  if  the  degrees  are  found  at  the  bottom,  the 
names  sine,  cosine,  &z,c.,77iust  also  be  found  at  the  bottom.  Then  opposite  to  the  number 
of  the  minutes  will  be  found  the  log.  sine,  log.  secant,  &c.  in  the  columns  marked  sine, 
secant,  &c.  respectively. 


EXAMPLE   I. 
Requu'ed  the  log.  suie  of  28°  37'. 

Find  28°  at  the  top  of  the  page,  directly 
below  which,  in  the  left-hand  column, 
find  37' ;  against  which,  in  the  column 
mai'ked  sine,  is  9.68029,  the  log.  suie  of 
tlie  given  number  of  degrees ;  and  in  the 
same  manner  the  tangents,  &c.  are  found. 


EXAMPLE   II. 
Required  the  log.  secant  of  126°  20'. 

Find  126°  at  the  bottom  of  the  page, 
du'ectly  above  which,  in  the  left-hand 
column,  find  20' ;  against  which,  in  the 
column  marked  secant,  is  10.22732  re- 
quu-ed. 


To  find  the  logarithm  of  the  sine,  cosine,  S^'c.for  degrees,  minides,  and  seconds, 

by  Table  XXVII. 

Find  the  numbers  corresponding  to  the  even  mmutes  next  above  and  below  the 
given  degi-ees  and  minutes,  and  take  their  diiference,  D ;  then  say.  As  60"  is  to  the 
number  of  seconds  in  the  proposed  number,  so  is  that  diflTerence,  D,  to  a  cori-ection,  d, 
to  be  ap])lied  to  the  number  corresponding  to  the  least  number  of  degi'ces  and  minutes ; 
additive  if  it  is  the  least  of  the  two  numbers  taken  from  the  table,  othenvise  subtractive. 


EXAMPLE  in. 
Required  the  log.  sine  of  24°  16'  38". 

Sine  of  24°  16' Log.  9.61382 

Sine  of  24   17 Log.  9.61411 

Difference D  rr  29 


EXAMPLE  IV. 
Requu-ed  the  log.  secant  of  105°  20'  16". 

Secant  of  105°  20' Log.  10.57768 

Secant  of  105  21   Log.  10.57722 

Difference D  =  46 

Then,  as  60"  :  16"  : :  46  :  12,  which, 
bemg  subtracted  from  the  nimiber  coixe- 
sponding  to  105°  20',  gives  10.57756,  the 
log.  secant  of  105°  20'  16". 

If  the  given  seconds  be  g-.^j  4>  -^,  or^,  or  any  other  even  parts  of  a  minute,  the  like 
parts  may  be  taken  of  the  difference  of  the  logarithms,  and  added  or  subtracted  as 
above,  which  may  be  frequently  done  by  inspection.  These  proportional  parts  may 
also  be  found  very  nearly  by  means  of  the  three  columns  of  differences  for  seconds, 
given,  for  the  first  time,  in  the  nintli  edition  of  this  Avork.  The  first  column  of 
J'fferences,  which  is  to  be  used  with  the  two  columns  marked  A,  A,  is  placed  between 


Tlien,  as  60"  :  38"  : :  29  :  18,  which, 
beuig  added  to  the  number  correspondmg 
to  24°  16',  gives  9.61400,  the  log.  sine  of 
24°  16'  38". 


34  LOGARITHMS. 

these  columns.  The  second  column  of  differences,  which  is  to  be  used  with  the 
two  cohuiais  B,  B,  is  placed  between  these  two  columns.  In  like  manner,  the  third 
column  of  differences,  between  the  columns  C,  C,  is  to  be  used  with  them.  The 
correction  of  the  tabular  logarithms  in  any  of  tlie  columns  A,  B,  C,  for  any  number 
of  seconds,  is  found  by  entering  the  left-hand  column  of  the  table,  marked  S'  at  the  top, 
and  finding  the  number  of  seconds  ;  opposite  to  this,  in  the  column  of  differences,  will 
be  found  the  corresponding  correction.  Thus,  in  tlie  table,  page  215,  which  contains 
the  log.  sines,  tangents,  &c.,  for  30°,  the  corrections  corresponding  to  25",  are  9  for 
the  columns  A,  A,  12  for  the  columns  B,  B,  3  for  the  columns  C,  C  ;  so  that,  if  it 
were  required  to  find  the  sine,  tangent,  or  secant  of  30°  12'  25",  we  must  add  these 
corrections  respectively  to  the  numbers  corresponding  to  30°  12' ;  thus. 

Col.  a.  Col.  B.  Col.  C. 

Logs,  for  30°  12' ....  Sine  9.70159  Tangent ....  9.76493  Secant ....  10.0C335 
Corrections  for  25"  in  S'        +  9  -f  12  -f  3 

Logs,  for  30°  12'  25" 9.70168  9.76505  10.06338 

these  corrections  being  all  added,  because  the  logarithms  increase  in  proceeding 
from  30°  12'  to  30°  13'.  Instead  of  taking  out  the  logarithms  for  30°  12',  and  adding 
the  correction  for  25",  we  may  take  out  the  logarithms  for  30°  13',  and  subtract  the 
correction  for  60"  —  25",  or  35",  found  in  the  margin  S' ;  thus, 

Logs,  for  30°  13' ... .  Sine  9.70180        Tangent ....  9.76522         Secant . . . .  10.06342 

Corr.  for  35"  in  col.  S',  ?  ,o  17  a 

or  25"  in  col.  G'  ....  $     ~  ^'^  ~^^  ~^ 

Logs,  for  30°  12'  25" ....  9.70167  9.76505  10.06338 

The  corrections  are  in  this  case  subtracted,  because  the  logaiithms  decrease  in 
proceeding  backward  35"  from  30°  13',  to  attain  30°  12'  25".  The  tangents  and 
secants,  in  this  example,  are  the  same  by  both  methods ;  the  sines  differ  by  one  unit, 
in  the  last  decimal  place,  and  this  will  frequently  happen,  because  the  difference  of 
the  logarithms  for  1',  sometimes  differ  one  or  two  units  from  the  mean  values  which 
are  used  in  the  three  columns  of  differences.  The  error  arising  from  this  cause  is 
generally  diminished  by  using  the  smallest  angle  *  S',  when  the  seconds  of  the  pro- 
posed angle  are  smaller  than  30" ;  or  the  greatest  angle  G',  when  the  number  of 
seconds  are  greater  than  30".  Thus,  in  the  above  example,  where  the  angle 
S'  =  30°  12',  and  the  angle  G'  r=  30°  13',  it  is  best  to  use  the  angle  S'  when  the  gifen 
angle  is  less  tlian  30°  12'  30",  but  the  angle  G'  when  it  exceeds  30°  12'  30".  thus, 
if  it  be  required  to  find  the  sine  of  30°  12'  51",  it  is  best  to  use  the  angle  G'=:30°  13', 
and  find  the  correction  by  entering  the  margin  marked  S',  with  the  difference 
60"  —  51"  =^9",  opposite  to  which,  in  the  column  of  diflerences,  is  3,  to  be  subtracted 
from  log.  sine  of  30°  13'  =  9.70180,  to  get  the  log.  sine  of  30°  12'  51"  =  9.70177.  To 
save  the  trouble  of  subtracting  the  seconds  from  60",  wc  may  use  the  right-hand 
margin,  marked  G',  and  the  correction  may  then  be  found  by  the  following  rules: — 

Rule  1.  When  the  smallest  angle  S'  is  used,  find  the  seconds  in  the  column  S', 
and  take  out  the  corresponding  correction,  which  is  to  be  applied  to  the  logarithm 
corresponding  to  S' ;  by  adding,  if  the  log.  of  G'  be  greater  than  the  log.  of  S'; 
otherwise,  by  subtracting. 

Rule  2.  When  the  greater  angle  G'  is  used,  find  the  seconds  in  the  column  G',  and 
take  out  the  corresponding  correction,  which  is  to  be  applied  to  the  logarithm 
corresponding  to  G';  by  adding,  if  the  log.  of  S' be  greater  than  the  log.  of  G'j- 
otherwise,  by  subtracting;  so  that,  in  all  cases,  the  required  logarithm  may  tall  be- 
tween the  two  logarithms  corresponding  to  the  angles  S'  and  G'. 

The  correctness  of  these  rules  Avill  evidently  appear  by  comparing  them  with  the 
preceding  exam])les  ;  and  by  the  inverse  process  we  may  find  the  angle  correspond- 
ing to  a  given  logarithm,  as  in  the  next  article. 

We  have  given  at  the  bottom  of  the  page,  in  this  table,  a  small  table  for  finding 
the  proportional  jjarts  for  the  odd  seconds  of  time,  corresponding  to  the  column  of 
Hours  A.  M.  or  P.  M. ;  to  facilitate  the  process  of  finding  the  log.  sine,  cosine,  ifcc- 
correspomling  to  the  nearest  second  of  time  in  the  column  of  hours,  or,  on  the  con- 
trary, to  find  the  nearest  second  of  time  corresponding  to  any  given  log.  sine,  cosine. 
&c.     Thus,  in  the  preceding  examples,   where  the   angle   8'  =  30°  12',   and   the 

*  If  wc  neglect  the  seconds  in  any  proposed  angle  whose  sine,  &c.  is  required,  we  get  llie  angle 
denoted  above  by  S',  and  this  angle  increased  by  1',  is  represented  by  G'' ;  so  that  the  proposed  angle 
falls  between  S'antI  G' ;  S'  being  a  smaller,  and  G'  a  crreater  angle  than  that  whose  log.  sine,  <^c.,  is 
required  ;  the  letters  S'and  G',  accented  for  minutes,  being  used  because  they  are  easily  rememtored 
as  tlse  viiitials  of  smalhr  and  greater 


1 


B. 

C. 

Tangent  9.7G493 

Secant  10.06335 

+  11 

+  3 

LOGARITHMS  35 

angle  G'r=30°  13';  the  times  corresponding  in  the  column  of  Hours  P.M.,  are 
S,_4h  jm  36s.  G'  =  i^  1-"  44';  and  if  we  wish  to  find  the  log.  sine,  cosine,  &c., 
corresponding  to  any  intermediate  time,  as,  for  example,  4''  1""  39%  which  differs  3* 
from  the  angle  S',  we  must  find  the  tabular  logarithm  corresponding  to  S',  and  apply 
the  correction  for  3%  given  by  the  table  at  the  bottom  of  the  page,  as  in  the  following 
examples : — 

A. 
Logs,  for  S'  r=  4"  1"  30^         Sine  9.70159 
Correction  for         -}-  3'  -|-  8 

Logs,  for 4h  1"^  39^         Sine  9.70167        Tangent  9.76504        Secant  10.06338 

Nearly  tlie  same  results  are  obtained  by  using  the  angle  G',  in  the  manner  we 
have  before  explained : — 

Logs,  for  G'  =  4"  1-"  44'  Sine  9.70180  Tangent  9.76522  Secant  10.06342 
Correction  for  —  5'  — 13  — 18  5 

Logs,  for 4"  1-"  39'         Sine  9.70167        Tangent  9.76504        Secant  10.06337 

These  corrections  must  be  applied  by  addition  or  subtraction,  according  to  the 
directions  given  above,  so  as  to  make  the  required  logarithm  fall  between  those 
which  correspond  to  the  times  S'  and  G'. 

The  inverse  process  will  give  the  time  corresponding  to  any  logarithm.  Thus, 
if  the  log.  sine  9.70167  be  given,  the  difference  between  this  and  9.70159,  corre- 
sponding to  S'  =  4''  1"  36',  is  8  ;  seeking  this  in  the  cohnnn  A,  in  the  second  line  of 
the  table  at  the  bottom  of  the  page,  it  is  found  to  correspond  to  3' ;  adding  this  to 
the  time  S'  =:4''  1"  36',  we  get  4''  l""  39'  for  the  required  time.  We  may  proceed 
in  the  same  manner  with  the  logarithms  in  the  columns  13,  C ;  using  the  numbers 
coiTesponding,  marked  B,  C,  respectively,  in  the  table  at  the  bottom  of  the  page. 

To  find  the  degrees,  minutes,  and  stconds,  corresponding  to  any  given  logarithm 
sine,  cosine,  t^'c.  hij  Table  XXVII. 
Find  the  two  nearest  numbers  to  the  given  log.  sine,  cosine,  &c.,  in  the  column 
marked  sine,  cosine,  &c.,  respectively,  one  being  greater,  and  the  other  less,  and  take 
their  difference,  D  ;  take  also  the  difference,  d,  between  the  given  logarithm  and  the 
logarithm  corrcsj)onding  to  the  smallest  number  of  degrees  and  minutes  ;  then  say,  As 
the  first  found  difference  is  to  the  second  found  difference,  so  is  60"  to  a  number  of 
seconds  to  be  annexed  to  the  smallest  number  of  degrees  and  minutes  befoi-e  found. 
The  three  columns  of  differences  may  also  be  used,  by  an  inverse  operation  to  that 
which  we  have  explained  in  the  preceding  article. 

EXAMPLE   V. 
Find  the  degrees,  minutes,  and  seconds  (less  than  90°),  corresponding  to  the  log. 
sme  9.61400. 

Next  less  log.  S'  ==24°  16'  9.61382  Log.  of  smallest  angle  S'  =  24°  16'  is  9.61382 
Greater G'  =  24   17   9.61411         Given  log 9.61400 

D=:29  d=l8 


Then  say.  As  29  :  18 ::  60":  38",  nearly ;  which,  annexed  to  24°  16',  give  24°  16' 38", 
answering  to  losr.  sine  9.61400.  Subtracting  24°  16'  38"  from  180°,  there  remain 
155°  43'  22",  the  log.  sine  of  which  is  also  9.61400.  The  quantity  38"  may  also  be 
found  by  inspection  in  the  side  column  S'  of  the  page  opposite  d=zl8,  in  the 
column  of  differences  between  the  two  columns,  A,  A.  If  we  use  the  angle  G',  we 
shall  have  (/'  equal  to  11,  the  difference  of  the  logarithms  9.61411  and  9.61400,  and 
the  corresjionding  number  of  seconds  in  column  G',  is  37",  making  24°  16'  37". 

To  find  the  arithmetical  complement  of  any  logarithm. 
The  arithmetical  complement  of  any  logarithm  is  what  it  wants  of  10.00000,  and  is 
used  to  avoid  subtraction.  For,  when  working  any  proportion  by  logarithms,  you 
may  add  the  arithmetical  complement  of  the  logarithm  of  the  first  term,  instead  of 
suljtracting  the  logarithm  itself,  observing  to  neglect  10  in  the  index  of  the  sum  of  the 
logarithms.  The  arithmetical  complement  of  any  logarithm  is  thus  found : — Begin 
at  tlie  index,  and  icnite  down  ichat  each  figure  ivants  of  9,  except  the  last  significant 
figure,  which  take  from  10.*  Thus,  the  arithmetical  complement  of  9.62595  is  0.37405  ; 
that  of  1.86567  is  8.13433;  and  that  of  10.33133  is  89.66867,  or  9.66867. 

*  When  llic  index  of  (he  given  log-arithm  is  greater  than  10,  as  in  some  of  the  numbers  of  Table 
XXV II.,  tlie  left-hand  figure  of  it  must  be  neglected  ;  and  when  there  are  any  ciphers  to  the  right  hand 
of  the  last  significant  figure,  j'ou  may  place  the  same  number  of  ciohers  to  the  right  hand  of  the  other 
^iRiuf^s  of  the  arithmetical  complement 


36 


PLANE    TRIGONOMETRY. 


Plane  Trigonometry  is  the  science  which  shows  how  to  find  the  measures  of  the 
sides  and  angles  of  plane  triangles,  some  of  them  being  akeady  known.  It  is  divided 
into  two  parts,  right-angled  and  oliique-angled ;  in  the  former  case,  one  of  the  angles 
is  a  right  angle,  or  90° ;  in  the  latter,  they  are  all  obhque. 

In  every  plane  triangle  there  are  six  parts,  viz.  three  sides  and  three  angles ;  any 
three  of  which  beuig  given  (except  the  three  angles),  the  other  three  may  be  found 
by  various  methods,  viz.  by  Gunter's  scale,  by  the  slidmg  rule,  by  the  sector,  by 
geometrical  construction,  or  by  arithmetical  calculation.  We  shall  explain  each  of 
these  methods ;  *  but  the  latter  is  by  far  the  most  accurate ;  it  is  perlbrmed  by  the 
help  of  a  few  theorems,  and  a  ti-igonometrical  canon,  exliibituig  the  natural  or  the 
logarithmic  smes,  tangents,  and  secants,  to  every  degree  and  minute  of  the  quadrant.t 
The  theorems  alluded  to  are  the  followuig : — 


THEOREM  1. 

In  any  riglit-angled  triangle,  if  the  hypoteniisc  ie  made  radius,  one  side  ivill  be  tlie  sine 
of  the  opposite  angle,  and  the  other  its  cosine ;  but  if  either  of  the  legs  be  made  radius, 
the  other  leg  will  be  the  tangent  of  the  opposite  art^le,  and  the  hypotenuse  will  be  the  secant 
of  the  same  angle. 


A  HiidiusC 


ATanffeni 


•D 


1st.  If,  in  the  right-angled  plane  triangle  ACB  (fig.  1),  we  make  the  hypotenuse  AB 
radius,  and  upon  the  centre.  A,  describe  the  arc  BE,  to  meet  AC  produced  in  E,  then 
it  is  evident  that  BC  is  the  sine  of  the  arc  BE  (or  the  sine  of  the  angle  BAC),  and  that 
AC  is  the  cosine  of  the  same  angle  ^  and  if  the  arc  AD  be  described  about  the  centre  B 
(fig.  2),  AC  will  be  the  sine  of  the  angle  ABC,  and  BC  its  cosine. 

2dly.  If  the  leg  AC  (fig.  3)  be  made  radius,  and  the  arc  CD  bo  described  about  the 
centre  A,  CB  will  be  the  tangent  of  that  arc,  or  the  tangent  of  the  angle  CAB  ;  and 
AB  will  be  its  secant. 

3dly.  If  the  leg  BC  (fig.  4)  be  made  radius,  and  the  arc  CD  be  described  about  the 
centre  B,  CA  will  be  the  tangent  of  that  arc,  or  the  tangent  of  the  angle  B,  and  AB 
will  be  its  secant. 

Now,  it  has  been  already  demonstrated  (in  Art.  55,  Geometiy)  that  the  sme,  tangent, 
secant,  &c.  of  any  arc  in  one  circle  is  to  the  sine,  tangent,  secant,  &c.  of  a  similar  arc 
in  another  cu'cle  as  the  radius  of  the  former  cu'cle  to  the  radius  of  the  latter.  And 
since  in  any  right-angled  triangle  there  are  given  either  two  sides,  or  the  angles  and 
one  side,  to  find  the  rest,  we  may,  if  we  wish  to  find  a  side,  make  any  side  radius ; 
then  say.  As  the  tabularnumbcr  of  the  same  name  as  the  given  side  is  to  the  given  side 
of  the  triangle,  so  is  the  tabidar  number  of  the  same  name  as  the  requii-ed  side,  to  the 
requh-ed  side  of  the  triangle.  If  we  wish  to  find  an  angle,  one  of  the  given  sides  must 
be  made  radius ;  then  say,  As  the  side  of  the  triangle  made  radius  is  to  the  tabidar 

*  It  will  not  be  necessary  to  add  any  furlher  description  of  the  uses  of  the  sector  or  sliding  rule; 
for  what  we  have  already  given  will  be  suHicicnt  for  any  one  tolerably  well  versed  in  the  use  of 
Gunter's  scale. 

t  See  Tables  XXIV.  and  XXVII. 


PLANE  TRIGONOMETRY. 


37 


radius,  so  is  the  other  given  side  to  the  tabular  sine,  tangent,  secant,  &c.  by  it  repre- 
sented ;  which,  bemg  sought  for  m  the  table  of  sines,  &c.,  will  coirespond  to  the 
degi'ees  and  minutes  of  the  required  angle. 

THEOREM  II. 

In  all  plane  triangles,  the  sides  are  in  direct  proportion  to  the  sines  of  their  opposite 
angles  (by  Art.  58,  Geometiy). 

Hence,  to  find  a  side,  Ave  must  say.  As  the  sme  of  an  angle  is  to  its  opposite  side,  so 
is  the  sine  of  either  of  the  other  angles  to  tlie  side  opposite  thereto.  But  if  we  v/ish  to 
find  an  angle,  we  must  say.  As  any  given  side  is  to  the  sine  of  its  ojiposite  angle,  so  is 
either  of  llie  other  sides  to  the  suie  of  its  opposite  angle. 

THEOREM  HI. 

In  every  plane  triangle,  it  will  be,  as  the  sum  of  any  two  sides  is  to  their  difference,  so  is 
the  tangent  of  half  the  sum  of  the  two  opposite  angles  to  the  tangent  of  half  their 
difference  (by  Art.  59,  Geometry). 

THEOREM  IV. 

As  the  base  of  any  plane  triangle  is  to  the  sum  of  the  two  sides,  so  is  the  difference  of  the 
two  sides  to  twice  the  distance  of  a  perpendicular  {let  fall  upon  the  base  from  the  opposite 
angle)  from  the  middle  of  the  base  (by  Aii.  GO,  Geometiy). 

THEOREM  V. 

In  any  plane  triangle,  as  the  rectangle  contained  by  any  tioo  sides  including  a  sought 
angle,  is  to  the  rectangle  contained  by  the  half  sum  of  the  three  sides  and  the  same  half 
sum  decreased  by  the  other  side,  so  is  the  square  of  radius  to  the  square  of  the  cosine  of 
half  the  sought  angle  (by  Ad.  61,  Geometiy). 


In  addition  to  these  theorems,  it  will  not  be  amiss  for  the  learner  to  recall  to  mind 
the  following  aiticles : — 

1.  In  eveiy  triangle,  the  greatest  side  is  opposite  to  the  greatest  angle,  and  the 
greatest  angle  opposite  to  the  gi-eatest  side. 

2.  In  eveiy  tiiangle  equal  sides  subtend  equal  angles.     {Aii,  39,  Geometrj\) 

3.  The  three  angles  of  any  plane  triangle  are  equal  to  180°.     {Art.  35,  Geometiy.) 

4.  If  one  angle  of  a  triangle  be  obtuse,  the  rest  are  acute ;  and  if  one  angle  be  right, 
the  other  two  together  make  a  right  angle,  or  90° ;  therefore,  if  one  of  the  acute  angles 
of  a  right-angled  triangle  be  known,  the  other  is  found  by  subtractmg  the  known  angle 
from  90°.  If  one  angle  of  any  triangle  be  knoAvn,  the  sum  of  the  other  two  Is  found 
by  subtracting  the  given  angle  from  180°  ;  and  if  two  of  the  angles  be  knoAvn,the  third 
is  found  by  subtracting  their  sums  from  180°. 

5.  The  complement  of  an  angle  is  lohat  it  wants  of  90°,  and  the  supplement  of  an  angle 
is  what  it  luants  of  180°. 


In  the  two  following  tables  we  have  collected  all  the  lailes  necessaiy 
for  solving  the  vai'ious  cases  of  Right-angled  and  Oblique-angled 
Trigonometry. 


FORMULAS   L\   RIGHT-ANGLED  TRIGONOMETRY. 


Case. 

Given. 

Sought. 

Solutions. 

1 

Hyp.  AC. 
Angles. 

Leg  BC. 
Leg  AB. 

Bad.  :  hyp.  AC  :  :  sine  A  :  leg  BC. 
Rad.  :  hyp.  AC  :  :  sine  C  :  leg  AB. 

2&3 

Leg  BC. 

Angles. 

Leg  AB. 
Hyp.  AC. 

Bad.  :  leg  BC  :  :  tang.  C  :  leg  AB. 
(  Rad.  :  leg  BC  :  :  sec.  C  :  hyp.  AC. 
(  Or,  sine  A  :  leg  BC  :  :  rad.  :  hyp.  AC. 

4  &5 

Hvp.  AC. 
Leg  AB. 

Angles. 
Leg  BC. 

Hvp.  AC  :  rad.  :  :  leg  AB  :  sine  C,  whose  corap.  is  A. 
Rad.  :  hyp.  AC  :  :  sine  A  :  leg  BC. 

6 

Both  legs. 
AB  &  BC. 

Angles. 
Hyp.  AC. 

Leg  BC  :  rad.  :  :  leg  AB  :  tang.  C,  whose  conip.  is  A. 
(  Sine  C  :  leg  AB  :  :  rad.  :  hyp.  AC. 
}  Or,  rad.  :  leg  BC  :  :  sec.  C  :  hvp.  AC. 

88 


RIGHT-ANGLED  TRIGONOMETRY. 


A     a  C       D    -^        D  (r    <^ 

FORMULAS  LN    OBLIQUE-ANGLED   TRIGONOMETRY 


Case. 

Given. 

Sought. 

Solutions. 

1 

The  angles  and 
side  AB. 

Side  BC. 
Side  AC. 

Sine  C  :  side  A6  :  :  sine  A  :  side  BC. 
Sine  C  :  side  AB  :  :  sine  B  :  side  AC. 

2&3 

Two  sides,  AB, 
BC,  and  angle 
0    opposite    to 
one  of  tiiem. 

Angle  A. 
Angle  B. 
Side  AC. 

Side  AB  :  sine  C  :  :  side  BC  :  sine  A,  which  added  to  C, 
and  the  sum  subtracted  from  180°,  gives  B. 
Sine  C  :  side  AB  :  :  sine  B  :  side  AC. 

4&5 

Two  sides,  AC, 
AB,  and  the  in- 
cluded angle  A. 

Angles  C 
and  B. 

Side  BC. 

Subtract  half  the  given  angle.  A,  from  90° ;  the  remainder 
is  half  the  sum  of  the  other  angles.  Then  say.  As  the  sum 
of  the  sides,  AC,  AB,  is  to  their  ditference,  so  is  the  tangent 
of  the  half  sum  of  the  other  angles  to  the  tangent  of  half  their 
difference;  which  added  to  and  subtracted  from  the  half 
sum,  will  give  the  two  angles  B  and  C  ;  the  greatest  angle 
being  opposite  to  the  greatest  side. 

Sine  B  :  side  AC  :  :  sine  A  :  side  BC. 

6 

All  three  sides. 

All  the  angles. 

Let  fallaperpendicular,BU,  opposite  to  the  required  angle; 
then,  as  AC  :  sum  of  AB,  BC  : :  their  difference  :  twice  UG, 
the  distance  of  the  perpendicular  from  the  middle  of  the 
base  ;  hence,  AD,  CD,  are  known,  and  the  triangle  ABC  is 
divided  into  two  right-angled  triangles,  BCD,  BAD;  then, 
by  Cases  IV.  and  V.  of  Right-angled  Trigonometry,  we  may 
tind  the  angle  A  or  C. 

Either  angle, 
as  A. 

Either  of  the  angles,  as  A,  may  also  be  found  by  the  follow- 
ing rule.  From  half  the  sum  of  the  three  sides  subtract  the 
side  BC  opposite  to  the  sought  angle  ;  take  the  logarithms  of 
the  half  sum  and  remainder,  to  which  add  the  arithmetical 
complements  of  the  logarithms  of  the  sides  AB,  AC  (including 
the  sought  angle) ;  half  the  sum  of  these  four  logarithms  will 
be  the  logarithmic  cosine  of  half  the  sought  angle. 

In  calculating  by  logarithms  by  any  of  the  pi-eceding  i-ules,  you  must  remember,  that 
Git  logarithm  of  the  first  term  of  the  analogy  is  to  be  subtracted  from  the  sum  of  the 
logarithms  of  the  second  and  third  terms ;  the  reviainder  will  he  the  logaiithm  of  the  sought 
fourth  term. 

When  the  first  temi  is  radius  (whose  logarithm  is  10.00000),  you  need  only  reject 
a  unit  in  the  second  left-hand  figure  of  the  mdex  of  the  sum  of  the  second  and  thu'd 
terms.  But  when  the  i-adius  occurs  m  the  second  or  third  term,  you  must  suppose  a 
unit  to  be  added  to  the  second  left-hand  figui-e  of  the  mdex  of  the  other  term,  and 
subtract  thei'efrom  the  logarithm  of  the  fii'st  term. 


RIGHT-ANGLED   TRIGONOMETRY. 


Solution  of  the  six  cases  in  Right-angled  Trigonometry. 

CASE  1. 

The  angles  and  hypotenuse  given,  to  find  the  legs. 
Given   the  hypotenuse  AC  250  leagues,  and  the  angle  C,  opposite  to   the  side 
AB,  z=  35°  SO',  to  find  the  base  CB,  and  perpendiculai*  AB. 

li\  PROJECTION. 
Draw  tlie  base  CB  of  any  length ;  with  an  extent  equal  to  the 
chord  of  60°,  and  on  C  as  a  centre,  describe  the  arc  DE  ;    from  E 
to  D  lay  off"  35°  30'  taken  from  the  line  of  chords ;  *  through  C  and  D 


iJ      J? 


*  In  all  projections  of  this  kind,  the  angles  are  measured  from  the  line  of  chords ;  the  radius  used  for 
describing  arcs  by  which  the  angles  are  to  '^e  measured,  being  equal  to  the  chord  of  C0°,  the  sides  of 


i 


RIGHT-ANGLED   TRIGONOMETRY. 


39 


draw  the  line  AC,  which  make  equal  to  250 ;  from  A  let  fall  the  perpendicular  AB  to 
cut  CB  in  B,  and  it  is  done ;  for  CB  will  be  203.5,  and  AB  equal  to  145.2. 

BY  LOGARITHMS. 
By  making  the  hypotenuse  CA  radius,  it  will  be, 


To  find  the  base  BC. 

As  radius 10.00000 

is  to  the  hypotenuse  AC  250. .  2.3!)794 

So  is  the  sine  angle  A  54°  30' . .  9.910G9 

To  the  base  BC  203.5 2.30863 


To  find  the  perpendicular  AB. 

As  radius 10.00000 

Is  to  the  hypotenuse  AC  250 . .     2.39794 
So  is  the  sine  angle  C  35°  30' . .     9.76395 

To  the  pei-pendicular  AB  145.2    2.16189 


BY   GUNTER'S  SCALE. 

In  all  proportions  which  are  calculated  by  Gunter's  scale,  when  the  first  and  second 
terms  are  of  the  same  kind,  the  extent  from  the  first  tenn  to  the  second  will  reach  from 
the  third  to  the  fourth. 

Or,  when  the  first  and  third  tenns  are  of  the  same  kind. 

The  extent  from  the  first  term  to  the  thu'd  will  reach  from  the  second  to  the  fourth ; 
that  is,  we  must  set  one  point  of  the  compasses  on  the  division  expressing  the  first 
term,  and  extend  the  other  point  to  the  division  expressing  the  third  term ;  then, 
without  altering  the  openuig  of  the  compasses,  we  must  set  one  point  on  the  division 
representuig  the  second  term,  and  the  other  point  will  fall  on  the  division  showing  the 
fourth  term  or  answer. 

In  the  present  example  the  Avork  is  as  follows : — 

Extend  from  radius,  or  90°,  to  54°  30'  on  the  line  of  sines ;  that  extent  will  reach 
from  250,  the  hypotenuse,  to  203.5,  the  base  on  the  line  of  numbers ;  and  th  e  extent 
from  radius  or  90°,  to  35°  30'  on  the  line  of  sines,  will  reach  from  250  to  145.2  on  tlie 
line  of  numbers. 

Observe  the  same  method  in  all  the  following  examples,  except  in  those  proportions 
where  the  word  secant  is  mentioned,  which  cases  must  be  virought  by  consideruig  the 
hypotenuse  radius,*  there  being  no  line  of  secants  on  the  common  Gunter's  scale, 
although  it  can  easily  be  marked  on  the  line  of  sines. 

JVote.  The  radius,  according  to  the  nature  of  the  proportion,  may  be  either  of  the 
following  quantities  : — 
8  points  on  the  line  of  rhumbs.  1      90°  on  the  Ime  of  smes. 

4  points  on  the  Ime  of  tangent  rhumbs.      |      45°  on  the  line  of  tangents. 


CASES  II.  A>D  III. 

Tlie  angles  and  one  leg  given,  to  find  the  hypotenuse  and  other  leg. 

The  angle  ACB  33°  15',  the  leg  BC  163  miles,  given,  to  find  the  hypotenuse  and 
the  other  leg. 

BY   PROJECTION. 

Draw  the  line  BC,  which  make  equal  to  163  miles ;  on  B  erect  the 
perpendicular  BA  ;  on  C,  as  a  centre,  with  the  chord  of  60°,  sweep 
the  arc  BD,  which  make  equal  to  33°  15' ;  draw  CD,  and  continue 
't  to  cut  AB  in  A,  and  it  is  done ;  for  AB  being  measured  on  the 
same  scale  that  BC  was,  will  be  106.9,  and  AC  194.9  miles. 

BY   LOGARITHMS. 
By  making  the  base  BC  radius,  it  will  be, 


To  find  the  perpendicular  AB. 

As  radius  45° 10.00000 

Is  to  the  base  BC  163 2.21219 

So  is  tangent  angle  C  33°  15'  . .     9.81666 

To  tlie  perpendicular  AB  106.9    2.02885 


To  find  the  hypotenuse  AC. 

As  radius  90° 10.00000 

Is  to  the  base  EC  163 2.21219 

So  is  secant  angle  C  33°  15'  . . .   10.07765 

To  the  hypotenuse  AC  194.9  . .     2.28984 


the  triangles  are  measured  b}-  scales  of  equal  parts,  as  was  before  observed.  Instead  of  using  the  line 
of  chords,  it  is  much  more  convenient  to  set  oft'  the  angles  by  means  of  a  protractor,  or  circular  arc,  ou 
which  the  degrees  are  marked.     Its  construction  is  so  simple  that  it  needs  no  explanation. 

*  Or  by  usuig  in  the  analogy,  radius  :  cosine  angle,  instead  of  secant  angle  :  radius ;  and  radius  :  sine 
angle,  instead  of  cosecant  angle  :  radius. 


40 


RIGHT-ANGLED   TRlGONOMETRTf . 


BY   GUNTER. 

Extend  from  45°  to  33°  15'  on  the  line  of  tangents ;  that  extent  will  reach  from  the 
base  1G3  to  the  pei-pendicular  106.9,  on  the  Ime  of  numbers. 

2dly.  Extend  from  56°  45'  to  radius  on  the  Ime  of  smes ;  that  extent  wiU  reach  from 
the  base  163  to  the  hypotenuse  194.9,  on  the  Ime  of  numbers. 

CASES  IV.  AND  V. 

The  hypotenuse  and  one  leg  given,  to  find  the  angles  and  other  leg. 

Given  the  leg  AB  91,  and  the  hypotenuse  AC  170,  being  to  find  the  angle  ACB 
BAC,  and  the  leg  BC. 

BY  PROJECTION. 

Draw  BC  at  pleasure  ;  on  B  erect  the  pei-pendicular  BA,  which 
make  equal  to  91 ;  take  170  in  your  compasses,  and,  with  one  foot 
on  A,  describe  an  arc  to  cut  BC  in  C ;  join  A  and  C,  and  it  is  done; 
for  the  angle  C  is  32°  22',  the  angle  A  57°  38',  and  BC  143.6. 

BY   LOGARITHMS. 
By  making  the  hypotenuse  radius,  we  shall  have, 


To  find  the  angle  C. 

As  the  hypotenuse  170 2.23045 

Is  to  radius 10.00000 

So  is  the  perpendicular  91 1.95904 

To  sine  ande  C  32°  22' 9.72859 


To  find  the  base  BC* 

As  radius 10.00000 

Is  to  the  hypotenuse  170 2.23045 

So  is  the  sme  angle  A  57°  38' . .  9.92667 

To  the  base  BC  143.6 2.15712 


BY   GUNTER. 

Extend  from  the  hypotenuse  170  to  the  pei-pendicular  91,  on  the  line  of  numbers ; 
that  extent  will  reach  from  radius  to  the  angle  C,  or  thecomplementof  angle  Ar:32°22' 
on  the  line  of  sines. 

2dly.  Extend  from  radius  to  the  angle  A  57°  38',  on  the  Ime  of  sines ;  that  extent 
will  reach  from  the  hypotenuse  170  to  the  base  143.6,  on  tlie  line  of  numbers. 


CASE  VI. 

The  legs  given,  to  find  the  angles  and  hypotenuse. 

Given  the  legs  AB  178,  and  BC  141,  to  find  the  angle  BAC  or  ACB,  and  the 
hypotenuse  AC. 

BY   PROJECTION. 

Make  BC  equal  to  141,  and  on  B  ei-ect  the  perpendicular  BA, 
which  make  equal  to  178 ;  join  AC,  and  it  is  done ;  for  the  angle  C 
is  51°  37' ;   consequently  tlie  angle  A  38°  23',  and  the  hypotenuse 


227.1. 


BY   LOGARITHMS. 
By  makmg  the  base  radius,  we  shall  have, 


To  find  the  angle  C. 

As  the  base  141 2.14922 

Is  to  radius 10.00000 

So  is  the  pei-pendicular  178  . . .  2.25042 

To  tangent  amrlc  C  51°  37'. . . .  10.10120 


To  find  the  hypotenuse  AC.f 

As  radius 10.00000 

Is  to  the  base  141 2.14922 

So  is  the  secant  angle  C  51°  37'  10.20696 

To  the  hji)otenuse  AC  227.1 . .     2.35618 


BY   GUNTER. 

The  extent  from  141  to  178  on  the  line  of  numbers  will  reach  from  radius,  or  45 
degi-ees,  to  the  angle  C  51°  37',  on  the  line  of  tangents. 

2dly.  The  extent  from  the  angle  C  51°  37'  to  radius,  or  90°,  on  the  line  of  sines, 
will  reach  from  the  perpendicular  178,  to  the  hypotenuse  227.1,  on  the  Ime  of  numbere. 

*  When  you  take  the  log.  sines,  or  tangents,  to  the  nearest  minute  only,  it  is  best  to  use  tliis  canon  for 
finding  BC,  wliich  is  more  correct  than  the  one  found  by  making  the  perpendicular  radius,  because  the 
variatirin  of  the  log.  sine  of  an  arc  is  less  than  the  corresponding  variation  of  the  log.  tangent. 

t  VVIion  finding  AC,  it  is  best  to  make  the  greatest  side  radius,  for  the  reason  mentioned  in  the  last  note; 
60  that  in  the  present  example  it  would  be  rather  preferable  lo  use  the  perpendicular  178  for  the  radi  js 


OBLIQUE   TRIGONOMETRY. 


41 


QUESTIONS 

To  exercise  the  learner  in  Right-angled  Plane  Trigonometry. 

Qiiestion  1.  The  hypotenuse  496  miles,  and  the  angle  opposite  to  the  base  56°  15', 
given,  to  find  the  base  and  perpendiculai*. 

Answer.  Base  412.4,  and  the  perpendiculai*  275.6  miles. 

Quest.  2.  The  perpendicular  275  leagues,  and  the  angle  opposite  to  the  base  56°  15', 
given,  to  find  the  hypotenuse  and  base. 
Ans.  The  hypotenuse  495,  and  base  411.6  leagues. 

Qiiest.  3.  The  base  33  yards,  and  the  angle  opposite  to  the  peipendicular  53°  26', 
given,  to  find  the  hypotenuse  and  perpendicular. 

Ans.  Hypotenuse  55.39,  and  the  perpendicular  44.49  yards. 

Quest.  4.  The  hypotenuse  575,  and  peipendiculai*  50  miles,  given,  to  find  the  base 

Ans.  Base  572.8  miles. 

Qiiest.  5.  The  hypotenuse  59,  and  the   base  33  miles,    given,  to  find  the  per- 
pentl'iculai'. 

Ans.   Pei-pendicular  48.9  miles. 

Quest.  6.  The  base  33,  and  pei-pendicular  52  leagues,  given,  to  find  the  hypatenuse 
Ans.  Hypotenuse  61.59  leagues. 


OBLIQUE    TRIGONOMETRY. 


CASE  1. 

Two  angles  and  one  side  given,  to  find  either  of  the  legs. 

Given  the  angle  BAG  =z  100°,  the  angle  AGB  =  54°,  and  the  leg  AB  =220,  to  find 
the  sides. 

BY   PROJECTION. 

Subti-act  the  sum  of  the  angles  A  and  C  fi'om  180°;  the  remamder  will  be  the 
angle  B  =  26°.  Draw  the  indefinite  line  BE,  also 
the  line  BH,  making  the  angle  EBH  =r  26°  ;  oit  BH 
set  off"  BA  220.  On  A  make  the  angle  BAG  100° ; 
tlien  AC  will  intersect  the  line  BE  in  the  pomt  C, 
which  completes  the  triangle,  and  BC  will  measure 
(on  the  same  scale  fi'om  which  BA  was  laid  down) 
268  nearly,  and  AC  119. 

BY  LOGARITHMS,  bv  Theorem  II. 


To  find  BC. 
As  the  sme  of  the  angle  G  54°. .  9.90796 

Is  to  the  side  AB  220 2.34242 

So  is  the  sine  of  the  angle  A  100°  9.99335 


12.33577 
9.90796 


To  the  nde  BC  267.8 2.42781 


To  find  AG. 

As  sine  angle  C  54° 9.90796 

Is  to  the  side  AB  220 2.34242 

So  is  the  sine  angle  B  26° 9.64184 

11.98426 
9.90796 


To  the  side  AC  119.2 2.07630 


BY   GUNTER. 
The  extent  from  the  angle  C  =:  54°  to  the  angle  A,  or  its  supplement  80°,  on  tlie 
smes,  will  reach  from  AB  =:  220  to  BC  :=  268,  on  the  line  of  numoers. 

2dly.    The  extent  from  the  angle  C=:54°  to  the  angle  Brr:26°,  on  the  sines,  will 
reach  from  AB  =  220  to  AC  :=  119,  on  the  luie  of  numbei-s. 
6 


42 


OBLIQUE   TRIGONOMETRY. 


CASES    II.   AND    III. 

Ttoo  sides,  mid  an  angle  opposite  to  one  of  them,  being  given,  to  find  the  other  angles,  and 

the  third  side. 

JVote.  It  may  be  proper  to  observe,  that  if  the  given  angle  be  obtuse,  the  angle 
sought  will  be  acute ;  but  when  the  given  angle  is  acute,  and  opposite  to  a  shorter 
given  side,  then  it  is  doubtful  whether  the  required  angle  be  acute  or  obtuse  ;  it  ouglit 
therefore  to  be  given  by  the  conditions  of  the  problem. 

EXAMPLE. 

Let  there  be  given  the  side  BC  137,  the  side  AB  213,  and  the  angle  A  23h°,  to  find 
the  otlier  side  AC,  and  the  angles  ABC,  BCA. 


BY    I'ROJECTION. 

Draw  the  indefinite  line  FE ;  make  the  angle  DAE  =  23^° ;  on  AD  set  off  AB =213 ; 
then  on  B,  with  137  in  your  compasses,  taken  from  the  same  scale,  describe  an  arr 
cutting  FE  in  the  points  C  and  G;  join  B(/, 
BG,  and  it  is  done  ;  for  the  triangle  may  be 
either  ACB  or  AGB,  according  as  the  angle  C 
or  G  is  acute  or  obtuse  ;  if  that  angle  be  acute, 
the  triangle  will  be  ABC  ;  the  side  AC  will 
measure  303,  the  angle  ACB  will  measure  38J°, 
and  the  angle  ABC  will  measure  118°  nearly ; 
but  if  the  angle  at  the  base  be  obtuse,  the  triangle  will  be  AGB ;  the  side  AG  will 
measure  88,  the  angle  AGB  will  measure  141  §°,  and  the  angle  ABG  15°,  nearly. 

If  die  side  BC  had  been  given  greater  than  AB,  there  could  have  been  only  one 
answer  to  this  problem;  for  m  that  case,  the  point  G  would  have  fallen  on  the 
continuation  of  the  line  CA  towards  F,  in  which  case  the  angle  A  of  the  triangle 
would  become  equal  to  FAB,  instead  of  being  equal  to  its  supplement,  as  is  required 
by  the  conditions  of  the  problem. 


BY   LOGARITHMS,  by  Theorem  II. 


To  find  the  angle  C  or  G, 

As  the  side  BC  137 2.13672 

Is  to  Uie  sine  of  angle  A  23.^° 9.G0070 

So  is  the  side  AB  213.....". 2.32838 


Subtract 61  49 

From 180    0 

Angle  ABC...  118  11 


11.9S?908 
2.13672 

or  G   141  41 

23  30 

9.79236 

or     165  11 

180    0 

ABG    14  49 

To  find  AC. 

As  sine  angle  C  38°  19' 9.79210 

Is  to  AB  213 2.32838 

So  is  sineangle  ABC  118°  11' 9.94319 


12.27357 
9.79240 


To  the  side  AC  302.8 2.48117 

To  find  AG. 

As  sine  angle  G  141°  41' 9.79240 

IstoAB  213 2.32838 

So  is  sine  angle  ABG  14°  49' 9.40778 


11.73616 
9.79240 


To  the  side  AG  87.9 1.94376 


BY   GUNTER. 

1st.  The  extent  from  BCzz:137  to  AB  =  213,  onthe  line  of  numbers,  will  reach 
from  A  =:  23.^°  to  38°  19',  on  tlie  line  of  sines,  which  is  equal  to  tlie  angle  C ;  its 
supplement,  141°  41',  being  equal  to  the  angle  G. 

2dly.  The  extent  from  the  angle  C  =  38°  19'  to  01°  49'  (the  sup])lement  of  the 
angle  ABC,  118°  11')  on  the  sines,  will  reacli  from  x\Br=213  to  303,  nearly,  on  the 
line  of  numbers ;  therefore  the  side  AC  =  303. 

Or,  the  extent  from  38°  19'  (the  supplement  of  the  angle  G)  to  the  angle  ABG:= 
14°  49',  on  the  sines,  will  reach  from  AB  :=:  213  to  88,  on  the  line  of  numbers ;  hence 
\G  =  88. 


OBLIQUE   TRIGONOMETRY. 


43 


CASES  IV.  AND  V. 

Two  sides  and  their  contained  angle  being  given,  to  find  either  of  the  other  angles  and  the 

third  side. 

Given  the  side  AB  HO  miles,  AC  80  miles,  and 
angle  BAG  96°  0',  to  find  the  angles  BCA  and  CBA 
and  the  side  BC. 

BY   PROJECTION. 

Draw  the  indefinite  right  line  AD,  on  which  set 
ofFAB  =  110;  make  the  angle  EABr=96°;  and  on 
AE  set  oft'  AC  rr  80  ;  join  BC,  and  it  is  done  ;  fiar  BC 
will  measure  on  the  fi^rmer  scale  143,  and  the  angles 

B  and  C  will  measme  33°  SS'  and  50°  5',  respectively,  yi'^ ^ 1„^  '   °^ jj 

on  the  line  of  chords. 


To  find  the  angles  B  and  C,  by  Theorem  III. 

As  sam  of  sides  AC  and  AB  190 2.27875 

Is  (o  their  ditTercnce  30 1.47712 

So  is  tang:.  A 


BY   LOGARITHMS. 

To  find  the  side  BC,  by  Theorem  II. 


sum  opp.  angles  /  .^o 
or  complement  of  ^  angle  A  ) 


9.93444 


11.43156 
2.27875 


To  tangent  of  half  difference.. .     8°  5'=  9.15281 

Sum  is  angle  C 50    5 

Difference  is  angle  B 3.'5  55 


As  sine  angle  B  33°  55' 9.74662 

Is  to  AC  80 1.90309 

So  is  sine  angle  A     96°  0'  ^  „  ornn 

or  its  supplement  84    0  J J.JJioi 

1.90070 
9.74662 

TosideBC  142.6 2.15408 


BY   GUNTER. 

1st.  The  extent  firom  the  sum  of  the  sides,  190,  to  their  difference,  30,  on  the  line 
of  numbers,  will  reach  from  the  half  sum  of  the  angles  B  and  C,  42°,  to  their  half 
difference,  8°  5',  on  the  line  of  tangents.  The  sum  of  this  half  sum  and  half 
difference  gives  the  angle  C  50°  5',  and  their  diffei'ence  the  angle  B  33°  55' ;  the 
greatest  angle  being  ojiposite  to  the  greatest  side. 

2dly.  The  extent  from  the  angle  B  33°  55',  to  the  angle  A  96°  (or  its  supplement, 
84°)  on  the  line  of  sines,  will  reach  from  the  side  AC  80,  to  the  side  BC  142.6,  on  the 
line  of  numbers. 

CASE  VI 

The  three  sides  of  a  plane  triangle  given,  to  find  the  angles. 
The  sides  AB  85,  BC  57,  AC  108,  given,  to  find  the  angles  ABC,  BAC,  BCA. 


BY   PROJECTION. 

Draw  the  line  AC,  and  make  it  equal  to  108 ;  take  85  iu  your  compasses,  and,  with 
one  foot  or  the  point  A,  describe  an  arc  ;  then  take  the  disiaMce 
57  in  your  compasses,  and,  with  one  foot  on  C,  describe  another 
arc  intersecting  the  former  arc  in  the  point  B  ;  join  AB,  CB, 
and   it  is   done ;     for   the   angle  A   being  measured   will   be 

found  =  3U°,  B  — 97°,  and  the  angle  C  — 5Ii°,  neai-ly.  ^\      q 

A  id 


BY   LOGARITHMS,  by  Theorem  IV. 
Suppose  BD  to  be  ch-awn  perpendicular  to  AC,  and  that  AG : 


GC. 


Side  AB  =  85 
Side  BC  =  57 

Sum  ot  the  sides 142 

Difference  of  the  sides 28 

HalfbaseAC 54 

DG 18.4 

Sum  is  greatest  segment  AD 72.4 

Difference  is  least  segment  DC 35.6 


As  the  base  AC  108  Log.  2.03342 

Is  to  the  sum  of  the  sides  AB  and 

BC  142 Log.  2.15229 

So  is  the  difference  of  the  sides  AB 

and  BC  28 Log.  1.44716 

3.59943 
2.03342 


To  twice  DG  36.8 Log.  1.56603 

Its  half  is  DG  18.4  


44 


OBLIQUE  TRIGONOMETRY 


Having  divided  the  triangle  into  two  right-angled  triangles,  the  hypotenuses  and 
bases  of  which  ai-e  given,  we  may  find  the  angles  by  Theorem  I. 


To  find  the  angle  BAD. 

As  the  hypotenuse  AB  85 Log. 

Is  to  radius  90° Log-. 

So  is  the  greatest  seg.  AD  72.4  . .  .Log. 

Tb  cosine  BAD  =  31°  36' Log. 


1.92942 

10.00000 

1.83974 


9.93032 


To  find  the  angle  BCD. 

As  the  hypotenuse  BC  57 Log.    1.75587 

Is  to  radius  90° Log.  10.00000 

So  is  the  least  segment  DC  35.6. .  .Log.    1.55145 

To  cosine  of  BCD  =  51°  21' Log.    9.79558 

BAD  =  31    36  '     


Sum 82    57 

Subtract  from 180    00 


Remains  angle  ABC    97    03 


BY   GUNTER'S   SCALE. 

1st.  The  extent  fi-om  the  base  AC  z=:  108,  to  the  sum  of  the  sides  142,  on  the  line 
of  numbers,  wiU  reach  fi-om  the  difference  of  the  sides  28,  to  twice  DG  36.8,  on  the 
same  line  of  numbers. 

2dly.  The  extent  from  the  hypotenuse  AB  =  85,  to  the  gi-eater  segment  AD  72.4, 
on  the  fine  of  numbers,  avUI  reach,  on  the  smes,  fi-om  the  radius  90°,  to  58°  24',  which 
is  the  complement  of  the  angle  BAD. 

3dly.  The  extent  from  the  hypotenuse  BC  57,  to  the  least  segment,  DC  35.6,  or 
the  Ime  of  numbers,  will  reach  on  the  sines  from  the  radius  90°,  to  38°  39',  which  is 
the  complement  of  the  angle  BCA. 

This  case  may  be  solved  without  dividing  the  ti-iangle  into  two  right-angled  triangles, 
by  Theorem  V. 

Having  the  angle  A,  we  may  find  the 
angle  C  by  Theorem  II. 

AsB<;67 Log.  1.75587 

Is  to  sme  angle  A  31°  36' Log.  9.71932 

So  is  AB  83 Log.  1.92942 

11.64374 
1.75587 


To  find  the  angle  A. 

BC=   57 

AB  =    85  Arith.  Comp.  Log.  8.07053 
AC  =  108  Arith.  Comp.  Log.  7.96658 

Sum "250 

Half  sum 125 Log.  2.09691 

Half  sum  less  BC    68 Log.  1.83251 

Sum )19.96658 

Half  sum....  15°  48'       Cosine       Log.  9.98329 


f  >oul  led  is  . .  31    36  =  angle  A. 


To  the  sine  of  angle  C  51°  23'  ....Log.  9.89287 


45 


ASTRONOMY    AND    GEOGRAPHY. 


Astronomy  is  the  science  which  treats  of  the  motions  and  distances  of  the  heavenly 
bodies,  and  of  the  appeai-aiices  thence  arising. 

Geography  is  the  science  wliich  treats  of  the  situations  and  distances  of  the  various 
pails  of  the  surface  of  tlie  earth. 

The  common  opinion  of  astronomers  of  the  present  day  is,  that  the  universe  is 
composed  of  an  mfinite  number  of  systems  or  worlds ;  that  in  every  system  there  are 
certain  bodies  moving  in  free  space,  and  revolving,  at  diiferent  distances,  round  a  sun, 
placed  hi  or  near  the  centre  of  the  system ;  and  that  these  suns,  and  other  bodies,  ai'e 
the  stars  which  are  seen  in  the  heavens. 

The  Solar  System,  so  called,  is  that  in  which  our  earth  is  placed,  and  in  which 
the  sun  is  supposed  to  be  fixed  near  the  centre,  with  several  bodies,  similar  to  our 
earth,  revolving  round  at  different  distances.  This  hypothesis,  which  is  fully  confii-med 
by  obsei-vation,  is  called  the  Copernican  System,  from  Nicholas  Copernicus,  a  Polish 
philosopher,  who  revived  it  about  tlie  year*  1500,  after  it  had  been  buried  in  oblivion 
many  ages. 

Stai"s  are  distinguished  into  two  kinds,  fixed  and  tvandering.  Tlie  fonrier  are 
supposed  to  be  suns  m  the  centres  of  their  systems,  shming  Avith  their  oavu  light,  and 
preservuig  nearly  the  same  situation  with  respect  to  each  other.  They  are  usually 
distinguished  by  thek  brightness,  the  largest  being  called  of  the  first  magnitude,  and 
the  smallest  visible  to  the  naked  eye  being  of  the  sixth  or  seventh  magnitude.  A 
Constellation  is  a  number  of  stars  which  appear  near  to  each  other  on  the  concave 
surface  of  the  heavens,  and  astronomers,  for  the  sake  of  remembering  them  with 
gi'eater  ease,  suppose  them  to  be  circumscribed  by  the  outlmes  of  some  anmial  or 
other  figure.  Wandermg  stars  are  tliose  bodies  within  our  systerh,  or  celestial  sphere, 
which  revolve  round  the  sun ;  they  appear  lummous  by  reflectmg  the  light  of  the 
sun,  and  are  of  three  kinds,  namely,  primary  planets,  secondary  planets,  and  comets. 

The  Primary  Planets  are  bodies  which  revolve  round  the  sun  as  the  centre  of  their 
com^ses,  the  motions  beuag  regularly  performed  m  tracks  or  paths,  called  orbits,  that 
ai"e  nearly  cu'cular  and  concentrical  with  each  other.  A  Secondai-y  Planet,  Satellite, 
or  Moon,  is  a  body  which,  AvhUe  it  is  carried  round  the  sun,  reitolves  also  round  a 
primaiy  planet.  Comets  are  bodies  which  move  round  the  sun  in  veiy  excentrical 
orbits,  with  vast  atmospheres  about  them,  and  tails  derived  from  the  same. 

There  are  seventeen  primary  planets,  which,  reckoned  in  order  from  tho  sun,  are  as 
follows : — Mercury,  Venus,  the  Earth,  Mars,  Vesta,  Juno,  Pallas,  Ceres,  Astrea,  Hebe, 
Iris,  Flora,  Metis,  Jupiter,  Saturn,  Uranus,  and  Neptune. 

Mercury  and  Venus  are  called  inferior  planets,  because  their  orbits  are  within  the 
earth's ;  the  others  are  called  superior  planets,  as  theu-  orbits  mclude  that  of  the  earth. 

The  Sun,  the  first  and  greatest  object  of  astronomical  knowledge,  is  placed  near  the 
centre  of  the  orbits  of  all  the  planets,  and  turns  round  its  axis  m  25;^  days.  Its  diameter 
is  88.3,000  English  miles,  and  its  mean  distance  from  the  earth  95  millions  of  miles. 

IMercury  is  the  least  of  all  the  planets  knoAvn  before  the  discovery  of  Vesta,  Juno, 
Pallas,  and  Ceres,  and  is  the  nearest  to  the  sun,  his  mean  distance  from  that  luminary 
being  37  millions  of  miles.  His  periodic  revolution  in  liis  orbit  round  the  sun  is 
performed  hi  87  days  23  hours,  and  his  diameter  is  about  3200  miles. 

Vexus  is  the  brightest  of  all  the  planets.  Her  diameter  is  7687  miles ;  her  mean 
distance  from  the  sun,  C9  millions  of  miles ;  and  her  periodic  revolution  is  performed 
m  224  days  17  hours.  When  this  planet  is  m  that  part  of  her  orbit  which  is  west  of 
the  sun,  she  rises  before  him  in  the  morning,  and  is  called  the  morning  star ;  v»'hen 
she  is  in  the  eastern  part  of  her  orbit,  she  shmes  m  the  evening,  after  he  sets,  and  is 
called  the  evening  star. 

The  next  planet  is  the  Earth,  the  diameter  of  which  is  7914  miles,  the  distance 
from  the  sun  95  millions  of  miles,  and  the  time  of  revolution  round  the  sun,  one  year. 
The  earth  turns  round  its  axis  from  west  to  cast  in  23  hours  Tij  ninutes,  which 
occasions  the  apparent  diurnal  motion  of  the  sun  and  all  the  J-    -  n?c  l>odies  romul  il 


4G  ASTRONOMY   AND   GEOGRAPHY. 

from  east  to  west  in  the  same  time,  and  is,  of  course,  the  cause  of  then-  rising  and 
setting,  of  day  and  night.  Tlie  axis  of  the  earth  is  inchned  about  23°  28'  to  the  plane 
of  its  orbit,*  and  keeps  nearly  in  a  dkection  parallel  to  itself,  throughout  its  annual 
course,  which  causes  the  retiu-n  of  spring  and  summer,  autuimi  and  winter.  Tlius  the 
diumal  motion  gives  us  the  grateful  vicissitude  of  night  and  day,  and  the  annual  motion 
the  regular  succession  of  the  seasons.  The  earth  is  attended  by  a  satellite  called  the 
x^IooN,  whose  diameter  is  2161  miles.  Her  distance  from  the  centre  of  the  earth  is 
240,000  miles.  She  goes  round  her  orbit  in  27  days  8  hours ;  but,  reckoning  from 
change  to  change,  m  29^  days.  Her  orbit  is  inclmed  to  the  ecliptic  in  an  angle  of 
5°  9',  cutting  it  in  two  points  diametrically  opposite  to  each  othei-,  called  her  nodes. 
As  the  moon  shmes  only  by  the  reflected  light  of  the  sim,  she  must  appear  different 
when  in  different  situations  with  respect  to  that  luminary.  When  she  is  m  conjunction 
with  the  sun,  her  dark  side  is  turned  towards  the  earth,  which  renders  her  invisible ; 
this  is  called  new  moon:  when  she  is  m  opposition,  her  light  side  is  wholly  visible  from 
tlie  earth ;  this  is  called  full  moon. 

If  at  the  time  of  new  moon  she  is  near  to  either  of  her  nodes,  she  may  intercept  a 
part  oft...  iun's  light,  and  thus  cause  an  eclipse  of  the  sun;  and  if  she  is  near  either 
of  her  nodes  at  the  time  of  full  moon,  she  may  pass  mto  the  shadow  of  the  earth,  and 
cause  an  eclipse  of  the  moon.  In  a  sunUar  manner,  when  the  moon  passes  between  an 
observer  on  the  earth  and  a  star,  it  is  called  an  occuUation  of  the  star.  The  instant 
when  the  moon's  limb  fii-st  covers  the  star  is  called  the  immersion,  and  the  moment  of 
its  reappearance  is  called  the  emersion.  When  IMercury  or  Venus  passes  between  the 
sun  and  an  observer,  and  appears  to  pass  over  the  sun's  disk,  it  is  called  a  transit  of 
Mercury  or  Venus.  Eclipses,  occultations,  and  transits,  ai-e  of  gi-eat  importance  in 
ascertahiing  the  longitudes  of  places  on  the  earth.  Eclipses  of  the  moon  furnish  a 
convincing  proof  of  the  rotundity  of  the  earth,  since  the  shadow  of  the  earth,  seen 
upon  the  moon  when  eclipsed,  is  always  circular.  This  is  further  confirmed  by  the 
appearance  of  objects  at  sea;  for  when  a  ship  is  makmg  towards  the  land,  the  mariners 
first  descry  the  tops  of  steeples,  trees,  &c.,  pomting  above  the  water ;  the  lower  parto 
being  hid,  by  reason  of  the  curvature  of  the  earth. 

The  earth  is  not  a  perfect  globe  or  sphere,  but  is  a  little  flattened  at  the  poles,  beuig 
nearly  of  the  figure  of  an  oblate  spheroid,  the  equatorial  diameter  being  about  2(3  miles 
longer  than  the  polar ;  but  since  this  difference  bears  but  a  small  comparison  to  the 
whole  diameter,  we  may,  for  all  the  practical  purposes  of  navigation,  consider  the 
earth  as  a  ])erfect  sphere,  as  will  be  done  in  the  I'est  of  this  work.  The  natural 
divisions  of  the  earth  will  be  given  hereafter. 

Mars  is  the  next  planet  to  tlie  earth.  His  diameter  is  4189  miles.  His  distance  from 
the  sun  is  144  millions  of  miles,  and  his  periodic  revolution  is  performed  in  about 
687  days.  He  revolves  round  his  axis  in  24  hours  40  minutes,  appearmg  of  a  dusky- 
reddish  hue,  and  is  supposed  to  be  encompassed  with  a  very  gi-eat  atmosphere. 

Between  Mars  and  Jupiter  are  situated  eleven planets,named  asteroids,  viz.  Vesta,  Juno, 
Pallas,  Ceres,  Astreft,f  Hebe,  Iris,  Flora,  Metis,  Hygeia  and  Parthenope. 

Vesta  was  discovered  by  Dr.  Olbers,  of  Bremen,  on  the  29th  of  3Iarch,  1807.  Its 
mean  distance  from  the  sun  is  about  224  millions  of  mUes.  Its  periodic  revolution  is 
j»ei"forn>8d  in  1325  days. 

Juno  was  discovered  by  Mr.  Harding,  of  Lilienthal  (near  Bremen),  on  the  first  of 
September,  1804.  It  appeai-s  like  a  star  of  the  eighth  magnitude.  Its  distance  from 
the  sun  is  about  254  millions  of  miles.  Its  periodic  revolution  is  performed  in  1.593 
days.  The  incluiation  of  its  orbit  to  the  ecliptic  is  13°  4',  and  the  excentricity  of  the 
orbit  t  0.25. 

Pallas  was  also  discovered  by  Dr.  Olbers,  Mai'ch  28, 1802.  Its  diameter,  according 
to  Dr.  Hcrschel,  is  only  110  miles.  It  appears  like  a  star  of  the  eighth  magnitude. 
Its  mean  distance  Irom  the  sun  is  about  263  millions  of  miles.  Its  periodic  revolution 
is  performed  in  1686  days.  The  incluiation  of  its  orbit  to  the  ecliptic  is  34°  35',  and 
the  excentricity  of  the  orbit  0.242. 

Ceres  was  discovered  by  Mr.  Piazzi,  of  Palerino,  on  the  first  of  Januaiy,  1801.  Its 
diameter,  according  to  Dr.  Herschel,  is  only  160  miles.  It  appears  like  a  star  of  the 
seventh  or  eighth  magnitude.  Its  distance  from  the  sun  is  about  263  millions  of  miles, 
and  its  periodic  revolution  is  performed  in  1685  daj-s,  being  at  nearly  the  same  distance 
from  the  sini  as  Pallas.     The  inclination  of  the  orbit  of  Ceres  to  the  ecliptic  is  10°  37', 

*  The  inclination  decreases  at  present  about  50"  in  lOOyenrs,  by  reason  of  the  attraction  of  the  planet* 
im  the  earth.    It  is  also  affected  by  the  nutation  given  in  Table  XLIII.,  which  sometimes  amounts  to  9" 
t  Aslrea  was  discovered  by  Mr.  Hencke,  of  Dresden,  Dec.   8,  1845. 

Hebe  do.  do.  do.  do.        July  4,  1847. 

Iris  do.  do.       Mr.  Hind,  London,  Aug.  13,  1847. 

Flora         do.  do.  do.  do.        Oct.  18,  1847. 

Metis         do.  do.       Mr.  Graham,      Sligo,       May,       1848. 

Hygcia         do.  do.        M.  Gasparis,        Naples,    April,     1849. 

Parthenope  do.  do.  do.  do.       May  11,  1850. 

t  In  estimating  the  excenlricities  of  the  planets,  their  mean  distance  from  the  sun  is  put  equal  to  unltj. 


ASTRONOMY   AND   GEOGRAPHY.  47 

and  the  excentricity  0.077.  The  situations  of  the  nodes  of  the  two  planets,  Ceres  and 
Pallas,  and  the  incluiations  of  then-  orbits,  are  very  different  from  each  other,  so  that 
when  those  planets  are  in  the  same  plane,  they  are  at  a  gi-eat  distance  from  each  other, 
notwithstanding  their  mean  distances  from  the  sun  are  nearly  equal.  It  has  been 
supposed  by  some,  that  these  small  bodies  are  fragments  of  a  fonner  planet. 

Jupiter  is  situated  still  higher  in  the  system,  and  is  the  largest  of  all  the  planets, 
bemg  easily  distmguished  from  them  by  his  peculiar  mamitudc  and  light.  His 
diameter  is  89,170  miles ;  his  distance  from  the  sun  494  millions  of  miles ;  and  the 
time  of  his  periodic  revolution  is  4332J  days.  Though  Jupiter  is  the  largest  of  all  the 
planets,  yet  his  diurnal  revolution  is  die  swiftest,  being  only  9  hours  and  5G  minutes. 

Jupiter  is  attended  by  four  satellites,  invisible  to  the  naked  eye;  but  through  a 
telescope  they  make  a  beautiful  appearance.  In  speaking  of  them,  we  distinguish 
them  according  to  their  places,  into  the  first,  second,  and  so  on ;  by  the  first  we  mean 
that  which  is  nearest  to  the  planet.  The  appearance  of  these  satellites  is  marked  in 
the  XlXtli  page  of  the  Nautical  Almanac  for  some  particular  hour  of  the  night ;  tlie 
times  wlien  they  are  ecUpsed,  by  passhig  into  the  shadow  of  Jujiiter,  are  also  given  in 
the  Nautical  Almanac ;  these  eclipses  are  of  some  use  in  determining  the  longitudes 
of  i)laces  on  the  earth. 

Before  the  discovery  of  the  planet  Uranus,  Saturn  was  reckoned  the  most  remote 
planet  of  our  system.  He  shines  with  but  a  pale  and  feeble  light.  His  diameter  is 
79,042  miles;  his  distance  from  the  sun  907  millions  of  miles;  and  his  j)enodic 
revolution  in  his  orbit  is  performed  in  about  29  years  1G7  days.  This  i)lanct  is 
surrounded  with  a  broad,  fiat  ring,  has  a  diurnal  revolution  round  its  axis,  and  is 
attended  by  seven  satellites. 

By  some  observations  made  by  Dr.  Herschel,  it  appeared  that  the  largest  diameter 
of  Saturn  corresponds  to  die  latitude  of  45° ;  but  from  later  obsei-vations  he  has  been 
induced  to  believe,  that  this  uTegularity  is  owing  to  an  optical  deception,  arising  from 
the  refraction  of  the  light  in  passing  through  the  atmosphere  of  the  ring. 

Uranus,  Herschel,  or  Georgium  Sidus,  was  discovered  in  the  year  1781,  by  Dr.  Her- 
schel, though  it  had  been  seen  several  times,  but  had  been  considered  as  a  fixed  star. 
Its  diameter  is  35,109  miles;  its  distance  from  the  sun  is  1823  millions  of  miles;  and 
its  periodic  revolution  in  its  orbit  is  performed  in  83 ^  years.  Dr.  Herschel  has  also  dis- 
covered six  satellites  attending  this  planet. 

Neptune,  the  most  remote  planet  of  our  system,  was  seen  by  Dr.  Galle,  of  Berlin,  Oct. 
23,  1846.  Its  mean  distance  from  the  sun  is  2867  millions  of  miles — its  diameter  is  34,750 
miles,  and  its  period  of  revolution  is  165|  years.   Mr.  Lascelles  has  discovered  one  satellite. 

The  astronomy  of  comets  is  yet  in  its  infancy.  The  return  of  one  of  them  in  the 
year  1758  was  foretold  by  Dr.  Ilalley,  and  it  happened  as  he  predicted  ;  and  it  apjieared 
again  in  18.35.  He  also  foretold  tlie  return  of  another  in  the  year  1790,  but  it  never 
appeared.  This  was  owing  to  the  inaccuracy  of  the  obsei-vations  of  the  comet  at  its 
former  appearance ;  for  Mr.  Mechain,  having  collected  all  the  observations,  and 
calculated  the  orbit  again,  found  it  to  differ  essentially  from  that  determined  by 
Dr.  Halley.  Olber's  comet,  which  appeared  in  1815,  has  a  revolution  of  72  years; 
and  Encke's  comet,  which  lias  been  observed  in  several  successive  approaches  to  the 
pprihehon,  com})letes  its  revolution  in  the  short  period  of  1204  days.  Biela's  comet 
has  also  been  observed  several  times,  with  a  periodical  revolution  of  about  G^  years. 

Comets  move  romid  the  sun  in  all  directions ;  but  the  planets  and  satellites,  except 
one  of  the  satellites  of  Uranus,  move  from  west  to  east  when  seen  from  the  sun  ;  but 
if  viewed  from  any  other  of  the  planets,  as  the  earth,  they  would  appear  to  revolve 
round  it  as  a  centre  ;  but  the  sun  would  be  the  only  one  that  moves  uniformly  the  same 
^vay,  for  the  other  jilaiiets  would  sometimes  appear  to  move  from  west  to  east,  and 
then  to  stand  still ;  tiien  they  would  seem  to  move  from  east  to  west ;  and,  after 
-Standing  some  time,  diey  would  again  move  from  west  to  east ;  and  so  on,  continually. 
The  motion  of  a  jjlanet  from  west  to  east  is  called  the  direct  motion,  or  according  to 
the  order  of  the  signs.  The  contraiy  motion,  from  east  to  west,  is  called  retrograde. 
When  the  planet  appears  to  stand  still,  it  is  said  to  be  stationary. 

To  illustrate  what  has  already  been  said  relative  to  the  motions  and  distances  of  the 
planets  and  satellites,  we  have  given  die  adjoining  Plates  III.  and  IV.,  which  require 
no  exjilaiiation. 

In  noting  die  situations  of  the  stars  and  planets,  astronomers  have  been  under  the 
necessity  of  imagining  various  lines  and  circles  on  the  sphere ;  and  geographers  have 
done  the  same  for  fixing  the  situation  of  [)laces  on  the  earth.  The  most  remarkable 
of  these  are  the  following: — 

A  great  circle  is  diat  whose  plane  passes  through  the  centre  of  the  sphere ;  and  a 
mnall  circle  is  that  whose  plane  does  not  pass  through  that  centre. 

A  diameter  of  a  sphere,  perpendicular  to  any  great  cuxle,  is  called  the  axis  of  that 
circle;  and  the  extremities  of  a  diameter  are  called  its  poles.  Hence  the  pole  of  a 
great  circle  is  90°  from  every  point  of  it  u[)on  the  surface  of  the  spliere ;  but  as  tlio 


48  ASTRONOMY  AND   GEOGRAPHY. 

axis  is  perpendicular  to  the  cu'cle  when  it  is  perpendicular  to  any  two  radii,  a  point  on 
the  surface  of  a  sphere  90°  distant  from  any  two  points  of  a  great  cuxle,  will  be 
the  pole. 

All  angular  distances  on  the  surface  of  a  sphere,  to  an  eye  at  the  centre,  are  measured 
by  arcs  of  great  cu-cles.  Hence  all  triangles  formed  upon  the  surface  of  a  sphere,  for 
the  solution  of  spherical  problems,  must  be  formed  by  the  arcs  of  great  cuxles. 

Secondaries  to  a  gi'eat  cu'cle  are  great  cu-cles  which  pass  through  its  poles,  and 
consequently  must  be  pei-pendicular  to  theu*  gi-eat  cu'cles. 

The  axis  of  the  earth  is  that  diameter  about  which  it  pei-forms  its  diurnal  motion 
and  the  extremities  of  this  diameter  are  called  the  poles. 

The  terrestrial  equator  is  a  great  cuxle  of  the  earth  perpendicular  to  its  axis.  Hence 
the  axis  and  poles  of  the  earth  are  the  axis  and  poles  of  its  equator.  That  half  of  the 
earth  which  lies  on  the  side  of  the  equator  in  which  Europe  and  the  United  States  of 
America  are  situated,  is  called  the  northern  hemisphere,  and  the  other  the  southern ;  and 
the  poles  are  respectively  called  the  north  and  south  poles. 

The  latitude  of  a  place  upon  the  earth's  siu-face  is  its  angular  distance  from  the 
equator,  measured  upon  a  secondaiy  to  it.  These  secondaries  to  the  equator  are  called 
vieridians. 

The  longitude  of  a  place  on  the  earth's  surface  is  an  arc  of  the  equator  intercepted 
between  the  meridian  passing  through  the  place,  and  another,  called  ihefirsi  meridian, 
passing  through  that  place  from  which  you  begin  tomeasui-e ;  or  it  is  the  angle  formed 
at  the  pole  by  these  two  meridians.  The  Americans  and  English  generally  place  the 
fii'st  meridian  at  Greenwich ;  the  Fi-ench  place  it  at  Paris,  the  Spaniards  at  Cadiz ; 
some  geographers  place  it  at  Teneriffe,  and  others  at  other  places.  Thi'oughout  this 
work,  Greenwich  wiU  be  reckoned  as  the  first  meridian.  The  longitude  is  counted 
from  the  first  meridian,  both  eastward  and  westward,  till  it  meets  at  the  same  meridian 
on  the  opposite  point;  therefore  the  longitude  (and  also  the  difference  of  longitude 
between  any  two  places)  can  never  exceed  180°. 

If  the  plane  of  the  terrestrial  equator  be  pi'oduced  to  the  sphere  of  the  fixed  stars,  it 
tnai'ks  out  a  cu'cle  called  the  celestial  equator ;  and  if  the  axis  of  the  earth  be  produced 
in  like  manner,  the  pomts  of  the  heavens,  to  which  it  is  produced,  are  called  poles, 
being  the  poles  of  the  celestial  equator.  The  star  neai-est  to  each  pole  is  called  the 
pole  star. 

Secondaries  to  the  celestial  equator  are  called  circles  of  declination ;  of  these  24, 
which  divide  the  equator  into  equal  parts,  each  containing  15°,  are  called  hour  circles. 

Small  ckcles  pai-allel  to  the  celestial  equator  are  called  parallels  of  decUnation. 

The  sensible  horizon  is  that  cucle  m  the  heavens  whose  plane  touches  the  earth  at 
the  spectator.  The  rational  horizon  is  a  great  circle  in  the  heavens,  passing  through 
the  earth's  centre,  parallel  to  the  sensible  horizon. 

If  the  I'adius  dra^vn  from  the  centre  of  the  earth  to  the  place  where  the  spectator 
stands  be  produced  both  ways  to  the  heavens,  the  point  vertical  to  him  is  called  the 
zenith,  and  the  jx)mt  opposite,  the  nadir.  Hence  the  zenith  and  nadir  are  the  poles  of 
the  rational  horizon. 

Secondaries  to  the  horizon  are  called  vertical  circles,  because  they  are  pei-pendicular 
to  the  horizon.    On  these  cu-cles,  therefore,  the  altitude  of  a  heavenly  body  is  measured. 

The  secondai-y  common  to  the  celestial  equator,  and  the  horizon  of  any  place,  is  the 
celestial  meridian  of  that  place.  This  meridian  corresponds  with  the  terrestrial  meridian 
of  the  same  place,  which  passes  through  the  poles  of  tiie  earth,  the  zenith  and  nadir 
crossing  the  equator  at  right  angles,  and  cutting  the  horizon  in  the  north  and  south 
pomts ;  that  poiiit  being  called  north  which  passes  through  the  north  pol(>,  and  the 
opposite  direction  is  called  south.  The  vertical  circle  which  cuts  the  meridian  of  any 
place  at  right  angles  is  called  the  prime  vertical ;  the  points  where  it  cuts  the  horizon 
are  called  the  east  and  ivest  points,  and  to  an  observer,  v/ith  his  face  directed  towards 
the  south,  the  east  point  will  be  to  his  left  hand,  and  the  tvest  to  his  right  hand.  Hence 
the  east  and  west  points  are  90°  distant  from  the  north  and  south.  These  four  are 
called  the  cardinal  points.  The  meridian  of  any  place  divides  the  heavens  into  two 
hemispheres,  lying  to  the  east  and  west ;  that  lyuig  to  the  east  is  called  the  eastern 
hemisphere,  and  the  other  the  ivestern  hemisphere.  When  the  sun  is  at  its  greatest 
altitude  on  the  meridian  of  any  place,  it  is  noon,  or  mid-daj'. 

The  azimuth  of  a  heavenly  "body  is  its  distance  on  the  horizon,  when  refen-ed  to  it 
by  a  secondary,  from  the  north  or  south  i)oints.  The  amplitude  is  its  distance  from  the 
east  or  west  points,  at  the  time  of  rising  or  setting. 

The  ecliptic  is  that  great  circle  in  the  heavens  which  the  sun  appears  to  describe  in 
the  course  of  a  year.  The  cchptic  and  equator,  being  gi-eat  cucles^  must  bisect  each 
other,  and  tlieir  angle  of  inclination  is  called  the  oUiquity  of  the  ecliptic  ;  and  the  points 
'vhcre  they  uiterscct  are  called  the  equinoctial  poiiits.    The  times  when  the  sun  comes 


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18G1 


ASTRONOMY    A.N'U    (JEOGRAl'HY.  4"J 

to  tlicse  points  are  called  the  equinoxes.  The  ccli[)tic  is  divided  hito  12  e<]iial  paits, 
(ViUed  signs: — viz.  Aries  ^^  Taurus  y,  Gemhii  n,  Cancer  Ed,  Leo  £i,,  Virgo  Trjj, 
Libra  ^,  Scorpio  tT|^,  Sagittarius  f,  Capricornus  >?»  Aipiarius  r.-,  PiscfS  X-  The 
order  of  these  is  according  to  the  a])parcnt  motion  ot'the  sun.  The  first  point  of  Arie* 
coincides  with  one  of  the  equinoctial  points,  and  the  tirst  point  of  Libra  with  the  otlur. 
The  first  sly  signs  are  called  noiihern,  lying  on  the  north  side  of  the  equator  ;  and  the 
last  six  are  called  southern  lying  on  the  south  side. 

The  zodiac  is  a  space  extending  eight  degrees  on  each  side  the  ecliptic,  within 
which  the  motion  of  all  the  planets  is  contained,  except  the  newly-discovered  planets. 
The  right  ascension  of  a  body  is  an  arc  of  the  equator  intercepted  between  the  first 
point  of  Aries,   and   a  circle   of   declination   passing  through   the   body,  measured 
according  to  the  order  of  the  signs. 

Right  ascension  of  the  meridian,  or  mid-heaven,  is  the  distance  of  the  meridian  from 
the  fij-.st  point  of  Aries,  and  is  found  by  addmg  the  apparent  time  past  noon  to  the 
sun's  right  ascension. 

The  ascensional  difference  of  any  oljject  is  the  difference  betN-v-wen  the  right  ascension 
of  the  object  and  that  point  of  the  equator  which  rises  or  sets  with  it. 

The  declination  of  a  star  or  any  celestial  object  is  its  angular  distance  from  the 
equator,  measured  upon  a  secondary  to  it  passing  through  the  object. 

The  longitude  of  a  star  or  any  celestial  olyect  is  an  arc  of  the  ecliptic  intercepted 
Ijetwccn  the  first  point  of  Aries,  and  a  secondary  to  the  ecliptic  passing  through  the 
body,  measured  according  to  the  order  of  the  ^gns.  If  the  observer  be  on  the  earth, 
the  longitude  is  called  the  geocentric  longitude ;  but  if  seen  from  the  sun,  it  is  called  the 
heliocentric  longitude ;  the  body  in  each  case  being  referred  perpendicularly  to  the 
ecliptic  in  a  plane  passing  through  the  eye. 

jYoimgesimal  degree  of  the  ecliptic  is  its  highest  point  at  any  given  tune,  and  is  90° 
from  the  |)oints  where  the  ecliptic  iut^-sects  the  horizon. 

The  latitude  of  a  star  or  any  celestial  object  is  its  angular  distance  from  the  ecliptic, 
measured  upon  a  secondary  to  it  drawn  through  the  body.  If  the  body  be  observed 
frojn  the  earth,  its  angular  distance  from  the  ecliptic  is  called  tlie  geocentric  latitude ; 
but  if  ol)served  from  the  sun,  it  is  called  the  heliocentric  latitude.  The  secondary  cu-cle 
drawn  peri)endicular  to  the  ecliptic  is  called  a  circle  of  latitude. 

The  tropics  are  two  parallels  of  declination  touching  the  ecliptic.  One,  toucliing  it 
at  the  beginning  of  Cancer,  is  called  the  tropic  of  Cancer ;  and  the  other  touching  it  at 
the  beginning  of  Capricorn,  is  called  the  tropic  of  Capricorn.  The  two  points  where 
the  tropics  touch  the  ecliptic  are  called  the  solstitial  points. 

Colures  are  two  secondaries  to  the  celestial  equator,  one  passing  through  the 
equinoctial  ])oints,  called  the  cquinocticd  colure ;  and  the  other  passing  through  the 
solstitial  points,  called  the  solstitial  colure.  The  times  when  the  sun  comes  to  the 
solstitial  j)oints  are  called  the  solstices. 

Aberration  of  a  star,  or  any  heavenly  body,  is  a  small  aj)parent  motion,  occasioned 
bv  the  progressive  velocity  of  light.  This  is  calculated  by^neans  of  Tables  XXXIX., 
XLL,  or  XLII. 

JVutation  is  a  small  apparent  motion  of  the  heavenly  bodies,  occasioned  by  a  red 
motion  of  the  earth's  axis,  arising  from  the  attractions  of  the  sun  and  moon  on  the 
spheroidal  form  of  the  earth.  The  effect  of  this  on  the  right  ascension  and  declination 
is  given  in  Ta!)le  XLIIL,  and  on  the  longitude  in  Table  "XL. ;  the  correction  in  this 
last  tal)le  being  generally  called  the  equation  of  the  equinoxes  in  longitude. 

Precession  of  the  equinoctial  points  is  a  small  motion  of  about  50i"  per  year, 
occasioned  by  the  same  cause  as  the  nutation.  By  this  motion  the  equinoctial  j)oi|^ 
are  carried  backward  from  east  to  west ;  consequently,  tiie  heavenly  l)odies  appear  to 
move  forward  the  same  quantity  front  west  to  east.  The  annual  variations  ol"  the 
places  of  the  stars  from  precession,  and  the  secular  equations  arising  from  the  change 
of  tiie  earth's  orbit  by  the  attraction  of  the  planets,  are  given  in  Tables  VIII.  and 
XXXVII. 

The  arctic  and  antarctic  circles  are  two  jiarallels  of  declination,  the  former  about  the 
north,  and  the  latter  about  the  south  pole,  the  distance  of  which,  from  the  two  [wies,  is 
equal  to  the  distance  of  the  tropics  from  the  equator,  which  is  about  2;F  28'.^  These 
arc  also  called  polar  circles.  Tlie  two  tropics  and  two  polar  circles,  when  referred  to 
the  earth,  divide  it  into  five  parts,  called  zones;  the  two  i)arts  within  t!ie  polar  circles 
are  called  the /ri^i^  zones;  the  two  parts  between  the  polar  circles  and  tropics  are 
called  the  temperate  zones;  and  the  part  between  the  tropics  is  called  the  torrid  zone. 

iiesides  the  imagi)iary  divisions  of  the  earth,  there  are  various  natural  divisions  of 
its  surface,  such  as  continents,  oceans,  islands,  seas,  rivers,  «S:-c. 

A  continent  is  a  large  tract  of  land,  wherein  are  several  p.nii)ires,  kingdoms,  and 
countries  conjoined  ;  as  Europe,  Asia,  Africa,  and  America. 
1 


50  ASTRONOMY   AND    GEOGRAPHY. 

An  island  is  a  jtart  of  tlie  earth  tliut  is  envu'oned  or  euconipasscd  roimd  l)y  tlie  sea 
as  Long  Island,  Block  Island,  &c. 

A  peninsula  is  a  portion  of  land  almost  suiTounded  with  water,  save  one  narrow 
neck  Avhich  joins  it  to  the  continent;  as  the  Morea. 

An  isthmus  is  a  narrow  neck  of  land  joining  a  peninsida  to  tlie  adjacent  land,  by 
whicli  the  people  may  pass  from  one  to  the  other ;  as  the  isthmus  of  Darien. 

A  promontory  is  a  high  part  of  land  stretching  itself  into  the  sea,  the  extremity  of 
which  is  called  a  cape  or  htadland. 

A  mountain  is  a  rising  part  of  dry  land,  overtopping  the  adjacent  coimtry. 

An  ocean  is  a  vast  collection  of  water,  separating  continents  from  one  another,  and 
washing  theii*  borders  or  shores  ;  as  the  Atlantic  and  Pacific  Oceans. 

A  sea  is  part  of  the  ocean,  to  whicli  we  must  sail  through  some  strait ,  as  the 
Mediterranean  and  Baltic  Seas.  This  term  is  sometimes  used  for  the  whole  body  of 
salt  water  on  the  globe. 

A  strait  is  a  narrow  jiart  of  the  ocean  lying  between  two  shores,  and  opening 
a  way  into  some  sea  ;  as  the  Straits  of  Gibraltar,  that  lead  into  the  Mediterranean 
Sea. 

A  creek  is  a  small  narrow  part  of  the  sea  or  river,  that  goes  up  but  a  little  way  mto 
the  land. 

A  bay  is  a  gi-eat  uilet  of  the  land ;  as  the  Bay  of  Biscay,  and  the  Bay  of  Mexico ; 
otherwise  a  bay  is  a  station  or  road  for  ships  to  anchor  in. 

A  river  is  a  considerable  stream  of  water  issuing  out  of  one  or  various  s}irings,  and 
continually  gliding  along  in  one  or  more  channels,  till  it  discharges  itself  uito  the 
ocean :  the  smaller  streams  are  called  rivulets. 

A  lake  is  a  large  collection  of  waters  in  an  inland  place ;  as  the  Lakes  Superior  and 
Huron  in  America. 

A  g""//is  a  part  of  the  ocean  or  sea,  neai-ly  suj^-ounded  by  the  land,  except  where  it 
communicates  with  the«ea;  as  the  Gulf  of  Venice. 

Thus  we  have  given  the  most  useful  definitions  of  Astronomy  and  Geogi-aphy,  and 
to  assist  the  learner  there  is  also  given  Plate  V.,  in  which  those  terms  are  exj)lained  at 
one  view.  We  may  further  observe,  that,  as  the  latitude  of  any  place  upon  the  earth 
is  counted  from  the  equator  upon  an  arc  of  the  meridian,  the  difference  of  latitude 
between  two  places,  both  north  or  both  south,  is  found  by  subtracting  the  less  latitude 
from  the  greater;  hut  if  one  latitude  he  north,  and  the  other  south,  the  difference  is  found 
oy  adding  both  latitudes  together. 

1.  Consequently,  if  a  ship  in  north  latitude  sails  northerly,  or  in  south  latitude 
southerly,she  increases  her  latitude  ;  but  in  north  latitude  sailing  southerly,  or  in  south 
latitude  sailing  northerly,  she  decreases  her  latitude,  because  she  sails  nearer  to  the 
equator,  from  whence  the  latitude  is  reckoned. 

2.  Wherefore,  in  north  latitude  sailing  northerly,  or  in  south  latitude  sailing  southerly, 
the  difference  of  latitude,  added  to  the  latitude  left,  gives  the  latitude  in. 

3.  In  north  latitude  sailing  southerly,  or  in  south  latitude  sailing  northerly,  the  difference 
of  latitude,  subtracted  from  tne  latitude  left,  gives  the  latitude  in. 

4.  JFhcn  the  latitude  decreases,  and  the  difference  of  latitude  is  greater  than  the  latitude 
sailed  from,  subtract  the  latitude  left  from  the  difference  of  latitude,  and  the  remainder  ivill 
he  the  latitude  in,  but  of  a  different  iiame,  for  it  is  evident,  in  this  case,  that  the  skip  has 
crossed  the  equator. 

5.  The  difference  of  longitude  between  two  i)laces,  being  both  east  or  west,  is  found 
by  subtracting  the  less  longitude  from  the  grecdcr  ;  but  if  one  be  in  east  longitude  and  the 
oti£r  in  west,  their  sum  is  the  difference  of  longitude,  tvhen  it  does  not  exceed  180^,  bid  if 
iPcxceeds  180°,  that  sum  must  be  subtracted  from  360°,  and  the  remainder  ivill  be  the 
difference  of  longitude. 

0.  Therefore" in  east  longitude  sailing  easterly,  or  in  west  longitude  sailhig  westerly, 
the  difference  of  longitude,  added  to  the  longitude  left,  gives  the  longitude  in,  when  that 
sum  does  not  exceed  180°;  but  if  it  exceeds  180°,  the  simi,  subtracted  *  from  8fi0°, 
leaves  the  longitude  in,  but  of  a  different  name  from  that-lcfV. 

7.  In  east  longitude  sailing  westerly,  or  in  west  longitude  sailing  easterly,  the 
difference  of  longitude,  subtracted  from  the  longitude  left,  gives  the  longitude  in  ;  bid  when 
tlie  difference  of  longitude  is  greatest,  the  longitude  left  must  he  subtracted  from  that 
difference,  and  the  rcviavnder  will  he  the  longitude  in,  but  of  a  different  name  from  the 
longitude  left. 

*  In  this  Rile  it  is  supposed,  that  the  sum  of  the  longitude  left,  and  tlie  difference  of  loniptude,  is  less 
than  360-^,  whicli  is  always  the  case  when  the  ditVereiice  of  longitude  is  less  than  180°,  which  we  have 
gr«nerally  supposed  to  be  the  case  in  these  rules. 


Flate  IT. 


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EX:G\\TRLTTXT. 
1801 


ASTRONOMY   AxND    GEOGRAPHY. 


51 


What  has  been  said  will   be   rendered 
exan)i)les : — 

EXAMPLE   1. 
What  is  the  difFei-ence  of  latitude  be- 
tween Boston,  in  the  latitude  of  42'"il'  N., 
and  Richmond  (Vu-gir;ia),  in  the  latitude 
of  37°  32' N.? 

From  Boston's  latitude 42°  21'  N. 

Subtract  Richmond's  latitude  37  32  N. 


Remains  the  difference  of  lat, 

In  miles 289 


4  49 
GO 


EXAMPLE   n. 
A  ship  from  latitude  59°  27'  S.,  sails 
southward  until  her  difference  of  latitude 
is  374  miles ;  v/hat  latitude  is  she  come  to  ? 

Latitude  sailed  from 59°  27'  S. 

Difference  of  lat.  374-;- GO—    G  14  S. 

Latitude  in 65  41  S. 


familiar  to   tlie   learner   by  the  follow  uig 

EXAMPLE   111. 
Required    the    difference    of   latitu<ie 
between  Georgetown  and  Cape  Frio. 

Georgetown's  latitude 33°  22'  N. 

Cape  Frio's  latitude 23     IS. 

Difference  of  latitude 56   23 

60 

In  miles 3383 


EXAiMPLE   IV. 
A  ship  from   latitude  28°  25'  N.,  sails 
south  1800  miles ;  what  latitude  is  siie  in  ? 

From  difference  of  latitude, 

1800  miles,  or 30°  00'  S. 

Subtract  latitude  left 28  25  N. 

Difference  is  the  latitude  in ,     1  35  S. 


In  the  last  example,  it  is  evident,  that  as  the  difference  of  latitude  is  more  than  the 
latitude  left,  the  ship  must  have  crossed  the  equator,  and  consequently  has  come  into 
south  latitude. 

JVole.  When  one  of  the  jilaccs  has  no  latitude,  or  is  on  the  equator,  the  latitude  of 
the  other  place  is  then*  difference  of  latitude. 


EXAMPLE   V. 
What  is  the  difference  of   longitude 
oetween  Cape  Ann  light-house  and  Lis- 
l)on? 

Cape  Ann  light-house's  long.  70°  34'  W. 
Lisbon's  longitude 9     9  W. 

Difference  of  longitude 61  25 

60 

In  miles 3685 


EXAMPLE   VI. 
A  shi])  from  Cape  Charles,  in  Virginia, 
siiiis  eastward  till  her  difference  of  longi- 
tude is  400  miles  ;  what  longitude  is  she 
m? 


76°  02'  W. 
=  6  '40  E. 


Cape  Charles's  longitude . . 
Diif.  of  lonjcitude,  400  miles : 


longitude  in 69  22  W. 


EXAMPLE  Vll. 
What  is  the  difference  of  longitude 
between  Barcelona  and  Salem  ? 

Barcelona's  longitude 2°  11'  E. 

Salem's  longitude 70  54  W. 


Difference  of  longitude 


73    5  W. 


EXAMPLE   Vlll. 
A  ship  from  15°  40'  E.  longitude,  sails 
westward  till  her  difference  of  longitude 
is  27°  15' ;  what  longitude  is  she  m  ? 

Longitude  left 15°  40'  E. 

Difference  of  longitude 27  15  W 

Longitude  in 11  35  W. 

EXAMPLE   IX.      , 
What   is   the   difference   of   longitude 
between  3Ianilla  and  New  York  light- 
house ? 

Manilla's  longitude 121°  02'  E. 

New  York  light-house 74  01  W. 

Sum  exceeds  180° 195  03 

Sul)tract  it  from 360  00 

Difference  of  longitude  ...  164  57 

EXAxMPLE  X. 

A  ship  from  longitude  160°  20'  W.,  sails 
westward  until  she  diftei-s  her  longitude 
4.1°  20' ;  what  longitude  is  she  in  ? 

Lon'^:aide  left 100°  20'  W. 

Dirt(3reuce  of  lonsritude  ...    41  20  W. 


201  40 
360  00 


Longitude  in 158  20  K 


In  the  last  example,  the  ship  has  crossed  the  opposite  meridian,  and  therefore  has 
come  into  a  longitude  of  a  different  name 


52 


PLANE    SAILING. 


'Plaise  Sailing  is  the  art  of  navigating  a  sliip  upon  principles  deduced  from  th(i 
supposition  of  tlie  earth's  beuig  an  extended  plane,  on  which  the  meridians  are  all 
parallel  to  each  other.  A  map  of  the  several  parts  of  the  earth,  constructed  u])on  these 
princi})les,  is  called  a  Plane  Chart.  When  the  parts  of  the  eardi  are  thus  delineated 
on  a  plane,  it  is  easy  to  see  the  track  by  which  a  ship  may  go  from  one  place  to 
another,  and  also  what  angle  this  track  makes  with  the  meridian.*  Ships  at  sea  are 
kept  m  this  tract  by  means  of  an  instrument  called  the  mariner'^s  compass. 

The  Mariner's  Compass  is  an  artificial  rejiresentation  of  the  horizon  of  any  place. 
It  consists  of  a  circular  piece  of  pa])er  (see  Plate  VI.  fig.  1),  called  a  card,  divided  (like 
the  horizon)  into  360  degrees,  or  32  points.  This  is  fixed  on  a  piece  of  steel,  called  a 
needle,  to  which  the  magnetic  virtue  has  been  communicated  by  means  of  a  loadstone, 
which  has  the  property  of  pointmg  steadily  towards  the  north,  and  carrying  the  card 
with  it,  when  turning  freely  on  a  pivot  or  any  thing  to  support  it.  Thus  all  the  points 
of  the  card  will  be  dh'ected  towards  their  corresponding  points  df  the  horizon ;  f 
consequendy,  by  help  of  the  compass,  a  ship  may  be  kept  m  any  proposed  track  or 
course. 

The  Course  is  the  angle  which  the  line  described  by  a  ship  makes  with  the 
meridian,  being  sometimes  reckoned  in  pomts,  half  points,  &c.,  and  sometimes  In 
degrees. 

Distance  is  the  way  or  length  a  ship  has  gone  on  a  direct  course  in  a  given  time. 
The  method  of  measuruig  this  distance  by  the  log  will  be  explamed  hereafter. 

Difference  of  Latitude  is  the  distance  which  the  ship  has  made  north  or  south 
of  the  place  sailed  from,  or  the  portion  of  the  meridian  contained  between  the  jjarallels 
of  latitude  sailed  from  and  come  to. 

Departure  is  the  east  or  west  distance  a  ship  has  made  from  the  meridian,  or  the 
whole  easting  or  westing  made  by  the  ship. 

If  a  ship  sails  due  north  or  south,  she  sails  on  a  meridian,  makes  no  departure,  and 
her  distance  and  diflference  of  latitude  are  the  same.  If  she  sails  due  east  or  west,  she 
goes  on  a  parallel  of  latitude,  makes  no  difference  of  latitude,  and  her  departure  and 
distance  are  the  same. 

The  difference  of  latitude  and  the  departure  make  the  legs  of  a  right-angled  triangle, 
the  hypotenuse  of  which  is  the  distance  the  shi[)  has  sailed ;  the  perpendicular  is  the 
difierence  of  latitude  counted  on  the  meridian  ;  the  base  is  the  departure,  which  is 
easting  or  westing  counted  from  the  meridian ;  the  angle  oj)posite  to  the  base  is  the 
course,  or  angle  that  the  ship  makes  witli  the  meridian ;  and  the  angle  ojjjjosite  the 
perpendicular  is  the  complement  of  the  course,  which  beuig  taken  together,  make 
always  8  pohits  or  90  degrees. 

In  constructing  figures  relating  to  a  ship's  course,  let  the  upper  part  of  the  paper,  or 
what  the  figin-e  is  drawii  upon,  always  represent  tlie  north ;  the  lower  part  Avill  be  the 
south  ;  the  right  hand  east,  and  the  lefl  west. 

Draw  the  nortli  and  south  line  to  represent  the  meridian  of  the  place  the  ship  sails 
from  ;  then,  if  the  ship's  course  is  to  the  southward,  mark  the  upper  end  of  the  line 
for  the  place  sailed  from  ;  but  if  the  course  is  northward,  mark  the  lower  end  for  that 
place.      • 

When  the  course  is  easterly,  describe  the  arc,  and  lay  off  the  course  and  departure 
on  the  right-hand  side  of  the  meridian  ;  but  when  westerly,  on  the  left-hand  side. 

When  the  course  is  given  in  degrees,  they  must  be  taken  from  the  protractor,  or 
from  the  line  of  chords;  but  when  in  points,  from  the  line  of  rhumbs,  and  must  always 
be  laid  off  upon  the  arc,  beginning  at  the  meridian. 

*  The  method  of  calculating-  this  angle  on  the  true  principles  of  sailing-  on  the  spherical  surface  of  the 
earth,  will  be  given  hereafter. 

t  It  is  here  supposed  that  the  needle  points  to  the  true  north,  but  if  it  varies  therefrom,  allowance  mus" 
be  made  for  the  variation  by  the  rules  which  will  be  given  in  this  work. 


I^hitc  ^. 


THE      CIRCLES.     ZOJVES,    JiCC:     OF 

TMU  AlRTIlFlUCIAli   dlLOlSIIii     OR      (SMIB]P^: 


]EXr]LAITATI1])IT    of     -BIEOGMAIPMICAjL  TIEHMSo 


1861 


I'LAxNE    SAILING. 


53 


WIie»  the  course  is  given  in  points,  the  log,  sine,  log.  cosine,  &c.,  may  be  found  ill 
Ta!)le  XXV.,  otherwise  in  Table  XXVII. 

In  all  cases,  where  the  complement  oi" course,  or  cosine,  &c.,  is  used,  the  degrees  or 
pouirs  put  down  ar(3  the  course  itself,  but  the  logariduns  belonging  to  the  complemen 
or  cosine,  &c.,  of  that  course  ai-e  taken. 

A  Table  of  the  Angles  which  every  Point  of  the  Compass  makes  with  the 

Meridian. 


iNorth. 

South. 

Points. 

D.M. 

North. 

South. 

k 

2.4!) 

h 

5.37 

I 

8.2() 

N.  by  E. 

S.  by  E. 

1 

11.15 

N.  by  W. 

S.  by  W. 

14.  4 

1G.52 
19.41 

N.  N.  E. 

S.  S.  E. 

2 

22.30 

N.  N.  W. 

S.  S.  W. 

2i 

25.19 

2.i 

23.  7 

n 

30.56 

N.  E.  by  N. 

S.  E.  by  S. 

3 

33.45 

N.  W.  by  N. 

S.  W.  by  S. 

H 

36.34 

3.i 

39.22 

\ 

3| 

42.11 

N.  E. 

S.  E. 

4 

45.  0 

N.  W. 

S.  w. 

H 

47.49 

i 

^ 

50.37 

4% 

53.26 

N.E.  by  E. 

S.  E.  by  E. 

5 

56.15 

N.  W.  by  W. 

S.  W.  bj  w. 

5i 

H 

5| 

59.  4 
61.52 
64.41 

E.  N.  E. 

E.  S.  E. 

6 

67.30 

W.  N.  W. 

w.  s.  w. 

H 

70.19 

6i 

73.  7 

Oil 

75.56 

E.  by  N. 

E.  by  S. 

7 

78.45 

W.  by  N. 

1        W.  by  S. 

n 

81.34 

7i 

84.22 

7| 

87.11 

East. 

8 

90.  0 

West. 

In  the  following  Table,  the  Rules  for  soloing  the  various  Cases  of  Plane 
Sailing  are  collected. 

PLANE,  SAILING. 


Case. 

Given. 

RcqiTIRED. 

Solutions. 

1 

Course 
and  distani:e. 

Diff.  of  latitude. 
Departure. 

Kadius  :  distance  :  :  cos.  course  :  difference  of  latitude. 
Radius  :  distance  :  :  sine  course  :  departure. 

2 

Course  and 
diff.  of  latitude. 

Distance. 
Departure. 

Cosine  course  :  diff.  of  latitude  :  :  radius  :  distance. 
Radius  :   diff.  of  latitude  :  :  tang,  course  :  departure. 

3 

Course 
and  departure. 

Distance. 
Diff.  of  latitude. 

Sine  course  :  departure  :  :  radius  :  distance. 

Radius  :  departure  :  :  cotang.  course  :  diff.  of  Iatiti.de. 

4 

Distante  and 
ditr.  of  latitude. 

Course, 
v  Departure. 

Distant  e  :  radius  :  :  diff.  of  latitude  :  cos.  course. 
Radius  :  distance  :  :  sine  course  :  departure. 

5 

Distance  and 
departure. 

Course. 
Diff.  ol  lat  tude. 

Distance  :  radius  :  :  departure  :  sine  course. 
Radius  :  distance  :  :  cos.  course  :  diff.  of  latitude. 

1         0 

Off.  of  latitude 
and  departure. 

Course. 
Distance. 

Diff.  of  latitude  :  radius  :  :  departure  :  tang,  course. 
I  Sine  course  :  departure  :  :  radius  :  di-^tEyice. 
/  Radius  :  diff.  of  latitude  :  :  secant  course  :  distance. 

54 


PLAiNE    SAlLJiNG. 


CASE  I. 

Course  and  distance  sailed  given,  to  Jlnd  the  difference  of  latitude  and  departure  from  tht 

meridian. 

A  ship  from  the  latitude  of  49°  57'  N.,  sails  S.  W.  by  W.  244  miles ;  requu-ed  the 
latitude  she  is  iii,  and  her  departure  from  tlie  meridian  sailed  from. 


BY   PROJECTION. 

Draw  the  line  CA,  to  rejjresent  the  meridian  of  the  place  C,  from  whence  the 
sailed.  With  the  chord  of  G0°  in  your  compasses, 
and  one  foot  in  C,  as  a  centre,  describe  the  compass 
W.  S.  E.  Take  5  points  in  your  compasses  from 
the  line  of  rhumbs  on  the  plane  scale,  and  set  it  off 
on  the  arc,  from  S.  towards  W.,  for  the  course; 
through  this  pomt  and  C  draw  the  line  CB,  and 
make  it  equal  to  the  distance  244 ;  th-aw  BA  parallel 
to  the  east  and  west  line  EW,  to  cut  the  meridian 
in  A.  Then  will  CA  be  the  difference  of  latitude 
135.6,  and  AB  the  departure  202.9. 

BY   LOGARITHMS. 
By  making  the  distance  radius. 


ship 


To  fold  the  departure. 

As  radius  8  points 10.00000 

Is  to  the  distance  244 2.38739 

So  is  the  sine  course  5  points  . .    9.91985 

T^he  departure  202.9 2.30724 


To  find  the  difference  of  latitude. 

As  radius  8  points 10.00000 

Is  to  the  distance  244 2.;}8739 

So  is  the  cosine  course  5  pomts.    9.74474 

To  the  differonct  of  hit.  135.6. .    2.13213 


Now,  as  the  ship  is  in  north  latitude  sailing  southerly, 

From  the  latitude  left 49°  57'N. 

Take  the  difference  of  latitude  135.6 2   16  S. 


Gives  the  latitude  in 47   41  N. 

And  the  departure  ftom  tlie  meridian  is  202.9  miles. 

BY   GUNTER. 

Extend  from  radius  or  8  points  *  to  5  pomts  on  tlie  line  marked  SR ;  that  extent 
mil  reach  from  the  distance  244,  to  the  departurr  202.9,  on  the  line  of  numbers. 

2dl3^    Extend  from  radius  or  8  points  to  3  points,  the  complement  of  the  course,  on 
the  line  SR ;  that  extent  will  reach  from  the  distance  244,  to  the  diffei-ence  of  latitude 
135.6,  on  the  Ime  of  numbers. 
»    Thus  may  all  the  operations  be  performed  in  the  several  cases  of  Navigation. 

By  this  case  are  calculated  the  tables  of  latitude  and  departure  (Tables  I.  and  II.) 
for  every  degree,  point,  and  quarter  point  of  the  mariner's  compass,  to  the  distance  of 
300  miles.  By  the  ins})ection  of  these  tables,  a  day's  work  may  be  calculated  ui  a  much 
more  expeditious  manner  than  by.  logarithms  or  by  Gunter's  scale.  Inconsequence 
of  this  facility,  the  method  by  mspection  is  generally  used  at  sea  in  preference  to  evei7 
otlier  method.  ^ 

BY   INSPECTION. 

Find  the  given  course  at  the  top  or  bottom  of  the  tables,  either  among  the  points  o\ 
degi'ees,  and  in  that  page,  against  the  distance  taken  in  its  column,  will  stand  the 
difference  of  latitude  and  departure  in  their  columns.f 

It  must  be  obsei"ved,  that,  in  using  these  tables,  the  names  Dist.  Lat.  Dep.  must  be 
found  at  the  top  if  the  course  is  found  there,  but  if  the  course  is  found  at  the  bottom, 
those  names  must  be  found  at  the  bottom. 

Thus  the  course  S.  W.  by  W.  or  5  points,  is  found  at  the  bottom  of  the  table  of 
difference  of  latitude  and  departure  for  points;  and  against  244  in  the  distance  column 
stands  135.6  for  the  difference  of  latitude,  or  202.9  for  the  depaiture. 

*  When  the  course  is  given  in  points,  make  use  of  the  hncs  market!  sine  i-lmmbs,  and  tangent  rhumbs 
on  the  upper  side  of  the  scale  ;  \\hcn  hi  degrees,  make  use  of  the  hncs  marked  sine  and  tangent. 

t  Wlicn  tlie  distance  is  too  great  to  be  found  in  the  tables,  you  must  divide  it  by  2,  3,  4,  or  any 
■envcnicnt  number^  the  numbers  corresponding  to  the  quotient  benig  multiplied  by  the  divisor  will  givu 
ihc  souirht  numbers. 


J'foffXL 


18G1 


I'LANE   SAILLNG. 


55 


CASE  II. 

Course  and  difference  of  hditude  given,  to  find  the  distance  run,  and  departure  from  tin 

meridian. 

A  sliij)  runs  S.  E.  by  E.  from  1°  45'  north  latitude,  and  then,  by  obsciTation,  is  in 
0°  31'  south  liititude ;   required  her  distance  and  depmture. 

In  this  case,  as  the  ship  has  crossed  the  equator,  the  sum  of  the  two  latitudes,  1°  45 
and  0°  31',  is  tlie  difference  of  latitude,  2°  IG'rr  13G  miles. 


BY  PROJECTION. 

Draw  BC  equal  to  136,  and  BA  making  an  angle  with 
BC  equal  to  the  course  5  points,  or  50°  15' ;  draw  CA 
peri)eM(licuIar  to  BC  to  cut  BA  in  A,  and  it  is  done  ;  for 
CA  will  be  the  departure  equal  to  203.5,  mid  AB  die 
(.listance  equal  to  244.8. 

BY   LOGARITHMS. 
By  making  the  difference  of  lat.  BC  radius, 
To  find  the  departure. 

As  radius  4  points 10.00000 

Is  to  difference  of  latitude  13G. .    2.13354 
So  is  tmigent  course  5  pouits. . .  10.17511 


•To  the  departure  203,5 2.308G5 


By  making  the  distance  AB  radius.* 
To  find  the  distance. 

As  cosine  course  5  points 9.74474 

Is  to  the  difference  of  latitude  13G    2.13-354 
So  is  radius 10.00000 

To  the  distance  244.8 2.38880 


Hence  tlie  ship's  distance  run  is  244.8  miles,  and  her  depaj-ture  from  the  meridian 
is  203.5  easterly. 

BY   GUNTER. 

Extend  from  radius  or  4  pomts  to  the  course  5  points  on  the  line  marked  TR;  that 
extent  will  reach  from  the  difference  of  latitude  136,  to  the  departure  203.5,  on  the  line 
of  numbers. 

2dly,  Extend  from  the  complement  of  the  couree  3  points  to  the  radius  8  points  on 
tlie  line  SR ;  that  extent  will  reach  from  the  difference  of  latitude  13G,  to  the  distance 
244,8,  on  the  line  of  numbers, 

BY   INSPECTION. 

Fuid  the  course  among  the  jioints  or  degi'ees,  and  the  difference  of  latitude  in  ith 
column,  against  which  will  stand  the  distance  and  departure  in  theh-  columns. 


CASE   III. 

Course  and  departure  from  the  meridian  given,  to  find  the  distance  and  d^erence  oj 

latitude. 

If  a  sliip  sails  N.  E.  by  E.  |  E.  from  a  jjoit  in  3°  15'  soudi  latitude,  until  she  depart 
froiri  her  first  meridian  203  miles,  requu-ed  the  distance  sailed,  and  the  latitude  she 
'ts  ui. 

BY   PROJECTION. 

Draw  the  meridian  AB,  upon  which  erect  the  perpendicular  BC,  and  set  off  thereon 
the  departure  203,' easterly  Iroiu  B  to  C ;  with  die  chord 
of  60°  on  C,  as  a  centre,  describe  an  arc,  and  set  off  thereon 
the  complement  of  the  course ;  through  diis  point  and  C 
draw  tlie  line  CA,  cuttmg  the  meridian  in  the  point  A  ; 
then  AC  measured  on  the  same  scale  before  used,  gives 
the  distance  224.6,  and  AB  96,  the  difference  of  latitude. 

By  making  BC  radius,  you  would  have,  radius  :  ditrcreiice  of  latitude  ::  secant  course  :  distance- 
■.  ji  ll'is  roiioi)  would  not  do  for  a  common  scale  on  which  there  is  no  lin»  of  secants.  The  sama 
h-.;r,;»  ;.«i  '■■  fj<!  observed  in  the  followin,?  case.s 


J)epar-fu,re203 


5f) 


PLANE    SAILIiNG. 


BY   LOGARITHMS 
By  iTiakuig  the  depaitiu'e  BC  radius. 

As  radius  4  points lO.OCOOO 

Is  to  tlie  departure  203 2.30750 

So  is  cotangent  coui-se  5|  poiuts    9.G7483 

To  the  difference  ol" latitude  96. 


1.98233 


By  making  the  distance  AC  radius. 

As  sine  course  5|  points 9.95616 

Is  to  the  departure  203 2.30750 

So  is  radius 10.00000 


To  the  distance  224.6. 


2.3513! 


P'rom  the  latitude  left 3°  15'  S. 

Subtract  the  difference  of  latitude  96  miles,  or 1    36  N. 

The  remainder  shows  that  the  ship  is  m  the  latitude  of  ... .   1°  3'J'  S. 

BY   GUNTElv. 

Extend  from  radius  or  4  poiuts  to  the  complement  of  the  course  '2^  points  on  the 
tine  marked  TR ;  that  extent  will  reacii  from  the  departure  203,  to  tlie  difference  of 
latitude  9(5,  on'the  line  of  numbers. 

2dly.  Extend  from  the  course  5:]  points  to  radius  on  tne  line  SR  ;  that  extent  will 
reach  from  the  departure  203,  to  the  distance  224.6  miles,  on  ;ho  line  of  numbers. 

BY   INSPECTION. 

Find  the  couree,  either  among  the  points  or  degrees,  and  the  departure  in  its 
colunui,  against  which  will  stand  the  distance  and  diffei'ence  of  latitude  in  their 
respective  columns.        , 

Thus  with  the  course  5|  points,  and  departure  203,  we  find  224.6  for  the  distance^ 
and  96.0  for  the  difference  of  latitude. 


CASE  IV. 

Distance  and  difference  of  latitude  given,  to  find  the  course  and  departure 

Suppose  a  ship  sails  244  miles,  between  the  south  and  the  east,  from  a  port  in  2°  52 
south  latitude,  and  then,  by  observation,  is  in  5°  08'  south  latitude ;  what  course  has 
she  steered,  and  what  departure  has  she  made  ? 

From  the  latitude  by  observation  5°  08',  take  2°  52',  the  latitude  left;  the  remainder, 
2"^  16' =  136  miles,  is  the  difference  of  latitude. 

BY   PROJECTION. 

Draw  the  meridian  AB:=136;  upon  which  erect  the 
peipendicidar  BC ;  take  244  in  your  compasses,  and  with 
one  foot  on 'A,  as  a  centre,  describe  an  arc  cutting  BC  in 
C  ;  join  A  antl  C  ;  then  will  BC  be  the  departure  202.6,  and 
the  angle  BAC  the  course,  equal  to  56°  08',  or  5  points, 
nearly. 

BY    LOGARITHMS. 


B  Departure 


*  To  find  the  course. 

As  the  distance  244 2.-38739 

Is  to  radius 10.00000 

So  is  the  difference  of  lat.  136. .    2.13354 

To  cosine  course  56°  08' 9.74615 


To  find  the  departure 

As  radius 10.00000 

Is  to  the  distance  244 2.38739 

So  is  the  shie  course  56°  08' ... .    9.91925 


To  the  departure  202.6. 


2.30664 


Hence  the  course  is  S.  E.  by  E.,  and  the  depart\u-c  202.6. 

BY   GUNTER. 

The  extent  from  the  distance  24  J,  to  the  difference  of  latitude  136,  on  the  line  of 
numl>crs,  will  reach  from  radius  or  90°  to  33°  52',  the  complement  of  the  course  on  the 
line  of  sines. 

And  the  extent  from  radius,  to  56°  08'  on  the  line  of  sines,  will  reach  from  the 
distance  244,  to  the  depaiture  202.6,  on  the  line  of  numbers. 

BY   INSPECTION. 

Seek  in  the  tables  till  against  the  distance,  taken  ui  its  column,  is  found  the  given 
difference  of  latitude  in  one  of  the  following  colunuis;  adjouiing  to  it  will  stand  the 


i'LANb:   SAILING. 


57 


departure  ;  wliich  if  less  than  the  difference  of  latitmle,  tlie  course  is  to  be  found  at 
the  top  ;*  bdt  if  greater,  the  conrse  is  to  be  found  at  tiie  bottom. 

Thus  the  distance  244,  and  tlie  difference  of  latitude  I'.iG,  are  found  to  correspond 
to  a  course  of  5  points,  or  S.  E.  by  E.,  and  to  the  departure  203.9,  nearly. 

CASE   V. 

Distance  and  departure  given,  to  Jind  the  course  and  difference  of  latitude. 

Suppose  a  ship  sails  244  miles  between  the  north  and  west,  from  the  latitude  of 
32°  25'  north,  until  her  departure  is  203  miles;  what  eourse  has  siie  steered,  and  what 
latitude  is  siie  in .'' 

BY   PROJECTION. 

Draw  the  line  AB  equal  to  the  departure  203;  and, 
perpondioular  thereto,  the  line  BC,  to  represent  die 
meridian  ;  then  take  the  distance  244  in  your  compasses, 
and,  fixing  one  foot  in  A,  as  a  centre,  describe  an  arc, 
cutting  BC  in  C ;  join  AC,  and  it  is  done ;  for  the  angle 
ACB  will  be  the  course,  and  BC  the  difference  of  latitude 

DY  LOGARITHMS. 


I)C77arfiire  203 


To  find  the  course. 

As  the  distance  244 2.38739 

is  to  raflius 10.00000 

So  is  the  departure  203 2.30750 

To  tlie  sine  of  course  56°  18' . . .    9.9201 1 


To  find  the  difference  of  latitude. 

As  radius 10.00000 

Is  to  the  distance  244  T 2.-38739 

So  is  cosme  course  5G°  18'  ....    9.74417 

To  the  difference  of  lat.  135.4. .    2.13156 


Hence  the  course  is  N.  50°  18'  W.,  or  N.  W.  by  W.  nearly. 

To  tlie  latitude  sailed  from  32°  25'  add  the  difference  of  latitude  135  or  2°  15' ;  the 
sum  34°  40'  is  the  latitude  the  ship  is  in. 

BY    GUNTER. 

Extend  from  the  distance  244,  to  the  departiu-e  203,  on  the  line  of  numbers ;  that 
extent  v/ill  reach  troni  radius  to  tlie  course  5G°  18'  on  the  line  of  sines. 

2dly.  Extend  from  radius  to  the  complement  of  the  course  33°  42',  on  the  line  of 
sines  ;  that  extent  will  reach  from  the  distiuice  244,  to  the  difterence  of  latitude  135.4, 
on  the  luie  of  numbers. 

BY   INSPECTION. 

Seek  in  the  tables  till  against  the  distance  taken  in  its  column  is  found  the  given 
departure  in  one  of  the  following  columns;  adjoining  to  it  will  stand  the  difference  of 
latitude ;  and  if  it  be  greater  than  the  deiiarture,  tlie  course  is  to  be  found  at  the  top ; 
but  if  less,  the  course  is  to  be  found  at  the  bottom. 

Thus  the  distance  244,  and  the  departure  203,  agree  to  a  course  of  5  points,  or 
N.  W.  by  W.,  and  a  difference  of  latitude  135.G  miles,  nearly. 


CASE  VI. 

Difference  of  latitude  and  departure  given,  to  find  the  course  and  distance. 

A  ship  sails  between  the  north  and  west  till  her  difference  of  latitude  is  136  miles, 
and  her  dej>aiture  is  203  miles ;  requu'ed  her  course  and 
distance. 

BY   PROJECTION. 

Draw  AB=:136,  and  perpendicular  to  it  BCz=203; 
join  C  and  A  ;  then  will  the  angle  CAB  be  the  course 
56°  11',  and  AC  the  distance  244.4  miles. 


Dcpartu-re 

-zos 

B 

VI 

"^ 

^^ 

<-> 

""""^^ 

^^ 

N.       c» 

*  It  may  also  be  known  whether  the  course  he  marked  at  the  top  or  bottom  of  the  table,  by  observing 

whether  the  difference  of  latitude  and  dcparinre  correspond  wiih  the  marks  ai  tlie  top  or  bottom.     Thus 

I  le  distance  2J4,  and  difference  ol  latitude  13G,  correspond  to  the  course  5  points,  because  the  column  iu 

/'hich  136  is  found,  is  marked  latitude  at  the  bottom  ;  the  same  may  be  observed  in  the  foUowiDg  case*. 

B 


5S 


PLANE   SAILING. 


BY   LOGARITHMS. 


To  find  the  distance. 

As  radius 1000000 

Is  to  the  difference  of  lat.  130. .    2.1.3354 
So  is  secant  of  course  56°  U'  . .  10.25451 

To  the  distance  244.4 2.38805 


To  find  the  couree. 
As  the  difference  of  latitude  136    2.13354 

Is  to  radius 10.00000 

So  is  the  departure  203 2.30750 

To  tangent  of  course  50°  11'. . .  10.17396 

Hence  her  course  is  N.  50°  11'  \V.,  or  N.  W.  by  W.,  and  the  distance  sailed  is 
244.4  miles. 

BY   GUNTER. 

Extend  from  the  difference  of  latitude  130,  to  the  departure  203,  on  the  line  of 
numbers;  that  extent  will  reach  from  radius  to  50°  11',  the  course  on  the  line  of 
tangents. 

2(lly.  For  the  distance  we  must  consider  it  as  nidius  (unless  there  is  a  line  of 
secants  on  the  scale),  and  extend  from  the  course  56°  11',  to  the  radius,  or  90°,  on  the 
line  of  sines  ;  that  extent  will  reach  from  the  depai'ture  203,  to  the  distance  244.4,  on 
the  luie  of  numbers. 

BY  INSPECTION. 

Seek  m  tlie  tables  till  the  given  dilfei-ence  of  latitude  and  departure  are  found 
together  m  their  respective  columns ;  then  against  them  will  be  the  distance  in  its 
column,  and  tlie  course  will  be  found  at  the  top  of  that  table  if  the  departure  be  less 
than  the  difference  of  latitude,  otherwise  at  the  bottom. 

Thus  with  tlie  difference  of  latitude  136,  and  the  departure  203,  enter  the  tables, 
and  these  luunbers  will  be  found  to  coiTcspond  nearly  to  5  points,  or  N.  W.  liy  W. 
course,  and  a  distance  equal  to  244  miles. 


QUESTIONS 

To  exercise  the  learner  in  tlie  foregoing  rules. 

Question  I.  A  ship  in  2°  10'  south  latitude,  sails  N.  by  E.  89  leagues ;  what  latitude 
is  she  ill,  and  what  is  her  departure  ? 

Answer.    Latitude  in  2°  12'  N.,  and  departure  17.30  leagues. 

Quest.  II.  A  sliij)  sails  S.  S.  W.  from  a  jjort  in  41°  30'  north  latitude,  and  then,  by 
observation,  is  in  30°  57'  north  latitude;  required  the  distance  run,  and  dejjarture. 

Ans.    Distance  run  98.5  leagues,  departure  37.7  leagues. 

Quest.  III.  A  ship  sails  S.  S.  W.  h  W.  from  a  port  in  2°  30'  south  latitude,  until  hei 
departure  be  59  leagues ;  requhed  the  distance  run,  and  latitude  in. 

Jlns.    Distance  run  125.2  leagues,  latitude  in  8°  1'  south. 

Quest.  IV.  If  a  ship  sails  .300  miles  south-westward  from  21°  59'  south  latitude,  until 
by  observation  she  be  in  24°  49'  south  latitude,  what  is  her  course  and  de])arture  ? 

Ans.  The  course  is  S.  W.  by  W.  k  W.,  or  S.  01°  49'  W.,  and  her  depaiture  from 
the  meridian  is  317.3  miles. 

Quest.  V.  Suppose  a  ship  sails  354  miles  noith-eastward  from  2°  9'  south  latitude, 
until  her  departiu-e  be  150  miles,  what  is  her  course  and  latitude  in  ? 

Ans.    Her  course  is  N.  25°  4'  E.,  or  N.  N.  E.  i  E.  nearly,  and  she  is  in  lat.  3°  12'  N. 

Quest.  VI.  Sailing  between  the  north  and  the  west,  from  a  port  in  1°  59'  south 
latitude,  and  then  arriving  at  another  port  in  4°  8'  north  latitude,  which  is  209  miles 
to  the  westward  of  the  first  port,  requu-ed  the  course  and  distance  from  the  fu-st  port 
to  the  second. 

Ans.  The  course  is  N.  29°  40'  W.,  or  N.  N.  W.  |  W.  nearly,  and  the  distance  of  the 
ports  is  422.4  miles,  or  140.8  leagues. 

Quest.  VII.  Four  days  ago  we  were  in  latitude  3°  25'  S.,  and  have  since  that  tune 
sailed  in  a  direct  course  N.  W.  by  N.  at  the  rate  of  8  miles  an  hour;  reauh-ed  our 
present  latitude  and  departure. 

Ans.    Latitude  in  7°  14'  N.,  departure  420.7  miles. 

Quest.  VIII.  A  ship  in  the  latitude  of  3°  52'  south,  is  bound  to  a  \)on  bearing 
N.  W.  by  W.  h  W.  in  the  latitude  of  4°  30'  north  ;  how  far  does  that  port  lie  to  tlie 
westward,  and  what  is  the  ship's  distance  from  it? 

Ans.   The  port  lies  939.2  miles  to  the  westward,  and  the  db-ect  distance  is  1065  miles. 

Quest.  IX.  A  sliip  from  the  latitude  of  48°  17'  N.,  sails  S.  W.  by  S.  until  she  has 
depressed  the  north  pole  2  degrees;  what  direct  distance  has  she  sailed,  and  how  many 
miles  has  slic  sailed  to  the  westward  ? 

Ans.    Diiitance  run  144.3  miles,  and  has  sailed  to  the  westward  80.2  miles. 


Ki 


TRAVERSE    SAILING. 


A  TRAVERSE  is  ail  irregular  track  which  a  sliip  makes  l)y  sailing  on  several  different 
courses  ;  these  are  reduced  to  a  single  course  by  means  of  two  or  more  cases  of  Plane 
Sailing,  either  by  geometrical  construction,  or  by  arithmetical  calculation.* 

The  geometrical  construction  is  performed  as  follows  : — Describe  a  circle  with  the 
chord  of  G0°,  to  represent  the  compass,  and  lay  off  on  its  circumference  the  various 
courses  sailed.  From  the  centre,  upon  the  first  course,  set  off  the  first  distance,  and 
mark  its  extremity  ;  through  this  extremity,  and  j)arallel  to  the  second  course,  di-aw 
the  second  distance  of  its  proper  length  ;  througli  the  extremity  of  the  second  distance, 
and  j)arailel  to  the  third  course,  draw  tlie  third  distance  of  its  proper  length ;  and  thus 
proceed  till  all  the  distances  are  drawn.  A  line,  drawn  from  the  extremity  of  the  last 
distance  to  the  centre  of  the  circle,  will  rej)resent  the  distance  made  good  ;  a  Ime, 
drawn  from  the  same  point,  perpendicular  to  the  meridian,  will  represent  the  departure , 
and  the  part  of  the  meridian  intercepted  between  this  and  the  centre,  will  represent  the 
difference  of  latitude. 

The  arithmetical  calculation  to  work  a  traverse  is  as  follows: — Make  a  traverse  table 
consisting  of  six  columns ;  title' them.  Course,  Distance,  N.,  S.,  E.,  W. ;  begin  at  the  left 
side,  and  write  the  given  coiu'ses  and  distances  in  their  respective  colunuis.  Find  the 
difference  of  latitude  and  departure  for  each  of  these  courses,  by  Gimter's  scale,  or  by 
Tables  I.  or  11.  (as  in  Case  I.  Plane  Sailing),  and  ^vl•ite  them  in  their  proper  columns ; 
that  is,  when  the  course  is  southerly,  the  difi'erence  of  latitude  must  be  set  in  the 
column  S. ;  when  northerly,  in  the  column  N. :  the  departure,  when  westerly,  m  the 
column  W. ;  and  when  easterly,  in  the  column  E.  Add  up  the  columns  of  northhig 
southing,  easting, and  westing;  take  the  difference  between  the  northing  and  southing, 
and  also  between  the  easting  and  westing ;  the  former  difference  will  be  the  difTerence 
of  latitude,  which  will  be  of  the  same  name  as  the  greater ;  and  the  latter  will  be  the 
departure,  which  will  be  also  of  the  same  name  as  the  gi-eater.  With  this  diffei'ence  of 
latitude  and  departure  the  course  and  distance  made  good  are  to  be  found  as  in  Case  VI. 
Plane  Sailing. 

EXAMPLE   I. 

Su]ipose  a  ship  takes  her  departure  from  Block  Island,  in  the  latitude  of  41°  10'  K, 
the  middle  of  if  bearing  N.  N.  W.,  distance  by  estimation  5  leagues,  and  sails  S.  E.  34, 
VV.  by  S.  1(),  W.  N.  W.  39,  and  S.  by  E.  40  miles; 
required  the  latitude  she  is  in,  and  her  bearing  and 
distance  from  Block  Island. 

BY   PROJECTION. 

Let  L  represent  the  nfiddle  of  Block  Island ;  draw 
the  meridian  LM,  and  on  L,  as  a  centre,  with  a  chord 
of  GO^,  describe  a  circle  to  represent  the  com])ass,  on 
whicl)  mark  the  various  courses  sailed,  and  the  bearing 
of  the  land  at  the  tune  of  taking  the  departure;  0])po- 
site  to  this  bearing  draw  the  S.  S.  E.  line  LA,  which 
make  equal  to  15  miles,  the  estimated  «listance  of  the 
land  ;  then  will  A  represent  the  place  of  the  ship  at  the 
time  of  taking  the  departure :  through  A  draw  AB 
equal  34  miles,  ])arallel  to  the  S.  E.  line  ;  then  will  B 
be  the  place  of  die  ship  after  sailing  her  first  course: 
in  like  manner  draw  BC  equal  to  16  miles,  jiarallel  to 
the  AV.  by  S.  line ;  CD  equal  to  39  miles,  parallel  to 

*  This  mclhod  of  reducing  compound  courses  to  a  single  one  is  perfectly  accurate  in  sailing  on  a  plane, 
and  is  nearly  so  in  sailing  a  sliori  distance  on  the  splicrica!  surface  of  the  earth  ;  and  though  in  this  case 
it  is  liable  to  a  small  error  in  high  latitudes,  yet  in  general  the  rule  is  sufficiently  accurate  for  educnig 
the  several  courses  -and  distances  sailed  in  one  day  to  a  single  course  and  distance. 


(JO 


TRAVERSE   SAILING. 


the  W.  N.  W.  liiie,  and  DE  equal  to  40  miles,  parallel  to  the  S.  by  E.  line  ;  then  wiL 
E  represent  the  place  of  the  ship  after  sailing  her  several  courses.  Join  EL,  and  draw 
EM  perj)endicular  to  LM ;  then  will  LE  be  the  distance  of  Block  Island,  (30.8  miles; 
and  the  angle  ELlMm  12°  16',  will  be  the  course  made  good  ;  LM  the  difference  of 
latitude,  and  EM  the  departure. 

TO   FliND   THE   SAME   BY   LOGARITHMS. 
For  the  first  course  S.  S.  E.  15  miles. 


To  find  the  difference  of  latitude. 

As  radius  90° 10.00000 

Is  to  cosine  course  2  pouits  ....    9.9()5(32 
So  is  distance  15 1.17G09 

To  difference  of  latitude  13.9  . .    1.14171 


For  depailiu-e. 

As  radius  90° '. 10.00000 

Is  to  shie  course  2  points 9.58284 

So  is  distance  15 1.17609 


To  departure  5.7 0.75893 


Second  course  S.  E.  34  miles. 


For  difference  of  latitude. 

As  radius  90° 10.00000 

Is  to  cosine  course  45° 9.84949 

So  is  distance  34 1.53148 

To  difference  of  latitude  24 ... .    1.38097 


For  departure. 

As  radius  90° 10.00000 

Is  to  sine  course  45° 9.84949 

So  is  distance  34 1.53148 


Third  course  W 
For  difference  of  latitude. 

As  radius  90° 10.00000 

Is  to  cosine  course  78°  45' 9.29024 

So  is  distance  16 1.20412 

To  difference  of  latitude  3.1  .. .    0.49436 


Fomth  course  W 
For  diffei'ence  of  latitude. 

As  radius  90° 10.00000 

Is  to  cosine  course  67°  30' 9.58284 

So  is  distance  39 1.59106 

To  difference  of  latitude  14.9  . .    1.17390 


To  departure  24 1.3S097 

by  S.  16  miles. 

For  departure. 

As  radius  90° 10.00000 

Is  to  sine  course  78°  45' 9.99157 

So  is  distance  16 _L20412 

To  departure  15.7 1.19569 

N.  W.  39  miles. 

For  depaiture. 

As  radius  90° 10.00000 

Is  to  sine  course  67°  30' 9.9()5G2 

So  is  distance  39 1.59106 

To  departure  36 1.55668 


Fifth  course  S.  by  E.  40  miles. 


For  difference  of  latitude. 

As  radius  90° 10.00000 

Is  to  cosine  course  11°  15' 9.99157 

So  is  distance  40 1.60206 


To  difference  of  latitude  39.2  . .    1.59363 


For  departure. 

As  radius  90° 10.00000 

Is  to  sine  course  11°  15'. . « 9.29024 

So  is  distance  40 1.60206 


To  departure  7.8 . 


0.89230 


Though  this  uiethod  of  finding  the  difference  of  latitude  and  departure  by  logarithms 
is  accurate,  yet  the  calculations  may  be  more  easily  made  by  the  tables  of  diflerence 
of  latitude  and  departure,  as  in  Case  I.  Plane  Sailing. 

Place  all  these  courees,  distances, 
&c.,  in  the  traverse  table ;  then  add  up 
all  tlie  westings,  eastings,  northings, 
and  soutlungs,  separately,  and  set 
dov^Ti  their  respective  sums  at  the 
bottom  of  each  column;  and  as  tlie 
westing  is  gi-eater  than  tlie  easting, 
subtract  the  casting  therefrom  ;  the 
difference,  14.2,  shows  that  the  ship's 
departm-e  is  so  much  west  of  her  first 
meridian. 

Again,  the  southing  being  greater 
than  the  northing,  subtract  the  north- 
ing from  it,  and  the  remainder,  6.5.3,  shows  how  far  the  ship  is  to  the  southward  of 
her  fii'st  place. 


TRAVERSE 

TABLE. 

Courses. 

Dist. 

Diff.  of  Lat. 

Departure. 

N. 

S. 

E. 

W. 

S.  S^E. 

S.  E. 

W.  by  S. 

W.  N.  W. 

S.  by  E 

15 

34 

IG 
31) 

40 

14.9 

13.9 

24.0 

3.1 

39.2 

5.7 
24.0 

7.8 

15.7 
36.0 

From  sum 
Rpmaiuder 

take . .  . 

14.9 

80.2 
14.9 

37.5 

51.7 
37.5 

C5.3 

14.2 

TRAVERSE    SAlLliNG. 


t)l 


To  fiud  the  direct  coui-se  or  bearing  of 

Block  Island  from  the  ship. 
As  the  difference  of  latitude  65.3    1.81491 

Is  to  radius  45^ 10.00000 

So  is  the  departure  14.2 1.15221) 

To  tangent  course  12°  IC D.33738 

Which,  because  the  difference  of 
latitude  is  soutlierlv,  and  tlie  departure 
westerly,  is  S.  12°  16'  W.  Whence  Block 
Island  beai-s  from  the  ship  N.  12°  W  E-, 
or  N.  by  E.  1°  1'  E. 


To  find  the  distance  of  the  islai^l. 

As  sine  of  course  12°  IG" 9.32728 

Is  to  the  departure  14.2 1.15229 

So  is  radius  90° 10.00000 

To  die  distance  66.8 1.82501 


BY   INSPECTION. 

Find  the  course  and  distance  by  Case  VI 
of  Plane  Sailinff. 


EXAMPLE  II. 

A  ship  from  Mount-Desert  rock,  in  the  latitude  of  43°  50'  N.,  sails  for  Cape  Cod,  in 
the  latitude  of  42'  3'  N.,  its  departure  from  the  meridian  of  Mount-Desert  rock  being 
supposed  to  be  84  miles  west ;  but  by  reason  of  contrary  winds,  she  is  obliged  to  sail 
on  the  following  courses,  viz.  south  10  miles,  W.  S.  W.  25  miles,  S.  VV.  30  miles, 
and  W.  20  miles.  Reipiired  the  bearing  and  distance  of  the  two  places,  the  course 
aiid  distance  sailed  by  the  ship,  and  the  bearing  and  distance  of  her  intended  port. 


BY   PROJECTION. 

Latitude  of  Mount-Desert  rock  43°  50'  N. 

Latitude  of  Cape  Cod 42     3  N. 

Difference  of  latitude 1   47  =r  107  jiiiles. 

Let  C  represent  Mount-Desert  rock ;  draw  the  meridian  CF,  which  make  ernial  to 
107  miles,  the  difference  of  latitude  between  the  two  places,  and  peri>endicular  thereto 
the  line  FE,  equal  to  the  departure,  84  miles  ;  then  is  E  the  place  of  Cape  Cod.  WitK 
the  chord  of  60°  sweep  about  the  centre,  C,  a  circle,  S.  W.,  to  represent  the  compass, 
and  upon  it  note  the  various  courses  sailed.  The  first  course  being  south,  the  distance 
10  miles,  is  set  off  from  C  towards  F  upon  the  meridian,  and  this  point  represents  th« 
place  of  the  ship  after  sailing  her  first  course;  continue  setting  off  the  various  course; 
and  distances  as  in  the  last  example,  viz.  W.  S.  W.  25  miles,  S.  W.  30  miles,  and 
west  20  milps.  to  tlie  point  A  ;   then  will   A   represent  the  jilaco  of  the  ship  after 


m 


TRAVERSE  SAILING. 


sailing  these  courses.  Join  CE,  AC,  AE ;  draw  AB  perpendicular  to  the  meridian 
CF,  and  AD  parallel  thereto;  then  will  AC  =  76.2  miles  be  the  distance  made  good; 
AEr=G9.1  miles,  the  distance  of  Cape  Cod  from  the  ship;  CE  the  distance  of  the  two 
places  =1 13G  miles ;  ACB  =  57°  36',  the  course  made  good ;  EAD  =  16°  34',  the  course 
to  Cape  Cod  ;  and  ECF  the  course  from  Mount-Desert  rock  to  Cape  Codr:38°  8',  &c 

BY   LOGARITHMS. 
To  find  the  bearuig  and  distance  of  the  two  places  by  Case  VI.  Plane  Sailmg. 


To  find  the  bearing. 
As  difference  of  latitude  107. . .    2.02938 

Is  to  radius  45° 10.00000 

So  is  departure  84 1.92428 

To  tangent  course  38^  8' 9.89490 


To  find  the  distance. 

As  radius  90° 10.00000 

Is  to  difterence  of  latitude  107.    2.029:38 
So  is  secant  course  38°  8' 10.10426 

To  the  distance  136 2.13364 


Whence  the  course  from  Mount-Desert  rock  to  Cape  Cod  is  S.  38°  8'  W.,  distance 
136  i.'.iles.     The  same  may  be  found  by  the  scale,  or  by  inspection. 

Tiie    difference   of    latitude     and  TRAVERSE   TABLE, 

departure  for  the  several  courses 
being  calculated,  by  Case  I.  Phme 
Sailing,  and  arranged  in  the  traverse 
table,  it  appears  that  the  difference  of 
latitude  made  good  by  the  ship  is 
40.8  miles,  and  the  departure  64.3 
miles ;  then,  by  Case  VI.  Plane  Sail- 
ing, these  numbers  are  found  to  cor- 
respond to  a  course  of  S.  57°  30'  W. 
and  distance  76.2  miles. 

Subtract  the  difference  of  latitude  made  good  by  the  ship,  40.8  miles,  from  the  whole 
difference  of  latitude,  107  miles,  and  there  remain  66.2  miles,  which  is  the  difference 
jf  latitude  between  the  ship  and  Cape  Cod.  In  the  same  manner,  by  subtracting  the 
ship's  departure,  64.3  miles,  from  the  whole  departure,  84  miles,  tliere  remain  19.7nule-! 
for  the  dej)arture  between  the  ship  and  Cape  Cod.  With  this  diftljrence  of  latitude 
66.2,  and  departure,  19.7,  the  bearing  of  Cape  Cod  is  found,  by  Case  VI.  Plane  Sailing 
S.  16°  34'  W.,  and  its  distance,  69.1  miles. 

All  the  pr(!ceding  calculations  may  be  made  by  logarithms,  by  the  scale,  or  by 
inspection.  But  v/e  shall  leave  them  to  exercise  the  learner,  anil  for  the  same 
pm-pose  shall  add  the  foUowmg  example. 


Courses. 

Dist. 

Diff.  of  Lat. 

Dcjjarture. 

N. 

S. 

E. 

W. 

South. 

w.  s.  w. 

s.  w. 

w. 

10 
25 
30 
20 

10.0 

9.C 

21.2 

23.1 
21.2 
20.0 

DifF.  of  lat,    40.8 

Depart.  G4.3 

EXAMPLE   III. 

A  ship  in  the  latitude  of  37°  10'  N.,  is  bound  to  a  i)or 
which  lies  180  miles  west  of  the  meridian  of  the  ship;  but 
she  sails  the  following  courses,  viz.  S.  W.  l)y  W.  27  mil 
W.  by  S.  25  miles,  VV.  by  N.  18  miles,  S.  S.  E.  32  m 
S.  by  E.  25  miles,  S.  31  miles,  and  S.  S.  E.  39  miles.  Re 
is  in,  and  her  departm-c  from  the 
meridian,  with  the  course  and 
distance  to  her  intended  port. 

The  difference  of  latitude  and 
departure  made  on  each  course, 
are  given  in  the  adjoined  traverse 
table;  hence  it  ai)pears  that  the 
difierence  of  latitLide  made  good 
is  169.4  miles ;  the  dc{)arture,  47.4 
miles  ;  and  by  Ciise  VI.  Plane 
Sailing,  the  course  S.  15°  38'  W., 
and  tlistance,  175.9  miles;  and 
the  course  to  the  intended  port, 
S.  58°  42'  W.,  distance  155.2 
miles  ;  the  latitude  being  in 
34°  21'  N. 


t  in  the  latitude  of  33°  0'  N., 
by  reason  of  contrary  wiiids, 
es,  W.  S.  W.  h  W.  30  miles, 
lifes,  S.  S.  E.  i  E.  27  miles, 
([uired  the  latitude  the  ship 


TRAVERSE   TABLE. 


Courses. 

Dist. 

Diff.  of  Lat. 

Departure. 

N. 

S. 

E. 

W 

S.  W.  by  W. 

w.  s.  w.  h  w. 

W.  by  S. 

W.  by  N. 
S.  S.  E.* 
S.  S.  E.  1  E. 
S.  by  E. 
South. 
S.  S.  E.* 

27 
30 
25 
18 
32 
27 
25 
31 
3'J 

3.5 

15.0 

8.7 
4.9 

20.G 
23.2 
24.5 
31.0 
3G.0 

12.2 

13'9 

4.9 

14.9 

22.4 
287 
24.5 
17.7 

3.5 

172.9 
3.5 

45.9 

93.3 
45.9 

Diff. 

of  lat.    169.4 

Depart.  47.4  | 

*  Iiisicad  of  piitliiif^  the  course  S.  S.  E.  32  miles,  and  S.  S.  E.  39  mile-s  you  might  make  me  enm 
only,  cailiiiij  it  .S.  S.  E.  'I  miles. 


63 


PARALLEL     SAILING. 


In  Plane  Sailing,  the  earth  is  considered  as  an  extended  plane  ;  but  tins  supposition 
Ls  very  erroneous,  because  the  em'th  is  nearly  of  a  spherical  figure,  in  which  the 
meridians  all  meet  at  '^the  poles ;  consequently  the  distance  of  any  two  meridians 
measured  on  a  parallel  of  latitude  (wliicii  distance  is  called  the  meridian  distance) 
decreases  in  proceedhig  from  the  equator  to  the  j)o1gs.  To  illustrate  this,  let  Pli 
represent  die  semi-axis  of  the  earth,  IJ  the  centre,  V  the  pole,  PCA 
a  quadrant  of  the  meridian,  AB  the  radius  of  the  equator,  and  CD 
(parallel  thereto)  the  radius  of  a  parallel  of  latitude.  Then  it  is 
evident  that  CD  will  be  the  cosine  of  AC,  or  the  cosine  of  the 
latitude  of  the  point  C,  to  the  radius  AB ;  now,  if  the  quadrantal  arc 
PCA  be  supposed  to  revolve  round  die  axis  PB,  the  point  A  will 
describe  the  circumference  of  the  equator,  and  C  the  circumference 
of  a  parallel  of  latitude  ;  and  the  former  circumference  will  be  to  the  latter  as  AB  to 
CD  (as  may  easily  be  deduced  from  Art.  55,  Geometry),  that  is,  as  radius  to  the  cosine 
of  the  latitude,  or  the  point  C  ;  hence  it  follows,  that  the  length  of  any  arc  of  the 
equator  intercej)ted  between  two  meridians,  is  to  die  length  of  a  corresponding  avc  of 
any  parallel  intercepted  between  the  same  meridians,  as  radius  is  to  the  cosine  of  the 
latitude  of  diat  parallel.     Hence  we  obtain  the  following  theorems. 

THEOREM  I.       ♦ 

Tfie  circumference  of  the  equator  is  to  the  circumference  of  any  other  parallel  of  latitude, 
ns  radius  is  to  the  cosine  of  that  latitude.  ' 

TIIE0RE3I  II.  j 

As  the  length  of  a  degree  of  the  equator  is  to  the  mendian  distance  corresponding  ta 
a  degree  on  any  other  parallel  of  latitude,  so  is  radius  to  the  cosine  of  that  parallel  of 
latitiuk. 

TlIEOREiM   III. 

As  radius  is  to  the  cosine  of  any  latitude,  so  are  the  miles  of  diference  of  longitude 
between  two  mcridinns  [or  their  distance  in  miles  upon  the  equator)  to  the  distance  of  these 
two  meridians  on  that  parallel  oflatiliule  in  miles. 

THEOREM   IV. 

As  the  cosine  of  any  latitude  is  to  rculius,  so  is  the  length  of  any  arc  on  that  parallel  of 
latitude  [intercepted  between  two  meridians)  in  miles  to  the  length  of  a  similar  arc  on  the 
equator,  or  viiles  of  difference  of  longitude. 

THEOREM  V. 

As  the  cosine  of  any  latitude  is  to  the  cosine  oj  any  other  latitude,  so  is  the  length  of 
any  arc  on  the  fust  parallel  of  latitude  in  miles,  to  the  length  of  tlie  same  arc  on  the  other 
in  miles. 


By  means  of  Theorem  IH.  the  following  table  was  calculated,  wliich  shoAVs  the 
meridian  distance  corrcsjionding  to  a  degree  of  longitude  in  every  latitude  ;  and  may 
be  made  to  answer  for  any  degree  or  minute  by  taking  |)roportional  parts. 


fil 


parallp:l  sailing. 


The  following  Table  shows  for  every  degree  of  latitude  how  many  miles  distant 
the  ttoo  meridians  are,  ichose  difference  of  longitude  is  one  degree. 


Lat. 

Miles. 

L.\T. 

Miles. 

Lat. 

Miles. 

Lat. 

Miles. 

Lat. 

Miles. 

1- 

59.99 

19° 

56.73 

37° 

47.92 

55° 

34.41 

73° 

17.54 

2 

59.90 

20 

56.38 

38 

47.28 

56 

33.55 

74 

16.54 

•3 

59.92 

21 

56.01 

3') 

46.63 

57 

32.68 

4  a 

15.53 

4 

59.85 

22 

55.63 

40 

45.96 

58 

31.80 

76 

14.52 

5 

59.77 

23 

55.23 

41 

45.28 

59 

30.90 

77 

13.50 

6 

59.C7 

24 

54.81 

42 

44.59 

60 

30.00 

78 

12.47 

7 

59.55 

25 

54.38 

43 

43.88 

61 

29.09 

79 

11.45 

8 

59.42 

26 

53.93 

44 

43.16 

62 

28.17 

80 

10.42 

y 

59.2G 

27 

53.46 

45 

42.43 

63 

27.24 

81 

9.39 

10 

59.09 

28 

52.98 

Aij 

41.68 

64 

2(1.30 
20.36 

82 

8.35 

11 

58.90 

29 

52.48 

47 

40.92 

65 

83 

7.31 

12 

58.G9 

30 

51.96 

48 

40.15 

66 

24.40 

84 

6.27 

13 

58.46 

31 

51.43 

49 

39.36 

67 

23.44 

85 

5.23 

14 

58.22 

32 

50.88 

50 

38.57 

63 

22.48 

86 

4.19 

15 

57.96 

33 

50.32 

51 

37.76 

69 

21.50 

87 

3.14 

16 

57.G8 

34 

49.74 

52 

36.94 

70 

20.52 

88 

2.09 

17 

57.38 

35 

49.15 

53 

36.11 

71 

19.53 

89 

1.05 

18 

57.06 

36 

48.54 

54 

35.27 

72 

18..54 

90 

0.00 

When  a  sliip  sails  east  or  west  on  the  surface  of  the  earth  sup])osefl  to  be  snjherical, 
she  describes  a  parallel  of  latitude,  and  this  is  called  Parallel  Sailing.  In  tliis  case, 
the  distance  sailed  (or  dej)artiire)  is  equal  to  the  distance  between  the  meridians  sailed 
from  and  arrived  at  in  that  ])arallel ;  and  it  is  easy,  by  Theorem  IV.  (preceding)  to 
find  the  difference  of  longitude  from  the  distance,  or  the  distance  from  the  difference 
of  longitude,  as  will  ajjpear  plain  by  the  following  examples.  » 


CASE   I. 

The  differtnce  of  longitude  between  two  places  in  the  same  parallel  of  latitude  being 
given,  to  find  the  distance  between  them. 

Suppose  a  ship  in  the  latitude  of  49°  30',  north  or  south,  sails  directly  east  or  west, 
until  her  difference  of  longitude  be  3°  30' ;  required  the  distance  sailed. 

•>  BY   PROJECTION. 

Take  the  sine  of  90°  from  the  plane  scale,  and,  with  one  foot  of  the  compasses  on 
(fig.  1)  as  a  centre,  describe  the  arc  EQ,*  with  the  difference  of  longitude,  210  miles, 
in  the  compasses,  and  one  foot  in  E,  J 

as  a  centre,  describe  an  arc  cutting 
Ea  in  Q;  jom  PE,  PQ.  Take  the 
sine  of  the  com{)lement  of  the  latitude 
40°  30'  in  your  compasses,  and  with 
one  foot  in  P,  as  a  centre,  describe  the 
arc  FG,  cutting  PE,  PQ,  in  F,  G ;  then 
the  length  of  tlie  chord  FG  being 
measured  on  the  same  scale  of  equal 
parts,  will  be  the  departure  13().4  miles. 

Or  this  projection  may  be  made  in 
the  following  manner.  Draw  AD 
(!ig.  2)  of  an  iiid(;finite  length  ;  make 
tlic  angle  DAC  eijual  to  tlie  latitude 
49°  30',  and  AC  c(|Hal  to  the  diflerence 


Fig.  2. 


of  longitude  210  miles  ;  draw  CD  perpendicular  to  AD;  then  will  the  line  AD  be  the 
distance  or  departure  reipiired. 

BY    LOGARITHMS. 
To  find  the  departiu'c  or  distance. 

As  radius  90° 10.00000 

Is  to  the  ditrereiK-e  of  longitude  210 2.32222 

So  is  cosine  latitude  49°  30' 9.81254 


To  the  (li.<t;mce  or  departure  13(14. 


2.13470 


PARALLEL   SAILING. 


05 


BY   GUNTER 
The  extent  from  radius  to  the  com])leiiient  of  the  latitude  40°  30'  on  the  line  of 
sines,  will  reach  from  the  difference  of  longitude  210,  to  the  distance  i;i(J.4,  on  the  Uiie 
of  numbers. 

BY  INSPECTION. 

Find  the  latitude  among  the  degrees  in  Table  II.,  and  in  the  distance  column  the 
difference  of  longitude,  opposite  to  which  in  the  colunni  of  latitude  will  be  the  distance 
required. 

In  the  present  example,  the  latitude  is  49°  30' ;  and  as  the  table  is  only  calculated  to 
suigle  degrees,  we  must  find  the  numbers  in  tlie  tables  of  49°  and  50°,  and  take  the 
mean  of  them  ;  the  former  is  137  8,  the  latter  135.0,  the  mean  of  which  is  the  sought 
distance  or  departure,  13G.4. 

CASE  II. 

The  distance  between  two  places  on  the  same  parallel  of  latitude  given,  to  find  their 

difference  of  longitude. 

Suppose  a  ship  in  the  latitude  of  49°  30'  N.  or  S.,  and  longitude  36°  40'  W.,  sails 
directly  west  136.4  miles ;  requked  the  difference  of  longitude,  and  longitude  in. 

BY   PROJECTION. 

With  the  sine  of  the  complement  of  the  latitude,  40°  30',  m  your  compasses,  and  one 
oot  in  P,  as  a  centre  (fig.  1,  of  the  preceding  case),  describe  the  arc  FG,  upon  which 
set  off  the  departure  136.4  milee,  upon  the  chord  FG,  and  through  the  points  F  and  G 
draw  the  lines  PE  and  PQ. ;  then,  with  the  sine  of  90°  in  the  compasses,  and  one  foot 
on  P,  as  a  centre,  describe  an  arc  to  cut  PE,  PQ,  in  E  and  Q  ;  then  the  chord  EQ 
being  measin-ed  upon  the  same  scale  of  equal  parts  that  the  departure  was,  will  be  the 
difference  of  longitude  210  miles. 

Or  thus ;  dra\v  the  line  AD  (fig.  2),  which  make  equal  to  the  given  distance  136.4 ; 
at  D  erect  DC  perpendicular  to  DA ;  make  the  angle  DAC  equal  to  the  latitude ;  tlien 
will  AC  be  the  sought  difference  of  longitude  210  miles. 

BY   LOGARITHMS. 


As  cosine  of  latitude  49°  30' ... .    9.81254 

Is  to  the  distance  136.4 2.13481 

So  is  radius 10.00000 

To  the  difference  of  long.  210. .    2.32227 


Longitude  left 36°  40/  W. 

Difference  of  longitude 3  30  W. 


Longitude  in 40  10  W. 


BY   INSPECTION. 

Look  for  the  latittide  among  the  degrees,  as  if  it  was  a  course,  and  the  departure  in 
the  column  of  latitude ;  against  which  will  stand  the  difference  of  longitude  in  tlie 
distance  colunni. 

Thus,  m  the  course  49°,  we  must  seek  for  136.4  in  the  latitude  column,  and  we  find 
it  con-esponds  to  the  distance  208 ;  and  in  the  course  50°,  we  find  it  nearly  corresponds 
to  212 ;  half  the  sum  of  208  and  212  is  210,  which  is  the  sought  difference  of  longitude. 


QUESTIONS 

To  exercise  the  learner. 


Question  I.  A  ship  in  the  latitude  of  32°  N.,  sails  due  east  till  her  difference  of 
longitude  is  384  miles ;  requh-ed  the  distance  saUed. 

Answer.    325.7  miles. 

quest.  II.  A  ship  from  the  latitude  of  53°  36'  S.,  longitude  10°  18'  E.,  sails  due  west 
236  miles ;  required  her  jjresent  longitude. 

Ans.    3°40'E. 

Quest.  III.  If  two  ships  in  the  latitude  of  44°  30'  N.,  distant  216  miles,  should  sail 
directly  south  until  they  were  in  the  latitude  of  32°  17'  N.,  what  distance  are  they  from 
each  other  ? 

Ans.    By  Theorem  V.,  256  miles. 

Quest.  IV.  A  ship  having  run  tlue  east  for  three  days,  at  the  rate  of  5  knots  an 
hour,  finds  she  has  altered  her  longitude  8°  16' ;  what  parallel  of  latitude  did  she 
sail  in  ? 

Ans.    43°  28' N.  or  S. 
9 


66 


MIDDLE    LATITUDE    SAILING. 


In  sailing  north  or  south  (or  on  a  meridian)  the  difference  of  longitude  is  notliaig, 
and  the  lUfference  of  latitude  is  equal  to  the  distance  sailed  ;  ijut  in  sailing  ea'^t  or  wes 
(or  on  a  parallel  of  latitude),  the  difference  of  latitude  is  nothing,  and  the  difference  of 
longitude  may  be  calculated  by  the  foregoing  theorems  of  Parallel  Sailing.  In  sailing 
on  any  other  course,  the  shij)  changes  both  her  latitude  and  longitude ;  in'this  case  the 
difference  of  latitude,  departure,  and  difference  of  longitude,  may  be  calculated  by 
a  proi)er  ajiplication  of  the  principles  of  Plane  Saihng  to  the  sailing  on  a  S])herical 
surface  ;  to  do  which,  the  surface  of  the  globe  may  be  sup])osed  to  be  divided  into  an 
indefinite  number  of  small  surfaces,  as  square  miles,  furlongs,  yards,  &ic.,  which,  on 
account  of  their  smallness,  in  comparison  with  the  wliole  surface  of  the  earth,  may  be 
esteemed  as  plane  surfaces,  and  the  difference  of  latitude  and  departure  (or  meridian 
distance)  made  in  sailing  over  each  of  these  surfaces,  may  be  calculated  by  the 
common  rules  of  Plane  Sailing;  and  by  summing  up  all  tlie  differences  of  latitude  and 
departures  made  on  these  different  planes,  we  shall  obtain  the  whole  difference  of 
latitude  and  departure  nearly.*  Now,  by  Case  I.  of  Plane  Sailing,  the  distance 
described  on  any  one  of  these  small  surfaces  is  to  the  corresj)onding  difference  of 
latitude  as  radius  is  to  the  cosine  of  the  course ;  and  as  the  course  is  the  same  on  all 
these  surfaces,  it  follows  that  the  sum  of  all  the  distances  described  thereon,  is  to  the 
Bum  of  the  corresponding  differences  of  latitude  as  radius  is  to  the  cosine  of  the 
course ;  that  is,  the  whole  distance  sailed  on  the  globe,  is  to  the  corresi)onding 
difference  of  latitude  as  radius  is  to  the  cosine  of  the  course.  In  a  similar  manner  it 
appears,  that  the  distance  described  on  the  globe  is  to  the  sum  of  all  the  corresponding 
departures  (or  meridian  distances)  described  on  these  different  surfaces,  as  radius  is  to 
the  sine  of  the  course  ;  so  that  the  canons  for  calculating  the  whole  difference  of 
latitude  and  departure  from  the  course  and  distance  are  the  same,  whether  the  earth 
be  esteemed  as  an  extended  plane  or  a  sjiherical  surface  ;  and  the  same  is  to  be 
obsei-ved  with  respect  to  the  other  cases  of  Plane  Sailing. 

We  shall,  tlierefore,  in  all  the  calculations  of  sailing  on  the  spherical  surface  of  the 
earth,  in  which  the  course,  distance,  difference  of  latitude  and  deiiarture,  occur,  make 
use  of  the  canons  already  taught  in  Plane  Sailing,  and  shall  construct  the  schemes 
exactly  in  the  same  manner.  The  only  additional  calculation  in  sailing  on  a  sjjherical 
surface,  consists  in  determining  the  longitude  from  the  departure;  for  in  sailing  on  a 
plane,  the  departure  and  longitude  are  the  same  ;  but  in  sailing  on  a  sjjherical  surface, 
the  ivhole  departure  {as  ivas  observed  above)  is  equal  to  the  sum  of  all  the  meridian  distances 
made  in  sailing  over  the  indefinite  number  of  small  surfaces,  into  luhich  loe  have  supposed 
the  spherical  surface  to  be  divided,  and  the  whole  difference  of  longitude  corresponding  is 
equal  to  the  sum  of  all  the  differences  of  longitude,  deduced  from  each  of  these  small 
meridian  distances  bj/  Theorem' \Y.  of  Parallel  Sailing.\  Several  metho'ds  have  been 
proposed  for  abridging  the  calculation  of  the  difference  of  longitude  from  the  dej)arture, 
the  tnost  noted  of  which  are  those  knowii  by  the  r\m\\es  ot  Middle  Latitude  Sailing 
and  Mercator's  Sailing;  the  latter  (which  will  be  hereafter  explained)  is  perfectly 
accurate  ;  J  the  former  is  only  an  approximation,  but  it  is  veiy  much  used  in  calculating 

*  The  error  arising  from  this  supposition  will  be  decreased  by  increasing  the  number  of  the  planes,  so 
that,  by  increasing  the  number  indefinitely,  the  error  may  be  made  less  than  an}'  assignable  ouantity. 

_t  Using  (in  estimating  the  difference  of  lono-iiude  corresponding  to  each  of  these  small  meridian 
distances)  the  latitude  corresponding  to  the  middle  point  of 'the  surface  on  which  these  small  meridian 
distances  are  respectively  made. 

t  This  is  tnie  in  theory,  and  would  be  so  in  practice,  if  the  meridional  difference  of  latitude  in 
Table  III.  were  given  to  a  sulficient  number  of  decimals  ;  but  being  only  given  to  the  nearest  mile  or 
minute,  the  error  arising  from  this  cause,  when  the  difference  of  latitude  is  small,  is  greater  than  the 
error  in  Middle  Latitude  Sailing ;  in  consequence  of  this,  the  method  by  middle  latitude  is  alm'^st 
exclusively  used  in  the  common  operations  on  shipboard. 


J 


MIDDLE   LATITUDE   SAILING.  67 

Fhfirt  runs  and  days'  works;  Imt  in  calculating  large  distances  across  distant  paralliels,  it 
is  liable  to  error.  The  principle  on  which  the  calculations  of  Middle  Latitude  Sailing 
are  foujided,  is  this: — Instead  of  calculating  the  difference  of  longitude  corresponding 
to  the  de[)arture  made  on  each  of  the  small  surfaces,  into  which  we  have  supposed  the 
sj)here  to  be  divided,  and  adding  then*  together,  the  whole  depai-ture  (or  sum  of  tlie 
meridiiui  distances)  is  calculatetl,  and  the  longitude  deduced  therefrom  by  the  rules  of 
Parallel  Sailing,  using  for  the  latitude  the  arithmetical  mean  between  the  latitude  sailed 
from  and  that  arrived  at.  On  this  supposition,  we  have  the  two  first  of  he  following 
theorems  for  calculating  the  departure  from  the  difference  of  longitude,  or  the 
difference  of  longitude  from  the  dei)arture,  which  ai-e  the  same  as  Theorems  IIL 
and  IV.  of  Parallel  Sailing,  excejjt  in  %viiting  departure  for  distance,  and  middle 
latitude  for  latitude :  the  other  theorems  are  easily  obtained  by  combining  the  two  first 
witii  the  common  theorems  of  Plane  Sailing  ;  observing  that  the  middle  latitude  is  half 
the  sum  cf  the  two  latitudes,  if  they  are  of  the  same  iiame,  or  half  their  difference  if  of 
conlrarij  names.  This  method  may  be  rendered  perfecdy  accurate  by  applying  to  tlie 
middle  latitude  a  correction  taken  from  the  table  following  Case  VII.  of  diis  article 
We  shall,  however,  in  the  following  exam[)les,  make  the  calculations  without  applying 
this  correction,  because,  in  most  cases  in  practice,  it  is  of  but  little  importance. 

THE0RE3I  I. 

As  radius  is  to  the,  coshu  of  the  middle  latitude,  so  is  the  difference  of  longitude  to  the 
departure. 

THE0RE3I  II. 

As  the  cosine  of  the  middle  latitude  is  to  the  radius,  so  is  the  departure  to  the  difference 
of  longitude. 

Now,  by  Case  I.  of  Plane  Sailing,  the  radius  is  to  the  sine  of  the  course,  as  the 
distance  sailed  is  to  the  depaiture,  and,  if  we  combine  this  analogy  with  Theorem  II., 
we  shall  have 

THEOREM   III. 

As  the  cosine  of  the  middle  latitude  is  to  the  sine  of  the  course,  so  is  the  distance  sailed 
to  the  difference  of  longitude. 

By  Case  II.  of  Plane  Sailing,  we  have  this  analogy ;  As  raduis  is  to  the  tangent  of 
the  course,  so  is  the  diffi^rence  of  latitude  to  the  departui-e ;  by  combining  this  with 
Theorem  II.,  we  have 

THEOREM   IV. 

As  the  cosi7ie  of  the  middle  latitude  is  to  the  tangent  of  the  course,  so  is  the  difference  of 
latitude  to  the  difference  of  longitude. 

Whence  we  easily  deduce  the  following, 

THEOREM  V. 

As  tlie  difference  of  latitude  is  to  the  difference  of  longitude,  so  is  the  cosine  of  the  middle 
latitude  to  the  tangent  of  the  course. 


By  means  of  the  preceding  theorems,  we  have  formed  the  following  table,  which 
contains  all  the  rules  necessary  for  solving  tlie  various  cases  of  Middle  Latitude 
Sailing. 


08 


MIDDLE   LATITUDE  SAILLNG. 


MIDDLE   LATITUDE   SAILING. 


Case. 

Given. 

Sought. 

Solutions. 

1 

Botli  latitudes 
and  longitude. 

Departure. 
Course. 

Distance. 

Radius  :  d^.  of  long.  ::  cosine  middle  lat.  :  departure. 
(  Difference  of  lat.  :  radius  : :  departure  :  tangent  course. 
(  Diff.  lat.  :  diff.  long.  : :  cosine  middle  lat.  :  tangent  course. 
(Radius  :  difference  of  latitude  ::  secant  course  :  distance. 
(  Sine  course  :  departure  :  :  radius  :  distance. 

2 

Both  latitudes 
and  departure. 

Course. 

Distance. 

Diff.  of  long. 

Difference  of  lat.  :  radius  ::  departure  :  tangent  course 

Sine  course  :  departure  ::  radius  :  distance. 

Cosine  middle  lat.  :  departure  ::  radius  :  diff.  of  long. 

3 

One  latitude, 

course,  and 

distance. 

Ditr.  of  latitude. 

Departure. 
Diff.  of  long. 

Radius  :  distance  ::  cosine  course  :  difference  of  latitude. 
Hence  the  other  latitude  and  middle  latitude  are  found. 

Radius  :  distance  : :  sine  course  :  departure, 
t  Cosine  middle  lat.  :  departure  : :  radius  :  diff.  of  long. 
(  Cosine  middle  lat.  :  sine  course  ::  distance  :  diff.  of  long. 

4 

Both  latitudes 
and  course. 

Departure. 
Distance. 

Diff.  of  long. 

Radius  :  diff.  of  lat.  ::  tangent  course  :  departure. 

Cosine  course  :  diff.  of  latitude  : :  radius  :  distance. 
(  Cosine  middle  lat.  :  departure  : :  radius  :  diff.  of  long. 
(  Cosine  middle  lat.  :  tangent  course  :  :  diff.  lat.  :  difl".  long. 

5 

Both  latitudes 
and  distance. 

Course. 
Departure. 
Diff.  of  long. 

Distance  :  radius  :  :  diff.  of  latitude  :  cosine  course. 

Radius  :  d, stance  ::  sine  course  :  departure. 

Cosine  middle  lat.  :  departure  : :  radius  :  diff.  of  long. 

6 

One  latitude, 
course,  and 
departure. 

Ditr.  of  latitude. 

Distance. 
Diff.  of  long. 

Radius  :  departure  :  :  cotangent  course  :  diff.  of  latitude. 
Hence  the  other  latitude  and  middle  latitude  are  known. 
Sine  course  :  departure  : :  radius  :  distance. 
Cosine  middle  lat.  :  departure  ::  radius  :  diff.  of  long. 

7 

One  latitude, 

distance,  and 

departure. 

Course. 
Diff.  of  latitude. 

Diff.  of  long. 

Distance  :  radius  : :  departure  :  sine  course. 
Radius  :  distance  : :  cosine  course  :  difference  of  latitude. 
Hence  we  obtain  the  other  latitude  and  middle  latitude. 
Cosine  middle  lat.  :  departure  ::  radius  :  diff.  of  long. 

We  shall  now  proceed  to  illustrate  these  rules,  by  workmg  an  example  in  every  case. 

CASE   I. 

Thelatitudes  and  longiludes  of  two  places  given,  to  find  their  beanng  and  distance. 

Required  the  bearing  and  distance  between  Cape  Cod  light-house,  in  the  latitude 
of  42^  3'  N.,  longitude  70°  4'  W.,  and  tlie  island  of  St.  Mary  (one  of  the  Western 
Islands),  in  the  latitude  of  36°  59'  N.,  and  longitude  25°  10'  W. 


Cape  Cod's  latitude..  42°    3' N. 
St.  Mary's  latitude. . .  36  59  N. 

Difference  of  latitude     5     4 
60 


42°    3' 
36  59 

79     2 


Middle  lat.  39  31* 


Longitude. 
Lonsjitude . 


70°    4'W. 
25   10  W. 


Sum. 


44  54 

60 


In  miles 304 


Diff.  of  long.  2694  miles 


Sf.Mary. 


BY   PROJECTIOiN. 
Draw  the  east  and  west  line  DC  ;  with  the  chord  of  60°  describe  the  arc  QS  about 
the  centre  D,  to  cut  DC  in  Q ;  upon  this  arc,  set  off,  from  Q  to  S,  the  middle  latitude 
39°  31' ;  through  D  and  S  draw  the 
line  DB,  which  make  equal  to  the       Cape CcrlXj^ 
difference  of  longitude  2694  miles ; 
from    B    let    fall    upon    DC     the 
peri)endicular  BC  ;    continue   this 
towards  A,  making  AC  equal  to  the 
difference  of  latitude   304  miles  ;f^ 
join  AD,  and  it  is  done.     For  by 
this  method  of  construction,  on  the 
princii)lcs  before  explained,  A  will 
be  the  situation  of  Cajie  Cod,  D  the 
situation  of  St.  Mary  ;  CD  will  be  tlie 
departin-c,  which,  bcis^.g  measured, 

*  The  rorrection  of  this  (|uantil y,  in  the  table  nl  tlie  end  of  Case  VII.,  is  3'  additive,  making  it  3D°  34', 
which  ran  lie  used  instead  of  3D°  31',  if^^real  accurarj'  lie  required. 

t  If  llic  place  A  be  to  the  southward  of  U,  the  line  AC  should  be  set  olT  upon  the  line  CL5,  from  C 
towards  B. 


MIDDLE   LATITUDE   SAILING. 


69 


will  be  found  to  be  2078  miles  ;  the  distance  will  be  represented  by  AD,  which,  being 
measured,  will  be  found  to  be  2102  miles,  and  the  course  from  Caj)e  Cod  to  St.  Mary 
will  be  rejjresented  by  the  angle  CAD  equal  to  81°  41' ;  therefore  the  course  will  bo 
S.  81°  41'  E.,  or  E.  %  S.,  nearly. 

JVute.  The  course  is  put  S.  81°  41'  E.  becaxise  St.  Mary,  being  in  a  less  northern 
latitude  than  Cape  Cod,  is  to  the  southward  of  it ;  it  is  also  to  tlie  eastwai'd  of  Cape 
Cod,  because  it  is  in  a  less  western  longitude. 


To  find  the  departure  (by  Theorem  L) 

As  radius  90° 10.00000 

Is  to  difference  of  long.  2G94 ; . .    3.43040 
So  is  cosine  middle  lat.  39°  31'  .    9.88730 


To  the  departure  2078 


BY   LOGARITHMS. 

To  find  the  course. 
As  difference  of  latitude  304  .. .    2.48287 

Is  to  radius  45° 10.00000 

So  is  the  departure  2078 3.31770 

To  tangent  of  course  81°  41'. . .  10.83483 


3.31770 


To  find  the  distance. 

As  radius  00° 10.00000 

Is  to  the  difference  of  lat.  304. .  2.48287 

So  is  secant  of  course  81°  41'  . .  10.83970 


To  the  distance  2102 3.32257 

JVote.  The  logarithm  of  the  departure 
above  found,  3.31770,  is  rather  greater 
flian  the  logarithm  of  2078  :r  3.31705  ;  but 
in  finding  the  course  by  the  departure,  I 
have  used  the  quantity  found  at  the  first 
operation,  and  shall  do  the  same  m  all 
future  calculations. 


JVbfe.  The  course  may  be  found  with- 
out the  departure,  by  Theorem  V.  Middle 
Latitude  Sailing. 


As  the  difference  of  latitude  304 
Is  to  the  difference  of  long.  2G94 
So  is  cosme  middle  lat.  39°  31'. 


To  tangent  of  couree  81°  41' 


2.48287 
3.43040 
9.88730 

13.31770 

2.48287 

10.83483 


BY   GUNTER. 

Extend  from  the  radius,  or  90°,  to  50°  29',  the  complement  of  the  middle  latitude,  on 
the  line  of  sines;  that  extent  will  reach  from  the  difference  of  longitude  2694,  to  the 
departure  2078,  on  the  line  of  numbers. 

2dly.  Extend  from  the  difference  of  latitude  304,  to  the  departure  2078,  on  the  line 
of  numbers ;  that  extent  will  reach  from  radius,  or  45°,  to  the  course  81°  41',  on  the 
line  of  tangents. 

3dly.  Extend  from  the  course  81°  41',  to  the  radius  90°,  on  the  line  of  sines;  that 
extent  will  reach  from  the  departure  2078,  to  the  distance  2102  miles,  on  the  line  of 
numbere. 

BY   INSPECTION. 

Rule.  Look  for  the  middle  latitude,  as  if  it  was  a  coni-se  in  Plane  Sailing,  and  the 
difference  of  longitude  in  the  distance  column,  ojjposite  to  which,  in  the  column  of 
latitude,  will  stand  the  departure  ;  having  the  difference  of  latitude  and  departure,  the 
course  and  distance  are  found  (as  in  Case  VI.  Plane  Sailing)  by  seeking  in  Table  II., 
with  the  difference  of  latitude  and  departure,  until  they  are  found  to  agree  in  their 
respective  columns ;  opposite  to  them  will  be  found  the  distance  in  its  column,  and 
tlie  coin-se  will  be  found  at  the  top  of  that  table,  if  the  departiu-e  be  less  than  the 
difference  of  latitude,  otlierwise  at  the  bottom. 

Thus,  with  one  tenth  of  the  difference  of  longitude  2G9.4  or  2G9, 1  enter  Table  II., 
and  opposite  to  it,  in  the  distance  colinnn  of  tlie  tables  of  39°  and  40°,  I  find  209.1,  and 
206.1  in  the  ktitude  colunm ;  now,  the  middle  latitude  being  nearly  Sd-^°,  I  take  the 
mean  of  these,  207.G,  for  die  departure,  which  being  multiplied  by  10,  gives  the  whole 
departure  2076.  Again,  I  enter  Table  I.  with  one  tenth  of  the  departure  207.6,  and 
one  tenth  of  the  difference  of  latitude  30.4,  and  find  that  they  agree  nearly  to  a  course 
of  7i  points,  and  a  distance  of  210,  which,  multiplied  by  10,  gives  the  sought  distance, 
2100  miles,  nearly. 


7ft 


MIDDLE   LATITUDE   SAILING. 


CASE  IL 

Both  latitudes  and  departure  from  the  meridian  given.,  to  find  the  course,  distance,  and 

difference  of  longitude. 

A  ship  in  the  latitude  of  49^57'  N.,  and  longitude  of  15°  16'  W.,  sails  south-westerly 
till  her  departure  is  194  miles,  and  latitude  m  AT  18'  N.  Requh-ed  the  course, 
distance,  and  longitude  ui. 

Latitude  left 49°  57'  N. 

Latitude  in 47   18  N. 


Difference  of  latitude . 


2  39  =  159mUe8. 


Sum  of  latitudes 97 

Middle  latitude 48 


15 

38* 


BY   PROJECTION. 

Draw  the  meridian  ACD,  on  which  take  AC  equal  to  the  dif- 
ference of  latitude  159  miles ;  draw  CB  peii^endicular  to  AC,  and 
make  it  equal  to  the  departure  194  miles  ;  about  B,  as  a  centre, 
describe  an  arc  ab,  on  which  set  oft'  the  middle  latitude  48°  38' ; 
through  B  and  b  draw  the  line  BD,  meeting  ACD  in  D ;  join  AB, 
and  it  is  done ;  for  AB  will  be  the  distance  sailed,  which,  being 
measured,  will  be  found  equal  to  250.S  miles ;  BD  will  be  the 
difference  of  longitude,  equal  to  293.5  miles ;  and  the  angle  CAB  will  represent  the' 
couree  from  the  meridian,  50°  40'. 

BY   LOGARITHMS. 


To  find  the  coin-se. 
As  the  difference  of  latitude  159    2.20140 

Is  to  radius  45° 10.00000 

So  is  the  departure  194 2.28780 

To  tangent  course  50°  40' 10.08040 

To  find  the  diffei-ence  of  longitude. 
As  cosine  middle  lat.  48°  38' . . .    9.82012 

Is  to  the  deimrture  194 2.28780 

So  is  radius  90° 10.00000 

To  difference  of  long.  293.5 ....    2.4G7G8 


To  find  the  distance. 

As  sine  course  50°  40' 9.88844 

Is  to  the  dejiarture  194 2.28780 

So  is  radius  90° 10.00000 


To  the  distance  250.8 . 


2.39936 


Longitude  sailed  from 15°  16'  W. 

Difference  of  long.  294  miles.     4  54  W. 

Longitude  in 20  10  W. 


BY   GUNTER. 

1st.  The  extent  from  the  difference  of  latitude  159,  to  the  departure  194,  on  the  line 
of  numbere,  will  reach  from  radius,  or  45°,  to  the  course  50°  40',  on  tlie  line  of  tangents. 

2dly.  The  extent  from  50°  40'  to  radius,  or  90°,  on  the  line  of  sines,  will  reach  from 
the  departure  194,  to  the  distance  251,  on  the  line  of  numbers. 

3dly.  The  extent  from  the  complement  of  middle  latitude  41°  22',  to  radius,  or  90°, 
on  the  line  of  shies,  will  reach  from  the  departure  194,  to  the  difference  of  longitude 
294,  on  the  line  of  numbers. 

BY   INSPECTION. 

Rule.  With  the  difference  of  latitude  and  dey)arture,  find  the  course  and  distance 
(as  in  Case  VI.  of  Plane  Sailing),  by  seeking  in  Table  II.  until  the  difference  of  latitude 
and  departure  are  found  to  correspond,  against  whicli,  in  the  distance  cohunn,  will  be 
the  distance  ;  and  if  the  departure  be  less  than  the  difference  of  latitude,  the  course 
will  be  found  at  the  top  of  that  table,  otherwise  at  the  bottom. 

Then  take  the  middle  latitude  as  a  com-se,  and  find  the  departure  in  tlie  latitude 
column ;  the  number  corresponding  in  the  distance  column  will  be  tlie  difference  of 
longitude. 

In  the  present  example,  with  the  difference  of  latitude  159,  and  the  departure  194, 
we   find   that  the  nearest  niunbcrs  to  these  are   158.0  and   195.1,  standing  together 


*Tlie  correction  of  lliis  latitiRlc  in  llie  laMc 
iniddle  lalilude  48°  39'. 


at  \\\c.  cml  oCCase  \'I].  is  about  1',  mnkiiiff  the  correcteo 


MIDDLE   LATITUDE   SAILING. 


71 


\ 


over  51°,  against  the  distance  251  ;  whence  the  course  by  inspection  is  S.  51°  W.,  and 
the  distance  251.  Then,  taking  as  a  course  49°  (which  is  the  nearest  to  the  middle 
latitude  48°  38'),  seek  for  the  departure  194  in  the  latitude  column  ;  the  nearest  number 
is  194.2;  opposite  to  this,  in  the  distance  cohunn,  is  29G,  for  the  difference  of  longitude; 
this  value  diffei-s  a  little  from  that  found  by  logarithms,  owing  to  the  miles  of  middle 
latitude  neglected ;  for  if  we  were  also  to  find  the  difference  of  longitude  for  the  midcUe 
latitude  48°,  and  proi)ortion  for  the  mhuites,  the  result  would  come  out  nearly  the 
sajne  as  by  logarithms. 

CASE  III. 

One  latitude,  course,  and  distance  given,  to  find  the  difference  of  latitude  and  difference  of 

longitude. 

A  sliip  in  the  latitude  of  42°  30' N.,  and  longitude  58°  51' W.,  sails  S.  E.  by  S. 
300  miles.     Required  the  latitude  and  longitude  in. 

BY  PROJECTION. 

Draw  the  meridian  ADE  (as  in  Case  I.  Plane  Sailing) ;  \\\m\\  A, 
as  a  centre,  describe  an  arc  with  the  chord  of  G0°,  and  upon  it  set  off, 
from  where  it  cuts  AD,  the  course  S.  E.  by  S.,  or  3  pouits;  through 
that  pouit  of  the  arc,  and  the  point  A,  draw  the  line  AC,  which 
make  equal  to  the  distance  300  miles ;  from  C  let  fall  upon  AD  the 
perpendicular  CD ;  then  will  CD  be  the  departure  166.7  miles,  and 
AD  the  difference  of  latitude  249.4  miles.  Hence  we  obtain  the 
latitude  arrived  at,  and  the  middle  latitude ;  draw  the  line  CE, 
making  an  angle  with  DC  of  40°  26'  equal  to  the  middle  latitude ; 
and  the  distance  CE  will  be  the  difference  of  longitude  219  miles ; 
lience  the  longitude  is  easily  obtained. 


BY   LOGARITHMS 
To  find  the  difference  of  latitude. 

As  radius  8  points 10.00000 

Is  to  the  distance  300 2.47712 

So  is  cosine  course  3  points. . . .    9.91985 

To  the  difference  of  lat.  249.4 


2.39697 


Latitude  left 42°  30'  N. 

Difference  of  latitude 4  09  S. 


Latitude  in .38  21  N. 

Sum  of  latitudes 80  51 

Middle  latitude 40  26  * 


Longitude  left 58°  51'  W. 

Difterence  of  lousitude  219. .     3  39  E. 


Longitude  in 55   12  W. 


To  find  the  departure. 

As  radius  8  points 10.00000 

Is  to  the  distance  300  ... , 2.47712 

So  is  siiie  course  3  points 9.74474 

To  the  departure  166.7 2.22186 

To  find  the  difference  of  longitude  with 

the  departure. 

As  cosine  middle  lat.  40°  26'. . .  9.88148 

Is  to  the  departure  166.7  f 2.22186 

So  is  radius  90° 10.00000 

To  difference  of  longitude  219  .  2.34038 

Without  the  dej)arture. 
As  cosine  mid.  lat.  40°  26'  Ar.  Co.    0.1 1852 

Is  to  sine  course  3  points 9.74474 

So  is  distance  300  ijiiles 2.47712 

To  difference  of  longitude  219.  2.34038 


BY   GUNTER. 

1st.  The  extent  from  radius  8  points,  to  the  complement  of  the  course  5  points,  on 
the  line  marked  SR,  will  reach  from  the  distance  300,  to  the  difterence  of  latitude  249, 
on  the  line  of  numbers. 

2(lly.  The  extent  from  radius  8  points,  to  the  course  3  points,  on  the  Ime  SR,  will 
reach  from  the  distance  300,  to  the  departure  167,  on  the  line  of  numbei-s. 

3dly.  The  extent  from  the  complement  of  middle  latitude  49°  34',  to  radius  90°,  cw» 
the  line  of  sines,  will  reach  from  the  departure  167,  to  the  difference  of  longitude  219, 
on  the  line  of  numbers. 


*  The  correction  of  this  latitude  in  the  table  at  the  end  of  Case  VII.  is  2',  making-  the  corrected  middle 
latitude  40^  "IW. 

t  The  logarithm  of  the  departure  was  found  by  the  preceding  canon  to  he  2.22186,  differing  a  liltlg 
from  the  'oiiarilhm  of  166.7. 


72 


MIDDLE   LATITUDE   SAILING. 


BY  INSPECTION. 

Rule.  Witli  tlie  course  and  distance,  find  the  difference  of  latit.  do  and  departure 
(as  in  Case  L  of  Plane  Sailing),  by  finding  the  given  course  at  the  top  or  bottom  of  the 
tables,  either  among  the  points  or  degrees ;  in  that  page,  and  opposite  to  the  distance 
taken  in  its  colunui.will  stand  thedifterenceof  latitude  and  departure  in  their  columns. 
Then  take  the  middle  latitude  as  a  course,  and  find  the  departure  in  the  latitude 
column ;  against  it,  m  the  distance  column,  will  stand  the  difference  of  longitude. 

Thus,  under  the  course  three  points,  or  S.  E.  by  S.,  and  against  the  distance  300, 
stand  the  difference  of  latitude  249.4,  and  the  departure  166.7.  With  the  middle 
latitude  40°  2G',  or  40°,  as  a  course,  and  the  departure  166.7,  found  in  the  latitude 
colutmi,  we  find,  in  the  distance  colunni,  the  difference  of  longitude  218. 

CASE  IV. 

Both  latitudes  and  course  given,  to  find  the  departure,  distance,  and  difference  of  longitude. 

Suppose  a  ship  sailing  from  a  place  in  the  latitude  of  49°  57'  N.,  and  longitude  of 
30°  W.,  makes  a  course  good  of  S.  39°  W.,  and  then,  by  observation,  is  m  the  latitude 
of  47°  44'  N. ;  requu-ed  the  distance  run,  and  the  longitude  in. 

Latitude  from 49°  57'  N. 

Latitude  by  observation 47  44  N. 

60 
Difference  of  latitude 133 


Sum  of  latitudes 97°  41' 

Middle  latitude 48  51* 


BY   PROJECTION. 

Draw  the  meridian  ACD,  on  Avhich  set  off  AC  equal  to  the 
difference  of  latitude  133  miles ;  draw  CB  peqiendicular  to  AC  ; 
draw  the  line  AB,  making  an  angle  equal  to  the  course  39°,  with 
AC,  and  meeting  BC  in  B  ;  thi-ough  B  draw  BD,  makuig  an 
angle  equal  to  the  middle  latitude  48°  51',  with  the  line  BC,  and 
it  is  done ;  for  AB  will  be  the  distance  171.1  miles,  BO  the 
departure  107.7  miles,  and  BD  the  difference  of  longitude  163.7 
miles. 

BY   LOGARITHMS. 


To  find  the  departure. 

As  radius  45° 10.00000 

Is  to  the  difference  of  lat.  133. .    2.12385 
So  is  tangent  of  course  39° 9.90837 

To  the  departure  107.7 2.03222 

To  find  the  distance. 

As  cosine  of  the  cours'e  39° 9.89050 

Is  to  the  difference  of  lat.  133. .  2.12385 

So  is  radius  90° 10.00000 

To  the  distance  171.1 2.23335 

To  find  t'.ie  longitude  in. 

Longitude  sailed  li-om 30°  00'  W. 

Difference  of  longitude  104..     2  44  W. 

Longitude  in 32  44  W. 


To  find  the  difference  of  longitude  by  the 

departure. 
As  cosine  middle  lat.  48°  51'. . .    9.81825 

Is  to  the  departure  107.7 2.03222 

So  is  radius  90° 10.00000 

To  the  difference  of  long.'  1G3.7    2.21397 

The  difference  of  longitude  may  he 
found  without  the  dejiarture  by  Theorem 
IV.  Middle  Latitude  Sailing  ;  thus. 

As  cosine  middle  lat.  48°  51'.    .    9.81825 

Is  to  tangent  of  course  39° 3.90837 

So  is  the  difference  of  lat.  133. .    2.12385 

12.03222 
9.81825 


To  the  difl'erence  of  long.  163.7    2.21397 


BY   GUNTER. 

1st.   The  extent  from  radius  4.5°,  to  the  course  39°,  on  the  line  of  tangents,  will  i-each 
from  the  difference  of  latitufle  133,  to  the  departure  107.7,  on  the  line  of  numbers. 


*  The  rorroctioii  of  this  lalilude  in  the  tal)le  at  the  end  of  Case  V'H.  Is  I',  niAkiiijj  tiic  corrccled 
luiddlc  latitude  4o°  b'Z' 


MIDDLE   LATITUDE   SAILING.  73 

2dly.  The  extent  from  tlie  coniplenient  of  the  course  51°,  to  the  radius  90°,  on  the 
line  of  sines,  will  reach  from  the  difference  of  latitude  13-3,  to  the  distance  171.1,  on 
the  Ime  of  numbei"s. 

3dly.  The  extent  from  tlie  complement  of  the  middle  latitude  41°  09',  to  radius  90°, 
on  the  line  of  sines,  will  reach  from  tlie  dejjarture  107.7,  to  the  diftcrcucc  of  longitude 
1(33.7,  on  the  line  of  numbei-s. 

BY  INSPECTION. 

Find  the  course  among  the  ])oints  or^degrees  (in  Table  I.  or  II.,  as  in  Case  II.  Plane 
Saihng),  and  the  difference  of  latitude  in  its  cohnnn,  against  which  will  stand  the 
distance  and  departure  in  then-  cohmnis;  then  take  the  middle  latitude  as  a  course, 
and  find  the  departure  in  the  latitude  column,  against  which,  in  the  distance  column, 
will  stand  the  difference  of  longitude. 

Thus,  with  the  com-se  39°,  and  die  difference  of  latitude  133,  I  enter  Table  11.-, 
the  nearest  numbA-  in  the  table  is  132.9,  which  corresponds  to  the  distance  171,  and  to 
the  departure  107.G  miles. 

Then  Avith  the  middle  latitude  48°  51',  or  49°,  as  a  course,  I  enter  Table  II.,  and  seek 
for  the  departure  107.G,  in  the  latitude  column,  which  corresponds  to  the  distance  1G4, 
or  the  difference  of  longitude. 


CASE  V. 

Both  latitudes  and  distance  given,  to  find  the  course,  departure,  and  difference  of  longitude. 

Suppose  a  ship  sails  300  miles  north-westerly  from  a  place  m  the  latitude  of  37°  N., 
and  the  longitude  of  32°  16'  W.,  until  she  is  in  the  latitude  of  41°  N. ;  requu'ed  her 
course  and  longitude  in. 


Latitude  left 37*^ 

Latitude  in 41 


0'  N 37°  0'  N. 

0         41    0 


4    0 

60 


Sum 

Middle  latitude. 


78    0 
39    0* 


Difference  of  latitude  240 


BY   PROJECTION. 

Draw  the  meridian  ACD,  on  which  set  off  DC  equal  to  the 
difference  of  latitude  240  miles ;  di*aw  the  line  Ci?  perpendicular 
to  DC  ;  take  the  distance  300  in  your  compasses..^  and,  with  one 
foot  in  D,  as  a  centre,  sweep  an  arc  cutting  CB  in  P;  join  DB ; 
make  the  angle  CBA  equal  to  the  middle  latitude  39°,  ana  draw  BA 
cutting  DCA  in  A,  and  it  is  done  ;  for  BC  is  the  dejjarture  18C  miles, 
BA  the  difference  of  longitude  231.6  miles,  and  the  angle  BDC 
represents  the  anffle  of  the  ship's  course  with  the  meridian,  whiclt  is 
therefore  N.  36°  52'  W. 

BY   LOGARITHMS. 


To  find  the  course. 

As  the  distance  300 2.47712 

Is  to  radius  90° 10.00000 

So  is  difference  of  latitude  240.    2.38021 

To  cosine  coui-se  36°  52' 9.90309 


To  find  the  departure. 

As  radius  90° 10.00000 

Is  to  the  distance  300 2.47712 

So  is  sine  course  36°  52' 9.77812 

To  the  departure  180.0 2.25524 


To  find  the  differenct  of  longitude  by  the 

departure. 
As  cosine  middle  latitufle  89°. .    9.89050 

Is  to  the  departure  180.0 f  2.25.524 

So  is  radius  90° 10.00000 


To  difference  of  lonjj.  231.6. 


2.3G474 


To  find  the  longitude  in. 

Longitude  left 32°  16'  W. 

Difference  of  longitude 3   52  W. 


Longitude  in 36     8  W 


*  The  correction  of  lliis  latitude  in  the  table  at  the  end  of  Case  VH.  is  2',  making  the  corrected 
middle  latitude  3;)°  2'. 

t  This  h>sariihin,  by  tlie  preceding;  operation,  was  found  cijual  to  2.25521,  diilering  a  little  from  the 
logarithm  of  180.0. 

10 


74 


MIDDLE   LATITUDE   SAILINO. 


BY    GUNTER. 

1st.  The  extent  from  the  distance  300,  to  tlie  difference  of  latitude  240,  on  the  line 
of  numbers,  will  reach  from  radius  90°,  to  the  complement  of  the  course,  equal  to  53°  8' 
on  the  line  of  sines. 

2tlly.  Tlie  extent  from  radius  90°,  to  the  course  36°  52',  on  the  line  of  sines,  will 
reach  from  the  distance  300,  to  the  departure  180,  on  the  line  of  numbers. 

3dly.  The  extent  from  the  complement  of  the  middle  latitude  51°,  to  the  radius  90° 
on  the  line  of  sines,  will  reach  from  the  departure  180,  to  the  difference  of  longitude 
231.6,  on  the  Ime  of  numbers. 

BY   INSPECTION. 

Find  the  course  (as  in  Case  IV.  Plane  Sailing)  by  seeking  in  Table  II.  till  against  the 
distance  taken  in  its  colimin  is  found  the  difference  of  latitude  in  one  of  the  tbllowhig 
cohunns;  adjoining  to  it  will  stand  the  departure  ;  which  if  less  than  the  difference  of 
latitude,  the  course  is  to  be  found  at  the  top  of  the  table,  but  if  greater,  at  the  bottom ; 
then  take  the  middle  latitude  as  a  course,  and  find  the  departure  in  the  column  of 
diflerence  of  latitude,  against  which,  in  the  distance  column,  will  stand  the  difference 
of  longitude. 

Thus  the  distance  300,  and  the  difference  of  latitude  240,  are  found  to  correspond 
nearly  to  a  course  of  37°,  and  a  departure  of  180.5  ;  tlien,  taking  the  middle  latitude  39° 
as  a  course,  I  seek  the  dejjarture  180.5,  in  the  latitude  column,  corresponding  to  which, 
in  the  distance  column,  is  tlie  difference  of  longitude  232. 


CASE   VI. 

One  latitude,  course,  and  departure  given,  to  find  the  difference  of  latitude,  distance,  and 

difference  of  longitude. 

A  ship  in  the  latitude  of  50°  10'  S.,  and  longitude  of  30°  00'  E.,  sails  E.  S.  E.  until 
her  de})arture  is  160  miles;  required  her  distance  sailed,  aiid 
latitude  and  longitude  hi. 

BY   PROJECTION. 

Draw  the  meridian  ACD,  and  parallel  thereto,  at  a  distance 
equal  to  the  departure  160  miles,  draw  the  line  EB  ;  make 
the  angle  CAB  equal  to  the  coiu'se  6  points,  and  draw  AB 
meeting  EB  in  B ;  from  B  let  fall  upon  AD  die  perpendicular 
BC  ;  then  is  AC  the  difference  of  latitude  66.3  miles,  and  AB 
the  distance  sailed  173.2  miles ;  having  thus  obtained  the 
middle  latitude  50°  43',  make  the  angle  CBD  equal  diereto,  arid 
draw  BD  meeting  ACD  hi  D  ;  then  will  BD  be  the  difference 
of  longitude  252.7  miles. 

BY   LOGARITHMS. 


A 

^^^«^. 

E 

S 

Departure  ^^^ 

B 

6 

A 

D 


To  find  the  difference  of  latitude. 

As  radius  4  points 10.00000 

Is  to  the  departure  160 2.204 12 

So  is  cotangent  coui*se  6  points.    9.61722 

To  the  difference  of  lat.  66.3. . .     1.82134 

Latitude  left 50°  10'  S. 

Difference  of  latitude  66 1   06  S. 

Latitude  in 51    1 6  S. 

Sum  of  latitudes lOI   26 

Middle  latitude 50  43  * 


To  find  the  distance. 

As  sine  course  6  points 9.96562 

Is  to  the  departure  160 2.20412 

So  is  radius  8  points 10.00000 

To  the  distance  173.2 2.23850 

To  find  the  difference  of  longitude. 
As  cosine  middle  latitude  50°  43'    9.80151 

Is  to  the  departure  160 2.20412 

So  is  radius  90° 10.00000 

To  the  difference  of  long.  252.7    2.40261 


Longitude  left 30°  00'  E. 

Difference  of  longitude  253 4   13  E. 

Longitude  in 34   13  E. 


The  correclion  of  ihis  lalilucle  in  llif  lable  at  tlie  end  of  Case  VII.  is  insensible. 


MIDDLE   LATITUDE   SAILING. 


75 


BY    GUNTER. 
1st    The  extent  from  the  course  G  points,  to  the  radius  4  points,  on  the  line  marked 
TR,  will  reach  from  the  departure  100,  to  tlie  difference  of  latitude  Gb.3,  on  the  Imo 

of  numbei-s.  ^      .  .i      v  i     i  cjr» 

2div    The  extent  from  6  points,  to  the  radius,  or  8  points,  on  the  line  marked  bK, 

will  reach  from  the  departure  160,  to  the  distance  173.2,  on  the  line  of  numbers. 
3dlv.  The  extent  from  the  complement  of  the  middle  latitude  39°  1/',  to  the  radiua 

90°,  on  the  sines,  will  reach  from  the  departure  lGO,to  the  dilFerence  of  longitude  2o2.7, 


pi.iu  n' 


REFRA^CTIOy 


G5      - 


FUf.2 


\ 


•■•3> 


% 


:ji-- 


:::::;-  5^- 


.V. 


PAHALL  AX 


So  is  cosine  coui-se  50°  58   ....     y./!Ji)l» 
To  the  difference  of  lat.  135.4. .    2.131 62 

To  find  the  difference  of  longitude. 
As  cosine  middle  lat.  48°  23' . . .    9.82226 

Is  to  the  departure  167 2.22272 

So  is  radius 10.00000 

To  the  difference  of  long.  251.5    2.40046 


N. 

'able  I.  or  Table  II.  (as  in  Case  III. 
•esponding  to  which,  in  the  columns 
and  the  distance  and  difference  of 
le  as  a  course,  seek  the  departure  iu 
the  distance  column,  will  stand  the 

nts,  and  seek  for  the  dejiarture  160, 
numbers  give  the  distance  173,  and 

13',  or  (51°  nearly)  as  a  course,  and 
n,  opposite  to  which,  in  the  distance 
254  miles,  nearl3\ 


he  vimdian  given,  to  find  the  course^ 
ence  of  longitude. 

ide  of  2.5°  0'  W.,  sails  south-easterly 
n  be  167  miles ;  required  the  course 
in. 


167  miles,  and 

ABC ;  take  an 
es,  and  with  one 

in  A ;  join  AD  ; 
miles,  and  BAD 

latitude  in,  and 
3  middle  latitude, 
je  the  difference 

,iMS. 


tudelefl 49°30'N. 

erence  of  latitude  135  . . .  2  15  S. 

itude  in 47   15  N. 

n  of  the  latitudes 96   45 

Idle  latitude 48  23* 


Longitude  lefl 25°  00'  W 

Difference  of  longitude  252. .     4   12  E. 

Longitude  m '-^0  48  W. 


*  The  correction  of  this  lalilude  m  the  table  is  V,  nuiking  tlie  corrected  inidille  latitude  48°  2V 


74 


MIDDLE   LATITUDE   SAILING. 


BY   GUNTER. 

1st.  The  extent  from  the  distance  300,  to  tlie  difference  of  latitude  240,  on  the  line 
of  numbers,  will  reach  from  radius  90°,  to  the  complement  of  the  course,  equal  to  53°  8' 
on  the  ILue  of  sines. 

2dly.  The  extent  from  radius  90°,  to  the  course  36°  52',  on  the  line  of  sines,  will 
reach  from  the  distance  300,  to  the  departure  180,  on  the  line  of  numbers. 

3dly.  The  extent  from  the  complement  of  the  middle  latitude  51°,  to  the  radius  90° 
on  the  line  of  sines,  will  reach  from  the  departure  180,  to  the  diffei'ence  of  longitude 
231.6,  on  the  line  of  numbers. 

BY  ; 

Find  the  course  (as  in  Case  IV.  Plar 
distance  taken  in  its  column  is  found  t 
columns ;  adjoining  to  it  will  stand  the 
latitude,  the  course  is  to  be  found  at  th 
then  take  the  middle  latitude  as  a  coi 
difference  of  latitude,  against  which,  ii 
of  longitude. 

Thus  the  distance  300,  and  the  difl    .•  •.■         • 

nearly  to  a  course  of  37°,  and  a  departu  .     ■ 

as  a  course,  I  seek  the  departure  180.5     .      .     •*.',,  ^  '       .       -,  -. 
in  the  distance  column,  is  the  differenc  ';  ,-     ■- 


One  latitude,  course,  and  departure  givi 

differei 

A  ship  in  the  latitude  of  50°  10'  S., 
her  de})arture  is  160  miles;  required 
latitude  and  longitude  in. 

BY   PROJECTI 

Draw  the  meridian  ACD,  and  paral 
equal  to  the  dei)arture  160  miles,  dra 
the  angle  CAB  equal  to  the  coin-se  i 
meeting  EB  in  B ;  from  B  let  fall  upo 
BC  ;  then  is  AC  the  difference  of  latit 
the  distance  sailed  173.2  miles ;  ha^ 
middle  latitude  50°  43',  make  the  angle 
draw  BD  meeting  ACD  m  D  ;  then  wL 
of  longitude  252.7  miles. 

BY  L( 
To  find  the  difference  of  latitude. 

As  radius  4  points 10.000( 

Is  to  the  departure  160 2.204. 

So  is  cotangent  course  6  points.    9.617' 

To  the  difference  of  lat.  66.3. . .     1.82U 

Latitude  left 50°  10'  t 

Difference  of  latitude  Qij 1   06  i 

Latitude  in 51    16  S. 

Sum  of  latitudes 101   26 

Middle  latitude 50  43  * 


lis,  x^woiiiv^  lumuic  lailiuue  OU"  4^3'      I'.BULtI 

Is  to  the  departure  160 2.20412 

So  is  radius  90'-' 10.00000 


To  the  difference  of  lonu:.  252.7    2.40261 


Longitude  left 30°  OO'  E. 

Difference  of  longitude  253 4   13  E. 

Longitude  in 34   13  E. 


*  The  correclion  of  this  latitude  in  thf  table  at  the  end  of  Case  VII.  is  insciisiWe. 


MIDDLE   LATITUDE   SAILING. 


75 


BY    GUNTER. 

1st.  The  extent  from  the  course  6  points,  to  the  radius  4  points,  on  the  line  marked 
TR,  will  reach  from  the  departure  160,  to  the  diffci-ence  of  latitude  66.3,  on  the  line 
of  niimbei-s. 

2diy.  The  extent  from  6  points,  to  the  radius,  or  8  points,  on  the  line  marked  SR, 
will  reach  from  the  departure  100,  to  the  distance  173.2,  on  the  line  of  numbers. 

3dly.  The  extent  from  the  complement  of  tlie  middle  latitude  39°  17',  to  the  radius 
90°,  on  the  sines,  will  reach  from  the  depai'tui-e  100,  to  the  difference  of  longitude  252.7, 
on  the  Ime  of  numbere. 

BY   INSPECTION. 

Find  the  coui-se  among  the  points  or  degi-ees.  Table  I.  or  Table  II.  (as  in  Case  III. 
Plane  Sailing),  and  the  departure  in  its  column,  corresponding  to  which,  in  the  colunms 
of  distance  and  difference  of  latitude,  will  be  found  the  distance  and  difference  of 
latitude  respectively  ;  then  with  the  middle  latitude  as  a  course,  seek  the  departure  in 
the  column  of  latitude,  coiTCsponding  to  which,  in  the  distance  column,  will  stand  the 
difference  of  longitude. 

Thus,  I  enter  Table  I.,  above  E.  S.  E.,  or  0  points,  and  seek  for  the  departure  100, 
the  nearest  to  which  is  159.8;  the  corresponding  numbcre  give  the  distance  173,  and 
the  difference  of  latitude  06.2  miles. 

Enter  Table  II.  with  the  middle  latitude  50°  43',  or  (51°  nearly)  as  a  course,  and 
seek  for  the  dejiarture  100,  in  the  latitude  column,  opposite  to  wliich,  in  the  distance 
column,  will  be  found  the  difference  of  longitude  254  miles,  nearl3\ 


CASE  VII. 

One  latitude,  distance  sailed,  and  departure  from  the  meridian  given,  to  Jind  the  course^ 
difference  of  latitude,  and  difference  of  longitude. 

A  ship  in  the  latitude  of  49°  30'  N,,  and  longitude  of  25°  0'  W.,  sails  south-easterly 
215  miles,  until  her  departure  from  the  meridian  be  167  miles ;  required  the  course 
Bteered,  and  the  latitude  and  longitude  the  ship  is  in. 

BY   PROJECTION. 

Draw  the  line  BD  equal  to  the  departure  107  miles,  and 
peiiJendicidar  thereto  draw  the  meridian  line  ABC ;  take  an 
extent  equal  to  the  distance  215,  in  your  compasses,  and  with  one 
foot  in  D,  as  a  centre,  describe  an  arc  cutting  AB  in  A ;  join  AD ; 
then  will  AB  be  the  difference  of  latitude  135.4  miles,  and  BAD 
the  course,  S.  50°  58'  E.  Hence  we  have  the  latitude  in,  and 
middle  latitude;  make  the  angle  BDC  equal  to  the  middle  latitude, 
and  draw  DC  cutting  ABC  in  C ;  then  DC  will  be  the  difference 
of  longitude  251.5  miles. 

BY   LOGARITHMS. 
To  find  the  course. 

As  the  distance  215 2.-33244 

Is  to  the  radius  90° 10.00000 

So  is  the  departure  107 2.22272 


To  sine  course  50°  58' 9.89028 


To  find  the  difference  of  latitude. 

As  radius 10.00000 

Is  to  the  distance  215 2.33244 

So  is  cosine  coui-se  50°  58' 9.79918 

To  the  difference  of  lat.  135.4. .  "2.131(32 

To  find  the  difference  of  longitude. 
As  cosine  middle  lat.  48°  23' . . .    9.82220 

Is  to  the  departure  107 2.22272 

So  is  radius 10.00000 


Latitude  left 49°  30'  N. 

Difference  of  latitude  135  .. .     2   15  S. 


Latitude  in 47   15  N. 


Sum  of  the  latitudes 96   45 

Middle  latitude 48  23* 


To  the  difference  of  long.  251.5    2.40040 


Longitude  left 

25°  00'  W 

Difference  of  longitude  252. 

.     4  12  E. 

Longitude  in 

.  20  48  W 

*  The  correctiou  of  this  latitude  in  the  table  is  1',  making  the  corrected  middle  latitude  48°  £J/ 


76 


MIDDLE   LATITUDE   SAILIiNG. 


BY   GUNTER. 

1st.  The  extent  from  the  distance  2L5,  to  the  departure  167,  on  the  line  of  niunbei-s, 
will  reach  from  the  radius  90°,  to  the  course  50°  58'  on  the  line  of  sines. 

Sdly.  The  extent  from  radius  90°,  to  the  com})lement  of  the  course  39°  02',  on  the 
line  of  sines,  will  reach  from  the  distance  215,  to  the  difference  of  latitude  135.4,  on 
the  line  of  numbers. 

3dly.  The  extent  from  the  complement  of  the  middle  latitude  41°  37',  to  the 
radius  90°,  on  the  Une  of  sines,  will  reach  from  the  depaitm-e  107,  to  the  difference  of 
longitude  251.5,  on  the  line  of  numbers. 

BY  INSPECTION. 

As  in  Case  V.  Plane  Sailing,  find  the  course  by  seeking  iii  Table  II.  till  against  the 
distance,  in  its  column,  is  found  the  given  departure  in  one  of  the  following  colunnis, 
adjoinmg  to  which,  in  the  other  column,  will  be  the  difference  of  latitude,  which  if 
gi-eater  than  the  departure,  the  course  will  be  at  the  top,  but  if  less  the  course  will  be 
found  at  the  bottom.  Then  take  the  middle  latitude  as  a  course,  and  find  the  departure 
in  the  column  of  difference  of  latitude,  against  which,  m  the  distance  column,  will  be 
found  the  difference  of  longitude. 

Thus  the  distance  215,  and  the  depaiture  167,  are  found  nearly  to  con-es])ond  to  a 
course  of  51  degrees,  and  a  difference  of  latitude  of  135.3  ;  then  with  the  iiiiddle  latitude 
48°,  as  a  course,  I  enter  the  table,  and  seek  for  the  departure  167,  in  the  latitude 
column;  the  distance  con-espouduig  250  is  the  difference  of  longitude  nearly. 

In  all  the  preceding  examples,  we  have  used  the  middle  latitude,  without  any 
coiTection,  in  computing  the  difference  of  longitude ;  but  when  absolute  accin-acy  is 
required,  this  latitude  must  be  corrected.  We  have  given  in  the  following  table  the 
value  of  this  correction  in  the  most  common  cases.  It  requires  no  particular  explana- 
tion :  one  examj)le  will  serve  to  show  its  use.  Suppose,  therefore,  tae  two  latitudes 
to  be  40°  and  C0°.  Here  the  middle  latitude  is  50°,  and  the  difference  of  latitude  20° ; 
tlie  tabular  correction  corresponding  to  these  numbers  is  57' ;  adding  this  to  50°,  we 
get  tlie  cori-ected  middle  latitude  50°  57',  which  is  to  be  used  instead  of  50°,  when 
great  accuracy  is  required.  We  have  inserted  in  the  notes  at  the  bottom  of  the  pages, 
in  the  preceding  examples,  the  values  of  this  correction,  but  have  not  introduced  it 
into  the  calculations,  because  it  is  generally  unnecessary  on  account  of  its  smallness 

TABLE. 


Tliis  Table  contains  the  correction,  in  minutes,  to  be  added  to  the  Middle  Latitude  to 
obtain  the  corrected  Middle  Latitude. 

Mid. 
Lat. 

Difference  of  Latitude. 

Mid. 
Lat.. 

o 

15 
18 
21 

24 
30 
35 

40 
45 
50 

1° 

1 

0 
0 
0 

0 
0 
0 

0 
.0 
0 

0 
0 
0 

0 
0 
0 

0 
0 
0 

2° 

1 
1 

3° 
2 

1 
2 
2 

2 
2 
2 

2 
2 
3 

4° 
/ 
3 
3 
2 

2 
2 
2 

2 
2 
2 

3 
3 
3 

3 
3 

4 

4 
4 
5 

5° 
/ 

5 
4 
4 

3 
3 
3 

3 
3 
4 

4 

4 
4 

5 
5 
5 

6 
6 

7 

6° 

/ 

7 
6 
5 

5 
5 
4 

5 
5 
5 

6 
6 
6 

7 
7 
8 

8 

9 

10 

7° 
1 
9 

8 
7 

7 
6 
6 

6 
6 

7 

8 
8 
9 

9 
10 
11 

12 
13 
14 

8° 

/ 

12 
10 

9 

9 

8 
8 

8 
8 
9 

10 
11 
11 

12 
13 
14 

15 
16 

18 

9° 
/ 

15 
13 
12 

11 

10 
10 

10 
11 
11 

13 
14 
14 

15 

16 

18 

19 
21 
23 

10° 
/ 

18 
16 
15 

14 
13 
12 

13 
13 
14 

16 
17 

18 

19 

20 
22 

24 

26 
29 

12° 
/ 

26 
23 
21 

20 

18 
18 

18 
19 
20 

22 
24 
26 

27 
29 
32 

34 
38 
42 

14° 

1 

36 
32 
29 

27 
25 
24 

25 
26 

28 

31 
33 
35 

37 
40 
43 

47 
52 

58 

16° 

/ 

47 
41 
37 

35 
32 
32 

32 
34 
36 

40 
43 
46 

49 
52 
57 

62 
68 
76 

18° 
/ 

59 
52 
47 

44 
41 
40 

41 
43 
46 

51 

55 

58 

62 
67 
72 

79 

88 
98 

20° 

/ 

72 
64 

58 

54 

50 
49 

50 
53 

57 

63 
68 
72 

77 
83 
90 

99 
110 
124 

o 

15 
18 
21 

24 

30 
35 

40 
45 
50 

55 
53 
GO 

55 

58 
60 

62 
64 
66 

68 
70 
72 

62 
64 
66 

68 
70 
72 

This  Table  is  to  be  entered  at  the  top  with  the  difference  of  the  two  latitudes,  and  at 
the  side  with  the  middle  latitude  ;  under  the  former,  and  opposite  to  the  latter,  is  the 
correction,  in  minutes,  to  be  added  to  the   middle  latitude,  to  obtain  the   corrected 
middle  latitude. 

MIDDLE    LATITUDE   SAlLl^<Jr.  77 

QUESTIONS   FOR  EXERCISE. 

Question  I.  Required  the  bearing  and  distance  between  two  ]ilaces,  one  in  the 
latitude  of  37°  55'  N.,  and  longitude  of  54°  23'  W. ;  the  other  in  the  latitude  of  32°  38'  N., 
and  longitude  of  17°  5'  W. 

.^7iswer.    S.  80°  9'  E.,  and  N.  80°  9'  W.,  distance  1854  miles. 

Qiiest.  II.  Required  the  direct  course  and  distance,  from  a  place  in  the  latitude  ol 
36°  55'  S.,  and  longitude  of  20°  C  E.,  to  another  place  in  the  latitude  of  32°  38'  S.,  and 
longitude  of  8°  54'  W. 

^ns.    N.  79°  46'  W.,  distance  1447  miles. 

quest.  III.  A  ship  from  the  latitude  of  37°  30'  S.,  and  longitude  of  60°  E.,  sails 
N.  79°  56'  W.  202  miles ;  required  the  latitude  and  longitude  in. 

Jlns.    Latitude  36°  55'  S.,  longitude  55°  50'  E. 

quest.  IV.  A  ship  from  the  latitude  of  34°  35'  N.,  and  longitude  of  45°  16'  W.,  sails 
S.  83°  36'  E.,  101  miles ;  required  her  latitude  and  longitude. 

Ans.    Latitude  34°  24'  N.,  longitude  43°  14'  W. 

quest.  V.  A  ship  in  the  latitude  of  49°  57'  N.,  and  longitude  of  15°  16'  W.,  sails 
south-westerly  till  her  dejjarture  is  789  miles,  and  latitude  in  39°  20'  N. ;  requu-ed  the 
com'se,  distance,  and  longitude  in. 

Am.    Course  S.  51°  05'  W.,  distance  1014  miles,  longitude  in  33°  45'  W. 

quest.  VL  A  ship  in  the  latitude  of  42°  30'  N.,  and  longitude  58°  51'  W.,  sails 
S,  E.  by  S.  591  miles;  required  the  latitude  and  longitude  in. 

A71S.    Latitude  34°  19'  N.,  longitude  51°  52'  W. 

quest.  VII.  Suppose  a  ship  sailing  from  a  place  in  the  latitude  of  49°  57'  N.,  and 
longitude  of  30°  W.,  makes  a  course  good  of  S.  39°  W.,  and  then,  by  observation,  is  ii7 
the  latitude  of  4p°  31'  N. ;  required  the  distance  run,  and  longitude  m. 

Ans.    Distance  342.3,  longitude  35°  20'  W. 

quest.  VIIT.  A  ship  in  the  latitude  of  50°  10'  S.,  and  longitude  of  30°  00'  E.,  sails 
E.  S.  E.  until  her  departure  is  957  miles ;  required  her  distance  sailed,  and  latitude 
and  longitude  hi. 

Ans.    Distance  1036  miles,  latitude  56°  46'  S.,  longitude  56°  48'  E. 

quest.  IX.  A  ship  in  the  latitude  of  49°  30'  N.,  and  longitude  of  25°  00'  W.,  sails 
soutii-eastcrly  645  miles,  until  her de])arture  from  the  meridian  be  500  miles;  required 
the  course  steered,  and  the  latitude  and  longitude  the  shij)  is  in. 

Am.    Course  S.  50°  49'  E.,  latitude  42°  42'  N.,  longitude  12°  59'  W 


78 


MERCATOR'S    SAILING. 


The  calculations  by  Middle  Latitude  Sailing  are  sufficiently  exact  for  a  short  run, 
or  a  day's  work,  and  ai'e  to  be  preferred  in  all  cases  where  the  difference  of  latitude  is 
small  in  comparison  with  the  difference  of  longitude ;  but  this  method  is  liable  to 
great  errors  in  calculating  the  situations  of  places  differing  greatly  in  latitude  and 
longitude,  [)articularly  in  liigh  latitudes.  To  remedy  this  inconvenience,  a  chart  was 
invented  and  published  in  the  year  15G6,  by  Gerard  Mercator,  a  Flemish  geographer, 
in  which  all  the  meridians  are  parallel  to  each  other,  but  proportionally  lengthened  so 
as  to  conform  to  the  spherical  figure  of  the  earth.  The  i)rinciples  on  which  this  chart  is 
constructed  were  fii"st  explained  in  the  year  1599,  by  Edward  Wright,  an  Englishman, 
and  are  as  folloA\'s : — 

By  Theorem  II.  of  Parallel  Sailing,  the  distance  of  two  meridians  coiTespouding  to 
a  degree  or  mileof  longitude,  in  any  latitude,  is  to  the  length  of  a  corresponding  degree 
or  mile  of  the  meridian,  as  the  cosine  of  the  latitude  is  to  the  radius,  that  is  {hy  Art.  56, 
Geometry),  as  radius  is  to  the  secant  of  the  latitude.  Hence,  if  the  meridians  are 
supposed  to  be  parallel  to  each  other,  or  the  distance  of  the  meridians  to  remain  the 
same  i.n  every  latitude,  the  degree  or  mile  of  latitude  must  be  increased  in  proportion 
to  the  secant  of  the  latitude.  Therefore,  if  the  radius  be  supposed  to  be  equal  to  one 
mile,  the  length  of  the  first  mile  of  latitude  from  the  equator  will  be  represented  by 
the  secant  of  1' ;  the  second  mile;,  by  the  secant  of  2' ;  the  third  mile,  by  the  secant 
of  3',  &c.  Therefore  the  length  of  the  exitanded  arc  of  the  meridian  may  be  found 
by  a  continual  addition  of  secants,  to  every  degree  and  minute  of  the  quadrant,  as  in 
Table  III.,  I)y  means  of  which  the  chart  (called  Mercator's  Chart)  may  be  constructed, 
and  all  the  cases  of  Mercator's  Sailing  may  be  projected  and  calculated.  * 

In  using  this  table,  the  degrees  are  to  be  foiuid  at  the  top  or  the  bottom,  and  the 
miles  at  the  side ;  in  the  angle  of  meeting  will  be  the  length  of  the  corresponding 
expanded  arc,  usually  called  the  mendional  parts.  If  you  wish  to  find  the  arc  of  the 
expanded  meridian  intercepted  between  any  two  parallels,  or,  as  it  is  usually  called, 
the  meridional  difference  of  latitude,  you  must,  ivhen  both  places  are  on  the  same  side  of 
the  equator,  subtract  the  mendional  parts  of  the  least  latitude  from  the  meridional  parts 
of  the  greatest ;  the  remainder  ivill  be  the  mendional  difference  of  latitude  :  bxd  if  they  are 
on  differt-nt  sides  of  the  equator,  the  sum  of  the  mendional  parts  of  both  latitudes  will  be 
the  meridiomd  difference  of  latitude  required. 

EXAMPLE   I. 

Required  the  meridional  parts  corresponding  to  the  latitude  of  42°  34'. 

Look  in  the  bottom  or  toj)  of  the  table  for  42°,  and  in  the  right  or  left  hand  column, 
marked  (1\1 ),  fi)r  34' ;  under  the  former  and  opposite  the  latter  stand  2828,  the  meridional 
parts  corresponding  to  42°  34'.  . 

EXAMPLE   IL 

Required  the  meridional  difference  of  latitude  between  Cape  Cod,  in  the  latitude  ol 
42°  03'  N.,  and  the  island  of  St.  Mary,  in  the  latitude  of  3G°  59'  N. 

Cape  Cod's  latitude 42°  03'  N.      Meridional  parts  278G 

St.  Maiy's  latitude 3G°  59'  N.      Meridional  parts  2391 

RIeridional  difference  of  latitude 395 

*  The  inniiner  of  conslrucling  lliis  cliart  will  he  parllriilarly  explained  hereafter.  It  may  be  observed, 
that  Ihc  siiiajjer  the  subdivisions  of  the  arc  of  the  meridian  are,  the  greater  will  be  the  accuracy  of  the 
calculated  lcii!;th  of  the  expanded  arc  of  the  meridian.  To  be  perfectly  accurate,  the  arc  ouajht  to  be 
subdivided  into  the  smallest  quantities  possible.  Attention  was  paid  to  tliis  circumstance  in  calculatmg 
\\\e  abovc-nieiilioncd  table. 


MERCATOR'S   SAILING. 
EXAMPLE   in. 


79 


Required  tlie  meridional  difference  of  latitude  between  a  place  in  the  latitude  of 
35°  12'  N.,  and  the  Cape  of  Good  Hope,  in  the  latitude  of  34°  22'  S. 


Latitude 35°  12'  N. 

Cape  of  Good  Hope's  lat.  34°  22'  S. 


Meridional  parts  2259 
Rleridional  parts  2198 


Sum  is  meridional  difference  of  latitude 4457 


From  these  principles  it  follows,  that  in  sailing  upon  any  course,  </!.e<nte  or  ;)ro/)er 
difference  of  latitude  is  to  the  departure  as  the  meridional  difference  of  latitude  is  to  the 
difference  of  longitude.  Hence  if  MI  (in  the  figure  of  Case  I.  following)  be  the  proper 
difference  of  latitude,  lO  the  departure,  MO  the  distance,  the  angle  IMO  the  course, 
and  we  take  MT  equal  to  the  meridional  difference  of  latitude,  and  draw  TH  parallel  to 
10  to  cut  MO  contuuied  in  H,  the  line  TH  will  represent  the  difierence  of  longitude; 
for  {by.M.5-3,  Geonietiy)  Ml  :  lO  : :  MT  :  TH.  Now,  in  the  triangle  IMTH,  by 
making  IMT  radius,  we  have  MT  :  radius  : :  TH  :  tangent  TMH ;  that  is,the  meridional 
difference  of  latitude  is  to  radius,  as  the  difference  of  longitude  is  to  the  tangent  of  the 
course,  liy  making  MH  or  TH  radius,  we  siiall  have  other  analogies,  which,  being 
combined  with  those  in  Plane  Sailing,  furnish  tiie  solutions  of  the  various  cases  of 
Mercator's  Sailiuff  contained  in  the  following  table. 


MERCATOR'S   SAILING. 


Case. 

Given. 

Sought.                                                      Solutions. 

1 

Both  latitudes 
and  longitudes. 

Course. 
Distance. 

Departure. 

Having  liotli  lats.  the  mer.  diff.  lat.  is  found  by  Table  III. 
Mer.  dilf.  of  lat.  :  radms  :  :  d.ff.  of  long.  :  tangent  course. 
I  Radius  :  proper  diff.  of  latitude  :  :  secant  course  :  distance. 
(  Cosine  course  :  prop.  diff.  of  latitude  :  :  radius  :  distance, 
i  Radius  :  proper  diff.  of  lat.  : :  tangent  course  :  departure. 
j  Bier.  diff.  of  lat.  :  dill",  of  long.  :  :  prop.  diff.  of  lat.  :  depart. 

2 

Both  latitudes 
and  dejiaiture. 

Course. 
Distance. 

Diff.  of  long. 

Merid.  diff.  of  hit.  being  found  by  Table  111.,  we  liave 
Troper  dill",  of  lat.  :  radius  :  :  departure  :  tangent  course. 
(  Radius  :  proper  diff.  of  latitude  : :  secant  course  :  distance. 
j  Sine  course  ;  departure  :  :  radius  :  distance. 
\  Radius  :  nierid.  d:ff.  of  lat.  : :  tangent  course  :  diff.  of  long. 
(  Prop.  diff.  of  lat.  :  det)arture  : :  mer.  diff.  of  lat.  :  diff.  long. 

3 

One  latitude, 

course,  and 

dlstani:e. 

Departure. 
Diff.  of  latitude. 

Diff.  of  long 

Radius  :  distance  ::  sine  course  :  departure. 
Radius  :  dist.  ::  cosine  course  :  prop.  diff.  of  lat.    Hence  we 
have  the  other  latitude  and  mer.  diff.  of  lat.  by  Table  III. 
Radius  :  inerid.  d;ff.  of  lat.  : :  tangent  course  :  diff.  of  long. 

4 

Both  latitudes 
and  course. 

Distance. 
Departure. 

Diff.  of  long. 

Cosine  course  :  proper  diff.  of  latitude  :  :  radius  :  distance. 
Radius  :  (iroiier  ditf.  of  lat.  :  :  tangent  course  :  departure. 

Mcrid.  diff.  of  lat.  being  found  in  Table  III.,  we  have 
Radius  :  uierid.  diff.  of  lat.  : :  tangent  course  :  diff.  of  long. 

5 

Both  latitudes 
and  distance. 

Course. 
Departure. 
Diff.  of  long. 

Distance  :  radius  : :  proper  diff.  of  latitude  :  cosine  course. 

Radius  :  distance  :  :  sine  course  :  departure. 

Radius  ;  inerid.  diff.  of  lat.  :  :  tangent  course  :  diff.  of  long. 

6 

7 

One  latitude, 
coursii,  and 
departure. 

Diff.  of  latitude. 

Distance. 
Diff.  of  long. 

Radius  :  departure  : :  cotangent  course  :  proper  diff.  of  lat. 
Hence  we  have  the  other  latitude  and  mend.  diff.  of  lat. 

Sine  course  :  departure  : :  radius  :  distance. 
(  Radius  :  inerid.  ditf.  of  lat.  :  :  tangent  course  :  diff.  of  long. 
i  Prop.  diff.  of  lat.  :  departure  :  :  mer.  diff.  of  lat.  :  diff.  long. 

One  latitude, 

distance,  and 

de[iarture. 

Course. 
Diff.  of  latitude. 

Diff.  of  long. 

Distance  :  radius  :  :  departure  :  sine  course. 

Radius  :  distance  :  :  cosine  course  :  diff.  of  lat.     Hence  we 

obtain  the  other  latitude  and  inerid.  difference  of  latitude. 

I  Radius  :  nierid.  diff.  of  lat.  :  :  tangent  course  :  ditf.  of  hmg. 

j  Prop.  diff.  of  lat.  :  de|iarture  :  :  mer.  diff.  of  lat.  :  dilf.  long. 

CASE  I. 

The  latitudes  and  longitudes  of  two  places  given,  to  find  the  direct  course  and  distance 

between  them. 

Required  the  bearing  and  distance  from  Cape  Cod  light-ho'  se,  m  the  latitude  of 
42°  03'  N.,  and  longitude  70°  04'  W.,  to  the  island  of  St.  Mary,  one  of  the  Western 
^slnjids,  hi  the  latitude  of  30°  59'  N.,  and  longitude  of  25°  10'  W. 


so 


MERCATOR'S   SAILING. 


Cape  Cod's  latitude  42°    3'  N. 
St.  Mary's  latitude.  36  59  N. 

5     4 

GO 


Difference  of  lat.  . .  304  miles. 


Meridional  parts  . .  2786 
Meridional  parts  . .  2391 

Merid.  diff.  of  lat.     395 


Longitude  70°    4'  W 
25   10  W. 

44   54 

60 


Diiference  of  lonff.  2694  miles. 


BY  PROJECTION. 


CapeCbd' 


Draw  the  meridian  MT  equal  to  tlie  meridional  difference  of  latitude  395  iuiles;  set 
off  also  upon  it  MI  equal  to  the  proper  difference  of  latitude  304  miles  ;  perpendicular 
to  MT  draw  TH  and  lO  ;  make  TH  equal  to  the  difference  of  longitude  2694  miles; 
draw  MH  cutting  TO  in  O  ;  then  will  the  angle  TMH  be  the  course  S.  81°  40^  E.,  and 
OM  the  distance  2098  miles. 

BY   LOGARITHMS. 


To  find  the  course. 
As  the  meri  d.  diff.  of  latitude  395    2.59660 

Is  to  radius  45° 10.00000 

So  is  the  difference  of  long.  2694    3.43040 

To  tangent  of  course  81°  40'. . .  10.83380 


To  find  the  distance 

As  radius  90° 10.00000 

Is  to  the  proper  diff  of  lat.  304.    2.48287 
So  is  secant  of  course  81°  40'  . .  10.83884 

To  the  distance  2098  mUes 3.32171 


BY    GUNTER. 

1st.  Extend  from  the  meridional  difference  of  latitude  395,  to  the  difference  ol 
longitude  2694,  on  the  line  of  numbers ;  that  extent  will  reach  from  the  radius  or  45°, 
to  the  course  81°  40',  on  the  line  of  tangents. 

2dly.  Extend  from  the  complement  of  the  course  8°  20',  to  radius  90°,  on  the  line  of 
sines ;  that  extent  will  reach  from  the  proper  difference  of  latitude  304,  to  the  distance 
2098,  on  the  Ime  of  nmiibers. 

BY  INSPECTION. 

With  the  meridional  difference  of  latitude  and  difference  of  longitude  used  as 
difference  of  latitude  and  departiu'e,  find  the  course,  by  inspecting  the  tables  until 
those  numbers  are  found  to  correspond ;  witli  this  course  and  the  proper  difierence  of 
latitude,  find  the  corresponding  distance. 

Thus  one  tenth  of  the  meridional  difference  of  latitude  and  difference  of  longitude 
ai-e  found  to  agree  nearly  to  a  course  of  7^  pouits;  this  course  and  one  tenth  of  the 
proper  difference  of  latitude  30.4,  is  found  to  con-espond  to  the  distance  207 ;  this 
multiplied  by  10  gives  the  distance  2070,  differing  a  little  from  the  result  by  logarithms, 
owuig  to  the  neglect  of  a  few  muiutes  hi  the  coui'se. 


CASE  II. 

Both  latitudes  and  the  departure  given,  to  find  the  course,  distance,  and  difference  of 

longitude. 

A  ship  in  the  latitude  of  49°  57'  N.,  and  longitude  of  15°  16'  W.,  sails  south-westerly 
until  her  departure  is  197  miles,  and  then,  l)y  observation,  is  m  the  latitude  of  47°  18'  N. ; 
requu-ed  her  course,  distance,  and  longitude  hi. 

Latitude  left 49°  57'  N.  IMeridional  parts 3470 

Latitude  in 47    18  N.  Meridional  parts  . . . .  3229 

Difference  of  latitude 2  39  =  159  miles.         Merid.  diff.  of  latitude    241 


MERCATOR'S   SAILING. 


81 


BY   PROJECTION. 
VV^ith  the  proper  difference  of  latitude  and  departure,  project,  as  in  Case  VI.  Plane 
Sailing,  by  drawing  the  meridian  AEB,  on  -which  take  AE  ^ 

equal  to  the  proper  difference  of  latitude  159  miles ;  erect 
ED  perpendicular  to  AE,  and  make  it  equal  to  the  departure 
197  miies  ;  join  AD,  and  continue  it  towards  C  ;  make  AB 
equal  to  the  mei-idional  difference  of  latitude  241  miles,  and 
draw  BC  perpendicular  to  AB,  to  cut  AC  in  C,  and  it  is 
done.  For  AD  will  be  the  distance  253.2  miles,  BC  the 
difference  of  longitude  298.6  miles,  and  the  angle  BAC  will 
be  the  course  S.  51°  00'  W. 

BY   LOGARITHMS. 


Diff.  Lonff. 


To  find  the  course. 

As  the  proper  diff.  of  lat.  159  . .  2.20140 

Is  to  radius  45° 10.00000 

So  is  the  departure  197 2.29447 

To  tangent  course  51°  06' 10.09307 

To  find  the  difference  of  longitude. 

As  radius  45° 10.00000 

Is  to  merid.  diff.  of  latitude  241    2.38202 
So  is  tangent  course  51°  00'.. .  *  10.09307 

To  difference  of  longitude  298.6    2.47509 


To  find  the  distance. 

As  radius 10.00000 

Is  to  proper  diff.  of  latitude  159    2.20140 
So  is  secant  course  51°  00' 10.20207 


To  the  distance  253.2 2.40347 


Longitude  left 15°  10'  W. 

Difference  of  longitude 4   59  W. 

Longitude  in 20    15  W. 

The  difference  of  longitude  may  also 
be  found  by  saying,  As  proper  difference 
of  latitude  :  departure  : :  meridional  differ- 
ence of  latitude  :  difference  of  lonjjitude. 


BY   GUNTER. 

1st.  The  extent  from  the  difference  of  latitude  159,  to  the  departure  197,  on  the  line 
of  niunbers,  will  reach  from  radius  45°,  to  the  course  51°  00',  on  the  line  of  tangents. 

2dly.  The  extent  from  the  course  51°  00',  to  i-adius  90°,  on  the  sines,  will  reach  from 
the  departure  197,  to  the  distance  253.2,  on  the  line  of  numbers. 

3dly.  Tlie  extent  fi'om  the  radius  45°,  to  the  course  51°  00',  on  the  line  of  tangents, 
will  reach  from  the  meridional  diffei-ence  of  latitude  241,  to  the  difference  of  longitude 
298.6,  on  the  line  of  numbers. 

BY   INSPECTION. 

Find  the  course  by  Plane  Sailing,  Case  VI.,  by  seeking  in  the  tables  with  the  proper 
difference  of  latitude  and  departure  till  they  are  found  to  agi-ee  in  their  respective 
columns,  corresponding  to  which  will  be  the  distance  in  its  column,  and  the  course 
will  be  found  at  the  top  of  that  column  if  the  departure  is  less  than  the  proper  difference 
of  latitude,  otherwise  at  the  bottom;  with  the  same  course  find  the  meridional  differ- 
ence of  latitude  in  the  latitude  column,  corresponding  to  which,  in  the  depai"tm-e 
column,  will  be  the  true  difference  of  longitude. 

Thus  with  the  true  difference  of  latitude  and  dcpartiu'e  159,  and  197,  I  find  the 
course  51°,  and  the  distance  253  ;  in  the  same  table,  opposite  to  half  of  the  meridional 
difference  of  latitude  120.5,  1  find  the  departure  148.8,  which,  being  multiplied  by  2, 
gives  the  difference  of  longitude  298  miles,  nearly. 


CASE   IIL 

One  latitude,  course,  and  distance  given,  to  find  the  difference  of  latitude  and  dlfferenct 

of  longitude. 

A  ship  in  the  latitude  of  42°  30'  N.,  and  longitude  of  58°  51'  W.,  sails  S.  W.  by  S. 
300  mUes  ;  required  the  latitude  and  longitude  hi.  71 

BY  PROJECTION. 
Draw  the  meridian  ABC  and  ADE,  making  an  angle  with  it 
equal  to  the  course  3  points ;  make  AD  equal  to  the  distance  sailed 
300  miles,  and  from  D  let  fall  upon  AB  the  perpendicular  BD  ; 
then  will  BD  be  the  departure,  and  AB  the  difference  of  latitude 
249.4  miles.  Hence  we  have  both  latitudes,  and  the  meridional 
difference  of  latitude,  to  Avhicli  make  AC  equal,  and  draw  CE 
I)arallei  to  BD,  meetuag  ADE  in  E ;  then  will  CE  be  the  difference 
of  loHfritude  218.5  miles. 


m/f.  Long. 


'  This  logarithm,  by  the  preceding  operation,  was  found  equal  to  10.09307,  differing  a  little  from  the  log.  tang,  of  iSl'  06 
11 


82 


MERCATOR'S   SAILING. 


BY   LOGARITHMS. 


To  find  the  difFerence  of  latitude. 

As  radius  8  points 10.00000 

Is  to  the  distance  300 2.47712 

So  is  cosine  course  3  pomts. . . .    9.91985 

To  proper  difF.  of  latitude  249.4    2.39697 


To  find  the  difference  of  longitude 

As  radius  4  points 10.00000 

Is  to  the  merid.  diff.  of  lat.  327.    2.51455 
So  is  tangent  coui'se  3  points. . .    9.82489 

To  difference  of  longitude  218.5    2.33944 


Latitude  left..  42°30'N. 
Diff.  of  lat.  249    4  09  S. 


Latitude  in . 


38  21  N. 


Meridional  parts  2822 

Meridional  parts  2495 
Mer.  diff.  of  lat.    327 


Longitude  left  ..  58°51'W. 
Diff:  of  long.  219    3  39  W. 

Longitude  m 


62  30  W. 


BY   GUNTER. 

1st.  The  extent  from  radius  8  points,  to  the  complement  of  the  course  5  points,  on 
the  line  marked  SR,  will  reach  from  the  distance  300,  to  the  difference  of  latitude  249.4, 
on  the  line  of  numbers. 

2dly.  The  extent  from  the  radius  4  points,  to  the  course  3  points,  on  the  line  marked 
TR,  will  reach  from  the  meridional  difference  of  latitude  327,  to  the  difference  of 
longitude  218.5,  on  the  line  of  numbers. 

BY  INSPECTION. 

As  in  Case  I.  Plane  Sailing,  find  the  course  at  the  top  or  bottom  of  the  tables,  either' 
among  the  points  or  degi-ees,  and  in  that  page,  opposite  the  distance,  will  be  found  the 
difference  of  latitude  and  departure  in  their  respective  columns.  Then,  in  the  same 
table,  find  the  meridional  difference  of  latitude,  in  the  latitude  column ;  cori-esponding 
to  which,  in  the  departure  column,  will  be  the  difference  of  longitude. 

Thus,  under  the  course  S.  W.  by  S.  or  3  points,  and  opposite  the  distance  300, 
stands  the  difference  of  latitude  249.4.  Tlien  under  the  same  course  find  half  of  the 
meridional  difference  of  latitude  in  the  latitude  column,  against  which  stands  109 
nearly,  in  the  departure  column ;  which,  multipUed  by  two,  gives  218,  the  diffei-ence 
of  longitude,  nearly. 


CASE  IV. 

Both  latitudes  and  course  given,  to  find  the  distance  and  difference  of  longitude. 

A  ship  from  the  latitude  of  49°  57'  N.,  and  longitude  of  30°  W.,  sails  S.  39°  W.,  till 
she  an-ives  in  the  latitude  of  47°  44'  N. ;  requu-ed  the  distance  run,  and  longitude  in. 

Meridional  parts  3470 
Meridional  parts  3268 


Latitude  left  . . 
Latitude  in  ... 

49° 
47 

57' N. 
44  N. 

Diff.  of  latitude 

2 

13  =  133  miles. 

BY 

PROJECTION. 

IMer.  diff.  of  lat.    202  mUes. 


Draw  the  meridian  AEB,  on  which  take  AE  equal  to  the 
proper  difference  of  latitude  133  miles,  and  AB  equal  to  the 
meridional  difference  of  latitude  202  miles  ;  make  the  angle 
BAC  equal  to  the  course  39°,  and  draw  ED,  BC,  perpendicular 
to  AB,  cutting  ADC  in  D  and  C ;  then  will  AD  be  the  distance 
171.1  miles,  and  BC  the  difference  of  longitude  163.6  miles. 


BY 
To  find  the  distance. 

As  the  cosine  course  39° 9.89050 

Is  to  the  proper  diff.  of  lat.  133.    2.12385 
So  is  radius  90° 10.00000 


To  the  distance  171.1 2.23335 


LOGARITHMS. 

To  find  the  difference  of  longitude. 

As  radius  45° 10.00000 

Is  to  merid.  diff.  of  latitude  202 .    2.30535 
So  is  tangent  course  39° 9.90837 

To  the  difference  of  long.  163.6    2.21372 


Longitude  left 30°    0' W. 

Difference  of  longitude 2   44  W. 

Longitude  in 32   44  W 


MERCATOR'S  SAILING. 


83 


BY   GUNTER. 

1st.  The  extent  from  the  complement  of  the  course  51°,  to  the  radius  90°,  on  tlie 
sines,  Avill  reach  from  the  proper  difference  of  latitude  133,  to  the  distance  171.1,  on 
the  line  of  numbei-s. 

2(Ily.  The  extent  from  radius  45°,  to  the  course  39°,  on  the  line  of  tangents,  will 
reach  from  the  meridional  difference  of  latitude  202,  to  the  difference  of  longitude  163.6, 
on  the  line  of  numbers. 

BY   INSPECTION. 

As  in  Case  II.  Plane  Sailing,  find  the  course  among  the  pomts  or  degi'ees,  and  the 
proper  diffei-ence  of  latitude  m  its  column,  adjomuig  to  which  wUl  be  the  distance  and 
departure  m  then-  respective  columns ;  then,  in  the  same  table,  find  the  meridional 
diffei-ence  of  latitude  in  the  latitude  column,  adjoining  to  which,  in  the  departure 
column,  will  be  the  difference  of  longitude. 

Thus,  under  the  course  39°,  and  opposite  the  difference  of  latitude  133  (the  nearest 
to  which  is  132.9),  stand  the  distance  171,  and  the  departure  107.6 ;  in  the  same  table, 
opposite  the  meridional  difference  of  latitude  202,  found  in  the  latitude  colunm,  stands 
163.6,  in  the  departure  column,  which  is  the  difference  of  longitude,  as  before. 


CASE  V. 

Both  latitudes  and  distance  given,  to  find  the  course  and  difference  of  longitude. 

A  ship  from  the  latitude  of  37°  N.,  and  longitude  of  32°  16'  W.,  sails  300  miles 
north-westerly,  until  she  is  in  the  latitude  of  41°  N. ;  required  the  course  steered,  and 
longitude  in. 


Latitude  left  ... .  37°  N. 
Latitude  in 41°  N. 


Meridional  pai-ts.  2393 
Meridional  parts.  2702 


Difference  of  lat.    4°  z=  240  miles.    Merid.  diff.  of  lat.    309  miles. 


BY   PROJECTION. 

Draw  the  meridian  ABC ;  make  AB  equal  to  the  proper 
difference  of  latitude  240,  and  AC  equal  to  the  meridional  dif- 
ference of  latitude  309  mUes ;  draw  BD  and  CE  perpendicular 
to  ABC  ;  with  an  extent  equal  to  the  distance  300  in  your 
compasses,  and  one  foot  in  A,  as  a  centre,  describe  an  arc  cutting 
BD  m  D ;  draw  AD,  and  continue  it  to  cut  CE  in  E,  and  it  is 
done  ;  for  the  angle  BAD  is  equal  to  the  course  of  36°  52',  BD  is 
the  departure,  and  CE  is  the  difference  of  longitude  231.7  miles. 


7j  Dif-f.Lort^ 


To  find  the  course. 

As  the  distance  300 2.47712 

Is  to  radius  90° 10.00000 

So  is  proper  diff.  of  latitude  240    2.38021 

To  cosme  course  36°  52'  . . . 


BY  LOGARITHMS. 

To  fuid  the  difference  of  longitude. 

As  radius  45° 10.00000 

Is  to  the  merid.  diff.  of  lat.  309.    2.48996 
So  is  tangent  course  36°  52' . . .    9.87501 


9.90309 


To  the  difference  of  long.  231.7    2.36497 


Longitude  left 32°  16'  W. 

Difference  of  longitude  232  =        3  52  W. 

Longitude  in 36  08  W. 


BY    GUNTER. 

1st.  The  extent  from  the  distance  300,  to  the  proper  difference  of  latitude  240,  on 
the  line  of  numbers,  will  reach  from  the  radius,  or  90°,  to  53°  8',  the  complement  of 
the  course  on  the  line  of  sines. 

2dly.  The  extent  from  radius  45°,  to  the  course  36°  52',  on  the  line  of  tangents, 
will  reach  from  the  meridional  difference  of  latitude  309,  to  the  difference  of  longitude 
"231.7,  on  the  line  of  nunibers. 


84 


MERCATOR'S  SAILING. 


BY  INSPECTION. 

As  in  Case  IV.  Plane  Sailing,  seek  in  the  table  till  against  the  distance,  taken  in  its 
column,  is  found  the  given  difference  of  latitude  m  one  of  the  following  columns ; 
adjoining  to  it  will  stand  the  departui'e,  which  if  less  than  the  diffei-ence  of  latitude, 
die  course  will  be  found  at  the  top,  othenvise  at  the  bottom  ;  in  the  same  table  find 
the  meridional  difference  of  latitude  in  the  latitude  column,  adjoining  to  which  in  tlie 
departure  column  will  stand  the  difference  of  longitude. 

Thus  the  distance  300,  and  the  difference  of  latitude  240,  are  found  to  con-espond  to 
t  course  of  37°,  and  a  departure  of  180.5  ;  and  in  the  latitude  column,  opposite  half 
tlie  meridional  difference  of  latitude  154.5  (the  neai-est  to  which  is  154.1),  stands  116.2 
in  the  departure  column,  which  doubled  gives  tlie  difference  of  longitude  232.4. 


CASE  VI. 

One  latitude,  course,  and  depaiiure,  given,  to  find  the  distance,  difference  of  latitude,  and 

difference  of  longitude. 

A  ship  from  the  latitude  of  50°  10'  S.,  and  longitude  of  30°  E.,  sails  E.  S.  E.  until 
her  departmre  is  160  miles ;  requu-ed  the  distance  sailed,  and  the  latitude  and  longi- 
tude in. 

BY  PROJECTION. 

Draw  the  meridian  ABC,  and  at  a  distance  from  it  equal  to  the  departure  160  miles, 
draw  the  line  FD  parallel  to  AlBC  ;  make  the  angle 
BAD  equal  to  the  course  6  points ;  draw  AD  to  cut 
FD  m  D  ;  from  D  let  fall  upon  AB  the  pei-pendicular 
DB ;  then  will  AD  be  the  distance  173.2  miles,  AB 
the  difference  of  latitude  66.3  miles ;  hence  we  have 
both  latitudes,  and  the  meridional  difference  of  lati- 
tude 104  niiles ;  malve  the  Une  AC  equal  thereto,  .  -^H^^ritce^on^itudc 
and  draw  CE  perpendicular  to  AC,  meetuig  AD 
continued  in  E  ;  then  will  CE  be  the  difference  of  longitude  251.1  miles. 


BY  LOGARITHMS. 


To  find  the  distance. 

As  the  sine  course  6  points 9.96562 

Is  to  the  departure  160 2.20412 

So  is  radius  8  pouits 10.00000 

To  the  distance  173.2 2.23850 

To  find  the  difference  of  latitude. 

As  radius  4  points 10.00000 

Is  to  the  departure  160 2.20412 

So  is  cotangent  course  6  points.    9.61722 

To  proper  diff.  of  lat.  66.3  miles    1.82134 


Latitude  left  . .  50°  10'  S.  Mer.  parts  3490 
Diff.  of  latitude    1  06  S. 

Latitude  in. . .  51   16  S.  Mer.  parts  3594 

Merid.  difference  of  latitude    104 

To  find  the  difference  of  longitude. 

As  radius  4  points 10.00000 

Is  to  the  merid.  diff.  of  lat.  104.    2.01703 
So  is  tangent  course  6  points  . .  10.38278 

To  diff.  of  long.  251  =  4°  11'  E.    2.39981 
Longitude  left 30   00  E. 

Longitude  in 34    HE. 


BY  GUNTER. 

1st.  The  extent  from  the  course  6  pouits,  to  radius  8  pouits,  on  the  line  marked  S.  R. 
will  reach  from  the  departure  160,  to  the  distance  173.2,  on  the  line  of  numbers. 

2dly.  The  extent  from  radius  4  points,  to  the  complement  of  the  course  2  points,  on 
the  line  marked  T.  R.,  will  reach  from  the  departure  160,  to  the  difference  of  latitude 
66.3,  oil  the  Ime  of  numbers. 

3dly.  The  same  extent  (from  the  radius  4  pomts  to  the  course  6  points  on  the  line 
marked  T.  R.)  will  reach  from  the  meridional  difference  of  latitude  104,  to  the 
difterence  of  longitude  251,  on  the  line  of  numbers. 


BY   INSPECTION. 

As  m  Case  III.  Plane  Sailing,  find  the  coui-se  either  in  Table  I.  or  Table  II.,  and  the 
departure  in  its  column,  coiTCspondhig  to  which  will  stand  the  distance  and  difference 
of  latitude  in  then'  respective  columns ;  in  the  same  table  find  the  meridional  difference 


MERCATOR'S   SAILING.  85 

of  latitude,  in  the  latitude  column,  con-esponding  to  which,  in  the  departure  column, 
will  be  found  the  difference  of  longitude. 

Thus,  over  the  course  E.  S.  E.  or  G  points,  and  against  the  departure  160,  stands  the 
distance  173  miles,  and  the  difference  of  latitude  66.2  miles.  Again,  m  the  same  table, 
find  the  meridional  difference  of  latitude  104,  in  the  latitude  column,  opposite  to  which, 
in  the  departure  column,  stands  the  difference  of  longitude  251.3  miles. 


CASE  VII. 

One  latitude,  distance  sailed,  and  departure  given,  to  Jind  tlie  course,  difference  of  latitude, 
and  difference  of  longitude. 

A  ship  m  the  latitude  of  49°  30'  N.,  and  the  longitude  of  25°  W.,  sails  south-easterly 
215  miles,  making  1G7  miles  departure;  requu-ed  the  com-se  steered,  and  tlie  latitude 
and  longitude  in. 

BY   PROJECTION. 

Draw  the  meridian  ABC,  and  on  any  point  of  it  draw  BD  perpendicular  thereto, 
and  make  it  equal  to  the  departure  167  miles ;  with 
an  extent  equal  to  the  distance  215  miles  in  your 
compasses,  and  one  foot  on  D,  as  a  centre,  describe 
an  ai-c  to  cut  AB  in  A  ;  join  AD ;  then  Avill  AB  be 
the  pro]ier  difference  of  latitude  135.4  miles,  and  the 
angle  BAD  will  be  the  course  50°  58' ;  hence  we  have 
the  otlier  latitude,  and  the  meridional  difference  of 
latitude,  to  which  make  AC  equal,  and  draw  CE 
parallel  to  BD,  meeting  AD  produced  in  E  ;  then  will 
CE  be  the  difference  of  longitude  250.4  mUes. 


To  find  the  course. 

As  the  distance  215 2.33244 

Is  to  the  radius  90° 10.00000 

So  is  the  departure  167 2.22272 


To  siiie  of  course  50°  58' 9.89028 

To  find  the  difference  of  longitude. 

As  radius  45° 10.00000 

Is  to  merid.  diff.  of  latitude  203.    2.30750 
So  is  tangent  course  50°  58' 10.09111 


BY   LOGARITHMS. 

To  find  the  difference  of  latitude. 

As  radius  90° 10.00000 

Is  to  the  distance  215 2.33244 

So  is  cosine  course  50°  58' 9.79918 

To  difference  of  latitude  135.4  .    2.13162 


To  difference  of  longitude  250.4    2.39861 


Or  thus. 

As  proper  diff.  of  latitude  135.4  2.13162 

Is  to  dei)arture  167 2.22272 

So  is  merid.  diff.  of  latitude  203  2.30750 

4.53022 
2.13162 


Latitude  left  . .  49°30'N.  Mer.  parts  3428 
Diff.  of  lat.  135    2  15  S. 

Latitude  in  ...  47   15  N.  Mer.paits  3225 

Merid.  difference  of  latitude    203 


Longitude  left 25°  00'  W. 

Difference  of  longitude  250. .     4   10  E. 

Longitude  in 20   50  W 


To  difference  of  loneitude  250.4    2.39860 


BY   GUNTER. 

1st.  The  extent  from  the  distance  215,  to  the  departift-e  167,  on  the  line  of  numbers, 
Vt'D  reach  from  the  radius  90°,  to  the  course  50°  58',  on  the  line  of  sines. 

2(ily.  The  extent  from  radius  90°,  to  the  complement  of  the  course  39°  02',  on  the 
line  of  sines,  will  reach  from  the  distance  215,  to  the  difference  of  latitude  135.4,  on 
the  line  of  numbers. 

3dly.  The  extent  from  the  radius  45°,  to  the  course  50°  58',  on  the  line  of  tangents, 
will  reach  from  the  meridional  difference  of  latitude  203,  to  the  difference  of  longitude 
250.4,  on  the  line  of  numbers.  Or,  the  extent  fi-om  the  proper  difference  of  latitude 
135.4,  to  die  departure  167,  will  reach  from  the  meridional  difference  of  latitude  203, 
to  the  difference  of  longitude  250.4,  on  the  line  of  numbei*s. 


86 


MERCATOR'S   SAILING, 


BY  INSPECTION. 

Find  the  course  and  difference  of  latitude,  as  in  Case  V.  Plane  Sailing,  by  seeking 
in  Table  II.,  till  the  distance  and  departure  are  found  to  correspond  in  then-  respective 
columns,  adjoining  to  which,  in  the  column  of  latitude,  will  be  found  the  true  difference 
of  latitude,  which,  if  gi-eater  than  the  departure,  the  course  will  be  found  at  the  top  , 
but  if  less,  the  course  will  be  found  at  the  bottom :  with  this  course  seek  the  meridiona, 
difference  of  latitude  in  the  latitude  column,  adjoining  to  which,  in  the  departure 
colunui,  will  be  found  the  difference  of  longitude. 

Thus  the  distance  215,  and  the  departure  167,  are  found  to  correspond  to  a  course 
of  about  51°,  and  a  difference  of  latitude  1?>5.3.  Find  in  this  table  one  half  the 
meridional  difference  of  latitude  101.5,  opposite  to  which,  in  the  departm-e  column, 
stands  125.1 ;  this  doubled  gives  250.2,  for  the  difference  of  longitude,  nearly. 


Having  explained  the  method  of  calculatmg  suigle  courses  by  Middle  Latitude  and 
Mercator's  Sailmg,  it  now  remains  to  explain  the  method  of  calculatmg  compound 
courses.  To  do  this,  you  must  consti-uct  a  traverse  table,  and  find  the  difference  of 
latitude  and  departure  for  each  com-se  and  distance,  as  in  Traverse  Sailing,  and  from 
thence  the  whole  difference  of  latitude,  departure,  and  latitude  in ;  with  the  departure 
and  latitudes,  find  the  difference  of  longitude  and  longitude  in,  as  in  Case  II.  of  Middle 
Latitude  or  Mercator's  Sailing. 

This  method  is  exact  enough  for  working  any  single  day's  work  at  sea,  except  in 
high  latitudes,  where  it  will  be  a  little  erroneous ;  in  this  case  the  difference  of  longitude 
and  longitude  in,  may  be  calculated  for  every  single  com-se  and  short  distance ;  but  in 
general  this  nicety  in  calculation  may  be  neglected. 

To  Ulustrate  the  method  of  working  compound  courses,  we  shall  here  work  an 
example  by  Middle  Latitude  and  Mercator's  Sailing. 


EXAMPLE. 

A  ship  from  Cape  Henlopen,  in  the 
latitude  of  38°  47'  N.,  longitude  75° 
5'  W.,  sails  the  following  true  courses, 
viz.  E.  by  S.  20  miles,  E.  N.  E.  15 
miles,  S.  E.  26  miles,  south  16  miles, 
W.  S.  W.  6  mUes,  N.  W.  10  miles,  and 
east  30  miles;  requu-ed  her  latitude 
and  longitude. 

By  constructing  the  ti'averse  table 
with  these  courses  and  distances,  it 
appears  that  the  ship  has  made  27.8 
miles  of  southing,  and  69.3  miles 
of  easting  ;  and  by  subtracting  the 
southuig  from  the  latitude  of  Cape 
Henlopen,  there  remams  the  latitude 
in  38°  19'  N. 

Cape  Henlopen's  latitude  38°  47'  N. 
Latitude  in 38    19  N. 

Sum  of  latitudes 77      6 

Middle  latitude 38    33 


TRAVERSE 

TABLE. 

Dist 

Diff.  of  Lat. 

Departure. 

N. 

S. 

E. 

W. 

E.  by  S. 

20 

3.9 

19.6 

E.  N.  E. 

15 

5.7 

13.9 

S.  E. 

26 

18.4 

18.4 

South. 

16 

16.0 

W.  S.  W. 

6 

2.3 

5.5 

N.  W. 

10 

7.1 

7.1 

East. 

30 

30.0 

12.8 

40.6 

81.9 

12.6 

12.8 

12.6 

DifF. 

of  lat. . 

.27.8 

69.3  Dep. 

Meridional  parts  2528 
Meridional  pai-ta  2492 

36 


By  inspection  of  Table  II.  it  appears  that  the  difference  of  latitude  27.8,  and 
departure  69.3,  con-espond  to  a  course  of  68°  nearly,  and  a  distance  of  75  miles ;  and 
in  the  same  page  of  the  table,  opposite  to  the  meridional  difierence  of  latitude,  found 
in  the  column  of  latitude,  stands  the  difference  of  longitude  89  miles  in  the  departure 
column ;  this  being  subtracted  from  the  longitude  of  Cape  Henlopen,  75°  5'  W.,  leaves 
the  longitude  ui  73°  36'  W.,  by  Mercator's  Sailing.  Or,  with  the  middle  latitude 
38°  33'  to  39°,  as  a  course,  find  the  departure  69.3,  m  the  latitude  column,  opposite  to 
which  is  89  in  •  the  distance  column,  which  is  the  difference  of  longitude  by  ftliddle 
Latitude  Sailmg ;  consequently  the  longitude  in  is  73°  36'  W.,  as  above. 

Thus  we  see  that  such  examples  are  performed  as  m  Traverse  Sailing,  and  Case  II. 
of  Mercator's  or  Middle  Latitude  Sailing,  cither  by  inspection,  as  above,  or  by  the 
scale  of  logaiithms. 


MERCATOR'S   SAILING.  87 

QUESTIONS  FOR  EXERCISE. 

Question  l.  A  ship  in  tiie  latitude  of  49°  57'  N.,  and  longitude  of  15°  IC  W.,  sails 
south-westerly  until  her  depai'ture  is  789  miles,  and  then,  by  observation,  is  in  tlae 
latitude  of  39°  20'  N. ;  requu-ed  her  coui-se,  distance,  and  longitude  in. 

Answer.    Course  S.  51°  05'  W.,  distance  1014  miles,  longitude  in  33°  50'  W. 

Quest.  II.  A  ship  in  the  latitude  of  42°  30'  N.,  and  longitude  of  58°  51'  W.,  sails 
S.  W.  by  S.  591  miles ;  tlie  latitude,  and  longitude  in,  ai'e  required. 

Ans.     Latitude  in  34°  19'  N.,  longitude  in  65°  51'  W. 

Quest.  III.  A  ship  from  the  latitude  of  49°  57'  N.,  and  longitude  of  30°  00'  W., 
sails  S.  39°  W.  till  she  arrives  in  the  latitude  of  45°  31'  N. ;  requijred  the  distance  run, 
and  longitude  in. 

Ans.    Distance  342.3,  longitude  in  35°  21'  W. 

Quest.  IV.  A  ship  from  the  latitude  of  50°  10'  S.,  and  longitude  of  30°  00'  E.,  sails 
E.  S.  E.  until  her  departure  is  957  miles ;  required  the  distance  sailed,  and  the  latitude 
and  longitude  in. 

Jlns.    Distance  1036  miles,  latitude  in  56°  46'  S.,  longitude  in  56°  50'  E. 

Quest.  V.  A  ship  in  the  latitude  of  49°  30'  N.,  and  the  longitude  of  25°  00'  W., 
sails  south-easterly  645  miles,  makuig  500  miles  depaiture ;  requned  the  course  steered, 
and  the  latitude  and  longitude  m. 

Ans.     Com-se  S.  50°  49'  E.,  latitude  in  42°  42'  N.,  longitude  m  12°  57'  W. 


Having  gone  through  the  necessaiy  problems  in  Mercator's  Sailing,  we  shall  now 
show  how  Alercator's  Chart  may  be  constructed  by  means  of  the  Table  of  Meridional 
Parts. 

To  construct  a  3Iercator' s  Chart  to  commence  at  the  equator. 

Suppose  it  was  required  to  construct  the  Chart  in  the  Plate  prefixed  to  this  work, 
which  begins  at  the  equator,  and  reaches  to  the  parallel  of  50  degi-ees,  and  contains  95 
degi'ees  of  longitude  west  from  the  meridian  of  Greenwich. 

Draw  the  line  AD  representing  the  equator ;  then  take  from  any  scale  of  equal  ])arts 
the  number  of  mmutes  contained  in  95  degrees,  viz.  5700,  which  set  off  from  A  to  D  ; 
subdivide  this  line  into  95  equal  parts,  representing  degi'ees  of  longitude.  Through  A 
and  D  draw  the  lines  AB,  DC,  perpendicular  to  AD,  and  make  each  of  tliem  equal  to 
3474,  which  are  the  meridional  parts,  correspondi)ig  to  50  degrees.  Join  BC,  which 
must  be  subdivided  in  the  same  manner  as  the  line  AD  ;  and  through  the  correspond- 
ing points  of  the  lines  AD,  BC,  must  be  drawn  (at  the  distance  of  10°  or  20°)  the  lines 
parallel  to  AB,  representing  meridians  of  the  earth  ;  these  lines  must  be  numbered  0, 
10,  20,  &c.,  beginning  at  the  luie  AB,  which  represents  the  meridian  of  Greenwich. 
Set  off  in  like  manner  upon  the  meridians  AB,  DC  (beginning  from  the  equator  AD), 
the  meridional  parts  corresponding  to  each  degree  of  latitude  from  0°  to  50° ;  and 
tlirough  the  corresponding  points  (at  the  distance  of  10°  or  20°)  draw  lines  parallel  to 
the  equator  AD,  to  represent  the  parallels  of  latitude.  Then  the  upper  part  of  the 
chart  will  represent  the  nortli,  the  lower  the  south,  the  right  hand  the  east,  and  the  left 
hand  the  west  (which  is  generally  supposed  in  charts,  unless  the  contrary  is  expressly 
mentioned). 

If  the  chart  does  not  commence  at  the  equator,  but  is  to  serve  for  a  certain  portion 
of  the  globe  contained  between  two  parallels  of  latitude  on  the  same  side  of  the  equa- 
tor, you  must  draw  the  meridians  as  directed  in  the  last  example  ;  then  subtract  the 
meridional  parts  of  the  least  latitude  of  the  chart  from  the  meridional  parts  of  the  other 
latitudes,  and  set  off  these  differences  on  the  extreme  meridians  ;  draw  lines  through 
the  corresponding  pomts,  and  they  will  be  the  parallels  of  latitude  on  the  chart. 

If  the  chart  is  to  be  bounded  by  parallels  of  latitude  on  different  sides  of  the  equator, 
you  must  draw  a  line  representing  the  equator,  and  perpendicular  to  it  draw  the  lines 
to  represent  the  meridians,  continuing  tliem  on  both  sides  of  the  equator ;  then  set  off 
the  pfiTallels  of  latitude  on  both  sides  of  the  equator,  in  the  same  manner  as  in  the  first 
example. 

Take  from  the  Table  of  Latitudes  and  Longitudes  of  places  the  latitude  and  longitude 
of  each  particular  place  contained  within  the  bounds  of  the  chart,  and  lay  a  rule  over 
its  latitude,  and  another  crossing  that  over  its  longitude  ;  the  point  Avhere  these  meet 
will  represent  the  proposed  place  upon  the  chart.  The  most  remarkable  {)omt  of  a  sea- 
coast  being  thus  laid  down,  lines  may  be  drawn  from  point  to  point,  which  will  form 
the  outlines  of  the  sea-coast,  islands,  &c. ;  to  which  may  be  annexed  the  depths  of  water 
expressed  in  common  Arabian  numbers,  the  time  of  high  water  on  the  full  and  change 
days  expressed  m  Roman  numbei-s,  the  setting  of  the  tide  expressed  b}  an  arrow,  and 
whatever  else  may  be  thought  convenient  for  the  chart  to  coutaui. 


88  MERCATOR'S   SAILING. 

This  chart  is  not  to  be  considered  as  a  just  representation  of  the  earth's  surface,  for 
the  figures  of  islands  and  countries  are  distorted  towards  the  poles,  as  is  evident  from 
the  construction ;  but  the  degi'ees  of  latitude  and  longitude  are  increased  m  the  riame 
proportion,  so  that  the  bearmgs  between  places  will  be  the  same  on  the  chart  as  on  the 
globe ;  and  as  the  meridians  are  right  lines,  it  follows,  that  the  rhumbs,  which  forai 
equal  angles  with  the  meridians,  will  be  straight  lines,  which  render  this  projection  of 
the  eaith's  surface  much  more  easy  and  proper  for  the  mai'mer's  use  than  any  other. 

Having  tlie  latitude  and  longitude  of  a  ship  or  place,  to  find  the  coi'responding 

point  on  the  chart. 

Rule.  Lay  a  ruler  across  the  chart  in  the  given  parallel  of  latitude  ;  take  in  your 
compasses  the  nearest  distance  between  the  given  longitude  and  the  nearest  meridian 
drawn  acros's  the  chart ;  put  one  foot  of  the  compasses  m  the  point  of  intersection  of 
the  ruler  and  meridian,  and  extend  the  other  along  the  edge  of  the  ruler  on  the  same 
side  of  the  meridian  as  the  place  lies,  and  that  point  will  represent  the  pkce  of  the  ship. 

If  the  longitude  on  the  chart  be  counted  from  a  different  meridian  from  that  you 
reckon  from,  you  must  i-educe  the  given  longitude  to  the  longitude  of  the  chart,  by 
adding  or  subtracting  the  diffei'ence  of  longitude  of  those  meridians,  and  then  mark  off 
the  ship's  place,  as  before  du-ected.  Or  you  may  draw  a  meridian  line  through  the 
place  you  reckon  your  longitude  from  ;  then  measure  off  the  ship's  longitude  on  the 
equator,  and  apply  it  to  the  edge  of  the  ruler  from  this  meridian,  and  you  will  obtain 
the  ship's  place. 

To  find  the  hearing  of  any  place  from  the  ship. 

Rule.  Lay  a  ruler  across  the  given  place  and  the  place  of  the  ship  ;  set  one  foot 
of  the  compasses  in  the  centime  of  some  compass  near  the  ruler,  and  take  the  neai*est 
distance  to  the  edge  of  the  ruler ;  slide  one  foot  of  the  compasses  along  that  edge, 
keeping  the  other  extended  to  the  greatest  distance  from  the  rulei*,  and  observe  what 
point  of  the  compass  it  comes  nearest  to,  for  that  will  be  the  bearmg  required. 

To  find  the  distance  of  any  place  from  the  ship. 

Rule.  Take  the  distance  between  the  ship  and  the  given  place  in  your  compasses, 
and  apply  it  to  the  side  of  the  chart  or  gi-aduated  meridian,  setting  one  foot  as  much 
above  one  place  as  the  other  is  below  the  other  place ;  the  number  of  degrees  betweea 
the  points  of  the  compasses  will  be  the  distance  nearly. 

When  the  places  bear  north  and  south  of  each  other,  this  rule  is  accurate ;  but  when 
they  bear  nearly  east  and  west,  and  the  distance  is  lai-ge,  it  will  err  considerably ;  but 
in  general  it  is  exact  enough  for  common  purposes ;  if  gi-eater  accuracy  is  required,  it 
is  best  to  find  the  distance  by  calculation. 

If  any  one  wishes  to  estimate  tlie  distance  accurately  by  the  chart,  he  must  proceed 
in  the  following  manner : — 

1.  If  the  place  be  in  the  same  longitude  that  the  ship  is  hi,  then  the  preceding  rule 
is  accurate. 

2.  If  the  place  be  in  the  same  latitude  as  the  ship,  or  bear  east  or  west,  the  distance 
cannot  be  obtained  without  calculating  it  by  Case  I.  of  Parallel  Sailing. 

3.  If  the  place  be  neither  m  the  same  latitude,  nor  in  the  same  longitude  as  the  ship, 
the  distance  must  be  found  in  the  followmg  manner : — Lay  a  ruler  over  both  places,  and 
draw  through  one  of  them  a  parallel  to  the  equator ;  take  the  difference  of  latitude 
between  both  places  in  your  compasses  f^om  the  equator  ;  slide  one  foot  on  that  par- 
allel, keeping  the  other  extended  so  that  both  points  shall  be  on  the  same  meridian, 
and  note  the  point  of  the  ruler  which  is  touched  by  the  other  foot  of  the  compasses ; 
take  the  distance  from  this  point  to  the  given  place  through  which  the  parallel  was 
drawn,  and  apply  it  to  the  equator,  and  you  will  have  the  sought  distance. 

The  bearing  rn;!  distance  of  any  two  places  from  each  other  may  be  found  in  the 
same  manner  as  ilic  bearbig  and  distance  of  any  place  from  the  ship. 

EXAMPLE. 

Required  the  bearing  and  distance  between  the  east  end  of  Long  Island  and  the 
north  part  of  Bermudas. 

A  ruler  being  laid  over  both  places,  as  du'ected  in  the  precedmg  rule,  it  will  be 
found  to  lie  parallel  to  the  N.  W.  by  N.  and  S.  E.  by  S.  line  ;  and  the  distance  between 
the  two  places  being  taken  in  the  compasses,  and  applied  to  the  graduated  meridian, 
will  measure  about  10  degrees  or  600  miles  ;  therefore  these  places  bear  from  each 
other  N.  W.  by  N.  and  S.  E.  by  S.,  and  their  distance  is  GOO  miles,  nearly. 


89 


PROBLEMS   USEFUL    IN   NAVIGATION  AND 

SURVEYING. 


PROBLEM  I. 

Coasting  along  shore,  I  saio  a  cape  of  land  bearing  jY.  JV*.  E.,  and  after  sailing  W.  JV.  W 
yO  miles,  it  bore  JV.  E.  by  E.  ;  required  the  distance  of  the  ship  from  the  cape  at  both 
staiioiis.  .  C 

BY   PROJECTION. 

Describe  the  compass  ESW,  and  let  its  centre  A 
represent  the  place  of  the  ship  at  the  first  station ;  draw 
the  W.  N.  W.  line  AB  equal  to  20  miles,  and  B  will 
represent  the  second  station.  Draw  the  N.  N.  E.  line 
AC,  of  an  indefinite  length,  and  the  line  BC  parallel  to 
the  N.  E.  by  E.  Ime  of  the  compass ;  the  point  of 
intei'section  C  will  represent  the  place  of  the  cape ;  and 
the  distance  BC,  being  measm-ed,  will  be  found  36  miles  ; 
and  AC  30  miles. 

BY  LOGARITHMS,  (by  Case  L  of  OBLiquE  Trigonometry.) 

The  difference  between  N.  N.  E.  and  W.  N.  W.  is  8  points  or  90°,  therefore  BAG 
is  a  right  angle  ;  also  the  difference  between  the  N.  E.  by  E.  and  N.  N.  E.  is  3  points, 
equal  to  the  angle  ACB ;  and  the  difference  between  the  N.  E.  by  E.  pomt  and  the 
point  opposite  to  W.  N.  W.  is  5  pomts,  equal  to  the  angle  ABC. 


To  find  the  distance  BC. 

As  sine  angle  ACB  3  pts.  Ar.  Co.  0.25526 

Is  to  the  distance  AB  20 1.30103 

So  is  sme  angle  BAC  8  points. .  10.00000 

To  the  distance  BC  36.0 1.5.5629 


To  find  the  distance  AC. 
As  sine  ACB  3  points  . .  .Ar.  Co.  0.25526 


Is  to  the  distance  AB  20 

So  is  sme  angle  ABC  5  points  . 

To  the  distance  AC  29.93 1.47614 


1.30103 
9.91985 


The  above  solutions  are  by  Case  I.  Oblique  Trigonometiy,  though  they  might  have 
been  done,  in  this  example,  by  Case  II.  of  Right-angled  Trigonometry,  because  the 
angle  BAC  is  a  right  angle. 

If  the  bearings  of  the  middle  pomt  C  of  an  island  (or  any  remarkable  peak)  be 
dctennined  in  this  manner,  we  may,  at  tlie  same  time,  find  the  limit  of  the  dimensions 
of  the  island,  by  measuring  with  a  quadrant  or  sextant, 
held  in  a  horizontal  position,  the  angular  distances  between 
that  middle  point  and  the  extremes  of  the  island.  For  by 
drawuig  the  lines  ADE,  AGF,  making  the  angles  Dx\C, 
GAG,  with  AC,  equal  to  the  angular  distances  observed  at 
A,  and  in  the  same  manner  by  drawing  the  lines  BDG, 
BEF,  making  angles  with  BC  equal  to  the  angular  distances 
obsei-ved  at  B,  you  would  obtain  the  quadrilateral  figure 
DEFG,  within  which  the  island  is  to  be  placed.  If  similar 
observations  could  be  procured  at  H,  they  woukl  in  general 
take  off  the  comers  at  D  and  F  ;  and  observations  at  I 
would  generally  take  off  the  comers  at  E  and  G  ;  and  by 
observing  the  pi'ojecting  pomts  and  coves  in  the   island, 

while  sailuig  round  it,  and  drawing  a  figure  conformable  thereto,  within  the  limiting 
ejiace  thus  found,  the  form  and  dimensions  of  the  island  may  be  obtoinerl  to  a  consid- 
'irable  degi-ee  of  accuracy. 
12 


90 


PRUBLEMS   USEFUL   IN   NAVIGATION  AND  SURVEYING. 


PROBLEM  IL 

bting  at  sea,  ive  saia  two  headlands,  ivhose  bearing  from  one  another  by  the  chart  tvas 
W.  by  JV.,  and  E.  by  S.,  and  the  distance  15  miles ;  the  westernmost  bore  from  us 
S.  S.  W.,  and  the  easternmost  S.  E.  by  E. :  required  our  distance  from  each  of  them. 


line 


■---—  D 


...K 


BY   PROJECTION. 

Draw  the  compass  NESW,  and  through  the  centre  A,  di-aw  the  E 
AR,  the  S.  S.  W.  line  AB,  and  the  S.  E.  by  E. 
line  AC,  and  contmue  the  two  latter  indefi- 
nitely ;  upon  the  fonner,  AJR,  take  AD  equal 
to  15  miles ;  througli  D  draw  DC  parallel  to 
AB,  to  meet  AC  in  C,  and  draw  CB  parallel 
to  AD.  Then  A  will  be  the  place  where  the 
headlands  B  and  C  were  obsei-ved ;  and  the 
distance  AB  of  the  westernmost  headland, 
being  measured,  is  found  to  be  5.8  miles,  and 
the  distance  AC  of  the  easternmost  headland 
15  miles. 

BY  LOGARITHMS, 

Between  the  S.  S.  W.  line  AB,  and  the  S.  E.  by  E.  line  AC,  are  7  points  r=  angle 
BAC ;  and  between  the  S.  E,  by.E.  line  AC,  and  the  E.  by  S.  Ime  AD,  are  2  points  =: 
angle  CAD  mangle  ACB  (because  AD,  BC,  are  parallel);  therefore  ACB4-BAC=:9 
points  ;  and  smce  aU  three  angles  ACB,  BAC,  ABC,  are  equal  to  16  pomts,  the  angle 
ABC  is  also  equal  to  7  points  ;  therefore  (by  AH.  39,  Geometry)  the  sides  AC,  CB,  are 
equal,  beuig  opposite  to  the  equal  angles  ABC,  BAC.  If  these  angles  had  not  been 
equal,  the  side  AC  might  have  been  calculated  in  the  same  manner  as  we  shall  now 
calculate  the  side  AB, 

To  find  the  side  AB. 

As  sine  BAC  7  points Arith.  Comp.  0.00843 

Is  to  BC  15  miles 1.17609 

So  is  sme  ACB  2  points 9.58284 

To  AB  5.85 0.76736 


This  problem,  or  the  first,  may  be  used  for  finding  the  distance  of  a  ship  from  any 
headland,  &c.,  when  taking  a  departure  from  the  land. 


PROBLEM  IIL 

Tivo  ships  sail  from  the  same  port;  the  first  sails  JV*.  E.  h  E.  16  miles ;  the  second  sails 
easterly  20  miles,  and  then  finds  that  the  first  bears  JV.  JV*.  W.  :  required  the  course  of 
the  second  ship,  and  the  distance  between  the  two  ships. 

BY    PROJECTION. 

Draw  the  compass  ESW,  and  let  its  centre  A  represent  the  port  sailed  from ; 
draw  the  N.  E.  J  E.  line  AB  equal  to  16  miles  ;  also 
through  B,  the  line  BC,  parallel  to  the  N.  N.  W.  line, 
and  continue  it  indefinitely ;  take  a  distance  repre- 
senting 20  miles  in  your  compasses,  and  putting  one 
foot  in  A,  describe  with  the  other  an  arc  cutting  the 
line  BC  in  C,  and  join  AC.  Then  B  will  be  the 
place  of  the  first  sliip,  C  that  of  the  second,  and  AC 
the  course  steered  by  the  second  ship,  which  will 
be  nearly  E.  S.  E.  ^  E.,  and  BC  the  distance  of  the 
sliips  17i  miles. 

BY  LOGARITHMS. 

The  course  from  B  to  C  is  S.  S.  E,  (opposite  to  N.  N.  W.),  and  from  B  to  A  is 
S.  W.  h  W.  (opposite  to  N.  E.  i  E.) ;  the  difference  between  these  bearings  is 
6h  points,  equal  to  73°  7',  equal  to  the  angle  ABC ;  having  this  angle  and  tlie  sides 
AB,  AC,  we  may  find  the  other  angles  and  side  by  Cases  II.  and  III.  of  01>lique 
Trigonometry,  as  follows  • — 


PROBLEMS   USEFUL   IN   NAVIGATION  AND   SURVEYING. 


91 


To  find  the  angle  C. 

As  the  side  AC  20  miles 1.30103 

Is  to  sine  ABC  73°  7' 9.98087 

So  is  side  AB  16  miles 1.20412 

11.18499 
Subtract 1.30103 

TosineaneleC     49°  57' 9.88396 

For  N.  N.  W.,  add  22  30  

Sum  is  N.  72°  27'  W.,  the  bearing 
of  A  from  C ;  whence  the  course  of  the 
ship  from  A  towai-ds  C,  is  S.  72°  27'  E.,  or 
E.  S.  E.  i  E.,  nearly. 


To  find  the  distance  of  the  ships  BC. 

Add  the  angle  C  =  49°  57',  to  tlie  angle 
B  73°  7',  we  obtain  the  sum  123°  4' ; 
subtracting  this  from  180°,  leaves  the 
angle  CAB  56°  56'. 

As  sine  angle  ABC  73°  7',  Ai-.  Co.  0.01913 

Is  to  the  side  AC  20  miles 1.30103 

So  is  sme  CAB  56°  56' 9.92326 

To  the  side  BC  17.5  mUes 1.24;M2 


PROBLEM  IV. 

Two  ships  sail  from  the  same  port,  the  one  JV.  W.  30  miles,  and  the  other  JV*.  E.  by  JV. 
40  miles ;  required  the  bearing  and  distance  of  the  ships  from  each  other. 

_^.C 

BY  PROJECTION. 

Draw  the  compass  NESW,  and  let  its  centi-e  A 
represent  the  port  sailed  from ;  draw  the  N.  W.  line  AB 
equal  to  30  miles,  and  the  N.  E.  by  N.  line  AC  equal 
to  40  miles ;  jom  BC,  which  will  be  the  bearing  and 
distance  of  the  two  ships  ;  whence  the  bearing  will  be 
found  to  be  W.  S.  W.  h  W-,  and  the  distance  45.1  miles, 
neai-ly. 


BY  LOGARITHMS,  (by  Cases  IV.  V.  of  Oblique  Trigonometry.) 

Between  the  N.  W.  line  AB,  and  the  N.  E.  by  N.  line  AC,  there  are  7  pohits,  equal 
to  angle  BAC  ;  half  the  supplement  of  this  to  180°  is  50°  37^',  equal  to  half  sum  of  the 
angles  C  and  B. 


To  find  the  angfes  B,  C. 
As  sum  of  AB  and  AC  70. .  Ar.  Co.  Log.    8.154-90 

Is  to  llieir  difference  10 1.00000 

So  is  tangent  half  sum  angles  50°  37^'..  10.08583 

To  tangent  half  diff.  of  angles    9   5%  . .    9.24073 

Sum  is  angle  B 60  30 

Difference  is  angle  C 40   45 


To  find  the  distance  BC. 
As  sine  angle  B....  60°  30'Ar.Co.Log.  0.06030 

Is  to  side  AC 40         1.60206 

So  is  sine  angle  A..  78    45 9.'J9157 

To  the  distance  BC  45.1 1.65393 


To  the  angle  C,  equal  to  40°  45',  add  the  angle  representing  the  course  from  C  to  A, 
equal  to  33°  45',  the  sum  is  74°  30',  which  is  the  bearing  of  B  from  C,  namely, 
S.  74°  30'  W.,  or  W.  S.  W.  h  W.,  nearly. 


PROBLEM   V. 

Two  ports  bear  from  each  other  E.  by  JV.  and  W.  by  S.,  distance  400  miles :  a  ship  from 
the  easternmost  sails  northerly  450.7  miles ;  another  from  the  westernmost  sails  300  miles, 
and  meets  thefrst  :  required  the  course  steered  by  each  ship. 

C 

BY  PROJECTION. 

Draw  the  compass  ESW,  and  let  the  centre  B 
represent  the  westernmost  port ;  draw  the  E.  by  N. 
line  BD  equal  to  400  miles,  and  D  will  be  the  eastern- 
most port ;  with  300  in  your  compasses,  and  one  foot 
in  B,  describe  an  arc ;  witli  450.7  in  your  compasses, 
and  one  foot  in  D,  describe  another  arc,  cutting  the 
former  in  C  ;  join  DC,  BC.  Then  BC  will  be  the 
course  sailed  by  the  westernmost  ship,  and  DC  the 
course  sailed  by  the  easternmost  ship. 


c(]>' 


92 


PROBLEMS  USEFUL   IN   NAVIGATION   AND   SURVEYING. 


BY   LOGARITHMS. 

To  find  the  angle  CBD 

By  Theorem  IV.  Trigonometiy. 

Divide  the  triangle  BCD  into  two  right-angled 
triangles  by  means  of  the  perpendicular  CA,  and 
bisect  BD  in  a  ;  then 
As  the  base  BD  400 Ar.  Co.  Log. 


Is  to  the  sum  of  BC,  CD 750.7 

So  is  difference  of  BC,  CD...  150.7 


To  twice  Aa , 


282.8. 


7.39794 
2.87547 
2.17811 

2.45152 


Half,  orAa 141.4 

HalfBD  =  Ba= 200 

Difference  is  BA 58.6 

Then,  in  the  triangle  ACB, 

As  hypotenuse  DC  300 2.47712 

Is  to  radius  90° 10.00000 

So  is  AB  58.6 1.76790 

To  cosine  CBD  78°  44' 9.29078 


By  Theorem  V.  Trigonometry. 

CD  =  450.7 

BD  =  400    . .  Ar.  Co.  Log.  7.39794 

BC  =300    . .  Ar.  Co.  Loa-.  7.52288 


Sum 1150.7 

Half  sum 575.35 Log.  2.75993 

Half  sum  less  CD  124.65 Log.  2.09569 

Sum 2)19.77644 


Half  sum..  39°  22', 
2 


.Cosine  9.88822 


Doubled  is  78    44=  angle  CBD.    Having  found 
this  angle,  we  may  find  either  of  the  others,  thus  : 

To  find  the  angle  CDB. 

As  CD  450.7 Arith.  Comp.  7.34611 

Is  to  sine  CBD  78°  44' 9.99155 

So  is  BC  300 2.47712 


To  sine  CDB  40°  45' 9.81478 


As  the  angle  CBD  is  78°  44',  or  7  points  nearly,  and  the  course  fi-om  B  to  D  is  E. 
by  N.,  the  course  fi'om  B  to  C  must  be  north.  The  course  fi"om  D  to  B  being  W^.  by 
S.,  or  W.  11°  15'  S.,  and  the  angle  BDC  equal  to  40°  45',  the  bearing  of  C  fiom  D 
must  be  W  29°  30'  N.,  because  40°  45'  —  11°  15'  =  29°  30' 


PROBLEM  VL 

Coasting  along  shore,  ice  saio  two  headlands ;  the  first  lore  from  us  JV*.  £.,  the  secona 
E.  JV.  E. ;  after  sailing  E.  by  S.  10  miles,  the  first  hore  JV.  by  E.,  and  the  second 
JV.  E.  by  JV. :  required  the  hearing  of  the  two  headlands  from  each  other,  and  their 
distance. 

BY   PROJECTION. 

Draw  the  compass  NESW,  and  let  its  centre  A  represent  the  place  of  the  ship 
at  the  first  station ;  draw  the  E.  by  S.  line 
AB  equal  to  10  miles,  and  B  Avill  be  the 
place  of  the  ship  at  the  second  station; 
draw  the  N.  E.  line  AC,  and  the  E.  N.  E. 
line  AD  ;  through  the  point  B  di-aw  the 
lines  BC,  BD,  parallel  to  the  N.  by  E.  and 
N.  E.  by  N.  lines,  and  the  points  C  and  D, 
where  they  intersect  the  lines  drawn  from 
A  to  the  same  headlands,  will  be  the  points 
representing  them  respectively ;  jom  the 
points  C  and  D ;  then  will  CD  be  the 
distance  of  the  two  headlands,  and  a  line 
drawn  through  A  parallel  to  CD  will  repre- 
sent the  beai-ing  of  those  places  from  each 
other  on  the  compass.  ^ 

BY  LOGARITHMS. 

In  the  triangle  ABC,  we  have  all  the  angles  and  the  side  AB  to  find  BC  ;  for  the 
bearings  of  B  and  C  from  A  are  E.  by  S.,  and  N.  E.,  the  difference  being  5  points, 
equal  to  BAC ;  and  the  bearings  of  B  and  A  from  C  are  S.  by  W.,  and  S.  W.,  the 
difference  being  3  points,  equal  to  the  angle  ACB. 

To  find  the  side  BC. 

As  sine  of  ACB  3  points Arith.  Comp.  0.2552G 

Is  to  the  side  AB  10 1.00000 

So  is  sme  angle  BAC  5  points 9.91985 

To  BC  14.97 1.17511 

In  the  triangle  ABD,  we  have  all  the  angles  and  the  side  AB  to  find  BD  ;  fi)r  the 
bearings  of  B  and  A  from  D  are  S.  W.  by  S.j  aiid  ^Y.  S.  W.,  the  difference  being 


PROBLEMS   USEFUL   IN   NAVIGATION   AND   SURVEYING. 


93 


3  points,  equal  to  BDA ;  aiid  the  bearings  of  B  and  D  from  A  are  E.  by  S.,  and 
E.  N.  E.,  the  difference  being  also  3  points,  equal  to  the  angle  BAD ;  therefore  the 
angle  BAD  =  BDA,  and  (by  ^rt.  39,  Geometry)  BD  =  AB  =  10  miles.  If  these 
angles  had  not  been  equal,  we  might  have  calculated  the  side  BD  in  the  same  manner 
asBC. 

Now,  in  the  triangle  CBD,  we  have  BD  =  10,  BC  =  14.97,  and  the  angle 
CBD  =:  22"  30' ;  for  the  bearings  of  C  and  D  from  B  are  N.  by  E.,  and  N.  E.  by  N., 
differing  2  points,  or  22^  30' ;  hence  we  may  find  tlie  other  angles  and  side  CD  as  in 
Case  IV.  of  Oblique  Trigonometry. 

To  find  the  angles  BCD,  BDC.  To  find  the  distance  CD. 

As  sum  of  BC,  BD,  24.97,  Arilh.  Comp.    8.6025S  As  sine  angle  BCD  33°  44',  Arith.  Comp.  0.2534o 

Is  to  their  diflerence    4.97 0.69636  Is  to  side  BD  10 1.00000 

So  is  tang-,  half  sum  op.  angles  78°  4o'..  10.70134  So  is  sine  angle  CBD  22°  30' 9.58284 

To  tangent  half  diff.  of  angles  45     1  . .  10.00023        To  the  distance  CD  G.89 0.83829 

Sum  is  angle  BDC  =  123   46 

Diflerence  is  angle  BCD  =  33  ^l,  or  nearly 
3  points  ;  and  as  tiie  bearing  of  B  from  C  is 
S.  by  W.,  the  bearing  of  D  from  C  must  be 
S.S.E. 


PROBLEM   VII. 

Jlie  bearings  and  distances  of  three  points  of  land,  A,  B,  C,  being  given,  together  tvith  the 
horizontal  angles  ADC,  CDB,  measured  in  a  boat  placed  over  a  shoal  at  the  point  D ; 
required  the  bearing  and  distance  of  the  shoal  from  any  one  of  the  points  A,  B,  C. 


BY   PROJECTION. 

The  sum  of  the  two  angles  ADC,  CDB,  is  equal  to  the  angle  ADB.  Make  the 
angles  BAF,  ABF,  each  equal  to  the  complement  of  the  angle  ADB,  and  di"aw  the  lines 
AF,   BF,  which    will    intersect 

each  other  in  the  point  F.    Upon  js 

F,  as  a  centre,  with  the  radius 
FA,  equal  to  FB,  describe  the 
circle  AEBD.  Then  any  pomt 
D,  of  this  circumference  ADB, 
may  be  taken  as  the  vertex  of  a 
triangle,  whose  base  is  AB,  form- 
ing an  angle  ADB,  which  will 
satisfy  the  condition  of  being 
equal  to  the  sum  of  the  meas- 
ured angles  ADC,  CDB.  In  the 
same  marmer  we  may  find  the 
centre  G,  of  a  circle  BCD,  whose 
circumference  will  contain  the 
vertex  D,  of  the  triangle  DCB, 
foiTning  an  angle  at  the  vertex 
equal  to  the  measured  angle 
CDB.  The  point  of  intersection 
of  these  two  circles  is  the  place 
of  the  shoal  at  D;  whence  we  easily  obtain  the  distances  AD,  BD,  CD;  also  the 
bearings  of  the  shoal  from  the  points  A,  B,  C.  Continue  the  line  DC  to  meet  the 
ch-cle  ADB  in  E. 

BY   LOGARITHMS. 

We  have  the  bearings  and  distances  of  the  points  A,  B,  C,  given  from  the  map,  or 
by  previous  observations,  so  that  all  parts  of  the  triangle  ABC  are  known.  In  the 
ti-iangle  AEB,  we  have  the  angle  EAB  equal  to  the  observed  angle  CDB  {Art.  41. 
Geometry),  also  angle  EBA  equal  to  the  observed  angle  CDA ;  the  smn  of  these  tvvo 
angles  subtracted  from  180°,  leaves  the  remaining  an'gle  AEB  of  the  triangle  AEB ; 
hence  we  have  all  the  angles,  and  the  bass  AB  of  this  triangle,  to  find  AE,  by  Case  I.  of 
Oblique  Trin■onometr}^  In  the  triangle  AEC,  we  have  AE  by  the  preceding  calcula- 
tion, and  AC  from  the  map,  also  the  angle  EAC  =.  CAB  -f  EAB  =r  CAB  -f-  EDB ; 
«y>  that  we  have  the  two  sides  AE,  AC,  and  the  included  angle  EAC,  to  find  the  anglo 


94 


PROBLEMS   USEFUL   IN   NAVIGATION  AND   SURVEYING. 


ACE,  by  Case  IV.  of  Oblique  Trigonometiy ;  the  supplement  of  this  angle  is  the  value 
of  the  angle  ACD ;  adding  this  angle  to  the  observed  angle  CDA,  and  subti-acting  the 
sum  from  180°,  we  get  the  angle  CAD.  Then  in  the  triangle  CAD  we  have  all  the 
angles,  and  the  side  AC,  to  find  CD,  AD,  by  Case  I.  of  ObUque  Trigonometiy.  In  like 
manner  we  may  find  BD,  in  the  triangle  CBD. 


EXAMPLE. 


Suppose  we  have  given,  by  the  map,  AB  zn  3200  feet,  BC  =  1330  feet,  AC  — 1990 
feet,  angle  BAC=12°  34' ;  also,  by  observation,  CDB=EAB=:25°,  CDA=EBA=:28° ; 
requued  ACD,  CAD,  AD,  CD,  BD. 


To  find  AE,  in  the  triangle  AEB. 


EAB  =     25° 
EBA=     28  . 


.Sine  9.67161 


Sum 


63 
180 


AEB=    127 Arith.  Comp.  Sine  0.09765 

AB    =3200 Log-.  3.50515 


AE    =1881   Log.  3.27441 

To  find  AEC,  ACE,  in  the  triangle  AEC. 


BAG  =  12°34.' 
EAB  =25  00 

Sum 


37  34=EAC 
180  00 


AC+AE=3871 
AC— AE=  109 

AEC+ACE  =142  26 

i(AEC-(-ACE)=  71°13' Tangent  10.46839 

AC  -j-AE    =  3871... Ar.  Co.  Log.    6.41218 
•    AC  — AE    =    109 Log.    2.03743 

i(AEC— ACE)=    4'' 44' ....Tangent    8.91800 
|(AEC+ACE)=71    13  

SumAEC=75   67 
Difference  ACE=  66    29;  the  supplement  of  this 
angle  ACE  is  equal  to  ACD  =  113°  31' 
CDA=  28    GO 


Sum 


CAD: 


141    31 

180    00 

:   38    29 


To  find  AD,  CD,  in  the  triangle  ACD. 

As    CDA  =  28°. . .  .Arith.  Comp.  Sine  0.32839 

Is  to  AC    =  1990 Log.  3.29885 

So  is  ACD  =  113°  31' Sine  9.96234 

3887 Lo?.  3.58958 


To     AD 


As    CDA  =  28°  . . .  .Arith.  Comp.  Sine  0.32839 

Is  to  AC     =1990 Log.  3.29885 

So  is  CAD  =  38°  29' Sine  9.79399 


To    CD     =2638 Log.  3.42123 


To  find  BD,  in  the  ti-iang.o  BAD. 


BAC  =  12°  34' 
CAD  =  38    29 


CDB  =  25°  00' 

CDA  =  28    00 


Sum  BAD       51    03 


ADB  =  53°  00 


As    ADB  =  53°  ... .  Arith.  Comp.  Sine  0.09765 

Isto  AB    =3200 Log.  3.50515 

SoisBAD  =  51°  03' Sine  9.89081 


To    BD     =3116 Log-.  3.49361 


This  method  becomes  defective  when  the  points  F,  G,  approach  very  near  to  each 
other ;  to  avoid  this,  we  must  be  careful  not  to  take  for  the  place  of  observation  any 
point  which  approaches  near  to  the  cu-cumference  of  a  circle  which  passes  through 
the  observed  points  A,  C,  B  ;  because  a  very  small  error  in  the  observed  angles  might 
then  produce  a  very  great  eiTor  in  the  result,  or  place  of  the  observer.  Care  must  also 
be  taken  to  have  both  the  angles  observed  at  the  same  point,  without  allowing  tlae  boat 
to  drift,  m  which  the  observations  are  made. 


PROBLEM  Vin. 

Being  96  fathoms  from  the  bottom  of  a  tower,  I  found  Us  altitude  above  the  horizontal  line 
draivn  from  my  eye  was  15°  10' ;  required  the  elevation  above  that  line. 


BY   PROJECTION. 


Draw  the  horizontal  line  AB  equal  to 
96  fathoms,  and  peqiendicular  thereto,  the 
line  BC;  make  the  angle  BAC  equal  to 
15°  10',  and  draw  AC  to  cut  BC  in  C ; 
then  will  BC  be  the  height  of  the  tower, 
26  fathoms. 


PROBLEMS  USEFUL  IN  NAVIGATION  AND  SURVEYING. 


95 


BY   LOGARITHMS. 

As  radius  90° 10.00000 

Is  to  the  distance  AB  96  fathoms L98227 

So  is  tangent  angle  A  15°  10' 9.43308 

To  the  height  BC  2C.0  fathoms 1.41535 


When  an  object,  whose  elevation  above  the  horizon  is  to  be  detennined,  is  at  a  very 
great  distance,  it  will  be  necessary  to  notice  the  correction  arising  from  tiie  curvature 
of  the  earth  and  the  refraction,  and  apply  that  con-ection  to  the  height  estimated  by 
the  above  method.  Thus,  if  the  angular  elevation  of  a  mountain  whose  base  was 
more  distant  than  the  limit  of  the  visible  horizon,  was  observed  by  an  instrument  of 
reflexion,  the  approximate  lieight  must  first  be  obtained,  as  in  the  preceding  example, 
and  then  the  correction  of  that  approximate  height  for  the  curvature  of  the  earth, 
refraction,  and  dip,  must  be  calculated  by  the  following  rule,  and  added  to  that  height ; 
tlie  sum  will  be  the  true  height  above  the  level  of  the  sea. 

Rule.  Find  in  Table  X.  the  number  of  miles  coiresponding  to  the  height  of  the 
observer  above  the  level  of  the  sea,  and  take  the  difference  between  that  number  and 
the  distance  of  the  mountain  from  the  observer  in  statute  miles  ;  with  that  difference 
enter  the  same  table,  and  find  the  height  in  feet  corresponding,  which  will  be  the 
correction  to  be  added  to  the  approximate  height  to  obtain  the  true  height  of  the 
mountain  above  the  level  of  the  sea. 


Example.  Suppose  the  distance  was  32  statute  miles  (or  168960  feet),  and  the 
obsei-ved  altitude  1°  2',  the  observer  being  18  feet  above  the  level  of  the  sea ;  required 
the  height  of  the  mountain  above  the  same  level. 


As  radius Log.  10.00000 

Is  to  distance  168960 Log.    5.22779 

So  is  elevation  1°  2' Tang.    8.25616 


Approximate  height  3048 . 
Correction 398 


.Log.    3.4a395 


Distance  of  mountain. 32 

Table  X.  18  feet 5.61 

Difference 26.39 

Corresponding  Corr.  Table  X.  . . . .  398ft. 


Sum 3446  is  the  true  height  above  the  level  of  the  sea. 


PROBLEM  IX. 

I  observed  the  altitude  of  the  top  of  a  lower  above  the  level  sand  on  the  sea-shore  to  be  59°; 
then,  measuring  directly  from  it  98  yards,  its  elevation  loas  found  to  he  44°  :  requirea 
the  height  of  the  toiver. 


Let  AB  represent  the  height  of  the  tower,  C  the 
first  station,  and  D  the  second ;  then  we  have  the 
angle  ACB  equal  to  59°,  the  angle  ADB  equal  to 
44°,  the  angle  DAC  —  59°  —  44°  =  15°. 


To  fiind  the  side  AC. 

As  DAC  15° Sme  9.41300 

Is  to  DC  98 Log.  1.99123 

So  is  ADC  44° Sine  9.84177 

11.83300 
9.41300 


To  AC  263.0 Log.  2.42000 


To  find  the  height  AB. 

As  radius Log.  10.00000 

Is  to   AC  263.0* Log.    2.42000 

So  is  ACB  59° Sine    9.93307 

12.35307 
10.00000 


To  AB  225.5 Log.  2.35307 


*  The  log.  AC,  by  tlie  preceding  operation  waa  found  to  be  2.42000,  differing  but  a  little  from  the  log.  of  363. 


9« 


PROBLEMS   USEFUL   IN   NAVIGATION   AND   SURVEYING. 


PROBLEM  X. 

By  observation,  I  found  the  angle  of  elevation  of  a  monument,  at  one  station,  to  be  21'', 
and  the  horizontal  angle,  at  this  station,  betiveen  the  spire  of  the  monument  and  the 
second  station,  ivas  I'd'' ;  the  horizontal  angle,  at  the  second  station,  between  the  spire 
and  the  first  station,  ivas  69° ;  the  distance  between  the  two  stations  being  139  yards  . 
required  the  height  of  the  monument.  ji 


Let  AD  represent  the  monument,  C  the  first 
station,  B  tlie  second ;  then  the  vertical  angle  DCA 
is  21°;  and  the  horizontal  angles  BCD  equal  to 
79°,  CBD  equal  to  69°  ;  the  sum  of  these  two 
angles  being  subtracted  from  180°,  leaves  BDC 
equal  to  32°. 


To  find  the  side  CD. 

As  BDC  32° Sme  9.72421 

Is  to  BC   199 Log.  2.14.301 

So  is  CBD  69° Sine  9.97015 

12.11316 
9.72421 

To   CD  244.9 2.38895 


c 

To  find  the  height  AD. 

As  radius Log. 

Is  to  CD  244.9* Log. 

So  is  ACD  21° Tang. 

To  AD  94 Log. 


10.00000 
2.38895 
9.58418 

1.97313 


PROBLEM  XL 

Sailing  towards  the  land,  I  discovered  a  light-house  just  appearing  in  the  horizon,  my  eye 
being  elevated  20  feet  above  the  sea;  it  is  required  to  find  the  distance  of  the  light-house, 
supposing  it  to  be  elevated  200  feet  above  the  surface  of  the  sea. 

The  solution  of  tliis  problem  depends  on  the  uniform  curvature  of  the  sea,  by  means 
of  which  all  terrestrial  objects  disappear  at  certam.  distances  from  the  observer.  These 
distances  may  be  computed  by  means  of  Table  X.,  in  which  the  elevation  in  feet  is 
given  in  one  column,  and  the  distance  at  which  it  is  visible  is  expressed  in  statute 
miles  in  the  other  column.  If  the  place  from  which  you  view  the  object  be  elevated 
above  the  horizon,  you  must  add  together  the  distances  corresjponding  to  the  height  of  the 
observer  and  the  height  of  the  object ;  the  sum  will  be  the  gi'eatest  distance  at  which  that 
object  is  visible  from  the  observer;  this  process  being  similar  to  that  in  Problem  VIII. 

In  the  present  example,  the  height  of  the  observer  was  20  feet,  and  the  height  c.f  the 
oljjcct  200  feet. 

In  Table  X.  opposite  20  feet  is    5.92  miles. 
200  feet      18.71 

Distance 84.63  statute  miles,  of  about  69^  to  a  degi'ee ;  t!ie 

distance  in  nautical  leagues,  of  20  to  a  degree,  being  about  7. 


PROBLEM  XII. 

Jl  man,  being  on  the  main-top-gallant-mast  of  a  man-of-icar,  200  feet  above  the  tvater,  sees 
a  100  gun  ship  she  had  engaged  the  day  before,  hull  to ;  how  far  were  those  ships 
distant  from  each  other  ? 

A  ship  of  100  guns,  or  a  first-rate  man-of-war,  is  about  60  feet  from  the  keel  to  the 
rails,  from  which  deduct  about  20,  leaves  40  for  the  height  of  her  quarter-deck  above 
water.     Now,  a  ship  is  seen  hull  to  when  her  upper  works  just  appear. 
In  Table  X.  opposite  200  feet  stand  18.71 
40  feet  8.37 


Distance 27.08  miles. 


TliC  log.  of  CD,  by  the  preceding  operation,  was  found  to  be  2.38895,  differing  but  a  little  from  the  los.  of  0)1  !' 


PROBLEMS   USEFUL  IN   NAVIGATION   AND   SURVEYING.  97 

PROBLEM  XIII. 

Upon  seeing  the  Jlash  of  a  gun,  I  counted  30  seconds,  hy  a  watch,  before  I  heard  the 
report ;  how  far  ivas  that  gun  from  me,  supposing  that  sound  moves  at  the  rate  of  1142 
feet  per  second  ? 

The  velocity  of  light  is  so  gi'eat,  that  the  seeing  of  any  act  done,  even  at  the  distance 
of  a  number  of  miles,  is  instantaneous ;  but,  by  observation,  it  is  found  that  sound 
moves  at  the  rate  of  1142*  feet  per  second,  or  about  one  statute  mile  in  4.6  seconds  ; 
consequendy  the  number  of  seconds  elapsed  between  seehig  the  flash  and  hearing  the 
report  being  divided  by  4.6,  will  give  the  distance  in  statute  miles.  In  the  present 
example,  the  distance  was  about  Gh  miles,  because  30  divided  by  4.6  gives  Qh  nearly. 

PROBLEM  XIV. 

To  find  the  difference  between  the  true  and  apparent  directions  of  the  wind. 

Suppose  that  a  ship  moves  in  the  direction  CB  from  C  to  B,  while  the  A. 

wind  moves  in  its  true  direction  from  A  to  B ;  the  effect  on  the  shij)  Avill 
be  the  same  as  if  she  be  at  rest,  and  the  wind  blow  iu  the  direction 
AC  with  a  velocity  represented  by  AC  ;  the  velocity  of  the  ship  being 
represented  by  BC.  In  this  case,  the  angle  BAG  will  represent  the 
difference  between  the  true  and  the  apparent  directions  of  the  wind  ;  the 
apparent  beuig  more  ahead  than  the  true,  and  the  faster  the  vessel  goes, 
the  more  ahead  the  wind  will  appear  to  be.  We  must,  however,  excejjt 
the  case  where  the  wind  is  directly  aft,  in  which  case  the  dii-ection  is  not 
altered.  ^*=^^ 

It  is  owing  to  the  difference  between  die  true  and  apparent  directions  of  the  wind, 
that  it  appears  to  shift  its  direction  by  tacking  ship  ;  and  if  the  difference  of  the  direc- 
tions be  observed  when  on  different  boards  (the  wind  on  both  tacks  being  suj^posed 
to  remain  constant,  and  the  vessel  to  have  the  same  velocity  and  to  sail  at  tlie  same 
distance  from  the  wind),  the  half  difference  will  be  equal  to  tlie  angle  BAC.  By 
knowing  this,  together  with  the  velocity  of  the  ship  BC,  and  the  angle  BCA,  we  may 
obtain  the  true  velocity  of  the  wind  ;  or,  by  knowmg  the  velocity  of  the  wind  and  of 
the  ship,  and  the  apparent  direction  of  the  wind,  we  may  calculate  the  difference 
between  the  true  and  the  apparent  directions  of  the  wind. 

Thus,  if  the  velocity  of  a  ship  represented  by  BC  be  7  miles  per  hour,  that  of  the 
wind  represented  by  AB  27  miles  per  hour,  and  the  angle  of  the  vessel's  course  with 
die  apparent  direction  of  the  wind  BCA  equal  to  7^  points ;  the  difterence  between 
the  true  and  apparent  directions  of  the  wind  will  be  obtained  by  drawing  the  line 
BC  equal  to  7  miles,  taken  from  any  scale  of  equal  parts,  and  making  the  angle  BCA 
equal  to  7h  points ;  then,  with  an  extent  equal  to  27  miles,  taken  from  the  scale,  and 
Avirh  one  foot  in  B,  describe  an  arc  to  cut  the  line  AC  in  A ;  join  AB  ;  then  the  angle 
BAC,  being  measured,  will  be  the  required  difference  between  tlie  true  and  apparent 
directions  of  the  wind. 

BY   LOGARITHMS. 

As  AB  27  miles Arith.  Comp.  Log.  8.56864 

Is  to  BCA  7h  points Log.  Sine  9.99790 

So  is  BC   7  miles Log.  0.84510 

To   BAC    14=  57' Log.  Sine  9.41164 

So  diat,  in  diis  case,  the  difference  between  the  true  and  apjjarent  directions  of  the 
wind  is  about  li  points ;  and,  by  tacking  ship  and  sailing  on  the  other  board,  as  above 
mentioned,  the  wmd  will  appear  to  cliange  its  dnections  above  2^  points. 

PROBLEM    XV. 

To  measure  the  height  of  a  mountain  by  means  of  the  heights  of  two  barometers,  taken  at 
the  top  and  bottom  of  the  mountain. 

Procure  two  barometers,  with  a  thermometer  attached  to  each  of  them,  in  order  to 
ascertain  the  temperature  of  the  mercury  in  the  barometers,  and  two  other  thermome- 
ters, of  the  same  kind,  to  ascertain  the  temperature  of  the  air.  Then  one  observer  at 
the  top  of  the  momitaiu,  and  another  at  the  bottom,  must  observe,  at  the  same  time, 

*  The  velocity  of  sound  at  2/2"  Fahrenheit  is  1090  feet  per  second,  and  for  cacli  additional  degree  of  li'-at  add 
0.9G  to  ttiis  velocity. 

13 


98 


PROBLEMS   USEFUL  IN   NAVIGATION   AND   SURVEYING. 


the  heig^ats  of  the  bai'ometers,  and  the  theiinometers  attached  thereto,  and  the  heights 
of  the  detached  thermometers,  placed  in  the  open  au-,  but  sheltered  from  tie  sun. 
Having  taken  these  obsei-vations,  the  height  of  the  upper  observer,  above  the  lower, 
may  be  determined  by  the  foUowuig  rule,  which  is  adapted  to  a  scale  of  English  inches 
and  to  Fahrenheit's  thermometer  : — 

Rule.  Take  the  difference  of  the  logarithms  of  the  observed  heights  of  the  barom- 
eters at  the  two  stations,  considermg  the  first  four  figures,  exclusive  of  the  index,  as 
whole  numbers,  the  remainder  as  decimals ;  to  this  difference  must  be  applied  the 
product  of  the  decimal  0.454,  by  the  diffei-ence  of  the  altitudes  of  the  two  attached 
thermometers,  by  subtracts  ,  if  the  thermometer  be  highest  at  the  lowest  station, 
otlierwise  adding :  the  sun  n-  difference  will  be  the  approximate  height  in  English 
fatlioms.  Multiply  this  b;  he  decimal  0.00244,  and  by  the  difference  between  the 
mean  of  the  two  altitudes  the  detached  thermometers  and  32° ;  the  product  will  be  a 
correction,  to  be  added  to  .le  approximate  height  when  the  mean  altitude  of  the  two 
detached  thermometers  exceeds  32°,  otherwise  subtracted  :  the  sum  or  difference  will 
be  the  true  height  of  tlie  upper  above  the  lower  observer  in  EngUsh  fathoms,  which, 
being  multiplied  by  6,  will  be  the  height  in  feet. 


EXAMPLE. 

Suppose  the  following  observations  were  taken  at  the  top  and  at  the  bottom  of  a 
•nountain  ;  required  its  height  in  fathoms. 


Attached  Thermometer. 

Jbs.  at  lower  station 57° 

upj)er  station 43 

Difference 14 


Detached 
Thermometer. 


.56 
.42 


Mean. 49 

32 

Difference ...  17 


Barometer. 

29.68  inches Log.  14724.6 

25.28 Log.  14027.8 

Difference 696.8 

0.454  X  14 ..6.4 

Approximate  height 690.4 

690.4  X  17  X  0.00244 28.6 


Height  in  fathoms 


.719.0 


99 


MENSURATION. 


PROBLE3I   I. 

To  find  the  area  of  a  parallelogram. 

Rule.     Multiply  the  base  by  the  peqiendiculai*  height ;  the  product  will  be  the  area. 

JVote.  If  both  dimensions  are  given  in  feet,  inches,  &c.,  the  product  will  be  the 
nrea,  expressed  in  square  feet,  scpiarc  inches,  &c.,  respectively.  If  one  of  tlie  dimen- 
sions be  given  in  feet  and  the  other  in  inches,  the  product,  divided  by  12,  will  be  the 
answer  in  squm-e  feet.  If  both  dimensions  are  given  in  inches,  the  proiluct  will  be 
square  inches,  which,  being  divided  by  144,  will  be  the  answer  in  square  feet.  The 
same  is  to  be  miderstood  in  finding  the  area  of  other  surfaces. 

Example  I.  Suppose  the  base  BC  of  the  rectangular  parallelogi-am  /i 
ABCD  is  7  feet,  and  the  perpendicular  AB  3  feet ;  required  the  area.         [ 

The  product  of  the  base  7  feet  by  the  perpendicular  3  feet  gives  the  L 
area  21  square  feet. 

Example  II.  Suppose  ABCD  is  a  board  whose  length  BC  is  22  feet,  and  breadth 
AB  is  14  inches  ;  requii'ed  the  niunber  of  square  feet. 

The  product  of  the  base  22  feet  by  the  l)readth  14  inches  is  308 ;  this,  divided  by  12, 
gives  25}  square  feet,  the  sought  area. 

Example  III.  If  BC  be  25  inches,  and  AB  20  mches,  requu-ed  the  area  in  square 
feet. 

The  product  of  the  base  25  inches  by  the  perpendicular  20  inches  gives  500,  which, 
divided  by  144,  gives  the  area  3.47  or  3^^^^^^  squai-e  feet. 

Example  lA''.     Given  the  base  AD  of  the  oblique  angular  J?^ 

parallelogram  ABCD,  equal  to  30  feet,  and  the  perpendiculai'  /  \ 

height  BE  15  feet ;  required  the  area  of  the  parallelogram.  /    \ 

Alultiply  the  base  30  feet  by  the  perpendicular  15  feet ;        /. L 

the  pi'oduct  450  is  the  area  in  square  feet.  -^       -^ 


PROBLEM    II. 

To  find  the  area  of  a  triangle. 

Rule.     Multiply  the  base  by  half  the  pei-pendicular  height,  and  the  product  will  be 
the  area  required. 

Example.  Given  the  base  AC  30  feet,  and  the  peq^endicular 
BD  20  feet ;  required  the  area  of  the  triangle. 

The  base  30  multiplied  by  half  the  perpendicular  10  gives 
the  area  300  squai'e  feet. 


Rule. 


PROBLE3I    III. 

To  find  the  area  of  any  regular  right-lined  figure. 
Reduce  the  figure  to  triangles,  by  drawing  diagonals  therein  ;  then  find  the 


area  of  each  triangle,  and  the  sum  of  tliem  will  be  the  area  of  the  proposetl  figure.  O., 
instead  of  finding  the  area  of  each  triangle  sej)aratelv,  you  may  find,  at  one  operation, 
the  area  of  two  triangles,  having  the  same  diagonal,  by  nndtiplying  the  diagonal  by 
half  the  sum  of  the  perpendiculars  let  fall  thereon. 


100 


MENSURATION. 


Example.    Required  the  area  of  the  figure  ABCDE,  in  which  CE 
BE  =  22  feet,  and  the  perpendicular  AF  z=  13  feet,  BG  =  14 
feet,  and  DH  r=  12  feet. 

The  diagonal  BE,  22  feet,  multiplied  by  half  the  pei-pen- 
dicular  AF,  6.5  feet,  gives  the  area  of  the  triangle  ABE,  143 
squai-e  feet ;  and  the  diagonal  CE,  33  feet,  multiplied  by  half 
the  sum  of  the  perpendiculars  BG,  DH,  13  feet,  gives  the  ai-ea 
of  the  figure  BCDE,  429  feet;  this,  added  to  the  triangle 
ABE,  143  feet,  gives  the  whole  area  572  square  feet. 


A'K--- 4.  -^ 


PROBLEM  IV. 

To  find  the  area  of  a  circle. 

Rule.  Multiply  the  square  of  the  diameter  of  the  circle  by  the  quantity  0.7854,  and 
you  will  have  the  souglit  area. 

A^oie.  Instead  of  multiplying  by  0.7854,  you  may  multiply  by  11  and  divide  by  14 ; 
the  quotient  will  be  the  area  nearly.  This  quantity,  0.7854,  represents  the  area  of  a 
circle  whose  diameter  is  1 ;  the  circumference  of  the  same  circle  being  3.1416  nearly. 
The  proportion  of  the  diameter  to  the  circumference  is  expressed  in  whole  numbers 
by  the  ratio  of  7  to  22  nearly,  or  more  exactly  by  113  to  355.* 

r 


Example.  Required  the  area  of  a  circle  ABCD,  whose 
diameter  BD  is  10.6  feet. 

The  diameter  10.6  multiplied  by  itself  and  by  0.7854  gives 
the  sought  area,  88.247544  squai-e  feet. 


PROBLEM    V. 

To  find  the  area  of  an  ellipsis. 

Rule.  Multiply  the  longest  diameter  by  the  least,  and  the  product  by  0.7854  ;  this 
last  product  will  be  the  area  requu-ed. 

Example.  Required  the  area  of  an  ellipsis  ABCD, 
whose  longest  diameter  AC  is  12  feet,  and  the  shortest 
diameter  BD  10  feet. 

The  product  of  the  two  diametei-s  is  12  X  10=:  120;  this, 
multiplied  by  0.7854,  gives  the  sought  area,  94.2480  square 
feet. 

The  area  of  a  sector  of  a  circle  may  be  found  by  means  of  die  whole  area  of  the 
circle  obtained  in  Problem  IV.,  by  saying.  As  360  degrees  is  to  the  angle  contained 
KotHreen  the  two  legs  of  the  sector,  so  is  the  whole  area  of  the  cuxle  to  the  area  of  the 
sector. 

There  are  various  regular  solids.  The  most  noted  are  the  following: — (1.)  A  Cube, 
whicli  is  a  figure  bounded  by  sLx  equal  squares.  (2.)  A  Parallelopiped,  which  is  a  solid 
terminated  by  six  quadiilateral  figures,  of  which  the  opposite  ones  are  equal  and 
parallel.  (3.)  A  Cylinder,  which  is  a  figure  formed  by  the  revolution  of  a  rectangular 
parallelogram  about  one  of  its  sides.  (4.)  A  Pyramid,  which  is  a  solid  decreasing 
gradually  from  the  base  till  it  comes  tc  a  point.  There  are  various  kinds  of  pyramids, 
accordmg  to  the  figure  of  their  bases.  Thus,  if  the  base  be  a  triangle,  the  solid  is 
called  a  triangidar  pyramid ;  if  a  parallelogram,  a  parallelogramic  pyramid ;  and  if  a 
circle,  a  circular  pyramid,  or  simply  a  cone.  The  point  in  which  the  pyramid  ends  ia 
called  the  vertex,  and  a  line  di'a^vn  from  the  vertex  perpendicular  to  the  base  is  called 
the  height  of  the  pyramid. 


*  This  ratio  may  be  easily  remembered  by  observing  tiiat,  if  the  first  three  odd  numbers,  1,  3,  5,  are 
repeated  twice,  they  will  produce  the  quantity  113355  5  the  three  first  figures  of  which  make  tl>e  first 
term  of  the  ratio,  and  the  three  last  the  last  term  of  the  ratio. 


MEiNSUIlATlON. 


101 


Rule. 


PROBLEM   VI. 

To  find  the  solidity  of  a  cube. 
Multiplying  the  length  of  a  side  of  the  cube  by  itself,  and  the  product 


by  the  *ame  length,  gives  the  solidity  required  ;  which  will  be  expressed  in  cubic  feci 
if  the  diuiensions  be  given  in  feet,  but  iu  cubic  inches  if  the  dimensions  be  given  m 
inches,  &:c, 


Example.  If  the  side  AB  of  the  cube  be  G.3  feet,  it  is 
required  to  determine  the  solidity. 

The  product  of  G.3  l)y  G.3  is  39.G9  ;  this,  muUiplied  again  by 
6.3,  gives  the  solidity  250.047  cubic  feet. 


JD 


A'^ 


PROBLEM   VIL 

To  fuid  the  solidity  of  a  redangidar  parallelopiped.:    •  :'!^  ', .. 

Rule.     Muhiply  the  length,  breadth,  and  depth,  into  each  othc ;  the  product 

be  the  solidity  required.  '  o        '      '      '  ' 


will 


JBy- 


EXAMFLE. 

Suppose,  in  the  parallelopiped  ABCDFGHE, 
the  length  EF  is  3G  feet,  the  breadth  EG  IG  feet, 
and  the  depth  DF  12  feet ;  it  is  required  to  find  the 
solidity. 

The  product  of  the  length  36  by  the  breadth  16 
is  576  ;  this,  multiplied  by  the  depth  12,  gives  the 
solidity  6912  cubic  feet. 


PROBLEM    VIII. 

To  find  the  solidity  of  a  cylinder. 

Rule.     Multiply  the  square  of  the  diameter  of  the  base  by  the  length,  and  tliia 
product  by  the  constant  quantity  0.7854  ;  the  last  product  will  be  the  solidity  required. 

Example.  Required  the  solidity  of  a  cylinder  ADHF, 
whose  length  DH  is  13  feet,  and  diameter  of  the  base 
AD  11  feet. 

The  diameter  11,  multiplied  by  itself  and  by  the  length 
13,  gives  1573,  which,  being  muhiiilied  by  0.7854,  gives  the 
solidity  in  cubic  feet  1235.4342. 


F                               A 

I' jwmmmnwmmm. 

]/      '                       B 

PROBLEM   IX.  \ 

To  find  the  solidity  of  a  grindstone. 

Grindstones,  in  the  form  of  cylinders,  are  sold  by  the  stone  of  24  inches  diameter, 
and  4  inches  thick.  The  number  of  stones  that  any  one  contains,  may  be  obtained  by 
the  following  rule. 

Rule.  ]\Iultiply  the  square  of  the  diameter  in  inches  by  the  thickness  in  inches, 
and  divide  the  product  by  2304,  and  you  will  have  the  number  of  stones  required. 

Exaimfle.  Required  the  number  of  stones  in  a  grindstone  whose  diameter  is  30 
inches,  and  thickness  8  inches. 

The  square  of  the  diameter  36  is  1296,  which,  being  multiplied  by  the  thickness  8 
gives  10368.     This,  divided  by  2304,  gives  4.5,  or  4i  stones,  the  solidity  required. 

This  problem  may  be  solved  by  means  of  the  line  of  numbers  on  Gunter's  Scale,  in 
a  ver}'  expeditious  manner,  by  the  following  rule. 

Rule.  Extend  from  48  to  the  diameter;  that  extent,  turned  over  twice  the  same 
way,  from  the  thickness,  will  reach  to  the  number  of  stones  required. 

Thus,  in  the  preceding  example,  the  extent  from  48  to  the  diameter  36,  turned  over 
twice,  from  the  thickness  8,  will  reach  to  4.5,  or  4i,  which  is  the  number  of  stones 
sought. 


102 


MENSURATION. 


PROBLEM    X. 

To  find  the.  solidity  of  any  pyramid  or  cone. 

RuLK.  Multiply  the  area  of  the  base  by  one  thu'd  of  the  perpen- 
dicular height  of  the  pyramid  or  cone;  the  product  will  be  the 
solidity  required. 

Example  I.  If  the  j)yramid  have  a  square  base,  the  side  of 
which  is  4  feet,  and  the  perpendicular  height  6  feet,  it  is  required  to 
determine  the  solidity. 

The  area  of  the  base  is  4  X  4  rr  16  square  feet ;  this,  being  mul- 
tiplied by  one  thhd  of  the  height,  or  2  feet,  gives  32  feet,  the  solidity 
requu'ed. 


E\AMPLK  .II.  If  the  diameter  of  the  base  of  a  cone  be  10.6  feet, 
and,  tlie  .p^ji-peijdicular  height  30  feet,  it  is  required  to  find  the 
splidity. 

.' .The.  area  of"  this  ^  base  was  found  in  Problem  IV.  equal  to 
88.347544  ;  this,  muftii)lied  by  one  tliird  of  the  height,  or  10  feet, 
gives  the  solidity  required,  equal  to  882.47.544  cubic  feet. 

Having  obtained,  by  the  foregoing  rules,  the  number  of  cubic  feet 
in  any  body,  you  may  find  the  corresponding  number  of  tons  by 
dividuig  the  number  of  cubic  feet  l)y  40,  which  is  the  number  of 
cubic  feet  contained  in  one  ton.  Thus,  the  solidity  of  the  above- 
mentioned  cone,  882.47544,  being  divided  by  40,  gives  22.06188C, 
which  is  the  number  of  tons  in  that  cone. 


PROBLEai    XI. 

To  find  the  tonnage  ofi  a  $hip. 

By  a  law  of  the  Congress  of  the  United  States  of  America,  the  tonnage  of  a  ship  is 
to  be  found  in  the  following  manner : — 

If  the  vessel  be  double-decked,  take  the  length  tliereof  from  the  fore  part  of  the 
main  stem  to  the  after  pai-t  of  the  stern-post  above  the  uj)per  deck  ;  the  breadth 
thereof  at  the  broadest  i)art  above  the  main  wales  ;  half  of  this  breadth  shall  be 
accounted  the  depth  of  such  vessel ;  then  deduct  from  the  length  three  fifths  of  the 
breaddi,  multiply  the  remainder  by  the  breadth,  and  the  product  by  the  depth  ;  divide 
this  last  product  by  ninety-fivCj  and  the  quotient  will  be  the  true  content  or  tonnage  of 
such  vessel. 

If  the  vessel  be  single-decked,  take  the  length  and  breadth  as  above  directed  in 
respect  to  a  double-decked  vessel,  and  deduct  from  the  length  three  fifths  of  the 
breadth,  and  taking  the  depth  from  the  uuder'side  of  the  deck-plank  to  the  ceiling  of 
the  hold,  multiply  and  divide  as  aforesaid ;  the  quotient  will  be  the  true  content  or 
tonnage  of  such  vessel. 

Example.  Suppose  the  length  of  a  doul)le-decked  vessel  is  80  feet,  and  the  breaddi 
24  feet,  what  is  her  tonnage  ? 

Three  fifths  of  the  breadth,  24  feet,  is  14.4  feet,  which,  being  siditracted  from  tlie 
length,  80  feet,  leaves  65.6.  This,  multii)lied  by  the  breadth,  24  feet,  gives  1574.4  ;  this 
multiplied  by  the  dcptli,  12  feet  (half  of  24),  gives  18892.8,  which,  being  divided  by 
95,  gives  the  tonnage  198.9. 

Car]>enters,  in  finding  tlie  tonnage,  midtiply  the  length  of  the  keel  by  the  breadth  of 
the  main  beam  and  the  depth  of  the  hold  in  feet,  and  divide  the  product  by  95;  the 
quotient  is  the  number  of  toi^s.  In  doidile-decked  vessels,  half  the  breadth  is  taken 
for  the  depth.  (^ 


103 


GAUGING. 


Having  found  the  number  of  cubic  inches  m  any  body,  by  the  preceding  rules,  you 
may  thence  determine  the  content  in  gallons,  bushels,  &c.,  by  dividing  that  number 
of  cubic  niches  by  the  number  of  cubic  inches  in  a  gallon,  bushel,  &c.,  respectively. 

A  tvine  gallon,  by  which  most  liquors  are  measured,  contains  231  cubic  inches.  A 
beer  gallon,  by  which  beer,  ale,  and  a  few  other  liquors,  are  measured,  contains  282 
cubic  inches.  A  bushel  of  corn,  malt,  &c.,  contains  2150.4  cubic  inches;  this  measure 
is  subdivided  into  8  gallons,  each  of  which  contains  2G8.8  cubic  inches. 

In  all  the  folloiviyig  i-ules,  it  ivill  be  supposed  that  the  dimensions  of  the  body  are  given 
in  inches,  and  decimal  parts  of  an  inch. 

PROBLEM    I. 

To  find  the  number  of  gallons  or  bushels  in  a  body  of  a  cubic  form. 

Rule.  Divide  the  cube  of  the  sides  by  231,  the  quotient  wHl  be  tlie  answer  in 
wine  gallons  ;  or  by  282,  and  the  quotient  will  be  tiie  answer  in  beer  gallons ;  or  bj 
2150.4,  and  the  quotient  will  be  the  number  of  bushels. 

Example.  Required  the  number  of  wine  gallons  contained  in  a  cubic  cistern,  the 
length  of  whose  side  is  G2  inches. 

Multii)lying  G2  by  itself,  and  the  product  again  by  G2,  gives  the  solidity  238328 
which,  being  divided  by  231,  gives  the  content  1031|  wine  gallons. 

PROIJLEM    II. 

To  find  the  number  of  gallons  or  bushels  contained  in  a  body  of  the  form  of  a  rectangular 
parallelopiped.     (See  figure  of  Problem  VII.  of  Mensuration.) 

Rule.  Rlultiply  the  length,  breadth,  and  depth,  together ;  divide  this  last  product 
by  231  for  wine  gallons,  by  282  for  beer  gallons,  or  by  2150.4  for  bushels. 

Example.  Required  the  number  of  wine  gallons  contained  in  a  cistern  ABCDFGHE 
(see  fig.  Prob.  VII.  of  Mensuration)  of  the  form  of  a  parallelopiped,  whose  length  EF 
is  GG  inches,  its  breadth  FG  35  inches,  and  its  depth  DF  24  inches. 

Multiplying  the  length  G6  by  the  breadth  35  gives  2310  ;  multiplying  this  by  the 
depth  24"gi\'es  the  solidity  55440,  which,  being  divided  by  231,  gives  240  wine 
gallons. 

PROIJLEM    III. 

To  find  the  mmiber  of  gallons  or  bushels  contained  in  a  body  of  cylindrical  form. 

Rule.  Multijjly  the  square  of  the  diameter  by  the  height  of  the  cylinder,  and 
divide  the  product  by  294.12  ;  the  quotient  will  be  the  number  of  wine  gallons.  If 
you  divide  l)y  359.05,  tlie  quotient  will  be  the  number  of  ale  gallons ;  and  if  you 
divide  by  2738,  the  quotient  will  be  the  number  of  bushels. 

mte.  These  divisors  are  found  by  dividing  231,  282,  and  2150.4,  by  0.7854 
respectively. 

Example.  Required  the  number  of  wine  gallons  contained  in  the  cylinder  AFHD 
(see  the  fig.  of  Problem  VIII.  of  Mensuration),  the  diameter  AD  of  its  base  being  26 
inches,  and  length  DH  18  inches. 

The  diameter  2G  multiplied  by  itself  gives  G7G ;  multiplying  this  by  the  length  18 
gives  the  solidity  121G8,  which,  being  divided  by  294.12,  gives  the  answer  41  wine 
gallons  nearly. 


liJ4 


GAUGING. 


PROBLEM   IV. 

To  find  the  number  of  gallons  or  bushels  contained  in  a  body  of  the  form  of  a  pyramid 
or  cone.    (See  figures  of  Problem  X.  of  Mensuration.) 

Rule.  Multiply  the  area  of  the  base  of  the  p}Tamid  or  cone  by  one  third  of  its 
perpendicular  height ;  the  product,  divided  by  231,  will  give  the  answer  in  wine 
gallons.  If  it  be  divided  by  282,  the  quotient  will  be  the  number  of  beer  gallons  ;  or 
by  2150.4,  the  quotient  will  be  the  number  of  bushels. 

Example.  Requii'ed  the  number  of  beer  gallons  contained  in  a  pyramid  DEFGK 
(see  fig.  Prob.  X.  Example  I.),  whose  base  is  a  square  EFGK,  a  side  of  which,  as  EF, 
is  equal  to  30  inches,  and  the  perpendicular  height  of  the  pyramid  is  60  mches. 

The  square  of  30  is  the  area  of  the  base  900  ;  this,  being  multiplied  by  one  thu'd  of 
the  ahitu'de  20,  gives  the  soUdity  18000,  which,  being  divided  by  282,  gives  the  answer 
in  beer  gallons  63.8. 

PROBLEM    V. 

To  find  the  number  of  gallons  or  busliels  contained  in  a  body  of  the  form  of  a  frustum 
of  a  cone.     (See  the  figure  below.) 

Rule.  Multiply  the  top  and  bottom  diameters  together,  and  to  the  product  add 
one  third  of  the  square  of  the  difference  of  the  same  diameters ;  multiply  this  sum  by 
the  perpendicular  height,  and  divide  the  product  by  294.12  for  wine  gallons,  by  359.05 
for  ale  gallons,  or  by  2738  for  bushels. 

a£- •■ -^-B 

Example.  Given  the  diameter  CD  of  the  bottom  of  a 
fifustum  of  a  cone  36  inches,  the  toj)  diameter  AB  :z=  27 
inches,  and  the  perpendicular  height  EF  50  inches ;  required 
the  contents  in  wine  gallons. 

The  product  of  the  two  diameters,  36  and  27,  is  972 ;  their 
difference  is  9,  which,  being  squared  and  divided  by  3,  gives 
27 ;  adding  this  to  972  gives  999,  which,  being  multiplied  by 
the  height  50,  gives  the  solidity  49950;  dividing  this  by  294.12 
gives  the  content  in  wine  gallons  169.8. 

PROBLEM  VI. 

To  gauge  a  cask. 

To  gauge  a  cask,  you  must  measure  the  head  diameters,  AF,  CD,  and  take  the 
mean  of  them  when  they  differ ;  measure  also  the  diameter  BE  at  the  bung  (taking 
the  measure  within  the  cask);  then  measure  the  length  of  the  cask,  making  due 
allowance  for  the  thickness  of  the  heads.  Having  these  dimensions,  you  may  calcu- 
late the  content,  in  gallons  or  bushels,  by  the  following  rule : — 

Rule.  Take  the  difference  between  the  head  and  bung  diameters ;  multiply  this 
by  0.62,  and  add  the  product  to  the  head  diameter ;  the  sum  will  be  the  mean  diameter ; 
multi])ly  the  square  of  this  by  the  length  of  the  cask,  and  divide  the  product  by  294.12 
for  wine  gallons,  by  359.05  for  beer  gallons,  or  by  2738  for  bushels. 

The  quantity  0.62  is  generally  used  by  gangers  in  finding  the  mean  diameter  of  a 
cask.  But  if  the  staves  are  nearly  straigiit,  it  will  be  more  accurate  to  take  0.55,  or 
less ;  *  if,  on  the  contraiy,  the  cask  is  full  on  the  quarter,  it  will  be  best  to  take  0.64 
or  0.65. 

Example.  Given  the  bung  diameter  EB  m  34.5  inches,  the 
head  diameter  AF  z=  CD  =r  80.7  inches,  and  the  length  59.3 
hiches  ;  required  the  number  of  wine  gallons  this  cask  will  hold. 

The  difference  of  the  two  diametere,  34.5  and  30.7,  is  3.8 ;  this 
being  multi])lied  by  0.62,  gives  2.4  nearly,  to  be  added  to  the  head 
diameter  30.7  to  obtain  the  mean  diameter  33.1.  The  square  of 
33.1  is  1095.61  ;  multiplying  this  by  the  length  59.3,  gives  the 
solidity  64969.673 ;  dividing  this  by  294.12,  gives  the  content  in  wine  gallons  220.9. 

*  In  the  example  to  Problem  V.  preceding  (which  may  be  esteemed  as  the  half  of  a  hog-shead  with 
staves  perfectly  straight),  the  multiplier  is  only  0.51.  For  this,  being  multiplied  by  9  (thrf  difference 
between  AB  and  CU),  produces  4.59  or  4.6  nearly  ;  adding  this  to  27  gives  31.G,  whose  square,  being 
luultiplied  by  60,  and  the  product  divided  by  294.12,  gives  170  gallons  nearly. 


GAUGING.  105 

To  gauge  a  cask  hy  means  of  the.  line  of  numbers  on  Gunler^s  Scale,  or  that  07i  the 
Callipers  used  by  gangers. 

Make  marks  on  the  scale  at  the  points  17.15,  18.95,  and  52.33,  whicli  are  the  square 
roots  of  294.12,  359.05,  and  2738,  respectively.  A  brass  pin  is  generally  fixed  on  the 
callipers  at  each  of  tliese  points,  which  are  called  the  gauge  points.  Having  prepared 
the  scale  in  this  manner,  you  may  calculate  the  number  of  gallons  or  bushels  by  the 
Allowing  rule : — 

Rule.  Extend  from  1  towards  the  left  hand  to  0.62  (or  less,  if  the  staves  be  nearly 
iti'aight) ;  that  extent  will  reach  from  the  difference  between  the  head  and  bung 
diameters  to  a  number  to  the  left  hand,  which  is  to  be  added  to  the  head  diameter  to 
get  the  mean  diameter ;  then  put  one  foot  of  the  compasses  upon  the  gauge  point 
(which  is  17.15  for  wuie  gallons,  18.95  for  ale  gallons,  and  52.33  for  bushels),  and 
extend  the  other  to  the  mean  diameter ;  this  extent,  turned  over  twice  the  same  way, 
from  the  length  of  the  cask,  will  give  the  number  of  gallons  or  bushels  respectively. 

In  the  preceding  example,  the  extent  from  1  to  0.62  will  reach  from  3.8  to  2.4  nearly, 
which,  being  added  to  30.7,  gives  the  mean  diameter  33.1  ;  then  the  extent  from  the 
gauge  point  17.15  to  33.1,  being  turned  over  twice  from  the  length  59.3,  will  reach  to 
220.8  wine  gallons. 

If  we  use  the  gauge  point  18.95,  the  answer  will  be  in  ale  gallons  ;  and  if  we  use 
52^,  the  answer  will  be  in  bushels. 
14 


106 


SURVEYING. 


Land  is  generally  measured  by  a  chain  of  66  feet  in  length,  divided  mto  100  equal 
parts  called  link^,  each  link  being  7.92  inches. 

A  pole  or  rod  is  16^  feet,  or  25  links,  m  length.  Hence  a  square  pole  contains  272^ 
square  feet,  or  625  square  links. 

An  acre  of  land  is  equal  to  160  squai'e  poles,  and  therefore  contams  43560  square 
feet,  or  100,000  square  links. 

To  find  the  number  of  square  poles  in  any  piece  of  land,  you  may  take  the  dimen- 
sions of  it  in  feet,  and  find  the  area  in  square  feet,  as  in  the  preceding  problems ;  then 
divide  this  area  by  43560,  and  the  quotient  will  be  the  number  of  acres  ;  or  by  272.25, 
and  the  quotient  wWl  be  tlie  number  of  square  poles.  If  the  dimensions  be  taken  in 
links,  and  the  area  be  found  in  square  links,  you  may  obtain  the  numl)er  of  acres 
by  dividing  by  100000  (that  is,  by  crossing  off  tlie  five  right-hand  figures),  and  the 
number  of  square  poles  may  be  obtamed  by  dividing  by  625. 

PROBLEM   L 

'"o  Jlnd  the  numher  of  acres  and  poles  in  a  piece  of  land  in  the  form  of  a  rectangular 

parallelogram. 

Rule.  Multiply  the  base  by  the  pei-pendicular  height,  and  divide  by  625  if  the 
dimensions  be  taken  in  Ihiks,  or  by  272.25  if  they  be  taken  in  feet ;  the  quotient  will 
be  the  number  of  poles.     Dividmg  tliis  by  160,  we  get  the  number  of  acres. 

Example.     Suppose  the  base  BC  (see  the  figure  of  Ex.  I.  Pro!).  I.  of  Mensuration) 
of  the  rectangular  parallelogi-am  ABCD  is  60  feet,  and  the  perpendicular  AB  25  feet 
required  the  area  m  poles. 

The  product  of  the  base  60  by  the  perpendicular  25,  gives  the  content  1500  square 
feet ;  and  by  dividuig  it  by  272.25,  we  obtain  the  answer  m  square  poles  5.5,  nearly. 

PROBLEM   II. 

To  find  the  numher  of  acres  and  poles  in  a  piece  of  land  in  the  form  of  an  ohlique-angidar 
parallelogram.    (See  the  figure  of  Prob.  I.  Ex.  IV.  of  Mensuration.) 

Rule.  This  area  may  be  found  in  exactly  the  same  manner  as  in  the  preceding 
problem,  by  midtiplying  the  base  AD  by  the  perpendicular  height  BE,  and  dividing 
ijy  625  when  the  dimensions  are  taken  in  links,  or  by  272.25  when  taken  in  feet ;  the 
quotient  will  be  the  answer  in  poles,  which,  being  divided  by  160,  will  give  the  answer 
in  acres. 

Example.  Suppose  the  base  AD  is  632  links,  and  the  perpendicular  BE  326  luiks; 
requii'ed  the  number  of  poles. 

Multiiily  tlie  base,  632  links,  by  the  perpendicular,  326  links ;  the  product  206032, 
divided  by  625,  gives  the  answer  in  poles  329.7. 

PROBLEM  III. 

To  find  the  numher  of  acres  and  poles  in  a  piece  of  land  of  a  triangxdar  form. 

Rule.  ]Multij)ly  the  base  by  the  pei-])Pndicular  height,  and  divide  the  product  by 
1250  when  the  dimensions  are  given  in  luiks,  or  by  544.5  when  tliey  are  given  in  feet ; 
the  quotient  will  be  the  answer  in  poles. 

JVote.  Instead  of  dividing  by  1250,  you  may  multiply  by  8  and  cross  oflF  the  four 
right-hand  figures. 


SURVEYliXG. 


107 


Example.  Given  the  base  AC  (see  figure  of  Problem  II.  of  Mensuration)  equal  to 
JOO  feet,  and  the  perpendicular  BD  150  feet ;  required  the  area  in  poles. 

Multiply  the  base  300  by  the  perpendicular  150;  the  product  45000,  divided  by 
544.5,  gives  tlie  answer  m  poles  82.G. 

PROBLEM   IV. 

To  Jind  the  number  of  acres  and  poles  in  a  piece  of  land  of  any  irregular  right-lined 

figure. 

Rule.  Fuid  the  area,  as  in  Pi-oblem  III.  o(  Mensuration,  by  drawing  diagonals,  and 
reducing  the  figure  to  triangles ;  the  base  of  each  triangle  being  nndtii)lied  by  the 
perpendicular  (or  by  the  sum  of  the  perpendiculars  falling  on  it),  antl  the  sum  of  all 
these  products  divided  by  1250  when  the  dimensions  are  given  in  links,  but  by  544.5 
when  in  feet,  will  give  the  area  of  the  figure  in  poles. 

Example.  Suppose  that  a  ])iece  of  land  is  of  the  same  form  as  the  figure  in  Prob. 
III.  of  Memuration,  and  that  BE  =  23  feet,  CE  =  33  feet,  AF  =:  13  feet,  BG  =  14 
feet,  and  Dll  zz:  12  feet;  it  is  required  to  find  the  area  in  poles. 

The  product  of  BE  22  feet,  by  AF  13  feet,  gives  double  the  triangle  ABE  286 
square  feet ;  and  the  diagonal  CE  33  feet,  multii)lied  by  the  sum  of  the  i)eri)endicular8 
BG,  DH,  26  feet,  gives  double  the  figure  BCDE,  858  square  feet ;  the  sum  of  this  and 
286,  being  divided  by  544.5,  gives  the  ai'ea  2.1  or  2-tj.  poles. 


To  find  the  content  of  afield  by  the  Table  of  Difference  of  Latitude  and  Departure. 

This  method  is  simple,  and  much  more  accurate  than  by  projection,  the  boundaries 
being  straight  Imes  whose  bearings  and  lengths  are  known.  The  rule  for  making 
these  calculations  is  as  follows : — 

RULE. 

I.  Begin  at  the  western  point  of  the  field,  as  at  the  point  A  in  the  figure  Prob.  III. 
of  Mensuration,  for  a  point  of  departure  ;  and  mark  down,  in  succession,  the  bearings 
and  lengths  of  the  boundary  lines  AB,  BC,  &.C.,  as  courses  and  distances  in  a  traverse 
table.  Fhid  the  corresponding  differences  of  latitude  and  departure  by  Table  I.  or  II. 
(or  by  logarithms),  and  enter  them  in  their  respective  columns  N.  S.  E.  W.  as  in  the 
adjoined  table. 


Courses. 

Dist. 

N. 

S. 

E. 

W. 

Mcr. 
Dist. 

M. 

Korlh 
Jircas. 

South 
Jircas. 

N.  58°  E. 

E.    6    S. 

S.  17    W. 

\V. 

N.  42°  35'  W. 

19. 
20. 
20. 
20. 
15.1 

10.1 
11.1 

2.1 
19.1 

16.1 
19.9 

5.8 
20.0 
10.2 

10.1 

36.0 

30.2 

10.2 

00 

16.1 
52.1 

0G.2 
40.4 
10.2 

1G2.61 
113.22 

109.41 

1264.42 

0 

21.2 

21.2 

36.0 

36.0 

275.83 
Half, 

1373.83 

275.83 

1098. 
549. 

2.  Find  the  departures  or  meridian  distances  of  the  points  B,  C,  &c.  from  the  point 
A,  by  ailding  the  departures  when  east,  but  subtracting  when  west,  and  mark  them 
respectively  against  the  bearings,  hi  the  column  of  meritlian  distance. 

3.  Place  in  the  fii-st  line  of  the  column  M  the  first  meridian  distance  16.1,  and,  in  the 
following  lines,  the  sum  of  the  meridian  distance  which  stands  on  the  same  line  and 
that  innnediately  above  it.  Thus  on  the  second  line,  I  put  52.1,  which  is  equal  to  the 
smn  of  16.1  and  36.0.     On  the  third  line,  66.2  =  36.0  +  30.2,  &c. 

4.  jMultiply  the  numbers  in  the  column  M  by  the  differences  of  latitude  in  the  same 
horizontal  line,  and  place  the  product  in  the  column  of  areas  marked  north  or  south, 
according  as  the  difference  of  latitude  is  north  or  south.  Thus  in  the  first  number  in 
the  column  M  is  16.1,  which,  being  multiplied  by  the  corresponding  difference  latitude 
10.1  N.,  produces  the  north  area  162.61.  The  second  value  of  M  .52.1,  multii)lied  by 
the  second  difference  of  latitude  2.1  S.,  produces  the  south  area  109.41.  The  third 
values  66.2  and  19.1  S.  produce  the  south  area  1264.42.     The  fourth  difference  of 


i08 


SURVEY  liS'O. 


latitude  is  0,  wliicli,  being  multiplied  by  the  foiu-th  nieridiun  distance  40.4,  ])rocluces  0 
for  the  corresponding  area,  as  is  the  case  whenever  the  bearing  is  east  or  west,  &c. 

5.  Add  up  all  the  north  and  all  the  south  areas ;  half  their  difference  will  be  the 
area  of  the  field  in  square  measures  of  the  same  name  as  those  made  use  of  in  meas- 
uring the  lines,  whether  feet,  links,  or  chains,  &c.  Thus  the  sum  of  all  the  nortli 
areas  is  275.83,  that  of  the  south  1373.83  ;  their  difference  is  1098,  half  of  which  is  549 
square  feet,  the  area  of  the  given  field. 

It  may  be  obsei-ved  that  the  hearings  and  lengths  of  the  boundary  lines  in  this 
example,  are  not  exactly  the  same  as  those  in  Problem  III.  of  Mensuration,  which  is 
the  reason  of  the  difference  between  the  area  above  calculated  and  that  found  in 
Problem  III.  by  dividing  the  field  into  triangles. 

If  it  be  necessary,  the  differences  of  latitude  and  departure  may  bo  taken  to  one 
decimal  place  farther,  by  entering  the  table  with  ten  times  the  length  19,  20,  &c.,  and 
taking  one  tenth  of  the  corresponding  differences  of  latitude  and  departure. 

In  the  above  calculations  we  have  supposed  the  survey  to  have  been  made  with 
accuracy,  in  which  case  the  simis  of  the  differences  of  latitude  in  the  columns  N.  S. 
must  be  equal  to  each  other;  also  the  sums  of  the  departures  in  the  columns  E.  W. 
This  is  the  case  in  the  above  example,  Avhere  the  sum  of  the  differences  of  latitude  is 
21.2,  and  the  sum  of  the  departures  36.0 :  but  it  most  frequently  happens  that  the 
numbers  do  not  agree  ;  in  which  case  the  work  must  be  carefully  examined,  and  if  no 
mistake  be  found,  and  the  error  be  great,  the  place  must  be  surveyed  again ;  but  if  the 
eiTor  be  small,  it  ought  to  be  apportioned  among  all  the  differences  of  latitude  and 
departure,  in  such  manner  as  to  produce  the  required  correction  with  the  least  possible 
changes  in  the  given  numbers.  The  method  of  doing  this  was  explained  by  me  in 
the  fourth  number  of  the  Analyst,  in  answer  to  a  prize  question  of  Professor  Patterson, 
and  is  as  follows  : — Find  the  error  in  latitude,  or  the  difference  between  the  sums  of 
southing  and  northing ;  also  the  sum  of  the  boundary  lines,  AB,  BC,  &c.  Then  say, 
As  this  sum  is  to  the  error  in  latitude,  so  is  the  length  of  any  particular  boundary  to 
the  correction  of  the  corresponding  difference  of  latitude,  additive  if  in  the  column 
whose  sum  is  the  least,  otherwise  subtractive.  The  corrections  of  the  departure  are 
found  by  the  same  rule,  except  changing  difference  of  latitude  into  departure.  Thus, 
in  the  adjoined  exam})le,  the  sum  of  the  boundary  lines  is  161.6,  the  error  of  latitude 
is  0.10,  and  of  departui-e  0.08  ; 


Bearings. 

Lengths. 

N. 

S. 

E. 

W. 

Corrections. 

Corrected  Values. 

N. 

0.02 
.02 
.02 
.02 
.02 

E. 

N. 

S. 

E. 

W. 

N. 45°  E. 
S.  30  W. 
S.    5    E. 

W. 
N.20    E. 

40.      . 

25. 

3G. 

29.6 

31. 

28.28 
29.13 

21.65 
35.86 

28.23 

3.14 

10.60 

12.50 
29.60 

0.02 
.01 
.02 
.01 
.02 

23.30 

0.02 
29.15 

21.63 
35.84 

28.30 
3.16 

10.62 

42.08 

12.49 
29.59 

1G1.6 

57.41 

57.51 
57.41 

42.02 

42.10 
42.02 

0.10 

0.08 

57.47 

57.47 

42.03 

Error, 

.10 

Error, 

.08 

and  the  corrections  of  the  difference  of  latitude  and  departure  are  found  by  the 
following  proportions : — 


161.6 


titudc. 

::  40 

0.02 

::  25 

0.02 

::  36 

0.02 

::  29.3 

0.02 

::  31 

0.02 

Departure. 


161.6  :  0.08 


40 

0.02 

25 

0.01 

36 

0.02 

29.6 

0.01 

31 

0.02 

The  first  correction  of  latitude  0.02  is  to  be  added  to  the  first  latitude  28.28,  because 
it  is  in  the  column  whose  sum  57.41  is  less  than  the  other  57.51,  so  that  the  first 

*  The  boundary  lines  in  this  example  are  so  nearly  of  an  equal  length,  that  the  corrertion  of  the 
differenoe  of  latitude  (taken  to  the  nearest  decimal)  is  0.02  for  each  of  them  ;  but  in  general  they  will 
be  different.  The  table  of  difference  of  latitude  and  departure  may  be  made  use  of  in  (inding:  these 
corrections,  thus  : — Seek  in  the  table  till  the  first  term  IGl.G  (or  1G2)  is  found  in  the  distance  column  to 
correspond  to  the  second  term  0.10  (or  10)  in  the  departure  column  ;  thus  opposite  the  third  term  40 
i'i,  36,  (Sic,  will  be  the  sought  corrections,  as  is  evident. 


SURVEYING.  109 

corrected  difference  of  latitude  is  28.30.  The  second  is  the  difference  between  21.65 
and  tlie  second  correction  0.02,  because  21.65  is  in  the  greatest  column  ;  the  corrected 
value  is  therefore  21.63.  The  third  is  found  in  the  same  manner  to  be  35.86  —  0.02 
:=  35.84.  The  fourth  corrected  difference  of  latitude  is  simply  the  fourth  correction 
0.02  placed  in  the  colunni  N,  because  tlie  sum  in  tliat  column,  57.41,  is  the  least,  and 
the  fourth  difference  of  latitude  in  the  original  table  is  0.  The  fifth  is  the  sum  of 
21).13,  and  the  fifth  correction  0.02,  making  29.15.  These  are  placed  in  their  proper 
colunms  in  the  corrected  vahies.  In  a  similar  manner  the  first  departure  is  equal  to 
the  sum  of  28.23  and  the  first  correction  0.02,  which  is  equal  to  28.30.  The  second 
is  the  difference  between  12.50  and  the  second  correction  0.01,  making  12.49  ;  and  so 
as  for  the  others,  taking  the  sum  when  the  departure  is  in  the  cohunn  whose  sum  is 
the  least  (which,  in  the  present  case,  is  the  east),  and  the  difference  when  in  the  other 
column.  In  the  traverse  table  thus  corrected,  the  sum  of  the  differences  of  latitude  is 
57.47  in  both  columns,  and  the  sum  of  the  departures  42.08.  Having  corrected  the 
vahies  of  this  traverse  table,  you  must  find  the  meridian  distances,  the  column  M,  the 
north  and  south  areas,  &c.,  as  in  tlie  former  example. 

In  projecting  a  survey  of  this  kind,  where  there  is  a  small  error,  you  must  plot  off  as 
usual  the  boundary  lines  AB,  BC,  CD,  &.C.,  and  it  will  be  found  that  the  termination 
of  the  last  line  AE  will  not  fall  exactly  in  the  point  A,  but  will  be  at  a  point  near  it, 
which  we  shall  call  a.  To  correct  this  error,  you  must  draw  through  the  points  B,  C, 
D,  &;c.,  lines  parallel  to  aA,  in  the  du-ection  from  a  to  A,  of  such  lengths  as  to  be  to 
Aff,  as  the  distances  of  those  points  respectively  from  A  (measm-ed  on  the  boundary 
ABCD,  &c.)  are  to  the  whole  length  of  the  boundary  line  ;  through  tiiese  points  draw 
the  corrected  lines  terminating  on  A. 


The  3Ianncr  of  Surveying  Coasts  and  Harbors. 

From  what  has  been  said  in  the  preceding  problems,  the  intelligent  reader  will 
readily  perceive  the  method  of  surveying  a  coast  or  harbor.  But  as  this  is  an  impor- 
tant subject,  we  shall  enter  more  fully  into  an  explanation  of  the  different  methods 
which  may  be  used. 

To  take  a  draught  of  a  coast  in  sailing  along  shore. 

Having  brought  the  ship  to  a  convenient  place,  from  which  the  principal  points  of 
the  coast  or  bay  may  be  seen,  either  cast  anchor,  if  it  is  convenient,  or  lie-to  as  steady 
as  possible ;  or,  if  the  coast  is  too  shoal,  let  the  observations  and  measures  be  taken 
in  a  boat.  Then,  while  the  vessel  is  stationarj',  take,  with  an  azimuth  com[)ass,  the 
bearings,  in  degrees,  of  such  points  of  the  coast  as  form  the  most  material  projections 
or  hollows.*  Write  down  these  bearings,  and  make  a  rough  sketch  of  the  coast, 
observing  carefully  to  mark  the  points,  whose  bearings  are  taken,  with  letters  or 
numbers,  for  tlie  sake  of  reference. 

Tlien  let  the  ship  or  boat  run  in  a  direct  line  (which  must  be  very  carefully  meas- 
ured l)y  the  log,  or  otherwise)  one,  two,  or  three  miles,  until  she  comes  to  another 
situation,  from  which  the  same  points,  before  observed,  can  be  seen  again  with  quite 
different  bearings.  Then  let  the  vessel  lie  steady,  as  at  the  former  station,  and  observe 
again  the  bearings  of  the  same  points,  and  make  a  rough  sketch  of  the  coast.  This 
sketch  may  !)e  made  more  accurately  while  the  vessel  is  running  the  base  line. 

To  describe  the  chart  from  these  observations,  you  must,  in  some  convenient  part 
of  a  sheet  of  paper,  draw  the  magnetic  meridian,  and  lay  off  the  several  bearings  taken 
at  the  first  station,  marking  them  with  their  proper  letters  or  numbers.  Lay  down  also 
the  bearings  taken  from  the  second  station.  Draw  a  line  to  represent  the  ship's  run 
both  in  length  and  course,  and  from  that  end  of  the  line  expressing  the  fii-st  station, 
draw  lines  parallel  to  the  respective  bearings  taken  from  that  end  ;  also  from  the  other 
enil  draw  lines  parallel  to  the  bearings  taken  at  that  end,  and  note  the  intei-section  of 
each  pair  of  lines  directed  to  the  same  point ;  and  through  these  intersections  draw  by 
hand  a  ciu'ved  line,  observing  to  wave  it  in  and  out  as  near  as  can  be  like  the  trending 
of  the  coast  itself  Then  mark  off  the  variation  of  the  compass  from  the  north  end  of 
the  magnetic  meridian,  towards  the  right  hand  if  it  be  west,  or  towards  the  left  hand  if 
it  be  east,  and  draw  the  ti-ue  meridian  through  that  point  and  the  centre  of  the  circle. 

*  Tn  taking  the  bcarinn^s,  if  the  vessel  has  much  motion,  the  mean  of  several  observations  should  be 
taken. 


110  SURVEYIiNG. 


eacli  part  draw  tlie  appearance  of  the  land  marked  in  the  sketclies,  distiii- 
lic  rocky  sliore,  highland,  l)each.  &c.,  as  in  Plate  V.  or  VIII.   Thus  the  sand 


Against 
guishing  the  ^  ,      ,.  . 

beaches  may  be  marked  asin  Plate  VIII.  ngure  8,  and  the  rocky  shore  as  in  figure  9, 
&c.  Put  in  the  saveral  soundings,  at  low  water,*  in  small  figures,  distinguishing 
whether  they  are  fathoms  or  feet.  Show  the  time  of  high  water,  on  the  full  and 
change  days,  by  Roman  figures,  and  note  the  rise  of  the  tide  in  feet.  The  direction  and 
velochy  of  the  flood  tide  are  to  be  observed  ;  which  may  be  done  by  heaving  the  log 
when  the  ship  or  boat  is  at  anchor,  and  the  direction  is  to  be  represented  by  an  arrow. 
Insert  a  compass  and  a  scale  of  miles  or  leagues,  such  as  the  vessel's  run  was  laid 
domi  by.  Add  the  name  of  the  place,  and  the  latitude  and  longitude,  as  true  as  can 
be  obtained. 

If  there  are  shoals  or  sands  on  the  coast,  let  them  be  observed  in  a  boat,  sailuig 
round  them,  keeping  account  of  the  courses,  distances,  and  soundings,  f  But  to  put 
them  in  tlie  draught,  the  observer  in  the  boat  must  take  the  bearings  of  two  pohits  on 
the  coasts  (tlie  bearings  of  which  have  been  taken  from  the  ship)  from  souie  part  of 
each  sand  or  shoal  so  sailed  round ;  or  the  bearing  of  the  boat  at  some  part  of  the 
shoal,  or  of  some  beacon  in  that  place,  must  be  taken  by  the  ship  at  each  of  the  stations 
where  the  bearings  of  the  shore  Avere  taken  from  the  ship ;  for  by  either  of  these 
means,  one  point  of  the  sand  being  obtained,  the  rest  of  it  can  be  laid  doAvn  from  the 
observations  taken  in  the  boat.  Rocky  shoals  may  be  marked  on  the  chart  as  in  Plate 
Vlll.figure  ll,and  sand-banks  as  in  figure  10. 

If  tlie  coast  be  a  bay  or  harbor,  winding  in  such  manner  that  all  its  parts  cannot 
be  seen  at  two  stations,  let  as  many  bases  or  lines  be  run  and  measured  exactly  as  may 
be  found  necessary,  observing  that  the  several  distances  run  should  join  to  one  another, 
in  the  nature  of  atraverse,  that  each  new  set  of  objects  or  points  observed  should  be 
taken  from  two  stations  at  the  ends  of  a  known  distance,  and  that  the  objects  whose 
Dearin"-s  are  taken  do  not  so  much  extend  beyond  the  Umits  of  the  base  as  to  make 
ano-les'^vith  it  less  than  about  h  or  %  of  a  point,  but  rather  resei-ve  such  objects  for  the 
next  measured  base  line;  for  when  lines  lie  very  obliquely  to  one  another,  their 
intersections  are  not  easily  ascertained. 

If  any  particular  parts  of  the  harbor  cannot  be  conveniently  seen  from  either  of 
the  stations,  take  the  boat  into  those  places ;  having  well  examined  them,  and  made 
sketches  thereof,  estimating  tlie  lengths  and  breadths  of  the  several  inlets,  either  by  the 
rowing  or  sailing  of  the  boat,  take  as  many  bearings,  soundings,  and  other  notes,  as 
may  be  thought  necessary;  then  annex  these  particular  views,  in  then-  proper  places, 
in  the  general  draught. 

If  there  are  any  dangerous  sands  or  rocks,  besides  inserting  them  in  their  proper 
places,  you  must  see  if  there  be  any  two  objects  ashore  (such  as  a  church,  mill,  house, 
noted  cliff.  Sec.)  which  appear  in  the  same  right  line  when  on  the  shoal,  and  these 
objects  must  be  noted  on  your  chart.  If  none  can  be  found,  you  must  take  the 
bearings  of  some  remarkable  points,  and  note  them  on  your  chart.  By  this  means  we 
may  know  how  to  avoid  the  danger. 

We  must  mark  in  the  draught  the  kind  of  bottom  obtained  in  sounding,  whether 
mud,  sand,  shells,  coral,  rocky  ground,  &c. ;  and  where  there  is  good  anchorage,  draw 
the  figure  of  au  anchor ;  also,  if  there  is  any  particular  channel  more  convenient  than 
another,  it  is  to  be  pointed  out  by  Unes  drawn  to  its  entrance  from  two  or  more  noted 
marks  ashore. 

The  ])ositioiis  of  objects,  taken  by  a  magnetic  compass,  being  liable  to  great  uncer- 
tainties, as  is  well  known  to  those  who  have  had  any  experience,  especially  at  sea, 
it  has  been  recommended  to  observe  only  the  bearings  of  the  station-lines  by  the 
compass,  and  then  measure  the  angles  which  the  other  objects  make  with  these  lines 
by  a  quadrant  or  sextant,  which,  for  this  purpose,  must  be  held  in  a  horizontal 
position. 

EXAMPLE    I.     (See  Plate  VII.  fig.  1.) 

Suppose,  in  a  sMp  at  A,  we  obsei-ve  the  bearings  of  the  most  remarkable  points  of  a 
bay,  C,  D,  E,  F,  G,  II,  and  I,  and  then  sail  S.  04°  E.  li  miles  to  B,  and  at  B  observe 
the  beaiin"-s  of  the  same  points ;  it  is  required  to  construct  the  chart. 


*  If  the  soiiiuliii<Ts  were  not  taken  at  low  water,  they  may  be  reduced  thereto  by  a  method  wliich-will 
be  explaiiiod  liercaiter. 

t  It  is  diiruull  to  ascertain  correctly  the  courses  and  distances  sailed  by  the  boat,  on  account  of  the 
currents  and  olher  causes.  This  inconven'ence  may  oe  obviated,  if  liie  sliip  be  at  anchor,  and  not  far 
from  the  bo;il,  by  observing-  in  the  boat  tiie  bearins^  of  the  sliip  by  compass,  and  by  measuring-,  with  a 
quadrant,  the  aii^le  contained  between  the  top-galianl-mast  iiead  and  that  part  of  tiie  ship  which  is  at 
the  samc'hei"ht  as  the  eye  of  the  observer  ;  for  by  this  angle  the  tlistance  of  the  boat  from  the  s'lip 
may  be  d'Uorinined,  as  will  be  explained  hereafter. 


SURVEYING. 

Bearing  of  C 

from  A, 

S.  2CP  W. 

Beai-ing 

of  C  fi 

om 

B, 

S.  89°  W 

D 

N.    9^W. 

D 

N.  48°  W. 

E 

N.  26°  E. 

E 

N.  24°  W. 

F 

N.  55°  E. 

F 

N.  13°  E. 

G 

East. 

G 

N.  47°  E. 

H 

S.  40°  E. 

H 

/ 

S.  38°  W. 

I 

S.  19°  E. 

I 

S.  46°  W. 

Ill 


Draw  the  line  AB,  S.  64°  E.  Ih  miles.  Through  the  points  A  and  B  draw  the 
lines  AC,  AD,  AE,  AF,  AG,  AH,  Al,  BC,  BD,  BE,  BE,  BG,  BH,  and  Bl,  with 
their  respective  bearings ;  and  where  the  corresponding  lines  cut  each  other,  will  be 
the  jjoints  C,  D,  E,  F,  G,  II,  and  I,  resj)ectively.  Through  these  points  the  different 
curvatm-ps  of  the  land  must  be  drawn,  corresponding  with  j'our  eye-draught.  In  this 
manner  may  a  chart  be  constructed  by  observations  taken  upon  the  water.  The 
manner  of  surveying  upon  land  is  exactly  similar. 


To  survcT/  a  harhor  hy  ohservations  on  shore. 

Make  an  eye-ckaught  of  the  place  to  be  surveyed,  and,  in  going  round  the  coast,  fix 
station-staves,  or  straiglit  jjoles,  tall  enough  to  be  seen  at  a  consitlerable  distance,  in  the 
most  remarkable  points  and  benduigs  of  tlie  shore  ;  but  if  at  any  of  those  places  there 
is  a  noted  tree,  house,  or  any  ether  remarkable  thing,  that  object  may  serve  instead  of 
a  station-staff;  and  it  will  be  convenient  to  black  the  staves,  and  tie  a  piece  of  white 
bunting  at  the  top  of  each  ;  then  in  the  eye-draught  put  letters  or  numbers,  at  the 
noted  points  or  marks,  for  the  sake  of  distinction. 

Choose  the  most  extensive  and  level  sj)ot  of  ground  you  can  meet  with  to  measure 
your  base  line  upon.  This  line  must  not  be  less  in  length  than  a  tenth  part  of  the 
distance  of  the  two  extreme  objects  which  are  to  be  observed ;  and  the  two  extreme 
points  of  it  must  be  so  situated  that  as  many  of  the  station-staves  as  possible  may  be 
seen  from  bodi  of  diem.  The  bearing  or  position  of  the  base  must  be  well  determined 
in  degrees  and  minutes,  and  the  length  accurately  measured,  either  by  a  measuring- 
chain  or  a  piece  of  log-line. 

From  each  end  of  the  base  obsei-ve,  with  an  azimuth  compass,  or  with  a  theodolite 
(if  it  can  be  procured),  the  bearings  of  each  of  the  station-staves;  or  else  widi  a  sextant 
measure  the  angles  contained  between  the  staves  or  remarkable  objects  and  the  other 
end  of  the  station-line,  and  write  them  down,  in  regular  ordei*,  hi  your  book.  These 
measures  and  angles,  being  plotted  down,  as  before  directed,  will  give  the  most 
conspicuous  points  of  the  shore.  The  intermediate  spaces  are  to  be  filled  up  from  the 
sketches  made  on  the  spot. 

But  if  any  one  of  these  objects  be  situated  so  far  beyond  the  limits  of  the  base  as  to 
appear  nearly  in  the  same  direction,  or  to  make  angles  not  exceeding  10°  ;  or  if  some 
of  the  remarkable  objects  be  visible  only  from  one  end  of  the  base;  then  let  the 
bearings  of  such  objects  be  taken  from  a  place  whose  position  has  been  determined 
from  both  ends  of  the  measured  base :  or,  if  there  are  several  remai-ked  objects  which 
cannot  be  seen  from  either  end  of  the  liase  lines,  let  the  bearings  of  such  objects  be 
taken  from  each  of  two  points  whose  positions  have  been  determined  by  bearings 
taken  from  both  ends  of  the  base :  or  it  may,  on  some  occasions,  be  proper  to  choose 
another  place  on  which  anotiier  base,  of  a  convenient  length,  may  be  measured,  and 
from  the  extremities  of  which  the  ends  of  the  first  base  may  be  seen,  and  as  many  as 
possible  of  the  remaining  objects  which  lay  too  obliquely,  or  which  could  not  be  seen 
from  the  first  base.  In  such  manner  proceed  until  the  bearings  are  taken  of  all  the 
points  judged  necessary  for  comj)leting  the  survey  of  the  limits  of  the  harbor. 

If  a  right  line  of  a  sufficient  length  for  a  base  line  cannot  be  measured,  it  may  be 
taken  in  two  adjoining  lines,  as  the  two  sides  of  a  triangle,  the  included  angle  being 
accurately  measured,  and  the  bearing  of  one  of  the  lines  observed. 

When  the  oudincs  or  limits  of  a  harbor,  bay,  road,  &c.,  are  delineated  by  the 
preceding  precepts,  let  a  small  vessel  go  out  to  sea  to  take  drawings  of  the  appearance 
of  the  lanil  and  its  bearings.  Sail  likewise  into  the  hai'bor,  and  draw  the  apiiearance 
of  its  entrance.  Take  particular  notice  if  there  be  any  false  resemblance  of  the 
entrance,  by  which  ships  may  be  deceived  and  run  into  danger;  or  if  any  two  objects, 
being  brought  in  a  line,  will  lead  uito  the  harbor  without  danger.  Search  for  the  best 
anchoring-])laces,  and,  if  possible,  denote  those  places  by  bringing  two  ol)jects  in  one; 
if  not,  take  the  exact  bearings  of  two  or  diree  other  objects,  so  that  the  places  may  be 
easily  determined.  After  drawing  the  chart,  we  must  insert  a  compass,  with  the 
variation,  and  scale  properly  fitted  to  die  plan.  Then  the  islands,  rocks,  sands,  &c., 
must  be  marked  in  their  proper  places,  with  then-  soundings  at  low  water;    the 


112 


SURVEYING. 


anchoring-places,  with  the  best  ti-ack  to  get  to  them ;  the  proper  sailing-marks  to 
avoid  dangers  ;  the  places  whei-e  fresh  water  can  be  obtained ;  the  name  of  the  place, 
that  of  the  countiy,  or  of  the  sea  ;  the  latitude  and  longitude  ;  a  sketch  of  the  appear- 
ance the  place  makes  at  sea,  upon  a  known  bearing,  and  at  an  estimated  distance ;  and 
whatever  else  a  judicious  seaman  may  think  proper  to  insert.  Then  will  the  plan  be 
fit  for  all  nautical  purposes,  and  may  be  embellished  with  proper  colors,  if  necessaiy. 

EXAMPLE  II.     (See  Plate  VII.  fig.  2.) 

From  each  end  of  a  base  line  AB  of  1200  fathoms,  were  observed  the  points  C,  D, 
E,  F,  and  G ;  and  as  the  points  I,  K,  and  L,  were  not  visible  from  the  extremities  of 
the  base  line,  another  base  line  was  measured,  from  the  point  D  to  H,  of  680  fatlioms, 
from  which  points  the  bearings  of  I,  K,  and  L,  wei*e  obsei-ved.  Hence  it  is  required 
to  construct  a  chai't  of  the  place. 


Bearing  of  B  from 

A, 

East. 

Bear 

Ulg 

of  C  from  B, 

N.  W.  b.  W. 

C 

North. 

D 

N.  N.  W. 

D 

N.  E.  b.  N. 

E 

North. 

E 

N.  E.  h  N. 

F 

N.  b.  E. 

F 

N.  E.  b.  E.  h  E. 

G 

N.  E. 

G 

E.  b.  N.  h  N. 

Beai-ing  of  H  from 

D, 

N.  W. 

Bear 

ing 

of  I  from  H, 

N.  E.b.N 

I 

N.  b.  W. 

K 

N.  E.  h  E. 

K 

N.  b.  E.  }>  E. 

L 

E.N.E 

L 

N.  N.  E.  h  E. 

Draw  the  east  line  AB  equal  to  1200  fathoms  ;  from  each  end  of  this  line  draw  the 
lines  AC,  AD,  AE,  AF,  AG,  BC,  &c.,  at  their  respective  bearings  ;  the  points  of 
intei-section  will  give  the  points  C,  D,  E,  F,  and  G.  From  the  point  D  (which  was 
found  in  this  manner)  draw  the  N.  W.  line  DH  equal  to  680  fathoms,  and  through 
these  points  draw  the  lines  DI,  DK,  DL,  HI,  &c.,  at  their  respective  bearings ;  the 
points  of  intersection  of  the  corresponding  lines  will  be  the  situation  of  the  points 
I,  K,  L.  Between  these  remarkable  pomts,  draw  the  outlines  of  the  land,  conformable 
to  your  rough  draught. 

In  order  to  determine  the  situation  of  the  point  M,  which  was  seen  too  obliquely 
from  the  bases  AB,  DH,  you  may  take  the  bearing  of  that  point  from  B,  and  then  from 
G  (whose  situation  has  been  determined  by  bearings  taken  from  the  points  A,  B) ;  the 
intersection  of  the  lines  BM,  GM,  will  determme  the  situation  of  M. 


Method  of  surveying  a  small  hank  or  shoal  ichere  great  accuracy  is  required. 

The  method  of  determining  the  extent  and  situation  of  shoal  gi'ound  by  sailing 
round  it,  and  keeping  an  account  of  the  courses  and  distances  sailed,  is  well  adapted  to 
the  taking  of  an  extensive  survey,  or  to  the  exploring  of  a  large  bank,  where  great 
accuracy  is  not  required.  But  the  difficulty  of  ascertaining  with  precision  the  courses 
and  distances  sailed  (which  are  liable  to  error  on  account  of  the  tides,  currents,  and 
the  different  velocity  of  the  boat  at  different  times,  owing  to  the  unsteadiness  of  the 
wind)  prevents  this  method  from  being  sufficiently  accurate  to  be  used  in  exploring  a 
dangerous  shoal  or  bank  at  the  entrance  of  a  narrow  channel  of  a  harbor,  or  any  other 
place  where  the  exact  form  of  the  shoal  is  to  be  found  ;  and  if  to  obtain  the  necessaiy 
degree  of  coirectness,  the  bearings  of  two  remarkable  objects  are  taken  at  every 
time  of  sounding,  the  time  expended  in  taking  the  observations,  if  there  be  only 
one  observer,  v/ill  be  increased  beyond  all  reasonable  bounds.  To  obviate  these 
difficulties,  we  may  use  either  of  die  following  methods,  by  which  the  necessaiy 
observations  for  determining  the  situation  of  the  boat,  can  be  made  as  fast  as  the 
soundings  are  taken. 

First  Method.  Procure  a  large  sail-boat  with  a  high  mast,  and  a  small  row-boaL 
Bring  the  sail-boat  to  anchor  on  the  bank  which  is  to  be  exjjlored,  and  take  accurately 
the  bearings  of  two  remarkable  points  of  land,  or  other  objects,  whose  situation  has 
already  been  determined  by  ol «sei-vations  taken  on  shore,  or  in  sailing  along  the  land. 
By  this  means  the  situation  of  the  sail-boat  may  be  accurately  marked  on  the  chart. 
Then  enter  the  small  boat,  and  row  from  the  other  in  any  particular  direction,  observing 
to  keep  the  mast  of  the  boat  to  bear  upon  any  point  of  the  compass,  or  (which  is  much 
more  accurate)  to  keep  the  mast  of  tlie  boat  to  range  on  any  particular  point  of  land, 


J'  IT  ji   1"  :e  :r  J  3^  &. 


P7<iliW\ 


ISCO 


SURVEYING.  113 

or  other  object  marked  on  tlie  cliart,  s^o  that  any  error  whicli  might  arise  in  tlie  courst; 
of  tJie  boat  may  be  prevented.  Wlii-lo  ])roceedinof  in  this  direction,  let  one  i)erson  take 
the  soinidings,  while  another  observes,  with  a  quadrant  or  sextant,  the  angular  elevation 
of  the  top  of  the  boat's  mast  above  the  horizontal  line  drawn  from  the  eye  of  the 
observer,  and  a  thuxl  person  notes  the  observations  in  the  minute-book,  and  the  time 
of  observation,  in  order  to  make  the  necessary  reduction  in  the  soundings,  to  reduce 
tliem  to  low  water.  Proceed  in  this  manner  from  the  sail-boat,  till  you  get  off  the 
bank  uito  deep  water,  or  till  the  elevation  of  the  mast  is  not  much  less  than  one 
degree  ;  then  row  across  the  bank  till  the  bearing  of  the  mast  is  altered  considerably 
or  till  it  appears  in  a  range  with  another  point  of  land,  at  a  considerable  angidai 
distance  from  the  point  with  which  the  mast  ranged  in  the  first  observations ;  then  row 
towards  the  boat,  sounding  and  observing  the  angular  elevation  of  the  mast  as  before 
Proceed  in  this  manner,  in  sounding  to  and  from  the  sail-boat,  till  you  have  jn-ocured 
a  sutKcient  number  of  soundings  in  every  direction.  Then  go  on  board  the  sail-boat, 
and  shift  her  birth  to  another  part  of  the  bank,  where  soundings  have  not  been  taken, 
and  proceed  to  sound  as  before.  Continue  sounding  and  shifting  the  situation  of  the 
boat,  till  the  whole  bank  has  been  ex])]ored,  and  then  the  observations  may  be  plotted 
off  by  the  directions  in  the  following  example. 

Let  ABC  (Plate  VIII.  fig.  1)  be  the  mast  of  the  sail-boat ;  D  the  situation  of  the 
eye  of  the  person  who  observes  the  angular  elevation  of  the  mast.  Draw  the  line  BD 
parallel  to  the  horizon,  and  join  AD.  Then  the  height  AB  must  be  measured* 
accurately,  and,  that  being  given  and  the  observed  angle  ADB,  the  corresponding 
distance  BD  may  be  obtained  iiy  the  usual  rules  of  trigonometry,  by  saying.  As 
radius  :  AB  ::  cotangent  ADB  :  BD.  Thus,  if  the  height  AB  be  30  feet,  and  the 
angle  ADB  1°,  the  distance  BD  will  be  17J9  feet  (being  57.3  times  as  great  as  AB). 
The  distances  coiTesponding  to  2",  3°,  &c.,  are  given  in  the  adjoined 
table,  by  examining  which  it  will  appear  that  the  distance  BD  corre- 
sponding to  any  angle  ADB  (less  than  30°)  may  be  obtained  nearly  by 
dividing  1719  by  the  angle  ADB  in  degrees.  Thus,  for  4  degrees,  by 
this  rule,  the  distance  would  be  J._7_.i.9  r:r  429]  nearly,  as  in  the  table. 
The  gi'eatest  difference  between  the  distances  determined  by  the 
rule  and  by  the  table  is  5  feet,  corresponding  to  the  angle  30° ;  for 
17  J.9  =z  57,  whereas  by  the  table  the  distance  is  52.  In  taking 
soundings  by  this  method,  it  will  be  very  rarely  necessary  to  measure 
ail  angle  so  great  as  30°  ;  so  that,  for  all  practical  purposes,  the 
distance  may  be  determined,  in  this  examjile,  to  a  sufficient  degree 
of  accuracy,  by  dividing  1719  by  the  observed  angular  elevation 
in  degi'ees.  On  these  principles  we  have  the  following  rule  for 
calculating  the  distance,  corresponding  to  a  mast  of  any  given  height,  and  to  any 
observed  angular  elevation. 

Rule.  Multiply  the  height  of  the  mast  above  the  eye  of  the  observer  by  57.3,  and 
the  product  will  be  a  constant  quantity,]  ivhich,  being  divided  by  the  observed  angle  of 
devalion,  expressed  in  degrees  and  decimals  of  a  degree,  the  quotient  ivill  be  the  sougJU 
distance  nearly. 

If  the  height  of  the  mast  be  exjjrcssed  in  equal  parts,  taken  from  the  scale  by  which 
the  chart  is  plotted  off,  the  distances  found  by  the  above  rule  will  be  expressed  in  the 
same  equal  parts  ;  so  that,  if  the  distances  thus  expressed,  con-esponding  to  1°,  2°,  3 
&c.,  be  calculated  and  marked  on  a  slip  of  paptr  (Plate  VIII.  fig.  2)  from  H  to  1". 
from  II  to  2°,  and  from  H  to  3°,  &c.,  respectively,  the  slip  H  1,  thus  marked,  will  be  a 
veiT  convenient  scale  for  plotting  oft"  such  distances. 

For  further  illustration  of  this  method,  we  have  given  an  example  in  Plate  VIII.  fii". 
I,  in  which  C  represents  the  place  where  the  sail-boat  is  at  anchor ;  A  and  B  the 

*  A  mark  may  be  made  at  B,  and  a  vane  placed  at  the  top  of  the  mast  at  A,  to  enable  the  observe- 
lo  (lisliiii,'uish  those  objects  when  at  a  great  distance.  If  the  height  of  the  observer  above  the  horizon 
be  small  in  comparison  with  the  height  of  the  mast,  tiie  angular  distance  ADE  between  the  .surface  oj 
the  sea,  near  the  boat,  and  the  top  of  the  boat's  mast  may  be  measured^  instead  of  ADB  ;  for,  if  tl.e 
distances  BC  and  CE  remain  the  same  in  all  observations,  it  will  be  immaterial  which  angle  is  meas- 
ured ;  observing,  however,  that  different  scales  must  be  used  for  plotting  off  the  angles  ADB  and  ADE. 

If  AD  represent  ihe  known  vertical  height  of  the  summit  of  an  island  above  the  eye  of  an  observe, 
the  distance  from  the  island  can  be  determnied  by  measuring  the  angular  elevation  ADB.  as  is  evideu. 
from  what  has  been  said  above. 

t  This  constant  quantity  may  be  determined  without  actually  measuring  the  altitude  AB,  if  tlie  angular 
elevation  can  be  measured  at  a  place  D,  where  the  distance  BD  is  known.  'I'liiis,  in  the  example 
'Plate  VIII.  iig.  4),  the  distance  AC  being  known,  and  the  angular  elevation  of  the  mast  at  C  behig 
observed  at  A  in  degrees  and  decimals  of  a  degree,  and  multiplied  by  the  distance  AC,  the  product 
will  be  the  constant  ciuantity  mentioned  in  the  rule.  This  method  may  be  used  in  dcicrminin?'  ihs 
dr.;tance  frimi  an  island  bv  the  method  mentioned  in  the  last  note. 
15 


ADB 

BD 

FEKT. 

1° 

1719 

o 

859 

3 

572 

4 

429 

5 

343 

10 

170 

20 

82 

30 

52 

114  SURVEYliNG. 

points  observed,  in  order  to  ascertain  tlie  position  of  the  boat  on  the  chart,  by  drawing 
thereon  the  hnes  AC,  BC,  in  opposite  directions  to  tlie  bearings  of  the  points  A,  B, 
observed  from  the  boat, — the  point  of  intersection  C  being  evidently  the  place  of  the 
boat  upon  the  chart.  Suppose,  now,  that  in  the  first  set  of  observations,  the  mast  of 
the  sail-boat  is  made  to  range  on  the  point  A ;  in  this  case  the  course  of  the  boat  must 
he  on  the  continuation  of  the  line  AC  towards  D  :  then  the  slip  H  I  (Plate  VIII.  fig. 2) 
is  to  be  laid  upon  the  line  CD  (Plate  VIII.  fig.  4),  with  the  point  II  upon  C  ;  and 
the  angular  elevation  being  foiuid  on  the  slip,  the  sounding  corresponding  (reduced  to 
low  water)  is  to  be  marked  on  the  line  CD,  immediately  under  the  mark  on  the  slip. 
Thus,  if  the  angle  be  4°,  the  point  corresponding  will  be  G.  Having  ])lotted  oflT  the 
sounilings  taken  in  the  direction  CD,  proceed  in  the  same  manner  with  the  othei-s,  viz. 
those  in  the  direction  CE,  found  by  keeping  the  boat's  mast  in  a  range  with  the  church 
at  11 ;  those  in  the  direction  CF,  found  by  keeping  the  boat's  mast  in  a  range  with  the 
point  B  ;  those  in  the  direction  CA,  found  by  keeping  the  mast  to  bear  E.  N.  E. ;  and 
so  on  with  the  other  observations.  When  all  the  soundings  are  marked  on  the  chart, 
dotted  lines  are  to  be  made  round  the  shoal  soundings ;  and  thus  the  true  figure  of  the 
shoal  part  of  the  bank  will  be  obtained. 

This  method  I  have  frequently  used  in  taking  a  sun^ey  of  the  part  of  the  coast  of 
Massachusetts  Bay  included  between  Manchester  and  Lynn.  The  heiglit  of  the  mast 
of  the  boat  used  on  the  occasion  was  about  30  feet ;  and  it  was  found  that  distances 
less  than  a  third  of  a  mile  could  be  obtained  in  this  manner  to  a  great  degree  of 
precision. 

SccoJid  Method.  This  method  of  determining  the  place  where  soundings  are  taken, 
consists  in  keeping  (while  sailing  in  a  boat  and  sounding)  a  particular  point  of  land,  or 
any  other  object,  to  bear  always  in  the  same  direction,  and  measuring  with  a  quadrant 
or  sextant,  held  in  a  horizontal  position,  the  angular  distance  between  that  object  and 
another  object  .making  a  considerable  angle  with  the  former;  for  by  this  means  the 
situation  of  the  boat  at  the  time  of  sounding  may  be  determined.  Instead  of  bringing 
the  object  to  bear  upon  a  particular  point  of  the  compass,  you  may  (when  it  can  be 
done)  liring  the  object  in  a  range  with  another  remarkable  object,  and  by  this  means 
you  will  avoid  the  error  which  nnght  arise  from  the  use  of  a  compass. 

For  an  exam])le  of  this  method,  suppose  that  a  survey  of  the  small  islands  A  B,  K 
(Plate  VIII.  fig.  3),  and  the  large  one  CGH,  has  been  taken  and  plotted  oflT  as  in  the 
figure.  Then  soundings  may  be  taken,  in  the  direction  BCD,  by  bringing  the  small 
island  B  in  a  range  with  the  southern  part  of  the  great  island,  and  measuring  the  angle 
CDG  formed  by  the  extremes  of  the  great  island ;  or  by  kee])ing  the  small  island  A  to 
range  with  the  northern  part  of  the  great  island,  and  measuring  the  angle  IIIK  formed 
by  tlie  northern  extreme  of  that  island  and  the  small  island  K  ;  or  by  running  in  the 
direction  KL,  so  as  to  keep  the  island  K  to  bear  W.  i  S.,  and  measuring  the  angle 
formed  by  that  island  and  the  northern  extreme  of  the  great  island,  &c. 

The  method  I  have  generally  used  for  plotting  oft'  such  angles,  is  by  means  of  a 
sector ;  and  as  that  instrument  is  more  easily  procured  than  others  better  adapted  to 
the  ])urpose,  I  shall  explain  the  method  by  showing  how  the  angle  CDG,  measured  as 
above,  may  be  i)lotted  off"  so  as  to  determine  the  point  D  where  that  angular  distance 
\vas  ol)ser\ed.  To  do  this,  you  must  draw  the  line  CD,  and  ojien  the  sector  till  the 
two  legs  form  with  each  other  an  angle  equal  to  the  observed  angle  CDG  ;  then  slide 
one  leg  of  the  sector  on  the  line  CD  till  the  other  leg  touches  the  northern  extreme  of 
the  island  at  the  point  G,  and  the  point  directly  under  die  centre  of  the  joint  of  the 
sector  will  be  the  point  of  observation.  As  this  point  cannot  be  exactly  marked,  on 
account  of  the  size  of  the  joint  of  the  instrument,  you  may  mark  with  a  j)encil  on  the 
line  CD  the  two  points  where  the  circumference  of  the  joint  touches  that  line,  and 
note  the  sounding  in  the  middle  between  those  two  marks. 

If  a  quadrant  of  a  circle  be  described  on  a  piece  of  paper,  with  a  radius  equal  in 
length  to  one  of  the  legs  of  the  sector,  and  then  divided  into  90°,  the  sector  may,  by 
means  of  that  (juadrant,  be  opened  to  any  angle  in  a  very  expeditious  manner. 

This  method  of  obtaining  distances  when  sounding,  I  have  frequently  used  with 
success. 

Tli'ird  Methoil — with  two  observers.  This  method  is  founded  upon  the  process 
ex])lained  in  Problem  VII.  page  03.  It  consists  in  finding,  at  the  siune  time,  by  means 
of  two  observers  furnished  with  sextants,  the  horizontal  angles  ADC,  BDC,  (figure 
Problem  VII.  i)iige  93)  formed,  at  the  point  D  of  the  shoal,  by  the  right  lines  DA,  DC, 
DB,  drawn  to  three  points  of  land  or  reniarUaltie  objt'cis.  A,  C,  B  whose  positions  are 
given  on  the  chart,  or  have  been  ascerTaincd  by  i)revious  observations.  Jn  this  way 
various  points  P  of  the  shoal  nr  bank  may  be  found,  while  the  boat  is  sailing  over  it; 


SURVEYING.  115 

and  the  corresponding  soundings  can,  at  tlie  same  time,  be  obsei'ved.  As  tlie  process 
of  projecting  and  computing  such  obsen'ations  has  alreadj'  been  explained  in  Problem 
VII.,  it  will  not  be  necessary  to  make  any  additional  remarks  in  this  place,  except  that 
great  care  must  be  taken  in  selecting  the  points  to  be  oliserved,  A,  C,  B,  so  as  not  to 
have  the  centres,  F,  G,  ©f  the  tAvo  intersecting  circles,  ABD,  BCD,  near  to  cacli  other; 
because,  in  that  case,  a  slight  error  in  either  of  the  observed  angles,  ADC,  BDC,  may 
produce  a  very  imi)ortant  error  in  the  situation  of  the  point  D  of  the  shoal,  corre- 
sponding to  the  intersection  of  these  circles;  it  being  evident  that  the  method  would 
wholly  fail  if  the  point  C  were  to  be  placed  at  E  upon  the  circumference  of  the  circle 
ABD,  because  the  centres  F,  G,  would  then  coincide,  and  there  woidd  be  no  single 
point  of  intersection  D,  since  any  point  whatever  of  the  circumference  of  the  circle 
BCD  would  satisfy  the  obsen'ations.  This  difficulty  is  inherent  in  this  method  of 
observation,  and  no  process  of  numerical  calculation  will  help  it ;  so  that  we  may  rest 
assured,  that  whenever  it  is  difficult  to  find  the  precise  point  of  intersection  D,  by  a 
geometrical  construction,  the  points  A,  C,  B,  have  not  been  well  selected ;  and  the 
observations  may  lead  to  a  very  incorrect  result,  except  the  angles  are  taken  with  the 
utmost  degree  of  accuracy. 

To  reduce  soundings  taken  at  any  time  of  the  tide  to  low  icatrr. 

The  soundings  at  low  water  are  always  to  be  marked  on  a  chart ;  and  if  they  are 
taken  at  any  other  time  of  the  tide,  a  con-ection  must  be  applied  to  reduce  them  to  low 
water.  This  allowance  may  be  made,  if  the  whole  vertical  rise  of  the  tide  from  low 
to  high  water  be  known,  with  the  time  of  high  and  low  water,  as  in  the  following 
example : — 

Supi)ose  the  vertical  rise  of  the  tide,  fi-om  low  to  high  water,  to  be  10  feet,  the  time 
of  low  water  5h.  A.  M.,  and  the  time  of  high  water  llh.  30m.  A.  IM. ;  required  the 
allowance  to  be  made  on  an  observation  taken  at  8,  A.  M. 

Draw  the  line  AC  (Plate  VIII.  fig.  5),  and  make  it  equal  to  the  whole  rise  of  the 
tide,  10  feet,  taken  from  any  scale  of  equal  parts,  and  divide  the  line  into  equal  parts, 
representing  feet,  at  the  points  1,  9,  3,  &c.  to  10,  the  mark  10  (corresjtonding  to  the 
whole  rise  of  the  tide)  being  at  the  point  C  ;  and  through  these  points  draw  lines 
11,  22,  33,  (Sec,  perjjendicular  to  AC,  to  meet  the  circumference  of  a  circle  drawn  on 
the  diameter  AC.  Divide  the  scmicircunifcrence  ABC  of  this  circle  into  a  number 
of  equal  parts  representing  the  number  of  hours  elapsed  from  low  to  high  water  * 
(which,  in  this  case,  is  6^h.),  the  hour  of  low  water  being  marked  at  A,  and  tiiat  of 
high  water  at  C,  the  intermediate  hoiu's  being  marked  in  succession,  as  in  the  figiu-e  ; 
then,  any  hour  being  found  on  the  arc,  the  number  of  the  line  drawn  perpendicular  to 
AC,  and  passing  through  the  hour,  will  rcj^resent  nearly  the  number  of  feet  to  be 
subtracted  from  a  sounding  taken  at  that  time,  to  reduce  it  to  low  water.  Thus  the 
number  of  feet  corresponding  to  8h.  is  between  4  and  5,  because  the  mark  Bh.  falls 
i)ctween  the  lines  marked  4  and  5;  therefore  the  reduction  is  between  4  and  5  feet,  on 
soundings  taken  at  8,  A.  ]\T.,  to  reduce  them  to  low  water,  on  the  day  of  observation 
and  if,  on  that  day,  the  tide  docs  not  ebb  so  much  as  on  a  sjjring  tide,  the  reduction 
nnist  be  increased  by  the  difference  in  the  ebbing  of  the  two  tides.  Thus,  if,  on  the 
day  of  observation,  the  tide  did  not  ebb  so  nuich  by  two  feet  as  on  a  spring  tide,  the 
reduction  corresponding  to  8h.  must  be  increased  two  feet,  and  will  therefore  be 
between  6  and  7  feet.  Allowance  may  be  made  for  this,  by  increasing  the  number 
of  feet  given  in  figure  5,  by  marking  2  feet  at  A,  3  feet  at  1,  4  feet  at  2,  &c.,  as  is. 
evident 

To  reduce  a  draught  to  a  smaller  sccde. 

With  a  black-lead  pencil,  draw,  on  the  draught  to  be  reduced,  cross  lines,  Hn-ming 
exact  squares ;  and  on  the  clean  paper  for  the  copy  draw  the  same  munber  of  squares^ 
making  their  sides  larger  or  smaller  in  proportion  to  the  intended  size  of  the  scale, 
such  as  h,  J,  &c.,  the  length  of  the  other.  Distinguish  by  a  stronger  mark  every  fillh  or 
sixth  row  of  squares  in  both,  so  that  the  several  coiTcsjionding  squares  may  be  readily 
perceived  ;  then,  in  each  of  the  squares  of  the  draught,  draw,  by  the  eye,  a  curve  on 
the  pai)er,  similar  to  that  in  the  square  of  the  copying-draught,  till  the  whole  is  copied, 
when  the  black-lead  lines  may  be  rubbed  out  with  bread  or  India-rubber. 


*Tliis  division  of  the  semicircle  may  be  made  by  means  of  a  line  of  chords  ;  the  number  of  deffreei 
corresponding-  to  one  hour  being  found  by  salving,  "As  the  whole  elapsed  time  from  low  to  high  water 
(G.I  hours)  is  to  180°,  so  is  one  hour  to  the  arc  corresponding  to  1  hour,  27°  42',  which,  being  taken 
from  a  line  of  chords,  and  laid  off  from  5h.,  will  reach  to  Gh  ,  &c. 


116 


SURVEYING. 


A  chail  may  also  be  reduced  in  the  following  manner,  thus  : — Suppose  you  would 
reduce  a  chart  in  the  ratio  of  the  line  MN  (Plate  VIII.  fig,  6)  to  HI.  Draw  the  line 
AC,  and  make  it  equal  to  HI ;  upon  A,  as  a  centre,  describe  the  arc  CF,  and  make 
tlie  chord  CF  equal  to  MN ;  join  AF ;  then,  if  you  take  any  distance,  AB,  you  wish 
to  reduce,  and,  upon  A  as  a  centre,  describe  an  arc  BD,  the  chord  BD,  intercepted  by 
the  lines  AC,  AF,  will  be  the  reduced  distance  corresponding  to  AB.  This  reduced 
distance  may  also  be  obtained  by  another  method,  which  is  more  simple  than  the 
former : — Take  any  extent  from  the  large  chart,  which  is  to  be  reduced  to  a  smaller 
scale,  and  apply  it  from  A  to  O  (Plate  VIII.  fig.  7) ;  take  in  your  compasses  the  corre  • 
spondjng  distance  on  the  small  chai't,  and,  with  one  foot  in  O,  sweep  an  arc  P ;  draw 
the  line  AP  just  touching  the  arc  in  P;  then,  if  you  take  any  distance  from  the  great 
chart,  and  apply  it  from  A  to  R,  and,  at  the  point  R,  sweep  an  arc  S  to  touch  the  line 
AP,  the  extent  RS  will  be  the  I'educed  distance  corresponding  to  the  line  AR. 


Comparison  of  the  French  Brasse  rvilh 
the  English  Fathom. 


Comparison  of  the  Spanish  Brxt. a  ivilh 
the  Enslish  Fathom. 


Brasses. 

Fathoms. 

Brasses. 

Fathoms. 

1 

0.9 

9 

8.0 

2 

1.8 

10 

8.9 

3 

2.7 

20 

17.8 

4 

3.6 

30 

26.6 

5 

4.4 

40 

35.5 

G 

5.3 

50 

44.4 

7 

6.2 

75 

66.6 

8 

7.1 

100 

88.8 

Brazos. 

P'athoms. 

Brazos. 

Fathoms. 

1 

0.9 

9 

8.3 

2 

1.9 

10 

9.3 

3 

2.8 

20 

18.5 

4 

3.7 

30 

27.8 

5 

4.6 

40 

37.1 

(3 

5.6 

50 

46.4 

7 

6.5 

75 

69.5 

8 

7.4 

100 

93.7 

Compayison  of  the.  Russian  Sashe  tvith 
the  English  Fathom. 


Comparison  of  the  Dutch  Palm   icith 
the  En":lish  Fathom. 


Sashes. 

Fathoms. 

Sashes. 

Fathoms. 

1 

1.2 

<) 

10.5 

2 

2.3 

10 

11.7 

3 

3.5 

20 

23.3 

4 

4.7 

30 

35.0 

5 

,5.8 

40 

46.7 

G 

7.0 

50 

58.3 

7 

8.2 

75 

87.5 

8 

9.3 

100 

116.7 

Palms. 

Fathoms. 

Palms. 
9 

Fathoms. 
0.492 

1 

0.055 

2 

0.109 

10 

0.547 

3 

0.164 

20 

1.094 

4 

0.219 

30 

1.641 

5 

0.274 

40 

2.188 

6 

0.328 

50 

2.735 

7 

0.383 

75 

4.103 

8 

0.438 

100 

5.470 

PI  ah-  Vlfl 


EA.<'AV.r.l,L-.VT. 
18GI 


117 


OF    WINDS. 


The  earth  is  surrounded  by  a  fine,  invisible  fluid,  called  atV,  which,  by  its  weight,  is 
capable  of  supporting  the  vapors  raised  by  the  sun,  and,  by  its  elasticity,  is  capable  of 
expanding  or  spreading  itself  so  as  to  fill  up  a  larger  space.  When  the  elasticity 
of  any  portion  of  the  aii'  is  changed,  by  the  heat  of  the  sun  or  by  other  causes,  the 
neighboring  i)arts  are  put  in  motion  to  restore  the  equilibrium.  In  this  manner  a 
current  of  air  is  formed,  called  the  Wind,  which  is  distinguished  by  several  names,  viz. 
trade  ivinds,  monsoons,  vmiahle  tvinds,  &c.  The  trade  tvinds  blow  constantly  from 
the  same  part ;  the  monsoo7is  blow  half  the  yeai-  one  way,  and  half  the  other  ;  and  the 
variahle  icimls  are  such  as  blow  without  any  regularity  either  as  to  time,  place,  or 
direction.  The  following  obsenations  on  the  wind  have  been  made  by  Dr.  Halley 
and  others. 

There  are  constant  trade  winds,  blowing  from  the  east,  in  most  parts  of  the  Atlantic 
and  Pacific  Oceans,  between  the  latitudes  of  30°  N.  and  30°  S.  Near  the  northern 
limits  of  these  winds,  they  blow  between  the  north  and  east ;  and  neai"  their  southeni 
liiiiits,  between  the  south  and  east. 

In  tlie  Atlantic  Ocean,  at  about  100  leagues  from  the  coast  of  Africa,  between  the 
latitudes  of  28°  and  10°  north,  there  is  generally  a  fresh  gale  of  wind  blowbig  from  the 
N.  E. 

Those  bound  to  the  Caribbee.  Islands,  across  the  Atlantic,  find,  as  they  approach 
the  American  side,  that  the  N.  E.  wind  becomes  easterly,  or  seldom  blows  more  than 
a  point  from  the  east,  either  to  the  northward  or  southward. 

These  trade  winds,  on  the  American  side,  are  sometimes  extended  to  30°,  31°,  or 
even  to  32°  of  north  latitude,  which  is  about  4°  farther  than  what  they  extend  to  on 
the  African  side  ;  also  to  the  southward  of  the  equator,  the  trade  winds  extend  3  or  4 
degrees  farther  towards  the  south,  on  the  coast  of  Brazil,  on  the  American  side,  than 
they  do  towards  the  Cape  of  Good  Hope,  on  the  African  side. 

But  we  nnist  not  conclude  that  the  above  limits  are  without  exception  ;  for  both 
their  extent  and  direction  vary  considerably  with  the  season  of  the  year.  When  the 
sun  approaches  the  tropic  of  cancer,  the  S.  E.  trade  winds  ])revail  farther  to  the  north- 
ward of  the  line,  and  incline  more  to  the  southward  of  S.  E ;  and  the  N.  E  trade  wind 
inclines  more  to  the  eastward ;  and  the  contraiy  at  the  opposite  season  of  the  year. 

On  the  African  coast,  from  Cape  Blanco  to  Sierra  Leone,  the  winds  m  genei-al  blow 
from  the  north,  inclining  from  the  westward  rather  than  from  the  eastward.  From 
Sierra  Leone  to  Caj)e  Palmas,  the  ordinary  course  of  the  winds  is  from  W.  N.  W.,  and 
beyond  Cajie  Palmas,  as  far  as  28°  soiuh  latitude,  from  S.  W.  to  S.,  hiclining  more  to 
the  southward  or  westward,  according  to  the  particular  situation  or  bearing  of  the 
shores  and  lands  ;  and  the  [)art  of  the  ocean  extending  along  this  coast,  to  the  distance 
of  80  or  100  leagues  from  the  shore,  is  much  troubled  with  frequent  calms,  and  \vith 
sudden  and  violent  gusts  of  wind,  knomi  by  the  name  of  tornadoes,  which  blow  from 
all  parts  of  the  horizon  The  reason  of  tliis  change  in  the  direction  of  the  trade  wind 
near  the  land,  is  prolw^'y  owing  to  the  nature  of  the  coast,  which,  being  violently 
heated  by  the  sun,  rarefies  the  air  exceedingly ;  consequently  tlie  cool  air  from  the  sea 
will  keej)  rushing  in  to  restore  the  equilibrium. 

In  the  Gulf  of  Guinea,  there  is  a  periodical  wind,  called  harmattan,  which  blows  in  a 
N.  E.  direction  from  the  interior  parts  of  Africa.  The  season  in  which  this  wind 
prevails,  is  during  the  months  of  December,  January,  and  Febrtiary. 

Between  the  4th  and  10th  degrees  of  north  latitude,  and  between  the  longitude  of 
Cape  Verd  and  the  easternmost  of  the  Cape  Verd  Islands,  there  is  a  tract  of  sea 
which  seems  to  be  very  liable  to  calms,  attended  with  much  thunder  and  lightning,  and 
frequent  rains.  The  cause  of  this  seems  to  be,  that  the  westerly  winds,  setting  m  on 
the  coast  of  Africa,  and  meeting  the  general  easterly  winds  in  this  tract,  balance  each 
other,  and  so  cause  the  calms ;  and  tlie  vapors,  carried  thither  by  each  wind,  meeting 
and  condensing,  occasion  the  almost  constant  rains. 

These  observations  show  the  reason  of  the  difficulty  which  shi[«  find  in  sailuig  t« 


lis  OF   WINDS. 

the  southward,  between  the  coasts  of  Guinea  and  Brazil,  particularly  in  the  months  of 
July  and  August,  notwithstanding  the  width  of  the  sea  is  more  tlian  500  leagues;  for 
the  S.  E.  winds,  at  that  tune  of  the  year,  commonly  extend  some  degi-ees  beyond  their 
ordinary  limits  of  4°  north  latitude,  and  become  more  southerl}^,  so  as  to  be  sometimes 
south,  or  a  point  or  two  to  the  west  of  south.  It  then  only  remains  to  ply  to  wind- 
ward ;  and  if,  on  the  one  side,  they  steer  W.  S.  W.,  they  get  a  wind  more' and  more 
easterly ;  but  then  there  is  danger  of  falling  in  with  the  coast  or  shoals  of  Brazil ; 
and  if  they  steer  E.  S.  E.  they  fall  into  the  neighborhood  of  the  coast  of  Guinea, 
whence  tliey  cannot  depart  without  running  easterly  as  far  as  the  island  of  St 
Thrmas. 

When  ships  depart  from  Guinea  for  Europe,  their  direct  course  is  northward  ;  but 
on  this  course  they  cannot  go,  because,  the  coast  trending  nem-ly  east  and  west,  the 
land  is  to  the  northward.  Therefore,  as  the  winds  on  this  coast  are  generally  between 
the  south  and  W.  S.  VV.,  they  are  obliged  to  steer  S.  S.  E.  or  south,  and  with  these 
courses  they  run  off  the  shore ;  but,  in  so  doing,  they  always  find  tlie  wind  more  and 
more  contrary,  so  that  though,  when  near  the  shore,  they  can  lie  south,  at  a  gi-eat 
distance  they  can  make  no  better  than  S.  E.,  and  afterwards  E.  S.  E.,  with  which 
courses  they  generally  fetch  the  island  of  St.  Thomas  or  Cape  Lopez,  where  findin" 
the  wind  to  the  eastward  of  the  south,  they  sail  westerly  with  it,  till,  comiu"-  to  the 
latitude  of  4  degrees  south,  they  find  the  S.  E.  wind  blowing  perpetually. 

On  account  of  these  general  winds,  all  bound  from  Europe  to  the  West  Indies,  or  to 
the  southern  States  of  America,  consider  it  most  advantageous  to  get  as  soon  as  they 
can  to  the  southward,  so  the*'  "lay  be  certain  of  a  fair  and  fresh  gale^to  run  before  it  to 
the  westward.  For  the  same  reason,  those  bound  from  the  southern  States  of  America 
to  Europe  endeavor  to  gain  the  latitude  of  30  degrees,  where  they  first  find  the  wind 
begin  to  be  variable,  though  the  most  ordinary  wuids  in  the  North  Atlantic  Ocean 
come  between  the  south  and  west. 

And,  for  the  same  reasons,  those  bound  to  India  from  America  run  to  the  eastward 
in  the  variable  winds,  so  as  to  be  in  the  longitude  of  35°  or  38°  W.  when  in  the  latitude 
of  30°  N.  From  thence  they  steer  south-easterly  towards  the  Cape  de  Verds,  passing 
4°  or  5°  to  the  westward  of  them,  unless  they  wish  to  stop  for  supplies.  Being  then  in 
the  com.mon  route  ofkhe  Ein-opean  Indiamen,  they  steer  southerly  to  cross  the  equator 
between  the  longitude  of  20°  W.  and  28°  W.,  where,  meeting  the  S.  E.  trade  winds, 
they  must  brace  up  and  sail  upon  a  wind  till  they  get  through  them  and  come  into 
the  variable  winds,  where  they  may  steer  to  the  eastward.  Near  the  equator,  the  trade 
wind  is  generally  stronger  to  the  westward  than  to  the  eastward  ;  and  were  it  not  for 
the  fear  of  falling  in  with  the  Brazil  coast,  a  ship  miglit  cross  the  line  even  farther 
to  the  westward.  Ships  homeward  bound,  fi-om  the  Cape  of  Good  Hope  towards 
America,  may  deviate  a  little  to  the  westward  of  their  straight  course,  and  cross  the 
equator  in  the  longitude  of  30°  W.,  or  even  as  far  as  33°  W.,  in  order  to  take  advantage 
of  this  fresher  trade  wind. 

Between  the  southern  latitudes  of  10°  and  30°  in  the  Indian  Ocean,  the  general  trade 
^^■inds  about  S.  E.  are  found  to  blow,  all  the  year  round,  in  the  same  manner  as  in  the 
like  latitudes  in  the  South  Atlantic  Ocean  ;  and  during  the  six  months  from  May  to 
November,  these  winds  reach  to  within  2  degrees  of  the  equator;  but  during  the  other 
six  months,  from  November  to  ]May,  a  N.  W.  wind,  called  the  little  monsoon,  blows  in 
tlie  tract  lying,  between  the  3d  and  10th  degrees  of  south  latitude,  in  the  meridian  of 
the  uortli  end  of  Madagascar,  and  between  the  2d  and  12th  degrees  of  south  latitude, 
near  the  longitude  of  Sumatra  and  Java. 

In  the  tract  between  Sumatra  and  the  African  coast,  and  from  3°  of  south  latitude 
quite  northward  to  the  Asiatic  coast,  including  the  Arabian  Sea  and  the  Bay  of 
Bengal,  the  monsoons  blow  from  Octo])er  to  April  on  the  N.  E.,  and  from  April  to 
October  on  tlie  S.  W.  In  the  former  half-year,  the  wind  is  more  steady  and  gentle, 
and  the  weather  clearer,  than  in  the  latter  six  months.  In  the  Red  Sea,  the  winds  blow 
nearly  nine  months  of  the  year  from  the  southward,  that  is,  from  August  to  May,  aufl 
the  rest  of  the  year  from  the  N.  and  N.  N.  W.  with  land  and  sea  breezes.  In  the 
Gulf  of  Persia,  from  October  to  July,  the  winds  blow  from  the  N.  W.,  and  about  three 
months  from  the  opposite  quarter  ;  these  winds  being  often  interrupted  by  gales  from 
the  S.  W.,  and  by  land  breezes. 

Between  the  island  of  Bladagascar  and  the  coast  of  Africa,  and  thence  northward  as 
far  as  the  equator,  there  is  a  tract  wherein,  from  April  to  October,  there  is  generally  a 
S.  S.  W.  wind,  and  a  contraiy  Avind  the  I'est  of  the  year,  with  regular  land  and  sea 
breezes  on  both  coasts. 

To  the  eastward  of  Sumatra  and  Malacca,  on  the  north  of  the  equator,  and  along 
the  coasts  of  Cambodia  and  China,  quite  through  the  Philippines  as  far  as  Japan,  the 
monsoons  blow  N.  E.  and  S.  W.,  the  N.  E.  setting  m  about  October  or  November 
and  the  S.  W.  about  3Iay 


r 


OF    WINDS.  119 

Between  Sumatra  and  Java  to  the  west,  and  New  Guinea  to  the  east,  there  are 
regular  monsoons.  The  N.  W.  monsoon  blows  from  October  to  April;  the  S.  E. 
monsoon  the  rest  of  the  year. 

The  monsoons  do  not  shift  suddenly  from  one  pouit  of  the  compass  to  the  opposite 
In  some  places  die  time  of  the  change  is  attended  with  calms,  in  others  by  variable 
wincis;  and  it  often  hai)i)ens,  on  the  shores  of  Coromandel  and  China,  towards  the  end 
of  the  monsoons,  that  there  are  most  violent  storms  called  tij-Jbo7igs,  greatly  resem- 
bling the  hurricanes  in  the  West  Indies,  wherein  the  wind  is  so  violent,  that  hardly 
any  thing  can  resist  its  force;  for  diis  reason,  it  is  more  dangerous  to  approach  these 
shores  at  tlie  tune  of  the  breaking  up  of  tlie  monsoon,  dmn  at  any  other  season  of 
the  year. 

The  land  and  sea  breezes  prevail  ])rincipally  between  the  tro])ics.  The  sea  breeze 
generally  sets  in  about  ten  in  the  forenoon,  and  continues  till  ai)out  five  or  six  in 
the  evening:  at  seven  the  land  breeze  begins,  and  contuuics  till  about  eight  in  the 
morning.  The  cause  of  these  winds  is  this  : — During  the  day,  the  sea  is  not  so  much 
heated  by  the  sun  as  the  laud,  nor  so  mucii  cooled  at  night.  Hence,  iu  the  day  time, 
tlie  cooler  air  from  the  sea  will  rush  towards  the  land,  to  supjily  the  deficiency 
occasioned  by  the  greater  rarefaction  of  the  air  ;  and  from  this  arises  the  sea  breeze. 
In  like  manner,  during  the  night,  the  air  at  land,  being  more  cooled  than  that  at  sea, 
will  therefore  blow  from  the  land  towards  the  sea,  and  occasion  a  land  breeze. 

A  ichirlwind  is  a  dangerous  phenomenon,  caused  by  the  atljacent  air  rushing  in  from 
all  parts  towards  a  centre  with  great  rapidity,  and  sometimes  destroying  every  oljject 
it  passes  over  in  its  ))rogressive  motion.  Waterspouts  and  whirlwinds  arise  from  the 
same  cause :  the  latter,  being  formed  at  land,  are  composed  principally  of  air ;  but  tht; 
former,  being  formed  at  sea,  are  composed  of  water. 

It  was  first  observed  by  Dr.  Franklin,  that  the  N.  E.  storms,  on  the  coast  of  the 
United  States  of  America,  frequently  begin  earlier  in  the  southern  States  than  in  the 
northern.  This  he  accounts  for  by  sup})osing  a  great  rarefaction  of  air  in  or  near  the 
Gulf  of  Mexico  ;  the  air  rising  thence  has  its  place  supplied  by  the  next  more  northern, 
and  therefore  denser  and  heavier  air ;  a  successive  cui-rent  is  thus  formed,  to  which 
the  coast  and  inland  mountains  give  a  N.  E.  direction. 

Experiments  have  been  made  by  several  persons  to  detennine  the  velocity  of  the 
wind,  by  observing  the  space  passed  over  by  a  cloud  or  any  light  substance,  and  by 
other  methods;  and  it  has  been  found  tiiat  tlie  velocity  of  tlie  wind,  in  a  violent  fjale, 
is  about  50  or  60  miles  per  hour. 


TIDES. 


The  tides  are  periodical  changes  of  level  of  the  water  occurring  generally  twice  in 
each  lunar  day.  Tlie  rise  of  the  tide  is  known  as  the  flood  and  its  fall  as  the  ebb,  the 
highest  rise  of  any  flood  being  called  high  tide  or  high  water  or  full  sea,  and  the  lowest 
fail  of  any  ebb  being  termed  low  tide  or  low  water.  Each  ebb  and  each  flood  occupies 
about  six  lunar  houi'S.  The  rise  and  fall  of  the  tide  is  the  difference  of  level  at  low  and 
high  Avater.  These  periodical  changes  of  level  known  as  the  tides  should  be  carefully 
distin"-uished  from  the  effects  which  they  produce,  known  as  tidal  currerds.  These 
refer  to  the  horizontal  motion  of  the  water, 

The  cause  of  the  tides  is  the  unequal  attraction  of  the  sun  and  moon  upon  different 
parts  of  the  earth ;  for  they  attract  the  parts  of  the  earth's  surface  nearer  to  them  with 
a  o-reater  force  than  they  do  its  centre,  and  attract  the  centre  more  than  they  do  the 
opposite  surface.  To  restore  the  equilibrium,  the  waters  take  a  spheroidal  figure, 
whose  longer  axis  is  directed  towards  the  attracting  body.  The  mean  force  of  the  sun 
in  raising  the  tide  is  to  that  of  the  moon  only  as  1  to  2^,  for  though  the  mass  of  the  sun 
is  vastly  greater  than  that  of  the  moon,  its  distance  causes  it  to  attract  the  different 
parts  of  tlie  earth  with  nearly  the  same  force.  A  small  inland  sea,  such  as  the  Medi- 
terranean or  Baltic,  is  little  subject  to  tides,  because  the  action  of  the  sun  and  moon  is 
always  nearly  equal  at  the  extremities  of  said  seas.  The  mathematical  theory  of  the 
tides  has  not  yet  reached  the  point  Avhere  the  tides  at  any  given  place,  or  even  the  changes 
from  tide  to  tide  at  the  same  place,  can  be  calculated  by  merely  knowing  the  position 
of  the  sun  and  m.oon  Avithout  resort  to  observation.  Nevertheless,  by  theory  combined 
with  observation,  we  are  enabled  to  predict  the  tides  Avithin  moderate  limits. 

High  Avater  occurs  on  the  average  of  the  twenty-eight  days,  comprising  a  lunar 
month,  at  about  the  same  inter\'al  after  the  time  of  the  moon's  crossing  (transit  over) 
the  meridian.  This  nearly  constant  interval,  expressed  in  hours  and  minutes,  is  knoAvn 
as  the  luni tidal  interval.  The  observed  interval  at  the  time  of  full  and  change  of  the 
moon  at  any  port  is  called  the  establishment  of  the  port,  a  word  which  is  in  common 
use  among  navigators,  and  the  amount  of  which  is  designated  on  the  charts  by  Eoman 
numerals  Tind  fractions.  Thus  (vii|),  near  Sandy  Hook,  on  a  chart  denotes  that  seven 
hours  and  a  half  after  the  moon's  transit  on  full  and  change  days  high  water  Avill  occur. 
The  average  of  all  the  lunitidal  intervals  in  a  month  which  gives  a  more  correct  result, 
taking  one  day  with  another  in  the  course  of  the  month,  has  been  termed  by  Mr. 
WheAvell  (one  of  those  Avho  have  recently  done  most  for  the  knoAvledge  of  the  tides) 
"■corrected  establishment^''  and  to  distinguish  the  other  number  it  is  called  the  "vulgar 
(or  common)  establishment."  In  our  tables  of  establishment,  the  corrected  ones  are 
specially  marked  and  are  for  the  ports  of  the  United  States,  the  same  with  those  given 
upon  the  Coast  Survey  charts. 

The  highest  tides  do  not  occur  at  the  precise  time  of  full  and  new  moon,  but  subse- 
quent to  full  and  change.  Upon  our  Atlantic  coast  they  occur  one  day  after,  and  on 
the  Atlantic  coast  of  Europe  tAVO  days  after,  but  on  our  Pacific  coast  nearly  at  full  and 
change.  The  highest  tides  are  called  spring  tides,  and  the  lowest,  occurring  Avhen  the 
njQSSas  near  the  fii:si_and  third  quarters,  are  called  nsia^  tides.  At  the  periods  of  full 
and  change  the  attraction  of  the  sun  and  moon  conspire  to  raise  the  tide  at  a  given  place; 
at  the  first  and  last  quarter  the  high  Avater  produced  by  the  moon  Avould  occur  at  the 
time  of  the  Ioav  Avater  caused  by  the  sun,  and  vice  versa,  so  that  the  two  actions  oppose 
each  other. 

By  fixing  a  st.iff  graduated,  say,  into  feet  and  inches,  against  the  vertical  face  of  a 
pier  or  Avharf,  and  observing  the  mark  Avhich  the  water  reaches  at  low  tvaier,  we  shall 
see,  after  some  minutes,  a  sIoav  rise  of  the  AA'ater  begin,  groAAmig  more  and  more  rapid 
for  about  three  hours,  then  gradually  slackening  for  three  more,  until,  as  it  nears  six 
hours  from  the  first  observation,  it  again  stands  for  some  minutes  Avhen  it  begins  to  fall 
toAvards  Ioav  Avater,  accelerating  as  before  for  three  hours,  and  then  slacking  off  again. 
The  two  periods  during  Avhicli  the  Avater  neither  rises  nor  falls  are  called  the  hir/h  water 
stand  and  low  water  stand,  or  sometimes  slack  water,  a  term  Avhich,  to  avoid  confu- 
sion, it  is  best  to  apply  only  to  tidal  currents.  The  A^arying  rate  of  rise  and  fall 
of  the  tide  differs  much  at  different  places.  It  is  shoAvn  at  New  York  and  Liver- 
pool in  the  diagrams,  Nos.  1  and  2  on  Plate  IX.  The  hours  are  placed  on  the  hori- 
zontal line,  and  the  heights  Avhich  the  Avater  reaches  upon  the  staff  on  the  vertical  line. 
The  curve  shoAvs  the  rate  of  rise  and  fall.     The  same  result  is  giA^en  in  Table  A,  wnere, 


I'l.ite    IX 


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TIDES. 


121 


TABLE   A. 

Showing  the  rate  of  rise  and  fall 

of  the  tide  at  New  York  and 

Liverpool. 

Ilours  before 

Height  of  tide. 

or  after 

Low   water. 

New  York.   Liverpool. 

brs. 

ft. 

ft. 

(G 

4.2 

18.9 

5 

3.7 

1G.2 

|. 

4 

2.9 

10.4 

3 

1.8 

6.2 

P 

2 

0.9 

3.0 

1 

0.2 

0.9 

0 

0.0 

0.0 

ri 

0.5 

1.8 

2 

1.6 

5.6 

t^ 

3 

2.7 

11.0 

<5 

4 

.    3.6 

16.1 

5 

4.1 

19.7 

6 

4.4 

20.7 

opposite  to  each  hour,  from  low  water  is  shown  the  height  which  the  level  of  the  water 
would  mark  upon  a  staff  the  0  of  Avhich  was  at  low  water. 

This  curve  or  this  table  will  enable  the  navi- 
gator to  conjecture  the  probable  rise  and  fall 
from  low  or  high  water  at  ports  where  the  rise 
and  fall  is  about  the  same  as  at  New  York  or 
at  Liverpool,  but  will  not  apply  to  others. 

If  watching  this  tide  staff  from  day  to  day  in 
some  port  upon  our  coast  we  should  note  the 
time  of  high  and  low  water,  and  the  height  be- 
ginning wilh,  say,  two  days  after  change  day  of 
the  moon,  and  continuing  for  a  lunar  month  or 
twenty-eight  days,  we  should  find  that  on  that 
day  the  lunitidal  interval  was,  nearly  the  average 
of  all  which  we  would  obtain  in  the  course  of 
the  month,  and  that  the  water  rose  higher  and 
fell  lower  than  at  any  otlier  high  and  low 
water.  These  are  spring  tides.  The  interval 
would  go  on  decreasing  until  two  days  before 
the  first  quarter,  when  it  would  reach  its  least 
value.  The  height  of  high  water  would  de- 
crease, and  of  low  water  increase,  until  one  day 
after  the  first  quarter,  when  the  one  would  reach 
its  least  and  the  other  its  greatest  height,  corre- 
sponding to  neap  tides^  or  least  rise  and  fall  of 
the  water.  From  the  period  of  its  least  value 
to  three  days  before  the  full  the  lunitidal  interval 
would  increase  and  then  decrease,  and  so  onward  to  two  days  af'ter  the  full,  when  the 
interval  would  have  its  average  value  again,  and  the  heights  would  again  correspond 
to  spring  tides.  The  corresponding  changes  in  the  lunitidal  intervals  and  heights  take 
place  from  the  full  to  change,  passing  through  the  moon's  third  quarter.  It  is  hardly 
necessary  to  remind  the  navigator  that  at  change  the  moon  and  sun  cross  the  meridian 
together,  or  the  hour  of  transit  is  0  hrs.,  and  that  at  the  first  quarter  the  hour  of  transit 
(moon's  southing)  is  6  hrs.,  at  the  full,  12  hrs.  This  change  in  the  lunitidal  interval 
runs  its  course  from  change  to  full  or  full  to  change,  that  is,  m  a  half  lunar  month ;  it  is 
'hence  called  the  half  monthly  inequality,  and  is  in  general  the  largest  of  the  changes  in 
the  lunitidal  interval,  wliich  must  be  taken  into  account. 

If  there  were  no  changes  in  the  lunitidal  interval,  it  would  be  very  simple  to  deter- 
mine the  time  of  high  or  low  water  at  a  place.  A  table  of  intervals  and  an  almanac 
showing  the  time  of  transit,  or  as  it  is  sometimes  called  in  the  almanacs  the  time  of  the 
moon's  southing,  would  be  all  that  is  necessary.  Suppose  we  wish  to  determine  the 
time  of  high  water  at  Boston  on  the  12  th  of  December,  1859.  From  the  table  of  es- 
tablishments. No.  LV.,  we  take  that  of  Boston,  llh.  27m.;  from  a  Boston  almanac,  the 
time  of  the  moon's  upper  transit  on  that  day  Ih.  59m.,  A.  M.,  adding  the  two  numbers 
we  have  13h.  26i-ri.,  or  Ih.  26m.,  P.  M.,  as  the  time  of  high  water.  The  corresponding 
low  water  is  6h.  after,  or  more  exactly  6h.  13m.  So,  if  the  heights  did  not  change, 
one  number  in  the  table  would  give  us  the  rise  and  fall.  This  supposes  that  we  had 
an  almanac  of  the  port  at  which  we  desired  to  know  the  time  of  high  water,  but  as 
this  would  usually  not  be  the  case,  we  must  take  our  result  from  the  Nautical  Almanac, 
with  which  we  are  provided.  This  referring  to  the  time  of  transit  of  the  moon  over  the 
meridian  of  Greenwich,  and  to  the  same  meridian  for  the  longitude,  2m.  must  be  added 
to  the  time  of  transit  at  Greenwich  for  every  hour  of  west  longitude,  and  subtracted 
for  every  hour  of  east  longitude.  The  same  result  may  be  had  from  the  table  B, 
where  the  numbers  to  be  added  to  the  time  of  the  moon's  transit  are  given  for  every 
ten  degrees  of  longitude. 

Rule  I.— Find  the  time  of  the  moon's  coming  to  the  meridian  of  Greenwich,  on  the 
given  day  in  the  Nautical  Almanac.  Enter  Table  B  and  find  the  longitude  of  tiie  given 
plaee  in  the  left  han;i  column,  corresponding  to  which  is  a  number  of  minutes  to  be  ap- 
plied to  the  time  of  passing  the  meridian  at  Greenwich,  by  adding  when  in  luest  longi- 
tude, but  subtracting  Avhen  in  east  longitude ;  the  sum  or  difference  will  be  nearly  the 
time  that  the  moon  passes  the  meridian  of  the  given  place. 

To  this  corrected  time  add  the  time  of  high  water  or  full  sea  from  Table  LV.  The 
sum  will  bo  the  time  of  high  water  on  that  day. 

Example  I. — Required  the  time  of  high  water  at  Charleston  (S.  C.),  November  19, 
1859,  in  the  afternoon,  civil  account.  From  the  Nautical  Almanac  we  find  the  moon's 
meridian  passage  at  Greenwich,  November  18,  at  19h.  26ui.,  which  corresponds  to  7h. 

16 


122 


TIDES. 


26m.,  A.  M.,  of  the  19th  day  by  civil  account.  From  Table 
LIV.  we  have  the  longitude  of  Charleston  79°  54'  W.,  Avhich, 
for  this  purpose,  may  be  assumed  as  80°.  Entering  Table  B 
with  80°,  we  find  the  correction  of  the  moon's  passing  the 
meridian  to  be  11  minutes,  which  is  to  be  added  as  the  longi- 
tude is  west.  The  moon's  meridian  passage  at  Charleston  is 
therefore  at  7h.  3Tm.,  A.  M.  Adding  to  this  the  lunitidal  in- 
terval 7h.  13m.  from  Table  LV.  we  obtain  14:h.  5Qm.,  or  2h, 
50m.,  P.  M.,  as  the  time  of  high  water  at  Charleston  in  the 
afternoon  of  November  19,  1859. 

Example  II. — Required  the  time  of  high  water  at  Portland 
(]\Iaine),  Deceml^er  13,  1859,  in  the  afternoon,  civil  account. 
The  jSTautical  Almanac  gives  the  moon's  meridian  passnge  at 
14h.  47m.  on  the  12th,  corresponding  to  2h.  47m,  A.  M.,  on 
the  13th.  The  longitude  of  Portland  is  70°  12'  W.,  in  time 
(Table  XXI.)  4h.  41m.  At  the  rate  of  two  minutes  for  every 
hour  of  west  longitude  we  should  add  9m.  to  the  Greenwich 
time  of  the  moon's  meridian  passage,  giving  it  for  Portland  at 
2h.  56m.  Adding  the  lunitidal  interval  from  Table  LV.  llh. 
25m.,  gives  14h.  21m.,  or  2h.  21m.j  P.  M.,  for  the  time  of  high 
water  on  December  13th. 

These  results  would  be  the  time  of  high  water,  did  not  the 
lunitidal  interval  vary. 

If  the  changes  of  lunitidal  interval  Irom  half  monthly  ine- 
quality were  the  same  for  all  ports,  it  would  be  easy  by  a 
table  of  a  single  column  to  apply  the  required  correction  to 
the  time  of  high  water  when  the  moon  was  not  at  full  or 
change  but  this  is  not  the  case.  It  has  been  found,  however,  that  the  general  law  of  this 
change' is  the  same,  and  that  by  knowing  the  greatest  and  least  lunitidal  interval  for  any 
port  we  can  determine  by  computation  the  change  of  interval.  The  ports  having  nearly 
the  same  difference  of  greatest  and  least  interval  are  grouped  together,  and  the  correction 
to  be  applied  to  the  establishment,  according  to  the  age  of  the  moon,  is  given  in  Table  C. 
The  ports  Avhich  may  thus  be  classed  together  are  the  following :  a.  The  ports  of 
Encfland  and  of  the  western  coast  of  Europe  in  general,  b.  The  ports  on  tiie  eastera 
or  Atlantic  coast  of  the  United  States,  c.  The  ports  of  the  western  coast  of  Florida 
and  of  the  western  or  Pacific  coast  of  the  United  States. 

This  table  is  arranged  on  the  supposition  that  the  corrected  estabhshment  is  used, 
■which  is  the  case  for  the  more  important  ports  in  Table  LV. 

In  other  parts  of 


TABLE  B. 

Longitude 

of 
the  place. 

CoiTc'Ction 
of  moon's 

passini   the 
iiieriiiian. 

dea. 

inin. 

0 

0 

10 

1 

20 

3 

30 

4 

40 

5 

56 

7 

60 

8 

70 

9 

SO 

11 

90 

12 

100 

14 

110 

15 

120 

16 

130 

18 

140 

19 

150 

20 

100 

22 

170 

23 

180 

24 

TABLE   C. 

Time  of 
Moon's 
transit. 

pioup 
(a.) 

group 

group 
(^'■) 

Oh 

add  41m 

add  19m 

Om 

1 

"     17 

"       6 

subt.  17 

2 

sul;t.    11 

subt.     8 

"     32 

3 

"     27 

"     16. 

"     44 

4 

"     40 

"     22 

"     47 

5 

"     47 

"     24 

'•     35 

0 

"     41 

"     19 

"       0 

7 

u     17 

«       6 

add  17 

8 

add  11 

a<ld     8 

"     32 

9 

"     27 

"     16 

"     44 

10 

"    40 

"     22 

"     47 

11 

"     47 

"    24 

"     35 

the  world  than 
those  mentioned  in 
the  groups  a,  b, 
c,  the  half-monthly 
inequality  is  Uttle 
known  ;  the  fol- 
lowing table,  form- 
ed by  averag- 
ing the  three  col- 
umns of  Table  C, 
will  probably  give 
a  sufficient  ap- 
proximation. The 
corrections  are  to 
be  applied  to  the 
vulgar  estdblish- 
ment. 

Thus,  in  Exam- 
ple I.,  given  before,  the  time  of  the  moon's  meridian  passage  being  7h.  37m.,  we 
enter"the  table  with  that  quantity  in  the  column  of  time  of  the  moon's  transit,  and 
under  the  head  of  group  6,  and  by  an  easy  proportion  we  find  the  correction  to  the 
lunitidal  interval  to  be,  "  add  3,"  that  is,  three  minutes  must  be  added  to  the  mean  luni- 
tidal interval  at  Charleston,  making  it  7h.  16m.,  which,  added  to  the  tmie  of  moons 
transit,  would  give  2h.  53m.,  P.  M.,  as  a  more  accurate  time  for  the  high  water  of 
November  19,  1859.  .„     ,      ,       „,     r^ 

In  Example  II.  Ave  had  the  time  of  the  moon's  transit  at  Portland  at  21i.  obm.,  en- 
terino-  Table  C  with  3h.  in  the  column  of  moon's  transit  (which  is  near  enough  for  this 
purpose),  we  find  in  the  column  of  group  b  a  correction  of  "subt  16m.,"  i.  e.,  sixteen 
minutes  must  be  subtracted  from  the  mean  lunitidal  interval,  makmg   it   llh.  9m., 


Time  of 

moon's 

Correction. 

transit. 

Oh 

Om 

1 

subt.  18 

2 

"     37 

3 

"     49 

4 

"     50 

5 

"     55 

6 

"     40 

7 

"     22 

8 

"       3 

9 

add     9 

10 

"     IG 

11 

"     15 

TIDES. 


123 


wliich  added  to  tbe  time  of  moon's  transit,  gives  14h.  5m.,  or  2h.  om.,  P.  M.,  for  the 
time  of  high  water  on  December  13  th. 

The  changes  of  the  moon  in  declination  cause  a  tide.once  in  twenty-four  lunar  hours, 
which  adds  itself  to  the  morning  high  water,  increasing  it,  and  subtracts  itself  from  the 
next,  or  afternoon,  high  water,  or  vice  versa.  This  is  called  the  diurnal  inequality.  It 
affects  the  time  and  the  height  of  both  high  and  low  water.  In,  most  of  the  ports  of 
the  G-ulf  of  Mexico  this  diurnal  tide  is  the  only  marked  one,  except  when  the  moon  is 
near  the  equator.  In  the  ports  of  Great  Britain  and  Ireland,  France  and  Spain,  the 
diurnal  inequality  in  height  is  marked,  but  in  time  is  inconsiderable.  On  the  Atlantic 
coast  of  the  United  States  it  is  small  both  in  time  nnd  height.  It  increases  in  passing 
along  the  straits  of  Florida  to  the  western  coast  of  the  Florida  peninsula,  and  the  semi- 
diurnal tides  almost  disappear  from  Cape  San  J51as  to  the  mouths  of  the  Mississippi,  re- 
appearing only  slightly  between  Isle  Derniere  and  Galveston,  and  again  being  merged 
in  the  diurnal  tide  from  Aransas  Pass  to  Vera  Cruz,  and  probably  southward.  The  small 
tide  of  the  day  is  frequently  called  by  navigators  a  half  tide,  and  in  speaking  of  the  large 
and  small  tides  of  the  day  they  say  the  tide  and  half- tide.  On  the  western  coast  of  the 
United  States  this  inequality  is  large  both  in  time  and  height,  amounting  at  San  Fran- 
cisco at  its  greatest  value  to  two  and  a  half  hours  of  time  and  four  feet  of  height.  It  is 
probably  large  on  the  whole  western  coast  of  South  America,  but  observations  are  want- 
ing to  give  information  in  regard  to  the  tii.les  of  these  localities. 

The  following  table  will  give  the  corrections  tor  the  daily  inequality  in  time  and  height 
for  the  Pacific  coast  of  the  United  States  to  within  about  eight  minutes  of  time  and 
and  three  inches  of  height. 

The  quantities  in  this  table  are  the 
corrections  to  be  applied  to  the  times 
of  high  or  low  water  obtained  by  means 
of  Rule  I  and  corrected  by  Table  C. 

Rule. — Find  from  the  Nautical  Al- 
manac the  number  of  daj'S  elapsed  since 
the  moon's  declination  was  greatest,  or 
if  before,  the  number  of  days  to  come 
to  that  time.  With  this  enter  Table 
D  in  the  first  column,  and  opposite  the 
number  find  the  correction  in  the  sec- 
ond colum.  "When  the  moon's  declina- 
tion is  north,  the  correction  is  to  be 
subtracted ;  when  south,  it  is  to  be 
added.  (V7hen  the  moon's  declination 
is  nothing,  the  correction  is  nothing. 
The  foui't'h  and  fifth  columns  give  the 
corrections  to  the  heights  of  mean  high 
water  and  mean  low  water  for  I  he  saffie 
days.  The  corrections  for  the  height  of  low  wati^TTollow  tTie  same  rulelxs  those  for 
the  times  of  high  water;  but  for  the  heights  of  high  Avater  they  are  the  contrary,  that 
is,  they  are  to  be  subtracted  Avhen  the  former  are  to  be  added,  and  vice  versa. 

The  effects  of  this  inequality  may  be  also  expressed  in  the  following  way:  The 
moon's  declination  being  north,  the  high  water  next  following  the  moon's  transit  will 
be  earlier  and  higher  than  the  average,  the  next  low  water  later  and  lower,  the  next 
high  water  later  and  lower,  and  the  next  low  water  earlier  and  higher;  when  the 
moon's  declination  is  south,  the  first  high  water  is  later  and  lower,  the  next  low  water 
earlier  and  higher,  the  next  high  water  earlier  and  higher,  and  the  next  low  water  later 
and  lower,  by  the  amounts  given  in  the  table. 

Example. — Required  the'  time  of  high  water  at  San  Francisco,  October  16,  1859. 
By  Rule  I.,  we  find  the  moon's  transit  to  happen  at  3h.  21m.,  A.  M.,  on  that  day.  The 
establishment  for  San  Francisco,  from  Table  LV.,  is  12h.  6m.,  which  added  to  3h.  21m., 
gives  loh.  27m.,  or  3h.  27m.,  P.  M.,  as  the  time  of  high  water,  uncorrected  for  tlie  half 
monthly  and  diurnal  inequalities.  The  former  is  obtained  from  Table  C,  group  c,  and 
is  45m.,  which  is  to  be  subtracted,  giving  2h.  42m.;  the  second  is  obtained  from 
Table  D.  By  referring  to  the  Nautical  Almanac,  we  find  that  on  the  given  day  the 
moon  had  her  greatest  declination  north.  Enterin.^,  therefore,  the  table  with  0  day 
fiom  greatest  declination,  we  find  corresponding  to  it  in  the  second  column  64m.,  to 
be  subtracted,  as  the  decUnation  is  north,  giving  Ih.  38m.  as  the  time  of  high  water. 
If  the  corrections  had  been  neglected,  we  should  have  been  nearly  two  houi-s  in  error. 
The  same  table  tells  us  in  the  other  columns  that  this  high  water  would  be  1.0  foot 
higher  than  an  average  high  water,  and  the  next  low  water  1.8  foot  lower.  Tlie  next 
high  water,  A.  M.,  of  the  17th,  would  be  one  foot  lower  than  the  average,  or  two  feet 
lower  than  the  above  high  Avater,  the  next  low  water  1.8  feet  higher  than  the  average, 
or  3.6  feet  higher  than  the  preceding  one. 


TABLE   p. 

Days  ft-oin 
moon's 

Lunitidal  intervals. 

Ileiglits. 

greijtest 

High 

Low 

High 

Low 

declination. 

water. 

water. 

water. 

wate/. 

min. 

min. 

tt. 

ft. 

0 

64 

38 

1.0 

1.8 

1 

62 

37 

0.9 

1.8 

o 

55 

35 

0.9 

1.6 

3 

45 

31 

0.8 

1.4 

4 

33 

23 

0.7 

1.0 

5 

22 

18 

0.4 

0.7 

6 

9 

6 

0.2 

0.3 

7 

0 

0 

0 

0 

124 
CURRENTS. 


A  CURRENT  is  a  progressive  motion  of  the  water,  causing  all  floating  bodies  to 
move  that  way  towards  which  the  stream  is  directed.  The  set  of  a  current  is  that 
point  of  the  compass  towards  which  the  waters  run,  and  its  drift  is  the  rate  it  runs 
per  hour.  The  most  usual  way  of  discovering  the  set  and  drift  of  an  unknown  cur- 
rent, is  the  following,  supposing  the  current  at  the  surface  to  be  much  more  pow- 
erful than  at  a  great  distance  below  the  surface : — 

Take  a  boat  a  short  distance  from  the  ship,  and,  by  a  rope  fastened  to  the  boat's 
stern,  lower  down  a  heavy  iron  pot  or  loaded  kettle  to  the  depth  of  80  or  100  fath- 
oms; then  heave  the  log,  and  the  number  of  knots  run  out  in  half  a  minute  will  be 
the  miles  the  current  sets  per  hour,  and  the  bearing  of  the  log  will  show  the  set  of  it. 

There  is  a  very  remarkable  current,  called  the  Gulf  Stream,  which  sets  in  an 
north-east  direction  along  the  coast  of  America,  from  Cape  Florida  tov/ards  the 
Isle  of  Sables,  at  unequal  distances  from  the  land,  being  about  75  miles  from  the 
shore  of  the  southern  States,  but  more  distant  from  the  shore  of  the  northern  States. 
The  width  of  the  stream  is  about  40  or  50  miles,  widening  towards  the  north. 

We  were  first  indebted  to  Doctor  Franklin,  Commodore  Truxton,  and  Mr.  Jon- 
athan Williams,  for  the  knowledge  we  possess  of  the  direction  and  velocity  of  this 
stream.  Its  general  course,  as  given  by  them,  is  marked  on  the  chart  affixed  to 
this  v/ork.  They  all  concur  in  recommending  the  use  of  the  thermometer,  as  the 
best  means  of  discovering  when  in,  or  near,  the  stream ;  for  it  appears,  by  their 
observations,  that  the  water  is  warmer  than  the  air  when  in  the  stream  ;  and  that 
at  leaving  it,  and  approaching  towards  the  land,  the  water  will  be  found  six  or  eight 
degrees  colder  than  in  the  stream,  and  six  or  eight  degrees  colder  still  when  on 
soundings.  Vessels  coming  from  Europe  to  America,  by  the  northern  passage, 
should  keep  a  little  to  the  northward  of  the  stream,  where  they  may  probably  be 
assisted  by  a  counter  current.  When  bound  from  any  southern  port  in  the  United 
States  of  America  to  Europe,  a  ship  may  generally  shorten  her  passage  by  keeping 
m  the  Gulf  Stream.  By  steering  N.  W.  you  will  generally  cross  it  in  the  shortest 
lime,  as  its  direction  is  nearly  N.  E.     (See page  6,  Notes  and  Correction's.) 

In  other  parts  of  the  Atlantic  Ocean,  the  currents  are  variable,  but  are  generally 
south-easterly  along  the  coast  of  Spain,  Portugal,  and  Africa,  from  the  Bay  of  Bis- 
cay towards  Madeira  and  the  Cape  de  Verds.  Between  the  tropics,  there  is  gen- 
erally a  current  setting  to  the  westward. 

There  is  also  a  remarkable  current  which  sets  through  the  Mozambique  Channel, 
between  the  Island  of  Madagascar  and  the  main  continent  of  Africa,  in  a  south- 
westerly direction.  In  proceeding  towards  Cape  Lagullas,  the  current  takes  a  more 
westerly  course,  and  then  trends  round  the  cape  towards  St.  Helena.  Ships  bound 
to  the  westward  from  India,  may  generally  shorten  their  passage  by  taking  advan- 
tage of  this  current.  On  the  contrary,  wnen  bound  to  the  eastward,  round  the  Cape 
of  Good  Hope,  they  ought  to  keep  far  tb  the  southward  of  it.  However,  there  ap- 
pears to  be  a  great  difference  in  the  velocity  of  this  current  at  difterent  times;  for 
some  ships  have  been  off  this  cape  several  days  endeavoring  to  get  to  the  west- 
ward, and  have  found  no  current;  others  have  experienced  it  setting  constantly  to 
the  westward,  during  their  passage  from  the  cape  towards  St,  Helena,  Ascension, 
and  the  West  India  Islands.  Instances  have  however  occurred,  where  an  easterly 
current  was  experienced  off  the  Cape  of  Good  Hope.  Off  Cape  Horn  there  is  a 
current  setting  N.  80°  E.,  at  the  rate  of  12  miles  the  24  hours,  during  the  summer 
months — during  the  autumn  months  it  is  accelerated  nearly  double,  and  sets  N. 
49^  E. 


The  following  is  compiled  from  a  communication  of  Lieut.  Bent  to  Mr.  G.  W.  Blunt, 
respecting  a  stream  of  warm  water,  tvhich  is  found  on  the  east  coasts  of  Formosa 
and  the  Japan  Islands. 

This  stream  has  its  origin  in  the  great  Equatorial  current  of  the  Pacific,  from 
which  it  is  separated  by  the  south  end  of  Formosa,  whence  it  is  deflected  to  fhe 
nortliward  along  the  east  coast  of  that  island,  until  reaching  the  parallel  of  SG'' 
north,  when  it  bears  off  to  the  northward  and  eastward,  washing  the  whole  souths 
east  coast  of  Japan  as  far  as  the  Straits  of  Sanger. 

Near  its  origin  the  stream  is  contracted,  and  seems  to  be  usually  confined  bo 


CURRENTS. 


125 


tween  the  islands  of  Formosa  and  Majico-Suica,  with  a  breadth  of  100  miles ;  but  to 
the  northward  of  the  latter  it  expands  rapidly  on  its  southern  limit  and  reaches  the 
Lew-Chew  and  Benin  groups,  attaining  a  width  to  the  northward  of  the  latter  of  500 
miles.  The  north-western  edge  of  the  stream  is  strongly  marlved  by  a  sudden  change 
in  the  temperature  of  from  10^  to  20°  ;  but  the  south-eastern  limit  is  less  distinctly 
defined.  Along  the  borders  of  the  stream,  and  also  in  its  midst,  where  whirls  and 
eddies  are  produced  by  islands  and  inequalities  in  its  bed,  strong  tide  rips  are  en- 
countered. 

The  average  strength  of  the  current  between  the  south  end  of  Formosa  and 
the  Straits  of  Sanger  is  from  35  to  40  miles  per  day.  Its  maximum  once  off  the 
Gulf  o?  Yedo  was  observed  as  high  as  72,  74  and  80  miles  respectively  per  day. 
A  cold  counter-current  may  exist  io  the  north  of  40°  and  long.  143°,  running  through 
the  Straits  of  Sanger;  but  to  the  loestward  of  a  line  connecting  the  north  end  of 
Formosa  and  the  south-western  extremity  of  Japan,  a  cold  current  sets  to  the 
southward,  through  the  Formosa  Channel,  into  the  China  Sea. 

This  current  is  well  known  to  the  navigators  trading  on  the  const  of  China, 
who  never,  in  the  north-east  monsoon,  attempt  to  beat  against  it,  but  make  the 
passage  usually  to  the  eastward  oi  Formosa. 

The  Japanese  call  this  warm  stream,  setting  along  their  southern  shores  to  the 

color, 
tempera- 
ture and  that  of  the  ocean  due  to  the  latitude  is  on  an  average  about  12°. 

There  is  ?io  counter-current  intervening  between  the  Kuro-Sicoo  and  the  coast 
of  Japan  south  of  the  Straits  of  Sanger,  consequently  the  large  bod}?-  of  warm  water 
which  washes  the  shores  of  the  island  must  essentially  contribute  in  modifying  its 
climate. 


northward  and  eastward,  the  Kuro-Sicoo.  or  Black  Stream,  from  its  deep  blue 
Its  maximum  temperature  is  about  86*,  and  the  difference  between  its  tem 


All  cases  of  sailing  in  a  current  are  calculated  upon  the  principle  that  the  ship 
is  affected  by  it  in  the  same  manner  as  if  she  had  sailed  in  still  water,  with  an  ad- 
ditional course  and  distance  exactly  equal  to  its  set  and  drift.  On  this  principle  the 
projection  and  calculation  of  any  problem  of  this  kind  may  be  easily  mad  3 


EXAMPLE. 

If  a  ship  sail  98  miles  N.  E.  by  N.,  in  a  current  which  sets  S.  by 
W.  27  miles,  in  the  same  time,  required  her  true  course  and  distance. 

BY    PROJECTION. 

Describe  the  compass  NESW ;  througli  tlie  centre 
A  draw  the  N.  E.  by  N.  line  AC  equal  to  98  miles ; 
through  C  draw  the  line  BC  parallel  to  the  S.  by  W. 
line,  make  BC  equal  to  27  miles,  and  join  AB.  Then 
AB  will  be  the  course  and  distance  made  good ;  and 
by  measuring,  we  find  the  course  to  be  N.  E.  |  N.,  the 
distance  74  miles. 

BY   CALCULATION 

The  shortest  method  of  calculating 
this  problem,  is  by  means  of  Table 
I.,  as  in  the  adjomed  Traverse  Table ; 
putting  in  it  the  coui-se  sailed  by  the 
ship,  and  the  set  of  the  current ;  then 
finding  the  difference  of  latitude  and 
departure  by  the  table.  The  course 
and  distance  made  good  is  then 
found  as  in  Case  VI.  of  Plane  Sail- 
ing. In  tlie  present  example,  the 
coui-se  is  N.  E.  i  N.,  and  the  distance 
74  miles  nearly. 


TRAVERSE 

TABLE. 

Courses. 

Dlst. 

N. 

S. 

E. 

W. 

N.E.by  N. 
S.  by  W. 

98 
27 

81.5 

26.5 

54.4 

5.3 

81.5 
2G.5 

2(J.5 

.54.4 
5.3 

5.3 

Di 

ff.  Lat.. 

.55.0 

Dep.  4n.l 

126 


OF    THE    LOG-LINE    AND    HALF-MINUTE 

GLASS. 


Various  methods  have  been  proposed  for  measurhig  the  rate  at  which  a  ship  sails; 
but  that  most  in  use  is  by  the  Log  and  Half-3Iinute  Glass. 

The  Log  is  a  flat  piece  of  thm  board,  of  a  sectoral  or  quadrantal  form  (see  Plate  VI 
fig.  3),  loaded,  on  the  cii'cular  side,  with  lead  sufficient  to  make  it  swim  upright  in  the 
water.  To  this  is  fastened  a  line,  about  150  fathoms  long,  called  the  log-line,  which 
is  divided  into  certain  spaces  called  knots,  and  is  wound  on  a  reel  (see  Plate  VL  fig.  4) 
which  tm-ns  ^'ery  easily.  The  Half-Minute  Glass  is  of  the  same  form  as  an  Hour 
Glass  (see  Plate  VI.  fig.  2),  and  contains  such  a  quantity  of  sand  as  will  run  tlu-ough 
tlie  hole  in  its  neck  in  half  a  minute  of  time. 

The  making  of  the  experiment  to  find  the  velocity  of  the  ship,  is  called  heaving  the 
log,  which  is  thus  performed: — One  man  holds  the  reel,  and  another  the  half-minute 
glass ;  an  ofiicer  of  the  watch  throws  the  log  over  the  ship's  stem,  on  the  lee  side,  and 
when  he  observes  the  stray  line  is  run  off  (which  is  about  ten  fathoms,  this  distance 
being  usually  allowed  to  carry  the  log  out  of  the  eddy  of  the  ship's  wake),  and  the  first 
mark  (which  is  generally  a  red  rag)  is  gone  off",  he  cries.  Turn;  the  glass-holder 
answers.  Done ;  and,  watching  the  glass,  the  moment  it  is  run  out,  says.  Stop.  The 
reel  being  immediately  stopped,  the  last  mark  run  off  shows  the  number  of  knots, 
and  the  distance  of  that  mark  from  the  reel  is  estimated  in  fathoms.  Then  the 
knots  and  fathoms  together  show  the  distance  the  ship  has  run  the  preceding  hour,  if 
the  wind  has  been  constant.  But  if  the  gale  has  not  been  the  same  during  the  whole 
hour,  or  interval  of  time  between  heaving  the  log,  or  if  there  has  been  more  sail  set  or 
handed,  a  jn-opcr  allowance  must  be  made.  Sometimes,  when  the  ship  is  before  the 
wmd,  and  a  great  sea  setting  after  her,  it  will  bring  home  the  log.  In  such  cases,  it  is 
customary  to  allow  one  mile  in  ten,  and  less  in  proportion  if  the  sea  be  not  so  great. 
Allowance  ought  also  to  be  made,  if  there  be  a  head  sea. 

This  jn-actice  of  measuring  a  ship's  rate  of  sailing,  is  founded  upon  the  following 
princij)le — that  the  length  of  each  knot  is  the  same  part  of  a  sea  mile,  as  half  a  minute 
is  of  an  hour.  Therefore  the  length  of  a  knot  ought  to  be  ^x^  of  a  sea  mile  ;  but,  by 
various  admeasurements,  it  has  been  found  that  the  length  of  a  sea  mile  is  about  GUyOj'^. 
feet ;  hence  the  length  of^a  sea  knot  should  be  51  feet.  Each  of  these  knots  is  divided 
into  10  fiithoms,  of  about  5  feet  each.  If  the  glass  be  only  28  seconds  in  running 
out,  the  length  of  the  knot  ought  to  be  47  feet  and  6  tenths.  These  are  the  lengtlis 
generally  recommended  in  books  of  navigation  ;  but  it  may  be  observed,  that,  in 
many  trials,  it  has  been  found  that  a  ship  will  generally  oveiTun  her  reckoning 
with  a  log-line  thus  marked ;  and,  since  it  is  best  to  err  on  the  safe  side,  it  has  been 
generallyrecommended  to  shorten  the  above  measures  by  3  or  4  feet,  making  the 
length  of  a  knot  about  7A  fathoms,  of  6  feet  each,  to  correspond  with  a  glass  that  runs 
28  seconds. 

In  heavin.g  the  log,  you  must  be  careful  to  veer  out  the  line  as  fast  as  the  log  will 
take  it ;  for  if  the  log  be  left  to  turn  the  reel  itself,  the  log  will  come  home  and  deceive 
you  in  your  reckoning.  You  must  also  be  careful  to  measure  the  log-line  pretty  often, 
iest  it  stretch  and  deceive  you  in  the  distance.  Like  regard  nnist  be  had  that  the  half- 
minute  glass  be  just  30  seconds  ;  otherwise  no  accurate  account  of  the  shij)'s  way  can 
be  kept.  The  glass  is  much  influenced  by  the  weather,  running  slower  in  damp 
weather  than  in  dry.  The  half-minute  glass  tnay  be  examined  by  a  watch,  with  a 
second  hand,  or  by  the  following  method: — Fasten  a  jjhunmet  on  a  line,  and  hang 
it  on  a  nail,  observing  tliat  the  distance  between  the  nail  and  middle  of  the  phnnmet 
be  39 J  inches;  tlien  swing  the  plummet,  and  notice  how  often  it  swings  while  the 
glass  is  running  out,  and  that  will  be  the  number  of  seconds  measured  by  the  glass. 


OF  THE   LOG-LINE   AND   HALF-MINTJTE   GLASS. 


127 


To  correct  tlie  distance  ichcn  the  log-line  and  half-minute  glass  are  faidty 

If  there  be  any  error  in  the  log-lhie  or  glass,  the  measured  distance  must  be 
corrected  m  the  following  manner,  supposing  that  a  30"  glass  requires  50  feet  to  a 
knot : — 

(L)  If  the  glass  only  is  faulty,  you  must  say,  As  the  seconds  run  by  the  glass  are  to 
30  seconds,  so  is  the  distance  given  by  the  log  to  the  tnie  distance.  Thus,  if  a  ship  sails 
8i  knots  ])er  hour,  by  a  glass  of  3G  seconds,  the  true  number  of  knots  i)er  hour  will  be 
7.1 ;  for  36  :  30  : :  8.5  :  7.1. 

(2.)  If  tlie  log-line  only  is  faulty,  you  must  say.  As  fifty  feet  is  to  the  distance  of  a 
knot  071  the  line,  so  is  the  distance  rim  by  the  log  to  the  true  distance.  Thus,  if  a  ship 
sails  7  knots  per  hour,  by  a  log-line  measuring  53  feet,  her  true  distance  will  be  7.4 
miles  per  hour;  because  50  :  53  ::  7  :  7.4. 

(3.)  If  the  log-line  and  glass  are  both  faulty,  you  must  say.  As  50,*  midtiplied  by  the 
length  of  the  glass,  is  to  30,  multiplied  by  the  length  of  the  line,  so  is  the  measured  to  the 
true  distance.  Thus,  if  a  ship  sails  G  knots  per  hour,  with  a  glass  of  24  seconds,  and 
a  log-iiue  of  GO  feet  per  knot,  her  true  velocity  will  be  9  miles  per  hour,  because 
50  X  24  :  30  X  60  ::  6  :  9. 

The  following  description  of  Massey's  Patent  Log,  used  in  surveying  operations,  is 
from  Ca],:.  Edward  Belcher's  treatise  on  Nautical  Surveying: — 

"  It  is  composed  of  a  brass  wedge-shaped  box,  having  within,  three  cogged  wheels, 
acting  on  each  other  in  such  proportion  that  a  total  revolution  of  one  completes  a  division 
of  the  next,  (or  one-twentieth,)  a  revolution  of  the  next  one-eighth,  registering  thus  from 
one  hundred  and  sixty  miles  to  tenths,  and  decimal  parts ;  the  action  is  by  the  rotation  of 
a  spindle  with  four  spirally  fixed  wings,  (termed  the  rotator,  or  fly,)  which  turns  an  end- 
less screw  in  the  box,  acting  directly  on  the  decimal  wheel.  It  is  towed  astern  by  a 
stout  lead  line  of  sixty  fathoms,  and  is  registered  every  time  the  course  is  changed,  angles 
taken,  &,c.,  but  should  not  be  reset  until  the  twenty-four  hours  have  elapsed,  the  ship 
anchors,  or  goes  less  than  three  knots — (when  it  becomes  uncertain  from  not  towing 
horizontally.)" 

*  Instead  of  multiplying'  the  length  of  the  glass  by  50,  and  the  line  by  30,  you  may  multiply  the 
former  by  5,  and  ihe  latter  by  3.  If  any  one  chooses  to  mark  the  log-line  at  less  than  50  Icct  for  a  glass 
ofTO  seconds,  he  must  put  his  estimated  length  of  the  knot,  instead  of  50,  in  all  the  above  rules. 


TABLE, 

Showing  tlie  length  of  a  mile  of  Longitude,  in  feet,  for  different  Latitudea 

[A  Geograpbical  or  Nautical  Mile  at  the  Equator  is  6086.4.] 


Lat. 

Eng. 
feet. 

Lat. 

Eng. 
feet. 

Lat. 

Eng. 

feet. 

Lat. 

Eng. 
feet. 

Lat. 

Eng. 
feet. 

Lat. 

Eng. 
feet. 

0 

6086 

15 

5880 

30 

5275 

45 

4311 

60 

3051 

75 

1580 

I 

6086 

16 

5852 

31 

5222 

46 

4235 

61 

2958 

76 

'477 

2 

6083 

17 

5822 

32 

5166 

47 

4158 

62 

2865 

77 

1374 

3 

6078 

18 

5790 

33 

51 10 

48 

4080 

63 

2771 

78 

1269 

4 

6072 

19 

5757 

34 

5051 

49 

4001 

64 

2675 

79 

1 165 

5 

6063 

20 

5722 

35 

4991 

50 

3920 

65 

2579 

80 

1&60 

6 

6053 

21 

5685 

36 

4930 

51 

3838 

66 

2482 

81 

955 

7 

6041 

22 

5646 

37 

4867 

52 

3755 

6-1 

2385 

82 

850 

8 

6028 

23 

5605 

38 

4S02 

53 

3671 

68 

2287 

83 

744 

9 

6015 

24 

5663 

39 

4736 

54 

3585 

69 

2188 

84 

638 

10 

5995 

25 

5519 

40 

4669 

55 

3499 

70 

2088 

85 

532 

II 

5975 

26 

5474 

41 

4600 

56 

3411 

71 

1987 

86 

426 

IZ 

5954 

27 

5427 

42 

4530 

57 

3323 

72 

18S7 

87 

320 

13 

5931 

28 

5378 

43 

4458 

58 

3233 

73 

1785 

88 

213 

H 

5907 

29 

5327 

44 

4385 

59 

3142 

74 

1683 

89 

107 

»5 

5880 

.30 

5275 

45 

4311 

60 

3051 

75 

1580 

90 

000 

128 


OESCIlIPTiON   AND    USE   OF  A  QUADRANT 

OF  REFLECTION. 


Mr.  John  Hadley  was  tlie  first  who  published  a  description  of  the  Quadrant  of 
Reflection,  for  measuring  angular  distances ;  and  the  instrument  still  bears  his  name, 
although  it  has  been  ascertained  that  Sir  Isaac  JVctvton  invented  a  similar  one  some 
years  before,  but  never  made  it  public.  One  of  our  countrymen,  Mr.  Thomas  Godfrey, 
of  Philadeli)hia,  had  also  contrived  an  instrument,  on  the  same  principles,  some  time 
before  Mi:  Hadley  made  known  his  discovery. 

Plate  IX.,  figure  1,  represents  a  quadrant  of  reflection,  the  principal  parts  of  which 
are,  the  frame  ABC,  the  graduated  arc  BC,  the  index  D,  the  nonius  or  veniier  scale 
E,  the  index  glass  F,  the  horizon  glasses  G  and  H,  the  dark  glasses  or  screens  I,  and 
the  sight  vanes  K  and  L. 

The  graduated  arc  BC  is  an  octant,  or  eighth  part  of  a  cuxle,  but,  on  account  of  the 
double  reflection,  is  divided  into  90%  numbered  from  0°  towards  the  left,  and  each 
degree  is  commonly  divided  into  three  equal  parts,  of  20  minutes  each.  The  gi'adua- 
tiou  on  the  limb  is  continued  a  few  degi-ees  to  the  right  of  0°.  This  portion  is  called 
the  arc  of  excess,  and  is  found  very  convenient  for  several  purposes. 

The  index  D  is  a  flat  bar,  commoidy  made  of  brass,  movable  round  the  centre  of 
the  instrument,  and  broader  towards  the  axis  of  motion,  where  is  fixed  the  index  glass 
F  ;  at  the  other  end  is  fixed  the  nonius  or  vernier  scale,  used  in  estimating  the 
subdivisions  of  the  arc  ;  at  the  bottom  or  end  of  the  index,  there  is  a  piece  of  brass 
which  leads  luider  the  arc,  having  a  spring  to  make  the  vernier  lie  close  to  the  limb, 
and  a  screw  to  fasten  it  in  any  position.  Some  quadrants  have  a  tangent  screw 
aflixed  to  the  lower  part  of  the  index  to  adjust  its  motion.  The  vernier  is  a  small, 
narrow  slij)  of  brass  or  ivoiy,  fixed  to  that  part  of  the  index  which  slides  over  the 
gi-aduated  arc,  and  usually  contains  a  space  equal  to  21  or  19  divisions  of  the  liujb,  and 
is  divided  into  20  equal  parts.  Hence  the  difference  between  a  division  on  the  limb, 
and  a  division  on  the  dividing  scale,  is  one  twentieth  of  a  division  of  the  limb,  or  one 
minute.  Therefore,  if  any  division  on  the  veniier  is  in  the  same  straight  line  with  a 
division  of  tlie  limb,  then  no  other  division  on  the  vernier  can  couicide  with  a  division 
of  the  limb,  the  extreme  divisions  excepted.  Some  time  ago,  it  was  usual  to  reckon  the 
divisions  on  the  vernier  from  its  middle  tov/ards  the  right,  and  from  the  left  towards 
the  middle  ;  Init,  tliis  being  found  inconvenient,  a  more  commodious  method  has  been 
introduced  of  numbering  from  right  to  left.  Hence  the  degree  and  minute  pointed  out 
oy  the  vernier,  may  be  found  thus: — Observe  what  minute  on  the  vernier  coincides 
U'ith  a  division  on  the  limb ;  then  this  minute,  being  added  to  the  degi'ee  and  parts  of 
a  degree  on  the  limb  immediately  preceding  the  first  division  on  the  vernier,  will  be 
the  degree  and  minute  required.  Thus,  suppose  10'  on  the  vernier  coincides  with  a 
division  on  the  limb,  and  tliat  the  division  on  the  limb  preceding  the  first  division  of 
the  vernier  is  8°  20' ;  the  division  pointed  out  by  the  vernier  will  be  8°  30'. 

The  index  glass  F  is  a  plane  syeculum  or  mirror  of  gliiss,  quicksilvered  and  set  in 
a  brass  frame.  It  is  so  placed  that  the  face  of  it  is  perpendicular  to  the  ])laue  of  the 
instrument,  and  is  fixed  to  the  index  by  the  screw  PtI ;  the  other  screw  N  serves  to 
replace  it  in  a  pei-pendicular  position,  if,  by  any  accident,  it  has  been  put  out  of  order. 
The  use  of  this  muTor  is  to  receive  the  rays  from  the  sim,  or  other  object  observed, 
and  reflect  them  towards  the  horizon  glasses. 

The  horizon  glasses  G  and  H  are  two  small  speculums.  G  is  called  the  fore  horizor. 
glass,  from  its  being  used  in  the  common  or  fore  ohservcfion,  where  the  observer's  face 
is  turned  towards  the  object;  and  II  the  hack  horizon  glass,  hc'wg  used  in  the  back 
obseiialion,  where  the  observer's  bade  is  turned  towards  the  object.  These  mirrors 
receive  the  reflected  rays  from  the  index  glass,  and  reflect  them  to  the  eye  of  the 
o!)servpr.  Tlie  horizon  glasses  are  not  entirely  quicksilvered.  The  fore  horizon 
glass  G  is  only  silvered  on  the  lower  half,  the  other  half  being  transparent,  and  the 
back  part  of  the  frame  cut  away,  that  the  horizon,  or  any  other  object,  may  be  seen 


USE   OF   A    QUADRAN'I    OF    REFLECTION.  129 

through  it.  Tlic  back  horizon  glass  H  is  silvered  at  both  ends ;  in  the  middle  »=>  a 
transparent  slit,  through  which  the  horizon  may  be  seen.  These  two  glasses  are  set 
in  brass  frames,  similar  to  that  of  the  index  glass,  and  fixed  on  movable  bases,  which 
are  adjusted  by  screws  so  as  to  set  the  glasses  in  their  true  positions.  In  general  there 
are  three  dark  glasses  or  screens,  I ;  two  red  ones,  of  different  shades,  and  one  green. 
Each  is  set  in  a  brass  frame,  which  turns  on  a  centre,  that  they  may  be  used  sei)arately 
or  together.  They  serve  to  defend  the  eye  from  the  rays  of  the  sun  during  an  obser- 
vation. The  green  glass  is  peculiarly  adapted  to  take  off  the  glare  of  the  moon,  but 
may  be  useil  for  the  sun  when  much  obscured  by  clouds.  When  these  glasses  are 
used  for  a  fore  observation,  they  are  to  be  fixed  as  in  figure  1 ;  but  when  used  for  a 
back  observation,  they  are  to  be  placed  at  O. 

The  sight  vanes,  K  and  L,  are  pieces  of  brass,  standing  perpendicular  to  the  j)lane 
of  the  instrument.  The  vane  K  is  called  the  fore  sight  vane,  and  L  the  back  sight  vane. 
There  are  two  holes  in  the  fore  sigiit  vane,  the  lower  of  which  and  the  u|)pcr  odire 
of  the  silvered  })art  of  the  fore  horizon  glass  are  equidistant  from  the  plane  of  the 
instrument,  and  the  other  hole  is  opposite  to  the  middle  of  the  transparent  part  of  that 
glass.  The  back  sight  vane  has  one  perforation,  which  is  exactly  opposite  to  the 
middle  of  the  traus])arent  slit  in  the  back  horizon  glass,  ,      g  n/. 

The  ailjusli)}g  lever  (fig.  2),  which  is  fixed  on  the  back  of  the  nuadrant,  serves  to  Ju./t^.r^ ' <^ ^ 
adjust  tlie  horizon  glass,  by  placing  it  parallel  to  the  index  glass.    .^Vhen  this  lever  is      /^^q^jc. 
to  be  made  use  of,  the  screw  B  must  be  first  loospned  ;  and  when,  by  the  adjuster  A»     7^    ^ 
the  horizon  glass  is  sufficiently  moved,  the  screw  (I3)must  be  fastened  agaui  ;  by  this    J^' "^ 
means  the  horizon  glass  will  be  kept  from  changing  its  position. 

To  adjust  a  quadrant. 

As  the  quadrant,  from  various  accidents,  is  liable  to  be  out  of  order,  it  is  necessary 
that  the  mariner  should  be  able  to  ascertain  the  errors,  and  re-adjust  the  several  parts, 
before  he  proceeds  to  make  his  observations.  For  this  ])urpose,  he  must  examine 
whether  the  index  glass  and  the  horizon  glasses  be  ])er|)endicular  to  the  plane  of  th.e 
instnnnent,  and  whether  the  plane  of  the  fore  horizon  glass  be  parallel,  and  that  of  the 
back  horizon  glass  perpendicular  to  the  plane  of  the  index  glass,  when  0  on  the  veniier 
stands  against  0  on  the  limb. 

1  St.     To  ascertain  whether  the  index  glass  be  perpendicidar  to  the  plane  of  the  quadrant. 

Place  the  index  on  the  middle  of  the  arc,  and  hold  the  index  glass  near  the  eye. 
Look  into  it,  in  a  direction  ])arallel  to  the  plane  of  the  instrument,  and  see  if  the 
reflected  arc  appear  exactly  in  a  line  with  the  arc  seen  direct,  or  if  the  image  of  any 
point  of  the  arc  near  B  ap[)ear  of  the  same  height  as  the  corresjionding  \rdvt  of  the 
arc  near  C  seen  direct;  if  so,  the  index  glass  is  perpendicular  to  the  ])lane  of  the 
quadrant;  if  not,  the  error  nuist  be  rectified  by  the  screws  on  the  base,  behind  the 
frame,  by  loosening  the  screw  M,  and  tightening  the  screw  N,  or  by  loosening  the 
screw  N,  and  tightening  the  screw  IM. 

2d.     To  ascertain  ivhether  the  fore  horizon  glass  be  perpendiculnr  to  the  plane  of  the 

quadrant. 

Having  adjusted  the  index  glass,  hold  the  instrument  in  a  veitical  position  Look 
through  tlie  tore  sight  vane,  and  move  the  index  till  the  reflected  and  direct  images  of 
the  horizon,  seen  in  the  horizon  glass,  coincide.  Then  incline  the  instrument  till  its 
plane  is  nearly  parallel  to  the  horizon  ;  if  the  images  still  coincide,  the  horizon  glass 
stands  perj)endicular  ;  otherwise  it  does  not,  and  must  be  adjusted  by  the  screws 
placed  before  and  behind  it,  loosening  one  of  them,  and  tightening  the  other. 

Tliis  adjustment  may  be  made  by  the  sun,  moon,  or  a  star,  by  holding  the  quadrant 
in  a  veitical  position,  and  observing  if  the  object  seen  by  reflection  appears  to  tiie  right 
or  left  of  the  object  seen  direct,  and  moving  the  screws,  as  above,  till  both  image? 
coincide. 

After  having  made  the  horizon  and  index  glasses  parallel,  according  to  the  directions 
in  the  following  article,  it  will  be  best  to  re-examine  this  ailjustment. 

3d.     To  make  the  horizon  glass  parallel  to  the  index  glass,  ivhen  0  on  the  vernier  stands 

on  0  on  the  arc. 

Having  fixed  the  index,  so  that  0  on  the  veniier  stands  on  0  on  the  arc,  look  at  any 
distant  obiect,  and  see  if  the  hnage  of  it  coincides  with  the  object  itself;  if  it  does,  the 
17 


130  USE   OF  A   QUADRANT   OF   REFLECTION. 

acljustment  is  complete  ;  if  not,  tliey  must  be  made  to  coiiiciile  by  means  of  the 
adjusting  level*.  The  horizon  may  be  used  for  this  purpose  in  the  following  manner: — 
[{old  the  plane  of  the  instrument  vertical ;  look  through  the  lower  hole  in  the  vane  K, 
and  direct  the  sight  through  the  transparent  part  of  the  glass  G  to  the  horizon ;  then  if 
the  horizon  line,  seen  in  the  silvered  and  transparent  part,  coincides,  or  makes  one 
straight  line,  the  horizon  glass  is  said  to  be  adjusted  ;  but  if  the  horizon  lines  do  not 
coincide,  slacken  the  screw  B  (fig.  2)  in  the  middle  of  the  adjusting  lever,  and  turn  the 
Jiorizon  glass  on  its  axis  until  the  horizon  lines  coincide  ;  then  fix  the  lever  firmly  bj"^ 
tightening  the  screw  B.  If  this  adjustment  be  again  examined,  it  will  perhaps  be  found 
impei-fect.  In  this  case,  therefore,  it  remains  either  to  repeat  the  adjustment,  or  find 
the  error  of  it  (usually  called  the  index  error),  which  may  be  done  thus: — Let  the 
hoi'izon  glass  remain  fixed,  and  move  the  index  till  the  image  and  object  coincide  ; 
then  the  difference  between  0  on  the  vernier  and  0  on  the  arc  is  the  index  error, 
which  is  to  be  added  to  the  angle  or  altitude  observed,  if  tlie  0  on  the  vernier  be 
to  the  right  hand  of  0  on  the  arc,  otherwise  to  be  subtracted.  Thus,  if  the  horizon 
is  used,  the  instrument  being  held  in  a  vertical  position,  you  must  look  through  the 
lower  hole  of  the  vane  K,  towards  the  horizon  ;  then  move  the  index  till  the  reflected 
and  direct  images  of  the  horizon  coincide ;  the  difference  between  0  on  the  vernier 
and  0  on  the  arc  will  be  the  index  error. 

4th.     To  adjust  the  back  horizon  glass,  that  it  may  be  perpendicidar  to  the  plane  of  the. 
index  glass,  when  0  on  the  vernier  stands  on  0  on  the.  arc. 

Set  the  index  as  fiir  to  the  right  of  0  on  the  arc,  as  twice  the  dip  of  the  horizon 
(talien  from  Table  XIII.) ;  hold  the  quadrant  in  a  vertical  position ;  look  towards  the 
horizon  through  the  hole  in  the  back  horizon  vane  L,  and  the  transparent  slit  of 
the  back  horizon  glass  II ;  then,  if  the  reflected  horizon,  which  will  appear  inverted, 
coincide  with  that  seen  direct,  the  glass  is  truly  adjusted ;  otherwise  the  screw,  in  the 
centre  of  the  lever  on  the  imder  side  of  the  quach-ant,  must  be  slackened,  and  the  glass 
timied  on  its  axis  till  both  horizons  coincide,  when  the  lever  should  be  fixed  by 
tightening  the  screw. 

5th.     To  adjust  the  back  horizon  glass,  that  it  may  be  perpendicular  to  the  plane  of  tJit 

quadrant. 

Put  the  index  on  0  ;  hold  the  quadrant  nearly  parallel  to  the  horizon  ;  look  through 
the  hole  on  the  back  sight  vane,  and  if  the  true  and  reflected  horizons  appear  in  the 
same  straight  line,  tlie  glass  is  perpendicular  to  the  plane  of  the  instrument ;  but  if 
they  do  not  coincide,  the  sunk  screws,  before  and  behmd  the  glass,  must  be  turned  till 
both  appear  to  form  one  straight  line. 

To  take  an  altitude  of  the  sun  by  a  fore  observation. 

If  the  sun  is  bright,  turn  down  one  or  more  of  thft  dark  glasses;  hold  the  instrument 
in  a  vertical  position  ;  apply  the  eye  to  the  upper  hole  in  the  fore  sight  vane,  when 
the  image  is  so  bright  as  to  be  seen  in  the  transj)arent  part  of  the  fore  horizon  glass, 
otherwise  to  the  lower  hole  ;  direct  the  sight  to  that  part  of  the  horizon  beneath  the 
sun,  and  move  the  index  till  you  bring  the  image  of  his  lower  limb  to  touch  the 
horizon  directly  under  it ;  but  as  this  point  cannot  be  exactly  ascertained,  the  observer 
should  move  the  instrument  round  to  tlie  right  and  left  a  little,  keejjing,  as  nearly  as 
possible,  the  sun  always  in  that  part  of  the  horizon  glass  which  is  at  the  same  distance 
as  the  eye  from  the  plane  of  the  quadrant;*  by  this  motion  the  sun  will  appear  to 
sweep  the  horizon,  and  must  be  made  to  touch  it  at  the  lowest  j;art  of  the  arc  ;  the 
degrees  and  minutes  pointed  out  by  the  hidex,  will  be  the  observed  altitude  of  the 
sun's  lower  limb  at  that  instant. 

To  take  an  altitude  of  the  moon  by  a  fore  observation. 

In  the  night,  Avhen  the  moon  is  bright,  her  image  may  be  seen  in  the  trans.parcni 
part  of  the  fore  horizon  glass,  and  the  observation  may  be  taken  exactly  in  the  same 

*  In  common  qiiadrants,  if  ihe  upper  hole  lie  looked  throiigli,  the  sun's  imai^'c  must  he  made  to  appeal 
in  the  middle  ol"  me  transparent  part  of  the  horizon  glass  ;  but  if  (he  louor  hole  be  looked  Uirovi<;h,  tl!3 
imag'e  must  be  made  to  appear  on  the  line  joiniiig  the  silvered  and  Ira.isparent  parts  of  the  horizon 
fjlass,  as  these  parts  of  the  horizon  glass  are  at  llie  same  distances  from  the  plane  of  the  instrument,  as 
(lie  holes  of  the  sight  vanes  respectively. 


USE   OF  A   QUADRANT   OF  REFLECTION.  131 

manner  as  an  observation  of  the  sun.  If  tlie  image  is  so  faint  as  not  to  l)e  seen  in  tlie 
transparent  part  of  the  horizon  glass,  you  must  set  the  index  to  0  ;  hold  the  plane  of 
tlie  quadrant  in  a  vertical  position ;  direct  the  sight  to  the  moon,  and,  at  the  same 
time,  look  for  her  reflected  image  in  the  silvered  part  of  the  horizon  glass ;  move  the 
index  forward  till  the  moon's  image  (which  will  appear  to  descend)  just  touches  the 
horizon  ;  then  sweep  the  quadrant  as  in  observing  the  sun,  and  bring  iier  round  limb 
in  contact  with  the  horizon,  whether  it  be  her  upper  or  lower.  The  degrees  and 
minutes  pointed  out  by  the  index,  will  be  the  observed  altitude  of  that  limb  which 
was  brought  in  contact  with  the  horizon. 

To  ta1<e  an  altitude  of  a  star  hy  a  fore  observation. 

This  is  done  exactly  in  the  same  manner  as  in  observing  the  ivioon's  altitude,  when 
her  image  is  so  faint  as  not  to  be  seen  in  the  transparent  i)art  of  the  horizon  glass. 

To  take  the  sun's  altitude  hy  a  hack  observation. 

Put  the  dark  glasses  in  the  hole  O,  and  turn  one  or  more  of  them  down,  according 
to  the  brightness  of  the  sun  ;  then,  holding  the  instrument  in  a  vertical  position,  look 
through  the  back  sight  vane  towards  that  part  of  the  horizon  ojjposite  the  sun  ;  move 
the  index  till  the  sun's  image  is  seen  in  the  silvered  part  of  the  glass;  give  the  quadrant 
a  slow  vibratory  motion,  and  the  sun  will  ap|)car  to  describe  an  arc  with  its  convex 
side  upward  ;  bring  the  upper  limb,  when  in  the  upper  part  of  this  arc,  in  contact  with 
that  part  of  the  horizon  seen  through  the  transparent  slit,  and  the  degrees  and  minutes 
pointed  out  by  the  index  will  be  the  altitude  of  the  sun's  lower  limb.  The  altitude  of 
tiie  moon,  or  a  star,  may  be  obtained  in  the  same  manner,  only  observing  to  bring  the 
round  edge  of  the  moon  to  the  horizon. 

The  back  oI>servation  is  but  little  used,  on  account  of  the  difficulty  of  adjusthig  and 
observing.  Various  remedies  have  been  proposed  for  these  defects,  but  none  have  yet 
been  generally  adopted.  The  back  observation  of  the  altitude  of  any  object,  is  useYid 
only  when  there  is  not  an  open  horizon  for  the  fore  observation  ;  but  even  in  that  case, 
the  fore  observation  may  often  be  used,  if  the  distance  of  the  horizon  be  known,  as 
will  be  explained  hereaftei". 

To  observe  the  meridian  altitude  of  any  celestial  object  by  a  fore  observation. 

Wlien  the  object  rises  and  sets,  it  comes  to  the  meridian  above  the  horizon  only 
once  in  24  liours,  and  is  then  at  its  greatest  altitude  ;  and  by  observing  it,  the  latitude 
may  be  easily  determined.  The  sim  comes  to  tlie  meridian  exactly  at  noon,  or  12 
o'clock  apparent  time ;  the  moon  and  stars  at  various  hours.  To  observe  the  meridian" 
altitude,  begin  TL  fewniinutes  before  the  time  of  passing  the  meridian  ;  bring  the  object 
to  sweep  the  horizon,  according  to  tlie  preceding  directions ;  this  operation  must  be 
repeated  until  the  object  begins  to  descend  below  the  edge  of  the  sea;  the  degrees  and 
minutes  then  sho^vn  by  the  index  will  be  the  meridian  altitude. 

If  the  object  does  not  set,  it  comes  to  the  meridian  below  the  pole,  and  is  then  at  its 
least  altitude ;  this  altitude  may  be  observed  as  above  directed,  with  this  diflerence, 
that  you  must  continue  sweeping  till  the  object  begins  to  rise  above  the  edge  of  the 
sea,  instead  of  descending  below  it. 

The  meridian  altitude  of  any  object  may  be  taken  in  a  similar  manner  by  a  back 
observation. 

Strictly  speaking,  this  method  of  finding  the  meridian  altitude  is  not  absolutely 
accurate,  except  the  ship  be  at  rest,  and  the  sun's  declination  constant.  For  if  the 
ship  is  sailing  towards  the  sun,  the  altitude  will  be  increased  ;  but  the  altitude  will  be 
decreased  in  sailing  from  the  sun.  The  correction  of  altitude  arising  from  this  source 
is  generally  very  small,  and  it  may  be  neglected  in  most  cases,  as  will  be  shown 
hereafter. 

Advice  to  seamen  in  the  choice  of  a  quadrant. 

The  joints  of  the  frame  must  be  close,  without  the  least  opening  or  looseness,  and 
the  ivory  on  the  arc  inlaid  and  fixed,  so  as  not  to  rise  in  any  place  above  the  plane  of 
the  instrument ;  all  the  divisions  of  the  arc  and  vernier  must  be  exceedingly  fine  and 
straight,  so  that  no  two  divisions  of  the  vernier  (except  the  first  and  last)  coincide,  at 
the  same  time,  with  the  divisions  of  the  arc.    All  the  glasses  belonging  to  the  quadrant 


132  USE   OF   A   QUADRANT   OF   REFLECTION 

should  liave  tlunr  surfaces  perfectly  [)l;uie,  and  their  fore  and  l)ack  surfaces  exactly 
parallel ;  this  may  be  verified,  iu  the  horizon  glass  and  index  glass,  l)y  means  of  two 
distant  objects,  in  the  following  manner: — IMove  the  index  till  both  objects  are  exactly 
in  contact,  at  the  u})per  edge  of  the  silvered  i)art  of  the  horizon  glass;  then  move  llie 
quadrant  in  its  omi  [)lane,  so  as  to  make  the  united  images  move  along  the  line, 
separating  tjie  silvered  from^the  transparent  part  of  the  horizon  glass;  and  if,  in  this 
motion,  the  images  continue  united,  the  reflecting  surfaces  are  good  jilanes,  otherwise 
the  ])lanes  are  imi)erfect.  To  examine  the  dark  glasses,  we  must  bring  the  image  of  a 
distant  object  to  coincide  witli  the  object  si.'en  directly ;  tlien  turn  the  colorcil  glass  so 
that  the  |)Iane  which  was  next  to  the  uidex  glass  may  now  be  next  to  the  horizon 
ghiss,  antl  if  the  direct  and  reflected  images  still  coincide,  the  surfaces  of  the  ghiss  are 
|>arallf-l. 


133 


DESCRIPTION    AND    USE    OF    A    SEXTANT 
OF    REFLECTION. 


A  Sexta.nt  is  constnictecl  on  tlie  same  princlijles,  and  may  be  used  for  measuring 
altitudes  in  tlie  same  manner,  as  a  quadrant.*  Tlie  arc  of  a  sextant,  as  its  name 
nuplii'S,  contains  60°,  l)ut,  hy  reason  of  the  double  reflection,  is  divided  into  120'^. 
Tbis  instrument  is  ]»articularly  intended  to  measure  the  distance  of  the  moon  from  the 
sun,  a  ])lanet,  or  a  rixcd  star ;  and  as  that  distance  is  wanted  as  accurately  as  possible, 
to  determine  tlie  longitude  of  the  place  of  observation,  the  instrument  is  constrnctccl 
with  more  care,  and  is  provided  with  some  adflitional  appendages  that  are  not  in  the 
fjUadrant.  Plate  IX.,  figure  3,  re|)rcsents  a  sextant,  the  frame  behig  generally  made  of 
brass,  or  otlier  hard  metal;  the  handle  at  its  back  is  made  of  wood.  When  observing, 
the  instrument  is  to  be  held  with  one  hand,  by  the  handle,  while  the  other  hand  moves 
the  index.  Tiie  arc  AA  is  dividetl  into  120°,  each  degree  into  3  parts  of  20  minutes 
each,  and  the  vernier  scale  is  in  general  so  divided  as  to  show  lialf  or  a  quarter  of  a 
mimite.  In  some  sextants,  the  degree  is  divided  into  six  equal  parts,  of  10'  each,  and 
the  vernier  shows  10". 

In  order  to  observe-  witii  accuracy,  and  make  the  images  come  precisely  in  contact, 
a  tangent  screw  B  is  fixed  to  the  index,  and  !)y  this  it  can  be  moved  with  greater 
regularity  than  it  can  be  by  iiand  ;  bnt  the  screw  IJ  does  not  act  until  the  index  is 
fixed  by  the  screw  C,  at  the  back  of  the  sextant.  Care  must  be  taken  not  to  force  the 
tangent  screw,  when  it  arrives  at  either  extremity  of  its  arc.  When  the  index  is  to  be 
moved  any  consideraI)le  quantity,  the  screw  C  nnist  be  loosened  ;  and  when  the  index 
is  brought  nearly  to  the  division  requiretl,  the  back  screw  C  must  be  tightened,  and 
then  the  index  moved  gradually  by  tlie  tangent  screw. 

In  many  sextants,  die  lower  ])art  of  the  index  glass,  or  that  next  tlie  jilane  of  the 
instrument,  is  silvered  as  usual,  and  the  back  surface  of  the  upper  part  painted  black  ; 
a  screen,  painted  black,  is  fixed  by  its  axis  to  the  base  of  the  index  glass,  and  may  be 
placed  over  the  silvere(l  ])art  when  the  rays  are  strong;  in  this  case,  the  image  is  to  be 
reflected  from  the  outer  surface  of  the  u|)per  part,  aiui  the  error  which  might  possibly 
arise  from  the  planes  of  the  glass  not  being  |)ara]li'l,  is  thereby  avoided. 

The  colored  glasses  are  similar  to  those  ai)])!ied  to  a  common  (piadrant,  and  are 
nsually  fi)ur  in  number,  jdaced  at  D,  to  screen  tlie  eye  from  the  solar  rays,  and  the 
glare  of  the  moon  ;  they  may  be  used  separately  or  together,  as  occasion  re(pures.  In 
addition  to  tlicso,  there  are  three  similar  glasses,  placed  behind  the  horizon  glass,  to 
be  used  in  finding  the  index  error  by  means  of  tlie  sun,  and  in  observing  the  sun's 
altitude,  by  an  artificial  horizon  on  land.  The  paler  glass  is  sometimes  used  in 
observing  altitudes  at  sea,  to  take  off  the  strong  glare  of  the  horizon  below  the  sun, 
arising  from  the  smi's  light,  reflected  iiTcgnlarly  from  the  small  rii)i)ling  waves — an 
appearance  which  has  lately  been  called  kumatnge. 

A  sextant  is  generally  fnrnished  with  a  tube  without  glasses,  and  two  telescopes,  the 
one  representuig  the  objects  erect  or  in  their  natural  situation,  the  other  inverting  them, 

*  There  is  not,  in  general,  any  apparatus  for  ilie  back  oljservalion  fixed  lo  a  sextant;  but  if  the 
altitude  of  any  celestial  object  be  greater  than  (iO°,  the  sui>plcmcnl  of  liie  altitude  may  be  obtaine<l  by 
aback  observation,  with  a  sextant,  with  case  and  accuracy;  and  as  this  method  may  be  often  used 
with  advantage,  when  a  fore  observation  cannot  be  olitaincd,  we  shall  here  point  out  the  method  of 
taking  the  ol)servalion,  and  shall  hereafter  give  the  calculations  for  determining  the  latitude  from  a 
meridian  observation,  taken  in  this  manner: — The  back  of  the  observer  being  turned  to  the  sun,  he  must 
move  the  index  till  the  image  of  the  sun  touches  tlie  edge  of  the  back  horizon,  and  dien  move  the  sextant 
a  little  lo  the  right  and  left  (as  in  a  fore  observation),  and  the  image  will  describe  an  arc  with  the  convex 
side  upward  ;  move  the  index  till  the  lower  limb  of  the  image,  wlsen  in  the  upper  part  of  the  arc,  just 
touches  tiie  horizon,  and  the  observation  will  be  complete;  observii;g  that,  if  the  telescope  be  used,  the 
image  must  be  brought  in  the  middle  between  the  two  parallel  wires  ;  but  if  the  telescope  be  not  used, 
the  image  of  the  sun  must  be  seen  in  the  horizon  glass,  at  the  same  distance  from  the  plane  of  tiie 
instrument  as  the  eye  of  the  observer.  The  altitude  thus  obtained  will  be  the  supplement  olthe  altitude 
of  the  sun's  upper  limb.  The  corrections  to  be  applied  to  obtain  the  true  central  altitude,  will  be  {;iveii 
hereafter. 


134  USE   OF   A   SEXTANT   OF   REFLECTION. 

the  eye-glass  being  fixed  in  a  movable  tube,  in  order  to  adjust  tlie  telescope  to  a 
proper  focus.  By  means  of  these  telescopes,  the  line  of  sight  may  be  rendered  parallel 
to  the  i)lane  of  the  instrument,  and  the  contact  of  the  limbs  of  any  two  objects  more 
accurately  observed.  The  tube,  or  either  telescope,  is  to  be  screwed  into  a  brass  ring, 
which  is  connected  with  another  brass  ring  by  means  of  two  screws ;  and  by  loosening 
one,  and  tightening  the  other,  the  axis  of  the  tube  or  telescope  may  be  set  parallel  to 
the  plane  of  the  instrument.  One  of  these  rings  is  fixed  to  a  brass  stem,  which  slides 
in  a  socket;  and  by  means  of  the  screw  L,  at  the  back  of  the  sextant,  it  may  be  raised 
or  lowered  so  sis  to  move  the  axis  of  the  telescope  to  point  to  that  i)art  of  the  horizon 
glass  judged  the  most  fit  for  observation. 

A  cucular  head,  containing  a  plate,  in  which  there  are  three  colored  glasses,  and  a 
I)art  that  is  open,  sometimes  accompanies  the  sextant;  this  Jiead  is  to  be  screwed  on 
the  eye  end  of  the  tube,  or  on  that  of  either  telescope.  Tlie  edge  of  the  plate  projects 
a  little  beyond  the  head  on  one  side,  and  is  movable  by  the  finger,  so  that  the  open 
ring,  or  any  of  the  colored  glasses,  may  be  brought  between  the  eye-glass  of  the 
telescope  and  the  eye  ;  this  answers  the  purpose  of  the  dark  glasses  placed  at  E,  in 
adjusting  by  the  sun,  or  observing  by  an  artificial  horizon  on  land. 

To  these  are  added  a  small  screw-driver,  to  adjust  the  screws,  and  a  magnifyhig 
glass,  to  read  off  the  observation  with  greater  accuracy. 

The  adjustments  of  a  sextant  are  similar  to  those  of  a  quadrant;  the  index  and 
horizon  glasses  must  be  peri)endicular  to  the  plane  of  the  instrument,  and  their  planes 
parallel  to  each  other  when  the  index  stands  on  0 ;  also  the  axis  of  the  telescope  must 
be  set  parallel  to  the  ])lane  of  the  instrunient ;  each  of  these  particulars^  must  be 
examined  before  an  observation  is  taken,  and  the  adjustments,  if  requisite,  made 
according  to  the  foUowmg  directions  : — 

list.      To  set  the  index  glass  prrpendicidar  to  the  plane  of  the  instrument. 

Rlove  the  index  forward  to  about  60°,  and  proceed  exactly  in  the  manner  prescribed 
for  the  adjustment  of  the  index  glass  of  a  quadrant,  page  129. 

2d.      To  make  the  horizon  glass  perjjendicular  to  the  plane  of  the  sextant. 

This  adjustment  is  made  exactly  in  the  same  manner  as  that  of  the  quadrant, 
described  in  page  129,  except  that,  instead  of  looking  through  the  sight  vane,  you  may 
use  the  tube,  or  a  telescope. 

To  malic  the  horizon  glass  and  index  glass  parallel  when  the  index  is  on  0. 

Having  made  the  foregoing  adjustments,  set  the  first  division  on  the  index  at  0  on 
the  limb;  fasten  the  index  in  this  position,  and  make  the  coincidence  of  these  divisions 
as  perfect  as  possible,  by  means  of  the  tangent  screw,  the  eye  being  assisted  by  the 
magnifying  glass ;  screw  the  tube,  or  telescoi)e,  into  its  support,  and  turn  the  screw  L, 
at  the  back  of  the  instrument,  till  the  line  Avhich  sei)aratcs  the  transparent  and  silvered 
parts  of  the  liorizon  glass  appears  in  the  middle  of  the  tube  or  telescope  ;  havi)ig  done 
this,  hold  the  plane  o'f  the  sextant  vertically,  and  direct  the  sight  through  the  tube  or 
telescope  to  the  horizon  ;  then,  if  the  reflected  and  true  horizons  do  not  coincide,  turn 
the  tangent  screw  at  the  back  of  the  horizon  glass  till  they  are  made  to  appear  in  the 
same  straight  line.     Then  will  the  horizon  glass  be  adjusted. 

After  the  screw  that  retains  the  liorizon  glass  in  its  place  is  fastened,  it  will  be 
proper  to  re-examine  this  adjustment ;  if  the  coincidence  of  the  horizons  is  not  perfect, 
the  adjustment  must  be  repeated  till  it  is  so;  but  as  it  is  ditficult  to  obtain  a  jierfect 
coincidence  by  diis  means,  the  horizons  may  be  brought  to  coincide  by  turning  the 
tangent  screw  of  the  index;  and  tlie  diflcrence  between  the  0  on  the  arc  and  the  0  on 
the  vernier  will  be  the  index  error,  which  is  additive  to  all  observations  if  the  0  of 
the  index  stand  on  the  extra  arc,  otherwise  subtractive.  The  index  error  may  also  be 
found  very  accurately,  by  measuring  the  diameter  of  the  sun  twice,  with  a  motion  of 
the  indexin  contrary  directions  ;  that  is,  first  bring  the  upjjcr  limb,  seen  by  reflection, 
to  coincide  with  the  lower  limb  seen  directly ;  then  bring  the  lower  limb  by  reflection 
to  coincide  v.ith  the  upj)er  seen  directly.  If  l)0th  these  measures  are  taken  either  to 
the  right  or  left  of  0  on  the  limb,  half  their  sum  will  be  the  index  error;  additive  if  to 
the  right  of  0,  sid)tractivc  if  to  the  left :  but  if  one  of  the  measures  be  taken  to  the 
right,  and  the  other  to  the  left  of  0,  half  their  diflTercnce  will  be  the  index  error,  which 
will  be  additive  when  the  diameter  measured  to  the  right  of  0  exceeds  that  measured 
to  the  left,  otherwise  subtractive.     Thus,  if  the  me:isures  were  38'  to  the  left  of  0  on 


USE   OF   A   SEXTANT   OF   REFLECTIOIN  135 

ihe  arc,  and  26'  to  tlie  right  *  on  the  extra  arc,  half  the  diflerence,  or  C,  would  be  the 
correction,  subtractive.  In  some  sextants,  the  horizon  glass  cannot  be  adjusted  ;  the 
index  error  must  in  that  case  be  found,  and  must  be  considered  as  a  constant  quantity 
to  be  applied  to  all  angles  measured  with  the  same  instrument. 

To  set  the  oris  of  the  telescope  parallel  to  the  plane  of  the  sextant. 

In  measuring  angular  distances,  the  line  of  sight,  or  axis  of  the  telescope,  must  be 
parallel  to  the  plane  of  the  instnmient,  as  a  deviation  in  that  resjiect,  in  measuring 
large  angles,  will  occasion  a  considerable  error.  To  avoid  this,  a  telescope  is  made  use 
of,  in  which  are  placed  two  wires,  parallel  to  each  other,  and  equidistant  from  the 
centre  of  the  telescope ;  by  means  of  these  wires,  the  adjustment  may  be  made  in  the 
following  manner: — Screw  on  the  telescope,  and  turn  the  tube  containing  the  eye-glass 
till  the  wires  are  parallel  to  the  plane  of  the  instrument;  then  select  two  objects,  as 
the  sun  and  moon,  whose  angular  distance  must  not  be  less  than  90°,  because  an  error 
is  more  easily  discovered  when  the  distance  is  great ;  bring  the  reflected  image  of  the 
sun  exacdy  in  contact  with  the  direct  image  of  the  moon,  at  the  wire  nearest  the  plane 
of  the  sextant,  and  fix  the  index  ;  then,  by  altering  a  little  the  position  of  die  instru- 
ment, make  the  objects  ajtpear  on  die  other  wire  ;  if  the»contact  still  remains  ])erfect, 
the  axis  of  the  telescope  is  in  its  right  situation  ;  but,  if  the  limbs  of  the  t\vo  objects 
appear  to  separate  or  lap  over,  at  the  wire  which  is  farthest  from  the  plane  of  the 
sextant,  the  telescope  is  not  parallel,  and  it  must  be  rectified  by  turning  one  of  the  two 
screws  of  the  ring  into  which  the  telescope  is  screwed  and  fixed,  having  previously 
unturned  the  other  screw  ;  by  repeating  this  operation  a  few  times,  the  contact  will  be 
precisely  the  same  at  both  wires,  and  the  axis  of  the  telescope  will  be  parallel  to  the 
plane  of  the  instrument.f 

In  order  to  estimate  the  error  committed  in  not  observing  the  contact  of  die  objects 
in  tlie  middle,  between  the  two  parallel  wires  of  the  telescope,  it  is  necessary  to  know 
the  angular  distance  of  diese  wires.  This  may  be  found  as  follows: — Turn  round 
the  eye-piece  of  the  telescope,  till  the  wires  are  perpendicular  to  the  plane  of  die 
instrument ;  hold  the  instrument  in  a  vertical  jiosition,  and  move  the  index  till  the 
direct  and  reflected  images  of  the  horizon  appear  in  the  same  line,  which  will  happen 
when  the  index  is  at  0,  if  the  instrument  be  well  adjusted  ;  then  move  the  index  till 
the  reflected  linage  of  the  horizon  be  at  one  wire,  and  the  direct  image  at  the  other; 
die  angle  moved  through  by  the  index,  as  shown  by  the  divisions  of  the  arc,  will  be 
the  angular  distance  of  the  two  wu-es.  This  angular  distance  being  obtained,  the 
observer  may,  by  means  of  it,  estimate,  at  each  observation,  how  much  the  place  where 
the  contact  is  observed  is  elevated  above,  or  depressed  below,  the  plane  jjassing  through 
die  eye  and  die  middle  line  between  the  two  parallel  wires ;  the  correction  in  Table 
XXXV.,  corresponding  to  this  angle,  is  to  be  subtracted  from  the  observed  angular 
distance  of  die  objects.  Thus,  if  the  distance  between  the  wires  be  3°,  one  of  them 
v/ill  be  elevated  above  die  plane  1°  30',  and  the  other  depressed  as  much  below  it ;  and 
if,  in  taking  an  observation,  the  point  of  contact  is  estimated  to  be  one  third  part  of  the 
distance  from  the  middle  towards  either  wire,  the  angle  of  elevation  or  depression  will 
be  one  third  part  of  1°  30',  or  30' ;  and  if  the  observed  distance  be  100°,  the  correction 
in  Table  XXXV.  will  be  19",  subtractive  from  the  observed  angle,  which  will  there- 
fore be  100°  — 19"  =r  99°  59'  41".  In  general,  it  will  not  be  necessary  to  attend  to  this 
correction. 

To  measure  ihe  distanee  hetioccn  the  sun  and  moon. 

Screw  on  the  telescope,  and  })lace  the  wires  parallel  to  the  jjlane  of  the  instrument ; 
then,  if  the  index  glass  is  half  silvered  and  half  blacked,  and  the  sun  very  bright,  raise 
the  plate  before  the  silvered  part  of  the  glass,  and,  widi  the  screw  L,  raise  the  telescope 

*  In  rending  ofl'the  measure  on  the  extra  arc,  you  must  reckon  the  minutes  on  the  vernier  from  left  to 
right,  counting  19'  as  1',  18'  as  2',  ifec.,  or  else  take  the  diflerence  between  tlie  iniiuues  denoted  by 
llie  vernier  and  20'.  Thus,  if  the  angle  on  the  extra  arc  appeared  by  the  nonius  to  be  11',  the  reaJ 
angle  would  be  only  G'. 

t  This  adjustment  may  be  made  in  a  manner  similar  to  that  by  which  the  graduation  on  the  frame 
of  the  telescope  of  a  circular  instrument  is  verified,  by  using  the  adjusting  tools  of  a  circle  or  a  ruler 
whose  surfaces  are  perfectly  parallel  to  each  other.  Thus,  lay  the  sextant  horizontally  on  a  tabls,  and 
place  the  ruler  on  the  limb  or  plane  of  the  instrument,  and,  at  about  12  or  15  foot  ilistance.  let  a  well- 
defincd  mark  be  ]jlaced  in  a  range  with  the  telescope,  so  as  to  be  in  the  same  straight  line  with  the  top 
of  the  ruler  ;  then  raise  or  lower  the  telescope,  by  means  of  the  screw  L,  till  the  centre  of  the  eye-piece 
of  the  telescope  be  at  the  same  height  as  the  top  of  the  ruler ;  then,  if  the  mark  be  seen  in  the  middle 
between  the  wires  of  the  telescope,  it  is  well  adjusted  ;  if  not,  it  must  be  altered  by  means  of  the  screws 
of  the  ring  into  which  the  telescope  is  screwed. 


136  USE   OF  A   SEXTANT  OF   REFLECTION. 

to  the  transparent  part  of  the  horizon  glass  ;  turn  (JoAvn  one  or  more  of  the  dark 
glasses,  acconhng  to  the  brightness  of  the  sun  ;  then  hold  the  sextant  so  that  its  plane 
may  pass  through  the  sun  and  moon  ;  if  tlie  sun  be  to  tlie  right  hand  of  the  moon,  the 
sextant  is  to  be  held  with  its  face  upwards  ;  if  to  the  left  hand,  the  face,  is  to  be  held 
downwards  ;  with  the  instrument  in  this  jjosition,  look  directly  at  the  moon  through 
the  telescope,  and  move  the  index  forward  till  the  sun's  image  is  brought  nearly  into 
contact  with  the  moon's  nearest  limb;  then  fix  the  index  by  the  screw  under  the 
sextant,  and  make  the  contact  perfect  by  means  of  the  tangent  screw  ;  at  the  same 
time,  move  the  sextant  slowly,  making  the  axis  of  the  telescope  the  centre  of  motion  ; 
by  this  means  the  objects  will  pass  each  other,  and  the  contact  be  more  accurately 
made  ;  observing  that  the  point  of  contact  of  the  limbs  must  always  be  observed  in  the 
middle  between  the  parallel  wires.  The  obscnation  being  thus  made,  the  index  will 
point  out  the  distance  of  the  nearest  limbs  of  the  sun  and  moon. 

To  measure  the  distance  between  the  i/ioori  and  a  star. 

Turn  down  one  of  the  screens,  if  the  moon  is  briglit,  and  direct  the  plane  of  tlie 
instrument  through  both  objects,  with  its  face  upwards,  if  the  moon  is  to  the  right  of 
the  star;  but  if  to  the  left,  the^face  is  to  be  held  downwards ;  look  at  the  star  through 
the  telesco{)e  and  transparent  part  of  the  horizon  glass,  and  move  the  index  till  the 
moon's  image  appears  nearly  in  contact  with  the  star  ;  fasten  the  index,  move  the 
sextant  round  tlie  axis  of  the  telescope,  as  in  measuring  the  distance  of  the  sim  and 
moon,  and  turn  the  tangent  screw,  till  tlie  coincidence  of  tlie  star,  and  the  enli<rhtene(l 
or  round  limb  of  tlie  moon  is  perfect ;  observing  that  the  ])oiiit  of  contact  of  the  limb 
of  the  moon  and  star  must  always  be  in  the  middle  between  the  pai'allel  wires.  The 
olisenaticn  being  thus  made,  the  index  will  jioint  out  the  distance  of  the  enlightened 
limb  of  tlie  moon  from  the  star,  whether  it  be  the  farthest  or  nearest  limb. 

Vcrijication  of  the  parallelism  of  tlie  index  glass. 

This  verification  is  to  be  made  ashore,  by  observing  the -angular  distance  of  two 
well-defined  objects,  whose  distance  exceeds  90°  or  100°  (having  previously  m'cU 
adjusted  tlie  iustrumont),  then  taking  out  the  central  mirror,  and  turning  it,  so  tiiat  the 
e<lge  wliich  was  formerly  uppermost  may  now  be  nearest  the  plane  of  the  iiistruinent ; 
rectify  its  position,  and  again  measure  the  distance  of  the  two  objects ;  half  the  differ- 
ence between  tlicse  two  distances  will  be  the  error  of  the  observed  angle  arising  from 
the  defect  of  j)arai!elism  of  the  central  mirror.  If  the  first  distance  exceeds  the  second, 
the  error  is  subtractive,  otherwise  additive,  the  mirror  being  in  its  first  [)osition ;  but 
the  contraiy  when  in  its  second  position.  Thus,  if  the  first  distance  was  119°  59'  21", 
and  the  second  120°  0'  39'',  the  error  would  be  39",  additive  when  theniirror  was  in 
its  first  ])osition,  subtractive  for  the  second.  The  error  for  any  other  angle  may  be 
fouuil  by  means  of  col.  2d  Table  XXXTV^.,  when  the  inclination  of  the  ])lane  of  the 
horizon  glass  to  tlie  axis  of  the  telescojje  is  80°,  by  saying,  As  the  tabular  correction 
corresponding  to  120°  (::=4'  5")  is  to  tlie  en-or  of  the  glass  .39",  so  is  the  tabular  error 
for  any  other  angle,  as  85°  {=i  1'  15"),  to  the  corresponding  error  of  the  glass  12".  In 
this  maiuii.'r  a  table  of  errors  may  be  made  for  all  angles.* 

The  angle  between  the  plane  of  the  horizon  glass  and  axis  of  the  telescope  produced 
!)eing  constant  in  all  observations  and  adjustments  of  the  sextant,  no  eiror  can  arise 
from  the  want  of  iiarallclism  of  its  surfaces.  , 

Verification  of  the  parallelism  of  the  surfaces  of  the  colored  glasses. 

Turn  down  the  glass  at  D  which  is  to  be  examined,  and  another  at  E  to  defend  the 
eye  from  the  stm  ;  direct  the  telescope  to  the  sun,  and  move  the  index  till  its  direct 
and  reflected  images  coincide  ;  then  turn  the  dark  glass  at  D  so  that  the  surface  which 
was  farthest  froni'^the  horizon  glass  may  now  be  nearest  to  it,  and  if  the  contact  of  the 
same  two  limbs  be  complete,  the  surfaces  of  this  glass  are  parallel;  but  if  they  lap 
over  or  s<'parate,  the  index  must  be  moved  to  bring  them  again  in  contact ;  then  half 
the  arc  pr.ssed  over  by  the  index  will  be  the  error  arising  from  the  want  of  jiarallelism 
of  the  glass  at  I).  If  a  defect  of  this  kind  is  found  in  any  one  of  these  colored  glasses, 
it  is  best  to  avoid  the  use  of  it  altogether. 

*TliR  niL'tliod  of  ralciilaling'  ihe  above  tabular  numbers,  ulion  llu;  nii^le  nfiiiflination  of  llie  telescope 
ami  horiziiu  jfhiss  ililfers  from  C0°,  is  g-iveii  in  tlic  cxiiliiiuuioii  of  Table  XXXIV.  pretixeJ  to  the  tables 


!  Capt;  FsENCir,  of  l]»e  Boston  Navy  Yard  is 
on  a  tour  thwmgh  Spriogield,  Albany  New- 
York.  New  Haven  and  Hartford,  to  test,  by 
obscrvatioas,  the  correctness  of  a  nei</ly  in. 
vented  sextant,  by  means  of  which,  the  iavea- 
tor  claims,  (he  latitu-e  and  longitude  can  b9 
obtained  wrthout  seeing  sun,  'm  jou  or  stars.  If 
he  was  out  in  the  late  snow  storm  ha  had  a 
good  chance  to  test  that  instrument.  /^  7^ 


-^^  ,;;^^^ 


J'^^. 


M 


^- 


137 


DESCRIPTION  AND  USES  OF  THE   CIRCLE 
OF  REFLECTION. 


The  Cirde  of  Reflection  vvjus  invented  by  the  celebrated  Professor  Mayer,  of 
C)-oiiingf'n,  and  has  since  been  greatly  improved  by  the  Chevalier  De  Borda,  ftlr. 
Troughlon,  and  I\Ir.  Mendoza  y  Rios.  In  its  present  improved  state,  it  has  a  decided 
su[)eriority  over  the  sextant,  in  measuring  the  distance  of  the  moon  from  the  sun  or 
a  star,  on  account  of  its  correcting,  in  a  great  measure,  the  errors  arising  from  a  faulty 
division  of  the  limb,  want  of  parallelism  in  the  surfaces  of  the  mirrors  and  colored 
glasses,  and  entirely  avoiding  the  error  which  might  arise  in  a  sextant  from  the  miiTors 
not  being  ])arallel  when  the  index  is  on  0. 

Figure  1,  Plate  X.,  represents  the  Circle  of  Reflection,  as  given  by  De  Borda.  In 
figm-e  2  is  a  section  of  the  same  instrument,  marked  with  the  same  letters  of  reference 
as  in  figure  1.  The  principal  parts  of  this  instrument  are,  the  circular  limb  LftlV ; 
the  central  uidex  EF  ;  the  horizon  index  MD  ;  the  central  glass  or  mirror  A  ;  the 
horizon  glass  or  mirror  R  ;  the  telescope  GH  ;  the  colored  glasses,  figures  3,  4  ;  the 
handle,  figure  5  ;  the  ventelle,  figure  6  ;  and  the  adjusting  tool,  figure  7. 

The  limb  of  the  instrumenl  LMV  is  a  complete  circle  of  metal,  and  is  connected 
with  a  perforated  central  plate  by  six  radii ;  it  is  divided  into  720°,  because  of  the 
double  reflection  ;  each  degree  is  generally  divided  into  three  equal  parts,  and  the 
division  is  carried  to  minutes,  or  lower,  by  means  of  die  verniers  of  the  two  indices. 

The  tico  indices  are  movable  round  the  same  axis,  which  passes  exactly  through  the 
centre  of  the  instrument ;  the  central  index  EF  carries  the  central  mirror  A  ;  and  the 
horizon  index  MD  carries  the  telescope  Gil  and  the  horizon  mirror  B  ;  both  indices 
are  furnished  with  verniers  and  tangent  screws  at  O  and  N. 

The  central  mirror  A  is  jilaced  on  the  central  index  immediately  above  the  centre  of 
the  instrument ;  the  plane  of  this  mirror  makes  an  angle  of  about  30°  with  the  middle 
line  of  the  index,  and  is  adjusted  pcri)endicular  to  the  plane  of  the  instrument,  by 
means  of  the  screws  j)laced  on  the  back  jiart  of  the  frame  of  the  mirror. 

The  horizon  glass  B  is  jilaccd  on  t!ie  horizon  index,  near  the  limb,  so  as  to  inteifere 
as  little  as  possible  with  the  rays  proceeding  from  objects  situated  on  the  ojjposite 
side  of  that  index  with  respect  to  the  central  mirror.  The  horizon  glass  is  adjusted 
j)erpendicular  to  the  plane  of  the  instrument,  in  a  similar  manner  to  that  of  the  horizon 
glass  of  a  sextant ;  and  in  some  circles,  this  mirror  is  movable  al)out  an  axis  perpen- 
dicular to  the  plane  of  the  instnuncnt ;  by  this  means  the  situation  v/ith  respect  to 
the  telescope  may  be  adjusted. 

The  telescope  GIT,  attached  to  the  other  end  of  the  horizon  index,  is  an  astronomical 
one,  inverting  the  observed  objects,  and  has  two  parallel  wires  in  the  common  focus 
of  the  glasses,  distant  from  each  other  between  two  and  three  degrees.  These  wires, 
at  the  time  of  observation,  must  be  i)laced  parallel  to  the  plane  of  the  instrument :  to 
eftect  this,  marks  are  made  on  the  eye-jiiece,  and  on  the  tube  at  G,  and  by  making 
them  coincide,  the  wires  may  be  brought  to  their  proper  position.  The  telesco|)e  may 
be  raised  or  depressed  by  two  screws,  I,  K,  so  as  to  be  du*ected  to  any  [)art  of  the 
horizon  glass ;  and,  by  ineans  of  the  gi-aduations  on  the  two  standai-ds,  i,  k  (Fig.  2), 
the  telescope  may  be  rendered  parallel  to  the  plane  of  the  instrument. 

There  are  two  sets  of  colored  glasses  (fig.  3,  4),  each  set  usually  containing  four 
glasses  of  different  shades ;  the  glasses  of  the  large  set  (fig.  4),  which  are  i)laced  before 
the  central  mhror  at  a,  a,  shouhl  have  each  about  half  the  degree  of  shade  with  which 
the  corresponding  glasses  (fig.  3)  of  the  other  set,  placed  at  C,  are  tinged,  because  the 
rays  from  the  luminous  object  pass  twice  through  the  colored  glass  placed  before  the 
central  mirror,  and  only  once  through  the  other.  The  glasses  placed  at  a,  a,  are  kept 
tight  in  their  jilaces  by  small  pressing  screws  at  their  ends,  or  by  slides  passing,  in 
front,  through  perforations  in  tlie  stems  of  their  frames  ;  when  fixed  for  obsei-vatioii, 
they  make  an  angle  of  about  85°  with  the  plane  of  the  nistrument ;  by  this  means,  tlie 
18 


138  CIRCLE   OF  REFLECTION. 

image  from  the  colored  glass  is  not  reflected  to  the  telescope.  When  the  angle  to  be 
measured  is  between  5°  and  35°,  one  of  the  large  set  is  to  be  fixed  at  a,  a ;  in  other 
cases,  one  of  the  small  set  is  to  be  placed  m  the  socket  C.  The  reason  of  using  the 
large  glass  is  this  : — when  the  small  glass  is  placed  at  C,  it  intercepts  tlie  direct  light  of 
the  luminous  object,  in  its  passage  towards  the  central  mirror,  if  the  object  happens  to 
be  situated  widiin  the  angular  space  included  by  the  lines  from  the  centre  A,  by  the 
sides  of  the  frame  of  the  glass  placed  at  C.    This  is  avoided  by  using  the  large  glasses. 

The  handle  (fig.  5)  is  of  wood,  and  is  fixed  to  the  back  of  the  instrument  immediatel}' 
under  the  centre.     By  this  it  is  held  during  the  time  of  obsen'ation. 

The  ventdle  (fig.  6)  is  used  in  terrestrial  obseiTations  to  diminish  the  light  of  the 
object  seen  directly,  to  render  it  equal  in  brightness  to  that  of  the  object  seen  by 
reflection  ;  this  is  performed  by  putting  the  ventelle  hi  tlie  socket  D,  and  raising  or 
depressing  it  till  the  objects  appear  of  equal  brightness. 

There  are  two  adjusting  tools,  of  the  form  represented  in  figure  7  ;  they  are  exactly 
of  the  same  size,  and  theu*  height  is  nearly  equal  to  tliiit  of  the  central  mirror;  they 
may  be  used  in  adjusting  the  central  mirror  perpendicular  to  the  plane  of  the  instru- 
ment, and  in  making  the  axis  of  the  telescope  paralk;!  to  that  plane. 

The  instrument,  as  we  have  now  described  it,  is  the  same  as  it  was  left  by  De 
Borda.  Mr.  Troughton  has  since  suggested  the  improvement  of  fixing  to  the  horizon 
index  the  arc  AVSPR,  and  providuig  it  with  two  sliding  pieces  U,  X,  in  order  to 
facilitate  the  fixhig  the  indices  at  their  proper  angles  with  each  other  in  taking 
successive  observations.  When  the  central  and  horizon  glasses  are  parallel,  the 
central  index  covers  the  space  SP  of  the  arc,  and  the  spaces  SVV,  PR,  are  each  divided 
into  degi-ees  from  S  to  W,  and  from  P  to  R,  and  numbered  0  at  S  and  P,  and  continued 
to  130°  towards  W  and  R.  The  use  of  this  ai'c  and  sliding  pieces  will  be  explained 
hereafter.* 

That  ingenious  mathematician  and  navigator,  Mr.  Mendoza  y  Rios,  has  further 
improved  the  circular  instrument,  by  the  substiuitiv/"  of  a  circular  ring  (moving  round 
the  centre  of  the  instriunent,  over  or  adjacent  lo  the  /anb  TMV)  for  a  vernier,  instead 
of  those  attached  to  the  indices  by  De  Borfla ;  and.  by  fixing  this  circular  vernier 
alternately  to  each  of  the  indices,  it  serves  as  a  vernier  for  both,  and,  after  any  number 
of  observations,  gives  the  compound  motion  of  both  indices  ;  and  thus  double  the 
number  of  distances  are  obtained  by  this  Jnfiiniiiient,  that  can  be  obtained  by  De 
Borda's  circle,  widi  the  same  number  of  observations.  Mr.  Rios  has  also  iaiproved 
the  form  of  the  handle  for  holding  the  iii.«h-<;r.ient.  In  theory,  the  instrument  as 
improved  by  Mr.  Rios  a])pears  to  be  superior  to  that  of  De  Borda ;  but  not  having 
used  one  of  the  former  kind,  I  cannot,  from  my  own  experience,  decide  whether  it 
is  so  much  superior  in  practice  ;  but  Mr.  Rios  says  that  he  found  it  answered  his 
expectations.  As  the  method  of  taking  the  observation  is  nearly  the  same  with  both 
instruments,  I  shall  confine  myself  to  the  explanation  of  the  uses  of  De  Borda's,  from 
whicli  the  method  of  using  the  other  will  be  easily  discovered. 

Adjustments  of  the  circle  of  refection. 

■Before  entering  ui)on  an  explanation  of  the  adjustments  of  this  instrument,  it  will 
be  proper  to  premise  that  there  are  three  diflerent  methods  of  observing  the  angular 
distance  of  two  objects  with  this  instrument,  viz.  (1)  by  what  is  called  an  observation 
to  the  right,  (2)  by  an  observation  to  the  left,  and  (3)  by  a  cross  observation. 

An  observation  to  the  right  is  that  where  the  object  whose  image  is  to  be  i-eflected, 
and  tlje  central  mirror,  are  on  the  same  side  of  the  telescope ;  an  observation  to  the 
left,  when  the  ol)ject  to  be  reflected  and  the  central  mirror  are  on  opposite  sides  of  tlie 
telescope,  which,  in  both  cases,  is  suj)poseil  to  be  directed  to  the  other  object;  and  a 
cross  observation  is  a  combinatiou  of  the  fore-mentioned  observations,  the  first  biiiig 
generally  taken  to  tlie  left,  and  the  second  to  the  right. 

The  adjustments  of  a  circle  consist  in  placing  the  inirrors  perpendicular  to  tin 
[ilane  of  the  instrument,  and  in  making  the  axis  of  the  telescojie  i)arallel  to  that  iilane. 

*  IMr.  'I'lono-htoii  suggested  another  alteration  in  the  circle;  but  (as  Mr.  Rios  justly  observes)  the 
instrument  tiuis  altered  inny  be  considered  as  a  sextant,  the  limb  of  which  is  completed  to  the  whole 
circumference.  A  circle  of  this  description  is  usually  funiishcd  with  tliree  indices  and  verniers,  by 
each  of  which  every  ol)servation  must  be  read  ofl".  This  is  very  troublesome,  particularly  in  the  night. 
It  is  true  tliat  this  method  corrects,  in  a  very  great  dos^rec,  the  error  of  not  bavin"-  the  index  exactly 
on  the  centre,  or  that  of  not  having  an  instrument  perfectly  circular  ;  but  errors  of  this  kind  in  Borda's 
circle  may  be  reduced  in  any  ratio  by  taking  a  number  of  observations,  and  the  error  will  in  general 
be  extremely  small  in  taking  a  sutVicient  number  to  bring  the  index  nearly  to  the  point  set  out  from  ,• 
so  that,  in  those  important  points,  I  should,  on  the  whole,  prefer  an  instrument  of  Borda's  const rurlLou 


CIRCLE   OF   REFLECTION.  139 

These  are  all  the  adjustments  necessary  in  measuring  an  angular  distance  by  cross 
observations ;  but  if  one  observation  only  be  taken  to  the  right,  or  to  the  loft,  it  will  be 
necessary  to  find  the  division  on  which  the  horizon  index  must  be  jjlaced,  to  make 
the  horizon  glass  parallel  to  the  central  glass,  when  the  central  index  stands  on  0 
These  adjustments  are  similar  to  those  of  a  sextant;  but  a  particular  explanation  of 
each  will  here  be  given. 

To  set  the  central  glass  perpendicular  to  the  plane  of  the  instrument. 

This  adjustment  may  be  made  by  placing  the  eye  in  fi-ont  of  the  central  glass  at  L, 
a  little  above  the  plane  of  the  instrument,  and  observing  if  tlie  reflected  image  of  that 
part  of  the  limb  nearest  the  eye  ai)pears  to  make  one  continued  circular  line  with  the 
parts  of  the  limb  towards  T,  seen  to  the  right  and  left  of  the  central  glass  ;  for,  in 
this  case,  the  glass  is  peiTiendicular  to  the  jjlane  of  the  instrument ;  otherwise  it  must 
be  adjusted  by  means  of  the  screws  till  the  two  images  coincide.* 

By  examining  this  adjustment  in  different  parts  of  the  limb,  it  will  be  known  if  the 
limb  be  in  the  same  plane.  If  any  difference  should  be  found,  the  central  glass  must 
be  so  fixed  that  the  reflected  image  of  the  limb  may  appear  as  much  above  the  direct 
image  in  some  places  as  below  it  in  othei-s. 

To  set  the  horizon  glass  perpendicular  to  the  plane  of  the  instrument. 

The  central  glass  being  previously  adjusted,  and  the  telescope  directeil  to  the  line 
separating  the  silvered  from  the  trans])arent  part  of  the  horizon  glass,  hold  the 
instrument  nearly  vertical,  and  move  either  index  till  the  direct  and  reflected  image 
of  the  horizon,  seen  through  the  telescope,  coincide  ;  then  incline  the  instrument 
till  it  is  nearly  horizontal,  and,  if  the  images  do  not  separate,  the  horizon  glass  is 
perpendicular  to  the  plane  of  the  instrument ;  but  if  they  do  separate,  the  position 
of  the  glass  must  be  rectified  by  means  of  the  screws  in  its  pedestal. 

This  adjustment  may  be  also  made  by  directing  the  sight  through  the  telescope 
to  any  well-defined  object ;  then  if,  by  moving  the  central  index,  the  reflected  image 
passes  exactly  over  the  object  seen  directly,  the  glass  is  perpendicidar  ;  otherwise  its 
position  must  be  adjusted  by  means  of  the  screws  attached  to  the  pedestal  of  the 
glass. 

A  planet,  or  star  of  the  first  magnitude,  will  be  a  good  object  for  this  purpose.  If 
the  sun  is  used,  one  of  the  colored  glasses  must  be  placed  at  C,  and  another  at  D. 

To  make  the  axis  of  the  telescope  parallel  to  the  plane  of  the  instrumeiit. 

The  telescope  may  be  raised  or  depressed  by  means  of  two  screws  attached  to  the 
standards  i,  k  (fig.  2),  and  passing  through  two  pieces  of  brass  connected  with  the 
tube  of  the  telescope.  On  each  of  these  pieces  is  a  mark  or  index,  by  which  tlie 
telescope  is  to  be  adjusted  ;  for,  by  bringing  the  indices  to  the  same  mark  on  each 
standard,  the  telescope  will  be  parallel  to  the  plane  of  the  instrument,  f 

To  fnd  that  division  to  ivhich  the  horizon  index  must  be  placed  to  render  the  mirrors 
parallel  tvhen  the  central  index  is  on  0. 

Place  the  central  index  on  0  ;  direct  the  telescop-e  to  the  horizon  glass,  so  that  the 
line  joining  the  silvered  and  transparent  parts  of  that  glass  may  appear  in  the  middle 
of  the  telescope  ;  hold  the  instrument  vertically,  and  move  the  horizon  index  till  the 
direct  and  reflected  horizons  agree,  and  the  division  shown  by  the  horizon  nidex  will 
be  that  required. 

This  adjustment  may  also  be  made  by  measuring  the  diameter  of  the  sun  Ln 

*  When  the  instrument  is  furnished  with  adjusting  tools,  this  adjustment  may  be  made  in  the  following 
manner : — Set  the  two  tools  on  opposite  parts  of  the  limb  at  T  and  L  ;  place  the  eye  at  e,  at  nearly  the 
same  hcin^ht  as  the  upper  edge  of  the  tools,  so  that  part  of  liie  tool  at  T  may  be  hid  by  the  central  glass  j 
move  the  central  index  till  the  reflected  image  of  the  tool  nearest  the  eye  appears  i^n  the  central  glass 
at  the  side  of  the  other  tool  seen  directly  ;  then,  if  the  upper  edges  of  the  tools  are  apparently  in  the 
same  straight  line,  the  central  glass  is  perpendicular  to  the  plane  of  the  instrument ;  otherwise  its 
position  must  be  adjusted  by  the  screws  at  the  back  of  the  frame. 

t  If  you  suspect  that  the  marks  on  the  standards  are  inaccurate,  you  may  examine  them  in  the  follow- 
ing manner: — Lay  the  circle  horizontally  on  a  table  ;  place  the  two  adjusting  tools  on  opposite  parts 
of  the  limb,  at  T  and  L  ;  and  at  about  12  or  1.5  feet  distance  let  a  well-defined  mark  be  placed,  so  as  to 
be  in  the  same  straight  line  with  the  tops  of  the  tools  ;  then  raise  or  lower  the  telescope  till  the  mark  is 
apparently  in  the  middle  between  the  two  wires  ;  then  the  axis  of  the  telescope  will  be  jwrallel  to  the 
plane  of  the  instrument,  and  the  difference  (if  any)  between  the  divisions  pointed  out  by  the  indices  on 
the  graduation  of  the  standards  i,  k  (fig.  2),  will  be  the  error  of  the  indices,  and,  this  being  known,  it  will' 
be  easy,  in  future  adjustments,  to  make  allowance  for  it. 


140  CIRCLE   OF   REFLECTION. 

contrary  directions  ;  thus,  the  central  index  bein^  fixed  on  0,  place  a  dark  glass  at  C, 
and  another  at  D  ;  direct  the  telescope  (througii  the  transparent  part  of  the  horizon 
glass)  to  the  sim,  and  move  the  horizon  index  till  liis  reflected  image  appear  in  tlie 
telescope  ;  bring  the  iijtper  edge  of  the  direct  image  to  coincide  with  the  lower  of  the 
other,  and  note  the  angle  shown  by  the  index ;  then,  by  moving  the  horizon  index, 
bring  the  lower  edge  of  the  direct  image  to  coincide  with  the  uj)i)er  edge  of  the 
reflected  one,  and  note  also  the  angle  pointed  out  by  the  index;  half  the  sum  of  these 
two  angles  will  l;e  tlie  point  of  the  limb  where  the  horizon  index  must  be  plnced  to 
render  the  mirrors  ])arallel.  Thus,  if  the  index,  in  the  first  obsei-vation,  is  on  473°  30', 
and,  in  tlie  second,  on  474°  34',  the  half  sum  of  tlie  two,  474°  2',  will  be  tlie  point 
wheie  the  horizon  index  must  be  placed  to  make  the  mirroi's  parallel. 

Thrse  art  all  the  arljiistments  necessary  to  he  made*  preparatory  to  ^measuring  any 
angular  distance.  When  the  angle  is  'measured  by  cross  observations,  the  error 
arising  from  the  want  of  j)arallelism  of  the  surfaces  of  the  mirrors  and  screens,  wi!l 
in  general  lie  very  small ;  however,  the  method  of  verifying  those  glasses,  and  making 
allowance  for  any  error  in  them,  will  be  given  hereafter. 

To  observe  the  meridian  altitude  of  any  crlrstial  object,  either  by  an  observation 
to  the  right  or  to  the  left. 

The  method  of  ol)serving  the  meridian  altitude  of  an  object  with  a  circle,  is  exactly 
similar  to  that  with  a  quadrant  or  sextant.  The  central  index  must  be  fixed  on  0, 
and  the  horizon  index  on  the  point  which  renders  the  two  mirrors  parallel  ;  then  the 
altitude  may  be  taken  either  by  an  observation  to  the  right  or  to  the  left ;  but  the 
former  method,  in  which  the  large  colored  glasses  are  not  necessary,  is  in  general  to 
be  preferred,  because  these  large  glasses  are  more  liable  to  cause  an  eiTor  in  the 
observation  than  the  small  ones. 

If  an  observation  to  the  right  is  to  be  taken,  a  small  dark  glass  must  be  placed  at  C,  if 
the  object  be  bright ;  then  hold  the  instrinnent  in  the  right  liand,  in  a  vertical  position  , 
move  the  central  index,  according  to  the  order  of  the  divisions  of  the  limb,  till  the 
reflected  image  of  the  object,  seen  in  the  telescope,  nearly  touches  the  direct  image 
of  the  horizon  ;  tighten  the  index  by  the  screw  at  the  back  of  the  instrument ;  make 
the  contact  comi)lete  in  the  middle  between  the  i)arallel  wires  of  the  telescope,  by  the 
tano-ent  screw,  and  by  s\veei)ing,  exactly  in  the  same  manner  as  when  observing  with 
a  quadrant,  and  the  central  index' will  ])oint  out  the  altitude  of  the  object. 

If  an  observation  to  the  Itjl  is  taken,  and  the  object  be  bright,  a  large  dark  glass  must 
be  placed  at  a,  a,  if  the  altitude  be  between  5°  and  3.")°,  otherwise  a  small  glass  at  C  •, 
liold  the  instrument  in  the  left  hand,  in  a  vertical  i)osition ;  move  the  central  index 
contrary  to  the  order  of  the  divisions,  and  bring  the  reflected  image  in  contact  with 
the  horizon  as  above  ;  the  angle  shown  by  the  central  index,  being  subtracted  from 
720°,  will  be  the  sought  altitude. 

In  both  these  methods  of  observing  the  meridian  altitude  of  an  object,  the  circle, 
the  radius  of  which  is  only  five  inches,  will  hardly  be  so  accurate  as  a  good  sextant 
of  a  laraer  radius  ;  but,  by  the  help  of  a  well-regulated  watch,  the  meridian  altitude 
may  be  obtained,  by  the  circle,  to  a  much  greater  degree  of  accuracy  than  by  a  sextant, 
by  observiug  in  the  following  manner: — A  few  mimites  before  the  object  passes  the 
meridian,  bei:in  to  observe  the  altitude  by  cross  observations  (in  the  manner  to  be 
described  in  the  next  article),  and  note  the  time  of  each  observation  by  the  watch-, 
continue  to  observe  till  a  few  minutes  after  the  object  has  jiassed  the  meridian  ;  then 
the  au'des  shown  by  the  central  index,  beiug  divided  by  the  whole  nmnber  of  obser- 
vations, will  give  tlie  ai)i)roximate  meridian  altitude  ;  the  correction  to  be  aiqilied  to 
it  to  obtain  the  true  meridian  altitude,  may  be  found  by  means  of  Tables  XXXII.  and 
XXXIJI.,  by  a  method  which  will  be  explained  hereafter,  when  treating  of  finding 
the  latitude  l)y  a  single  altitude  of  the  sun. 

In  this  article,  the  meridian  altitude  only  has  been  spoken  of,  though  it  is  evident 


I 


! 


*  In  some  instruments,  there  is  an  adjustment  of  the  horizon  fr'ass,  to  place  it  at  its  proper  anijle  witli 
the  axis  of  tlie  telescope  ;  if  an  adjustment  of  this  kind  is  necessary,  it  ought  to  be  made  before  thc^oih.  i 
adjustments,  in  such  manner  that  if  a  colored  glass  be  fixed  at  C,  none  of  the  rays  from  the  cenir.d 
glass  can  be  reflected  to  ll>e  telescope  from  the  horizon  glass,  without  passing  the  colored  glass.  'I'm 
eflect  tills,  the  renlelle  must  be  placed  at  I),  and  hnvcrcd  so  as  to  intercept  the  direct  light  ejitircly  ;  then 
place  the  colored  gl.-ss  at  C,  and  direct  the  telescope  to  the  silvered  part  of  tne  horizon  glass  ;  move  the 
central  index,  and  if  no  nncolored  images  appear  (reflected  from  the  central  glass),  but  all  have  the 
same  tinge  as  tliat  of  the  colored  glass  used,  the  horizon  glass  is  in  its  proper  position;  otherwise  it  must 
Be  turned  on  its  axis  till  the  uncolored  images  disappcnr. 


Times  of 

otx. 

4li. 

20in.  Os. 

4 

21 

10 

4 

q.T) 

1j 

4 

23 

0 

4 

21 

45 

4 

25 

30 

Angle. 

6)26 

16 

40 

6  )  G0°  24' 

4 

22 

47 

10      4 

CIRCLE   OF   REFLECTION.  141 

tnat  the  method  is  applicable  to  an  object  not  on  tlie  nieriilian  ;  bnt,  in  tiiis  case,  the 
cross  ol)servations,  wliicJi  give  to  tlie  circle  all  its  advantages,  may  be  used,  and  the 
mean  of  the  altitudes  taken  instead  of  a  single  altitude.  This  method  is  peculiarly 
adapted  to  the  taking  of  altitntles  for  regulating  a  watch  ;  for  this  reason  it  will  be 
particularly  explained  in  the  following  article  : — 

To  take  altitudes  of  the  sun,  or  any  crlcstiol  object,  hy  cross  observations,  for 

rrgulatiuff  a  watch. 

Fix  the  central  index  on  0,  and  if  the  object  be  bright,  and  the  altitude  between  5° 
and  35°,  place  a  large  colored  glass  jjefore  the  central  glass 
at  a,  a,  otherwise  a  small  one  at  C;  hold  the  instrument 
m  the  Itsft  hand,  in  a  vertical  position  ;  move  the  horizon 
index  till  the  image  of  the  reflected  object  be  brought  in 
com])lete  contact  with  the  horizon,  in  the  middle  between 
the  two  parallel  wires  of  the  telescope,  as  directed  in  the 
preceding  article,  and  note  the  time  of  ol)servation  by  the 
watch  ;  then  fasten  the  horizon  index;  hold  the  instrument 
in  the  right  hand,  in  a  vertical  position  ;  move  the  central 
index  according  to  the  order  of  the  divisions,  till  the  reflect- 
ed image  be  again  brought  into  complete  contact  with  the 

horizon  *  as  above,  and  note  the  time  of  observation.  Then  half  the  sum  of  the  times, 
and  half  the  angle  shown  by  the  index,  will  be  a  mean  time,  and  a  mean  altitude 
corresponding  thereto. 

If  greater  accuracy  be  required,  the  observation  must  be  repeated,  setting  out  from 
the  points  where  the  indices  then  are,  and  observing  in  the  same  manner  by  moving 
first  the  horizon  index,  then  the  central  one  ;  continue  taking  as  many  of  these  cross 
observations  as  are  judged  necessary,  and  note  the  times  of  each  observation;  then  the 
sum  of  the  times,  divided  by  the  whole  inmibcr  of  observations,  will  be  a  mean  time ; 
and  the  angle  shown  by  the  central  index,  divided  by  the  number  of  observations,  will 
be  a  mean  altitude  corresponding  thereto.  Thus,  if  sixf  observations  were  taken,  and 
the  times  noted  as  in  the  adjoined  table,  the  angle  shown  by  the  index  being  G0°  24', 
the  mean  time  would  be  obtained  by  dividing  the  sum  of  the  times,  26h.  16m.  40s.,  by 
6,  and  the  mean  altitude  by  dividing  60°  24'  by  6  ;  therefore  the  mean  time  would 
be  41).  22m.  47s.,  and  the  mean  altitude  corresponding  10°  4'. 

To  measure  the  distance  bctioecn  the  sun  and  moon  hy  a  circular  instrument. 

The  instrimient  being  well  adjusted,  fix  the  central  index  on  0,  and,  if  the  object  be 
bright,  place  a  small  dark  glass  at  C  ;  hold  the  instrument  so  that  its  plane  may  be 
directed  to  the  objects  with  its  face  tiownwards  when  the  sun  is  to  the  I'ight  of  the 
moon  ;  otherwise,  with  its  face  upwards  ;  direct  the  sight  through  the  telescope  to  the 
moon;  move  the  horizon  index,  according  to  the  order  of  the  divisions  of  the  limb,  till 
the  reflected  image  of  the  stm  appears  in  the  telescope,  and  the  nearest  limbs  of  the 
sun  and  moon  are  almost  in  contact ;  fasten  the  index,  and  make  the  coincidence  of 
the  limbs  perfect,  in  the  middle  between  the  two  jjarallel  wires  of  the  telescope,  by 
means  of  the  tangent  screw  of  the  horizon  glass,  and  note  the  time  of  observation; 

*  The  arc  described  on  ihe  limb  by  Ihe  central  index,  will  be  equal  to  twice  Ihe  altitude  of  the  object, 
or  twice  the  angle  passed  over  by  the  oilier  index  ;  if  more  cross  observations  be  taken,  each  of  the 
indices,  when  moved,  will  describe  an  arc  equal  to  double  the  allitude  of  the  object;  the  same  is  to  be 
observed  in  measuring  any  other  angular  distance.  If  the  instrument  is  furnished  with  the  arc  WSll, 
and  sliding  pieces  U,  X,  you  must  bring  the  slide  X  to  the  central  index,  after  taking  the  first  observa- 
tion to  the  left,  and  place  the  slide  U  at  the  same  degree,  on  the  arc  SW,  that  X  is  on  the  arc  PR; 
then,  in  the  next  observation,  tlie  central  index  is  to  be  brought  to  touch  the  slide  U  ;  in  the  next 
observation  to  the  left,  the  slide  X  is  to  be  brought  to  the  central  index,  and  so  on  for  the  other 
observations.  Thus,  by  means  of  the  slides,  the  indices  may  be  placed  at  nearly  their  proper  angles 
with  each  other  at  the  beginning  of  the  observation,  which  will  save  considerable  time.  After  being 
thus  fixed,  the  contact  must  be  completed  by  means  of  the  tangent  screw  of  the  index,  which  is  to  be 
moved. 

t  The  number  G  is  a  convenient  number  to  use,  because  the  remainder  of  the  division  of  the  hours  by 
6  gives  the  first  figure  of  the  minutes  ;  and  the  remainder  of  the  division  of  the  minutes  by  G  gives  the 
first  figure  of  the  seconds.  Thus,  in  the  above  example,  in  dividing  2Gh.  by  6,  we  get  4h.,  and  the 
remainder  2  is  set  down  immediately  for  the  first  figure  of  the  minutes  ;  the  second  figure  of  the  minutes  is 
the  quotient  2,  found  by  dividing  16m.  by  6,  and  tlie  remainder  4  of  this  last  division  is  the  first  figure  of 
the  seconds.  We  may  remark  that,  as  the  term  4h.  20ni.  is  common  to  all  the  6  observations,  it  maybe 
neglected  ;  then  adding  the  minutes  in  the  column  of  units,  and  the  seconds,  the  sum  becomes  16m.  JOs  ; 
■I'viding  this  by  6  gives  2m.  47s.,  to  be  connected  with  4h.  20m.,  making,  as  above,  4h.  22n3  4.T'J 


143 


CIRCLE    OF   REFLECTlvON. 


then  invert  the  instrument,  and  move  the  central  index,  according  to  tlie  ord-er  of  the 
divisions  of  the  Hmb,  by  a  quantity  equal  to  twice  the  arc  passed  over  by  the  hoi'izon 
index  (or  twice  the  distance  of  the  sun  and  moon);*  direct  the  plane  of  the  instrument 
to  the  objects ;  look  directly  at  the  moon,  and  the  sun  will  be  seen  in  the  field  of 
the  telescope  ;  fasten  the  central  index,  and  make  the  contact  of  their  nearest  limbs 
complete,  in  the  middle  between  the  two  jiarallel  wires  of  the  telescope,  by  means  of 
the  tangent  screw  of  the  central  index,  and  note  the  time  of  observation  ;  then  half  the 
arc  shown  by  the  central  index  will  be  tlie  distance  of  the  nearest  liuibs  of  the  sun  and 
moon,  and  half  the  sum  of  the  times  will  be  the  mean  time  of  observation. 

Having  finished  these  two  observations,  two  others  may  be  taken  in  the  same 
manner,  setting  out  from  the  points  where  the  indices  then  are,  and  moving  first  the 
horizon  index,  then  the  central  index  :  proceed  thus  till  as  many  observations  as  are 
judged  necessary  be  taken,  alwcys  observing  that  the  number  of  them  be  even  ;  then  the 
angle  shown  by  the  central  index  (or  that  angle  increased  by  720^  or  1440°,  &c.,  if 
the  index  has  been  moved  once  or  twice,  &.c.,  roinid  the  limb),  bciUg  divided  by  the 
whole  number  of  observations,  will  give  the  mean  distance  ;  and  the  sum  of  all  the 
times,  divided  in  like  manner,  will  be  the  mean  time  of  observation. 


7^0  measure  the  distance  between  the  moon  and  star  hy  a  circular  instrumint. 

Fix  the  central  index  on  0,  and,  if  the  moon  be  bright,  and  the  distance  between  5° 
and  35°,  place  a  large  green  glass  before  the  central  mirror  at  a,  a,  otherwise  a  small 
one  at  C  ;  hold  the  instrimient  so  that  its  plane  may  be  directed  to  the  objects  with  its 
face  downwards  when  the  moon  is  to  the  right  of  the  star,  otherwise  with  its  face 
upwards  ;  direct  the  sight  through  the  telescope  to  the  star;  move  the  horizon  index, 
according  to  the  order  of  the  divisions  of  the  limb,  till  the  reflected  image  of  the  moon 
appears  in  the  telescope,  and  the  enlightened  limb  of  the  moon  be  nearly  in  contact 
with  the  star  ;  fasten  the  index,  and  make  the  coincidence  perfect,  in  the  middle 
between  the  parallel  wires  of  the  telescope,  by  means  of  the  tangent  screw  belonging  to 
that  index,  and  note  the  time  of  observation ;  then  invert  the  instrument,  and  move  the 
central  index,  according  to  the  order  of  the  divisions  of  the  limb,  by  a  quantity  equal 
to  twice  the  arc  passed  over  by  the  horizon  index  ;  *  direct  the  pl^ne  of  the  instrument 
to  the  oljjects ;  look  directly  at  the  star,  and  the  moon  will  be  seen  in  the  field  of  the 
telescope  ;  fasten  the  central  index,  and  make  the  contact  of  the  enlightened  limb  of 
the  moon  and  the  star  complete,  in  the  middle  between  the  two  parallel  wires  of  the 
telescope,  by  means  of  the  tangent  screw  of  that  index,  and  note  the  time  ;  then  half 
the  arc  sliown  by  the  central  index  will  be  the  distance  of  the  star  from  the  enlight- 
ened limb  of  the  moon,  and  half  the  smn  of  the  times  will  be  the  mean  time  of 
observation  ;  these  two  observations  being  completed,  others  may  be  taken  in  the 
same  manner,  according  to  the  directions  above  given  for  measuring  the  distance  of 
the  sun  fioin  the  moon. 

In  continuing  to  take  these  cross  observations  by  a  circle  furnished  with  the  arc 
VVSR,  and  slides  U,  X,  it  will  be  very  easy  to  bring  the  reflected  image  into  the 
field  of  tlie  telescope ;  but  if  the  instrument  is  not  thus  furnished,  it  will  be  often 
difiicult  to  bring  the  image  into  the  field  of  the  telescope,  and  much  time  will  be  lost, 
and  the  observations  rendered  tedious  by  that  means;  to  remedy  this,  a  small  table  of 
the  angles,  at  which  each  index  should  be  ])laced,  ought  to  be  made  before  beginning 
the  observation  ;  this  table  is  easily  formed,  as  follows: — Find  roughly,  according  to 
the  directions  heretofore  given,  the  point  at  which  tlie  horizon  glass  must  be  placed  to 
be  parallel  to  the  central  glass,  when  tlie  central  index  is  on  0  ;  then  find  what  point 
of  the  arc  the  horizon  index  stands  upon,  after  measuring  the  first  distance,  as  directed 
above;  the  difference  between  these  two  points  will  be  the  angular  distance  of  the 
objects ;  the  doidile  of  this  distance,  being  successively  added  to 
0°,  and  to  th.e  aii'/l"  i)ointed  out  by  the  horizon  index  after  the 
first  observation,  w  ill  give  the  points  of  the  arc  where  the  indices 
must  be  placed  at  the  2d,  3d,  4th,  &c.  observations.  Thus,  if  the 
point  of  parallelism  is  471°,  and  the  point  where  the  horizon 
index  is  at  the  first  observation  is  525°,  the  difference,  or  54°,  will 
be  the  angular  distance  ;  the  double  of  this,  or  108°,  being  added 
to  525°,  gives  033°,  which  is  the  point  of  the  arc  where  that  index 
must  be  placed  at  the  third  observation  ;  633°  added  to  108°  gives 
741°  or  21°  (because  the  divisions  recommence  at  720°),  which  is 
the  point  where  the  index  must  be  y)laced  at  the  fifth  observa- 
tion, &c.,  as  in  the  adjoined  table.     The  central  index  being  at 

*  This  may  be  clone  expeditiously  by  means  of  the  slides  U,  X.  as  Is  explained  in  the  precediii"  no*e. 


Ccyitral 

Horizon 

Index. 

Index. 

0° 

525 

108 

033 

21G 

21 

324 

|..>I» 

432 

237 

540 

&c. 

&c. 

CIRCLE   OF   REFLECTION.  143 

first  on  0°,  after  the  second  observation  it  will  be  on  108°,  at  the  fourth  on  108° -|- 108° 
=  216°,  at  the  sixth  on  216°  -|-  108°  =  324°,  &c.  Thus,  by  constantly  adding  108°,  or 
twice  the  distance  of  the  objects,  the  angles  at  which  the  indices  must  be  placed  will 
be  obtained  ;  and  by  fixing  them  at  these  angles,  the  reflected  image  will  be  brought 
into  the  field  of  view  without  any  trouble.* 


Having  explained  the  methods  of  adjusting  and  using  tlie  circle  of  reflection,  it 
remains  to  show  how  to  calculate  the  error  arising  from  not  observing  the  contact  of 
the  objects  in  the  middle  between  the  parallel  wires  of  the  telescope,  and  also  to 
estimate  the  eiTors  arising  from  the  want  of  parallelism  of  the  mirrors  and  colored 
glasses.  These  verifications  are  much  more  necessary  in  a  sextant  than  in  a  circle, 
and  they  may  be  m  general  neglected  in  a  circle. 

To  estimate  the  error  arising  from  not  observing  the  contact  of  the  objects  in  the 
middle  between  the  parallel  wires  of  the  telescope. 

To  estimate  this  error,  it  is  necessary  to  know  the  angular  distance  of  the  wires  of 
the  telescope,  which  may  be  thus  determined : — 

Turn  round  the  eye-j)iece  of  the  telescope  till  the  wires  are  perpendicular  to  the 
plane  of  the  instrument,  and  put  the  central  index  on  0  ;  direct  the  telescope  to  any 
well-defined  object,  at  least  12  feet  distant,  and  move  the  horizon  index  till  the  direct 
and  reflected  image  of  the  object  coincide  ;  then  make  one  of  the  wires  coincide  with 
the  object,  and  turn  the  central  index  till  the  reflected  image  of  the  object  coincides 
with  the  other  wire — and  the  arc  passed  over  by  that  index,  will  be  the  angular 
distance  between  the  wires.  This  angle  being  obtained,  the  observer  must,  by  means 
of  it,  estimate,  at  each  obsen'ation,  how  much  the  place  where  the  contact  is  observed 
is  elevated  above,  or  depressed  below,  the  plane  passing  through  the  eye  and  the 
middle  line  between  the  two  parallel  Avires  of  the  telescope  :  the  con-ection  in  Table 
XXXV.,  con-esponding  to  this  angle,  is  to  be  subtracted  from  the  observed  angular 
distance  of  the  objects :  thus,  if  the  distance  between  the  wires  is  2°,  one  of  them 
will  be  elevated  above  that  plane  1°,  and  the  other  depressed  below  it,  by  the  same 
quantity ;  if,  in  taking  an  observation,  the  point  of  contact  is  estimated  to  be  one  thu-d 
part  of  the  distance  from  the  middle  towards  either  wire,  the  angle  of  elevation  or 
depression  will  be  one  third  part  of  1°,  or  20' ;  and  if  the  observed  distance  is  120°,  the 
correction  m  Table  XXXV.  will  be  12",  subtractive  from  the  observed  distance. 

The  correction  for  each  observed  distance  being  ascertained,  in  the  above  manner, 
the  sum  of  them  must  be  subtracted  from  the  whole  angle  shown  by  the  central  index, 
and  the  remainder,  divided  by  the  whole  number  of  observations,  will  be  the  mean 
distance. 

Verification  of  the  parallelism  of  the  surfaces  of  the  central  mirror. 

This  verification  is  to  be  made  ashore,  by  observing  the  angular  distance  of  two 
well-defined  objects,  whose  distance  exceeds  90°  or  100°,  having  previously  well 
adjusted  the  instrument :  after  taking  several  cross  observations,  and  finding  the  mean 
distance,  take  out  the  central  mirror,  and  turn  it  so  that  the  edge  which  was  formerly 
uppermost  may  now  be  nearest  the  ])laue  of  the  instrument ;  rectify  its  position,  and 
take  an  equal  number  of  cross  observations  of  the  angular  distance  of  the  same  two 
objects;  half  the  difference  betv/een  the  mean  of  these  and  that  of  the  former,  will  be 
the  error  of  the  observed  angle,  arising  from  the  defect  of  parallelism  of  the  central 
mirror.  If  the  first  mean  exceeds  the  second,  the  error  is  subtractive,  otherwise 
additive,  the  mirror  being  in  its  first  position  ;  but  the  contrary  when  in  its  second 
position.  Thus,  if  10  observations  are  taken  at  each  operation,  and  in  the  fii-st  the 
angle  shoAAii  by  the  mdex  is  1199°  53.V,  and  in  the  second  1200°  6^',  by  dividing  bv 
10  tlie  mean  angles  are  found  to  be  lf9°  59'  21"  and  120°  0'  39",  and  their  diflference 
is  78"  ;  the  half  of  it,  or  39",  is  the  error  of  the  muTor,  additive  when  it  is  in  its  first 
position,  subtractive  in  the  second.  The  error  for  any  other  angle  may  be  found  by 
Col.  4,  Table  XXXIV.,  when  the  inclination  of  the  plane  of  the  horizon  glass  to  the 
axis  of  the  telescope  is  80°,  by  saying.  As  the  tabular  error  corresponding  to  120°, 
that  is,  1'  30",  is  to  the  en-or  found  in  the  glass  39",  so  is  the  tabular  error  for  any 

*  If  the  distance  of  the  object  vanes  durino;  the  observation,  these  angles  will  require  correction  as 
you  proceed  with  the  observations.  Thus,  if  Ihe  distance  was  increasing,  and  at  tiie  sixth  observation 
il  was  found  tiiat  the  central  index  was  on  3'2G°  instead  of  324°,  the  increase  being  2°,  you  must  add  2° 
to  the  rest  of  the  numbers  in  the  table,  and  place  the  horizon  index,  at  the  seventh  observation,  on 
129°  4-  2°  =  131°,  and  the  central  index,  a\.  the  eighth  observation,  at  432°  -j-  2°  =  43i°,  &c. 


144  CIRCLE   OF   REFLECTION. 

other  angle  85°,  which  is  (X  28",  to  tlie  error  of  the  glass  coiTesi)onding  12" ;  and  in 
this  manner  a  tahle  of  eiTors  may  be  made,  not  only  for  the  cross  observations,  but 
also  for  observations  to  the  right  or  to  the  left.* 

It  may  be  remarked  that  the  errore  are  much  less  in  the  cross  observations  than  in 
the  observations  to  the  right,  which  are  those  made  witli  a  quadrant  or  sextant ;  so  that 
the  circle  has,  in  this  respect,  greatly  the  advantage  of  those  instruments. 

The  angle  between  the  plane  of  the  horizon  glass  and  axis  of  the  telescope  produced 
being  nearly  the  same  in  all  observations  and  adjustments  of  the  circle,  no  sensible 
error  can  arise  from  the  want  of  parallelism  in  the  surfaces  of  that  glass. 

Verification  of  the  paralleUsm  of  the  colored  glasses. 

Place  one  of  the  dark-colored  glasses  at  C,  and  another  at  D  ;  fix  the  central  index  at 
0,  direct  the  telescojie  to  the  sim,  and  move  the  horizon  index  till  the  limbs  of  the 
direct  and  reflected  image  coincide  ;  then  turn  the  dark  glass  placed  at  C,  so  that  the 
sui-face  which  was  farthest  from  the  horizon  glass  n)ay  now  be  nearest  to  it,  and  if  the 
contact  of  the  same  two  limbs  be  complete,  the  surfaces  of  the  glass  y^laced  at  C  are 
parallel ;  but  if  the  limbs  lap  over  or  separate,  the  central  index  must  be  moved  to 
brhig  them  again  in  contact;  then  half  the  arc  passed  over  by  that  index  will  be  the 
error  arising  from  the  want  of  parallelism  of  the  glass  C.  If  great  accuracy  is  required, 
the  operation  may  be  repeated  by  setting  out  from  the  point  where  the  indices  then 
are,  and  taking  4  or  6,  &c.,  obsen^ations ;  then  the  arc  passed  over  by  the  central 
index,  being  divided  by  4,  6,  &c.,  will  be  the  sought  error.  The  other  small  glasses 
may  be  verified  in  the  same  manner  ;  and,  by  placing  one  of  the  larger  glasses  before 
the  central  index  at  a,  a,  and  one  of  the  smaller  ones  at  D,  the  former  may  be  verified 
as  above.  The  gi-een  glasses  may  be  verified  by  observing  the  diameter  of  the  full 
moon,  or  by  some  bright  terrestrial  object. 

It  may  be  remarked,  as  one  of  the  greatest  advantages  of  the  circle,  that,  in  measur- 
ing an  angle  by  the  cross  observations,  no  error  can  arise  from  the  want  of  parallelism 
in  the  surfaces  of  the  smaller  dark  glasses  ;  for  if  these  glasses  give  too  gi'eat  an  angle 
by  an  observation  to  the  right,  they  will  give  too  little  by  the  same  quantity  by  an 
observation  to  the  left.  It  is  not  so  with  the  large  glasses  placed  at  a,  a,  because  the 
incidence  of  the  rays  on  these  glasses  is  more  oblique  in  one  observation  than  in  the 
other,  so  that  the  errors  do  not  wholly  balance  each  other;  .however,  as  these  glasses 
are  used  only  in  measuring  angles  less  than  35°,  where  the  errore  are  nearly  the  same 
as  if  the  incidence  of  the  rays  wei'e  perpendicular,  the  errors  of  these  glasses  will  also 
nearly  compensate  each  other  in  the  cross  obsen^ations  ;  and  if  such  observations 
only  are  used,  it  v/ill  be  unnecessary  to  verify  the  dark  glasses.  Even  when  taking 
observations  to  the  right,  or  observations  to  the  left,  the  error  of  the  dark  glasses  will 
be  destroyed,  if  the  glass  is  turned  at  each  observation,  and  the  number  of  observa- 
tions is  even  ;  but  there  are  some  cases  in  which  an  angle  can  only  be  measured  by 
one  observation  ;  then  it  will  be  necessary  to  allow  for  the  error  of  the  dark  glass,  if 
the  distance  is  required  to  be  found  withiii  a  few  seconds. 

*  If  ilie  incliiiaiion  of  llie  plane  of  the  horizon  glass  and  the  axis  of  tlie  telescope  fliffer  from  80°,  you 
may  fiiKJ  the  tabular  numbers  by  the  method  given  in  the  explanation  of  Table  XXXIV.  aflixed  to  the 

table*. 


Plate  XI 


Fiffl 


Fig.6  Figvr 


SI 


I  HOI 


DESCRIPTION   AND   USE   OF   A   PORTABLE 
TRANSIT    INSTRUMENT. 


A  Transit  Instrument  is  of  no  service  on  board  of  a  vessel,  but  is  much  used 
arbore,  in  seaports,  for  regulating  chronometers  for  sea  voyages,  and  in  making 
observations  to  determine  the  longitude.  We  have,  therefore,  thouglit  it  would  be 
useful  to  give  a  brief  description  of  it,  with  the  methods  of  adjustment ;  particularly 
as  it  may  be  considered  as  a  valuable  accession  to  the  apparatus  of  a  good  navigator, 
who,  while  remaining  in  port  a  few  days,  can,  by  means  of  it,  adjust  and  fix  the  rate 
of  going  of  his  chronometer  with  ease  and  accuracy,  and  also  obtain  the  best  data  for 
determining  the  longitude  of  the  place,  by  observing  the  times  of  tlie  moon's  transit 
or  passage  over  the  meridian. 

The  figure  in  Plate  XI.,  figure  1,  represents  this  instrument,  according  to  the 
usual  construction  of  Mr.  Troughton,  with  a  telescope  of  about  twenty  inches  foca 
length.  The  telescope  tube  AA  is  in  two  parts,  connected  together  by  a  sphere  B 
which  also  receives  the  larger  ends  of  the  two  axes  C,  C,  placed  at  right  angles  to  the 
direction  of  the  telescope,  and  forming  the  horizontal  axis.  This  axis  terminates 
in  two  cylindi-ical  jjivots,  which  rest  in  Y's  fixed  at  the  upper  end  of  the  vertical 
standards  D,  D.  One  of  the  Y's  possesses  a  small  motion  in  azimuth,  communicated 
I)y  tuniing  the  azimuth  screw  a.  In  these  Y's,  the  telescope  turns  upon  its  pivots ; 
but,  that  it  may  move  in  a  vertical  circle,  the  pivots  must  be  precisely  on  a  level  with 
each  other ;  otherwise  the  telescope  will  revolve  in  a  plane  oblique  to  the  horizon, 
instead  of  being  perpendicular  to  it.  The  levelling  of  the  axis,  as  it  is  called,  is  there- 
fore one  of  the  most  important  adjustments  of  the  instrument,  and  is  effected  by  the 
aid  of  a  spirit  level  E,  which  is  made,  for  this  purpose,  to  stride  across  the  telescope, 
and  rest  on  two  pivots. 

The  standards  DD  are  fixed  by  screws  upon  a  brass  circle  F,  which  rests  on  three 
screws  b,  c,  d,  forming  the  feet  of  the  instriunent,  by  the  motion  of  which  the  operation 
of  levelling  is  performed.  The  two  oblique  braces  GG  are  for  the  purpose  of  steady- 
ing the  supports,  it  being  essential  for  the  telescope  to  have  not  only  a  free  but  a 
steady  motion.  On  the  extremity  of  one  of  the  pivots,  which  extends  beyond  its  Y, 
is  fixed  a  circle  II,  which  turns  with  the  axis,  while  the  double  vernier,  ee,  remain.s 
stationary  in  a  horizontal  position,  and  shows  the  altitude  to  which  the  telescope  is 
elevated.  The  verniers  are  set  horizontal  by  means  of  a  spirit  level  f,  which  is 
attached  to  them,  and  they  are  fixed  in  their  position  by  an  arm  of  brass  g,  clamped 
to  the  supports  by  a  screw  h;  the  whole  of  this  apparatus  is  movable  with  the 
telescope,  and,  when  the  axis  is  reversed,  can  be  attached,  in  the  same  manner,  to  the 
opposite  standard. 

Near  the  eye-end,  and  in  the  principal  focus  of  the  telescope,  is  placed  the  diaphragm, 
or  ivire-plate,  which  has  five  vertical  and  two  horizontal  wires.  The  centre  vertical 
wire  ought  to  be  fixed  in  the  optical  axis  of  the  telescope,  and  perpendicular  to  a  line 
drawn  through  the  pivots  of  the  axis.  It  will  be  evident,  upon  consideration,  that 
these  wires  are  rendered  visible,  in  the  day-time,  by  the  rays  of  light  passing  down  the 
telescope  to  the  eye  ;  but  at  night,  except  when  a  very  luminous  object  (as  the  moon) 
is  observed,  they  cannot  be  seen.  Their  illumination  is  therefore  effected  by  piercing 
one  of  the  pivots,  and  admitting  the  light  of  a  lamp  fLxed  on  the  top  of  one  of  the 
standards,  as  sho\\ii  at  I.  This  light  is  directed  to  the  wires  by  a  reflector  placed 
diagonally  in  the  sphere  B.  The  reflector,  having  a  large  hole  in  its  centre,  does  not 
interfere  with  the  rays  passing  down  the  telescope  from  the  object,  and  thus  the 
observer  sees  distinctly  the  wires  and  the  object  at  the  same  time.  When,  however, 
the  object  is  very  faint  (as  a  small  star),  the  light  from  the  lamp  would  overpower 
its  feeble  rays.  To  remedy  this  inconvenience,  the  lamp  is  so  constructed  that,  by 
turning  a  screw  at  its  back,  or  inclining  the  o];)ening  of  the  lantern,  more  or  less  light 
may  be  admitted  to  the  telescope,  to  suit  the  circumstances  of  the  case. 

The  telescope  is  furnished  with  a  diagonal  eye-piece,  by  which  stars  near  the 
Kenith  may  be  observed  without  inconvenience. 
19 


14G  PORTABLE   TRANSIT   INSTRUMENT. 

Acljusiments  of  a  transit  insirumen' 

In  fixing  the  instrument,  it  should  be  so  placed  that  the  telescope,  when  level, 
should  point  north  and  south  as  near  as  can  possibly  be  ascertained.  This  can  at  first 
be  done  only  in  an  approximate  manner,  as  the  correct  determination  of  the  meridian 
can  only  be  obtained  by  observation,  after  the  other  adjustments  are  completed. 

To  adjust  the  line  of  coUimation. 

The  first  adjustment  is  that  of  the  line  of  collimation,  or  line  of  sight.  Direct  the 
telesco]ie  to  some  distant,  well-defined  object  (the  more  distant  the  better),  and  bisect 
it  Avitii  the  middle  of  the  central  wire  ;  tlien  lift  the  telescope  veiy  carefully  out  of  its 
angular  bearings  or  Y's,  and  replace  it  with  the  axis  reversed  ;  ])oint  the  telescope 
again  to  the  same  object,  and,  if  it  be  still  bisected,  the  collimation  adjustment  is 
correct;  if  not,  move  the  wires  one  half  the  error,  by  turning  the  small  screws  which 
hold  the  diaphragm  near  the  eye-end  of  the  telescope,  and  tlie  adjustment  will  be 
accomplished  ;  but  as  half  the  deviation  may  not  be  correctly  estimated  in  moving  the 
wires,  it  becomes  necessary  to  verify  the  adjustment  by  moving  the  telescope  the 
other  half,  which  is  done  by  turning  the  azimuth  screw  a ;  this  gives  the  small 
azimutlial  motion  to  the  Y,  before  spoken  of,  and  consequently  to  the  pivot  of  the  axis 
which  it  carries.  Having  thus  again  bisected  the  object,  reverse  the  axis  as  before, 
and,  if  half  the  error  was  correctly  estimated,  the  object  will  be  bisected  upon  the 
telescope  being  directed  to  it ;  if  not  quite  correct,  the  operation  of  reversing  and 
correcting  half  the  error,  in  the  same  manner,  must  be  gone  through  again,  until,  by 
successive  approximations,  the  object  is  found  to  be  bisected  in  both  positions  of  the 
axis  ;  the  adjustment  will  then  be  perfect. 

To  adjust  the  ivires  in  the  telescope. 

It  is  desirable  that  the  central  or  middle  wire  (as  it  is  usually  termed),  should  be 
truly  vertical,  as  we  shall  then  have  the  power  of  observing  the  transit  of  a  star  on 
any  part  of  it,  as  well  as  the  centre.  We  may  ascertain  whether  it  is  so,  by  elevating 
and  depressing  the  telescope ;  for  when  directed  to  a  distant  object,  if  it  is  bisected  by 
every  part  of  the  wire,  the  wire  is  vertical ;  if  otherwise,  it  should  be  adjusted  by 
turning  the  inner  tube  carrying  the  wire-plate  until  the  above  test  of  its  being  vertical 
be  obtained,  or  else  care  must  be  taken  that  observations  are  made  near  the  centre 
only.  The  other  vertical  wires  are  ;>laced,  by  the  maker,  equidistant  from  each  other 
and  parallel  to  the  middle  one  ;  therefore,  when  the  middle  one  is  adjusted,  the  others 
are  so  too  ;  he  also  places  the  two  transverse  wires  at  right  augles  to  the  vertical  middle 
wire.  These  adjustments  are  always  ])erformed  by  the  maker,  and  are  but  little  liable 
to  derangement.  When,  however,  they  happen  to  get  out  of  order,  and  the  observer 
wishes  to  correct  them,  it  is  done  by  loosening  the  screws  which  hold  the  eye-end 
of  tlie  telescope  in  its  place,  and  turning  the  end  round  a  small  quantity,  by  the  hand, 
until  the  error  is  removed.  But  this  operation  requires  very  delicate  handling,  as  it  is 
liable  to  remove  the  wires  from  the  focus  of  the  object-glass. 

To  Jix  the  axes  or  arms,  upon  which  the  telescope  revolves,  in  a  ho  -izontal position 

The  axes  on  which  the  telescope  tui-ns,  must  then  be  set  horizontal.  To  do  this,  ai)])ly 
the  level  to  the  pivots ;  bring  the  air-bubble  to  the  centi-e  of  the  glass  tube,  by  turning 
the  foot-screw  h,  which  raises  or  lowers  that  end  of  the  axis,  and  consequently  the 
level  resting  upon  it ;  then  reverse  the  level,  by  turning  it  end  for  end,  and,  if  the  air- 
bubble  still  remain  central,  the  axes  will  be  horizontal ;  but  if  not,  half  the  deviation 
nnist  be  corrected  by  the  foot-screw  h,  and  the  other  half  by  turning  the  small  screw 
i,  at  one  end  of  the  level,  which  raises  or  lowers  the  glass  tube  (containing  the  air- 
bubble)  relative  to  its  sn])i)orts,  which  rest  upon  the  pivots.  This,  like  most  of  the 
adjitstmmls,  freqnentb/  requires  several  repetitions  before  it  is  accomplished,  on  account  of 
the  dificidti/  of  estimating  exactly  half  the  error. 

This  adjustment  may  also  be  made  by  means  of  the  polar  star ;  first  by  observing 
direcdy  its  transit  over  any  one  of  the  vertical  wires  of  the  telescojje,  and  immediately 
afterwards  observing  the  reflected  image  of  the  same  star  from  a  basin  of  quicksilver. 
For  if  the  star  a])pear  on  the  same  wire,  the  axis  is  properly  adjusted  ;  if  not,  you 
must  bring  the  wire  half  way  towards  it  by  the  small  screw  i,and  then,  by  the  azimuth 
screw  a,  bring  it  upon  the  wire  again.     This  being  completed,  you  nnist,  as  soon 


rORTABLE   TRANSIT   INSTRUMENT.  147 

as  possible,  loolc  directly  towards  the  star,  and  if  it  appear  on  the  same  wire,  the 
adjustment  is  arcurate  ;  if  not,  repeat  the  operation  till  it  is  so ;  observing  that  the 
motion  of  the  pole-star  is  so  very  slow,  that  it  will  not  be  sensibly  altei-ed  in  the 
interval  of  taking  its  ti'ansit  directly  and  by  reflection.  The  farther,  however,  you 
observe  the  star  from  the  meridian,  the  more  accurate  will  the  observation  be,  since 
the  motion  of  the  star  in  a  direction  parallel  to  the  horizon  will  then  be  the  least ;  and 
when  it  is  at  its  greatest  azimuth,  the  horizontal  motion  is  nothing. 

To  fix  the  inslrume.nl  so  thai  the  line  of  collimalion  of  the  telescope  may  move  accurately  in 

the  plane  of  the  meridian. 

Having  set  llie  axis,  on  which  the  telescope  turns,  parallel  to  the  horizon,  and 
proved  the  correct  position  of  the  central  wire,  or  line  of  collimation,  making  it 
descril)e  a  vertical  great  circle,  perpendiculai*  to  the  axis,  we  must,  in  the  last  place, 
fix  the  instrument  so  that  this  vertical  circle  may  be  the  meridian  of  the  place  of 
observation. 

We  have  supposed  the  instrument  to  be  nearly  in  the  meridian.  It  may  bo  so 
placed,  with  a  great  degi'ce  of  accuracy,  at  the  v^ery  first  operation,  by  means  of  a 
well-regulated*  and  accin-ate  time-keeper,  by  whii.'i  we  can  detei'mine  very  nearly 
the  exact  instant  of  the  transit  of  the  pole-star  over  the  meridian,  either  above  or 
below  the  pole.  A  few  minutes  before  the  time  of  the  transit,  we  must  direct  the 
telescope  towards  the  star,  and,  by  turning  the  azimuth  screw  a,  bring  the  star  upon 
the  middle  wire  of  the  telescope.  The  apparent  motion  of  this  star  is  so  very  slow, 
that  we  can,  by  a  very  small  and  gentle  motion  of  the  azimuth  screw  a,  keep  the  star 
constantly  bisected  on  the  middle  vertical  wire  of  the  telescope,  till  the  moment  of  this 
^•ansit,  as  indicated  by  the  time-keeper,  has  arrived ;  then  the  instrument  will  be  very 
learly  in  the  plane  of  the  meridian,  and  the  final  con-ections  must  be  made  in  the 
.  ollowing  manner  : — 

First  Method.  Make  the  observations  of  the  transits  of  the  pole-star,  above  and 
lelow  the  pole,  at  three  successive  transits,  and  note  the  times  of  observation  by  an 
iccurate  time-keeper.  Then,  if  the  interval  of  time  between  the  first  and  second 
'ransits  is  equal  to  the  interval  between  the  second  and  third  transits,  the  instrument 
will  be  truly  fixed  in  the  plane  of  the  meridian.  In  this  case,  each  of  the  intervals 
will  be  equal  to  12  hours,  sideral  time,  corresponding  nearly  to  ll**  58'"  2%  as 
shown  by  an  accurate  chronometer,  regidated  to  mean  solar  time,  f  It  is  very  impor- 
tant, in  this  operation,  that  the  rate  of  the  time-keeper  should  be  perfectly  imiform 
during  both  intervals ;  but  it  is  not  necessary  that  its  I'ate  or  regulation  should  be 
previously  known.  For,  in  the  preceding  example,  if  the  time-keeper  move  too  fast 
for  mean  solar  time,  and  gain,  for  example,  10'  in  each  of  the  above  intervals,  making 
them  equal  to  11''  58""  12^  by  the  time-keeper,  their  equality  would  prove  the  accuracy 
of  the  adjustment  to  the  plane  of  the  meridian,  with  the  same  degree  of  certainty  as  if 
the  time-keeper  were  regulated  to  mean  solar  or  sideral  time.  However,  it  is  much 
more  convenient  to  have  it  well  regulated. 

Sujipose,  now,  that  the  intervals,  instead  of  being  equal  to  each  other,  are  found 
to  differ.  In  this  case,  tlie  instrument  is  not  placed  accurately  in  the  plane  of  the 
meridian  ZM??tII  (Plate  XI.  fig.  2,  3),  but  the  motion  of  the  telescope  is  in  some 
vertical  circle,  as  ZSsT,  which  cuts  the  horizon  in  the  point  T,  situated  to  the 
west  of  the  meridian  H,  in  figure  2,  or  to  the  east,  in  figure  3  ;  the  distance  from 
the  meridian  being  measured  on  the  horizon  by  tlie  arc  of  azimuth  HT.  If  we 
now  su])i)ose  that  MWmE  is  the  small  circle  described  by  the  star  in  its  diurnal 
motion,  M  will  be  the  place  of  the  star  at  its  upper  transit  over  the  meridian,  and  ?n 
its  place  at  the  lower  transit,  when  well  adjusted  ;  but  when  the  vertical  motion  of 
the  telescojie  is  in  the  vertical  circle  ZSsT,  the  upper  observed  transit  will  be  at 
S,  and  the  lower  observed  transit  at  s ;  the  observed  intervals  of  times  being  propor- 
tional to  the  arcs  SWs,  sES.     Now,  it  is  evident,  from  the  inspection  of  figures  2,  3, 


*  Tliis  reg^ulation  can  be  made  by  equal  altitudes  of  the  sun,  observed  with  a  sextant  ;  or  by  a  siiifjle 
altitude,  when  the  latitude  of  the  place  is  known  ;  or  by  similar  observations  with  a  known  star.  Tlie 
method  of  obtaining  the  time  from  such  observations,  will  be  explained  hereafter. 

t  If  the  sun  be  supposed  to  move  uniformly  in  the  plane  of  the  equator,  rtie  interval  of  two  suscessive 
transits  of  the  sun  over  the  upper  meridian,  will  be  equal  to  24  hours,  mean  solar  time,  and  it  is  for  this 
mean  solar  time  that  chronometers  are  usually  adjusted.  The  interval  between  two  successive  transits 
of  a  fixed  star  over  the  same  meridian,  is  very  nearly  equal  to  SS"*  oG™  4',  mean  solar  time  ;  but  it  is 
found  very  convenient,  in  many  astronomical  and  nautical  calculations,  to  divide  it  into  24  hours,  which 
are  called  hours  of  sideral  time,  and  they  are  divided,  as  usual,  into  minutes  and  seconds.  We  iiave 
pvcn,  in  our  collection  of  tables,  two  tables  for  facilitating  the  reduction  of  the  one  of  these  limes  to 
the  other. 


148  PORTABLE  TRANSIT   INSTRUMENT. 

that  t\.^  deviation  is  alwaj's  towards  that  side  of  the  jneridian  where  the  least  interval 
is  observed ;  as,  for  example,  in  figure  2,  wliere  the  telescope  describes  the  vertical 
circle  ZSsT,  to  the  west  of  the  meridian,  the  western  interval  SWs  is  the  least 
The  correction  of  this  adjustment  is  made  by  means  of  a  slight  motion  of  the  azimuth 
screw  a;  and  the  quantity  of  this  motion  depends  on  the  difference  of  the  two 
intervals.  Suppose,  for  example,  that  one  of  the  intervals  is  ll**  58"'  2%  and  the 
other  ll*"  58'"  22",  which  differ  20  seconds  of  time ;  the  half-difference,  10  seconds, 
represents  the  time  required  by  the  star  to  pass  over  both  the  small  arcs  MS,  sm  ; 
and,  in  the  case  of  the  pole-star,  where  the  polar  distance  PM,  or  P?n,  is  very  small, 
the  arcs  IMS,  sm,  are  very  nearly  equal  to  each  other,  so  that  each  of  these  arcs  will 
be  described  in  about  one  half  of  10',  or  5'  ;  or,  in  other  words,  the  time  required  to 
describe  the  arc  MS,  or  sm,  is  very  nearly  equal  to  one  quarter  part  of  the  difference 
between  the  two  intervals,  which,  in  the  present  example,  is  ^  X  20'  :=  5'.  To  correct 
this,  we  must  watch  the  pole-star,  as  it  approaches  towards  tlie  lower  transit  s,  if  the 
deviation  be  to  the  west  of  the  meridian,  or  as  it  approaches  towards  the  upper  transit 
S,  if  the  deviation  be  to  the  east  of  the  meridian  ;  and,  the  moment  the  star  is  bisected 
by  the  middle  wire  of  the  telescope,  we  must  begin  to  count  these  five  seconds  of 
time,  and,  by  a  very  gentle  motion  of  the  azimuth  screw  a,  keep  the  star  constantly 
bisected  by  the  wire  until  the  expiration  of  the  time  of  5  seconds,  or  the  quarter  of 
the  difference  of  the  intervals.  Then,  if  every  part  of  the  operation  has  been  done 
accurately,  and  the  time-keeper  be  perfectly  coiTcct,  the  instrument  will  be  accurately 
adjusted  in  the  plane  of  the  meridian  ;  but  as  this  is  one  of  the  most  important  and 
delicate  adjustments,  it  will  be  best  to  repeat  again  the  observations  of  the  three 
transits,  to  ascertain  whether  the  first  and  second  inteiTals  of  the  successive  transits 
are  equal ;  and,  if  a  slight  difference  should  still  be  found,  it  must  be  corrected  by 
repeating  the  operation  in  the  manner  Ave  have  already  explained. 

This  method  of  adjusting  the  transit  instrument  (by  means  of  the  pole-star)  is 
preferable  to  any  other  whatever.  Delambre,  who  had  much  practical  experience, 
says  there  is  no  advantage  in  using  two  stars ;  and  that,  with  a  single  star,  the  prefer- 
ence is  to  be  given  to  the  pole-star  ;  after  this  he  recommends  the  stars  d,  §,  y,  of  the 
Little  Bear,  and  y  Cephei.  These  stars  being  more  distant  from  the  Jjole,  it  may 
become  necessary  to  make  a  small  correction  in  the  quarter  part  of  the  difference  of 
the  intervals,  to  correct  for  the  difference  of  the  arcs  SM,  sm.  This  correction  is 
made  by  means  of  the  Table  A,  page  151,  which  gives  the  correction  for  the  pole- 
star,  and  for  other  stars  where  polar  distance  is  le^s  than  40'',  supposing  the  difference 
of  the  two  intervals  to  be  1000  seconds  of  time.  Thus,  if  the  polar  distance  of  the 
star  be  20°,  and  the  latitude  42°,  the  tabular  correction  is  82%  which  is  to  be  applied 
to  one  quarter  part  of  the  assumed  difference  of  the  intervals,  1000%  that  is,  250% 
making  250'  -f-  82'  =  332'  for  the  distance  of  the  star  from  the  meridian,  at  the 
time  of  the  lower  transit,  and  250'  —  82'  =  168'  for  the  distance  of  the  star  from 
the  meridian  at  the  upper  transit.  These  times,  332',  168',  must  be  reduced  in  the 
same  ratio  as  the  actual  difference  of  the  two  intervals  bears  to  the  tabular  difference 
1000'.  Thus,  if  the  observed  difference  of  the  two  intervals  were  205',  instead  of 
1000',  you  must  say.  As  1000'  is  to  205%  so  is  332'  to  68',  and  so  is  168'  to  34' , 
so  that  the  correction  to  be  applied  to  the  lower  transit  is  68%  and  to  the  upper  transit 
34'.  Tlierfifore,  if  the  star  be  approaching  towards  the  meridian  at  the  time  of  the 
lower  ti'ansit,  you  must  proceed  according  to  the  former  direction  relative  to  the  pole 
star,  and  keep  the  star  constantly  bisected  by  the  middle  wire  of  the  telescope,  by  a 
slight  and  gentle  motion  of  the  azimuth  screw  a,  from  the  time  of  its  first  transit  by 
that  wire,  till  you  have  counted  68'  by  the  time-keeper.  But  if  the  star  be  a])in-oach- 
iiig  towards  the  meridian  at  the  upper,  transit,  you  must  adjust  the  instrmnent  by 
means  of  the  next  upper  transit,  making  an  allowance  of  34'  for  the  distance  from 
the  meridian,  and  keeping  the  star  constantly  bisected,  from  the  time  of  its  transit 
by  the  middle  wire,  by  means  of  the  azimuth  screw  a,  until  the  termuiation  of  the 
time  of  34'. 

Before  closing  our  remarks  on  this  method  of  adjustment,  we  may  observe,  that  if 
the  angular  value  of  one  revolution  of  the  azinuith  screw  be  knouii,  or  the  instrument 
possess  an  azimuth  circle,  by  which  the  motion  of  the  telescope  may  be  accurately 
estimated,  we  may  correct  the  adjustment  by  estimating  the  correction  in  azimiuh  by 
means  of  Table  B,  whei'e  the  variations  of  azinuith,  in  seconds  of  a  degree,  are  given 
for  a  supposed  variation  of  1000  seconds  in  the  difference  of  the  two  intervals.  Thus, 
in  the  i)receding  example,  where  the  polar  distance  of  the  star  is  20°,  latitude  42° 
diffeiT'uce  of  the  two  intervals  205' ;  the  tabular  coirection  for  1000'  (difference  of 
the  two  intervals)  being  30'  42",  we  have  1000'  :  205'  : :  30'  42"  :  &  18"  ;  therefore 
the  correction  of  azimuth  is  6'  18",  to  bring  it  iiUo  the  plane  of  the  meridian 

After  the  instriunent  has  been  completely  adjusted  to  the  plane  of  the  meridian,  it 


PORTABLE   TRANSIT  INSTRUMENT.  149 

is  usual  to  fix  a  meridian  mark  on  some  distant  object  to  the  north,  and  another  to  the 
soutii ;  and,  by  mean  of  these  marks,  the  observer  can  ascertain,  with  much  certainty, 
whelher  the  instrumjnt  has  been  altered  in  its  adjustments,  from  any  accidental  cause, 
since  the  last  time  it  was  used.  Sometimes,  with  an  additional  glass  to  correspond  to 
the  distance  of  the  mark,  and  a  scale  of  seconds  of  azimuth  made  near  the  meridian 
mark,  we  may  correct  the  instrument  for  a  few  seconds'  motion  in  azimuth,  when 
correcting  the  adjustment  in  the  manner  we  have  just  been  speaking  of.  We  may 
here  n-niark,  that  the  instrument  ought  to  be  fixed  on  some  very  stable  support  (as, 
for  exaniiiie,  a  stone  block,  imbedded  in  the  ground  five  or  six  feet),  and  in  as  retired 
a  situation  as  possible,  to  avoid  the  tremulous  action  from  the  motion  of  carriages,  &c. 
It  will  also  be  extremely  convenient,  as  well  as  conducive  to  accuracy,  to  have  the 
instrument  covered  by  a  low  building,  with  slits  in  the  roof  on  the  north  and  south, 
fixed  with  movable  shutters,  so  that  the  particular  part  of  the  northern  or  southern 
sky,  where  the  observed  star  is  situated,  may  be  visible,  while  the  rest  is  covered  over, 
to  prevent  the  entrance  of  too  much  stray  light  to  the  eye,  when  obsei-ving  in  the 
twilight,  or  in  the  day-time.  As  a  greater  security  from  the  interference  of  this  kind 
of  light,  th^  observer  may  j)lace  a  thick  cloth  over  his  head,  with  a  part  of  it  very  near 
the  eye  end  of  the  telescope,  which  will  serve  very  well  to  protect  the  eye  from  any 
other  light  except  that  which  passes  through  the  telescope. 

Second  Method.  This  method  of  adjusting  the  instrument  to  ;he  plane  of  the 
meridian,  is  by  means  of  two  well-known  circnmpolar  stars,  of  nearly  the  same 
declination,  and  differing  nearly  twelve  hours  in  right  ascension,  by  observing  the 
one  above,  and  the  other  below  the  pole.  Then  it  is  evident  that  any  deviation  in  the 
instrument  from  the  meridian,  will  produce  contrary  effects  upon  the  observed  times 
of  transit,  exactly  as  in  the  upper  and  lower  transits  of  the  same  star.  Here  the  time 
which  ehijjses  between  the  two  observations,  will  differ  from  the  time  which  would 
elapse  according  to  the  catalogue,  by  the  sum  of  the  effects  of  the  deviation  upon  the 
two  stars.  We  have  given,  in  Table  C,  at  the  end  of  this  article,  the  corrections  in 
the  times  of  the  upper  and  lower  transits  of  stars,  for  various  declinations,  and  in  dif- 
ferent latitudes,  supposing  the  instrument  to  be  16'  40",  or  1000",  in  azimuth  from  the 
plane  of  the  meridian.  Thns,  if,  in  the  latitude  of  40°,  we  make  an  observation  of  the 
upper  transit  of  a  star  whose  polar  distance  is  25°,  and,  at  about  the  same  time,  the 
lower  transit  of  another  star  whose  polar  distance  is  30°,  we  shall  find  from  the  table 
that  the  correction  of  the  ujiper  transit  is  66%  and  of  the  lower  131%  for  1000"  of 
azimuth.  If  the  deviation  of  the  instrument  were  to  the  east  of  the  meridian,  by  the 
quantity  1000",  the  upper  transit  would  be  observed  too  early  by  GQ^,  and  the  lower 
too  late  by  13P  ;  consequently,  the  difference  between  the  observed  transits,  and  the 
times  of  passing  the  meridian  given  by  the  tables,  would  be  G'6'  -|-  131'  =  197^ 
Suppose,  now,  by  actual  observation  it  was  found  that  this  difference,  instead  of  being 
197%  was  only  50' ;  we  should  obtain  the  corresponding  correction  of  azimuth  by 
saying.  As  197"  is  to  50%  so  is  1000''  to  254"  ;  and,  to  correct  this  error,  we  must 
move  the  azimuth  screw  a  so  as  to  give  the  instrument  an  increase  of  254"  north- 
westerly azimuth.  In  like  manner  we  find  the  corrections  of  the  times  of  the 
transit,  by  saying.  As  197'  is  to  50%  so  is  66'  to  17',  the  correction  of  the  upper 
transit ;  or.  As  197'  is  to  50',  so  is  131'  to  33%  the  correction  of  the  lower  transit ; 
and  we  must  use  these  numbers  for  correcting  the  position  of  tlie  instrutnent,  in  the 
same  manner  as  we  have  before  directed.  Thus,  in  the  above  example,  the  star 
which  was  observed  approaching  towards  the  meridian,  at  the  up])er  transit,  was  17' 
from  the  meridian  in  time  ;  therefore,  at  the  next  upper  transit  of  the  same  star,  we 
must  observe  it  passes  the  middle  wire  of  the  telescope,  and  then,  by  means  of  the 
azimuth  screw  a,  keep  the  star  constantly  bisected  by  the  wire  during  17  seconds  of 
time,  and  then,  if  the  observation  has  been  accurately  made,  the  instrutnent  will  be  in 
the  plane  of  the  meridian. 

In  determining  the  direction  of  the  deviation,  it  must  be  recollected,  that  when  the 
deviation  is  to  the  east,  the  star  above  the  pole  passes  too  eariy,  and  that  below  the 
pole  too  late  ;  therefore,  if  the  upper  star  precedes,  the  interval  by  oliservation  will 
exceed  the  trne  interval,  between  the  passages  of  the  two  stars ;  but  if  tlie  lower  star 
precedes,  the  interval  by  obsei-vation  will  be  less  than  the  true  intei-val.  The  con- 
trary takes  i)lace  when  the  deviation  is  to  the  west  of  the  meridian.  This  method 
may  be  used  advantageously  with  8  Ursae  Minoris,  and  Cephei  51  Hev.,  which  are  given 
in  the  Nautical  Almanac.  In  like  inanner,  the  pole-star  may  be  combined  with  the 
stars  of  the  Great  Bear. 

Third  Method.  This  method  consists  in  observing  the  transits  of  any  two  stars, 
differing  from  each  other  considerably  in  declination,  and  but  little  in  right  ascension. 
The  nearer  the  obsei-vations  of  the  stars  are  to  each  other,  the  better,  as  this  prevents 


150  PORTABLE   TRANSIT   INSTRUMENT. 

the  possibility  of  any  eiTor  arising  from  a  change  in  the  rate  of  the  time-IceL'j)er 
And,  as  the  ai)parent  places  of  one  hundred  j)rincipal  stars  are  now  given  in  the 
Nautical  Almanac,  for  every  tenth  day,  it  will  be  better  to  select  two  stars  from  that 
work.  The  principle  upon  which  this  third  method  is  grounded,  is,  tliat  a  high  star 
is  less  affected  by  a  deviation  of  the  instrument  from  the  plane  of  the  meridian,  than  a 
low  star  ;  hence  it  is  evident  that  if  the  obsei'ved  differences  of  the  transits,  reduced 
to  sideral  time,  be  exactly  equal  to  the  difference  of  the  computed  right  ascensions,  the 
instrument  will  be  correctly  placed  in  the  plane  of  the  meridian  ;  if  not,  by  repeated 
operations,  by  methods  similar  to  those  before  explained,  the  adjustments  must  be 
completed.  The  restricted  limits  of  this  article  do  not  allow  us  to  go  into  many 
minute  details  which  are  used  in  large  observatories.  What  we  Irave  here  given  will 
be  sufficient  for  all  the  purposes  to  which  a  portable  transit  instrument  is  usually 
applied. 

To  observe  the  transit  of  any  heavenly  body  over  the  meridian. 

Having,  by  means  of  the  previous  adjustments,  made  the  line  of  coUimation  describe 
a  great  circle,  passing  through  the  zeniUi  of  the  place,  and  the  north  and  south  points 
of  the  horizon,  the  instrument  will  be  in  a  fit  state  for  making  the  observations.  We 
have  said  that  the  telescope  contains  five  vertical  and  two  horizontal  wires,  placed  a 
short  distance  from  each  other.  These  last  are  intended  to  guide  the  observer  in 
bringing  the  object  to  pass  across  the  middle  wire  of  the  field,  by  moving  the  telescope 
till  it  appear  between  them.  The  central  vertical  wire  is  in  the  meridian,  and  the  instant 
of  passing  this  wire  will  be  the  time  of  the  [)assage  on  the  meridian  by  that  star :  but 
as,  in  noting  the  time,  it  will  not  often  ha])pen  that  an  exact  second  will  be  sho\Mi  by 
the  clock,  when  the  star  is  bisected  by  the  wire,  but  it  will  pass  the  wire  in  the 
interval  between  two  successive  seconds ;  the  observer  must,  therefore,  whilst  watching 
the  star,  listen  to  the  beats  of  his  clock,  and  count  the  seconds  as  they  elapse  :  he  will 
then  be  able  to  notice  the  space  passed  over  by  the  star  in  every  second,  and,  conse- 
quently, its  distance  from  the  wire  at  the  second  before  it  arrives  at,  and  the  next 
second  after  it  has  passed,  the  wire  ;  and,  with  a  little  practice,  he  will  be  able  to 
estimate  the  fraction  of  a  second  at  which  the  star  was  on  the  wire,  to  be  added  to  the 
previous  second.  Thus,  if  the  observation  of  passing  the  wire  was  midway  between 
the  7th  and  8th  seconds,  the  time  of  the  transit  would  be  7". 5;  but  if  it  ajjpeared 
more  distant  on  the  one  side  than  on  the  other,  it  would  be  7'. 3,  or  7\7,  &c.,  accord- 
ing to  its  apparent  relative  distance  from  the  wire. 

This  kind  of  observation  must  be  taken  at  each  of  the  five  wires,  and  a  mean  of  the 
whole  taken,  which  will  represent  the  time  of  the  star's  passage  over  the  mean  or 
meridional  wire.  The  utility  of  having  five  wires,  instead  of  the  central  one  only,  will 
be  readily  understood  from  the  consideration  that  a  mean  result  of  several  observa- 
tions is  deserving  of  more  confidence  than  a  single  one. 

In  observing  the  su7i,  the  times  of  passing  of  both  the  first  and  second  limbs  over  the 
wires,  are  to  be  observed  and  set  down  as  distinct  observations  ;  the  mean  of  both 
observations  gives  the  time  of  the  passing  of  the  centre  across  the  meridian.  The 
wires  of  the  instrument  are  generally  placed,  by  the  maker,  at  such  a  distance  from 
each  other,  that  the  first  limb  of  the  sun  passes  all  of  them  before  the  second  limb 
arrives  at  the  first  wire,  and  the  observer  can  thus  take  the  observations  without  hurry 
or  confusion. 

The  round  Ihnh  only  of  the  moon  can  be  obsei-ved,  except  ivithin  an  hour  or  two  of  the 
fidl  moon.  In  observing  the  larger  planets,  the  first  and  second  limb  may  be  obsei-vcd 
alternately  over  the  five  ivires  ;  that  is,  the  first  limb  must  be  observed  at  the  1st, 
3d,  and  5th  wires,  and  the  second  limb  at  the  2d  and  4th  wires  ;  and,  by  reducing 
these  observations  in  the  same  manner  as  those  of  the  sun,  we  obtain  the  meridional 
passage  of  the  centre.  When  an  obsei-vation  at  one  or  more  of  the  wires  has  been 
lost,  it  is  impossible  to  take  the  mean  in  the  same  way  as  in  a  perfect  observation.  If 
the  centre  wire  is  the  one  that  is  deficient,  the  mean  of  the  other  four  may  be  taken  as 
the  time  of  the  meridional  passage  ;  or  the  mean  of  any  two,  equally  distant  on  each 
side  of  the  centre,  supposing  the  intervals  of  the  wires  to  be  equal.  But  when  any  of 
the  side  wires  are  lost,  and,  indeed,  under  any  circumstances  of  deficiency  in  the 
observation,  the  most  correct  metho/1  of  proceeding  is  as  follows : — Find,  by  a  consider- 
able number  of  careful  observations,  over  all  the  wires,  the  equatorial  interval  between 
each  side  wire,  and  the  central  one.  These  intervals  are  to  be  set  down  for  future 
use.  Then,  when  part  of  the  wires  only  are  observed,  each  wire  is  to  be  reduced  to 
the  mean,  l)y  adding  to,  or  subtracting  from,  the  time  of  observation,  as  the  case  may 
be,  the  equatorial  interval  between  that  wire  and  the  centre  wire,  multipUed  by  the 
secant  of  the  declination  of  the  star 


PORTABLE  TRANSIT  INSTRUMENT. 


151 


We  shall  hereafter  show  the  use  of  the  transit  instrument  in  regulating  a  chronom- 
eter ;  and  for  determining  the  longitude,  by  means  of  the  observations  of  the  transits 
of  the  moon  and  moon-culminating  stars. 


TABLE    A. 

Correction,  in  seconds  of  time,  to  be  applied  to  one  fourth  part  of  the 
difference  of  the  two  intervals,  supposing  the  whole  difference  to  be 
1000'  of  time. 

This  correction  is  suhlraclive  from  the  quarter  interval,  at  the  upper  transit;    additive  to  the 
quarter  interval  at  the  lower  transit. 

Lat. 

o 

0 

10 

20 
30 

40 
42 

44 

40 

48 

50 
52 
54 

50 

58 
60 

Pole 
Star. 

Polar  Distance  of  the  Star. 

0° 

5° 

10° 

15° 

20° 

25° 

30° 

35° 

40° 

s. 
0 
1 
2 
4 

5 
6 
6 

7 
7 
8 

s. 
0 
0 
0 
0 

s. 
0 
4 
8 

13 

0 

8 

16 

25 

s. 
0 
12 
24 
34 

s. 
0 

16 
33 
53 

s. 
0 
21 
42 

67 

s. 
0 
25 
53 

83 

s. 

0 

31 

64 

101 

s. 

0 

37 

76 

121 

0 
0 
0 

15 
18 

20 

31 
37 

40 

47 
56 
60 

64 
76 

82   ■ 

82 

98 

105 

101 
121 
130 

123 
147 
158 

147 
176 

189 

0 
0 
0 

21 
23 
24 

43 
46 
49 

65 
69 
74 

88 

94 

101 

113 
121 
130 

139 
149 
160 

169 
181 

194 

8 
9 
9 

0 
0 
0 

26 
28 
30 

52 
56 
60 

80 
86 
92 

108 
116 
125 

139 
149 
160 

172 

185 

199 

10 
11 
12 

0 
0 
0 

33 
35 

38 

65 

70 
76 

99 
107 
116 

135 
146 

158 

173 

187 

The  difference  of  the  two  intervals  actually  observed,  is  to  be  multiplied  by  the  number 
given  by  this  table,  and  the  product  divided  by  1000  (which  is  the  same  as  to  cross  olil'  the 
llirce  righi-haud  figures) ;  the  quotient  is  the  correction  to  be  applied  to  one  fourth  part  of 
the  difference  of  the  intervals. 

TABLE    B. 

Correction  of  tlie  azimuth,  in  minutes  and  tenths  of  a  minute  of  space,  corre- 
sponding to  a  difference  in  the  two  intervals  of  1000  seconds  in  time. 

Lat. 

o 

0 

10 

20 

30 

35 
40 
42 

44 
46 

48 

50 
52 
54 

56 
58 
60 

Pole 
Star. 

Polar  Distance  of  the  Star. 

0° 

5° 

10° 

15° 

20° 

25° 

30° 

35° 

40° 

1    II 
1.00 
1.43 
1.48 
1.57 

0 
0 
0 
0 

/    II 
5.29 
5.34 
5.50 
6.20 

/     // 
11.02 
11.13 
11.46 
12.46 

/     // 
16.47 
17.03 
17.52 
19.23 

/     II 
22.47 
23.10 
24.16 
26.20 

1     II 
29.13 
29.40 
31.06 
33.45 

1     II 
36.11 
36.44 
38.30 
41.47 

/     // 
43.53 
44.33 

46.42 
50.40 

1     II 
52.35 
53.23 
57.16 
60.43 

2.04 
2.12 
2.16 

0 
0 
0 

6.42 
7.09 
7.23 

13.29 
14.25 
14.52 

20.30 
21.55 
22.36 

27.51 

29.46 
30.42 

35.40 
38.09 
39.19 

44.10 
47.14 
48.41 

53.34 
57.17 
59.03 

64.11 

08.38 
70.45 

2.21 
2.26 
2.31 

0 
0 
0 

7.37 
7.. 54 
8.12 

15.22 
15.54 
16.31 

23.21 
24.10 
25.06 

31.42 
32.50 
34.05 

40.37 
42.04 
43.40 

50.18 
52.05 
54.04 

61.00 
63.10 
05.35 

2.38 
2.45 
2.52 

0 
0 
0 

8.32 
8.54 
9.20 

17.11 
17.57 

18.48 

26.07 
27.16 
28.34 

35.29 
37.03 

38.48 

45.28 

47.28 
49.43 

56.17 
58.46 
61.33 

3.01 
3.11 
3.23 

0 
0 
0 

9.48 
10.21 
10.58 

19.46 
20.51 
22.06 

30.02 
31.04 
33.35 

40.47 
43.03 
45.37 

52.16 
55.09 

58.27 

152 


PORTABLE  TRAiNSlT   INSTRUMENT. 


TABLE  C. 

Con'ection,  in  seconds  of  time,  for  1000  seconds  of  space  of  deviation  in 
azimuth  from  the  plane  of  the  meridian,  to  be  applied  to  the  time  of  the 
transit  of  the  star  observed  by  the  transit  instrument. 

Upper  Transit. 

Lower  Transit. 

Lat. 

o 

0 

5 

10 

15 
20 
25 

30 
35 
40 

45 
50 
55 

60 

Pole 
Star. 

s. 

24C7 
2451 
2418 

Polar  Distance. 

Pole 
Star. 

Polar  Distance. 

Lat. 

0 
0 
5 

10 

15" 

20 

25 

30 
35 
40 

45 

50 
55 

60 

s. 

763 
751 
737 

717 

691 
66] 

625 

584 
539 

490 
438 
381 

10° 

383 
370 
360 

347 
332 
314 

293 
271 
246 

220 
191 
162 

15° 

s. 

257 
241 
233 

222 

210 
197 

182 
165 
147 

128 
109 

88 

20° 

194 
176 
168 

159 
149 
137 

125 
HI 

97 

"82 
66 
50 

25° 

s. 

157 
136 
129 

121 
111 
101 

~90 
79 
66 

~54 
41 
27 

30° 

133 
109 
102 

"94 
85 
76 

~66 
56 
45 

"34 
23 
12 

35° 

s. 
116 

89 
82 

m 

58 

49 
40 
30 

20 
10 

40° 
s. 
103 

73 
66 

59 
52 
44 

~35 
27 

18 

~9 

5° 
s. 

764 
760 

751 
737 
717 

691 
661 
625 

584 
539 
490 

438 

10° 
s. 

333 

381 
377 
370 

360 
347 
332 

314 
293 
271 

246 

15° 

s. 

257 
256 
253 

248 
241 
233 

222 

210 
197 

182 

20° 
s. 

194 
194 

191 

188 
183 

176 

168 
159 

149 

25° 

5. 

157 

157 
155 
152 

148 
143 
136 

129 

30° 
s. 

133 
132 
131 

128 
125 
121 

US 

35° 
s. 

116 
115 

Il4 
112 

109 

105 

40° 
s. 

103 

103 
102 
100 

~97 

s. 

2463 
2441 

2365 
2295 
2208 

2103 

1982 
1847 

1697 
1535 
1360 

.1176 

2400 
2341 

2264 

2170 
2059 
1932 

1791 
1637 
1469 

322 

131 

66 

34 

14 

1291 

FlateYIL 


1861 


153 


ON  PARALLAX,  REFRACTION,  AND  DIP  OF 
THE  HORIZON. 


Parallax  (or  diurnal  parallax)  is  tJie  difference  between  tht  true  altitude  of  the  sun, 
moon,  or  star,  if  it  loere  observed  at  the  centre  of  the  earth,  and  the  apparent  altitude 
observed,  at  tlie  same  instant,  by  a  spectator,  at  any  point  on  the  surface  of  the  earth. 

Thus,  in  Plate  XII.,  figure  3,  let  ABC  be  the  earth,  C  its  centre,  A  the  place  of  a 
spectator,  ZAK  a  vertical  plane,  passing  through  the  place  D  of  the  moon,  or  tlie 
place  d  of  a  planet;  EDF,  edG,  circular  arcs  draAvn  about  C  as  a  centre,  and  KZ 
part  of  the  starry  heavens.  Then,  if  at  any  time  the  moon  be  at  D,  she  will  be  referred 
to  the  point  H,  by  a  spectator  supposed  to  be  placed  at  tlie  centre  of  the  earth,  and 
this  is  called  the  true  place  of  the  moon  ;  but  the  spectator  at  A  will  refer  the  moon 
to  the  point  b,  and  tliis  is  called  the  apparent  place  of  the  moon  ;  the  difference  H6 
(or  the  angle  HD6i=  ADC)  is  called  the  moon's  ^;ara/Zar  in  cdtitude,  which  is  evidently 
greatest  wlien  the  moon  is  in  the  horizon  at  E,  being  tlien  equal  to  the  arc  KI,  and 
it  decreases  from  the  horizon  to  the  zenith,  and  is  there  nothing.  The  parallax  is 
less  as  the  objects  are  farther  from  the  earth  :  thus  the  parallax  of  a  planet  at  d  is 
represented  hylah,  being  less  than  that  of  the  moon  at  D  ;  and  the  horizontal  parallax 
Kyof  the  planet  is  less  than  the  horizontal  parallax  KI  of  the  moon.  As  the  parallax 
makes  the  objects  appear  lower  than  they  really  are,  it  is  evic^nt  tliat  the  parallax 
must  be  added  to  the  apparent  altitude  to  obtain  the  true  altitude.  Having  the  hori- 
zontal parallax,  the  parallax  in  altitude  is  easily  found  by  the  following  rule: — ^s 
radius  is  to  the  cosine  of  tlie  apparent  altitude,  so  is  Vie  horizontal  parallcux  to  the  parcdlax 
in  cdtitude.  This  rule  may  be  easily  proved  ;  for  in  the  triangle  CAE  we  have  CE  : 
CA  ::  radius  :  sine  CEA  ;  and  in  the  triangle  CDA  we  have  CD  (or  CE)  :  CA  ::  sine 
CAD  :  sine  CDA  ;  hence  we  have  radius  :  sine  CEA  : :  sine  CAD  :  sine  CDA ;  but  CEA 
r=  horizontal  parallax,  CDA  :=z  parallax  in  altitude,  and  sine  CAD  =  cosine  app.  alt. 
Hence  we  have  radius  :  cosine  app.  alt.  ::  sine  hor.  par.  :  sine  par.  in  alt. ;  but  the 
parallaxes  of  the  heavenly  bodies  being  very  small,  the  sines  are  nearly  proportional 
to  the  parallaxes;  hence  we  may  say.  As  radius  :  cosine  app.  alt.  ::  hoi*.  j)ar.  :  par. 
in  alt. 

The  sun's  mean  parallax  in  altitude  is  given  in  Table  XIV.,  for  each  5^  or  10°  of 
altitude.  The  moon's  horizontal  parallax  is  given  in  the  Nautical  Almanac,  for  every 
noon  and  midnight  at  the  meridian  of  Greenwich  ;  also  that  of  the  sun  for  every  ten 
days,  and  the  parallaxes  of  Venus,  Mars,  Jupiter,  and  Saturn,  for  every  five  days, 
throughout  the  year. 

Refraction  of  the  heavenly  bodies. 

It  is  known,  by  various  exjieriments,  that  the  rays  of  light  deviate  from  their 
rectilinear  course,  in  passing  obliquely  out  of  one  medium  into  another  of  a  different 
density;  and  if  the  density  of  the  latter  medium  continually  increase,  the  rays  of  light, 
in  passing  through  it,  will  deviate  more  and  more  from  the  right  lines  in  which  they 
were  projected  towards  the  perpendicular  to  the  surface  of  the  medium.  This  may 
be  illustrated  by  the  following  experiment : — Make  a  mark  at  the  bottom  of  any  basin, 
or  other  vessel,  and  place  yourself  in  such  a  situation  that  the  hitlier  edge  of  the  basin 
may  just  hide  the  mark  from  your  sight ;  then  keep  your  eye  steady,  and  let  another 
person  fill  tlie  basin  gently  with  water ;  as  the  basin  is  filled,  you  will  perceive  the 
mark  come  into  view,  and  appear  to  be  elevated  above  its  former  situation.  In  a 
similar  manner,  the  light,  in  passing  from  the  heavenly  bodies  through  the  atmosphere 
of  the  earth,  deviates  from  its  rectilinear  course.  By  this  means  the  objects  appear 
higher  than  they  really  are,  except  when  in  the  zenith.  This  ap;)arent  elevation  of 
the  heavenly  bodies  above  their  true  places,  is  called  the  refraction  of  those  bodies 
To  illustrate  this,  let  ABC  (Plate  XII.,  fig.  2)  represent  the  atmosphere  suiTounding 


154  PARALLAX,   REFRACTION,   &c. 

the  eartli  DEF,  and  let  an  observer  be  at  D,  and  a  star  at  a ;  then,  if  there  were  no 
refraction,  the  observer  would  see  the  star  according  to  the  direction  of  the  right  line 
Da;  but  as  the  light  is  i-efracted,  it  will,  when  entering  the  atmosphere  near  A,  be 
bent  from  its  rectilinear  course,  and  will  describe  a  curve  line  from  A  to  D,  and, 
at  entering  the  eye  of  the  observer  at  D,  will  appear  in  the  line  D  h,  which  is  a 
tangent  to  the  cm"ve  at  the  point  D,  and  the  arc  ab  will  be  the  refraction  in  altitude, 
or,  simply,  the  refraction,  which  must  be  subtracted  from  the  observed  altitude  to 
obtain  the  true. 

At  the  zenith,  the  refraction  is  nothing  ;  and  the  less  the  altitude,  the  more  obliquely 
the  rays  will  enter  the  atmosphere,  and  the  greater  will  be  the  refraction :  at  the 
horizon,  the  refraction  is  greatest.  In  consequence  of  the  refraction,  any  heavenly 
body  im  y  be  actually  below  the  horizon  when  appearing  above  it.  Tims,  when  the 
sun  is  at  T  below  the  horizon,  a  ray  of  light  TI,  proceeding  from  T,  conies  in  a  right 
line  to  I,  and  is  there,  on  entering  the  atmosphere,  turned  out  of  its  rectilinear  course, 
and  is  so  bent  down  towards  the  eye  of  the  observer  at  D,  that  the  sun  appears  in  the 
direction  of  the  I'efracted  ray  above  the  horizon  at  S. 

The  mean  quantity  of  the  refraction  of  the  heavenly  bodies  is  given  in  Table  XII 
All  observed  altitudes  of  the  sun,  moon,  planets,  or  other  heavenly  bodies,  must  be 
decreased  by  the  numbers  taken  from  that  table  corresponding  to  the  observed 
altitude  of  the  object.  The  refraction  varies  Avith  the  temperature  and  density  of  the 
air,  increasing  by  cold  or  greater  density,  and  decreasing  by  heat  and  rarity  of  the 
atmosphere.  The  corrections  to  be  ap])lied  to  the  numbers  taken  from  Table  XII., 
for  the  different  heights  of  Fahrenheit's  Thermometer  and  the  Barometer,  are 
given  in  Table  XXXVL*  Thus,  if  the  refraction  be  required  for  the  apparent 
altitude  5°,  when  the  thermometer  is  at  20°,  and  the  barometer  at  30. G7  inches,  we 
shall  have  the  mean  refraction  by  Tal)le  XII.  equal  to  9'  53",  and  by  Table  XXXVI. 
the  correction  corresponding  to  the  height  of  the  thermometer  20°  equal  to  -(-  48'', 
and  for  the  barometer  30.67  equal  to  -\-  22";  hence  the  true  refraction  will  be 
9'  53"  +  48"  -1-  22"  =  11'  3". 

There  is  sometimes  an  irregular  refraction  near  the  horizon,  caused  by  the  vapors 
near  tlie  surface  of  the  earth;  the  only  method  of  avoiding  the  error  arising  from  this 
source,  which  is  sometimes  very  great,  is  to  take  the  obsei'vations  at  a  time  when  the 
object  which  is  observed  is  more  than  10°  above  the  horizon. 

The  refraction  makes  any  terrestrial  object  appear  more  elevated  than  it  really  is. 
The  quantity  of  this  elevation  varies,  at  different  times,  from  ^  to  -i^  of  the  angle 
formed,  at  the  centre  of  the  earth,  between  the  object  and  the  observer ;  but,  in 
general,  this  refraction  is  about  J^  of  that  angle. 


Dq}  of  the  horizon. 

Dip  of  the  horizon  is  the  angle  of  depression  of  the  visible  horizon  below  the  true 
or  sensible  horizon  (touching  the  earth  at  the  observer),  arising  from  the  elevation  of 
the  eye  of  the  observer  above  the  level  of  the  sea.  Thus,  in  Plate  XII.,  figure  1,  let 
ABC  represent  a  vertical  section  of  the  earth,  whose  plane,  being  produced,  passes 
through  the  observer  and  the  object,  and  let  AE  be  the  height  of  the  eye  of  the 
observer  above  the  surface  of  the  earth  ;  then  FEG,  drawn  parallel  to  the  tangent  to 
the  surface  at  A,  will  represent  the  true  horizon,  and  EIH,  touching  the  earth  at  I, 
will  represent  the  apparent  horizon  ;  therefore  the  angle  FEH  will  be  the  dip  of  the 
horizon.  Let  M  be  an  object  whose  altitude  is  to  be  observed  by  a  fore  observation 
by  bringing  the  image  in  contact  with  the  apparent  horizon  at  H ;  then  will  the  angle 
flIEH  be  the  observed  altitude,  which  is  greater  than  the  angle  MEF  (the  altitude 
independent  of  the  dip)  by  the  quantity  of  the  angle  FEH ;  so  that,  in  taking  a  fore 
observation,  the  dip  must  be  subtracted  from  the  observed  altitude  to  obtain  the 
altitude  corrected  for  the  dip.  In  a  back  observation,  the  apparent  horizon  is  in  the 
dii'ection  EK ;  and,  by  continuing  this  line  in  the  direction  EL,  Ave  shall  have  the 
observed  altitude  MEL  ;  and  it  is  evident  that  to  this  the  dip  LEF  (=  KEG)  must 
be  added  to  obtain  the  altitude  corrected  for  the  dip. 

In  Table  XIII.  is  given  the  dip,  for  every  probable  height  of  the  observer,  expressed 
in  feet.  In  calculating  this  table,  attention  is  paid  to  the  terrestrial  refraction,  which 
decreases  the  dip  a  little,  because  IE  becomes  a  cune  line  instead  of  a  straight  one, 
and  EH  is  a  tangent  to  tliat  curve  in  the  point  E. 

*  This  talile  is  to  be  entered  with  the  height  of  tlie  thermometer  or  barometer  at  the  tof),  and  the 
apparent  altitude  at  the  side ;  under  tlie  former,  and  opposite  the  latter,  will  be  the  correction  corre- 
sponding- to  the  thermometer  or  barometer,  uhicli  is  to  be  apphed  to  the  mean  refraction,  by  addition  o» 
subtraction,  according  to  tlie  signs  at  the  top  of  the  columns  rcsDectively. 


PARALLAX,   REFRACTION,   &c.  155 

What  has  been  said  concerning  the  dip  of  the  liorizon,  suj)poses  it  free  from  all 
encumbrances  of  land  or  other  objects;  but,  as  it  often  happens,  when  sliips  arc  sailing 
along  shore,  or  at  anchor  in  a  harbor,  that  an  oljservation  is  vwiiitcd  when  the  sun  is 
over  the  land,  and  the  shore  nearer  the  shij)  tlian  the  visible  horizon  would  be  if  it 
were  imconfined,  in  this  case,  the  dip  of  tlie  hoi-izon  will  be  different  from  what  it 
otherwise  would  have  been,  and  greater  the  nearer  the  ship  is  to  that  part  of  the 
shore  to  which  the  sun  is  brougiit  down.  For  this  reason  Table  XVI.  has  been 
inserted,  which  contains  the  dip  of  the  sea  at  different  heights  of  the  eye,  and  at 
diftl'i-ent  distances  of  the  ship  from  the  land.  This  table  is  to  be  entered  at  the  top 
with  the  height  of  the  eye  of  tiie  observer  above  the  level  of  the  sea  in  feet;  and  in 
the  left-hand  side  column,  with  the  distance  of  the  ship  from  the  laml  in  sea  miles 
and  parts.  Under  the  former,  and  oj)])osite  the  latter,  stands  the  dip  of  tiie  horizon, 
which  is  to  be  subtracted  from  the  altitude  observed  by  a  fore  observation,  instead 
of  the  numbers  in  Table  XIII. 

The  distance  of  the  land  requisite  in  finding  the  dip  from  Table  XVI.,  may  be 
found  nc'arly  in  the  following  manner: — Let  two  observers,  one  placed  as  high  on 
the  main-mast  as  he  can  conveniently  be,  and  the  other  on  the  deck  immediately 
beneath  him,  observe,  at  the  same  instant,  the  altitude  of  the  sun  or  other  object  that 
may  be  wanted,  and  let  the  height  of  the  eye  of  the  upper  observer  above  that  of  the 
lower  be  measured  in  feet,  and  multiplied  by  0.56;  then  the  product,  being  divided 
by  the  difference  of  the  observed  altitudes  of  the  sun  in  minutes,  will  be  tlie  distance 
in  sea  miles,  nearly. 

Thus,  if  the  eye  of  the  upper  observer  was  G8  feet  higher  than  that  of  the  loAver, 
and  the  two  observed  altitudes  of  the  sun  20°  0'  and  20°  12',  the  distance  of  the  land, 
in  sea  miles,  would  be  3.2.  For  C8  X  0.5G z=:  38.08,  and  this,  being  divided  by  the 
difference  of  the  two  observed  altitudes  of  the  sun  12',  gives  3.2,  nearly.  Now,  if  the 
lower  obsei-ver  be  25  feet  above  the  level  of  tiie  sea,  the  dip  corresponding  to  this 
height  and  the  distance  3.2  miles  will  l)e  C,  which,  being  subtracted  from  20°  O', 
leaves  19°  54',  the  altitude  corrected  for  the  dip. 

The  dip  may  be  calculated,  in  this  kind  of  observations,  to  a  sufficient  degree  of 
accuracy,  without  using  Table XVI.,  in  the  following  manner: — Divide  the  difference 
of  the  heights  of  the  two  observers  in  feet,  by  the  difference  of  the  observed  altitude 
in  minutes,  and  reserve  the  quotient.  Divide  the  height  of  the  lower  observer  in  feet 
by  tills  reserved  number,  and  to  the  quotient  add  one  quarter  of  the  reserved  number, 
and  the  sum  will  be  the  dip  in  minutes  corresponding  to  the  lower  observer.  Thus, 
in  the  above  example,  £S-=z5'.C  is  the  reserved  number,  and  ^z=4.4;  to  this  add 
one  fourth  of  5'.G  or  l',4,  and  the  sum  will  be  the  dip  5'.8,  or  nearly  6',  corresponding 
to  the  lower  observer,  being  the  same  as  was  found  by  the  table. 


156 


TO    FIND    THE   SUN'S    DECLINATION. 


The  declination  of  the  sun  is  given,  to  the  nearest  minute,  in  Table  IV..  for  every 
noon,  at  Greenwich,  from  the  year  1833  to  1848  ;  and  this  table  will  answer  for  some 
years  beyond  that  period,  without  any  material  error.  If  great  accuracy  is  required, 
the  declination  may  be  taken  from  the  Nautical  Almanac*  This  declination  may  be 
reduced  to  any  other  meridian,  by  means  of  Table  V.,  in  the  following  manner  :— 

To  find  the  sun's  declination,  at  noon,  at  any  place. 

RULE. 

Take  out  the  declination  at  noon,  at  Greenwich,  from  Table  IV.,  or  from  the 
Nautical  Almanac ;  then  find  the  longitude  from  Greenwich  in  the  top  column  of 
Table  V.,  and  the  day  of  the  month  in  the  side  column ;  under  the  former,  and 
opposite  to  the  latter,  is  a  correction,  in  minutes  and  seconds,  to  be  applied  to  the 
declination  taken  from  Table  IV. ;  to  know  whether  this  correction  be  additive  or 
subtractive,  you  must  look  at  the  top  of  the  column  where  you  found  the  day  of  the 
month,  and  you  Avill  see  it  noted  whether  to  add  or  subtract,  according  as  the  lon- 
citude  is  east  or  west.  This  correction  being  applied,  you  will  have  the  declination 
at  noon  at  the  given  place. 

EXAMPLE    I. 

Required  the  declination  of  the  sun,  at  the  end  of  the  sea  day,  October  10,  1848,  in 
the  longitude  of  130°  E.  from  Gi-eenwich. 

Sun's  declination,  October  10,  at  Greenwich,  at  the  end  of  the  sea  day,  or 

beginning  of  die  day  in  the  N.  A.,  by  Table  IV G°  48'  S. 

Variation  of  dec,  Table  V.,  October  10,  in  130°  E.  long sub.  _0 8 

True  dec  noon,  October  10,  in  long.  130°  E Aj^O.  ^• 

EXAMPLE   II. 

Required  the  sun's  declination  at  noon  ending  the  sea  day  of  March  12,  1848,  in 
the  longitude  of  65°  W.  from  Greenwich. 

Sun's  declination,  March  12,  by  Table  IV 3°  9'  S 

Var.  Table  V.,  March  12,  long.  C5°  W sub. 4 

True  declination,  noon,  March  12,  long.  65°  W 3    5  S 

The  preceding  correction  ought  always  to  be  applied  to  the  declination  used  in 
working  a  meridian  observation  to  determine  the  latitude,  though  many  mariners  are 
in  the  habit  of  neglecting  it. 

*  In  finding  the  declination,  or  any  oilier  quantity,  in  the  Nautical  Almanac,  you  must  be  careful  to 
note  the  difference  between  the  civil,  nautical,  and  astronomical  account  of  time.  The  civil  day  begins 
Ht  midnight,  and  ends  the  following  midnight,  the  interval  being  divided  into  2-1  hours,  and  is  reckoned 
in  numeral  succession  from  1  to  12,  then  beginning  again  at  1  and  ending  at  12.  The  nautical  or  sea  day 
begins  at  noon,  12  hours  before  the  civil  day,  and  ends  the  following  noon  ;  the  first  12  hours  are  mark- 
ed P.  31.,  the  latter  A.  M.  The  astronomical  day  begins  at  noon,  12  hours  after  the  civil  day,  and  24 
hours  after  the  sea  day,  and  is  divided  into  2-1  hours,  numbered  in  numeral  succession  from  1  to  2-1, 
beginning  at  noon,  and  ending  the  following  noon.  All  the  calculations  of  the  Nautical  Almanac  are 
adapted  to  astronomical  time;  the  declination  marked  in  the  Nautical  Almanac,  or  in  Table  IV.,  is 
adapted  to  the  beginning  of  the  astronomical  day,  or  to  the  end  of  the  sea  day,  it  being  at  the  end  of 
the  sea  day  when  mariners  want  the  declination  to  determine  their  latitude.  It  would  be  much  better 
if  seamen  would  adopt  the  astronomical  day,  and  wholly  neglect  the  old  method  of  counting  by  the 
Eca  day 


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1861 


TO   FIND  THE   SUN'S   DECLINATION.  157 

To  find  the  sun's  declination,  at  any  time,  under  any  meridian. 

RULE. 

Reduce  tlie  sun's  declination  at  noon  at  Greenwich,  to  noon  under  the  given  me- 
ridian, by  the  preceding  rule  ;  then  enter  Table  V.  with  the  time  from  noon  at  the 
top,  and  the  day  of  the  month  in  the  side  column;  under  the  former,  and  opposite  the 
latter,  will  be  the  correction  to  be  api)lied  to  tliat  reduced  declination.  To  know 
whether  this  correction  be  additive  or  subtractive,  you  must  look  at  the  top  of  the 
colunni  Aviiere  you  found  the  day  of  the  month,  and  you  will  find  it  noted  whether  to 
add  or  subtract,  according  as  the  time  is  before  or  after  noon. 

EXAMPLE    in. 

Required  the  sun's  declination  October  10,  1848,  sea  account,  at  S**  21""  in  the 
forenoon,  in  the  longitude  of  130°  E,  from  Greenwich. 

Sun's  declination  Oct.  10,  at  Greenwich,  at  noon,  by  Table  IV 6°  48'  S. 

Variation  for  130°  E.  long subtract  8 

Declination  at  noon,  October  10,  in  long.  130°  E 6  40    S. 

Variation  of  dec.  for  S*"  39""  from  noon,*  Oct.  10, subtract  3 

True  dec.  Oct.  10,  sea  ace.  in  long.  130°  E.  at  Si^  21">,  A.  ]M 6  37   S. 

EXAMPLE   IV. 

Required  the  sun's  declination  May  10,  1836,  sea  account,  at  5''  30"',  P.  M.,  in  the 
longitude  of  35°  45'  E.  from  Greenwich. 

Variation  of  declination,  May  10,  in  long.  35°  45'  E subtract  V  38" 

Variation  of  declination  for  5^  30™,  P.  M add  3  44 

Dift".  is  additive,  because  the  greatest  number  is  so 2  00 

May  10,  sea  account,  is  May  9,  by  N.  A.,  at  which  time  the  sun's 

declination 17°  26  27 

True  dec.  May  10,  5''  30",  P.  M.,  sea  account,  in  long.  35°  45'  E 17  28  33   N. 

EXAMPLE   V. 

Required  the  sun's  declination  March  26,  1836,  sea  account,  at  3'',  P.  jM.,  in  the 
longitude  of  140°  W.  from  Greenwich. 

Vm-iation  of  declination,  March  26,  in  long.  140°  W add  9'  08" 

Variation  for  3",  P.  M , add  2  50 

Sum 12  04 

March  26,  sea  account,  is  iMarch  25,  by  N.  A.,  .it  v/hich  time  the  sun's 

declination 1°  56  41     N 

Ti-ue  dec.  March  26,  3^  P.  M.,  sea  account, 2  08  45     N 

*  In  tlie  present  example,  Ihe  time  is  Oct.  10,  StZl™,  A,  M.,  which  evidently  \vant5  3''  SO""  of  the  emi 
o*"  tire  spa  day,  Oct.  10,  for  which  time  the  declination  is  marked  in  Table  IV. 


158 


VARIATION    OF    THE    COMPASS. 


It  was  many  years  after  the  discovery  of  the  compass,  before  it  was  suspected  that 
the  magnetic  needle  did  not  point  accurately  to  the  north  pole  of  the  world ;  but, 
about  the  middle  of  the  sixteenth  century,  obsen'ations  were  made  in  England  and 
France,  which  fully  proved  that  the  needle  pointed  to  the  eastward  of  the  true  north. 
This  difference  is  called  the  variation  of  the  coinpass,  and  is  named  east  when  the 
north  point  of  the  compass  (or  magnetic  north)  is  to  tlie  eastward  of  the  true  north,  but 
icest  when  the  north  point  of  the  compass  is  to  the  westward  of  the  true  north.  The 
quantity  of  the  variation  may  be  found  by  observing,  with  a  compass,  the  bearing  of 
any  celestial  object  when  in  the  horizon,  (or,  as  it  is  called,  the  magnetic  amplitude  ;) 
tiie  difference  between  this  and  the  true  amplitude,  found  by  calculation,  will  be  the 
variation.  Tlie  same  may  be  obtained  by  observing  the  magnetic  azimuth  of  any 
celestial  object,  (that  is,  its  bearing  by  a  compass  when  elevated  above  the  liorizon ;) 
the  difference  between  tbis  and  the  true  azimuth,  found  by  calculation,  will  be  the 
variation. 

Some  years  after  the  discovery  of  the  variation,  it  was  found  that  it  did  not  remain 
constant ;  for  the  easterly  variation,  observed  in  England,  gradually  decreased  till  the 
needle  pointed  to  the  true  north,  and  then  increased  to  the  westward,  and  is  now 
above  two  points. 

As  all  the  com'ses  steered  by  a  compass  must  be  corrected  for  the  variation,  to 
obtain  the  true  courses,  it  is  of  great  importance  to  the  navigator  to  know  how  to  find 
the  vai'iation  at  any  time.  To  do  this,  it  is  necessary  to  find  the  magnetic  amplitude 
or  azimuth  of  a  celestial  object,  which  may  be  done  as  follows  : — 

To  observe  an  amplitude  by  an  azimuth  compass* 

When  the  centre  of  the  sun  is  about  one  of  his  diameters  f  above  the  horizon,  turn 
the  compass  round  in  the  box,  until  the  centre  of  the  sun  is  seen  through  the  naiTOW 
slit  which  is  in  one  of  the  sight-vanes,  exactly  on  the  thread  which  bisects  the  slit  in 
the  other :  J  at  that  instant  pus'h  the  stop,  which  is  in  the  side  of  the  box,  against  the 
edge  of  the  card,  and  the  degi-ee  and  parts  of  a  degree  which  stand  against  the  middle 
line  on  the  top,  will  be  the  magnetic  aini)litude  of  the  sun  at  that  time,  which  is  gen- 
erally reckoned  from  the  east  or  west  point  of  the  com})ass. 

To  observe  an  azimuth  by  an  azimuth  compass. 

Turn  the  compass  round  in  the  box  until  the  centre  of  the  sun  is  seen  through  the 
narrow  slit  which  is  in  onelbf  the  sight  vanes,  exactly  on  the  thread  which  bisects  the 
slit  on  the  other,  or  until  the  shadow  of  the  thread  falls  directly  along  the  line  of  the 
horizontal  Itar ;  f  the  card  is  then  to  be  stopped,  and  the  degree  and  parts  of  a  degree 
which  stand  against  the  middle  line  of  the  stop,  will  lie  the  magnetic  azimuth  of  the 
sun  at  that  time,  which  is  generally  reckoned  from  the  north  in  north  latitude,  and 
from  the  south  in  south  latitude.  §  At  the  time  of  making  this  observation,  you  must 
also  observe  the  altitude  of  the  sun,  in  order  to  obtaiji  the  true  azimuth. 

What  is  here  said  of  the  sun,  is  alike  a])plicable  to  the  moon,  planets,  and  stars. 

*  The  figure  of  an  azimuth  compass,  furnished  wiih  sight-vanes,  is  given  in  Plate  VI.,  figure  5.  Tlie 
card  of  this  compass  is  similar  to  that  of  a  common  compass. 

t  The  observation  is  to  be  taken  at  that  altitude  on  account  of  the  dip,  refraction,  and  parallax,  the 
correction  of  altitude  depending  on  these  causes  being,  in  general,  nearly  equal  to  the  sun's  diameter. 

X  If  the  instrument  is  furnished  with  a  magnifying  glass  fixed  to  one  of  the  vanes,  you  may  (instead 
of  proceeding  as  above)  turn  tlie  compass  box  until  the  vane  is  directed  towards  the  sun,  and  when  the 
bright  speck  (or  rays  of  the  sun  collected  b}-  the  magnifying  glass)  falls  upon  the  slit  of  the  other  vane, 
or  upon  the  line  in  the  horizontal  bar,  the  card  is  to  t)e  slopped,  and  the  divisions  read  oft'  as  above. 

§  If  the  compass  vibrate  considerably  at  the  time  of  making  the  observations,  it  would  be  conducive 


VARIATIOiN   OF   THE   COMPASS.  159 

To  find  the  true  amplitude. 

RULE. 

By  Logarithms. — To  the  log.  secant  of  the  latitude  {rejecting  10  in  the  index)  add  the 
log.  sine  of  the  siui's  declination;  *  the  sum  will  he  the  log.  sine  of  the  true  amplitude,  or 
dista7ice  of  the  su7i  from  the  east  or  west  point,  towards  the  north  in  north  declination,  but 
towards  the  soidh  in  soidh  declination. 

By  Lvspectiox. — Find  the  declination  at  the  top  of  Table  VII.,  and  the  latitude  in  the 
side  column  ;  under  the  former,  and  opposite  the  latter,  will  he  the  true  amplitude.  When 
great  accuracy  is  required,  you  may  proportion  for  the  minutes  of  latitude  and  decli- 
nation. 

EXAMPLE    L 

Required  the  sun's  true  amplitude,  at  risin":,  in  tlie  hititude  of  39°  0'  N.,  on  the  22d 
of  December,  1848  when  his  decHnation  was  23°  28'. 

BY    LOGARITHMS. 
Latitude  39°    0'     Log.  Sec.  0.10950 

Sun's  dcclin.  23    28    Log.  Sine,  9.00012 

True  ampli.    30    49    Log.  Sine,  9.709G2 


BY  INSPECTION. 


Under  tlie  declination  23°  28',  and  op- 
posite the  latitude '39°,  stands  the  true 
amplitude  30°  49'. 


Hence  the  true  bearing  or  amplitude  of  the  sun  at  rising  is  E.  30°  49'  S.,  and  at 
setting  it  is  W.  30°  49'  S. 

EXAMPLE   II. 

Required  the  moon's  true  amplitude  at  setting,  in  the  latitude  of  35°  8'  N.,  when 
her  declination  is  13°  N. 


BY    LOGARITHMS. 

Latitude  35°   8'   Log.  Sec.  0.08734 

Moon's  declin.  13    0  Log.  Sine,  9.35209 

True  ampli.      15  58  Log.  Sine,  9.43943 


BY  INSPECTION. 
Under  the  declination  13°,  and  opposite 
the  latitude  35°,  stands  15°  56',  which  ia 
nearly  the  true  amplitude  ;  the  exact  value 
may  be  found  by  finding  the  amplitude  for 
36°  latitude,  and  proportioning  the  differ- 
ence for  the  miles  in  the  latitude. 

Hence  the  true  amplitude  at  setting  is  W.  15°  58'  N.,  and  at  rising  E.  15°  58'  N. 

EXAMPLE    III. 

Required  the  sun's  true  ami)litude  in  the  latitude  of  42°  30'  N.,  when  his  declinatiou 
was  20°  S. 


BY   LOGARITHMS. 
Latitr.de  42°  30'     Log.  Sec.  0.13237 

Sun's  declin.  20   00     Log.  Sine,  9..53405 


True  ampli.    27   38     Log.  Sine,  9.66642 


BY    INSPECTION. 

Under  the  declination  20°,  and  opposite 
the  latitudes  42°  and  43°,  stand  27°  24' 
and  27°  53';  the  mean  of  these  gives  the 
true  ami^Iitude  for  the  latitude  of  42°  30' 

z=  27°  38'. 

Hence  the  amplitude  at  setting  is  W.  27°  38'  S.,  and  at  rising  E.  27""  38'  S. 

To  find  the  true  azimuth  at  any  time. 

At  the  time  of  observing  the  magnetic  azimuth,  you  must  also  observe  tlie  altitude 
of  the  object ;  this  altitude  must  be  corrected  as  usual  for  the  dip,  parallax,  refraction,f 
&c.,  in  order  to  obtain  the  true  altitude  ;  you  must  also  find  the  declination  of  the 

to  accurac3'  to  lake  several  azimuths  and  altitudes,  and  to  lake  the  mean  of  all  the  azimuths  and  all  the 
latitudes,  and  work  the  observation  with  the  mean  azimuth  and  altitude.  The  same  is  to  be  observed 
in  taking  an  amplitude. 

*  The  declination  of  the  sun  at  noon  is  given  in  the  Nautical  Almanac,  and  in  Table  IV.,  and  must  be 
corrected  for  the  longitude  of  the  ship  and  the  hour  of  the  day,  by  means  of  Table  V. 

t  In  observations  of  the  altitude  of  the  sun's  lower  limb  by  a  fore  observation,  it  is  usual  to  add  12' 
for  the  effect  of  dip,  parallax,  and  semi-diameter.  The  refraction  is  to  be  subtracted  from  the  sum, 
and  the  remainder  will  be  the  true  altitude,  nearly. 


100  VARIATION   OF  THE   COMPASS. 

object,*  and  the  latitude  of  the  place  of  observation,  and  then  the  true  azimuth  may 
be  calculated  by  the  following  rule : — 

RULE. 

Add  together  the  polar  distance,  f  the  latitude,  and  the  true  altitude  ;  take  the  dif- 
ference between  the  half-sum  and  the  j)olar  distance,  and  note  the  remainder.  Then 
add  together  the  log.  secant  of  the  latitude,  tlie  log.  secant  of  the  altitude,  (rejecting  10 
in  each  index,)  the  log.  cosine  of  the  half-sum,  and  the  log.  cosine  of  the  remainder ; 
half  the  sum  of  these  four  logarithms  will  be  the  log.  cosine  of  half  the  true  azimuth, 
which,  being  doubled,  will  give  the  true  azimuth,  reckoned  from  the  north  in  north 
latitude,  but  from  the  south  in  south  latitude. 

EXAMPLE   I. 

fn  latitude  51°  32'  N.,  the  sun's  true  altitude  was  found  to  be  39°  28',  his  declination 
being  then  16°  38'  N. ;  required  the  true  azimuth  ? 

Polar  distance 73°  22' 

Latitude 51   32     Secant 0.20617 

Altitude 39  28      Secant 0.11239 

Sum 164  22 


Half-sum 82  11  '  Cosine 9.13355 

Polar  distance , .     73  22 

Remainder 8  49      Cosine 9.99484 


2)19.44695 


Half-sum Log.  Cosine     58°  4'  9.72347 

2  


True  azimuth IIG  8   from  the  north. 

The  logarithm  9.72347  of  this  example  is  also  the  cosine  of  121°  56',  which,  being 
doubled,  gives  another  azimuth  243°  52',  die  former  being  116°  8'.  One  of  these 
corresponds  to  an  obseiTation  in  the  forenoon,  the  other  to  an  afternoon  observation. 

EXAMPLE   II. 

In  latitude  42°  16'  S.,  the  sun's  true  altitude  was  found  to  be  18°  40',  his  declination 
being  then  7°  38'  N. ;  required  the  true  azimuth. 

Polar  distance 97°  38' 

Latitude 42   16      Secant 0.13076 

Altitude 18  40      Secant 0.02347 

Sum 158  34 


Half-sum 79   17      Cosine 9.26940 

Polar  distance 97  38 

Remainder 18  21      Cosine 9.97734 


Sum 19.40097 


Half-sum . . .  Log.  Cosine      59°  53'  9.70048 

Q  


True  azimuth 119  46    from  the  south. 


QUESTIONS  TO  EXERCISE   THE   LEARNER. 

Question  I.  Given  the  sun's  altitude,  corrected  for  dip,  refraction,  &c.,  20°  46',  his 
declination  17°  10'  S.,  and  the  latitude  of  the  place  40°  38'  N. ;  required  the  true 
azimuth. 

Ansiver.     137°  50'  from  the  north. 

*  The  (leclinntion  is  to  be  found  according-  to  the  directions  in  the  note  in  the  last  page. 

t  The  polar  distance  of  tiie  sun,  moon,  or  star,  is  tiie  distance  from  the  elevated  pole,  and  is  found 
by  subtracting  the  declination  of  tlie  object  from  90°  when  the  latitude  and  declination  are  of  the  same 
name,  but  bj'  adding  the  declinaljon  to  90°  when  of  different  names. 


VARIATION   OF   THE  COMPASS. 


161 


Q^uesf.  II.  What  is  the  sun's  azimuth  in  the  jatitude  of  20°  30'  N.  in  the  forenoon, 
when  his  correct  central  altitude  is  24°  28',  and  his  declination  22°  40'  N.  ? 

Ans.     75°  44'  from  the  north. 

Q_uest.  III.  At  the  island  of  St.  Helena,  the  sun's  true  central  altitude  was  found 
to  be  30°  23'  in  the  forenoon,  his  declination  being  then  22°  58'  S. ;  required  the 
azimuth  at  that  tiuie. 

Ans.     72°  21'  from  the  south. 

Quest.  IV.  What  point  of  the  compass  did  the  star  Aldel)aran  bear  on,  in  the 
latitude  of  34°  23'  S.,  on  January  1,  1836,  when  the  correct  altitude  of  that  star  was 
22°  26'  ? 

.^ns.     130°  23'  from  the  south. 


Having  fhe  true  and  the  magnetic  amplitude  or  azimuth,  to  find  the  variation. 

Having  found  the  true  and  magnetic  amplitude  or  azimuth,  the  variation  may  be 
easily  deduced  therefrom  by  the  following  rule,  in  which  the  amplitude  is  reckoned 
from  the  east  or  west  point  of  the  horizon,  and  is  called  north  when  to  the  northward 
of  those  points,  but  south  when  to  the  southward.  The  azimuth  is  reckoned  from 
the  north  in  north  latitudes,  but  from  the  south  in  south  latitudes,  and  is  named  east 
when  falling  on  the  east  side  of  the  meridian,  otherwise  west.  If  the  observed  and  true 
amplitudes  be  both  north  or  both  south,  their  difference  loill  be  the  variation ;  but  if  one 
be  north  and  the  other  south,  their  sum  will  be  the  variation.  If  the  true  and  observed 
azimuths  be  both  east  or  both  xoest,  their  difference  loill  be  the  variation,  otherwise  their  sum ; 
and  the  variation  will  be  easterlij  when  the  point  representing  the  true  bearing  is  to  the  right 
hand  of  the  point  representing  the  magnetic  bearing,  but  westerly  when  to  the  left  hand ; 
Vie  observer  being  supposed  to  look  directly  towards  the  point  representing  the  magnetic 
bearing. 

EXAMPLE    I. 

Suppose  the  sun's  magnetic  amplitude  at  rising  is  E.  26°  12'  N.,  and  the  true 
amplitude  E.  14°  20'  N. ;  required  the  variation. 

From  the  greater E.  26°  12'  N. 

Take  the  less E.  14  20  N. 

Remains  variation 11    52  E. 

The  variation  in  this  example  is  easterly,  because  the  true  amplitude  falls  to  the 
jight  of  tlie  magnetic. 


EXAMPLE  II. 

The  moon's  true  amplitude  at  rising 
was  found  to  be  E.  15^  20'  N.,  and  her 
magnetic  amplitude  E.  10°  0'  S. ;  recjuired 
the  variation. 

True  amplitude E.  15°  20'  N. 

Magnetic  amplitude E.  10     0  S. 

Sum  is  the  variation 25  20  W. 

EXAMPLE  III. 

The  sun's  true  azimuth  being  N.  80° 
E.,  and  his  magnetic  azimuth  N.  60°  E., 
it  is  required  to  find  the  variation. 

True  azimuth N.  80°  E. 

Magnetic  azimuth N.  60   E. 

DifF.  is  the  variation 20   E. 


EXAMPLE  IV. 

The  star  Aldebaran  was  observed  at 
rising  to  bear  by  compass  E.  N.  E.,  when 
the  true  amjilitude  was  N.  E.  by  E. ;  re- 
quired the  variation. 

True  amp..  .N.  E  by  E.  or  E.  33°  45'  N. 
Mag.  amp E.  N.  E.  or  E.  22  30  N. 

DifF.  is  the  variation 11   15  W. 

EXAMPLE  V. 

The  true  amplitude  of  the  planet  Jupiter 
was  E.  10°  N.  when  his  magnetic  ami)li- 
tude  was  E.  20°  S. ;  required  the  variation. 

True  amplitude E.  10°  N. 

Magnetic  amplitude E.  20   S. 

Sum  is  the  variation 30  W 


To  calculate  the  variation  hy  observing  the  swi's  azimuth  lohen  at  equal  altitudes 
in  the  forenoon  and  afternoon. 

The  variation  of  the  compass  may  also  be  determined  by  observing  the  magnetic 
azimuths  of  the  sun,  in  the  mornhig  and  evening,  when  at  the  same  altitude,  the 


162  VARIATION   OF   THE    COMPASS. 

observer  being  suy)posed  to  be  at  the  same  place  at  both  observations ;  for  it  is  evident 
that  if  the  dechnation  of  the  sun  do  not  vary  during  the  time  elapsed  between  the 
observations,  the  middle  point  of  the  compass  between  the  two  bearings  will  be  the 
bearing  of  the  true  north  or  south  point  of  the  horizon,  at  the  place  of  observation, 
and  the  difference  between  that  bearing  and  the  north  or  south  point  of  the  comj)ass 
will  be  the  variation. 

In  this  kind  of  observations,  it  will  be  convenient  always  to  estimate  the  magnetic 
azimuths  from  the  south  point  of  the  compass,  calling  them  east  or  west,  as  before 
directed ;  and  this  method  is  supposed  to  be  made  use  of  in  the  following  rule.  Then, 
if  one  azimuth  be  east  and  the  other  west,  half  their  difference  will  be  the  variation, 
otherwise  their  half-sum,  and  the  variation  will  be  of  the  same  name  as  their  gi-eater 
azimuth,  excepting,  however,  where  the  half-sum  is  taken  and  exceeds  90°,  in  which 
case  its  supjjjemcnt  will  be  the  variation,  of  a  different  name  from  the  azimuth  ;  the 
variation  being  always  supposed  less  than  90°. 

If  the  declination  of  the  sun  varies  during  the  elapsed  time  betvv'een  the  observations 
(as  is  generally  the  case),  an  allowance  may  be  made  for  that  variation  by  apjdying  a 
correction  to  the  afternoon  azimuth,  calculated  by  the  following  rule  : — 

RULE. 

Find,  from  Table  IV.,  the  daily  variation  of  the  sun's  declination  on  the  day  of 
observation.  Then  to  the  constant  logarithm  9.1249  add  the  log.  cosine  of  the  latitude  of 
the  place,  the  log.  sine  corresponding  to  the  elapsed  time  between  the  observations  found  in 
the  column  P.  M.,  the  Prop.  Log.  of  the  daily  variation  of  the  sun^s  declination,  and  the 
Prop.  Log.  of  the  elapsed  time,*  estimating  hours  and  minutes  as  mimdes  and  seconds ; 
the  sum,  rejecting  30  in  the  index,  ivill  be  the  Prop.  Log.  of  the  correction  to  be  applied  to 
the  western  azimuth,  by  subtracting  lohtn  the  sun  is  approaching  toivards  the  noHhem 
hemisphere,  otherwise  by  adding.^  The  azimuth,  thus  corrected,  is  to  be  used  in 
estimating  tlie  variation  instead  of  the  observed  azimuth. 

It  is  not  necetjsary,  in  this  calculation,  to  find  the  latitude  or  declination  to  any  great 
degree  of  accuracy,  which  is  the  greatest  advantage  of  the  method  ;  another  of  the 
advantages  consists  in  being  able  to  take  a  great  number  of  observations,  and  apply- 
ing the  correction  at  one  operation  to  the  variation  deduced  from  the  mean  of  all 
the  observations,  so  that,  when  gi'eat  accuracy  is  required  (as  in  taking  observations 
ashore),  this  method  may  be  used  with  success  ;  and  it  is  evident  that  it  is  alike 
applicable  to  the  moon  or  any  heavenly  body  ;  but  the  observations  must  be  taken  in 
the  same  place,  as  it  would  increase  the  calculation  considei'ably  to  make  an  allow- 
ance for  the  change  of  place,  as  well  as  for  tlie  cliange  of  declination  ;  and  it  would  be 
better,  in  this  case,  to  calculate  each  observation  separately  by  the  rules  before  given.  • 

EXAINIPLE. 

Su])pose  that,  on  the  10th  of  April,  1848,  in  the  latitude  of  42°  29'  N.,  longitude  50° 
W.,  the  sun's  morning  azimuth  is  observed  to  be  S.  .54°  24'  D.,  and  in  the  evening', 
when  the  sun  is  at  the  same  altitude,  is  S.  39°  46'  W.,  the  elajjsed  time  between  the 
observations  being  C  20™  ;  required  the  variation. 

Constant  logarithm 9.1249 

Latitude  42°  29' Cosine  9.8677 

Elapsed  time  6^  20™ Sine  9.8676 

Daily  variation  of  declination  22'  P.  L 9128 

Elapsed  time  6"  20",  taken  as  6'  20"  P.  L.. . .  1.4536 

Con*,  western  azimuth       11'  nearly  P.  L.  —  1.2266 
Western  azimuth     S.  39  46   W.  

Corrected  azimuth  S.  39  35    W. 
Morning  azimuth     S.  54  24    E. 

Difference S.  14  49    The  half  of  which,  7°  24',  is  the  varia- 
tion, which  IS  easterly,  because  the  gi-eater  azimuth  S.  54°  24'  E.  is  easterly. 

*  The  elapsed  time  may  be  tletennined  by  any  common  watch  ;  but  if  none  be  used  in  the  observa- 
tions, it  maj'  be  determined  as  follows  : — If  one  of  the  observed  azimuths  be  east  and  the  other  west, 
take  half  their  sum,  otherwise  half  their  diflerence,  and  to  the  log.  sine  of  this  half-sum  (or  half-din"ercnce) 
add  the  log',  secant  of  the  sun's  declination,  and  the  log.  cosine  of  the  sun's  correct  allitude  at  the  time 
of  taking  the  azimuth  ;  the  sum,  rejecting  20  in  the  index,  will  be  the  log.  sine  to  be  used  in  the  abo\e 
calculation,  and  this  logarithm  will  correspond  to  the  elapsed  time  marked  in  the  column  P.  M.  of 
Table  XXVII. 

t  In  this  rule  it  is  supposed  that  the  bearing  of  the  sun   by  the  afternoon  observation,  i.«  to  the  west- 


VARIATION   OF  THE   COMPAS?.  163 

The  variation,  tliiis  found,  is  to  be  allowed  on  all  courses  steered  by  the  con  pass,  to 
obtain  the  true  courses.  To  tnake  this  allowance,  you  must,  look  towards  tlie  point 
of  the  conii)ass  tiie  ship  is  sailing  upon,  and  allow  the  variation  from  it  ioivards  the 
rxs;ht  hand  if  the  variniion  be  east,  hut  to  the  left  hand  if  the  variation  be  ivest.  Tlius,  if  a 
shij)  steer  S.  E.  with  one  point  westerly  variation,  the  true  course  will  be  S.  E.  by  E 
If  tl.e  variation  be  one  point  easterly,  the  course  will  be  S.  E.  by  S. 

The  variation  in  Cambridge  (Mass.),  in  1708,  was  9°  W. ;  in  1742,  8°  W. ;  in  1782, 
[>^  4(5  W.  ;  decreasing  aljout  IJ  minutes  per  year.  At  Salem  (Mass.),  in  1808,  it  was 
5°  20  W. :  in  London,  in  1580,  11°  15'  E. ;  in  1672,  2°  30'  W. ;  in  1780,  22°  41'  W. :  at 
Paris,  in  1.550,  8°  E. ;  in  1660,  0°  ;  in  1769,  20°  W.  Hence  it  appears  that,  at  London 
and  Paris,  the  \ariation  formerly  increased  10  or  11  mimites  per  year ;  but,  by  some 
late  observations  made  in  London,  it  ai)pears  to  be  nearly  stationar3^  Off  the  Cape 
of  Good  Hope,  the  annual  increase  is  about  7  minutes. 

Besides  this  annual  change  of  the  variation,  there  is  also  a  small  diurnal  change, 
which,  at  London,  Paris,  and  Cambridge  (Mass.),  is  from  10'  to  15'  By  this  quantity 
the  absolute  variation,  at  tliose  ])laces,  increases  from  about  8,  A.  IM.,  to  2,  P.  ]\1.,  when 
the  needle  becomes  stationary  for  some  time  ;  after  that,  the  variation  decreases,  and 
the  needle  comes  back  again  to  its  former  situation,  or  nearly  so,  in  the  night,  or  by 
the  next  morning. 

In  addition  to  the  observations  contained  in  the  preceding  table,  it  may  be  obsei'ved 
that  the  variation,  which,  at  present,  is  less  than  ^  ])oint  W.  near  Cape  Cod,  decreases 
in  going  to  the  westward  along  the  coast  of  the  United  States  of  America,  so  that  near 
Cape  Hatteras  it  is  scarcely  sensible,  and  farther  to  the  westward  becomes  easterly. 
In  the  leeward  West  India  Islands  it  is  about  ^  point  E. ;  and  in  the  windward 
islands  i  point  E.  Along  the  noi'thern  shore  of  the  Brazils  there  is  a  small  easterly 
variation,  which  decreases  in  proceeding  to  the  eastward,  and  becomes  westerly  neai" 
Cape  Roque,  where  it  is  ^  point  AV.  In  proceeding  farther  to  the  southward,  along 
the  coast  of  America,  the  easterly  variation  increases  so  as  to  be  ai)ove  2  points  E. 
near  Cape  Horn,  and  from  thence  gradually  decreases  along  the  coast  of  Chili  and 
Peru,  so  as  to  be  about  1  point  E.  under  the  equator  near  Quito  ;  but  in  proceeding 
to  the  nortiiward  towards  the  N.  W.  coast  of  America,  the  easterly  variation  increases 
to  more  than  2  points. 

On  the  contrary,  in  proceeding  to  the  eastward  of  the  United  States  of  America,  the 
westerly  variation  increases,  being  nearly  1  point  W.  a  little  to  the  eastward  of  Cape 
Sable  (Nova  Scotia),  and  about  2|  points  W  on  the  E.  part  of  Newfoundland,  and  at 
the  Western  Islands.  At  the  Orkney  Islands  it  is  2^  ])oints  westerly,  and  is  nearly 
the  same  in  the  English  Channel,  and  on  the  coasts  of  England,  Scotland,"  and  Ire- 
land. On  the  coast  of  Holland,  it  is  from  1:^  to  2  points  W. ;  in  the  Cattegat  and 
Sound,  about  1-^  points  W. ;  in  the  western  part  of  the  Baltic,  about  1:^  points;  at 
the  entrance  of  the  Gulf  of  Finland,  1  ])oint  W. ;  in  the  Bay  of  Biscay,  about  2i 
[loints  W. ;  near  Cajie  St.  Vincents,  2  points  W. ;  in  the  Mediterranean,  from  1  to 
II  pohits  W. ;  near  Cape  Verd  (Africa),  1^  points  W. ;  and  from  thence  gradually 
increases  along  the  western  shore  of  Africa  towards  the  Cape  of  Good  Ilope,  and 
is  there  a!)ove  2  points  W.,  and  from  thence  increases  towards  Cape  Lagullas,  and 
a  little  to  the  eastward,  to  2i  points  or  25  points  W.,  and  then  decreases  in  proceeding 
along  the  eastern  shore  of  Africa,  and  is  about  |  point  westerly  at  the  entrance  of 
the  Red  Sea.  In  the  Arabian  Sea,  Bay  of  Bengal,  Java  Sea,  China  Sea,  and  off  the 
coast  of  Sumatra,  it  is  very  small,  and  on  the  S.  E.  part  of  New  Holland  is  about 
I  point  E. 

Before  the  introduction  of  the  method  of  finding  the  longitude  by  lunar  oliservatjons, 
and  the  improvements  in  the  construction  of  chronometers,  and  their  introduction 
into  connnon  use,  it  was  profjosed  to  find  the  longitude  by  means  of  the  observed 
variation,  and  charts  were  constructed  for  tliis  purjjose  ;  but  this  method  is  now 
wholly  given  uj),  because  there  is  always  a  gi-eat  uncertainty  in  observations  of  the 
variation,  since  it  is  not  uncommon  to  find  2  or  3  degrees  difference  between  an 
azimuth  in  the  morning  and  evening,  when  the  shi]),  during  that  time,  has  been  nearly 
stationary  ;  the  same  difference  will  sometimes  be  found  merely  from  making  the 
observation  when  tlie  ship  is  on  a  different  tack.  This  is  owing  to  the  iron  in  the 
ship,  \\hich  attracts  the  compass  by  a  force  which  is  generally  situated  in  a  point 
near  the  centre  of  the  ship.     When  this  point  and  the  compass  are  in  the  magnetic 

ward  of  the  ineridian  by  compass  ;  but  il  there  be  a  great  variation,  that  bearing  mig-ht  be  to  the 
eastward  of  the  meridian  by  tiie  compass,  and,  in  that  case,  the  correction  of  the  western  azimuth  must 
be  app'ied  in  i  contrary  manner  to  the  above  directions 


164  VARIATION   OF  THE   COMPASS. 

meridian  of  .ne  compass,  the  true  variation  is  obtained  ;  but  as  soon  as  the  position  of 
the  ship  is  changed,  so  as  to  bring  this  point  to  th-^.  eastward  or  westward  of  the 
magnetic  meridian  passin"  through  the  compass,  a  corresponding  change  or  altera- 
tion in  the  variation  to  the  eastward  or  westward  is  immediately  perceived.  This 
deviation  sometimes  amounted  to  8°  or  9°  in  the  surveys  of  New  Holland.  This  has 
since  been  confirmed  by  various  observations  in  different  places,  particularly  in  the 
voyages  towards  the  north  pole,  lately  made  by  order  of  the  English  government. 
The  method  wliich  v/as  at  first  used  to  correct  this  error,  which  is  sometimes  of 
considerable  importance  in  nautical  surveys  where  great  accuracy  is  required,  was 
to  -place  the  compass  always  in  the  same  part  of  the  ship,  and  to  find,  by  actual  observa- 
tion, the  greatest  deviation  arising  from  tiiis  local  attraction,  which  is  when  the  ship's 
head  is  directed  east  or  west.  The  deviation,  when  the  ship's  head  is  in  any  other 
direction,  is  found  by  entering  Tahle  I.  or  Table  II.  in  the  page  corresponding  to  that 
direction  as  a  course,  and  witli  that  greatest  error  in  minutes  in  the  distance  column, 
the  corresponding  number  in  the  dei>arture  colunui  will  be  the  required  correction 
nearly.  Thus,  if  the  deviation  was  2°  8'  (or  128')  when  the  ship's  head  was  directed 
towards  the  east,  the  deviation,  when  in  the  direction  of  one  point  from  the  meridian, 
(that  is,  N.  by  E.,  N.  by  W.,  S.  by  E.,  or  S.  by  W.),  would  be  found  by  entering 
Table  I.  in  the  page  for  one  point,  or  with  the  distance  128',  the  corresponding  de])art- 
ure  25'  would  be  the  correction  to  be  applied  on  all  bearings  taken  by  the  compass 
when  in  that  situation.  Mr.  Barlow  has  invented  a  method  of  correcting  this  error, 
making  use  of  a  curious  property  of  the  attractive  force  of  iron  on  the  compass,  it 
having  been  found  that  this  force  depends  on  the  attractive  surface,  and  not  wholly  on 
the  quantity  of  iron  ;  so  that  a  solid  globe  of  iron,  30  inches  in  diameter,  would  affect 
the  coni])ass  exactly  in  the  same  maimer  as  a  holloiu  shell  of  the  same  diameter, 
made  of  sheet  iron  only  one  tenth  of  an  inch  in  thickness,  though  this  shell  could 
not  contain  but  one  hundredth  part  the  quantity  of  iron  which  the  globe  does.  Mr. 
Barlow  therefore  proposed  to  have  a  sheet  of  iron  placed  abaft  the  compass,  cf  such 
dimensions,  and  at  such  a  distance,  as  should  be  found  by  experiment  to  bring  the 
needle  back  to  the  magnetic  meridian  when  the  ship's  head  was  east  or  west ;  then, 
keeping  the  iron  in  that  position,  it  would  correct  the  error  of  the  local  attraction  of 
the  ship  in  every  direction  of  the  ship's  head.  This  method  has  been  tested  by 
e.xperiment,  and  found  to  succeed  admirably.  It  has  also  been  attended  with  the 
great  advantage  of  leaving  the  compass  free  to  act  by  the  natural  magnetism  of  the 
earth  in  high  latitudes,  where  the  force  is  much  enfeebled  by  the  oblicpiity  of  its 
direction  on  account  of  the  greatness  of  the  dip.  In  the  voyages  above  named,  it 
was  found  that  the  compasses  thus  furnished  traversed  freely  and  accurately,  when 
those  of  the  conmion  form  moved  very  irregularly,  and  were,  in  some  cases,  almost 
useless. 

The  Transactions  of  the  Royal  Society  of  London  for  the  year  1833,  contain  a 
valuable  chai't,  by  P.  Barlow,  upon  which  are  marked  the  magnetic  lines  of  equal 
variations,  as  they  have  been  observed  in  late  vojages  of  discovery,  surveys,  &c.  We 
expect  to  give,  in  the  collection  of  tables,  a  few  numerical  results  from  this  chart 

On  the  dip  of  the  magnetic  needle. 

If  the  needle  of  a  compass  be  exactly  balanced  on  its  point  in  a  horizontal  position, 
and  then  the  magnetic  virtue  be  communicated,  the  needle  will  point  towards  the 
north,  and  will  also  be  inclined  to  the  horizon,  the  north  point  of  the  needle  tending 
down,\\anls,  and  the  south  point  upwards,  in  northern  climates,  and  the  contrary 
in  southern  climates.  This  inclination  of  the  needle  to  the  iiorizon  is  called  the 
dip  of  the  magnetic  needle,  which  is  different  in  different  places,  though  it  has  bewi 
found  to  remain  nearly  the  same  in  the  same  place,  since  its  discovery  in  the  year 
157(5,  in  which  year,  at  London,  the  dip  was  71°  50' ;  in  1723,  it  was  74°  or  75° ;  and, 
ut  i)resent,  is  about  72-^°.  31essrs.  Humboldt  and  Biot  published  a  method  by  which 
the  dip  may  be  calculated  for  any  given  ])lace,  in  north  latitudes,  to  a  considerable 
degree  of  accuracy.  This  method  is  explained  in  the  22d  vol.  of  Tilloch's  Magazine, 
and  is  in  substance  as  follows : — 

According  to  their  theory,  there  are  two  magnetic  poles,  one  in  the  latitude  of  79°  1' 
N.,  and  in  the  longitude  of  27°  42'  W.*  from  Greenwich,  the  other  diametrically 
opposite,  in  the  latitude  of  79°  1'  S.,  and  in  the  longitude  of  152°  18'  E.     The  great 

*  Capt.  Ross,  in  liis  voyage  to  the  north,  found  the  northern  pole  to  be  in  the  latitude  of  70°  5'  17 
N.,  and  in  the  longitude  of  yG°  46'  45"  VV. 


VARlATlOiN    OF   THE    COMPASS 


165 


circle  of  the  earth  90^  distant  from  tliese  poles  is  called  the  magnetic  equator.  On  tho 
ma.^netic  equator  the  dij)  is  nothing,  and  at  the  poles  is  90°;  at  any  other  point  on  the 
surface  of  the  earth,  the  dip  varies  with  the  distance  from  the  magnetic  pole.  This 
distance  may  be  calculated  by  connnon  si)herical  trigonometrj',  or  (which  is  much 
more  siinj)Ie,  and  sufficiently  accurate  for  this  purpose)  by  measuring  the  tlistance  on 
a  terrestrial  globe  from  the  magnetic  pole  to  the  place  for  which  the  dip  is  to  be 
calculated  ;  then  to  the  log.  cotangent  of  this  distance  add  the  constant  logarithm 
0.3010:j  ;  tiie  sum  will  be  the  log.  tangent  of  the  dip.  The  dip  was  calculated,  on 
these  prinr-iples,  for  twenty-eigiit  places  in  Euro])e,  Asia,  Africa,  and  America,  and  in 
ten  places  the  theory  did  not  differ  1°  from  actual  observations,  and  in  five  places  did 
not  differ  T ,  but  at  Spitzborgen  the  difference  wad  between  4°  and  5°. 

(See  page  459.) 


/     THE     MARINER'S     COMPASS'.??. 


166 


TO  FIND  THE  LATITUDE  BY  OBSERVATION. 


The  latitude  of  a  place,  being  its  distance  from  the  equator,  is  measured  by  an  arc 
of  the  meridian  contained  between  the  zenith  and  the  equator ;  hence,  if  the  distance 
of  any  heavenly  botly  from  the  zenith  when  on  the  meridian,  and  the  declination  of 
the  object,  be  given,  the  latitude  may  be  thence  found. 

The  meridian  zenitli  distance  of  any  object  may  be  found  by  observing  its  altitude 
when  on  the  meridian,  or  by  oljserving  one  altitude  taken  at  a  given  hour  from  pass- 
ing the  meridian,  or  by  two  altitudes  taken  out  of  the  meridian  and  the  elapsed  time 
between  the  observations.  Each  of  these  methods  will  be  explained  by  proper 
examples. 

Altitudes  of  the  sun  and  moon,  taken  at  sea,  require  four  corrections  in  order  to 
obtain  the  true  altitude  of  their  centres  ;  these  are  for  semidiameter,  dip,  refraction, 
and  parallax.*  When  a  planet  or  star  is  observed,  the  corrections  for  dij)  and  refrac- 
tion only  are  to  be  applied,  as  the  semidiameter  and  parallax  of  a  planet  are  but  a  few 
seconds,  and  may  be  neglected  in  finding  the  latitude  at  sea. 

In  a  fore  ohservation  ivith  a  quadrant,  sextant,  or  circle,  the  scmidiaineter  is  to  be 
added  if  the  lower  limb  is  observed,  but  subtracted  if  the  upper  limb  is  observed. 
The  dip  and  refraction  are  to  be  subtracted,  and  the  parallax  to  be  added,  and  the 
true  central  altitude  will  be  thus  obtained,  which,  being  subtracted  from  90°,  will  give 
the  true  zenith  distance. 

In  a  back  ohsei-vation  ivith  a  quadrant,  the  semidiameter  is  to  be  subtracted  if  the 
lower  liud)  is  observed,  but  added  if  the  upper  limb  is  observed.  Tlie  dip  and  paral- 
lax are  to  be  added,  and  the  refraction  subtracted,  and  the  central  altitude  will  be 
obtained,  which,  being  subtracted  from  90°,  will  give  the  true  zenith  distance. 

In  a  hack  observation  ivith  a  sextant  or  circle,  by  measuring  the  su])i)lement  of  the 
altitude,  (by  bringuig  the  lower  limb  of  the  image  of  the  object  to  touch  the  back 
horizon,)  the  senfidiameter  and  refraction  must  be  added  to  the  true  altitude  given  by 
the  instrument,  and  the  dij)  and  jiarallax  subtracted  therefrom,  and,  by  subtracting 
90°  from  the  remainder,  the  true  zenitli  distance  will  be  obtained. 

To  find  the  latitude  by  the  meridian  altitude  of  any  object. 

Having  obtained  the  true  meridian  zenith  distance  by  either  of  these  methods,  you 
must  then  find  the  declination  of  the  object  at  the  time  of  observation.  This  may  be 
found  for  the  sun  by  the  Nautical  Almanac,  or  by  means  of  Tables  IV.  and  V.,  in  the 
manner  before  explained.  The  declination  of  a  fixed  star  may  be  easily  found  by 
inspection  in  Table  VIII.,  or  from  the  Nautical  Almanac.  The  declination  of  the 
moon  or  a  planet  may  be  found,  in  the  Nautical  Almanac,  in  a  manner  wliich  will 
be  hereafter  explained.  Having  the  meridian  zenith  distance  and  declination,  the 
latitude  is  to  be  found  by  the  following  rules. 

CASE  I. 

Jflicji  the  object  rises  and  sets. 
RULE. 
If  the  object  bear  south  when  upon  the  meridian,  call  the  zenith  distance  noTth  ;  | 
but  if  the  bearing  be  noiih,  you  must  call  the  zenith  distance  south.     Place  the  zenith 

*  The  semidiameter  of  the  sun  may  be  found  in  the  Nautical  Ahnanac,  and  is  nearly  16'.  The  sun's 
parallax  is  found  in  Table  XIV. ;  the  refraction  in  Table  XII. ;  the  dip  in  Table  XIII."  The  semidiam- 
eter and  parallax  of  the  moon  may  be  found  from  the  Nautical  Almanac,  as  will  be  explained  ncreafler. 
It  may  also  be  observed,  tlial  it  is  usual  to  add  12'  for  the  correction  for  semidiameter,  dip,  and  parallax, 
in  a  fore  observation  of  the  sun's  lower  limb,  taken  upon  the  deck  of  a  common-sized  vessel ;  and,  by 
subtractinsT  the  retraction  from  the  sum,  the  true  altitude  will  be  obtained,  nearly;  and  it  ouj:ht  alwaj'S 
to  he  kept  in  mind,  that  the  refraction  at  low  altitudes  is  of  too  much  importance  to  be  neglected. 

t  In  tliis  rule,  the  sun  is  supposed  to  be  the  fixed  point,  and  the  zenith  is  referred  to  it.  Thus,  if  the 
sun  bears  south  from  an  observer  (or  from  his  zenith).  Uie  zenith  bears  north  from  the  sun ;  and  it  is  this 
Intfer  bearing  which  is  used  in  the  rule. 


TO   FIND   THE   LATITUDE  BY   OBSERVATION. 


167 


distance  under  the  declination,  and,  if  they  are  of  the  same  name,  add  them  together 
but  if  they  are  of  different  names,  take  their  difference;  this  sum  or  dificreuce  will  be 
the  latitude,  which  will  be  of  the  same  name  as  the  greatest  number. 

CASE  11. 

J f Tien  the  object  does  not  set,  hut  comes  to  the  meridian  above  the  horizon  ttvice  in  24  hours. 

IMany  stars  are  always  above  the  horizon  of  certain  places  of  the  earth,  and,  in  high 
latitudes,  t!ie  sun  is  sometimes  above  the  horizon  for  several  days,  in  which  case  the 
meridian  altitude  may  be  observed  twice  in  24  hours ;  that  is,  once  at  the  greatest 
height  above  the  pole,  and  again  at  the  lowest  height  upon  the  meridian  below  the 
pole.  In  the  former  case,  the  latitude  is  to  be  found  by  the  preceding  rule,  but  in  the 
latter  by  the  following : — 

RULE. 

Add  the  complement  of  the  declination  to  the  meridian  altitude ;  the  sum  will  be 
the  latitude,  of  the  same  name  as  the  declination. 

Note. — When, the  sun  or  star  is  on  the  equator,  or  has  no  declination,  the  zenith 
distance  will  be  equal  to  the  latitude  of  the  {)lace,  which  will  be  of  the  same  name  as 
the  zenith  distance.  When  the  sun  or  star  is  in  the  zenith,  tlie  declination  will  be 
equal  to  the  latitude,  and  it  will  be  of  the  same  name  as  the  declination. 


To  find  the  latitude  bi/  the  meridian  altitude  of  the  sun  or  star. 


*  EXAMPLE   I. 

Suppose  that,  at  the  end  of  the  sea  day, 
June  21, 1848,  in  the  longitude  of  G0°  W., 
the  meridian  altitude  of  tiie  sun's  lower 
limb,  bearing  south,  was  found  by  a  fore 
observation  to  be  40°  6' ;  required  the 
latitude,  supposing  the  correction  of  the 
observed  altitude  for  pai'allax,  dip,  and 
semidiameter,  to  be  twelve  miles. 

Observed  altitude 40°  OC 

Par., dip, and semidiam.  . .  .add        12 

Sum 40   18 

Refraction subtract  1 


True  altitude 40   17 

Subtract  from 90  00 

True  zenith  distance 49  43  N. 

Sun's  declination,  Table  IV.  . .  23  27  N. 

Latitude 73   10  N. 

EXAMPLE  II. 
Suppose  that,  at  the  end  of  the  seadaj^, 
April  14,  1848, in  tiie  longitude  of  140° 
E.  from  Greenwich,  the  altitude  of  the 
sun's  lower  limb,  by  a  fore  observation, 
was  00°  2.3'  when  on  the  tneridian  and 
bearing  south,  the  coi-rection  for  dip, 
semidiameter,  and  parallax,  being  twelve 
miles;  reciuired  the  latitude. 

Observed  altitude G0°  2;7 

Correction add        12 

True  altitude* CO  37 

Subtract  from 90  00 


True  zenith  distance 29  23  N. 

Sun's  declination.  Table  IV. 
cor.  by  Table  V.  for  long. 


9  25  N. 


Latitude 38  48  N. 


EXAMPLE  111. 
Suppose  that,  at  the  end  of  the  sea  day. 
May  15,  1848,  in  the  meridian  of  Green- 
wich, the  meridian  altitude  of  the  sun's 
lower  limb,  bearing  north,  was  found  by 
a  fore  observation  to  be  30°  GG',  the  cor- 
rection for  parallax,  dip,  and  semidiameter, 
being  twelve  miles  ;  required  the  latitude. 

Observed  altitude 30°  06' 

Par.,  dip, and  semidiam.. .  .add         12 

Sum 30   18 

Refraction subtract  2 


True  altitude 30   16 

Subtract  from 90   00 


True  zenith  distance 59  44  S. 

Sun's  declination 18  58  N. 


Latitude 40  46  S. 


EXAMPLE  IV. 
Suppose  that,  at  the  end  of  the  sea  day, 
Nov.  17,  1848,  in  the  longitude  of  80°  E. 
from  Greenwich,  by  a  fore  observation, 
the  meridian  altitude  of  the  sun's  lower 
limb  was  50°  OG',  bearing  south,  the  eye 
of  the  observer  being  seventeen  feet 
above  the  surface  of  the  sea ;  required 
the  latitude. 

Observed  altitude 50°  06' 

Sun's  semidiam add        16 

,        50  22 
Subtract  dip  and  refraction  ...  5 

True  altitude! 50   17 

Subtract  from 90  00 

True  zenith  distance 39  43  N 

Sun's  dec.  cor.  by  Table  V.. . .  19  03  S. 

Latitude 20  40  N 


*  The  refraction,  being-  small,  is  here  neglected. 

t  The  parallax,  being  small,  is  here  neglected,  and  the  sun's  semidiameter  is  supposed  to  be  16'. 


168 


TO   FIND   THE  LATITUDE   BY   OBSERVATION. 


EXAMPLE  V. 
By  a  fore  obsei-vation,  the  meridian 
altitude  of  tlie  sun's  lower  limb  was  found 
to  be  40°  20',  bearing  south  of  the  ob- 
server, the  declination  being  9"  56'  N., 
and  the  eye  twenty-six  feet  above  the 
horizon ; — required  the  latitude  of  the 
place. 

Observed  altitude 40°  20' 

Semidiameter add        16 

40  30 
Dip  5',  refraction  1'. .  .subtract  6 

True  alt.  of  the  sun's  centre  *     40  30 
Subtract  from 90  00 

Zenith  distance 49  30  N. 

Declination 9  56 N. 

Latitude 59  26  N. 

EXAMPLE  VL 

By  a  back  observation  with  a  quadrant 
of  reflection,  the  meridian  altitude  of  the 
sun's  lower  limb  was  25°  12',  when  the 
declination  was  21°  14'  S.,  and  the  eye 
of  the  observer  forty  feet  above  the  hori- 
zon, the  sun  bearing  south  ;  required  the 
latitude  of  the  place  of  observation. 

Obsei-ved  altitude 25°  12' 

Semidiameter subtract        16 

24  56 
Dip add        06 

25  02 
Refz'action subtract        02 

True  alt.  of  the  sun's  centre  *    25   00 

True  zenith  distance 65  00  N. 

Declination 21    14  S. 

Latitude 43  46  N. 

EXAMPLE  VII. 

Suppose  that,  on  January  1,  1830,  an 
observer,  seventeen  feet  above  the  water, 
finds  by  a  fore  observation  that  the  alti- 
tude of  Sirius  is  53°  33'  when  passing  the 
meridian  to  the  southward  ;  required  the 
latitude  of  the  place  of  observation. 

Observed  altitude 53°  33' 

Dip  of  the  horizon . . .  .subtract  4 

53  29 
Refraction subtract        01 

53  28 

True  zenith  distance 36  32  N. 

Sirius  declin.  Table  VIILf. . .  16  29  S. 

Latitude 20  03  N. 


EXAMPLE  VIII. 

Suppose  that,  on  the  13th  June,  1848, 
sea  account,  an  observer,  in  a  high  north- 
ern latitude,  and  in  the  longitude  of  65° 
W.  from  Greenwich,  his  eye  l>eiug  twenty 
feet  above  the  surface  of  the  water,  ob- 
served by  a  fore  observation  the  altitude 
of  the  sun's  lower  limb  on  the  meridian 
below  the  pole  8°  14' ;  required  tlie  lat' 
tude. 

The  sun  being  below  the  pole  at  12 
hours  before  the  end  of  the  sea  day  June 
13,  the  correction  of  declination  corre- 
sponding in  Table  V.  is  —  1'  46",  and  the 
correction  in  65°  W.  long,  is  -|-  0'  38'' ; 
hence  both  corrections  make  nearly  1', 
to  be  subtracted  from  the  declination  at 
noon  23°  15'  N.,  which  gives  the  declina- 
tion at  the  time  of  observation  23°  14'  N., 
the  comp.  of  which  is  06°  46'. 

Observed  alt.  sun's  lower  limb     8°  14' 
Semidiameter add         16  • 

8  30 
Dip subtract        04 

8  26 
Refraction subtract        00 

True  alt.  of  the  sun's  centre  *      8  20 
Comiilement  of  declination ...  00  40  N. 

Latitude 75  00  N. 


EXAMPLE   IX. 

Suppose  that,  by  a  back  observation 
with  a  sextant,  the  lower  limb  of  the 
sun's  image  was  brought  to  the  back 
horizon,  and  the  angle  shown  by  the 
index  was  110°  10',  the  sun  being  then 
on  the  meridian  and  bearing  south,  tlte 
declination  being  20°  5'  N.,  the  sun's 
semidiameter  10',  and  the  observer  20 
feet  above  the  horizon  ;  required  the  lat- 
itude. 

Observed  angle 110°  10' 

Semidiameter add  10 

110  20 
Dip sub.  4 

110  22 
Subtract 90  00 

Zenith  distance  J 20  22  IN 

Declination 20  05  N. 

Latitude 40  27  N 


*  The  parallax,  being  small,  is  here  neglected,  ami  tlic  sun's  semifliameter  is  supposed  to  be  IG'. 
t  The  declinations  ofthese  bright  stars  are  given  for  every  10  days  in  the  Nautical  Almanac.    Wlien 
great  accuracy  is  required,  these  declinations  slio4jld  be  used  instead  of  the  numbers  in  Table  VIII. 
i  The  refraction  and  parallax,  being  only  a  few  seconds,  are  neglected. 


TO  FIND  THE   LATITUDE   BY   OBSERVATION. 


IGO 


EXAMPLE  X. 

Suppose  tJiat,  on  January  10,  1830,  an 
obsen'er,  eighteen  feet  above  the  water, 
finds  the  altitude  of  the  north  star,  when 
on  the  meridian  below  the  pole,  to  be 
36°  23'  by  a  fore  obsei-vation  ;  required 
the  latitude  of  the  place  of  observation. 

Observed  altitude 36°  23' 

Subtract  dip  4',  ref.  1' 5 

True  altitude 36   18 

Comp.  declin.  Table  VIII.  *. . .    1   36  N. 

Latitude 37  54  N. 


EXAMPLE  XI. 
Suppose  that,  by  a  back  ooservation 
with  a  sextant,  the  lower  limb  of  the  sun's 
image  was  brought  to  the  back  horizon, 
and  the  angle' sliown  by  the  index  was 
106°  12',  the  altitude  of  the  observer 
being  twenty-two  feet,  and  the  correction 
for  semidiameter,  jiarallax,  and  dip,  being 
(as  usual)  about  12';  required  the  true 
latitude,  supposing  the  declination  to  be 
20°  S.,  and  that  the  sun  bore  north  at  the 
time  of  observation. 

Observed  angle 106°  12' 

Dip  and  seniidiam add  12 

106  24 
Subtract 90  00 

Zenith  distance  f 16   24  S. 

Sun's  declination 20  00  S. 

Latitude 36  24  S. 


We  have  observed,  in  the  directions  for  finding  the  meridian  altitude  of  an  object, 
that  an  error  will  arise  if  the  slii^i  be  in  motion,  or  the  sun's  declination  vary.  The 
amount  of  this  correction  may  be  estimated  in  the  following  manner: — 

Find  the  number  of  miles  and  tenths  of  a  mile  northing  or  southing  made  by  the 
ship  in  one  hour,  and  also  the  variation  of  the  sun's  declination  in  an  hour,  expressed 
also  in  miles  and  tenths.  Add  these  together,  if  they  both  cons{)ire  to  elevate  or 
depress  the  sun;  otherwise  take  their  difference,  which  call  the  arc  A.  Find,  in 
Table  XXXII.,  the  arc  B,  expressed  in  seconds,  corresponding  to  the  latitude  and 
declination  ;  then  the  arc  A,  divided  by  twice  the  arc  U,  will  express  the  time  in 
minutes  from  noo?r,  when  the  greatest  (or  least)  altitude  is  observed.  Moreover,  the 
square  of  the  arc  A,  divided  by  four  times  tlie  arc  B,  will  be  the  number  oi'  seconds  to 
|ie  applied  to  the  observed  altitude  to  obtain  the  true  altitude,  which  would  have  been 
observed  if  the  ship  had  been  at  rest. 

Thus,  if  the  ship  sail  towards  the  sun  south  11  miles  per  hour,  and  the  declination 
increases  northerly  1'  per  hour,  we  shall  have  A  =  11  -[-  1  =  12.  If  the  latitude  is 
42°  N.,  and  the  declination  2°  S.,  we  shall  have  by  Table  XXXII.  B=:2".  In  this 
case,  tlie  time  from  noon  is  Jt^-=z2  minutes,  and  the  correction  of  altitude  -l|4  =r  18 
seconds  only. 


*  Tlie  declination  of  tills  star  is  given  for  every  clay  in  the  Nautical  Almanac;  when  great  accuracy 
is  required,  this  declination  should  be  used  instead  of  that  in  Table  VIII. 
t  Tl^e  refraction,  being  small,  is  neglected. 

22 


170 


TO  FIND  THE  LATITUDE  BY  A  MERIDIAN 

ALTITUDE  OF  THE  MOON. 


The  latitude  may  be  found  at  sea,  by  the  moon's  meridian  altitude,  more  accurately 
tiian  by  any  other  method,  excej't  by  the  meridian  altitude  of  the  sun;  but  to  do  tliis, 
it  is  necessary  to  find  the  tiJne  of  her  passing  the  meridian,  and  her  declination  at 
that  time.  To  lacilitute  these  calculations,  we  have  given  the  Tables  XXVIII.  and 
XXIX.,  the  uses  of  which  will  evidently  appear  i'rom  the  following  rules  and 
examples. 

To  find  the  mean  time  of  the  7iioon's  passing  the  meridian. 

Find,  in  the  Nautical  Almanac,  the  time  of  the  moon's  coming  to  the  meridian  of 
Greenwich  for  one  day  earlier  than  the  sea  account,*  and  also  the  time  of  her  coming 
to  the  meridian  of  Greenwich  the  next  day,  when  you  are  in  west  longitude,  but  the 
preceding  day  when  in  cast  longitude  ;  take  the  difference  between  these  times,  w-ith 
which  you  nnist  enter  the  top  column  of  Ta!)Ie  XXVIII.,  and  against  the  ship's 
longitude  in  the  side  cohnnn  will  be  a  number  of  minutes  to  be  a])plied  to  the  time 
taken  from  the  Nautical  Almanac,  for  the  day  immediately  preceding  the  sea  account, 
by  adding  when  in  west  longitude,  but  subtracting  when  in  east  longitude;  the  sum 
or  difference  will  be  the  true  time  of  passing  the  meridian  of  the  given  })lace. 

EXAMPLE. 

Required  the  time  of  me  moon's  passing  the  meridian  of  Philadelphia,  April  19, 
1836,  sea  account. 

The  day  preceding  the  sea  account  is  April  18  ;  on  this  day,  the  moon  passed  the 
meridian  of  Greenwich  at  1''  55'".6,  and,  being  in  west  longitude,  we  find  the  time 
of  her  passing  the  meridian  the  next  day  2^  43'".0.  The  difference  between  these 
two  times  is  47™.4,  which  is  to  be  found  at  the  top  of  Table  XXVIII. ;  the  nearest 
tabular  nmnher  is  48'"  ;  under  this,  and  opposite  75°,  (the  longitude  of  Philadelphia,) 
is  the  con-ection  10™,  nearly,  to  be  added  to  1''  55^.6,  to  obtain  the  time  of  passing 
the  meridian  at  Philadelphia,  April  19'  2''  OS^.G,  sea  account,  or  April  18''  2''  05"\(), 
P.  M.,  civil  account. 

To  find  the  mooti's  declination  ivhen  on  the  meridian. 

Find  the  time  of  the  moon's  coming  to  the  meridian  as  above;  turn  the  ship's 
longitude  into  time  by  Table  XXI.,f  and  add  it  thereto  if  in  west  longitude,  but 
subtract  it  in  east ;  the  sum  or  difference  will  be  the  time  at  Greenwich.  Take  out 
the  moon's  declination  from  the  Nautical  Almanac,  for  the  nearest  hour  preceding 
the  Greenwich  time,  |  and  also  the  variation  for  10  minutes  in  the  next  cohunn. 

*  Takiiiij  llie  time  one  day  earlier  than  ihe  sea  arcount,  reduces  it  to  astronomical  lime  used  in  t!ie 
Nautical  Almanac.  Wc  may  observe  tiiat  llie  time  of  the  moon's  coming'  to  the  meridian,  is  <;ivcn  in 
the  Nautical  Almanac  to  tenths  of  a  minute,  instead  of  seconds  of  time.  This  is  done  to  facilitate  the 
calculation  of  the  right  ascension  and  declination,  by  using  common  decimal  fractions  instead  of  se.\a- 
gesimals. 

t  Longitude  may  be  turned  info  time,  without  the  help  of  Tabic  XXL,  by  multiplying  llie  degrees 
and  minutes  of  the  longitude  by  4,  and  considering  the  product  as  minutes  and  seconds  of  time  respec- 
tively ;  and,  by  the  inverse  process  of  dividing  liy  4,  we  may  turn  time  into  degrees,  &c.  Thus, 
80°  X  4  =:  320"  =  511  20'"  ;  and  15°  IG' x  4  =  GI-"  Ol' =  li^  ff"  4'.  In  like  manner,  l^  SO-"  or  80"', 
being  divided  by  4,  gives  20°,  and  IDH'",  being  divided  by  4,  gives  49°,  which  agree  witli  the  fable 
If  the  ship  be  furnished  with  a  chronometer,  regulated  for  mean  time  at  Greenwich,  we  may  avoid  the 
labor  of  tliis  part  of  the  operation  by  taking  the  time  at  Greenwich,  as  shown  by  the  chronometer,  at  the 
very  moment  when  the  meridian  altitude  of  the  moon  is  observed. 

}  If  the  time  at  Greenwich  fall  exactly  upon  any  hour,  the  declination  can  then  be  taken  /rom  the 
Nautical  Almanac,  by  mere  inspection,  without  any  reduclion.  We  may  also  remark,  that  the  reduc- 
tion of  the  declination  for  the  minutes  and  tenths  of  a  minute  of  time,  can  be  found  by  means  of  Table 
XXX  :  but  il  is  better  to  do  it  by  the  process  of  muUiplicalion,  as  in  the  rule  ffiven  above. 


TO  FIND  THE   LATITUDE   BY   THE   MOOiN.  171 

This  variation  is  to  be  imiltiplieil  by  the  niiiuites  and  tenths  of  a  minute  which  oecuf 
in  the  time  at  Greenwicli ;  the  product,  being  divided  by  10,  gives  tiie  correction  of 
the  declination  taken  from  the  Nautical  Almanac,  additive  if  that  declination  be 
increasing,  subtractive  if  decreasing  ;  the  sum  or  difference  will  be  the  true  declina- 
tion at  the  time  of  passing  the  meridian. 

NOTES. 

1.  By  the  above  rule,  the  day  of  the  month  on  which  the  moon  passes  the  merid- 
ian must  be  taken  one  less  than  the  sea  account.  When  the  longitude,  turned  into 
time,  is  added  to  the  time  of  passing  the  meridian,  and  the  hours  of  the  same  exceed 
24'',  you  nnist  subtract  24'',  and  add  one  to  the  day  of  the  month  ;  if  the  longitude  be 
sul)tractive,  and  greater  than  the  time  of  passing  the  meridian,  you  nuist,  before  the 
subtraction,  add  24  hours  to  the  time  of  passing  the  meridian,  and  subtract  one  from 
the  day  of  the  month ;  the  sum  or  difference  will  be  the  time  at  Greenwich. 

2.  When  the  declination,  taken  from  the  Nautical  Almanac  for  the  nearest  hour 
preceding  the  time  at  Greenwich,  is  decreasing,  and  the  correction  to  be  subtracted 
exceeds  this  declination,  the  difference  of  the  two  quantities  will  be  the  required 
declination,  with  a  different  name  from  that  of  the  declination  taken  from  the  Nau- 
tical Almanac. 

3.  In  the  same  manner  we  may  find  the  declination  for  any  other  tiuie  of  the  day, 
by  making  use  of  the  given  time  instead  of  the  time  of  the  moon's  passing  the  merid- 
imi.     In  all  these  rules,  the  second  differences  of  the  moon's  motion  are  neglected. 

EXAMPLE. 

Required  the  moon's  declination  at  the  time  of  her  passing  the  meridian  of  Phila- 
delphia, April  19,  1836,  sea  account. 

The  time  of  passing  the  meridian, of  Philadelphia  was  found,  in  the  preceding 
example,  to  be  April  19'  2^  5"" .6  sea  accoiuit,  or  x\pril  18''  2''  5"'.G  by  astronomical 
account;  adding  this  to  the  longitude  of  Philadelphia,  in  time  5''  1'"  nearly,  we  obtain 
the  time  at  Greenwich,  April  18'  7^  G"\G.  The  declination  in  the  Nautical  Almanac 
for  April  18^  7^  is  21°  13'  52"  N.,  and  the  variation  89"  for  10  minutes  of  time 
nearly ;  multiplying  this  by  6'".6,  and  dividing  by  10'",  we  get  59",  to  be  added  to 
21°  13'  52",  because  the  declination  is  increasing,  and  we  obtain  21°  14'  51"  N.  for  the 
required  declination  at  the  time  of  the  moon's  passing  the  meridian  of  Philadelphia. 

To  Jiiid  the   latitude    by   tlie   nnon's   rtieridian    altitude,    obtained  hij   a.  fore 

observation. 

At  the  time  of  the  moon's  passing  the  meridian,  the  altitude  of  her  round  limb 
must  be  observed,  whether  it  be  the  upper  or  lower  limb.  This  altitude  must  be 
corrected  for  the  semidiameter,  dip,  parallax,  and  refraction,  in  order  to  obtain  the 
central  altitude  ;  with  which,  and  tlie  declination,  we  may  find  the  latitude  by  tlie 
same  rules  as  we  have  used  in  finding  die  latitude  from  the  sun's  meridian  altitude. 
In  making  these  calculations,  we  must  find,  from  the  Nautical  Almanac,  the  moon's 
semidiameter  and  horizontal  parallax,  corresponding  to  the  time  of  ol)servation, 
reduced  to  the  meridian  of  Greenwich,  which  was  used  in  computing  the  declination. 
The  moon's  semidiameter  is  to  be  increased  by  the  correction  in  Tal)le  X\^,  and  this 
augmented  semidiameter  is  to  be  added  to  the  observed  altitude,  if  the  moon's  lower 
limb  be  observed;  but  if  the  upper  limb  be  observed,  we  must  subtract  this  augment- 
ed semidiameter  from  the  moon's  observed  altitude,  to  obtain  the  central  altitude. 
I'roin  this  central  altitude  you  must  subtract  the  dip  of  the  horizon,  found  in  Table 
XIII.,  to  obtain  the  apparent  altitude.  The  correction  for  parallax  and  refraction  is 
likewise  to  be  added ;  this  correction  is  easily  found  by  means  of  Ta!)le  XIX.,  by 
subtracting  the  tabidar  number  corresponding  to  the  moon's  altitude  and  horizontal 
parallax  from  59'  42"  ;  the  remainder  will  be  the  correction  for  ])arallax  and  refrac- 
tion,* which  is  to  be  added  to  the  ai)parent  central  altitude,  to  obtaiii  the  true,  altitude  ; 
and,  by  subtracting  this  true  altitude  from  90°,  we  obtain  the  true  zenith  distance. 
With  this  and  tlie  declinat'.on,  we  deduce  the  latitude  by  the  usual  rules,  similar  to 
those  given  for  the  sun  in  pages  1G6,  1G7. 

*  111  computing'  this  tabic,  the  mean  refraction  is  used ;  but,  wlien  very  great  accuracy'  is  required, 
ilie  true  rofraction  ought  to  be  used.  The  corrections  arising  from  this  cause  may  be  obtained  from 
Table  XXXVI.,  and  are  to  be  applied  to  the  above-found  zenith  distance,  with  the  same  signs  as  in 
tiiis  table. 


172 


TO  FIND  THE   LATITUDE   BY    THE   MOON. 


EXAMPLE    L 

Suppose  tliaf,  on  the  27th  of  June,  183G,  sea  account,  in  the  longitude  of  80°  W. 
from  Greenwich,  the  meridian  altitude  of  the  moon's  upper  liujb  was  observed  to  be 
40°  0',  bearing  south,  the  eye  of  the  observer  being  elevated  nineteen  feet  above  the 
surface  of  the  sea;  retpiired  the  true  latitude. 

June  27th,  sea  account,  is  June  2Gth  by  the  Nautical  Almanac  ;  on  this  day  tlte 
moon  passes  the  meridian  of  Greenwich  at  9''  55"'.9,  mean  time,  and  the  next  day  at 
10''  59"'.8,  the  daily  difference  being  G3'".9.  In  Table  XXVIIL,  umler  64'",  (which  is 
the  nearest  number  in  the  table  to  G3'".9,)  and  opposite  to  the  longitude  80°,  stand 
14'" ;  adding  tiiis  to  9''  55'".9,  we  get  10''  09'".9  for  tiie  time  of  passing  the  meridian 
at  the  place  of  observation. 


J)  passes  the  merid June  20'  10''  10"" 

Ship's  long.  80°  W.,  in  time,  5   20 

Time  at  Greenwich. ..  .June  26   15  30 

:D's  decli.  June  26''  15"      23°  37'  43''.2  S. 
Cor.  for  30 "'  is  30  X  8".798  -f     4  23  .9 

Required  declination 23  42  07  .1  S. 

Here  the  variation  of  the  declination  f(3r 
10""  is,  bv  the  Nautical  Almanac,  87".98, 
or  8".798Yor  1'".  ]\Iultiplying  this  l)y  30, 
we  <ret  the  correction  for  30"",  equal  to 
263''r94,  or  4'  23".9,  as  above.  This  is 
additive,  because  the  declination  is  in- 
creasing. For  the  same  time  at  Green- 
wich, we  find  3)'s  hor.  par.  60'  58',  and 


3)'s  scmidiameter  by  N.  A.,         16'  37'' 
;})'s  augmented  somidiam.  16  47 

Alt.  2)'s  upper  limb 40  00   00 

j)'s  semidiameter sub.         16  47 


3)'s  central  altitude 39  43  13 

Dip,  T.XIII.  for  19  feet,  sub.  4  17 

J)'s  apparent  altitude 39  38  56 

59'  42" 

Cor.  T.  XIX. —13  52  diff.add  45  50 

2)'s  true  aUitude 40  24  46 

3)'s  zenith  distance 49  35  14  N 

3)'s  declination 23  42  07  S. , 

Latitude 25  53  07  N. 


EXAMPLE  II. 

Suppose  that,  on  the  27th  September,  1836,  sea  account,  in  the  longitude  of  90°  E., 
th-e  meridian  altitude  of  the  moon's  lower  limb  was  observed  to  be  50°  0',  bearing 
south,  the  eye  of  the  oliserver  being  seventeen  feet  above  the  surface  of  the  sea ; 
required  the  true  latitude. 

Sept.  27th,  sea  account,  is  Sejit.  26th,  astronomical  account ;  on  this  day  the  moon 
passed  the  meridian  of  Greenwich  at  13''  28'".0,  and  the  preceding  day  at  12''  42"'.8, 
differing  45"'.2.  In  Table  XXVIIL,  under  46'",  (which  is  the  nearest  tabular  number,) 
and  O))posite  to  90°,  are  11'",  which,  being  subtracted  from  13''  28'",  leaves  13''  17'" 
for  the  time  of  passing  the  meridian  of  the  place  of  observation.  Subtracting  the 
longitude  6'",  gives  the  corresponding  time  at  Greenwich  Sept.  26 '  7''  17'". 


Sept. 26'  7",  ])'s  declination 

by  N.  A 8°  47'  27"  N. 

Cor.tbrl7'"isl7Xl4".482,  4  06 

Required  declination 8   51   33    N. 

J)'s  hor.  par.  by  N.  A 56'  49" 

3)'s  semidiam.  by  N.  A 15    29 

})'s  aug.  semidiam 15    41 


Obs.  alt.  ])'s  lower  limb. .  50°  00'  00" 

2)'s  semidiam add  15  41 

3>'s  central  altitude 50  15  41 

Dip,  Ta.  XIII.,  for  17  feet,  4  03 

2)'s  aj>parcnt  altitude 50  11  38 

59'  42" 

Cor. T. XIX.— 24    7  diff.add  35  35 

;])'s  true.ahitude 50  47  13 

5's  zenith  distance 39  12  47  N. 

j)'s  declination 8  51  33  N. 

Latitude 48  04  20  N. 


The  latitmle  may  also  be  obtained  from  the  moon's  meridian  altitude,  by  the 
following  ap|)roximative  method,  which  will  vary  but  very  little  Irom  the  truth,  except 
when  the  horizontal  parallax  and  scmidiameter  are  very  large  or  very  small : — 

AhriJged  approximative  method  of  Jinding  the  latitude  by  the  ?noon's  meridian 
altitude,  obtained  by  afore  observation. 

To  the  observed  altitude  of  the  moon's  lower  limb  add  12';  but  if  her  upper  limb 
be  obsei'V'ed,  subtract  20'.     With  this  corrected  altitude  enter  Table  XXIX.,  and 


TO   FIND  THE   LATITUDE   BV   THE    MOON. 


173 


take  out  the  corresponding  number  of  minutes,  which  are  to  be  added  to  tlie  cor- 
rected ahitude  ;  the  sum  will  be  nearly  e(iiial  to  the  true  altitude  of  the  moon  ;  its 
complement  is  the  zenith  distance,  which  is  to  be  used,  as  before,  with  the  moon's 
decUnation,  in  finding  the  latitude,  as  by  a  meridian  altitude  of  the  sun. 

EXAMPLE  III. 

Suppose  that,  on  the  29th  of  November,  1836,  sea  account,  in  the  longitude  of  150" 
VV.,  tlie  meridian  altitude  of  the  moon's  upper  limb  was  observed  00°  2G',  bearing 
north  ;  recjuired  tlie  true  latitude. 

Nov.  29th,  sea  account,  is  Nov.  28th  by  the  Nautical  Almanac  ;  on  this  day  the 
moon  j)asse(i  the  meridian  of  Greenwich  at  16''  33'".1,  and  the  next  day  at  17''  18'".6, 
differing  45'".5.  In  Table  XXVIII.,  under  46"",  (the  nearest  tabular  nundicr,)  and 
opposite  the  longitude  150°,  stands  l^""  ;  adding  this  to  16''  33'",  we  get  16''  52'"  for 
the  time  of  i)assing  the  meridian  of  tiic  ])lace  of  observation  nearly. 

])  passes  the  meridian 28'  16''  52" 

Long.  150°  W.,  in  time 10  00 

Time  at  Greenwich ....  Nov.  29  02  52 


3)'s  dec,  Nov.  29',  2^  . . .  20°  41'  06"  N. 
Cor.  for  52'"  is  52  X  9".6,    —     8   19 

Required  declination 20   32   47    N. 


Obs.  alt.  ])'s  upper  limb 60°  26' 

Subtract 20 


Apparent  altitude 60   06 

Cor.  Table  XXIX add  28 

])'s  true  altitude 60  34 

J)'ti  zenith  distance 29  26  S. 

J)'8  declination 20   33  N. 

Latitude 8  53  S. 


In  this  example,  the  moon's  horizontal  parallax  is  54'  23"  ;  with  this,  and  the 
altitude  60°  6',  we  lind  the  correction  in  Table  XIX.  is  33'  8"  ;  subtracting  this  from 
59'  42",  we  get  the  correction  of  altitude  26'  34",  instead  of  28'  found  above  from 
Table  XXIX.,  making  the  corrected  latitude  8°  54'  26"  S. 

We  shall  now  work  Exami)les  I.  and  II.  by  this  approximative  method. 


EXAMPLE  IV. 

[Su.mo  as  Example  I.] 

Alt.  3)'s  upper  limb 40°  00' 

Subtract 20 

])'s  central  altitude 39   40 

Cor.  Table  XXIX add         43 


3)'s  true  altitude 40  23 

2)'s  zenith  distance 49  37  N. 

2)'s  declination 23  42  S. 

Latitude 25  55  N. 


Differing  about    2'   from   the   correct 
method  "of  calculation  in  Example  I. 


EXAMPLE  V. 

[Same  as  Example  II.] 

Alt.  D's  lower  limb 50°  00- 

Add 12 


J)'s  central  altitude 50   12 

Cor.  Table  XXIX add         36 


])'s  true  altitude 50  48 

3)'s  zenith  distance 39  12  N 

;])'s  declination 8  52  N. 

Latitude 48  04  N 


IJeinj 
pie  II 


nearly  the  same  as  in  Exam- 


174 


TO   FIND  THE  LATITUDE  BY  A  MERIDIAN 
ALTITUDE  OF  A  PLANET. 


The  latitude  may  frequently  be  obtained,  with  great  accuracy,  (particularly  in  tlie 
morning  and  evening,  when  the  horizon  is  well  defined,)  by  observing  the  meridian 
altitude  of  Venus,  Mars,  Jupiter,  or  Saturn.  From  these  altitudes  we  may  find  the 
latitude  by  similar  methods  to  those  we  have  already  given  for  the  sun.  The  times 
of  passing  the  meridian  of  Greenwich,  and  the  declinations  of  these  planets,  are 
inserted  in  the  Nautical  Almanac,  at  every  noon,  at  Greenwich  ;  and,  as  the  daily 
variations  of  these  quantities  are  small,  we  can  find,  by  inspection,  to  a  sufficient 
degree  of  exactness  for  most  nautical  purposes,  the  corresponding  times  of  transit 
and  declijiations  at  the  place  of  observation,  and  thence  the  latitude,  as  in  the  follow- 
ing rule : — 

RULE. 

Find,  in  the  Nautical  Almanac,  the  time  of  passing  the  meridian  en  the  day  nearest 
to  that  in  which  the  observation  is  made  ;  this  will  be  nearly  the  tiniC  of  passing  the 
meridian  at  tlie  place  of  observation.*  Turn  the  ship's  longitude  into  time,  and  add 
It  to  the  time  of  passing  the  meridian,  Avhen  in  west  longitude,  but  subtract  it  in  east ; 
the  sum  or  difterence  will  be  the  time  at  Greenwich,  ncarly.f  Talce,  from  the  Nau- 
tical Almanac,  tiie  planet's  declination  for  tiie  noon  immediately  preceding,  and  for 
that  immediately  following,  the  time  of  observation,  and  note  the  difference  of  the 
declinations  when  they  are  of  the  same  name,  but  their  sum  when  of  different  names; 
this  sum  or  diflercnce  will  be  the  daily  variation  of  declination.  Then  say.  As  24 
hours  are  to  the  daily  variation  of  declination,  so  are  the  hours  and  minutes  of  the 
time  at  Greenwich  to  the  correction  of  the  declination  ;  to  be  applied  to  the  first  dec- 
lination taken  from  the  Nautical  Almanac,  additive  if  the  declination  be  increasing, 
subtractive  if  decreasing ;  the  sum  or  difference  will  be  the  declination  of  the  planet 
at  the  time  of  observation.  But  you  must  observe  that,  if  the  correction  of  declination 
be  greater  than  the  declination  first  marked  in  the  Nautical  Almanac,  their  difference 
will  be  the  sought  declination,  wJiich  will  be  of  a  different  name  from  the  first 
declination. 

From  the  observed  altitude  of  the  planet,  taken  by  a  fore  observation,  subtract  the 
refrartion  and  dip,  the  latter  being,  in  general,  about  4'.  The  remainder,  being 
subtracted  from  U0°,  will  give  the  true  zenith  distance  nearly,  t  with  which,  and  the 
declination,  we  may  find  the  latitude,  as  by  an  observation  of  the  sun. 

EXAMPLE   L 

Suppose  that,  on  the  2:3d  of  October,  1836,  sea  account,  in  the  longitude  of  C5°  \Y., 
J ujjiter  passed  the  meridian  to  the  southward;  the  meridian  altitude  of  his  centre, 
being  observed,  was  45°  20',  and  the  dip  4';  required  the  true  latitude. 

Oct.  23d,  sea  account,  is  Oct.  22d  by  the  Nautical  Almanac  ;  and  on  that  day 
Jupiter  passed  the  meridian  at  19''  5'",  nearly;  adding  the  longitude  G5°,  timied  into 

*  If  we  wish  to  find  the  lime  of  passing'  the  meridian  more  accurately,  we  must  take  a  proportional 
part  of  the  dillerence  of  the  times  of  coming  to  the  meridian  given  in  the  Nautical  Almanac,  in  like 
manner  as  in  finding  the  declination  of  the  planet ;  always  keeping  in  mind,  that  the  time,  according 
to  the  astronomical  compulation,  is  used  in  the  Nautical  Almanac,  and  is  one  day  less  than  the  sea 
account. 

t  This  part  of  the  operation  may  he  avoided,  if  we  have  a  chronometer  regulated  for  Greenwich 
time,  and  note  hy  it  the  time  of  observation. 

t  To  be  strictly  accurate,  we  ought  to  subtract  the  parallax  in  altitude  from  this  zenith  distance.  This 
is  found  in  Table"  X.  A.  Thus,  if  die  horizontal  parallax  of  the  jilanet  be  20",  and  the  altitude  60°,  the 
parallax  in  altitude  bj-  this  table  is  10",  to  be  added  to  tlie  observed  altitude,  or  subtracted  from  the 
observed  zeiiilh  liislaiicc.  'i'hc  centre  of  the  planet  being  observed  there  is  no  correction  for  the  semi- 
diameter  of  the  planet 


TO   FliND   THE   LATITUDE   BY    A    I'LAiNET.  175 

time,  (that  is,  4''  20"",)  we  get  the  correspoiuling  time  at  Greenwich,  l)y  the  Nautical 
Ahnanac,  Oct.  22'  23''  25'"  ;  and,  for  tliis  time,  we  tind  tlie  declination  of  the  planet, 
by  mere  inspection  of  the  Nautical  Almanac,  to  be  1G°  45'  N.,  nearly. 

From  Jupiter's  observed  altitude 45°  2(y 

Subtract  dip  4',  refraction  1' 5 

Leaves  the  true  altitude 45  15 

Whence  the  true  zenith  distance  is 44  45  N. 

Jupiter's  declination 16  45  N. 

Latitude 61   30  N. 

In  this  example  we  have  found,  by  inspection,  the  time  of  passing  the  meridian,  or 
the  declination.  If  greater  accuracy  is  required,  we  must  take  proportional  parts  of 
the  daily  variations,  corresponding  to  the  longitude  of  the  place,  and  the  time  of 
o!)servation.  Tiius,  the  time  of  passing  the  meridian  on  Oct.  22,  by  tiic  Nautical 
Almanac,  is  19''  5"'.4,  and  on  Oct.  23  is  19"  2"'.0,  decreasing  3"'.4  daily,  or  fur  360°  ol 
longitude.  Then,  by  proportion,  we  have  360°  :  3"'.4  : :  65°  :  0"'.6 ;  so  that  the  cor 
rection  of  the  time  of  passing  the  meridian  for  65°  W.  longitude  is  0'".6,  to  be 
subtracted  from  19"  5"' .4,  to  o!)tain  tlie  time  of  passing  the  meridian  in  the  place  o) 
observation,  19"  4"'.8.  Adding  to  this  the  longitude,  turned  into  tiiue,  4"  20'",  we  get 
the  corresponding  time  at  Greenwich,  22'  23"  24 '".S.  Now,  by  the  Nautical  Almanac, 
tlie  declination,  Oct.  22d,  is  16°  47'  17".2  N.,  at  noon,  and  the  next  dav,  16°  45'  18''.1  N., 
at  noon,  differing  1'  59".l,  or  119".l.  Then  say,  As  24"  :  119".!  :':  23"  24'".8  :  116 ' 
or  1'  56",  to  be  subtracted  from  16°  47'  17".2,  to  obtain  the  true  declination,  16°  45'  21' 
nearly,  at  the  time  of  observation.  The  horizontal  parallax,  by  the  Nautical  Almanac 
is  1".56,  which  is  wholly  insensible  ;  and  the  semidiameter  is  18",  whicli  must  b< 
neglectetl  because  the  central  altitude  was  observed.  Hence  we  see  that  these  correc. 
tions  in  the  calculations  produce  but  very  little  change  in  the  resulting  latitude,  an. 
that  die  process  by  inspection  is  sufliciently  accurate ;  and  this  will  be  found  generail; 
to  be  the  case  with  the  planets  Jupiter  and  Saturn. 

EXAMPLE   II. 

Suppose  that,  on  the  17th  of  September,  1836,  sea  account,  in  the  longitude  o* 
75°  E.,  Venus  passed  the  meridian  to  the  northward ;  the  meridian  central  altitude, 
being  observed,  was  26°,  and  the  dip  4' ;  required  the  true  latitude. 

Sept.  17th,  by  sea  account,  is  Sept.  16th  by  the  Nautical  Almanac  ;  and  on  this  day 
Venus  passed  the  meridian  at  20"  59'",  nearly;  subtracting  the  longitude  75°:=  5",  we 
get  Sept.  16^^  15"  59""  for  the  corresponding  time  at  Greenwich.  Now,  by  the  Nau- 
tical Almanac,  the  declination  of  Venus,  at  noon,  Sept.  16',  was  14°  49'  33".7  N.,  and 
the  next  day  14°  44'  22".l  N.,  differing  5'  11".6.  Then  we  have  24"  :  5'  11".6 : :  15"  59'" : 
3'  27".5 ;  su1)tractiiig  this  from  14°  49'  33".7,  we  get  14°  46'  06"  N.,  nearly,  fcr  the 
planet's  ieclination  at  the  time  of  observation. 

From  the  observed  central  altitude  of  Venus. ..  26° OO' 
Subtract  dip  4',  refraction  2' 6 

Leaves  the  true  altitude  nearly 25  54 

Whence  the  true  zenith  distance  is 64  06  S. 

Declination  of  Venus 14  46  N. 

Latitude 49  20  S. 


176 


TO    FIND    THE    LATITUDE    BY    DOUBLE 
ALTITUDES. 


Form  I. — Hy  double  altitudes  of  the  sun. 

When  (by  reason  of  clouds,  or  from  other  causes)  a  meridian  altitude  cannot  be 
olitained,  the  latitude  may  be  found  by  two  altitudes  of  the  sun,  taken  at  any  time  of 
the  day,  the  interval  or  elapsed  time  between  the  obsei-vations  being  measured  by  a 
good  watch  or  ciu-onometer,  noticing  the  seconds,  if  possible,  or  estimating  the  times 
to  a  third  or  a  quarter  of  a  minute,  if  the  watch  is  not  furnished  with  a  second-hand. 
The  observed  altitudes  of  the  sun  must  be  corrected,  as  usual,  for  the  semidiameter, 
dip,  refraction,  and  parallax,  in  the  same  manner  as  in  finding  the  latitude  by  a  merid- 
ian altitude.  When  great  accuracy  is  required,  the  declination  must  be  found  at  tlie 
time  of  each  observation,  using  the  third  method  of  solution  hereafter  given  ;  but 
when  the  sun's  declination  varies  slowly,  or  the  elapsed  time  is  small,  it  will  in 
general  be  sufliciently  accurate  to  find  the  sun's  declination  for  the  middle  time  between 
two  observfttio7is,  and  to  consider  it  as  invariable  during  the  observations,  computing 
the  latitude  by  the  first  or  second  method. 

Tills  manner  of  finding  the  latitude  is,  in  general,  most  to  be  depended  upon  where 
the  sun's  meridian  zenith  distance  is  great.  If  the  sun  passes  the  meridian  near  to 
he  zenitli,  much  greater  care  must  be  taken  in  measuring  the  altitudes  and  noting 
the  times,  than  would  be  necessary  under  other  circumstances.  The  nearer  the  sun 
is  to  the  meridian,  at  the  time  of  one  of  the  observations,  the  more  correct  the  result 
will  connnoiily  be.  In  general,  the  elajised  time  ought  to  be  as  great,  or  greater,  than 
the  time  of  the  nearest  observations  from  noon.  Similar  remarks  may  be  made  upon 
every  one  of  the  following  forms. 

In  all  these  observations  it  is  supposed  that  the  watch  moves  uniformly  according 
to  apparent  time,  measuring  twenty-four  hours  from  the  time  of  the  sun's  passing  the 
meridian  on  two  successive  days  at  the  same  place  of  observation.  If  the  watch 
gain  or  lose  on  apparent  time,  supposing  the  observer  to  be  at  rest,  a  correction  must 
be  apjilied  for  the  gain  or  loss  during  the  time  elapsed  between  the  observations,  so 
as  to  ol)tain  accurately  the  elapsed  time  or  hour  angle.  It  is  not  required  that  the 
watch  shoidd  be  regulated  so  as  to  give  precisely  the  ^our  of  observation ;  the  only 
thing  ree^uired  is  to  find  the  elapsed  time  with  all  possible  accuracy. 

Form  TI. — Double  altitudes  of  a  star. 

Doulde  altitudes  of  a  fixed  star  may  be  used  in  finding  the  latitude,  and  the  calcu- 
lation is  almost  identical  with  that  of'double  altitudes  of  the  sun  ;  the  only  difference 
consists  in  adding  a  small  correction  to  the  elapsed  mean  solar  time  between  the 
observations,  on  account  of  the  daily  acceleration  of  3'  5G"  in  the  time  a  star  comes 
to  the  meridian  on  successive  days  ;  in  other  words,  the  elapsed  time  (or  hour  angle) 
must  be  reckoned  in  sideral  time,  of  which  we  have  already  spoken  in  the  second 
note  on  page  147.  Now,  as  a  chronometer  is  usually  adjusted  to  mean  solar  time, 
and  the  observations  marked  by  it,  we  must  add  to  the  mean  time,  elapsed  between 
the  observations,  the  correction  given  in  Table  LI.,  to  reduce  it  to  sideral  time. 
Thus,  if  the  interval  in  mean  solar  time  be  3'',  the  corresponding  correction  in  this 
table  is  -{-  2'J\G,  making  the  interval  in  sideral  time  (or  the  correct  hour  angle) 
3''  00""  2[)'.G,  which  is  to  be  used  in  the  rest  of  the  calculation. 

In  observations  of  a  fixed  star,  the  altitudes  are  to  be  corrected  for  dip  and  refrac- 
tion, as  in  finding  the  latitude  by  a  meridian  altitude.  The  declination  of  the  star  is 
to  be  found  in  Table  VIII.*     With  these  altitudes,  the  declination,  and  the  hour 

*  Or  more  acciiraicly  in  the  NaiUical  Almanac,  if  any  one  of  the  bright  stars  is  observed  whosa 
place  is  given  in  liial  worL'. 


TO   FIND   THE    LATITUDE   BY   DOUBLE   ALTITUDES.  177 

aii."-lc,  the   calculation   is   to   be   made   by  cither   of  the   three   mctliods   hereafter 
given. 

The  chief  difficulty,  in  observations  of  this  kind,  with  a  fixed  star,  is  the  want  of  a 
good  horizon  in  tlie  night-time.  The  method,  however,  migiit  sometimes  be  used 
with  success,  soon  after  the  dawn  of  day,  or  hue  in  the  evening  twilight,  at  a  time 
when  the  horizon  is  well  defined,  and  the  star  sufficiently  bright  to  bring  its  reflected 
image  lo  the  horizon.  Sqinetimes  a  good  horizon  is  produced  by  the  aurora  borealis, 
\n  which  case  a  good  observation  might  be  made  with  stars  in  the  northern  horizon  ; 
but  a  single  observation  of  the  polar  star  will  answer  the  same  purpose,  and  will  be 
nmch  more  simjjle. 

Form  III. — Double  altitudes  of  a  planet. 

Double  altitudes  of  a  planet  (particularly  Jupiter  and  Venus,  on  account  of  their 
great  brightness)  may  sometimes  be  used  with  success.  The  observed  altitudes  must 
be  corrected  for  dip  and  refraction.  The  parallax  and  scmidiameter,  being  small, 
may  be  neglected,  except  in  cases  where  extreme  accuracy  is  required.  The  declina- 
tion of  the  planet  is  to  be  found,  in  the  Nautical  Almanac,  ibr  the  sujiposcd  time  at 
Greenwich.  The  daily  variation  of  the  time  of  coming  to  the  meridian  is  also  to  be 
found  in  the  same  page  ;  and  thus  the  time  elapsed  between  the  passage  of  the  ])lanet 
over  the  meridian  on  two  successive  days  is  found  ;  then  the  corrected  elaiKsed  time, 
or  hour  angle,  is  obtained  by  the  following  rule : — 

Rule,  ^s  the  interval  of  time  between  two  successive  passages  of  the  ohjed  over  the 
mendian  is  to  twenlijfour  hours,  so  is  the  elapsed  mean  time  between  the  observations  lo 
the  corrected  elapsed  time,  or  hour  angle. 

With  this  hour  angle,  the  declination,  and  corrected  altitudes,  the  latitude  may  be 
found  by  either  of  the  three  following  methods  of  calculation. 

Form  IV. — Douhle  altitudes  of  the  moon. 

Double  altitudes  of  the  moon  may  also  be  used  in  finding  the  latitude.  These 
observations  may  be  easily  and  very  accurately  made  ;  but  the  calculation  is  much 
more  complex  than  any  of  the  preceding  methods,  on  account  of  the  great  change  in 
the  moon's  declination  and  right  ascension  during  the  elapsed  time  betw^een  the 
observations.  If,  however,  by  the  times  of  observation,  and  the  longitude  of  the  ship, 
(or  else  by  a  chronometer,)  the  time  at  Greenwich  can  be  obtained  within  a  few 
minutes,  we  may,  from  the  Nautical  Almanac,  find  the  corresponding  declination, 
semidiamcter  and  horizontal  parallax  of  the  moon  for  each  of  these  observations. 
With  the  horizontal  parallax,  and  the  moon's  apparent  altitude,  find  the  correction  in 
Table  XIX.,  which,  being  subtracted  from  59'  42",  leaves  the  correction  of  the  moon's 
altitude  for  parallax  and  refraction  ;  *  this  is  to  be  added  to  the  corresponding  observ- 
ed altitude,  corrected  for  scmidiameter  and  di]),  to  obtain  the  moon's  correct  central 
altitude.  This  is  to  be  done  at  each  observation.  Lastly,  the  time  of  the  moon's 
passing  the  meridian  on  successive  days,  given  in  the  Nautical  Almanac,  shows  the 
interval  of  tune  between  two  successive  passages  of  the  moon  over  the  meridian,!  and 
this  time  is  to  twenty  four  hours  as  the  elapsed  time  between  the  observations  is  to  the 
corrected  elapsed  time  or  hour  angle.  With  this  hoiH*  angle,  the  correct  central  altitudes, 
and  tiie  declinations,  the  latitude  may  be  found  by  the  fourth  of  the  folJovviiiic  methods 
of  calcidation,  it  being  very  rare  that  the  other  methods  can  be  used,  on  account  of 
the  great  change  in  the  moon's  declination. 

FoRJi  V. — Uy  altitudes  of  two  different  objects,  taken  at  the  same  time. 

The  latitude  may  be  obtained  by  observing,  at  the  same  moment  of  time,  the  altitudes 
of  two  heavenly  bodies  ;  as,  for  example,  (I)  The  sun  and  moon  ;|  (2)  The  moon  and 
a  fixed  star  or  planet ;  J  (3)  A  planet  and  a  fixed  star ;  (4)  Two  planets  ;  (5)  Two  fixed 

*  When  cxtipmc  accurac}'  is  not  required,  we  may  find  the  correction  for  parallax  and  refraction 
from  Table  XXJX.,  wliicli,  if  the  altitudes  are  large,  will  not  var^'  much  from  the  truth. 

t  This  time  is  "(iven  to  tenths  of  a  minute,  which  in  g-encral  is  sufficient,  because,  if  the  elapsed  time 
be  small,  the  ellect  of  tliis  correction  will  be  only  a  few  seconds.  It  might  be  obtained  more  accurately 
by  means  of  the  right  ascensions  of  the  sun  and  moon,  using  the  second  differences,  as  taught  in  the 
Appendix. 

X  A  particular  case  of  this  method  occurs  in  taking  n  lunar  observation,  which  will  be  treated  of  sep 
arately,  because,  the  distance  of  the  two  bodies  being  known,  liie  calculation  becomes  more  simple. 

23 


178  TO   FIND   TPIE   LATITUDE   BY   DOUBLE   ALTITUDES 

stars.  Ill  tliese  metliods  tlie  altitudes  are  to  he  corrected,  as  in  the  preceding  Forms, 
for  dip  and  refraction  ;  also  for  parallax  and  seniidiameter  when  necessary,  as  is 
always  tlis  case  in  observations  of  the  moon  and  sim.  The  declinations  of  t!ie  bodies 
are  to  be  found  for  the  supposed  time  of  observation,  reduced  to  the  meridian  of 
Greenwich,  by  means  of  the  Nautical  Almanac,  or  by  Table  VIII.  for  the  fixed  stars, 
as  before  taught.  Then  the  difference  of  the  right  ascensions  of  the  bodies  (or  that 
difference  subtracted  from  24  hours,  if  it  exceed  12  hours)  will  bo  the  hour  angle, 
which  is  to  be  used,  with  these  declinations  and  coirected  altitudes,  in  finding  the 
latitude,  by  either  of  the  three  first  methods,  if  the  declinations  shoidd  be  equal,  or 
differ  but  one  or  two  minutes ;  otherwise  by  the  fourth  method,  which,  in  fact,  njay 
be  considered  as  the  only  method  to  be  used  in  this^  kind  of  observations,  because,  in 
almost  all  cases,  the  declinations  of  the  objects  differ  considerably. 

For.M  VI. — JBi/  altitudes  of  two  different  ohjccts,  talccn  within  aflio  ?.':iniitcs  of 
each  other,  hy  one  observer. 

It  may  sometimes  happen,  for  want  of  two  good  instrun)ents,  or  from  not  having 
two  observers,  that  the  preceding  Form  V.  cannot  be  employed.  In  this  case  the 
whole  of  the  observations  may  be  made  by  one  person,  noticing  the  interval  between 
the  observations,  and  making  the  calculation  as  in  the  following  Form  VII.  But  it  is 
in  general  much  better  to  make  the  observations  as  near  to  each  other  as  possible, 
and  then,  by  a  very  simple  process,  the  calculation  may  be  reduced  to  that  of  Form  V., 
in  which  the  observations  are  taken  at  the  sftme  momtnt.  This  is  done  by  observing 
tlie  first  object  twice,  before  and  after  observing  the  second  object.  For  if  the  intervals 
of  time  between  these  three  observations  be  equal,  (as,  for  example,  one  minute,  or 
two  minutes,)  the  half-sum  of  the  two  altitudes  of  the  first  object  jnay  be  taken  for 
the  altitude  corresponding  to  the  time  of  observing  the  second  altitude,  and  the 
calculation  may  then  be  made  as  in  Form  V.  Thus,  suppose  at  10''  2"',  A.  M.,  per 
watch,  tlie  altitude  of  Sirius  was  17°  54',  at  10''  4™  per  watch  the  altitude  of  Capella 
GO-  45',  and  at  10''  G™  per  watch  the  altitude  of  Sirius  was  again  observed  and  found 
to  be  17°  58'.  In  this  case,  the  intervals  of  time  are  exactly  two  minutes  ;  therefore 
tlie  hall-sum  of  the  altitudes  of  Sirius  is  to  be  taken  17°  5G',  and  combined  witli  the 
altitude  of  Ca])e]la  C0°  45',  supposing  both  to  have  been  observed  at  10''  4'"  per 
watch.  This  is  the  most  simple  form  in  which  an  observation  of  this  kind  can  be 
made  by  one  observer. 

If,  from  any  cause  whatever,  the  observations  cannot  be  taken  at  exactly  equal 
intervals,  the  altitude  of  the  first  object,  at  the  time  of  observing  the  second  object 
may  be  found  by  proportion,  supposing  the  altitudes  to  vary  uniformly  during  the 
few  minutes  of  the  observations.  Thus,  in  the  preceding  example,  supjiose  the 
altitudes  and  the  two  first-noted  times  to  remain  unaltered,  but  the  last  observation 
of  Sirius  to  have  been  at  10''  10""  per  watch,  instead  of  10''  G"".  In  this  case,  during 
the  eight  minutes  of  time  elapsed  between  lO"*  2'"  and  10''  10"",  Sirius  would  have 
risen  4',  (from  17°  54'  to  17°  58' ;)  tlierefore,  by  proportion,  it  is  found  that  in  two 
minutes  (the  time  elapsed  between  10''  2'"  and  10''  4"')  the  star  would  have  risen  1', 
and  the  altitude  would  liave  increased  from  17°  o4'  to  17°  55';  therefore,  at  the  time 
10''  4"'  per  watch,  the  altitude  of  Sirius  must  be  taken  at  17°  55',  the  altitude  of  Ca- 
pella G0°  45',  and  with  these  quantities,  considered  as  observed  at  this  last-mentioned 
time  10''  4'",  the  calculation  must  be  made  as  iji  Form  V. 

There  are  several  advantages  attending  these  tv/o  last  forms  V.,  Yl.,  since  no 
allowance  is  necessary  for  the  change  of  place  of  the  ship;  the  observations  can  be 
immediately  made,  in  a  short  interval  of  fair  weather,  when  the  common  method  of 
dou!)le  altitudes  might  fail  from  the  intervention  oC  clouds  ;  the  time  can  also  be 
obtained  at  the  same  operation,  &c.  . 

Form   VII. — Bj/  altitudes  of  tiro  diffrre:it  ohjcrts,  inkcn  at  different  times. 

This  method  differs  but  very  litde  from  the  two  last.  The  altitudes  are  to  be 
corrected,  in  the  same  manner,  for  dip  and  refraction ;  also  for  parallax  ami  semi- 
diameter,  when  necessary.  The  right  ascension  and  declination  of  each  object  is  to 
be  found  for  the  supposed  time  of  observing  that  object  reduced  to  the  mci'idian  of 
Greenwich.  Then  the  apparent  elapsed  time  between  the  observations,  is  to  be 
turned  into  sidcral  time,  which  may  be  done,  as  in  Form  II.,  by  adding  the  correct 
tion  in  Table  LI.  corresponding  to  this  time ;  add  this  sidereal  time  to  tlie  right 
ascension  of  the  body  first  observed ;  the  difference  between  this  sum  and 
the  right  ascension   of  the  body  last  observed  is  the  hour  angle.*     This,  witli  the 

*  If  tlijs  Qifference  exceed  1*2  hours.  s:iMr:iri  ii  frmn  "^l  !:oi;rs.  nnd  u?e  the  remainder  as  in  Form  V 


TO   FIND   THE   LATITUDE   BY   DOUBLE  ALTITUDES.  179 

declinations  and  corrected  altitudes,  is  to  be  vised  in  finding  the  latitude  by  the  third 
or  fourth  of  the  following  methods  of  calculation,  it  being  very  rarely  the  case  that 
the  first  or  second  methods  can  be  used,  on  account  of  the  ditference  of  the  ileclina 
tions.  These  three  last  forms,  when  a  fixed  star  or  planet  is  used,  are  restricted  very 
much  from  the  want  of  a  good  horizon  in  the  night ;  they  are  best  adapted  to  the 
morning  and  evening  twilight. 


GENERAL  RE3IARKS. 

Having  thus  explaine-d  several  of  tlie  different  forms  of  making  tliese  ol)servation;», 
and  the  manner  of  finding  in  each  Ibrm  tlic  hour  migle,  the  dedincUions,  v.m\  tJie  correct 
central  altitudes,  we  shall  now  give  foiu*  difterent  methods  of  calculating  the  latitude, 
and  shall  illustrate  the  rules  by  proper  examples.  In  the  frst  and  second  methods,  the 
declination  is  supposed  to  be  the  same  at  both  observations,  which  is  true  as  it  respects 
observations  of  a  fixed  star,  and  is  in  general  sufficiently  correct  for  common  observa 
lions  of  double  altitudes  of  the  sun.  The  first  of  these  methods  is  direct  and  simple, 
not  embarrassed  with  much  variety  of  cases,  requiring  only  ten  openings  of  the  Table 
XXVIL,  without  any  hah  ing  or  doubling  of  the  logarithms,  or  the  use  of  natural  or 
versed  sines.  This  method  is  in  fact  nearly,  if  not  fully,  as  short  as  the  second  or 
approximative  method  invented  by  IMr.  Douwcs,  and  which  was  exclusively  used  in 
the  former  editions  of  this  work.  Tiiis  second  (or  Douwes')  method  is  liable  to  the 
objection  that  the  calculation  must  sometimes  be  repeated  several  times  befoi-e  a  true 
solution  can  be  obtained,  and  then  it  becomes  extremely  troublesome.  This  difficulty 
does  not  occur  in  the  first  method  ;  and  on  this  account,  as  well  as  for  its  remarkable 
simplicity,  the  first  method  is  always  to  be  preferred. 

The  third  method  is  ai)plicable  to  cases  where  there  is  a  small  variation  in  the 
declination  of  the  object,  during  the  elapsed  time  between  the  observations,  as  most 
commonly  happens  when  the  sun  is  used.  This  mctliod  is  short  and  simple,  and  is 
much  facilitated  by  the  use  of  Table  XLVI.,  which  I  have  computed. 

The  fourth  method  embraces  the  general  solution  of  the  problem  in  the  case  where 
any  variation  whatever  of  declination  is  noticed.  This  increases  the  labor  consid- 
erably, and  renders  the  solution  more  complex  in  its  cases.  It  i?:,  however,  believed, 
that  this  method,  drawn  up  in  its  present  form  by  the  author  of  this  work,  will  be 
easily  understood  by  navigators,  and  that  they  will  thus  be  enabled  to  determine  the 
latitude  with  considerable  accuracy  in  cases  where  it  might  be  of  the  utmost  impor- 
tance to  know  it,  and  where  other  methods  could  not  be  resorted  to  on  account  of  bad 
weather.  This  method  is  nearly,  if  not  quite,  as  short  as  that  published  by  Dr. 
Brinkley  in  the  Nautical  Almanac  of  1825,  and  does  not  require,  like  his  method,  a 
second  or  third  (or  even  a  greater  number)  of  operations. 

If  the  observer  should  change  his  place  or  station,  during  the  elapsed  time  between 
the  observations,  a  correction  must  be  applied  to  one  of  the  altitudes  on  this  account. 
The  manner  of  doing  tliis  is  shown  in  the  following  examples. 

It  may  be  observed  that  in  like  manner  as  there  are  two  latitudes  corresponding  to 
tiie  same  meridian  altitude  of  the  sim,  according  as  the  zenith  is  north  or  south  of  the 
Sim  when  on  the  meridian,  so  in  double  altitudes  there  are  generally  two  latitudes, 
corres]>onding  to  the  proposed  altitudes,  according  as  the  zenith  and  north  pole  are  on 
the  same  side,  or  on  different  sides,  of  the  arc  or  great  circle  passing  through  the 
two  observed  bodies,  or  through  the  two  places  of  the  game  bpdy;  and  it  therefore 
becomes  necessar}^  to  notice,  at  the  time  of  observation,  how  tlie  zenith  and  north 
j)ole  are  situated  with  respect  to  this  great  circle. 


To  estimate  the  effect  of  small  errors  in  the  observations- 

When  running  in  with  the  land,  or  crossing  a  dangerous  parallel  with  no  other 
means  of  obtaining  the  latitude  than  by  double  altitudes,  it  becomes  a  matter  of  great 
imjiortance  to  ascertain  the  possible  error  of  the  latitude  thus  coniputed,  arising  from 
suj)posed  errors  in  the  observed  altitudes,  or  in  the  elapsed  time.  The  differential 
expressions  in  spherical  trigonometry  afford  mediods  of  doing  this  ;  but  they  are  not 
adapted  to  the  nature  of  this  work,  on  account  of  the  complication  and  variety  of 
cases.  The  following  method,  though  long,  is  general  and  infallible,  and  was  once 
used  by  the  writer  in  a  case  of  gi-eat  anxiety  and  danger. 

Rule.  After  having  computed  the  latitude  by  either  of  the  four  following 
methods,  using  the  observed  altitudes  *  and  elapsed  time,  repeat  the  operation,  varying 

*  Tli:il  is,  the  observed  altitudes,  corrected  as  usual  for  dip,  refraction,  parallax,  and  semidiametcr, 
If  uecessarj-. 


180 


TO  FIND   THE   LATITUDE   BY   DOUBLE   ALTITUDES. 


the  altitude  you  suspect  may  be  erroneous  by  2'  or  3',  (or  whatever  you  suppose  tlie 
limit  of  the  error  in  that  altitude  msiy  be  ;)  the  difference  between  this  second  latitude 
and  that  first  computed,  is  the  effect  of  the  suj)posed  error  in  that  altitude.  If  you 
suspect  the  second  altitude  also  to  be  erroneous,  the  operation  may  be  again  repeated, 
varying  this  second  altitude  2'  or  3',  (or  whatever  the  limit  may  be  sii|jposGd,)  but 
using  the  first  observed  altitude  and  elajjsed  time  ;  comparing  this  third  comjjuted 
latitude  with  the  Jirst,  the  difference  is  the  effect  of  this  supposed  error  in  the  second 
altitude.  Finally,  if  the  elapsed  time  is  suj:)posed  to  be  erroneous,  the  o])eration  may 
be  again  repeated,  using  the  observed  altitudes  and  varying  the  elapsed  time  by  20  or 
30  seconds,  (or  whatever  the  limit  of  this  error  may  be  supposed  ;)  the  difference 
between  this  fowth  latitude  and  that  _^rsf  computed  is  the  effect  of  this  su})j)0scd  error 
of  the  elai)sed  time. 

Thus,  snjipose  the  first-computed  latitude  was  30^,  the  second  30°  1',  the  third 
30°  3',  the  fourth  30°  2' ;  tlie  error  arising  from  the  first  altitude  would  be  V,  tliat  from 
the  second  altitude  3',  and  that  from  the  elajjsed  time  2'.  If  all  these  errors  existed  at 
the  same  time,  the  greatest  limit  of  the  error  would  be  the  sum  of  these  quantities  (or 
G'),  so  that  the  true  latitude  would  be  30°  ±  6',  or  between  21)°  54'  and  30°  G'.  In  this 
way  the  limit  of  the  error  may  be  obtained  in  any  case,  and  the  degree  of  confidence 
that  may  be  placed  in  the  observation  obtained.  This  examination  is  sometimes  very 
necessary,  because  the  objects  may  be  so  situated,  that  a  small  error  in  the  observa- 
tions might  produce  a  considerable  change  in  the  comjJUted  latitude.  It  may  be 
observed  tiiat  tlie  error  of  one. observation  is  frequently  corrected,  in  whole  or  in  part, 
by  the  error  of  the  other  ;  the  one  tending  to  increase  the  latitude,  the  other  to 
decrease  it. 


FIRST    METHOD. 

To  find  the  latitude  by  double  altitudes  of  the  sun,  or  any  other  object,  the 
declination  being  invariable. 

In  this  method,  the  log.  sines,  cosines,  &c.,  of  Table  XXVII.  are  used  ;  atid,  for 
brevity,  the  word  log.  is  omitted  in  the  rule.  For  the  convenience  of  writing  down 
at  once,  in  the  same  line,  all  the  logarithms  which  occur  at  the  same  opening  of  the 
book,  they  are  arranged  in  three  columns,  as  in  the  following  formula  ;  and  it  will  be 
very  convenient  to  have  one  of  these  blanks  prejiared  at  the  connnencement  of  the 
operation,  and  then  the  logarithms  may  be  written  down,  in  their  proper  places,  with 
great  raj)idit3^ 

FORMULA. 

CoL.  1.  Col.  2.  Col.  3. 

Elapsed  time,  [p.  ji.]  Cosec. 

.Coscc. 


Declination Secant  

A Coscc.  Cosine 

Half-sum  alts Cosine  Cosec. 

Half-diff.  alts Sine  Sec. 

• 

C Sine  Cosine 

[Z  less  tliin  90°  north  or  south,  like  the  bearing  of  zenith.]  SoC. 

\y.  is  the  sum  of  B,  Z,  ifof  the  same  name  ;  difference,  ifof  a  different  name.] 


.  Cosine 
Cosec. 


(B  less  tlian  90°,  liks 
liccliaaliun   N.  or  S.] 


Latitude 


,  Cosine 

Sine 
Sine 


RULE.     (Using  Table  XXVII.) 

1.  Find  the  elaj)sed  time*  in  column  P.  M. ;  take  out  the  corresponding  cosecant, 
and  put  it  in  Col.  I. 

2.  Put  die  secant  of  the  declination  in  Col.  1 :  its  cosecant  in  Col.  .3. 

3.  The  sum  of  the  logarithms  in  Col.  1  (rejecting  10  in  the  index)  is  the  cosecant 
of  the  angle  A,  whose  cosine  is  to  be  put  in  Col.  2  and  Col.  S.f 

4.  The  sum  of  the  logarithms  in  Col.  3  (rejecting  10  in  the  index)  is  the  cosecant 
of  the  angle  13,  (less  than  1)0°,)  which  is  to  be  named  noHh  or  south,  like  the  declination. 

*  If  any  ollior  ohjcct  tlian  llic  sun  is  observed,  the  corrected  elapsed  time,  or  Jiour  angle,  found  as 
before  tnug^lii,  is  to  i)e  used. 

t  Tlic  cosines  of  A  and  C  arc  each  wriUen  down  twice,  which  reduces  the  number  of  logarithms  in 
eacji  example  from  17  to  13. 


TO   FIND   THE   LATITUDE  .BY   DOUBLE  ALTITUDES. 


181 


5.  Find  liulf  tlie  5iim  of  tlie  two  altitudes;  place  its  cosine  in  Col.  1,  its  cosecant 
in  Col.  2.  Find  also  half  the  difl'erence  of  the  two  altitudes  ;  place  its  sine  in  Col.  1, 
its  secant  in  Col.  2. 

G.  The  sum  of  tlie  three  lower  logarithms  of  Col.  1  (rejecting  20  in  the  index)  is 
the  sine  of  the  angle  C,  whose  cosine  is  to  be  placed  in  Col.  2  and  Col.  3.* 

7.  The  sum  of  the  logm-iduns  in  Col.  2  (rejecting  30  in  the  index)  is  the  secant  of 
the  zenith  angle  Z,  which  is  to  be  taken  out  (less  than  90°)  and  placed  under  B,  in 
Qol.  3,  naming  it  north  if  the  zenith  and  north  pole  be  situated  on  the  same  side  of  the 
arc  or  great  circle  pa.ssing  through  the  two  observed  places  (or  objects),  but  south  if 
the  zenith  and  north  jioie  be  situated  on  different  sides  of  that  great  circlcf 

8.  The  angle  E  is  found  by  taking  the  sum  of  the  angles  B,  Z,  if  they  are  of- the 
same  name,  or  their  difference  il'  of  different  names,  marking  E  north  or  south,  like  the 
greatest  of  the  two  angles  B  or  Z.| 

9.  Put  the  sine  of  E  in  Col.  3,  and  the  sum  of  the  two  last-written  logarithms  of 
Col.  3  (rejecting  10  in  the  index)  is  the  sine  of  the  latitude,  of  the  same  name  as  E. 

If  the  time  of  observation  were  requu-ed,  it  might  be  foimd  by  the  following  rule, 
6tJll  using  Table  XXVII.:— 

Rule.  Add  the  tangent  of  C  to  the  secant  of  E  ;  the  sum  (rejecting  10  in  the 
index)  is  the  tangent  of  an  angle.  Take  out  half  the  corre.s])onding  time  in  Col. 
P.  M.,  (or  in  Col.  A.  M.,  increas,jd  by  12  hours,)  and  this  will  represent  the  horary 
distance  of  the  object  from  the  meridian  (uj)|)er  or  lower)  at  the  nfiddle  time  between 
the  two  observations.  Take  the  sum  and  dift'erence  between  this  and  half  the  elapsed 
time,  or  horn-  angle,  and  they  will  be  the  hours  and  minutes  distance  from  the  meridian 
corresponding  to  both  observations,  expressed  in  apparent  solar  time  if  the  sun  be 
observed,  sideral  time  if  a  star  is  observed,  &c. 

EXAMPLE   I. 

Being  at  sea,  in  latitude  40°  30'  N.  by  account,  when  the  sun's  declination  was 
11°  17'  N.  at  10''  2'"  per  watch,  in  the  forenoon,  the  sun's  correct  central  altitude  was 
46°  55',  and,  at  ll*"  27'",  per  Avatch,  in  tlie  forenoon,  the  correct  central  altitude  was 
54°  y  ;  re(juired  the  true  latitude. 

Subtracthig  10"  2'"  from  11"  27™  gives  the  elapsed  time  1"  25™. 


CoL.  1. 

El.  time  [p.m.]  1"25'",  Cosec.  10.73429 
Declination  11°  17'  N.   Sec.  10.00848 


Col.  2. 


CoL  3. 


A Cosec.  10.74277 

I  sum  alts.  50  32.. Cosine  9.80320 
Adifllalts.  3  37  ...Sine  8.79990 
C Sine  9.34587 


(Z  less  tl.an  90°,  ai  (1  N.  or  S.  like  bearing  of  zenilh.)  ScCaUt     10.09509        Z  3G    33  N 

[K  !s  IIk'  eum  of  B,  Z,  if  of  Ihc  same  name ;  dij'erence  if  of  a  dlferent  name.] 


Cosine  9.99278 

Cosec.  10.11239 

Secant  10.00087 

Cosine  9.98905 


.Cosec.  10.70850 
.Cosine    9.99278 


B  1  r  28' N.  Cosec.  10.70128 

[B  Ifss  ihan  90",  named 
N.  orS.  likcdecliii.J 

Cosine    9.98905 


E48  01  N.   Sine     9.87119 


Latitude  46  27  N.   Sine    9.86024 

If  the  sun  had  passed  the  meridian  to  the  north  of  the  observer,  Z  would  have 
been  3(i°  33"  S.,  and  E  =:  25°  5'  S.,  whose  sine  9.G2730,  added  to  cos.  C  9.98905, 
gives  the  sine  of  the  latitude  9.GIG3.5,  coi-responding  to  24°  25'  S. 

Li  the  iirst  case,  (in  north  latitude,)  the  tangent  of  C  9.35682,  added  to  the  secani 
E  10.174().3,  gives  9.53145,  which,  in  the  tangents,  corresponds  to  2''  30'"  12%  nearly, 
whose  half,  1"  15"'  6%  is  the  time  of  the  middle  observation  from  noon  ;  adding  and 
subtracting  half  the  elap.sed  time,  42'"  30%  gives  the  times  of  the  observations  from 
noon  1"  57'"  36'  and  O"  32'"  36'. 


*  'J'lie  cosines  of  A  and  C  arc  eacli  writleii  clown  twice,  which  reduces  the  number  of  logarithms  in 
each  example  from  17  to  15. 

t  In  observalions  of  the  sun,  the  angle  Z  may  in  general  be  called  north,  if  the  zenith  be  north  of  the 
Eun  when  ou  the  meridian  at  its  greatest  altitude  ;  but  south  if  the  zenith  be  then  south  of  the  sun. 
When  the  object  passes  the  meridian  near  the  zenith,  it  may  be  doubtful  whether  il  be  noiili  or  south, 
ill  wliicli  case  the  latitude  niay  be  computed  upon  both  suppositions,  and  that  one  selected  which 
agrees  best  with  the  estimated  place  of  the  ship;  and  this  e.xira  labor  is  very  small.  Rut  observations 
on  an  object  passing  near  the  zenith  are  liable  to  great  errors,  and  had  better  be  rejected, 

X  This  case  is  easily  remembered,  because  s  is  the  first  letter  of  same  and  staii,  and  d  the  first  V ttcr 
of  different  aiid  difference. 


182 


TO    FIND    THE    LATITUDE -BY    DOUBLE    ALTITUDES. 


EXAMPLE   n. 

At  sea,  in  tlie  latitude  of  47°  19'  N.  by  account,  when  the  sun's  declination  was 
12°  IG'  N.,  at  10''  24'"  A.  M.,  per  watch,  the  sun's  correct  central  altitude  was  49°  tX  ; 
at  1''  14'"  P.  M.,  per  watch,  his  correct  central  altitude  was  51°  59' ;  required  the 
latitude. 

Subtracting  lO*"  24"^  from  l""  14"'  increased  by  12'',  leaves  the  elapsed  time  2''  50'". 


Col.  1. 
El.  time  [p.m.] 2"  50'^',  Cosec.  10.44077 
Declination  12°  IC  N.   Sec.  10.01003 

A Cosec.  10.45080 

^  sum  alts.  50  34  .  .Cosine  9.80290 
h  diff.  alts.  1  25  . . .  .Sine  ^.39310 
C Sine     8.G4G80 

|Z  Ies3  than  90°  and  N.  or  S.,  like  bearing  of  zeuilh.] 


Col.  2. 


Col.  3. 


Cosine  9.97089 

Cosec.  10.11218 

Secant  10.00013 

Cosine  9.99958 


Secant  10.08278 


Cosec.  10.67272 

Cosine    9.97089 

B13°08'N.Cosec.  10.G43G1 

(Tl  lesslhan  Bli".  mjn.i' 
N.  orS.  lilie  declcu.l 

Cosine     9.99958 


Z  34  16  N. 


'.  of  B,  Z,  if  of  the  same  name  ;  difference  if  of  a  different  name.] 


E47  24N.   Sine     9.8GG94 
Latitude  47  20  N.    Sine    9.8GG52 


If  the  sun  had  passed  the  meridian  to  the  north  of  the  observer,  Z  would  have 
been  34°  16'  S.,  E  r=  21°  08'  S. ;  its  sine  9.55695,  added  to  cosine  C  9.99958,  gives 
9.55653,  the  sine  of  the  latitude  21°  7'  S. 

If  the  observed  object,  in  this  example,  had  been  a  fixed  star,  with  the  same  dwli- 
nation  12°  16'  N.,  the  same  altitudes  49°  9',  51°  59',  but  the  elapsed  time  2''  49'"  32% 
the  calculation  would  have  been  exactly  as  above.  For,  by  adding,  according  to  the 
rule  in  I'age  176,  the  correction  in  Table  LI.,  28%  to  reduce  it  to  sidera!  time,  ^^■e 
shall  ol)tain  the  corrected  elapsed  time,  or  hour  angle,  2''  50"',  and  every  part  of  tiio 
work  will  be  as  above. 

If  the  ])lanet  Venus  had  been  observed,  at  the  same  corrected  altitudes,  on  tlie 
I3th  of  ]\laroh,  1836,  in  a  place  where  his  declination  at  the  middle  time  between  the 
two  ol)servations  was,  by  the  Nautical  Almanac,  12°  16'  N.,  and  the  elapsed  time 
2''  50'"  03".5,  the  calculation  would  still  be  the  same.  For,  by  the  Nautical  Alma- 
nac, it  appears  that  Venus  passes  the  meridian  on  the  13th  and  14th  of  March,  at 
2h  27m  12s  a^j^jj  2''  27'"  42'  respectively,  increasing  30%  so  that  the  interval  of  two 
successive  transits  is '24''  00'"  30*.  Then  saying,  As  this  interval  is  to  24'',  so  is  tlie 
elaj)sed  time  2''  50"'  03'.5  to  the  corrected  elapsed  time,  or  hour  angle,  2''  50'"  00% 
which  is  to  be  used  as  above,  all  the  rest  of  the  work  being  the  same.  We  may 
proceed  in  the  same  manner,  if  the  moon  be  observed  at  a  time  when  the  declination 
varies  but  little. 


EXAMPLE    III. 

IJenig  at  sea,-  in  latitude  50°  40'  N.  by  account,  when  the  sun's  declination  was 
20°  0'  S.  at  10''  17™  A.  M.,  per  watch,  the  sun's  correct  central  altitude  was  found  to 
be  17°  13',  at  11''  17"",  per  watch,  the  correct  central  altitude  was  found  to  be  19°  41'; 
required  the  latitude. 

Subtracting  10''  17"  from  ll*"  17'",  gives  the  elapsed  time  1''. 


Col.  1. 

El.  time  [p.m.]  1"  0'",  Cosec.  10.88430 
Declination  20°  00'  S.  Sec.  10.02701 

A Cosec.  10.91131 

h  sum  alts,  18  27    Cosine     9.97708 

h  diff:  alts.      1   14  , .  ,Sine     8.33292 

C Sine    9.22131 


Col,  2, 


CoL.  3. 


[Z  lem  than  90",  and  N.  or  S.  like  bearing  of  zcniili.] 


Cosine  9.99670 
Cosec.  10.49966 
Secant  10.00010 
Cosine    9.99390 


Secant  10.49036 


[E  U  the  turn  of  B,  Z,  if  of  the  same  name  ;  dijercnce,  if  of  a  dijercnt  name.] 


Cosec.  10.46595 

Cosine    9.99G70 

B20°10'S.  Cosec.  10.46265 


Z  71  08  N. 


[B  less  thnn  90°,  mmed 
N.  orS.  likeUeclin.l 

.Cosine  9.99390 


E50  58N,     Sine    9.S903Q 
Latitude  50  00  N,     Sine    9.88420 


TO  FIND   THE   LATITUDE   BY   DOUBLE   ALTITUDES. 


183 


If  the  sun  liad  passed  the  meridian  to  the  north  of  the  observer,  Z  would  have 
been  71°  OS'  S.,  and  E=r:91°  18'  S.,  whose  sine  9.99989,  added  to  9.99390,  gives  the 
sine  of  tlie  latitude  9.99379,  corresponding  to  80°  20'  S. 


EXAMPLE    IV. 

Being  at  sea,  in  the  latitude  of  C0°  0'  N.  by  account,  when  the  sun  was  on  the 
equator  (or  had  no  declination)  at  l""  0"'  P.  M.,  per  watch,  hi.s  correct  central  altitude 
was  28°  53',  and  at  3'>  0'"  P.  M.,  per  watch,  the  correct  central  altitude  was  20°  42' ; 
required  the  true  latitude. 

CoL.  1.  CoL.  2.  CoL.  3. 

El.  time  [p.m.]  2'^  0"^,  Cosec.  10.58700 
Declination  0 Secant  10.00000 

A 15°  00'  Cosec.  10.58700 

i  sum  alts.  24  47^  Cosine  9.95801 

4  diff.  alts.     4     5h      Sine  8.85340 

C Sine  9.39841 

[Z  less  llun  90'',  anj  N.  or  S.  like  bearing  of  zenith.] 


Cosine  9.98494 
Cosec.  10.37745 
Secant  10.00110 
Cosine  9.98594 
Secant  10.34943 


[E  is  the  sum  of  B,  Z,  if  of  tlie  same  nam^  ;  dtference,  if  of  a  dijerent  came.] 


, [Cosec.  Injinite.] 

[Cosine  9.98494] 

B  00°  00'    [Cosec.  Injinite.] 

[B  less  limn  DO',  named 
N.  or  S.  lilie  dtclin.] 

Cosine  9.98594 

Z63_26  N. 
E  G3_2G  N.     Sine  9.95154 
Latitude  59  59  N.     Sine  9.93748 


•  The  calculations  v/ould  have  been  the  same  for  south  latitude,  which  would  be 
59°  59'  S.  The  computation  of  A  and  B  might  have  been  dispensed  with,  for  when 
the  declination  is  nothing,  B  is  notliing,  and  A  is  equal  to  half  the  elapsed  time  (P) 
turned  into  degrees  by  Table  XXL,  being,  in  this  example,  15°  ;  in  this  case,  all  the 
logarithms  included  between  the  brackets  []  may  be  omitted. 

In  tRe  preceding  examples,  both  altitudes  were  supposed  to  be  taken  at  the  same 
place  or  station  ;  but  as  that  is  seldom  the  case  at  sea,  the  necessary  correction  for 
any  change  of  place  must  be  made  in  the  following  manner : — 

Let  the  bearing  of  the  sun  be  observed,  by  the  compass,  at  the  instant  of  the  first 
observation  ;  take  the  number  of  points  between  that  bearing  and  the  sliij)'s  course, 
(cori-ectcd  for  lee-way,  if  she  makes  any,)  with  which,  if  less  than  eight,  or  with  what 
it  wants  of  sixteen  points,  if  more  than  eight,  enter  the  traverse  table,  and  take  out 
the  difference  of  latitude  corresponding  to  the  distance  run  between  the  observations, 
Md  this  difference  of  latitude  to  the  first  altitude,  if  the  number  of  points  between 
the  sun's  bearing  and  the  ship's  course  be  less  than  eight ;  but  suhtrad  the  difference 
of  latitude  from  the  fi.-st  altitude,  if  the  number  of  points  be  more  than  eight,  and  that 
altitude  will  be  reduced  to  what  it  would  have  bSen  if  observed  at  the  same  place 
where  the  second  was.*  This  corrected  altitude  is  to  be  used  with  the  second  observed 
altitude  in  finding  the  latitude  by  the  above  rule.  The  latitude  resulting  will  be  that 
of  the  ship  at  the  time  of  taking  the  second  altitude,  and  must  be  reduced  to  noon  by 
means  of  tlie  log. 

EXAMPLE   V. 

In  a  ship,  running  N.  by  E.  |  E.  per  compass,  at  the  rate  of  nine  knots  per  hour, 
at  10''  0'"  A.  M.,  per  watch,  the  sun's  correct  central  altitude  was  found  to  be  13°  18', 
bearing  S.  |  E.  by  compass ;  and  at  1''  40""  P.  IM.,  per  watch,  the  sun's  central  altitude 
was  found  to  be  14°  15' ;  the  latitude  by  account  being  49°  17'  N.,  and  the  sun's 
declination  23°  28'  S.    Required  the  true  latitude. 

*  This  is  the  only  correction  necessary  to  make  ful!  allowance  for  the  run  of  the  ship  ;  and  the  inex- 
perienceil  cnlculator  must  lake  care  not  lo  fall  into  the  error  of  applying^  a  correction  to  the  elapsed 
time,  as  is  directed  in  several  works  of  note,  particularly  in  the  "Complete  Navigator,"  by  Dr.  Mackay. 
This  will  appear  evident  by  supposing,  in  the  above  Example  V.,  that  a  second  observer,  with  a  watch, 
regulated  exactly  like  that  used  by  the  first,  was  at  rest  at  the  place  of  the  second  observation.  Thej, 
at  the  first  observation,  at  the  same  moment  of  time  by  both  watches,  the  first  observer  would  find  the 
sun's  altitude  13°  18',  and  the  second  observer  12°  49'.  At  the  second  observation,  the  times  and 
altitudes  would  be  alike,  so  thai  the  elapsed  time  found  by  both  observers  would  be  the  same,  and  the 
observations  would  require  no  correction,  except  what  arises  from  reducing  the  altitude  from  13°  18 
to  12°  "iy,  because  the  second  observer  is  supposed  to  be  at  rest,  and  his  observation  requires  no  cor- 
rection. 


184 


TO    FIND   THE  LATITUDE    BY    DOUBLE    ALTJTUUEa. 


The  coiredion  to  the  first  altitude. 
The  time  elapsed  between  the  observations  was  3'*  40'",  and  in  tliat  time  tlie  ship 
sailed  33  iniles  upon  tlie  course  N.  by  E.  \  E.,  which  makes  an  angle  of  13.J  |)oints 
with  the  sun's  bearing  at  the  first  observation  S.  \  E.,  the  complement  of  wliich  to  16 
points  is  2J  points.  Now,  in  Table  I.,  the  course  2^  points,  and  distance  33'",  give 
29  miLs  dillerence  of  latitude,  whicli  must  be  subtracted  from  the  first  altitude 
13°  18',  ijecause  the  ship  sailed  above  eight  |)oints  irom  the  sun  ;  therefore  the  first 
altitude  corrected  will  be  12°  49',  which  inust  be  used  in  the  rest  of  the  work. 

CoL.  1.  CoL.  2.  Col.  3. 

El.timc[i>.;i.]3''40">,Cosec.  10.33559 

Declination  23°  28'  S.  Sec.  10.03749 

A Cosec.  10.37308 

h  sum  alls.   13  32   Cosine  9.98777 

i  diff.  alts.      0  43  .  ..Sine  8.09718 

C Sine  8.45^03 

\Z  lees  lliu 


lid  X.  or  S.  like  bciring  of  zeuiih.] 


Cosine  9.95704 
Cosec.  10.G307G 
Secant  10.00003 
Cosine    9.99982 


Z  Sec.  10.587G5 


.Cosec.  10.39988 
Cosi'ne    9.95704 


B  2G°  05'  S.  Cos3c.  1 0.35G92 


Cosine    9.99982 


Z 75  01  N. 


(E  ;s  i!  e  sum  ofB,  Z,  if  of  the  sanu  i 


! ;  diffcTcnce,  if  of  a  different  i 


E  48  5G  N. 


Latitude  48  .54  N. 


Sine 
Sine 


9.87734 
9.87716 


If  the  sun  had  passed  the  meridian  to  the  north  of  the  observer,  Z  would  have  been 
75°  or  S.,  and  E  =  101°  00'  S.,  Avhose  sine  9.99180,  added  to  9.99982,  gives  the  sine 
of  the  latitude  9.991G2  corresponding  to  78°  47'  S. 


EXAMPLE  VL 

Sailing  N.  E.  h  E.  by  compass,  at  the  rate  of  nine  knots  an  hoin*,  at  0-  31""  40' 
P.  J\L,  per  watch,  the  altitude  of  the  sun's  lower  limb  was  28^  20'  above  the  korizon 
of  the  sea,  the  eye  being  elevated  twenty  feet  above  the  surface  of  the  water,  and  the 
Sim's  bearing  by  coni])ass  S.  by  W. ;  and  at  2''  58'"  20'  P.  M.,  by  watch,  the  altitude 
of  the  sun's  lower  limb  was  1G°  41'  above  the  horizon,  the  eye  being  elevated  as 
before,  t!ie  latitude  by  account,  at  the  time  of  the  last  observation,  48°  0'  N.,  and  the 
dechnation  13°  17'  S.    Required  the  true  latitude  at  taking  the  last  observation. 

The  correction  of  these  altitudes  for  semidiameter,  parallax,  and  dip,  was  twelve 
miles,  (additive,)  which  makes  tJiem  28°  32',  and  1G°  53'.  The  refraction  corre- 
.sponding  to  the  first  was  2  miles,  and  for  the  second  3  miles  :  and,  by  siil)tracting 
these  quantities,  we  have  the  true  central  altitudes,  28°  30',  and  10°  50'.  Now,  the 
elapsed  time  between  the  observations  was  2''  20'"  40^,  diu'ing  which  the  s'lip  sailed 
tvveniy-two  miles  (at  nine  miles  ])er  hour)  in  the  direction  of  N.  E.  h.  E.  per  com})ass; 
the  bearing  of  the  sun  at  the  first  observation  S.  by  W.  being  \2h  points  distant  from 
the  shi|)'s  course  ;  and  as  12^  [)oints  want  3^  of  IG  points,  we  iriust  enter  Table  I., 
and  find  the  course  3^  ])oints,  and  distance  22,  corresponding  to  which  in  the  latitude 
column  is  17  miles,  which,  being  suinractcd  from  the  first  altitude  28°  30',  leaves  the 
corrected  first  altitude  28°  13' ;  with  this,  and  the  second  altitude  1G°  50',  the  latitude 
is  found  in  the  following  manner: — • 


CoL.  1. 
EI.time[p.M.]  2"2G'M0%Cosec.  10.50232 
Declination  13°17'S.  Secant  10.01178 


CoL.  2. 


CoL.  3. 


A Cosec.  10.51410 

h  sum  alts.  22  3U  ..Cosine    9.90553 
A  diff  alts.     5  41i  ...  .Sine   8.99640 


Cosine  9.97801 
Cosec.  10.41070 
Secant  10.00215 
Cosine   9.97902 


C Sme   9.47003 

•  

SZ  less  ih:in  90°,  ;in(l  N.  or  ."5.  like  bearing  of  zeniili.] 

£  U  the  $um  of  }I.  7,  if  ol  the  tam    lamc  ;  difference,  if  of  a  different  name.] 


.  Cosec.  10.0387  J 
.Cosine   9.978G1 


B  13°  58'  S.    Cosec.  10.01732 

[B  less  linn  90°,  namc-l 
N.  or  S.  like  ilcclio.] 

Cosine    9.97962 


Z  Sec.  10.37708  Z  G5  11  N. 
E  51  13  N. 


Sine   9.89183 


Latitude  48  03  N.       Sine   9.87145 


TO   FIND   THE   LATITUDE   1}Y    DUUliLE   ALTITUDES. 


185 


If  tlie  snn  Iiad  passed  the  meridian  to  tlie  north  of  tlie  observer,  Z  would  have 
oeen  (io°  11'  S.,  and  K  =  79"  OS)'  S.,  whose  sine  U.!}'J217,  added  to  cosine  of  C  9.97902. 
gives  the  sine  of  the  latitude  9.97179,  corresponding  to  (J9°  34'  S. 


EXAMPLE   VII. 

[Same  as  Dr.  Briiiklcy's,  in  the  Nautical  Almanac  for  1800.] 

The  latitude  by  account*  G°  30'  N.,  sun's  decfniation  5°  30'  N.,  the  siui's  correct 
central  altitudes  35°  21',  and  70°  01',  elapsed  time  between  the  ob.servations  2'' 20"" ; 
required  tiie  laiitude,  the  sun  pas.sing  the  meridian  south  of  the  observer. 

El.time[p.M.]2'^20"',Cosec.  10.52186 
Declination  5°30'N.  Sec.  10.00200 

A Coscc.  10.5238G 

i  sum  alts.  52  41  Cosine  9.782G3 
h  diff.  alts.  17  20  Sine  9.47411 
C Sine    9.780G0 

[Z  less  lliaii  90",  njul  N.  or  S.  like  bearing  of  zenilh.] 

(E  is  the  cam  of  B,  Z,  if  of  ihe  same  name,  diferencc,  i(  of  a  dijferent  n.iine.] 


Cosine  9.979G2 
Cosec.  10.09947 
Secant  10.02018 
Cosine  9.90170 
Z  Sec.  10.00097 


Co.scc.  11.01843 
Cosine    9.979G2 


B  5°4G'N. . 

V-ZW.    iliW           ».    .<      .    t^v^/^ 

.  .Cosec.  10.99805 

[E  less  itiiTi  90°,  iKinied 
N.  or  S.  li!,e  Jcxlin.l 

.  Cosine    9.90170 

2  3  50  N. 

E  9  3G  N. . 

....Sine    9.22211 

Lat.  7J38  N Sine    9.12381 

If  the  smi  had  passed  to  the  meridian  north  of  tlie  observer,  Z  would  have 
been  3°  50'  S.,  and  E  =r:  1°  5G'  N.,  whose  sine  8.52810,  added  to  the  cosine  of  C 
9.90170,  is  8.42980,  which  is  the  sine  of  the  other  latitude  1°  32'  N.,  so  that  in  this 
example  both  latitudes  are  north. 


SECOND  METHOD 

Of  finding  the  latitude  by  double  altitudes  of  the  sun,  tohcn  the  variation  of 
declination  is  neglected. 

This  method  of  finding  the  latitude  depends  on  a  set  of  tables  (marked  XXIII.,  in 
this  collection,)  (irst  prepared  by  Mr.  Douwes,  containing  three  logariihins,  titled  half 
elapsed  time,  middle  time,  and  log.  risiiig.  The  two  former  are  arranged  together  as 
far  as  six  hours  ;  the  latter  is  placed  at  the  end  of  the  table,  and  is  extended,  in  the 
present  edition,  as  fir  as  twelve  hours.  The  table  with  the  })roi)er  title  must  be 
entered  at  the  top  with  the  hour,  at  the  side  with  the  minute,  and  in  the  colunm 
marUed  at  the  top  with  the  seconds;  the  corresponding  number  will  be  the  sought 
logarithm,  to  whiidi  must  be  prefixed  the  index  of  the  log.  under  0"  in  the  same 
horizontal  line.  Thus,  to  the  time  3''  52'"  10^  corrcsi)oiid  the  log.  half  elapsed  time 
0.07138,  log.  middle  time  5.229G5,  and  log.  rising  4.G7274,  In  general  it  will  be 
sufficiently  exact  to  take  these  logarithms  to  the  nearest  10  seconds,  particularly  when 
the  sun's  /eiiitii  distance  is  great;  but  if  the  log.  to  the  nearest  second  is  required,  it 
may  be  f)und  by  taking  the  difterence  of  the  tabular  logarithms  coriespondiiig  to  the 
next  greater  and  next  less  time,  and  saying.  As  10'  is  to  that  diflerence,  so  are  the 
odd  seconds  of  time  to  the  correction  of  the  first  tabular  logarithm,  additive  if 
increasing,  suhtractive  if  decreasing.  Thus,  if  the  log.  half  elapsed  time  correspond- 
ing to  3''  52'"  18'  were  required,  the  logs,  corresponding  to  3''  52'"  10'  and  3''  52'"  20' 
are  0.07138  and  0.07119,  whose  difference  is  19;  then  10'  :  19::  S'-  :  15;  this,  stib- 
trrK'tci!  ti-oin  0.07138,  leaves  0.07123,  the  sought  log;n-ithm.  By  inverting  the  process, 
■we  may  find  the  nearest  second  corres])onding  to  any  given  logaritiim.  We  shall 
now  give  the  rule  lor  calculating  the  latitude,  adapted  to  double  altitudes  of  the  sun. 

RULE.    • 

To  the  log.  secant  of  the  latitude  by  account  (Table  XXVII.)  add  the  log.  secant  of 
the  sun's  declination,  (Table  XXVII.,)  rejecting  10  in  each  index ;  the  stun  is  to  be 
called  the  log.  ratio. 
24 


18G  TO   FliND  THE   LATITUDE   BY   DOUBLE   ALTITUDES. 

From  the  natural  sine  of  the  greatest  altitude  (Table  XXIV.)  subtract  the  natural 
sine  of  the  least  altitude,  (Table  XXIV.;)  find  the  logarithm*  of  their  difference,  (in 
Table  XXVI.,)  and  place  it  undet*  the  log.  ratio. 

Subtract  the  time  of  taking  the  first  observation  from  the  time  of  taking  the  second, 
having  previously  increased  the  latter  by  twelve  hours  when  the  observations  are  on 
different  sides  of  noon  by  the  watch  ;  take  half  the  remainder,  which  call  hall"  tlie 
elapsed  time. 

With  half  the  elapsed  time  enter  Table  XXIII.,  and  from  the  cokniin  of  half 
elapsed  time  take  out  the  logarithm  answering  tliereto,  apd  write  it  under  the  log. 
ratio. 

Add  these  three  logarithms  together,  and  with  their  sum  enter  Table  XXIII.  in 
the  coUunn  of  middle  time,  where,  having  found  the  logarithm  nearest  thereto,  take 
out  the  time  corresponding,  and  put  it  under  half  the  elapsed  time.  The  difierence 
between  these  times  will  be  the  time  from  noon  when  the  greater  altitude  was 
taken. 

With  this  time  enter  Table  XXIII.,  and,  from  the  cojunm  of  log.  rising,  take  out 
the  logarithm  corresponding,  from  which  logarithm  subtract  tke  log.  ratio  ;  th(! 
remainder  will  be  the  logaritlim  of  a  natural  number,  which,  being  found  in  Table 
XXVI.,f  and  adfled  to  the  natural  sine  of  the  greater  altitude,  will  give  the  natural 
cosine  of  the  sun's  meridian  zenith  distance,  which  may  be  found  in  Table  XXIV. 
Hence  the  latitude  inay  be  obtained  by  the  rules  of  pages  IGG,  167. 


1.  If  this  computed  latitude  should  differ  considerably  from  the  latitude  by  account, 
it  will  be  proper  to  repeat  the  opei-ation,  using  the  latitude  last  found  instead  of  the 
latitude  by  account,  till  the  result  gives  a  latitude  nearly  agreeing  with  the  latitude 
used  in  the  comi)utation. 

2.  This  methoc<  is  best  suited  to  situations  where  the  sun's  meridian  zenitli  distance 
is  not  much  less  than  half  the  latitude  ;  for  in  latitudes  where  the  sun  approaches 
near  to  the  zenith,  the  observations  must  be  taken  much  nearer  to  noon  ;  and  the  pre- 
ceding rule,  instead  of  approximating,  will  in  some  cases  give  the  results  of  successive 
operations  wider  and  wider  from  the  truth.  To  remedy  this  difficulty,  a  set  of  tables 
was  published,  by  Dr.  Brinkley,  at  the  end  of  the  Nautical  Almanac  for  1799;  but  the 
great  variety  of  cases  incident  to  his  metliod,  will  hinder  it  from  being  generally  used. 
Instead  of  Dr.  liriukley's  method,  we  may  generally  use  the  method  of  arithmetical 
computation,  called  Double  Position,  which  will  frequently  give,  in  a  more  simple 
manner,  the  required  latitude,  as  will  be  shown  in  Example  X.;  and,  in  general,  it 
may  be  observed,  that  where  Douwes's  rule  does  not  approximate,  the  ol)ject  is  ujost 
commonly  so  situated  as  not  to  furnish  the  necessary  observations  to  obtain  a  correct 
latitude,  whatever  method  of  computation  might  be  used. 

3.  The  operation  is  the  same  whether  the  sun  has  north  or  south  declination  ;  and 
also  whether  the  ship  is  in  north  or  south  latitude.  When  the  sun  has  no  declination, 
the  log  secant  of  the  latitude  (rejecting  10  in  the  index)  will  be  the  log.  ratio  ;  and 
when  the  latitude  by  account  is  nothing,  the  secant  of  the  declination  (rejecting  10  in 
the  index)  will  be  the  log.  ratio.  This  rule,  as  well  as  the  former,  is  iounded  on  the 
supposition  that  the  declination  is  taken  for  the  middle  time  between  the  o!)scrva- 
tions,  and  that  it  does  not  vary  during  the  elajised  time,  which,  however,  rarely 
happens,  and  a  correction  ought  to  be  ai)plied  to  the  latitude  on  this  account.  F»ut 
this  correction  is  generally  small ;  and  if  it  is  large,  the  third  method  must  be  used ; 
and  when  the  declinations  differ  very  much  from  each  other,  we  must  use  the  fourth 
method. 

*  The  index  of  this  logarithm  being,  as  usual,  one  less  than  the  number  of  figures  contained  in  tlie 
difference  of  these  natural  sines ;  ol)serving,  also,  that  tlie  ahitudes  to  be  used  are  the  correci 
central  eiltitudes  j  that  is,  the  observed  altitudes  corrected  for  dip,  semidiameter,  parallax,  and 
refraction. 

t  Taking,  as  usual,  a  number  of  figures  equal  to  the  index  of  that  logarithm  increased  by  unity. 


TO   FIND   THE   LATITUDE  BY   DOUBI-E   ALTITUDES.  187 

EXAMPLE  VIIL 

[Same  as  Exaimple  L,  prccetling.] 

Being  at  sea,  in  latitude  46°  30'  N.  by  account,  when  the  sun's  declination  was 
11°  17'  N.  at  10''  2'"  in  the  forenoon,  the  sun's  correct  central  altitude  was  4G"  55' , 
and  at  11'"  27'"  in  the  forenoon,  his  correct  central  altitude  was  54°  9';  required  the 
tnie  latitude,  and  true  time  of  the  day  when  the  greater  altitude  was  taken. 

Times.  Jilt.     Nut.  Si.  Lat.  by  ace 46°  30' Sec.  0.16219 

2obscr.  IP  27'" 00"     54°  9'    81055  Dec ...11    17 Sec.  0.00848 

1  obser.  10 2 0^     46  55     73036  Log.  ratio 0.17067 

Elap.  time         1    25     0  Diir.  n^..  sine».    8019  Log.  difT.  Nat.  Sines 3.90412 

i  elap.  time       0   42   30  Log.  ^  elap.  time 0.73429 

Middle  time 1"  15"- 10*     4.80908 

^  elap.  time 42    30 

2  obs.  from  noon      0   32   40      Its  log.  rising 3.00608 

Log.  ratio  sub 0.17067 

Nat.  numb 685 corresponding  to  log.  2.83541 

Nat.  sine  greatest  alt 81055 

Sum  is  nat.  cosine  ©'s  zen.  dist.  8l740. .  .equal  to  35°  10'  N. 
©'s  declination "11  17   N. 

Lat.  in 46  27   N. 

The  latitude  46°  27'  (differing  only  3'  from  the  latitude  by  account)  may  be  assumed 
as  the  true  latitude. 

By  means  of  the  time  of  the  second  observation  from  noon  above  found  32'"  40', 
the  error  of  the  watch  may  be  found  ;  for,  in  the  i)resent  example,  by  subtracting 
32"^  40^  from  12'',  we  have  the  time  of  the  second  observation  11''  27'"  20^;  but  the 
time  of  the  watch  was  11''  27'"  0^ ;  tlierefore  the  watch  was  twenty  seconds  too  slow; 
a  small  tlifference  would  be  fouad  in  these  numbers,  if  we  were  to  jirojjortion  the 
logarithms  of  Table  XXIII.  to  seconds.  In  the  same  manner,  the  error  of  the  watch 
may  be  found  in  the  following  examples.* 

EXAMPLE   IX. 
[Same  as  Example  V.,  before  given.] 

In  this  example  the  latitude  by  account  is  49°  17'  N. ;  the  sun's  declination  2-3°  28'  S. 
the  first  altitude  corrected,  as  before,  12°  49' ;  the  second  altitude  14°  15',     Ilequired 
the  true  latitude. 

.4//.       A„t.  SL     Lat.  l>y  ace 49°  17' Sec.  0.1 8554 

2  obsor.         13"  40'"    0'     14°  15'      24615     Declination  ...  .23    28 Sec.  0.03749 

1  obser.         10     0     0       12  49       22183    Log.  ratio 0.22303 

Elap.  time      3   40     0  DifT.  nat.  si.    2432     Its  log 3.38596 

h  elaj).  time    1    50     0  Its  log 0.33559 

Mid.  time       0    10    10  Time  corresponding  to 3.94458 

5 obser.  from  noon,    1    39    50    Its  log.  iu  col.  of  risiug  is 3.97028 

Log.  ratio 0.22303 

,5588  Nat.  number  of Log.  .3.74725 

Nat.  sine  greatest  alt 24615 

Nat.  cosine  ©'s  nier.  zen.  dist 30203  =  72°  2.5'  N. 

Declination 23  28  S. 

Latitude 48  57  N. 


*  When  the  middle  time  is  greater  than  half  the  elapsed  time,  both  observations  are  on  the  same  side 
of  the  meridian  ;  ollicrwise,  on  dilTerent  sides  ;  whence  it  is  easy  to  determine  wiieliier  the  greater 
altitude  be  observed  before  or  after  noon 


188  TO  FIND  THE   LATITUDE   BY  DOUBLE   ALTITUDES. 

But  as  the  latitude  by  computation  differs  considerably  from  that  by  account,  the 
work  must  be  repeated. 

Lat.  last  found. . .  48° 57'  . .  .Sec.  0.18262 
Declination 23  28  ...  Sec.  0.03749 

Log.  ratio 0.22011 

Diti:  N.  sine  2432 Irs  log.  3.38596 

h  elapsed  time l"*  50™  0»     Its  log.  0.33559 

Middle  time 0    10    0      Its  log 3.94166 

Time  from  noon 1    40    0      Its  log.  in  col.  of  rising 3.97170 

Log.  ratio 0.22011 

5644      Nat.  number  of Log.  3.75159 

Nat.  sine  greatest  ahitude 24615  

30259     Nat.  cos.  mer.  zen.  distance  . . .  72=23'  N. 
Declination 23  28  S. 


True  latitude 48  55  N. 


This  latitude  (differing  only  two  miles  from  that  which  is  used  in  tlie  compTitation) 
may  be  depended  upon  as  the  true  latitude  of  the  sliip,  at  the  time  of  tlie  second 
observation.  If  the  first  altitude  had  not  been  corrected,  the  computed  latitude  would 
liave  been  Ibuud  =  48°  40'  N.  * 

EXAMPLE  X. 

[Same  as  Example  VII.,  before  given.] 

The  latitude  liy  account  6°  30'  N.,  sun's  declination  5°  30'  N.,  the  sun's  correct 
central  altitudes  35°  21'  and  70°  01',  elapsed  time  2''  20",  are  given  to  find  the  true 
latitude. 

Making  the  calcidations  wuh  the  latitude  by  account  6°  30',  the  computed  latitude 
by  the  fir^t  operation  will  be  8°  16'.  Repeating  the  operation  with  the  latitude  8°  16', 
tlie  second  oiieration  will  give  7°  10'.*  This  must  be  used  for  a  third  operation ;  and 
by  repeating  the  calculation  accurately  to  seconds,  it  Vili  require  more  than  a  dozen 
operations  to  obtain  the  true  latitude  7°  38',  which  was  founil,  by  the  first  method,  by 
a  single  operation.  Dr.  Brinkley  made  the  latitude  7°  30',  differing  8'  from  a  strict 
calculation  by  spherical  trigonometry.  The  detail  of  this  calculation  is  not  here 
given,  but  is  left  to  exercise  the  learner.  The  object  of  the  ])resent  example  is  to 
show  liow  tlie  number  of  operations  might  be  decreased  by  the  arithmetical  method 
of  ilouble  position  before  mentioned. 

Take  the  error  or  difference  between  the 
fii-st  assumed  latitude  6°  30',  and  the  first 
computed  latitude  8°  16',  equal  to  106' ;  also 
the  error  or  difference  between  the  second 
assumed  latitude  8=  16',  and  second  comput- 
ed latitude  7°  10',  which  is  66'.  Multiply 
theni  crosswise,  as  in  the  adjoined  scheme,  according  to  the  usual  rule  of  double 
position  ,-f  dividing  the  sum  of  the  products  1305°  16',  by  the  sum  of  the  errors  172, 
gives  the  corrected  latitude  7°  35'  N.  The  sum  of  the  products  is  taken  in  this  case, 
because  one  of  the  assumed  latitudes  was  greater,  and  the  other  less,  than  its  corre- 
sponding comjjiited  latitude.  II"  both  computed  latitudes  had  bi;en  greater,  or  both 
less,  than  the  corresponding  assiuned  latitudes,  the  differences  of  the  errors  and  of  the 
products  ought  to  have  been  taken.  It  will  rarely  hajjpen  that  more  than  one  pro- 
cess of  this  kind  will  be  refjuircd  to  give  a  correct  result.  In  the  present  instance, 
however,  it  will  be  necessary ;  for,  by  repeating  the  operation  with  the  assumed 
latitude  7°  35',  the  resulting  computed  latitude  is  7°  41.i',  and  the  third  error  6h'. 
Repeating  anew  the  compiUation,  with  this  and  the  second  latitude  8°  \6',  and  second 
error  66',  the  resulting  latitude  is  7°  38',  the  same  as  was  foimd  by  the  direct  compu- 
tation by  the  first  method,  and  as  accurately  as  could  be  obtained  by  repeating  the 
operations  aljout  fourteen  times  by  the  second  method. 

In  general,  when  SMch  a  largo  number  of  operations  are  required  to  produce  a 
correct  result,  it  is  a  sure  proof  that  the  situation  of  the  ol)ject  is  not  Avell  adapted  to 


I, 

lis. 

Errors. 

Prod, 

cs. 

6" 

30' 

X 

106: 

=  876° 

16' 

8 

16 

66-. 

=  429 

00 

172) 

1305 

16 

7° 

35'. 

*  Slioht  clifTereDros  will  be  found  in  tlie.se  calculations,  by  using  logarithms  to  seven  places  of  figures, 
and  making  ilie  calculation  accurately  to  seconds. 

f  If  the  degrees  of  both  latitudes  are  alike,  the  minutes  only  may  be  retained  in  these  multiplications. 


TO   FIND   THE   LATITUDE  BY   DOUBLE   ALTITUDES  189 

obtain  an  accurate  latitude;  and  it  would  be  lost  labor,  and  lead  to  great  n)istakes,  to 
attempt  it.  Thus,  in  tlie  present  e.vatnple,  if  the  greatest  altitude  had  been  decreased 
only  12'  42",  tnaUing  it  1)<J°  48'  18",  leaving  unaltered  the  other  altitude  35°  21',  and 
the  interval  2''  20"',  the  latitude  of  the  j)lace  of  observation  would  bo  0,  (or  under 
the  eipiator,)  as  is  easily  ])roved  by  computing  the  altitudes  of  the  sun  for  tlie  times 
1"  17"'  50'.8,  and  3''  37'"  50^8,  under  the  equator,  when  the  declination  is  5°  30'  N., 
by  the  rides  hereafter  given.  Hence  it  a|)pears  that  a  change  of  13'  42"  in  the 
greatest  altitude,  would  alter  tlie  computed  latitude  from  7°  38'  to  0°,  whif  h  makes 
an  error  of  one  degree  of  latitude  for  an  error  of  Ig  nfiles  in  that  altitude;  and  aa 
errors  in  the  altitudes  of  this  magnitude  are  easily  conuiiitted  at  sea,  even  by  very 
good  oiiservers,  it  shows  very  clearly  the  def(!ct  of  the  method  of  double  altitudes 
when  the  sun  ap])roaches  near  to  the  zenith.  This  does  not  arise  from  any  defect  of 
the  method  of  computation,  but  is  an  inherent  defect  of  the  method  iiself,  which  no 
process  of  s|)herics  can  remedy;  and  there  is  no  other  resource  left,  in  such  cases, 
than  to  make  use  of  another  object  to  determiue  the  latitude. 


THIRD   METHOD 

Of  Jin  ding  the  latitude  by  two  altitudes  of  any  heavenly  body,  noticing  the 
I       change  in  the  declination  during  the  time  between  the  two  observations. 

To  determine  the  latitude  accurately,  reducing  the  change  in  the  declination  of  the 
object,  we  have  comjuited  Table  XLVI.,  by  means  of  which  the  correction  of  either 
one  of  the  observed  altitudes  can  be  com])uted  for  the  change  of  declination  of  the 
observed  object  during  the  elaiised  time  between  the  observations,  and  thus  the 
problems  of  double  altitudes  of  the  sun,  moon,  planet,  or  fixed  star,  can  bo  reduced 
to  the  case  of  the  declination,  being  invariably  the  same  as  at  the  time  of  the  obser- 
vation of  the  ahitudes  which  is  not  corrected,  and  then  the  problem  comes  under  the 
frst  (or  second)  method  of  solution,  which  is  much  more  simple  and  free  from  cases 
than  the  general  solution  by  theybu;-//i  metliod.  This  process  of  correcting  the  alti- 
tude is  somewhat  similar  to  that  before  taught,  for  making  allowance  for  the  run  of 
a  ship  during  the  time  elapsed  between  the  observations;  and  the  same  altitude, 
which  is  corrected  for  the  run  of  the  shij),  can  also  be  corrected  for  the  change  of 
declination.  This  method  of  correcting  one  of  the  altitudes  is  particularly  applicable 
to  the  case  where  both  observations  are  made  on  the  same  heavenly  body,  and  the 
declination  does  not  vary  but  few  minutes,  or,  in  extreme  cases,  more  than  one  or  two 
degrees  ;  but  the  same  process  may  be  used  when  two  different  objects  are  observed, 
•j)rovided  their  declinations  are  nearly  equal,  or  do  not  ditier  more  than  one  or  two 
degrees. 

As  either  one  of  the  altitudes  may  be  corrected,  the  problem  admits  of  two  differ- 
ent ways  of  solution.  For  the  sake  of  precision,  the  altitude  which  is  selected  to  be 
corrected,  will  be  called  x\\e first  altitude  ;  and  the  corresponding  declination,  the  first 
declination  ;  tlie  other  altitude,  which  is  not  corrected,  will  be  called  the  second  altitude, 
and  the  corresponding  declination,  the  5fcortcZ  declination;  these  ivrms,  first  and  sec- 
ond, having  no  reference  to  the  order  in  which  these  observations  are  taken,  since  the 
altitude  here  defined  as  X\ie first  aliilnde,  may  be  actually  observed  either  before  or  afer 
the  other  observation. 

The  proposed  table  gives  for  various  declinations,  altitudes,  and  latitudes,  the  change 
of  the  first  altitude,  corresi)onding  to  a  variation  of  100"  in  the  first  declination.  Thus, 
with  the  latitude  50°  N.,  the  sun's  altitude  30°,  and  the  declination  14°  N.,  the  table 
gives  77"  for  the  variation  of  that  altitude  arising  from  a  change  of  100"  in  the  decli- 
nation. If  the  actual  change  of  declination  is  greater  or  less  than  100',  the  tabular 
number  77"  must  be  increased  or  decreased  in  the  same  proportion.  Thus,  if  the 
change  of  declination  be  200",  the  change  of  altitude  will  be  200"  X -j-Vij  =  l''^'*". 
IC  the  change  of  declination  be  GO",  the  change  of  altitude  will  be  GO"  X  t </<;  — ^  ^*^"" 
The  correction  of  th'is  first  altitude  having  been  found,  it  is  to  be  applied  to  the  first 
altitude,  corrected  as  usual,  for  dip,  refraction,  semidiameter,  and  |)arallax,  and  the 
corrected  first  altitude  will  be  obtained,  such  as  it  would  have  been,  if  the  declination 
at  the  time  of  observing  tliat  altitude  had  been  equal  to  the  second  declination.  With 
this  corrected  first  altitude,  the  second  altitude  and  second  declination  without  cor- 
rection, and  the  observed  elapsed  time,  or  hour  angle,  the  computation  of  the  latitude 
may  be  made  by  the  First  Method,  explained  iu  page  180,  or  by  the  Second  Mctlmd,  in 
page  185.  f 

This  table  is  calculated  for  every  2°  of  declination,  from  0°  to  2G°.  If  the  change 
'■  f  declination   is  not  very  great   during   the   elapsed   tiine,  it  will   in   general    be 


190  TO  FIND  THE   LATITUDE   BY   DOUBLE   ALTITUDES 

sufficiently  exact  to  enter  the  table  with  the  nearest  declination,  and  take  proportional 
parts  for  the  degrees  of  altitude  and  latitude.  The  latitude  by  account  is  to  be  used 
in  finding  the  luunbers  from  this  table,  it  being  sufficiently  accurate,  since  an  error 
of  1°  of  latitude  rarely  produces  more  than  2"  change  in  the  nimibers  of  tiie  table. 
Suppose,  now,  that  the  tabular  number  is  required,  when  the  latitude  is  37°  N.,  the 
Jirst  altitude  28°,  the  ly-st  declination  G°  25'  S.  In  this  case,  using  the  declination  G°, 
and  the  altitude  20°,  the  tabular  numbers  corresponding  to  the  latitudes  30°  S.  and 
40°  S.  avfi,  respectively,  57"  and  73'',  whose  diffijrence  16''  corresi)onds  to  a  change 
of  10°  of  latitude,  and  by  proportion,  the  change  corresponding  to  7°  .<^"  latitude  is? 
16"  X  yrr^^-^^"'^'  ^'"^  being  added  to  57",  gives  the  correction  corresponding  to  the 
alt'tnde  20°,  and  the  latitude  37°  S.  equal  to  6S".2.  Repeating  now  tlie  same  opera- 
tion with  the  altitude  30°,  the  two  tabular  numbers  are  G4"  and  81",  whose  difference 
17",  being  multiplied  by  ^^,  gives  11".9  to  be  added  to  64"  to  get  75".9,  the  correction 
corresponding  to  the  altitude  30°  and  the  latitude  37°  S.  Hence  it  api)cars  that  by 
changing  tlie  altitude  from  20°  to  30°,  the  correction  changes  from  ()8".2  to  75".9, 
increasing  7".7,  by  an  increase  of  10°  in  the  altitude;  the  corresponding  increase  for  a 
char;:  •  of  8°  in  the  altitude  is  equal  to  7".7  X  y8j=:6".2,  nearly.  This  being  added 
to  68  '.2,  gives  74".4,  for  the  tabular  number  corresponding  to  the  declination  6°,  the 
altitude  28°,  and  the  latitude  37°  S.  If  the  same  calculation  be  repeated,  using  tlie 
declination  8°,  the  tabular  number  will  be  76".2,  instead  of  74".4,  increasing  only  1".8 
Ifjr  an  increase  of2°:=120'  in  the  declination,  and  the  corresjioiiding  correction  for 
the  25' of  the  first  declination  is  1".8  X -f 7,7  =  0".4,  nearly.  This  being  added  to 
74"4,  gives  the  correct  tabular  number  74".8)  or  75",  nearly,  corresponding  to  the 
proposed  latitude,  37°  S.,  altitude  28°,  or  declination  G°  25'  S.  The  correction  for  the 
minutes  of  declination  is  in  this  case  small, and  in  genei'al  it  Avill  be  so;  and  when  tl\e 
change  of  declination  during  the  elapsed  time  is  only  a  few  nanutes,  it  will  be 
sufficiently  exact  to  take  out,  according  to  the  above  directions,  the  numbeis  corre- 
sponding to  the  nearest  declination  in  the  table.  As  there  is  nothing  ])eculiar  in  this 
method  of  finding  the  corrections  for  tlie  intermediate  degrees  of  altitude  and  latiuide 
(several  tables  in  the  work  having  been  arranged  upon  a  somewhat  similar  ])lan,)  it 
will  not  be  necessary  to  go  into  any  further  detail  relative  to  the  manner  of  finding  the 
number  from  the  table  corresponding  to  any  proposed  declination,  altitude,  or  latitude. 
The  use  of  tliese  numbers  in  finding  the  correction  of  the  first  altitude,  is,  for  the  sake 
of  easy  reference,  drawn  up  in  the  following  rules. 

RULE. 

1.  If  the  two  declinations  are  of  the  same  name,  take  their  difference ;  if  they  are  of 
different  names,  take  tl^eirsum;  and  this  difference,  or  sum,  will  be  the  change  of  declination 
coiresponding  to  the  two  ohservatioiis,  or  two  objects. 

2.  Find  in  Table  XLVI.  the  number  corresponding  to  the  frsl  declination,  the  frs 
altitude,  and  the  latitude  by  account.  Multiply  this  by  the  change  of  declination,  in 
seconds,  between  the  two  observations ;  the  product,  rejecting  the  two  right-hand  figures, 
will  be  the  number  of  seconds  to  be  applied  to  the  first  altitude,  with  the  same  sign  as  in  the 
table,*  if,  at  the  second  observation,  the  object  is  nearer  to  the  elevated  pole  than  at  the  frsl 
obscnation ;  but  with  a  different  sign  from  the  table,  if,  at  the  second  obsci-imtion,  the 
object  is  farther  from  the  elevated  pole  than  at  the  first  observation. 

Thus,  in  the  above  exam[)le,  where  the  tabular  correction  is  75",  if  the  second 
yjtitude  is  48'  and  the  second  declination  6°  15'  S.,  which  is  10'  or  600"  less  than  tin; 
Jlrst  declination  6°  25"  S.,  the  product  of  600"  by  75  (rejecting  the  two  right-hand 
figures)  is  450"  =  7' 30",  being  the  correction  to  be  added  to  the  first  altitude  28", 
making  it  28°  7' 30",  because  the  second  declination  is  nearest  to  the  elevated  ])o!<'. 
If  tlie  second  declination  be  6°  35'  S.,  instead  of  6°  15'  S.,  the  correction  7' 30"  will 
bo  subtractivp,  making  it  27°  52'  30". 

It  may  bo  observed,  that  the  metiiod  of  correcting  one  of  the  altitudes  c/ocsno<a//«-<Ae 
horary  angles  in  any  ivay  tvhatevcr,  ai  d  the  regulation  of  the  watch  used  in  t!ic  obser- 
vation is  calculated  in  exactly  the  same  manner  as  if  tiie  correction  had  not  been 
made,  and  whichever  altitude  is  corrected,  the  result  will  be  very  nearly  the  same ;  a 

*  The  signs  in  the  table  arc  positive  except  in  a  few  places  between  the  tropics.  In  all  cases  with- 
out the  tropics,  when  the  distance  from  the  elevated  polo  decreases,  the  altitude  is  to  be  increased,  and 
when  the  polar  distance  increases,  the  altitude  is  to  be  decreased.  TUe  contrary  takes  place  in  those 
latitudes  between  the  tropics  where  the  tabular  numbers  have  the  sign  —  prefixed.  It  may  also  be  ob- 
served, that  tiic  talnilar  number,  corresponding  to  any  possible  situation  of  the  object,  ciuinot  exceed 
100";  it  was,  however,  found  convenient  to  insert  a  few  numbers  exceeding  100",  for  the  purpose  of 
finding  more  accurately  the  proportional  parts  for  the  intermediate  degrees  of  altitude  or  Jaliiudc  cor- 
responding to  possible  cases. 


TO  FIND   THE   LATITUDE   BY    DOUBLE   ALTITUDES. 


191 


difference  of  a  few  seconds  will  sometimes  be  found,  owing  to  the  small  quantities 
neglected. 
To  illustrate  this,  the  following  exanip  es  are  given  : — 


EXAMPLE    XI. 

The  sun's  correct  central  altitude  was  32°  25',  his  declination  17°  N.  Eight  hours 
aftcrward.s,  hy  a  watch,  his  correct  central  altitude  was  30°  8',  and  declination  1G°  55' 
N.    Re(iViired  the  latitude,  supposing  the  latitude  by  account  to  be  53°  20'  N. 

The  tahidar  correction  correspondnig  to  the  first  altitude  32°  25',  declination  17°  N., 
and  latitude  liy  account  53°  20'  N.,  is  80".  Multiplying  diis  hy  the  diftereuce  of  the 
declination  17°  —  1G°  55' =:  5' =r 300",  the  product  (rejecting  the  two  right-hand 
figures)  is  240".00=r4',  the  correction  of  altitude.  This  is  to  be  subtracted  from 
32°  25',  because  the  sun  recedes  from  tlie  elevated  pole,  while  the  declination  changes 
from  17°  N.  to  1G°  55"  N. ;  therefore  the  corrected  jjrsi  altitude  is  32°  21".  Using  this 
with  tlic  second  altitude  30°  8',  the  second  declination  1G°  55',  and  the  elapsed  time  8 
hour.'*,  the  calculation  may  be  thus  made  by  tho first  method,  as  follows: — 


CoL.  1, 
El.  time  8"  ',p.  :,i.]     Cosec.  10.00247* 
Declination  1G°  55'  N.  Sec.  10.01921 

A Cosec.  10.081G8 

h  sum  alts.  31  14i  Cosine  9.93196 
h  diff.  alts.  1  6h  Sine  8.28650 
C 1     9       Sine     8.30014 


CoL.  2. 


CoL.  3. 


Cosine  9.74812 

Cosec.  10.28512 

Sec.  10.00008 

Cosine  9.99991 

Z  sec.  10.03323 


(Z  less  ihaii  90",  iiameil  N.  or  S.  Vue  tLe  beariiij  of  the  zenith.  ] 

[E  is  the  sum  of  B,  Z,  if  of  the  same  name  ;  dlfcrence,  if  o  a  dijcr,  nt  name.] 


.Cosec.   10.53614 
Cosine     9.74812 


B31°18'N.Cosec.  10.28426 

[B  less  than  90°,  named 
N.  or  S.  lilietliedeclin.] 

Cosine     9.99991 


Z22    8  N. 


E53  2G  N.    Sine    9.90480 


Latitude  53  25  N.    Sine    9.90471 


As  it  is  entirely  arbitrary  which  altitude  is  considered  as  the  first,  or  the  one  to  be 
corrected,  it  may  not  be  amiss  to  repeat  the  operation,  considering  30°  8'  as  tho  first 
altitude,  and  16°  55'  as  the  frst  declination.  The  tabular  number  corresponding  to 
these  quantities,  and  the  latitude  by  account,  is  79",  which,  being  multi))lied  by  tho 
chanife  of  declination  300",  (rejecting  the  two  right-hand  figures,)  gives  237" z=  3' 57", 
or  4'"neai-ly.  This  is  to  be  added  to  30°  8'  to  get  the  corrcctedyi/-s<  altitude  30°  12', 
because  the  sun  approaches  the  elevated  pole,  while  his  declination  changes  from 
16°  .55'  to  17°.  Assuming,  thei'efore,  the  corrected  first  altitude  as  30°  12',  tlie  second 
altitude  32°  25',  the  second  declination  corres])onding  thereto  l7°  N.,  and  the  elapsed 
time,  as  befare,  8  hours,  the  calculation  may  be  then  made  as  follov/s: — 


CoL.  1. 

El.  time  8"  [p.  m.]  Cosec.  10.0G247 
Declination  17°      N.  Sec.  10.01940 

A Cosec.  10.08187 

h  sum  alts.  31°  18^  Cosine  9.93165 
h  diff.  alts.  1  6h  Sine  8.28650 
C 1     9       Sine    8.30002 

[Z  less  ilian  90°,named  N.  or  S.  liltcthe  bearing  of  the  zenith.] 


CoL.  2. 


/CoL.  3. 


Cosine  9.74820 
Cosec.  10.23429 
Sec.  10.00008 
Cosine     9.99991 


ZScc.   10.03278 


.Cosec.  10.53406 
.Cosine     9.74850 


B  31°  27'  N.  Cosec.  10.28256 


[K  is  the 


of  B,  Z,  if  of  the  some  name  ;  difference,  if  of  a  different  mme.] 


Cosine     9.99991 

Z21  .59  N. 

E  53  26  N.     Sine   9.90480 


Latitude  53  25  N.     Sine    9.90471 


So  that  the  latitude  is  exacdy  the  same  by  both  methods. 

If  tlie  middle  time  between  the  two  observations  be  required,  it  would  be  obtained 
by  adding  the  log.  tangent  of  C  8.30263,  to  the  log.  secant  of  E  10.22493  ;  the  sunr, 
rejecting  10  in  the  index,  is  8..52756,  .which,  being  sought  for  in  iho  log.  tangents, 
correspond  in  the  Col.  P.  M.  to  O"  15™  26%  whose  half  0"  7'"  43'  is  the  middle  time 
between   tlie   two   observations.     Taking  the  sum  and  difference  of  tins  and  half 


10?J 


TO  FIND   THE   LATITUDE   BY   DOUBLE   ALTITUDES. 


the^elafiscd  time,  4*',  gives  the  times  from  noon  when  the  ohseivations  v. ere  made, 
4h  7m  433  y„j  31,  50m  175^  ti,g  Q„g  l)eiiig  before  noon,  the  otljer  afternoon.  The  same 
result  is  obtained  whichever  akitude  is  corrected. 

EXAMPLE    XII. 

Suppose  we  have,  at  the  same  moment  of  time,  the  moon's  correct  central  altitnde 
55°  20',  the  mooji'bMleciination  0°  3<J'  N. ;  the  sun's  correct  central  altitude  37°  40',  his 
dechiialion  0°  17'  S;  tlie  hour  angle,  or  difference  of  the  ri<rht  ascensions  of  the  sun 
and  moon,  as  given  hy  tlie  Nautical  Almanac,  ^hours;  required  die  true  latitude,  the 
latitude  by  account  being  23°  20'  N. 


The  tabular  correction  corresponding  to  the  latitude  by  account  23°  20'  N.,  the  sun's 

altitude  37°  40',  (considered  as  the  frsl  altitude,)  and  the  declination  0°  17'  S.,  is  50", 

the  cliange  of  the  two  declinations  from  0°  17'  S.  to  0°  30'  N.  is  (53' =)  3180 

being  muitii)lied  by  50,  and  the  two  right-hand  figures  rejected,  gives  the  corre 

of  altitude  1590"  =  20' 30"  ;  this  is  to  be  added  to  the  altitude  37°  40',  because  tl 


and  the  change  of  the  two  declinations  from  0°  17'  S.  to  0°  30'  N.  is  (53' =)  3180''  , 
this  being  muitii)lied  by  50,  and  the  two  right-hand  figm-cs  rejected,  gives  the  correc- 
tion of  altitude  1590"  =  20' 30"  ;  this  is  to  be  added  to  the  altitude  37°  40',  because  the 
change  of  the  sim's  declination  from  0°  17'  S.  to  0°  3G'  N.,  approaclies  the  sun  to  the 
elevated  pole  ;  therefore  the  sim's  corrected  altitude  is  38°  0' 30",  or  simply  38°  6'. 
Using  this  with  the  moon's  altitude  55°  20',  the  moon's  declination  0°3()'  N.,  and  the 
hour  angle  5  hours,  the  latitude  may  be  lbun*l  by  the  first  mdhcd,  in  the  following 
manner : — 

Col.  1.                                      CoL.  2.                     '       Coi..  3. 
Elapsed  time  5^  P.M.  Cosec.  10.21555 
Declination  0°  30'  N.    Sec.  10.00002     Cosec.  1 1.97998 

A Cosec.  10.21557  Cosine  9.89947     Cosine     9.89947 

h  sum  alts.  40  43  .  .Cosine    9.83008  Cosec.  10.13789 

ildifT.  alts.      8  .37  ...Sine     9.17558  Secant  10.00493 

C Sine     9.22723  Cosine  9.99372 


(Z  less  ihan  90",  iiameJ  N.or  S.  like  the  be.irinj  of  llie  zenith.] 


Z  Sec.  10.03001 


B  0°45'iN.  Cosec.  11.87945 

|R  l^is  th:w  90",  named 
N.urS.  like  ihediclin.l 

Coshie    9.99372 


[E  is  the  sum  of  13,  Z,  if  of  Ihs  eame  name ;  difference  if  of  a  different  name.] 


Z23    0^  N. 

E  23  40  N.  Sine 


Latitude  23  24  N.   Sine 


9.60532 
9.59904 


If  the  moon's  altitude,  55°  20',  be  considered  as  X\iq  first  altitude,  and  coi-rected,  the 
tabular  number  corresponding  to  this  altitude,  the  moon's  declination  0°  30'  N.,  and 
the  latitude  l)y  account  23°  20'  N.  will  be  70".  Multiplying  this  by  the  change  of 
declination  3180",  and  neglectuig  the  two  right-hand  figures,  gives  the  correction  of 
altitude  2220"  =  37' 0",  or  simjdy  37',  which  is  to  be  subtracted  from  the  moon's 
altitude  55°  20'  to  obtain  the  corrected  altitnde  54°  43',  because  tiie  change  from 
0°  30'  N.  to  0°  17'  S.  makes  the  moon  recede  from  the  elevated  pole.  Using  the 
corrected  altitude  54°  43',  the  sun's  declination  0°  17'  S.,  and  the  sun's  altitude  37°  40', 
with  the  hour  angle  5'',  the  latitude  may  be  found  by  the  first  method,  in  the  following 
manner : — 

Col.  1.                                     CoL.  2.                            Col.  3. 
Elapsed  time  5^  P.M.  Cosec.  10.21555  , 
Declination    0°  17'  S.    Sec.  10.00001     Cosec.  12.30583 


A Cosec.  10.21556 

A  sum  alts.  40  11^*  Cosine  9.81020 
d  diff.  alts.  8  3ia  . .  .Sine  9.17097 
C Sine    9.22079 


|Z  less  than  90°,  named  N.orS.  lilte  the  boarinj  of  the  zenith. 


Cosine  9.89947 
Cosec.  10.14167 
Secant  10.00482 
Cosine    9.99374 


Z  Sec.  10.03970    Z  24    7^  N, 


Cosine    9.89947 


B  0°21'i  S.Coscc.  12.20530 

[71  less  than  90",  nnmcd 
N.or  S.lilvCtlie  dcclin.) 

Cosine    9.99374 


',E  is  the  sum  of  B,  Z,  if  of  the  saTne  name  ;  d'fjf'erence  M  of  a  fli^crent  name.] 


E23  40N.   Sine     9.00532 
Latitude  23  24  N.    Sine    9.59900 


Which  agi-ees  with  the  preceding  calculation. 


"  111  taking  tlie  lia!f-sum  and  lialf-ditTerence  of  llie  altitudes,  it  will  be  cf)nvenient  to  prove  the  accu- 
racy of  the  caleiilalion  l)y  addinj;  tliis  half-sum  to  the  half-dilTercnce,  for  tlie  sum  will  he  the  j^reatcr 
akitiidf.  The  dillereiice  of  tlie  same  nuiiihcrs  will  he  the  least  altitude.  Tiiiis.  in  the  present  e.vample, 
4G°  11'^  -I-  8"  31'.i  — 54°  4'5',  the  greater  altitude,  and  4C°  ll'^  — 8°  3l'.{=:z31°  40',  the  least  altitude. 


TO    FIND   THE   LATITUDE   BY   DOUBLE   ALTITUDES.  193 


EXAMPLES  FOR   EXERCISE  IN  THIS  THIRD  METHOD. 

1.  The  sun's  correct  central  altitude  was  41°  33'  12",  his  declination  14°  N.  After 
an  interval  of  l*"  30"",  his  correct  central  altitude  was  50°  1'  12",  and  declination 
13°  58'  38"  ;  latitude  by  account  52°  5'  N.    Required  the  true  latitude. 

The  tabular  number  corresponding  to  the  altitude  41°  33'  12"  is  87",  and  this  being 
taken  for  the  first  altitude,  is,  when  corrected,  41°  32'  0" ;  the  second  altitude  is 
50°  1'  12",  the  elapsed  time  P  30-",  and  the  declination  13°  58'  38"  N.  These  make 
the  latitude  52°  5'  N. 

Or,  by  taking  50°  1'  12"  foi-  the  first  altitude,  and  using  the  corresponding  declina- 
tion, the  tabular  number  is  95",  the  carrected^rsi  altitude  becomes  50°  2'  30" ;  using 
tliis^  with  the  second  altitude  41°  33'  12",  the  declination  14°  N.,  and  the  elapsed  time 
l*"  30'",  we  find  that  the  latitude  becomes,  as  before,  52°  5'  N. 

2.  Given  the  correct  central  altitude  of  the  moon  53°  43',  her  declination  14°  IG'  N. 
After  an  interval,  in  which  the  hour  angle  was  1''  44"'  15%  her  correct  central  altitude 
was  42°  29',  and  declination  13°  52'  N. ;  the  latitude  by  account  48°  54'  N.  Required 
the  true  latitude. 

With  the  first  altitude  and  first  declination  the  tabular  number  is  98",  and  the 
coirected  first  altitude  53°  19'  28",  the  second  altitude  42°  29';  with  wliich  the  decli- 
nation 13°  52'  N.,  and  the  corrected  elapsed  time  or  hour  angle  I''  44™  15%  we  find 
that  the  latitude  is  48°  55'  N. 

Or,  by  taking  42°  29'  for  the  first  altitude,  and  13°  52'  N.  for  the  first  declination, 
the  tabular  correction  will  be  83",  and  the  corrected  first  altitude  42°  49' ;  using  this, 
and  the  second  altitude  53°  43',  the  corresponding  second  declination  14°  16'  N.,  and 
the  hour  angle  l*"  44 ""  15%  we  find  the  latitude  to  be  48°  54'  N.,  nearly ;  agreeing 
with  the  former  calculation. 

3.  Given  the  correct  central  altitude  of  the  moon,  55°  38',  her  declination  0°  20'  S. 
After  an  interval  in  which  the  hour  angle  was  5''  30™  49%  her  correct  central  altitude 
was  29°  57',  and  her  declination  1°  10'  N. ;  the  latitude  by  account  23°  25'  S.  Re- 
quired the  true  latitude. 

With  X\\Q  first  altitude  55°  38',  and  the  first  declination  0°  20^  S.,  the  tabular  coirec- 
tion  is  71",  and  the  first  corrected  altitude  54°  34'  6".  Using  this  with  the  second 
altitude  29°  57',  the  second  declination  1°  10'  N.,  and  the  hour  angle  5^  30"  49%  the 
true  altitude  will  be  found  23°  23'  S. 

Or,  by  taking  29°  57'  for  the//-s<  altitude,  and  1°  10'  N.  for  the  fi.rst  declination,  the 
tabular  correction  will  be  45",  and  the  first  corrected  altitude  30°  37'.  Using  this  with 
the  second  altitude  55°  38',  the  second  declination  0°  20'  S.,  and  the  hour  angle 
5'"  30"'  49%  the  true  latitude  will  be  found  to  be  23°  24'  S.,  nearly  agreeing  with  the 
preceding  calculations. 

In  making  the  calculations  of  these  three  examples,  the  seconds  ivere  noticed,  ivhich  is 
alivays  best  to  be  done,  particularly  ivhen  the  altitudes  are  nearly  equal ;  some  difference 
migiit  be  found  in  the  above  results  if  the  nearest  minutes  only  were  taken.  Thus, 
Example  XL,  calcidating  to  the  nearest  minute,  gives  the  latitude  53°  28'.  If  the 
calculation  be  made  as  in  page  191,  it  becomes  53°  25',  differing  3'.  This  difference 
would  be  avoided  by  taking  the  angles  to  seconds,  and  in  some  extreme  cases  it  would 
require  the  use  of  G  or  7  places  of  decimals. 


FOURTH  METHOD. 

To  Jiad  the  latitude  by  double  altitudes  of  the  same  or  different  objects,  the 
declinations  being  different. 

This  method,  like  the  first,  requires  only  the  use  of  Table  XXVIL ;  and  the  words 
sine,  cosine,  &c.,  are  written  for  log.  sine,  log.  cosine,  &c.  The  logarithms  are  aiTanged 
in  these  columns  as  in  the  first  method,  according  to  the  following  formula,  which 
ought  to  be  written  down  before  the  calculation  is  commenced ;  this  will  sim}>lify  the 
operation,  and  may  prevent  mistakes.  In  this  formula  it  is  said  that  C  is  of  the  same 
affection  as  B;  the  meaning  of  which  is,  that  if  B  is  less  than  90°,  C  also  is  less  than 
90° ;  and  if  B  is  greater  than  90°,  C  also  is  greater  than  90°.  Likewise  A  is  of  the 
same  affection  as  the  hour  angle  H,  meaning  that  if  the  hour  angle  is  less  than  6 
hours  or  90°,  A  will  be  less  than  90° ;  and  if  the  hour  angle  exceed  6  houi-s,  the  angle 
A  will  exceed  90°. 

25 


194 


TO   FIND  THE   LATITUDE   BY  DOUBLE   ALTITUDES. 


Col.  1. 
RoarangleH[  F.M.]..Sec. 

Decli.  d  [at  gr.  alti.J  Tan. 

A  [diff.  name  from  d.]  Tan. 

D.Dec.[at  least  alt.] 

B 

C Cosec. 


Least  altitude.... 

Sec 

Greatest  altitude.. 

Sum,  3  last  num. 

J  Sum 

S  S— g.  alt.=Rera. 

Sine 

Sum  of  4  logs. 

2) 

SZ 

Sine 

FORMULA. 

CoL.  2. 

Col.  3. 
Tan 

Sine 

A  [same  affection  as  HJCosec. 

...Cosine 

F 
Z 

C  [same  affection  as  B]  Cosine 

Cotan. 

[F  les.  than  90°,dlfl. 

G Sine 

G 

Sine 

[I  less 
[I  nau 

I                                         Tan. 

than  90°]  Sec 
led  as  G.] 

Dec.  D                     [at  least  alt.] 

K 

Latitude 

Sine 

[Z  named  N.  or  S.,  like  the  bearing  of  the  zenith.] 

In  some  late  works  on  navigation,  no  notice  is  taken  of  tlie  cases  where  the  hour 
angle  exceeds  90°,  or  the  distance  of  the  objects  exceeds  90°,  and  on  that  account  the 
rules  appear  less  subject  to  different  cases  than  the  following  rule,  which  embracee 
all  possible  cases,  and  the  apparent  simplicity  of  the  rules  referred  to,  arises  from 
their  imperfections  and  incompleteness. 


RULE. 

1.  Find  the  hour  angle  H,*  and  take  out  the  corresponding  secant,  which  put  in 
Col.  1,  and  its  tangent  in  Col.  3. 

2.  Take  the  declination  d,  coiresponding  to  the  greatest  altitude,  place  its  tangent 
in  Col.  1,  its  sine  in  Col.  2. 

3.  The  sum  of  the  two  logarithms  in  Col.  1  (rejecting  10  in  the  index)  is  the  tan- 
gent of  the  angle  A,  which  is  less  than  90°  if  the  hour  angle  is  less  than  6  hours,  (or 
90°,)  but  greater  than  90°  if  the  hour  angle  is  greater  than  6  hours.  This  angle  is  to 
be  marked  noHh  and  south,  with  a  different  name  from  the  declination  d,  at  the  greatest 
altitude.     The  cosecant  of  A  is  to  be  placed  in  Col.  2,  its  cosine  in  Col.  3. 

4.  Place  the  declination  D,  corresponding  to  the  least  altitude,  below  the  angle  A, 
and  if  they  are  of  the  f  same  name,  take  their  sum,  but  if  of  different  names,  take  their 
difference,  and  call  this  sum,|  or  difference,  the  angle  B,  making  it  north  or  south,  like 
the  greatest  of  the  two  quantities  A,  D.  The  cosine  of  B  is  to  be  placed  in  Col.  2, 
its  cosecant  in  Col.  3. 

5.  The  sum  of  the  three  logarithms  in  Col.  3  (rejecting  20  in  the  index)  is  the 
cotangent  of  an  angle  F,  (less  than  90°,)  which  is  to  be  taken  out  and  marked  north  or 
soidh,  with  a  different  name  from  B. 

6.  The  sum  of  the  three  logarithms  in  Col.  2  (rejecting  20  in  the  index)  is  the 
cosine  of  the  angle  C,  which  is  to  be  taken  less  than  90°  if  B  is  less  than  90°,  but 
greater  than  90°  if  B  is  greater  than  90°.  The  angle  C,  and  its  cosecant,  are  to  be 
placed  in  Col.  1. 

7.  Place  the  altitudes  below  C,  take  the  half-sum  of  these  three  quantities,  subtract 
the  greatest  altitude  from  the  half-sum,  and  note  the  remainder.  Place  the  secant  of 
the  least  altitude  in  Col.  1,  its  cotangent  in  Col.  2,  its  sine  in  Col.  3 ;  the  cosine  of 
the  half-sum  in  Col.  1,  and  the  sine  of  the  remainder  in  Col.  1.     The  sum  of  the  four 

*  The  hour  aii^le  is  the  same  as  the  elapsed  time  in  double  altitudes  of  the  sun.  This  time  is  turned 
mlo  deoj-rees  by  Table  XXI.,  but  it  is  more  simple  to  double  the  hour  angle,  and  find  it  in  Col.  P.  M., 
Table  XXVII.,  and  take  out  its  corresponding  tangent.  If  this  double  angle  exceeds  ISi",  reject  IS^, 
and  find  the  remainder  in  Col.  a.  m.,  and  take  out  its  corresponding  tangent.  lu  the  following  exam- 
ples this  double  angle  is  marked  with  the  letters  P.  M.  annexed. 

t  This  rule  is  easily  remembered  in  three  places  in  which  it  occurs,  from  the  circumstance  that  s  is 
the  first  letter  o(  sum  and  same,  and  d  the  first  letter  o(  difference  and  different. 

X  If  the  sum  be  taken  to  find  B,  and  it  exceed  180°,  subtract  it  from  360°,  and  call  the  remainder  I^ 
with  a  different  name  from  that  of  A,  D. 


TO   FIND   THE   LATITUDE   BY  DOUBLE   ALTITUDES. 


195 


last  logarithms  of  Col.  1,  (rejecting  20  in  the  index,)  being  divided  by  2,  gives  the  sine 
of  an  acute  angle,  which  being  found  and  doubled,  gives  the  zenith  angle  Z,  which  is 
to  be  named  Jiorth  if  the  zenith  and  north  pole  are  on  the  same  side  of  the  are  or 
^eat  circle,  passing  through  the  two  objects,  (or  the  two  observed  places  of  the  same 
object,)  but  south  if  the  zenith  and  south  pole  ai'e  on  the  same  side  of  that  great 
circle.* 

8.  Take  the  sum  of  the  angles  Z  and  F  if  they  are  of  the  same  name,  but  their 
difference  if  of  different  names  ;  this  sum  or  difference  is  the  angle  G,  to  be  marked 
north  or  south,  like  the  greatest  of  the  angles  Z,  F.f  The  sine  of  G  is  to  be  placed 
in  Col.  2. 

9.  The  sum  of  the  two  lower  logarithms  of  Col.  2  (rejecting  10  in  the  index)  is  the 
tangent  of  an  angle  I,  which  is  to  be  taken  out  (less  than  90°)  and  marked  north  or 
south,  like  G.     The  secant  of  I  is  to  be  placed  in  Col.  3. 

10.  Write  the  declination  D,  corresponding  to  the  least  altitude  below  I,  take  their 
sum  if  of  the  same  names,  their  difference  \{  of  different  names.  This  sum  or  difference 
is  the  angle  K,  of  the  same  name  as  the  greater  of  these  two  quantities.  The  sine 
of  K  is  to  be  placed  in  Col.  3. 

11.  The  sum  of  the  three  last  logai-ithms  in  Col.  3  is  the  sine  of  the  required  lati- 
tude, of  the  same  name  as  K. 


EXAMPLE    XIII. 

Given  the  sun's  correct  central  altitude  41°  33',  and  his  declination  14°  N.  After 
an  interval  of  l"*  30",  by  watch,  his  correct  central  altitude  was  50°,  and  his  declina- 
tion 13°  58'  N.  Required  the  latitude,  the  sun  being  south  of  the  observer  when  on 
the  meridian. 


Col.  '.- 

Hour  ang.  H  Ih  33m  [p.  m.  3h]  Sec.  10.03438 
Decli.  d.  [at  gr.  alti.]  13° 58'  N.  Tan.  9.395C9 
A  [dif.name  from  d.]  15  04  S.  Tan.  9.43007 
D  Dec.  [at  least  alt.]  14  00  N. 

B 1  04  S. 

C 21  49  Cosec.  I0.420S8 


Least  altitude 41  33  Sec.  10.1258s 

Greatest  altitude 50  00 

Sum 113  22 

JSum 56  41  Cosine  9.73978 

J  S.— gr.  alt.  =  Rem.    6  41  Sine  9.0n589 

Sum  4  logs.  2)19.36143 

JZ 28  39  Sine  9.68071 


CoL.  2. 


Sine  9.38266 

A  [same  aff.  as  H.]  Cosec.  10.58512 


.Cosine  9.99992 


C  [same  aff.  as  B.]  Cosine  9.96770 

G Sine  9.93738 

Cotan.  10.0.5243 

I  44''20'N.        Tan.  9.98981 


Dec.D.  14  00  N.  [at  least  alt.] 
K  58  20  N. 

Latitude 52 


CoL.  3. 

Tan.    9.61722 


.Cosine  9.98481 


.Cosec.  11.73012 


F  2M0'N.  Cotan.  11.33215 

Z  57  18  N.  [F  lei 
G  59  53  N. 


1 90°,diff. 
mB.] 


Sine    9.82169 

llessthan90'']Sec.l0.14552 
1  named  as  G.] 


.Sine   9.92999 


7  N.       Sine    9.89720. 


57  18  N.  [named  like  bearing  of  zenith.] 


If  the  latitude  had  been  south,  Z,  instead  of  being  57°  18'  north,  would  be  57°  18' 
south ;  G  =  54°  38'  S.,  I  =  42°  37'  S.,  K  =  28°  37'  S.,  and  the  latitude  25°  34'  S.  The 
labor  of  making  this  extra  calculation  is  but  little,  and  where  any  doubt  exists  of  the 
name  of  Z,  it  is  best  to  make  the  computation  both  ways;  this,  however,  will  rarely 
happen.  The  calculations  of  this  example,  and  most  of  the  following  ones,  are  made 
to  the  nearest  minute ;  where  great  accuracy  is  required,  it  will  be  proper  to  take  the 
logarithms  and  angles  corresponding  to  seconds. 


*  This  case  occurs  also  in  the  first  and  second  methods  of  solution,  and  it  must  be  determined  on  the 
spot  by  the  situation  of  the  objects.  In  double  altitudes  of  the  sun,  moon,  or  planets,  when  the  elapsed 
time  is  not  very  great,  the  angle  Z  is  generally  to  be  marked  with  the  bearin"^  of  the  zenith  from  the 
observed  object,  when  at  its  greatest  altitude  on  the  meridian,  which  in  north  latitudes,  without  the  trop- 
ics, is  in  general  north;  in  south  latitudes,  without  the  tropics,  south.  Sometimes,  when  the  sun  passes 
the  meridian  near  the  zenith,  it  may  be  doubtful  whether  the  zenith  be  north  or  south  ;  in  which  case 
the  problem  may  be  solved  for  both  cases,  (which  increases  the  labor  but  little,)  and  that  one  of  the  two 
computed  latitudes  selected  which  agrees  best  with  the  ship's  reckoning;  but  it  is  generally  safest  not 
to  use  observations  of  this  kind,  which  are  generally  liable  to  great  errors  from  small  mistakes  in  the 
altitudes. 

t  If  the  stim  be  taken  to  find  G,  and  it  exceed  180°,  subtract  it  from  360°,  and  call  the  remainder  G,. 
with  a  different  name  from  Z  or  F 


196 


TO  FIND   THE   LATITUDE   BY   DOUBLE   ALTITUDES. 


EXAMPLE   XIV. 

The  sun's  correct  central  altitude  was  32°  25',  his  declination  17°  0'  S. ;  8  hours 
afterwards,  by  a  watch,  the  sun's  correct  central  altitude  was  30°  8',  and  declination 
16°  55'  S.,  the  observer  being  in  a  high  soutli  latitude ;  required  the  latitude. 


Col.  L 

Hour  H  8li  [p.m.  lGh=4ti  a.  m.]  Sec.  10.30103 
Decli.d.[atgr.alti.]  17'00'S.  Tan.  9.48534 
A[dif.name  from  d.]  148  33  N.  Tan.  9.78637 
D  Dec.  [at  least  alt.]    16  55  S. 

B 131  33  N. 

C Ill  51  Cosec.  10.03238 


Least  altitude 30  08     Sec.  10.06305 

Greatest  altitude....  32  25 

Sum .174  24 

iSum 87  12  Cosine  8.68886 

^S.— gr.alti.=  Rem.  54  47      Sine  9.91221 
Bum  4  logs. 

iZ 12  53 

Z 


Col.  2. 


2)18.69650 


Sine  9.34825 


Sine    9.46594 

A  [same  aff.as  H.]  Cosec.  10.28253 


.  Cosine   9.82240 


Cfsameaff.  as  B.]  Cosine   9..57087 

G Sine  9.90005 

Cotan.  10.23623 

I  53°51'S.       Tan.  10.13628 


Dec.  D  16  55  S.  [at  least  alt.] 

K  70  46  S 

Latitude 53  28  S 


Col.  3. 

Tan.  10.23856 


.Cosine   9.93100 


.Cosec.  10.12644 


F  26°50'S.  Cotan.  10.29600 


Z25  46  S.[F  less  than  90°,cUff. 

name  from  B.] 

G  52  36  S. 

Sine   9.70072 


llesstlian90°]Sec. 10.22922 
I  named  as  G.] 


.Sine   9.97506 


Sine   9.90500 


25  46  S.  [named  like  bearing  of  zenith.] 

This  latitude  differs  3'  from  the  calculation  in  Example  XL,  page  191,  on  account 
of  not  noticing  the  seconds  in  the  angles. 

If  the  zenith  had  been  north  of  the  great  circle  passing  through  the  sun  and  moon, 
we  should  have  Z  =:25°  4G'  N.,  G  =  1°  04'  S.,  1  =  1°  50'  S.,  K  =:  18°  45'  S.,  and  the 
latitude  9°  18'  S. 


EXAMPLE   XV. 

Suppose,  at  the  same  moment  of  time,  the  moon's  correct  central  altitude  was 
55°  20',  the  moon's  declination  0°  3G'  N.,  the  sun's  correct  central  altitude  37°  40',  the 
sun's  declination  0°  17'  S. ;  the  hour  angle,  or  difference  of  the  right  ascensions  of 
the  sun  and  moon,  being,  by  the  Nautical  Almanac,  5  hours,  or  75°.  Requii'ed  the 
latitude,  supposmg  it  to  be  north. 


Col.  1. 

Hour  angle  H  5h  [p.  m.  lOh]  Sec.  10.58700 
Decli.  d.  [at  gr.  alt.]  0''36'N.  Tan.  8.02004 
A[dif.namefromd.]  2  19  S.  Tan.  8.C0704 
D  Decl.  [at  least  alt.]  0  17  S. 

B 236  S. 

C... 75  00    Cosec.  10.01506 


Col.  2. 


Least  altitude 37  40 

Greatest  altitude...  55  20 


Sec.  10.10151 


Sum 168  00 

J  Sum 84  00  Cosine   9.01923 

4  S.— gr.alt.=Rem.  28  40  Sine  9.68098 
Sum  of  4  logs.  2)18.81678 
14  50  Sine   9.40839 


Sine    8.02002 

A  [same  aff.as  II.]  Cosec.  11.39338 


.Cosine    9.999.55 


C  [same  aff.  as  B.]  Cosine  9.41295 


G Sine  9.70375 

. . .  i Cotan.  10.11241 

I  33"'13'N.      Tan.    9.81616 


Sine    9.78609 

rilesstlian90°]  Sec.10.07748 

„      ».    „  ._  ^  r     .  ■■  [InamedasG.l 

Dec.  D.    0  17  S.  [at  least  alt.]  "•  -" 

K  32  56  N.  Sine    9.73533 

Latitude 23°  24' N.      Sine    9.59890 


Col.  3. 

Tan.  10.57195 


.Cosine    9.99904 


.Cosec.  11.34330 


F    0°42'N.  Cotan.  11.91489 


Z  29  40  N.  [Flessllian90'',diff. 
uanw  from  B.] 


4Z 

Z 29  40  N.  [named  like  bearing  of  zenith.] 

This  latitude  agr-ces  with  the  calculation  in  Example  XIL,  page  192 

If  the  zenith  had  been  south  of  the  great  cu-cle  passing  through  the  objects,  we 
should  have  Zz=29°  40'  S.,  G  =  28°58'  S.,  I  =  32°G'  S.,  K=:32°23'  S,  and  the 
latitude  22°  44'  S. 


TO   FIND   THE   LATITUDE   BY   DOUBLE   ALTITUDES. 


197 


EXAMPLE    XVI. 
Given  the  moon's  correct  central  altitude  47°  37',  the  moon's  declhiation  17°  29'  S, 
the  sun's  correct  central   altitude,  at  the  same  time,  27°  22',  the  sun's  declination 
8°  28'  S.,  the  hour  angle,  or  difference  of  right  ascensions  of  the  sun  and  moon, 
5h  40ni  28»,  or  85°  7' ;  required  the  latitude,  supposing  it  to  be  north. 


Col.  L 

Hr.  HSo"?'  [p.m.  Ill'  201  5C"]  Sec.  11.0G993 
Decli.  d.  [at  gr.  alt.]  17°  29'  S.  Tan.  9.49828 
A  [dif.nariie  from  d.]  74  53  N.  Tan.  10.56821 
D  Decl.  [at  least  alt.]   8  23  S. 

B 66  25  N. 

C 82  51    Cosec.  10.00339 


Col.  2. 


Least  altitude 27  22 

Greatest  altitude...    47  37 


Sec.  10.05155 


8um 157  50 


Sine    9.47774 

A  [same  aff.  as  H.]  Cosec.  10.01529 


Cosine   9.60215 

C  [sameaff.asB.]  Cosine   9.09518 

G Sine   9.58497 

Cotan.  10.28599 

I              3G''37iN.    Tan.    9.87096 
Dec.  D.     8  28  S.  [at  least  alt.] 
K  28  09  N 


Col.  3. 

Tan.   11.00835 


.Cosine    9.41628 


.Cosec.  10.03788 


F  16°  43'  S.  Cotan.  10.5*251 


Z  39  20  N.  [Flesstha.i90°,diB-. 

Dame  from  B.] 

G  22  37  N. 


Sine   9.06246 

rilessthan90°]Sec.lO  09548 
[I  named  asG.] 


.Sine  9.07374 


Latitude  15°  41'  N.         Sine  9.43168 


i  Sum 78  55  Cosine    9.28384 

J  S,— gr.  aIt.=Rem.   31  18      Sine    9.71560 
Bum  of  4  logs.  2)19.05438 

\7. 19  40      Sine    9.52719 

Z 39  20  N.  [named  like  the  bearing  of  zenith.] 

If  the  zenith  had  been  south  of  the  gi-eat  circle  ))assing  through  the  objects,  we 
should  have  Zrr39°20'S.,  G=:56°3'  S.,  I  =  58°  2'  S.,  K  =  66°  3(y  S.,  and  the 
latitude  52°  46'  S. 

FIFTH  METHOD. 
To  Jiiid  the  latitude  from  the  altitudes  and  distances  fotind  in  taking  a  lunar 

observation. 

This  is  a  particular  case  of  Form  V.,  and  is  more  simple  than  the  general  solution, 
because  the  true  distance  of  the  objects,  computed  in  working  the  lunar  observation, 
may  be  used  to  shorten  the  calculation  of  the  latitudes ;  we  shall  therefore  give  a 
particular  rule  for  this  method. 

Having  the  apparent  altitudes  and  distance  of  the  objects,  find,  by  any  of  the 
methods  of  working  a  lunar  observation  hereafter  given,  the  true  distance.  Find  also 
the  true  altitudes,  by  correcting  the  apparent  altitudes  for  parallax  and  refraction. 
The  correction  of  the  moon's  altitude  is  equal  to  the  difference  between  59'  42"  and  the 
correction  already  found  from  Table  XIX.,  in  working  the  lunar  observation ;  this 
difference,  added  to  the  moon's  apparent  altitude,  gives  her  true  altitude.  In  like  man- 
ner the  correction  of  the  sun's  altitude  is  equal  to  the  difference  between  GO'  and  the 
correction  already  found  in  Table  XVIII.  (or  in  Table  XVII.  if  a  star  or  ])lanet  is  used) ; 
this  difference  is  to  be  subtracted  from  the  sun's  (or  star's)  apparent  altitude,  to  obtain 
its  true  altitude.  The  time  at  Greenwich,  corresponding  to  the  true  distance,  having 
been  found  in  working  the  lunar  observation,  take  from  the  Nautical  Almanac,  for  this 
time,  the  declinations  of  the  sun  and  moon,  as  is  taught  in  pages  156,  171.  If, 
instead  of  the  sun,  a  star  is  used,  its  declination  may  be  obtained  from  Table  VIII., 
or  more  accurately  from  ttie  Nautical  Almanac,  if  it  be  one  of  the  100  bright  stare 
whose  places  are  now  given  for  every  ten  days  in  that  work.  If  a  planet  is  used,  its 
declination  is  to  be  found  in  the  Nautical  Almanac,  From  these  declinations,  the 
north  polar  distances  must  be  found,  by  adding  the  declinations  to  90°  if  south,  or 
subtracting  from  90°  if  north. 

Having  thus  obtained  the  true  distance,  the  true  altitudes,  the  declinations  and  north 
polar  distances,  the  latitude  may  be  computed  by  the  following  rule,  adapted  exclu- 
sively to  Tal)le  XXVII.,  writing,  as  before,  sine,  cosine,  &c.,  for  log.  sine,  log.  cosine, 
&c.,  the  logarithms  being  arranged  in  three  columns,  as  in  the  former  methods. 

RULE. 
1.  Place  in  Col.  1  the  true  distance  and  the  polar  distances.  Take  their  half-sum, 
subtract  from  this  half-sum  the  polar  distance  of  the  object  which  had  the  greatest 
altitude,  and  note  the  remainder.  Put  in  the  same  column  the  cosecant  of  the  true 
distance,  the  cosecant  of  the  polar  distance  of  the  object  having  the  least  altitude,  the 
sine  of  the  half-sum,  the  sine  of  the  remainder.  The  sum  of  tliese  four  logarithms 
(rejecting  20  in  the  index)  being  divided  by  2,  gives  the  sine  of  an  acute  angle,  wliich 
being  found  and  doubled,  is  to  be  called  the  angle  F. 


j98 


TO  FIND   THE   LATITUDE   BY   DOUBLE  ALTITUDES. 


2.  Place  in  Col.  1  the  true  distance  and  the  true  altitudes.  Take  their  half-sum^ 
and  also  the  remainder  or  difference  between  the  half-sum  and  the  gi-eatest  altitude. 
Place  in  the  same  column  the  cosecant  of  the  distance,  (before  found,)  the  secant  of 
the  least  altitude,  the  cosine  of  the  half-sum,  the  sine  of  the  remainder.  The  sum  of 
these  four  logarithms  (rejecting  20  in  the  index)  being  divided  by  2,  gives  the  sine  of 
an  acute  angle,  which  being  found  and  doubled,  is  to  be  called  the  angle  Z. 

3.  If  the  zenith  and  north  pole  be  situated  on  the  same  side  of  the  great  circle, 
passing  through  the  two  objects,*  take  the  sum\  of  the  angles  F  and  Z  for  the  angle 
G  ;  but  if  the  zenith  and  nodh  pole  be  situated  on  different  sides  of  that  great  circle, 
take  their  difference  for  the  angle  G.     Place  the  cosine  of  G  in  Col.  2. 

4.  Write  in  Col.  2  the  cotangent  of  the  least  altitude,  and  its  sine  in  Col.  3.]:  The 
sum  of  the  two  logarithms  in  Col.  2,  is  the  tangent  of  the  angle  I,  which  is  to  be  taken 
less  tlian  90°,  and  marked  south  if  the  angle  G  is  less  than  90°,  but  noHh  if  G  is  more 
than  90°.     Place  the  secant  of  I  in  Col.  3. 

5.  Place  the  declination  corresponding  to  the  least  altitude,  below  I ;  take  their 
sum  if  of  the  same  name,  but  their  difference  if  of  diffehnt  names  ;  call  this  sum  or 
difference  the  angle  K,  and  mark  it  with  the  same  name  as  the  greatest  of  the  two 
quantities.     Place  the  sine  of  K  in  Col.  3. 

6.  The  sum  of  the  three  logarithms  in  Col.  3  (rejecting  20  in  the  index)  is  the  sine 
of  the  latitude,  of  the  same  name  as  K. 

Having  found  the  latitude,  the  hour  may  be  obtained  by  means  of  the  true  altitude 
and  declination  of  the  sun,  star,  or  planet,  by  any  of  the  usual  methods  hereafter  given 
for  tliat  purpose ;  but,  if  the  last  of  the  observed  altitudes  was  that  of  the  sun,  star,  or 
planet,  the  horary  distance  of  that  object  from  the  meridian  might  be  obtained  more 
simply  by  the  following  rule,  adapted  to  Table  XXVII. 

Rule.  Add  the  tangent  of  the  angle  G,  the  sine  of  the  angle  I,  the  secant  of  the 
angle  K ;  the  sum,  rejecting  20  in  the  index,  is  the  tangent  of  an  angle  ;  take  out  the 
corresponding  time  in  the  column  P.  M.  or  in  the  column  A.  M.  increased  by  12 
hours;  half  of  either  of  these  times  is  the  horary  distance  of  the  lowest  observed  object 
from  the  upper  or  lower  meridian,  whence  the  hour  may  be  obtained  directly  if  it  be 
the  sun,  but  if  it  be  the  star,  a  planet,  (or  the  moon,)  it  is  obtained  by  apj)lying  its 
horary  distance  to  the  hour  of  passing  the  meridian,  according  to  the  usual  methods  of 
finding  the  time  from  an  altitude  of  a  fixed  star  or  the  moon. 

EXAMPLE  XVII. 
[Same  as  Dr.  Brinkley's,  in  the  N  A.,  1825.] 
May  19 '  S**  6™,  P.  M.,  in  the  longitude  of  7"  23'"  west,  it  was  found,  by  working  a 
lunar  observation,  that  the  correct  distance  of  the  centres  of  the  sun  and  moon  was 
90°  57'  20";  ti-ue  altitude  of  the  sun's  centre  11°  33'  12";  true  altitude  of  the  moon's 
centre  27°  32'  18".  At  the  same  time,  by  the  Nautical  Almana'*,  the  sun's  declination 
was  19°  56'  48"  N.,  the  moon's  declination  13°  55'  48 '  N  Required  the  latitude  and 
hour  by  this  observation. 

Col.  1  Col.  2.  Col.  3. 


True  distance 
P.  dist.at  le.  alt 
P.  dist.  at  gr.  alt 

90°  57' 20" 
.  70  03  12 
.  76  04  12 

Cosec.  10.00006 
Cosec.  10.02687 

Sum 

;  Sum 

S — p.d.atgr.a 

237  04  44 

118  32  23 
.42  28  10 

52  36  00 

Sine    9.94374 
Sine    9.82943 

G  is  sum  of  F,  Z 
great  circle 

Z        61°  36*  52" 

if  north  pole  an 
,  but  their  differe 

d  zenith  are  on  same  side  of 
ice  if  on  dilTerenl  sides. 

iF 

2 ) 19.80010 
Sine    9.90005 

I  is   less  than  90°,   named 
south  if  G  IS  less  than  90', 
north  if  G  is  more  than  90°. 

Angle  F 

105  12  00 

Cosec.  10.00006 
Sec.  10.00888 

F 
G 

105  12  00 
166  48  52 

Cosine  9.98840 
Cotan.  10.68947 

Tan.  10.67787 

Tnie  distance. 

90  57  20 
11  33  12 

27  32  18 

.     Sine    9  30163 

Greatest  alt.... 

I 

78  08  33  N. 

Sec.  10.68723 

130  02  50 

Dec 

19  56  48  N. 

(at  least  alt.) 

J  Sum 

C5  01  25 
37  29  07 

Cosine    0.62557 
Sine    9.78430 

K 

98  05  21  N 

Sine    9.99566 

i  Sum — gr.  alt 

Lati 

lude74°48'N.   Sine    9.98452 

2)19.41881 

4Z 

Angle  Z 

30  48  26 
61  36  52 

Sine    9.70940 

*  In  places  without  the  tropics,  the  sum  is  used  generally  in  northern  latitudes,  and  the  difference  in 
southern  latitudes. 

t  If  this  sum  should  exceed  180°,  subtract  it  from  .360°,  and  call  the  remainder  the  ansjlc  G. 

X  Both  these  logarithms  may  bo  taken  out  at  the  same  time  when  the  sine  of  the  euigie  was  found  in 
the  computation  of  the  angle  Z 


TO  FIND   THE   LATITUDE   BY   DOUBLE   ALTITUDES.  199 

To  fnd  the  hour. 

G Tan.    9.36974 

I Sine     9.99063 

K Sec.    10.85166 

Hour  P.  M.  7h  47ra  42«,  or  A.  M.  +  lOh  =  iGh  12m  18s Tan.  10.21203 

Divided  by  9,  gives  the  horary  distance  of )  „^  -3^  -,,  „^  „.    „„, 

the  lowest  object  from  the  meridian,       j  J"  3^™  »i'.  or  tjn  Obm  UJ«. 

The  sun  being  at  the  lowest  altitude,  his  distance  from  the  upper  meridian  waa 
gh  gm  9>^  being  the  hour  of  the  day,  and  the  sun's  distance  from  the  lower  meridian, 
or  midnight,  was  S*"  53™  51'.  '' 

ADDITIONAL  QUESTIONS  FOR  EXERCISE. 

In  the  following  questions  the  sun's  semidiameter  is  supposed  to  be  16',  and  the 
pai'allax  nothing. 

1.  Being  at  sea,  in  latitude  by  account  39°  28'  N.,  when  tlie  sun's  declination  was 
20°  41'  N.,  at  ll''  30™  15%  A.  M.,  per  watch,  the  altitude  of  the  sun's  lower  limb  was 
observed  to  be  68°  18'  45",  and  at  12"  26™  28»  P.  M.  was  70°  58',  the  height  of  the  eye 
beuig  21  feet  above  the  surface  of  the  sea.     Requii-ed  the  true  latitude  of  the  ship. 

Answer,  39°  28'  N. 

2.  Being  at  sea  in  latitude  50°  40'  N.  by  account,  at  10'^  17™  30%  A.  M.,  per  watch, 
the  altitude  of  the  sun's  lower  limb  was  observed  to  be  17°  A'\,  and  at  11''  17™  30^  was 
\Q°  31'i,  the  declination  being  20°  S.,  and  the  height  of  the  eye  21  feet  above  the  sea. 
Required  the  latitude  in.  Answer,  50°  1'  N. 

3.  Suppose  a  ship  at  sea,  in  latitude  47°  34'  N.  by  account,  and  that  at  9''  55™  30% 
by  watch,  the  jaltitude  of  tlie  sun's  lower  limb  was  17°  24',  bearing  by  compass  S.  by 
E.  k  E.,  and  at  12''  54™  10'  the  altitude  of  the  same  limb  was  21°  45'i,  the  declination 
being  19°  30'  S.,  the  height  of  the  eye  20  feet  above  the  sea,  and  the  ship's  course  by 
compass  E.  h  S.,  at  the  rate  of  7  knots  per  hour.     What  was  the  true  latitude  ? 

Answer,  47°  24  N. 

4.  At  ll"*  28™  20%  A.  M.,  per  watch,  the  altitude  of  the  sun's  lower  limb  was 
28°  18',  the  sun  bearing  S.  by  W.  by  compass.  At  2'>  58™  20%  P.  M.,  the  altitude  of 
the  same  limb  was  16°  40',  the  height  of  the  eye  20  feet,  his  declination  13°  17'  S., 
and  the  latitude  by  accoimt  47° 50'  N.,  the  ship's  course  during  the  elapsed  time  N.  E., 
with  her  larboard  tacks  on  board,*  sailing  at  the  rate  of  6  knots,  and  making  half  a 
point  lee-way.     What  latitude  was  she  in  when  the  last  altitude  was  taken  ? 

Answer,  48°  9'  N. 

*  The  larboard  side  of  a  ship  is  the  left  side,  when  the  observer  is  aft,  lookins;-  towards  Iier  head,  and 
the  starboard  is  (he  right  side.  When  a  ship  is  sailing  with  her  larboard  tacks  on  board,  the  lee-way  is 
allowed  to  the  right  hand ;  but  if  her  starboard  tacks  are  on  board,  to  the  left  hand. 

In  calculating  the  answers  to  these  questions,  proportional  parts  were  taken  for  the  seconds ;  a  small 
dlfibrence  woukl  be  found  if  the  nearest  logarithms  only  were  taken. 


200 


TO  FIND  THE  LATITUDE  BY  ONE  ALTI- 
TUDE OF  THE  SUN  TAKEN  NEAR  NOON, 
HAVING  THE  TIME  OF  OBSERVATION. 


When  the  sun  does  not  pass  near  tlie  zenith,  the  meruhan  ahitude  and  the  latitude 
of  the  place  may  be  accurately  determined  by  observing  his  altitude  when  near  the 
meridian,  and  noting  the  time  by  a  watch  regulated  thejjreceding  morning  or  follow- 
ing evening,  by  either  of  the  methods  given  in  this  work.*  To  this  time  by  the  watch 
must  he  a])plied  a  correction  equal  to  the  difference  of  longitude  made  by  the  ship 
(tiumed  into  time)  in  the  interval  between  the  regulation  and  the  observation  near  the 
meridian,  by  adding  ivhen  the  place  of  regxdation  is  to  the  loestward  of  the,  place  of  taking 
the  other  ohservntion,  otherwise  oij  subtracting ;  the  sum  or  difference  will  be  the  time  of 
taking  the  observation ;  whence  the  time  from  noon  will  be  obtained ;  with  which, 
and  the  observed  altitude,  (corrected  for  semidiameter,  dip,  &c.,  as  usual,)  the  sim's 
declination,  (found  in  Table  IV.,  or  in  the  Nautical  Almanac,  and  corrected  for  the 
longitude  oi'  the  ship,)  and  the  latitude  by  account,  the  latitude  by  observation  may 
be  found  as  follows  : — 

RULE. 

Jlddtogether  the  log,  cosine  of  the  latitude  by  account,  {Table  XXVII.)  the  log.  cosine  of 
the  declination,  [Table  XXVII.)  the  logarithm  in  the  column  of  rising,  [Table  XXIII.)  cor- 
responding to  the  apparent  time  from  noon  lohen  the  observation  ivas  taken  ;  reject  20  in  the 
index;  the  natural  number  of  the  remainder  being  foxmd,  [in  Table  XXVI.)  and  added  to  the 
naiural  sine  of  the  observed  altitude,  ( Table  XXIV.)  the  sum  tvill  be  the  natural  cosine  of 
Oie  meridian  zenith  distance,  from  ivhich  the  latitude  may  be  obtained  by  the  common  rules. 

If  the  computed  latitude  differs  considerably  from  the  latitude  by  account,  it  is 
best  to  rejjeut  the  operation,  using  the  latitude  last  found  instead  of  the  latitude  by 
account.  This  method  of  finding  the  latitude  by  a  single  altitude  of  the  sun,  may  be 
applied  to  any  other  celestial  object. 

EXAMPLE   I. 

Being  at  sea,  in  latitude  49°  50'  N.  by  account,  when  the  sun's  declination  was 
20°  S.,  at  11''  29'"  20%  A.  M.,  apparent  time,  per  watch,  regulated  the  preceding  morn- 
ing, in  a  ]ilace  20  miles  of  longitude  to  the  eastward,  the  sun's  correct  central  a.titude 
was  19°  41',f  bearing  south.     Required  the  true  latitude. 

Time  per  watch 11"  29"-  20» 

2(y  in  time  by  Tab.XXL         1    20 

Time  of  observation..  11  28      0      Latitude  49°  SO'  Cosine  9.80957 

12  Declin.     20     0  Cosine  9.97299 


App.  time  from  noon . .       32      0 Log.  rising  2.98820 

Nat.  Num.  590  log.  2.77076 
Central  altitude  19°  41'  Nat.  Sine      33682 


Mer.  zen.  dist.  69  57  N.  Nat.  Cosine  34272 
Declination     20    OS. 
Latitude....  49  57  N. 


*  Tlie  best  time  for  regulating  a  watch  is  when  the  sun  bears  nearly  east  or  west,  and  is  above  10° 
from  the  horizon. 

t  The  observed  altitude  of  the  lower  limb  being  19°  32',  ©'s  semidiameter  16',  dip  4',  refraction  3', 
parallax  too  small  to  be  noticed. 


TO   FIND   THE   LATITUDE   BY    AN   ALTITUDE   NEAR  NOON.       201 

EXAMPLE  II. 

At  sea  ill  die  latitude  of  60°  N.  by  account,  the  sun  being  on  the  equator,  at 
C  59"  0%  P.  M.,  per  watch,  regulated  to  apparent  time  the  preceding  morning  in  a 
place  15  miles  in  longitude  to  the  westward,  the  sun's  correct  central  altitude  *  was 
28°  53',  bearing  south.     RcquLced  the  latitude. 

App.  time  per  watch     0^'  59™    0'  Latitude        60°  N.  Cosine    9.69897 

15' long,  in  time 1      0  Declination     0  Cosine  10.00000 

App.  time  from  noon     1     0     0 Corresponding  log.  rising      3.53243 

Nat.  Numb..    1704     Log.     3.23140 
Central  altitude ...  .28°  53'  Nat.  Sine . . .  48303  


Rler.  zenith  distance  60    0  N.  Nat.  Cosine.  50007 

Declination 0    0 

Latitude 60     ON. 


When  the  observation  is  taken  a  few  minutes  before  or  after  noon,  the  correction 
to  be  applied  to  the  altitude,  to  obtain  the  meridian  altitude,  may  be  found  Ijy  means 
of  Tables  XXXIL  and  XXXIU.,  the  first  of  which  contains  the  variation  of  the  alti- 
tude for  one  minute  from  noon,  expressed  in  seconds  and  tenths ;  the  other  contains 
the  square  of  the  minutes  and  seconds  of  a  minute  contained  in  the  top  and  the  side 
columns.  By  these  tables  the  correction  of  the  observed  altitude  may  be  found  by 
the  following  rule  : — 

RULE. 

Eyiter  Table  XXXIL,  and  find  the  latitude  hy  account  in  the  side  column,  and  the 
declination  at  the  top,  opposite  the  former,  and  under  the  latter,  ivill  he  the  change  of  altitude 
in  seconds  and  tenths  for  one  minute  from  noon :  then  enter  Table  XXXIIL,  and  find  the 
minides  of  the  apparent  time  from  noon  in  the  top  column,  and  the  seconds  in  the  side 
column;  under  the  former,  and  opposite  the  latter,  loill  be  a  number  lohichis  to  be  midtiplied 
hi  the  number  taken  from  Table  XXXIL,  and  the  product  tvill  be  the  sought  change  of 
altitude,  expressed  in  seconds  and  decimals. 

In  making  use  of  Table  XXXIL,  proportional  parts  may,  if  necessary,  be  taken  for 
the  miles  of  latitude  and  declination.  The  numbers  in  both  these  tables  are  expressotl 
in  Avhole  numbers  and  tenths. 

EXAMPLE    III. 

Being  at  sea  in  the  latitude  of  40°  N.  when  the  sun's  declination  was  21°  N.,  at  8™ 
past  noon,  apparent  time,  the  sun's  correct  central  altitude  f  was  70°  58'.  Required 
the  meridian  altitude  and  latitude. 

In  Table  XXXIL,  opposite  40°  lat.,  and  under  21°  dec,  is  4".3,  and  the  number  in 
Table  XXXIIL  corresponding  to  8™  is  64.0.  Multiplying  64.0  by  4".3,  we  get  the 
correction  275" .2  (or  5  nearly).  This  quantity,  being  added  to  70°  58',  gives  the 
meridian  altitude  71°  3' ;  and  the  latitude  deduced  therefrom  is  39°  57'  N. 

By  observing  several  altitudes  of  the  sun  when  near  the  meridian,  and  noting  the 
times,  the  meridian  altitude  may  bo  obtained,  by  the  above  method,  to  a  great  degree 
of  accuracy.  For  by  using  this  method,  many  observations  may  be  taken  on  the 
same  daj^,  and  the  mean  of  the  meridian  altitudes  deduced  therefrom  will  in  general 
be  much  more  correct  than  that  obtained  by  a  single  observation,  by  the  usual 
method.  To  obtain  the  correction  to  be  applied  to  the  mean  of  all  the  observed 
altitudes,  proceed  thus  : — 

Take  from  Tai)lc  XXXIIL  the  number  corresponding  to  each  time  from  noon, 
(the  minutes  being  found  at  the  top  and  the  seconds  at  the  side,  the  correction  being 
under  the  former  and  opposite  the  latter,)  and  divide  the  sum  of  these  tabular  num- 
bers by  the  number  of  observations ;  the  quotient,  being  multiplied  by  the  number 
taken  from  Table  XXXIL,  will  be  the  correction  to  be  applied  to  the  mean  of  the 
obsei"ved  altitudes,  to  obtain  the  meridian  altitude. 

EXAMPLE   IV. 

Being  at  sea  in  the  latitude  of  50°  N.  by  account,  when  the  sun's  declination  was 
22°  N.,  observed  with  a  sextant,  the  altitudes  of  die  sun's  lower  limb  (bearing  nearly 

*  The  observed  altitude  of  the  sun's  lower  limb  being  28°  43',  ©'s  S.  D.  16',  dip  4',  refraction  2', 
parallax  too  small  to  be  noticed. 

f  The  observed  altitude  of  the  sun's  lower  limb  being  70°  46',  semidlameler  16',  dip  4',  parallax  and 
refraction  too  small  to  be  noticed. 

26 


202       TO   FIND   THE  LATITUDE   BY   AN  ALTITUDE   NEAR  NOON. 

south)  as  in  the  following  table ;  the  correction  for  semidiameter,  dip,  refraction,  &c., 
being  12'  additive.    Required  the  meridian  altitude  and  latitude. 

The  mean  of  the  numbers  from  Table 
XXXIII.  is  17.5 ;  this  being  multiplied  by  the 
number  of  seconds  from  Table  XXXJI.,  viz. 
2" .5,  gives  the  correction  43''.75,  or  44",  which, 
being  added  to  the  mean  of  the  observed  alti- 
tudes, 61°  46',  gives  the  meridian  altitude  of  the 
sun's  lower  limb,  61°  46' 44",  or  61°  4?  nearly; 
to  this  add  12'  for  semidiameter,  &c.,  and  we 
get  61°  59'  for  the  correct  central  meridian 
altitude,  whence  the  latitude  is  50°  ]'  N. 

If  the  above  altitudes  had  been  taken  with  a  circle,  the  calculation  would  have 
>  been  exactly  the  same,  except  that  each  altitude  would  not  have  been  given,  but  the 
sum  of  all  of  them,  247°  4',  would  have  been  shown  by  the  central  index  after  finishing 
tlie  observations. 


Obs.  Alt. 
©L.  L. 

App.  Time 
from  Noon. 

m        » 

6    10 
4    15 
3      2 
2    10 

Numbers 
Tab. 

38.0 

18.1 

9.2 

4.7 

70.0 

O        1 

61.45 
61.46 
61.46 
61.47 

Sum  247.04 

Mean  61.46 

17.5 

EXAMPLE   V. 

Having  regulated  my  watch,  I  found  it  to  be  6""  2»  too  slow  for  apparent  time.  I 
then  sailed  to  the  southward  and  eastward  till  the  ship  had  made  60'  difference  of 
longitude,  and  was  by  account  in  the  latitude  of  40°  N.,  the  sun's  declination  being 
20°  S.  The  sun  being  then  nearly  on  the  meridian,  I  observed  ten  altitudes  of  his 
lower  limb  by  a  circle  of  reflection,  and  noted  the  times  by  the  watch  as  in  the  follow- 
ing table ;  and  the  sum  of  all  the  altitudes  taken  from  the  circle  was  298°  20'.  Required 
the  true  latitude,  supposing  the  dip  to  be  4'  and  the  semidiameter  16'. 

When  it  was  12  o'clock  by  the  watch,  it  was  12''  6""  2'  apparent  time  at  the  place 

where  the  watch  was  regulated,  and  12''  10™  2* 
apparent  time  at  the  place  where  the  altitudes 
were  taken  to  determine  the  latitude,  because 
the  former  place  was  60'  or  4""  in  time  to  the 
westward  of  the  latter;  consequently  the  watch 
was  10""  2°  too  slow  for  apparent  time  at  the  place 
of  taking  the  altitudes  for  determining  the  lati- 
tude. Hence  we  may  determine  the  time  from 
noon  of  taking  each  obsei-vation,  as  in  the  second 
column  of  the  adjoined  table,  and  find  the  num- 
bers corresponding  in  Table  XXXIII.,  the  mean 
of  which  is  6.97;  this,  multiplied  by  the  number 
in  Table  XXXII.  corresponding  to  the  latitude 
40°  N.  and  declination  20°  S.,  viz.  1".6,  will  give 
11".152  or  11",  which  is  the  correction  to  be 
added  to  the  mean  of  the  obseiTed  altitudes  to 
obtain  the  meridian  altitude. 

Now  the  sum  of  all  the  altitudes  298°  20',  being  divided  by  10, 

the  number  of  observations,  gives 29°  50'  0" 

Add  semidiameter  16'  and  the  above  correction  11" -|-  1^  ^ 

Add  parallax  found  in  Tal)le  XIV -)-  § 

Subtract  dip  4'  and  refraction  1'  39"  —    5  39 

Central  altitude 30    0  40 

Zenitli  distance. 59  59  20  N. 

Declination 20     0    OS. 

Latitude 39  59  20  N. 

When  the  meridian  altitude  of  the  object  is  small,  the  correction  of  altitude  may  be 
found  by  this  method,  for  12  or  15  minutes  from  noon,  to  a  great  degree  of  accuracy; 
but  when  the  sun  passes  near  the  zenitli,  the  time  of  obsei-vation  must  be  proportion- 
ally nearer  to  noon.  Thus,  in  Example  I.,  preceding,  the  time  from  noon  was  32', 
and  as  the  numbers  in  Table  XXXIIL  are  the  squares  of  the  number  of  minutes,  it 
follows,  that  the  number  corresponding  to  32'"  would  be  the  square  of  32,  or  1024.0. 
Thi.s,  being  multiplied  by  the  number  1".3  of  Table  XXXII.,  corresponding  to  the 
latitude  50°  N.  and  declination  20°  S.,  will  give  the  correction  1331".2,  or  neai-ly  22', 


Time  per 

App.  Time 

Numbers 

Watch. 

froniNoon. 

Tab.  XXXIII. 

18.1 

11.4543 

4'  15" 

46.58 

3    0 

9.0 

47.52 

2    6 

4.4 

48.50 

1     8 

1.3 

49.28 

0  30 

0.2 

50.48 

0  50 

0.7 

51.10 

1  12 

1.4 

52.13 

2  15 

5.1 

53.  8 

3  10 

10.0 

54.23 

4  25 

19.5 

Sum    69.7 

Mean  6.97 

TO  FIND  THE   LATITUDE   BY   AN   ALTITUDE   NEAR   NOON.       203 


which,  being  added  to  19°  41',  will  give  20°  3'  for  the  meridian  altitude,  or  G9°  57'  for 
the  zenith  distance,  being  the  same  as  in  that  example. 

It  is  very  advantageous  in  this  method  to  observe  as  many 
altitudes  in  the  afternoon  as  before  noon,  and  at  nearly  the  same 
distances  from  noon  ;  for  in  this  case  a  small  error  in  the  regu- 
lating of  the  watch  will  not  materially  affect  the  calculation. 
This  will  appear  evident  by  supposing,  in  the  preceding  example, 
that  the  watch  was  11™  2'  too  slow,  instead  of  10™  2' ;  by  this 
means  the  times  and  numbers  will  be  as  in  the  adjoined  table, 
and  the  mean  of  all  the  numbers,  taken  from  Table  XXXIIL, 
will  be  8.15,  which,  being  multiplied  by  1",6,  will  give  13'  nearly, 
for  the  correction,  instead  of  11",  so  that  in  this  case  an  error  of 
one  minute  in  the  regulation  of  the  watch  would  only  cause  an 
error  of  2  seconds  in  the  meridian  altitude. 

But  it  must  be  carefully  observed,  that,  in  using  this  method, 
you  must  not  take  the  observation  more  than  2  or  3  minutes  from 
noon,  wlM3n  the  sun  passes  within  10°  or  12°  of  the  zenith. 


Times. 

In  Tab. 
XXXIIL 

3.15 

lO.G 

2.00 

4.0 

1.06 

1.2 

0.08 

0.0 

0.30 

0.2 

1.50 

3.4 

2.12 

4.8 

3.15 

10.6 

4.10 

17.4 

5.25 

29.3 

Sum 

81.5 

Mean     8.15 

204 


TO  DETERMINE  THE  LATITUDE  ON  SHORE 

BY  MEANS  OF  AN  ARTIFICIAL  HORIZON. 


It  frequently  happens  that  the  latitude  of  a  place  on  shore  cannot  be  determined 
by  the  usual  methods,  by  a  quadrant,  sextant,  or  circle,  on  account  of  not  having  an 
open  horizon.  In  this  case  it  is  customai-y  to  make  use  of  an  artificial  horizon  formed 
by  the  surface  of  a  vessel  filled  with  mercury,  water,  Barbadoes  tar,  very  clear  mo- 
lasses, or  any  other  fluid  of  sufiicient  consistency  not  to  be  affected  by  the  wind.* 
With  this  apparatus  an  observation  may  be  taken  on  shore  when  the  altitude  of  the 
object  does  not  exceed  60°,  with  as  much  ease  as  at  sea.  Thus,  if  an  altitude  of  the 
sun  was  required  to  be  taken,  the  observer  must  place  the  vessel  containing  the  mer- 
cury (or  other  fluid)  in  a  firm  position  on  the  ground,  and  in  a  few  minutes  the  surface 
of  the  liquor  will  attain  a  horizontal  situation  ;  the  obsei'ver  must  then  place  himself 
in  a  situation  so  as  to  see  the  image  of  the  sun,  formed  by  the  fluid,  which  image  will 
evidently  be  depressed  as  much  below  the  horizon  as  the  sun  is  elevated  above  it,  so 
that,  to  obtain  the  double  of  the  sun's  altitude,  it  is  only  necessary  for  the  observer  to 
bring  the  image  of  the  sun,  formed  by  tiie  instrument,  down  to  the  image  formed  by 
the  artificial  horizon,  and  the  angle  then  pointed  out  by  the  index  will  be  double  of 
the  altitude  of  the  sun ;  the  half  of  which  will  be  the  apparent  altitude.  If  the  nearest 
limbs  of  the  two  images  are  brought  in  contact,  the  half  of  the  angle  obtained  by  the 
instrument  will  be  the  altitude  of  the  sun's  lower  limb,  but  if  the  farthest  limbs  are 
brought  in  contact,  the  half  angle  will  be  the  altitude  of  the  upper  limb.  The  alti- 
tude thus  obtamed  must  be  corrected  for  semidiameter,  parallax,  and  refraction,  as 
usual,  but  not  for  dip,  because  a  truly  horizontal  surface  is  obtained  by  means  of  the 
artificial  horizon.f  In  this  manner  the  altitude  of  the  sun,  or  any  other  bright  object 
may  be  obtained  when  the  altitude  is  less  than  G0°;  at  higher  altitudes  the  angle  cor- 
responding would  be  above  120°,  which  cannot  be  measured  by  a  sexlant  on  account 
of  the  length  of  the  arc,  nor  by  any  other  instrument  of  reflection,  in  a  convenient 
mannci*,  with  a  sufficient  degree  of  accm^acy.  To  illustrate  this  method  we  shall  here 
add  the  following  examples : — 

EXAMPLE  II. 

The  angular  distance  of  the  farthest  limbs  of  the 
two  images  of  the  sun,  when  on  the  meridian,  was 
obtained  by  the  above  method,  and  found  to  be  Si" 
0',  when  the  declination  was  10°  N.,  and  the  semi- 
diameter  16';  the  sun  bearing  north  of  the  observ- 
er.    Required  the  latitude  : — 

Half  of  34°  0'  is  the  obs.  alt 17°  0 

Subtract  semidiameter 16 


EXAMPLE   L 

The  angular  distance  of  the  nearest  limbs  of  the 
two  images  of  the  sun  was  found  by  the  above 
method  to  be  68°  10',  when  the  declination  was 
10°  S.,  and  the  sun's  semidiameter  16',  the  sun 
bearing  south  of  the  observer.  Required  the  lati- 
tude : — 

Half  of  68°  10'  is  the  obs.  alt 34°  5' 

Add  semidiameter 16 


34  21 
1 


Subtract  refraction 


True  altitude 34  20 


Zenith  distance 55  40  N. 

Declination 10    OS. 


Latitude 45  40  N. 


16  44 
3 


Refraction  sub 

True  altitude  16  41 


Zenith  distance 73  19  S. 

Declination 10    0  N. 


Latitude 63  19  S. 


*  In  case  the  wind  blows  fresh,  you  must  use  a  screen  formed  of  two  plates  of  talc  or  glass  whose  sur- 
faces are  ground  perfectly  parallel,  and  connected  together  in  a  frame  so  as  to  make  an  angle  of  about 
90°  with  each  other.  This  frame  is  to  be  placed  over  the  box  containing  the  fluid,  and  the  rays  of  the 
sun,  passing  through  one  of  the  plates,  are  reflected  from  the  surface  of  the  liquor,  and  pass  through  the 
other  plate  to  the  eye  of  the  observer.  The  use  of  these  plates  is  to  be  avoided,  when  it  can  possibly 
be  done,  on  account  of  the  defect  of  parallelism  of  the  surfaces.  This  error  is  generally  greatest  near 
tlie  border  of  the  glass,  so  that  it  has  been  recommended  to  cover  the  edge  of  the  glass  with  a  paper  or 
some  paint,  to  the  distance  of  ^  or  ^  inch  from  the  frame.  If  the  surfaces  of  the  glass  are  perfectly 
parallel,  the  observed  angle  will  be  the  same  as  if  the  screen  had  not  been  used.  Instead  t>(  using  the 
screen  we  may  place  one  of  the  glasses  of  the  screen  upon  the  surface  of  the  fluid,  which  will  prevent  it 
from  being  agitated  by  the  wind,  or  other  similar  causes.  If  the  reflecting  fluid  is  molasses,  air-bubbles 
will  sometimes  rise  on  the  surface  by  the  sun's  heat ;  this  ma^'  in  some  measure  be  avoided  by  heating 
Ihe  molasses  before  using  it. 

t  If  the  instrument  has  an  index  error,  it  must  be  applied  to  the  observed  angle,  or  the  half  of  the  index 
error  must  be  applied  to  the  sun's  altitude 


TO  DETERMINE   THE   LATITUDE   OiN   SHORE,   &c. 


205 


The  latitude  may  be  determined  on  sliore  by  this  method  to  a  great  degree  of  accu- 
racy by  means  of  a  circle  of  reflection,  by  taking  several  altitudes  a  few  minutes  before 
and  after  the  sun  passes  the  meridian,  and  estimating  the  correction  to  be  applied  to 
the  altitude  by  means  of  Tables  XXXII.  and  XXXllJ.  Thus,  if,  in  the  exanjple  page 
202,  the  obseiTations  had  been  taken  in  this  manner,  the  number  of  degrees  denoted 
by  the  circle  after  taking  ten  observations,  woulil  have  been  595°  20' ;  this,  being  divided 
by  20,  (twice  the  number  of  observations,)  will  give  for  tlie  observed  altitude  29°  46, 
and  by  adding  the  semidiameter  IG',  parallax  8",  and  the  correction  Ibund  by  Tables 
XXXII.  and  XXXIII.,  viz.  1]  seconds,  and  subtracting  the  refraction  1'  39",  the  cen- 
tral altitude  will  be  obtained,  30°  0'  40",  as  in  the  page  before  mentioned. 

Altitudes  may  be  observed  in  this  way  in  taking  an  azimuth  for  determining  the 
variation,  or  for  regulating  a  watch,  in  the  manner  explained  in  this  work  ;  observing, 
in  all  cases,  that  the  half  of  the  observed  angle  is  to  be  corrected  for  refraction, 
parallax,  and  semidiameter,  but  not  for  the  dip  of  the  horizon,  and  that  half  the 
index  error  only  is  to  be  applied. 


TO  FIND  THE  POSITION*  OF  A  SHIP  ON  A  LINE  OF  BEARING. 

CASE  I. 
"When  the  position  of  a  ship  is  unknown,  the  latitude  by  account  being  uncertain,  assume 
two  or  more  latitudes,  and  work  out  the  longitudes  corresponding  thereto.  A  line  drawn 
on  a  chart  through  the  two  points  tlnis  determined,  will  represent  (he  line  of  equal  altitudes. 
The  place  of  the' ship  will  be  somewhere  on  this  line  ;  and  if  it  passes  through  the  land, 
the  bearing  of  the  land  will  be  known.  If  the  coast  should  run  parallel  to  this  line,  you 
will  have  the  distance  of  the  ship  from  the  land,  but  of  course  not  tlie  absolute  position. 

EXAMPLE. 

December  I7th,  1837.— The 
latitude,  by  account,  being  51° 
37' N.,  the  "GreeiAvich  time  lOh. 
47ra.  13s.  A.  M.,  the  true  altitude 
of  the  sun's  centre  was  found  to 
be  12°  10'.  Required  the  true 
bearing  of  the  land. 

Let  the  assumed  latitudes  be 
51°  and  52°,  sun's  dechnation  23° 
23'  S.,  and  the  equation  of  time 
—  3m.  37s. — The  longitude  corresponding  to  51°  latitude  will  be  about  8°  42'  W. 
The  longitude  corresponding  to  52°  latitude  will  be  about  4°  50'  W. 
A  line  drawn  througli  these  positions  A  A',  will  represent  the  line  of  equal  altitudes,  and 
will  also  pass  through  "Small  Lights,"  and  run  parallel  to  the  S.  E.  coast  of  Ireland. 

The  light  was  seen  in  tlie  course  of  an  hour,  and  the  error  in  latitude  ascertained  to  tc 
8',  C  being  the  position  of  the  ship. 

CASE  IL 

When  a  douZle  altitude  is  taken,  the  jiosition  of  the  shi])  may  be  found  by  Avorking  the 
longitude  for  each  altitude,  au  in  Case  I.,  and  then  drawing  two  lines  of  equal  altitudes 
through  the  four  points  A  A'  and  B  B'  thus  determined.  The  point  of  intersection  of  said 
lines  will  give  the  position  of  the  ship.  The  necessary  correction  for  the  change  of  poai'ioyi, 
when  the  second  altitude  was  taken,  must  be  made  as  explained  on  page  183,  or  by  moving 
the  line  A  A'  projected  (parallel  to  itself)  along  the  course  and  distance  made  good  oy  the 
ship.  Thus,  suppose  between  the  observations  the  ship  had  sailed  E.  N.  E.  25  miles. 
Then  move  the  first  line  A  A'  parallel  to  itself  on  this  course  25  miles,  and  draw  a  line 
whose  intersection  with  the  second  line  B  B'  will  give  the  position  required. 

It  is  evident,  that  when 
the  two  lines  cross  each  oth- 
er at  about  right  angles,  the 
point  of  intersection  is  more 
easily  found. 

A  line  drawn  perpendicu- 
lar to  the  line  of  equal  alti- 
tude shows  the  direction  of 

the   sun,  and  consequently   _^ 

the  azimuth.  ^^  B\ 

The  assumed  latitude  must  be  near  the  truth,  to  give  val  je  to  this  method. 
"When  the  altitude  is  high,  an  error  in  the  assumed  latitude  is  of  greater  importance  than 
when  it  is  low. 


•  From  SuiDDer's  work. 


&fff 


TO  FIND  THE  LATITUDE  BY  AN  ALTI- 
TUDE OF  THE  POLE  STAR. 


Find 

in  the   s 

de  column,  the  sum  of  the 

apparent  time  ol 

observation,  and  the  sun's 

right  ascension; 

the  corresponding  number, 

in  the 

middle  co 

lumn,  will   be  the  correc- 

tion  of  the  tiue 

Eiltitude, 

on  account  of  the 

distance  of  the  star  from 

the  meridian. 

If  th 

B  time  is 

If  the  time   is 

found  ir 

eitherof 

Correc- 

found   in    either 

these    columns, 

tion    of 

of  these  columns, 

the  correction  is 

the  alti- 

the correction  is 

subtraclive. 

tude. 

additive. 

H.  M. 

H.  M. 

O        1 

H.  M. 

H.   M. 

1    08 

1    08 

1  26 

13  08 

13  08 

1  13 

1  03 

1  26 

13  03 

13  13 

1  23 

0  53 

1  26 

12  53 

13  23 

1  33 

0  43 

1  25 

12  43 

13  33 

1  43 

33 

1  25 

12  33 

13  43 

1  53 

23 

1  24 

12  23 

13  53 

2  03 

13 

1  24 

12  13 

14  03 

2  13 

8 

1  23 

12  03 

14  13 

2  23 

23  53 

1  21 

11  53 

14  23 

2  33 

23  43 

1  20 

11  43 

14  33 

2  43 

23  33 

1  19 

11  33 

14  43 

2  53 

23  23 

1  17 

11  23 

14  53 

3  03 

23  13 

1  15 

11  13 

15  03 

3  13 

23  03 

1   14 

11  03 

15  13 

3  23 

22  53 

1  12 

10  53 

15  23 

3  33 

22  43 

1  09 

10  43 

15  83 

3  43 

22  33 

1  07 

10  33 

15  43 

3  53 

22  23 

1  05 

10  23 

15  53 

4  03 

22  13 

1  02 

10  13 

16  03 

4  13 

22  03 

59 

10  03 

16  13 

4  23 

21  53 

57 

9  53 

16  23 

4  33 

21  43 

54 

9  43 

16  83 

4  43 

21  33 

61 

9  33 

16  43 

4  53 

21  23 

48 

9  23 

16  53 

5  03 

21  13 

45 

9  13 

17  03 

5  13 

21  03 

41 

9  03 

17  13 

5  23 

20  53 

38 

8  53 

17  23 

5  33 

20  43 

35 

8  43 

17  33 

5  43 

20  33 

31 

8  33 

17  43 

5  53 

20  23 

28 

8  23 

17  53 

6  03 

20  13 

24 

8  13 

18  03 

6  13 

20  03 

20 

8  03 

18  13 

6  23 
6  33 

19  53 

17 

7  53 

18  23 

19  43 

13 

7  43 

18  33 

6  43 

19  33 

9 

7  33 

18  43 

6  53 

19  23 

6 

7  23 

18  53 

1  7  03 

1  7  08 

19  13 

o 

7  13 

19  03 

19  08 

n 

7  08 

19  08 

In  northern  climates,  the  latitude  may  be 
determined  by  means  of  an  observed  altitude 
of  the  pole  star  ;  provided  the  apparent  time 
of  observation  can  be  ascertained  within  a  few 
minutes.*  This  method  might  be  frequently 
used  at  sea,  -when  the  horizon  is  well  defined, 
if  that  star  were  of  the  first  magnitude  ;  but 
being  only  of  the  second  or  third  magnitude, 
it  is  sometimes  so  dim  that  it  is  rather  difficult 
to  determine  the  altitude  with  precision.  How- 
ever, as  there  are  times  when  it  would  be  of 
great  importance  to  determine  the  latitude 
■within  8  or  10  miles,  it  was  thought  advisable 
to  explain  this  method,  which  may  be  used 
when  observations  of  the  sun  or  moon  cannot 
be  obtained. 

Having,  therefore,  the  apparent  time  of  ob- 
servation (which  miist  be  reckoned  from  noon 
to  noon  in  numerical  succession,  that  is,  6'',  A. 
M.,  must  be  called  18'',  (fee),  and  the  observed 
altitude  of  the  star  determined  by  a  fore  ob- 
servation, you  mnst  subtract  from  the  altitude 
the  dip,  which  is  in  general  4  minutes,  and  the 
refraction,  and  you  will  obtain  the  true  alti- 
tude of  the  star.  Then  the  sun's  right  ascen- 
sion corresponding  to  the  given  day,  must  be 
found  in  Table  VI. ,t  and  added  to  the  appa- 
rent time  of  observation  (rejecting  24  hours 
when  the  sum  exceeds  24  hours) ;  with  that 
sum  enter  the  adjoined  table,  and  take  out  the 
corresponding  correction,  which  must  be  added 
to,  or  subtracted  from,  the  true  altitude,  ac- 
cording to  the  directions  in  the  table  ;  the  sum 
or  difference  will  be  the  latitude  of  the  place 
of  observation. 


*  If  the  star  be  not  far  from  the  meridian,  an  error  of  half  an  hour  in  the  time  would  not  affect  the 
altitude  above  1  or  2  miles. 
t  It  is  accurate  enough  to  take  the'numbers  from  Table  VI.;  but  in  strictness  the  right  ascension  ought 


TO  FIND   THE   LATITUDE   BY   AN  ALTITUDE   OF   THE   POLE   STAR.  2U7 

EXAMPLE    I. 

At  7"  9"  P.  M.,  June  3, 1848,  the  observed  altitude  of  the  polo  star  was  IG"  10',  tlie 
dip  4'.    Required  the  latitude  of  the  place  of  observation. 

Obsei-ved  altitude 16°  W 

Hour  of  observation 7"  9"°        Sub.  dip  4',  refrac.  3' 7 

©'3  right  ascension 4  M  T,.^,e  altitude 16     3 

Sum 11  .53 Correction  corresponding  add     1  21 

Latitude 17  24  N. 

EXAMPLE  II. 

On  the  14th  September,  1848,  at  2*'  S™  A.  M.,  the  altitude  of  the  pole  star  was 
24°  16',  wlien  the  dip  was  4'.     Required  the  latitude. 

Observed  altitude 24°  IC 

Hour  of  ol)s.  2^  2""  A.  M.,  or . . .    14"    2"        Dip  4',  refrac.  2',  sub 6^ 

©'s  right  ascension H   28  True  altitude 24  10 

Sum,  rejecting  24'' 1   30 Corresponding  correction  sub.     1   25 

Latitude 22  45  N. 


EXAMPLE  III. 

At  5"  P.  M.,  December  5,  1848,  the  observed  altitude  of  the  pole  stai-  was  25°  15', 
the  dip  4'.     Required  the  latitude  of  the  place  of  observation. 

Observed  altitude 25°  15' 

Hour  of  observation 5"  GO™         Sub.  dip  4',  refrac.  2' 6 

©'s  right  ascension 16  47  Xrue  altitude 25  09 

Sum 21  47 Correction  con-espondingsub.    0  54 


Latitude 24  15  N. 


to  be  taken  from  tlie  Nautical  Almanac,  for  the  hour  of  observation,  reduced  to  Greenwich  time,  by 
adding  or  subtracting  the  longitude  turned  into  time. 

This  table  will  require  a  correction  after  a  few  years,  on  account  of  the  variation  of  declination,  and 
right  ascension  of  the  star.  It  correspoiuis  nearly  to  the  year  1860 ;  for  every  year  after  that  time  you 
must  add  one  quarter  of  a  minute  to  the  times  in  the  side  columns,  and  decrease  the  tabular  corrections 
of  altitude  about  -^-^  part.  Thus  for  the  year  1872  the  times  must  be  increased  3">  for  the  12  ysars, 
»o  that  l*"  0E">  must  be  called  l""  11">,  and  all  the  corrections  of  altitude  must  be  decreased  g^  part, 
M  tbal  1°  15'  must  ke  1°  12'  nearly,  and  0°  35'  must  be  0"  33<i  nearly. 


208 


TO    FIND    THE  TIME  AT  SEA,  AND    REGU- 

LATE  A  WATCH,  BY  THE    SUN'S 

ALTITUDE. 


We  have  already  noticed  the  difference  between  the  civil,  astronomical,  and  nauti- 
cal computation  of  time ;  but  as  it  is  a  subject  of  great  importance,  it  may  not  be 
unnecessary  again  to  i-epeat,  that  a  civil  day  is  reckoned  from  midnight  to  midnight, 
and  is  divided  into  24  hours;  the  first  12  hours  are  marked  A.  M.,  the  lattei-  12  hours 
P.  M.  being  reckoned  from  midnight  in  numeral  succession  from  1  to  12,  then 
beginning  again  at  1  and  ending  at  12.  Astronomers  begin  their  computation  at  the 
noon  of  the  civil  day,  and  count  the  hours  in  numeral  succession  from  1  to  24,  so 
that  the  morning  hours  are  reckoned  from  12  to  24.  Navigators  begin  their  compu- 
tation at  noon,  12  liours  before  the  commencement  of  the  civil  day,  (and  24  hours 
before  the  commencement  of  the  astronomical  day,)  marking  their  hours  from  1  to 
12  P.  M.  and  A.  M.,  as  in  tlie  civil  computation. 

There  are  two  kinds  of  lime,  mean  and  apparent.  Mean  time  is  that  shown  by  a 
chronometer,  which  is  always  regulated  to  mean  solar  time.  Jlpparent  time  is  that 
shown  by  the  sun,  estimating  the  apparent  noon  to  commence  at  the  passage  of  his 
centre  over  the  meridian  of  any  place.*  There  is  sometimes  a  difference  of  a  quarter 
of  an  hour  between  mean  and  apparent  time,  owing  to  the  unequal  motion  of  the 
earth  in  its  orbit,  and  the  inclination  of  its  axis.  This  difference  is  called  the  equation 
of  time,  which  is  given  in  Table  IV.,  A.,  or  more  accurately  in  the  Nautical  Almanac. 
It  is  always  necessary  to  take  notice  of  the  equation  of  time  when  regulating  a  cIn"o- 
nom.eter  to  mean  solar  time,  by  means  of  an  altitude  or  transit  of  the  sun. 

We  may  obtain  the  apparent  time  at  sea,  when  the  ship  makes  no  way  through  the 
water,  by  observing  an  altitude  of  the  sun  in  the  morning,  and  again  in  the  afternoon 
when  at  the  same  altitude,  and  noting  the  times  by  a  chronometer;  for  the  middle 
time  between  these  two  observations  will  be  nearly  the  apparent  time  of  the  sun's  pas- 
sage by  the  meridian  ;  hence  the  error  of  the  chronometer  may  be  foimd.  A  small 
correction  is  necessary  for  the  variation  of  the  sun's  declination  during  the  interval 
between  the  observations,  and  the  method  of  calculating  this  correction  will  be  given 
in  this  work,  but  this  method  cannot  often  be  made  use  of  at  sea,  by  reason  of*  the 
motion  of  the  vessel. 

The  best  method  of  obtaining  the  apparent  time  at  sea,  is  by  observing,  by  a  fore 
observation,  the  altitude  of  the  sun's  lower  limb  when  rising  or  falling  fastest,  or  when 
bearing  nearly  E.  or  W. ;  to  this  altitude  we  must  add  the  semidiameter  and  ))arallax, 
and  subtract  the  dip,  (or,  instead  of  these  three  corrections,  add  12' ,f  which  will  answer 
very  well  for  an  observation  taken  on  the  deck  of  a  connnon-sized  vessel ;)  subtract 
also  the  reiraction,  taken  from  Table  XII.,  and  the  remainder  will  be  the  correct 
altitude.  The  ship's  latitude  must  be  Ibund  at  the  time  of  observation  by  carrying 
the  reckoning  forward  to  that  time.}:  The  declination  must  be  taken  from  Table  IV., 
or  from  the  Nautical  Almanac,  and  corrected  for  the  ship's  longitude  §  and  the  time 

*  There  is,  as  vvc  have  already  observed  in  a  note  on  pnj^e  147,  anotlier  melliod  of  computing-  the  time, 
made  use  of  by  astronomers,  called  Sideral  time,  in  which  llie  interval  between  two  successive  transits 
of  a  fixed  star  over  the  meridian  is  estimated  at  24-  liours,  commencing'  the  day  at  the  time  the  first  point 
of  Aries  is  on  the  meridian,  so  that  the  hour  in  sideral  time  is  the  same  as  the  right  ascension  of  the 
meridian. 

f  The  semidiameter  is,  in  general,  about  16',  the  parallax  never  exceeds  9",  and  the  dip  is  about  4'; 
and  as  the  two  former  corrections  are  additive,  and  the  latter  subtractive,  the  cficct  of  all  three  cor- 
rections will  not  difi'er  materially  from  12'  additive. 

I  This  must  be  carefully  attended  to,  because,  when  the  ship  is  sailing-  in  a  northerly  or  southerly 
direction,  the  latitude  at  tlie  tune  of  regulating -the  chronometer,  may  vary  considerably  from  the 
observed  latitude  at  noon. 

§  The  declination  may  also  be  found  very  easily,  by  taking  it  out  for  the  time  at  Greenwich,  an 
shown  by  a  chronometer  regulated  to  Greenwich  time.    See  Tab.  LVII. 


TO   FIND   THE   TIME   AT   SEA   BY   THE   SUN'S   ALTITUDE  209 

of  tlie  sun  from  the  meridian  by  Table  V.  Then,  if  the  latitvde  and  declination  be  both 
north  or  both  south,  subtract  the  declination  from  90°,  and  you  will  have  the  polar  distance; 
but  if  one  be  north  and  the  other  south,  add  the  declination  to  90°,  and  i/ou  ivill  have  tlie 
polar  distance. 

Having  thus  found  the  correct  altitude,  latitude,  and  polai*  distance,  the  apparent 
time  of  observation  may  be  found  by  either  of  the  tin-ee  following  methods,  of  which 
the  first  is  the  most  simple,  since  it  does  not  require  the  table  of  natural  sines,  all  the 
logarithms  being  found  in  Table  XXVII.  This  method  is  abridged  by  means  of  the 
tahle  of  hom-s  affixed  to  the  table  of  log.  sines  ;  in  using  which  you  must  observe, 
that,  if  the  sine  or  cosijie  of  the  logarithm  sought  is  marked  at  the  top  of  the  table, 
tlie  title  "  Hour  a.  m."  or  "  Hour  p.  m."  is  also  to  be  found  at  the  top,  and  the  contrary 
if  the  sine  or  cosine  is  marked  at  the  bottom. 


FIRST  METHOD. 

w3«W  together  the  correct  altitude  of  the  sun's  centre,  the  latitude  and  the  polar  distance ; 
from  the  half-sum  subtract  the  sun's  altitude,  and  note  the  remainder.  Then  add  together 
the  log.  secant  of  the  latitude,  (this  and  all  the  other  logs,  being  found  in  Table  XXVII.) 
the  log.  cosecant  of  the  polar  distance,  [rejecting  10  in  each  index,)  the  log.  cosine  of  the  half- 
sum,  and  the  log.  sine  of  the  remainder ;  half  the  sum  of  these  four  logarithms,  being  sought 
for  in  the  column  of  log.  sines,  will  correspond  to  the  apparent  time  of  the  day  in  one  of  the 
hour  columns.  To  this  apparent  time  loe  must  apply  the  equation  of  time,  taken  from 
Table  IV.,  A.,  or  from  the  JVautical  Almanac,  and  we  shall  obtain  the  mean  time  of  the 
observation. 

EXAMPLE   1. 

Suppose  that,  on  the  10th  of  October,  1848,  sea  account,  at  8''  21",  A.  M.,  per 
watch,  in  the  latitude  51°  30'  N.,  and  longitude  130°  E.  from  Greenwich,  by  account, 
the  altitude  of  the  sun's  lower  limb  by  a  fore  observation,  was  13°  32',  the  cor- 
rection for  semidiameter,  parallax,  and  dip,  12'.  Required  the  apparent  time  of 
observation. 

By  Example  III.,  page  157,  the  declination  was  G°  37'  S. ;  this  added  to  90°  gives 
the  polar  distance  90°  37'.  To  the  sun's  observed  altitude  13°  32',  I  add  12  minutes 
and  subtract  the  refraction  4' ;  the  remainder  is  the  con-ect  altitude,  13°  40^. 

©'s  correct  altitude . .  13°  40^ 

Latitude 51  30         Secant 0.20585 

Polar  distance 96  37         Cosecant 0.00290 

Smn 161  47 


Half-sum 80  53         Cosine 9.19988 

©'s  altitude 13  40 

Remainder 67  13         Sbe 9.96472 

2)19.37335 

Sine 9.68668  corresponding  to  whicli, 

in  the  column  marked  A.  M.,  is 8''  7""  2 P,  the  apparent  time  of  observation. 

Equation  of  time,  sub 12  53 

INIean  time  of  observation 7  54  18 

Time  per  watch 8  21  00 

Watch  too  fast 26  42 


EXAMPLE   II. 

buppose  that,  on  the  10th  of  May,  1836,  sea  account,  at  5''  30™  P.  M.,  per  watch,  in 
latitude  39°  54'  N.,  longitude  by  account  35°  45'  E.  from  Greenwich,  the  altitude  of 
tlie  sun's  lower  limb,  by  a  fore  observation,  was  15°  45',  the  correction  for  dip, 
parallax,  and  semidiameter,  being  12  minutes,  consequently  the  correct  altitude 
15°  54'  Required  the  apparent  time  of  observation. 
27 


210 


TO   FIND   THE   TIME   AT   SEA   BY  THE   SUN'S   ALTITUDE. 


By  Example  IV.,  page  157,  the  sun's  declination  was  17°  29'  N.,  which,  being 
subtracted  from  90°,  leaves  the  polar  distance  72°  31'. 

©'s  altitude 15°  54' 

Latitude 39  54 

Polar  distance 72  31 


Secant 0.11511 

Cosecant....  0.02054 


Sum 128  19 

Half-sum 64  10 

(v)'s  altitude 15  54 

Remainder 48  16 


Cosine 9.63924 

Sine 9.87288 

2 )  19.64777 


m  the  column  P.M.,  is... 
Equation  of  time,  sub.. 


Sine 9.82388  corresponding  to  which, 

5h  34m  28»^  tiie  apparent  time  at  the  place  of  observatioiu 
3    49 


IMean  time  of  obsei-vation  5  30   39 
Time  per  watch 5  30   00 


Watch  too  slow 0  00   39 


EXAMPLES   TO    EXERCISE   THE   LEARNER. 

1.  In  latitude  36°  39'  S.,©'s  declination  9°27'N.,  die  altitude  of  the  ©'s  lower  limb 
in  the  morning  was  observed  10°  33'  ;*  required  the  apparent  time.  Answer,  7''  23™  51'. 

2.  In  latitude  36°  21'  S.,  0's  declination  8°  44'  N.,  altitude  ©'s  lower  limb  in  the 
moniing  10°  48';*  required  the  apparent  time.  Answer,  7''  22'"  ll^ 

3.  In  latitude  29°  25'  N.,  ©'s  declination  23°  20'  N.,  observed  altitude  of  ©'s  lower 
limb  in  the  afternoon  14°  58';*  required  the  apparent  time.  Answer,  5''  41". 

4.  In  latitude  3°  31'  S.,  ©'s  declination  20°  3'  S.,  observed  altitude  ©s  lower  limb 
38°  41'  *  in  the  afternoon  ;"required  the  apparent  lime.  Answer,  3''  18'"  47'. 

5.  In  latitude  13°  17'  N.,  ©'s  declination  22°  10'  S.,in  the  morning  observed  altitude 
of  ©'s  lower  limb  30°  26' ;  *  required  the  apparent  time.  Answer,  9''  17"'  8\ 

6.  In  latitude  21°  36'  S.,  ©'s  declination  3°  37'  S.,  in  the  morning  observed  altitude 
of  ©'s  lower  limb  35°  48' ;  *  required  the  apparent  tim.e.  Answer,  8'^  29"'  50'. 

SECOND  METHOD. 

Find,  as  in  the  former  method,  the  sun's  correct  altitude,  the  ship's  latitude,  and 
the  polar  distance;  thence  the  sun's  correct  zenith  distance,  and  the  complement  of 
latitude  ;  tiien  add  together  the  zenith  distance,  co-latitude,  and  polar  distance;  from  half 
their  sum  subtract  the  zenith  distance,  and  note  the  remainder.  Add  together  the  log. 
cosecant  of  the  co-latitude,  {this  and  all  the  other  logs,  being  found  in  Table  XXVII.,)  the 
log.  cosecant  of  the  polar  distance,  [rejecting  10  in  each  index,)  the  sine  of  the  half-sum  and 
the  sine  of  the  remainder ;  half  the  sum  of  these  four  logarithms,  being  found  among  the 
log.  cosines,  will  correspond  in  one  of  the  adjoined  columns  to  the  apparent  time  of  day. 
This  may  he  reduced  to  mean  time,  by  applying  the  equation  of  time  found  in  Table  IV.,  A., 
or  in  the  JVautical  Almanac. 

The  [jreceding  examples  I.  and  II.  are  thus  worked  by  this  method : — 


90°  0' 
©'s  cor. alt...  13  40 

Zen.  distance  76  20 
Co-latitude  . .  38  30 
Polar  distance  96  37 

Sum 211  27 


EXAMPLE    I. 

90°   0' 

90°  (y 

Latitude.. 

.  51  30 

©'s  dec.  . 

.     6  37 

Co-latitude   38  30      Polar  dist..  96  37 


Half-sum....  105  43 
Zen.  distance  76  20 

Remainder..  29  23      Sine, 


(^secant  0.20.585 
Cosecant  0.00290 


Sine. 


9.98345 

9.69077 

2 )  19.88297 

Cosine . .  9.94148  coiresponding  to  which  in  the  column  A.  M 
is  S**  7""  20',  the  apparent  time  of  day,  wnich  agrees  nearly  with  the  other  method. 

*  The  rorrertion  for  dip  and  semidiameter  being  12'  additive,  the  correction  for  refraction  is  also  to 
be  applied  as  usual. 


TO   FIND   THE  TIME  AT   SEA  BY  THE   SUN'S  ALTITUDE.         211 


EXAMPLE   II. 

90°  (y 

©'s  cor.  alt. . .  15  54 

90°  C 
Latitude...  39  54 

©'s  dec.  . . 

90°  0' 
,  17  29 

Zen.  distance  74    6 
Co-latitude  . .  50     6 
Polar  distance  72  31 

Co-latitude  50    6 
Cosecant  0.11511 
Cosecant  0.02054 

Polar  dist.. 

72  31 

Sum 19G  43 

Half-sum....  98  22 
Een.  distance    74    6 

Sine....  9.99535 
Sine....  9,61382 

Remainder  . .  24  16 

2 )  19.74482 

Cosine . .  9.87241  corresponding  to  which,  in  the  column 
P.  M.,  is  5*'  34™  25%  the  apparent  time  of  day,  which  agrees  nearly  with  the  fii-st 
method. 

By  the  preceding  method  you  may  find  the  beginning  or  ending  of  the  twilight,  by 
calculating  the  hour  when  the  sun's  zenith  distance  is  108°,  (or  when  the  sun  is  18° 
below  the  horizon  ;)  for  by  observation  it  has  been  found  tliat  the  twilight  begins  or 
ends  when  tjie  sun  is  at  that  distance  from  the  zenith. 


EXAMPLE  III. 

Required  the  time  of  beginning  and  ending  of  tlie  twilight,  in  the  latitude  of 
42°  23'  N.,  when  the  declination  is  23°  27'  N. 


Zenith  distance 108°  0' 

Co-latitude 47  37 

Polar  distance 66  33 


Sum 222  10 


Half-sum Ill     5 

Zeuiih  distance 108    0 


Remainder 


3     5 


Cosecant 0.13156 

Cosecant 0.03744 


Sine 9.90991 


Sine 8.73069 


Sum 18.86960 


Half-sum cosine  9.43480  which 

corresponds  to  2^  6""  20'  A.  31.,  and  9''  53™  40'  P.  M.  Therefore,  the  first  appearance 
of  tlie  twilight  in  the  morning  was  at  2''  6™  20',  and  tlie  end  of  it  in  the  evening  at 
9^  53'"  40",  apparent  time. 


THIRD  METHOD. 

If  the  sun's  declination,  and  the  latitude,  be  both  north  or  both  south,  take  their 
difference,  but  if  one  be  north  and  the  other  south,  take  their  sum,  and  from  the 
natural  cosine  of  this  difference,  or  sum,  subtract  the  natural  sine  of  the  true  altitude, 
(both  being  found  in  Table  XXIV.;)  find  the  log.  of  their  difference,  (in  Table  XXVL 
add  thereto  the  log.  secant  of  the  latitude  (from  Table  XXVII.)  and  the  log.  secant  of 
the  sun's  declination,  (from  the  same  table,)  rejecting  10  in  each  index ;  the  sum  of 
these  three  logarithms  being  found  in  the  column  of  rising  (Table  XXIII.)  the  hours, 
minutes,  and  seconds,  corresponding,  will  be  the  apparent  time  from  noon  ;  and  by 
applying  the  equation  of  time  to  the  apparent  time,  we  get  the  mean  time. 

The  preceding  examples  I.  and  II.  aie  thus  worked  by  this  tliird  metliod: — 


212         TO   FIND  THE  TIME   AT  SEA  BY   THE   SUN'S  ALTITUDE 


EXAMPLE    I. 

Latitude SPSCKN;  Secant..  0.20585 

Declination  . . .     6  37  S.  Secant. .  0.00290 

Sum 58    7  Nat.  cosine 52819 

Sun's  cor.  alt..  13  40  Nat.  sine 23627 

Difference 29192        Log 4.46526 

4.67401       cor  re 

spending  to  which  in  the  column  of  rising,  is 3^  52"'  32" 

12 

Subtracted  from  12*',  leaves  the  apparent  time.  8     7    28 

Equation  of  time sub.  12    53 

Mean  time  of  observation 7  54    35 

Time  per  watch 8   21    00 

Watch  too  fast  per  mean  time 26    25     agreeing    nearly 

with  the  other  methods. 


EXAMPLE  II. 

Latitude 39°54'N.  *  Secant..  0.11511 

Declination  . . .  17  29  N.  Secant. .  0.02054 

Difference 22  25  .  Nat,  cosine 92444 

Cor.  altitude  . .  15  54  Nat.  sine 27396 

Difference 65048        Log 4.81324 

4.94889       corre 
spending  to  which,  in  column  rising,  is  apparent  time  5^  34™  30' 
Equation  of  time sub.         3   49 

Mean  time  of  obsei*vation 5  30   41 

Time  per  watch 5   30    00 

Watch  too  slow 41      agreeing    nearly 

with  the  other  methods.  The  differences  between  the  results  of  the  different 
methods  arise  chiefly  from  not  taking  notice  of  the  seconds  in  the  angles,  and  some- 
times from  not  having  the  natural  sines  and  cosines  to  6  or  7  places  of  decimals ;  and 
we  remark  generally,  that  it  is  always  best  to  retain  the  seconds  in  the  calculation. 
This  is  easily  done,  in  the  first  and  second  methods,  by  means  of  the  columns  A,  B, 
of  proportional  parts  in  Table  XXVII. ;  and  by  i-etaining  tlie  seconds,  we  are  sure  to 
obtain  a  more  correct  result  in  the  calculation. 


213 


TO    FIND    THE  TIME    AT   SEA   BY   THE 
MOON'S    ALTITUDE. 


Having  a  chronometer  which  is  pretty  well  regulated  to  Greenwich  time,  we  can 
use  the  moon's  altitude  for  finding  the  mean  solar  time  at  the  ship,  which  is  required 
in  determining  the  longitude.  For,  in  the  present  improved  state  of  the  Nautical 
Almanac,  we  can  easily  find  the  moon's  right  ascension  and  declination  for  that  time 
at  Greenwich,  without  the  very  troublesome  operation  of  interpolating  for  the  second 
and  third  dift'erences,  as  was  necessary  in  the  former  arrangement  of  that  ephemeris. 
Even  without  a  chronometer  thus  regulated,  the  time  at  Greenwich  can  be  obtained, 
if  we  know  the  longitude  of  the  ship,  as  well  as  the  mean  time  at  the  place  of  obser- 
vation, by  a  watch  that  will  give  it  with  a  considerable  degree  of  accuracy  ;  because, 
by  adding  the  longitude  in  time  to  the  time  by  the  watch,  if  the  longitude  be  west,  or 
subtracting  it,  if  the  longitude  be  east,  we  shall  obtain  the  corresponding  time  at 
Greenwich.  We  must,  however,  always  keep  in  mind,  that  the  accuracy  of  an  obser- 
vation of  this  kind  depends  on  the  certainty  with  which  the  time  at  Gi'eenwich  is 
computed  ;  because  an  error  in  this  estimate  affects  the  moon's  right  ascension  and 
declination,  which  frequently  vary  rapidly,  as  may  be  seen  by  the  inspection  of  the 
Nautical  Almanac,  where  we  shall  find  that  in  a  minute  of  time  the  right  ascension 
may  vary  more  than  2%  and  the  declination  more  than  15". 

When  we  wish  to  ascertain  the  time  by  this  method,  we  must  observe,  with  a  fore 
observation,  the  altitude  of  the  moon's  round  limb,  and  at  the  same  instant  the  time 
by  the  watch  or  chronometer,  which  is  supposed  to  be  regulated  to  Greenwich  time. 
With  this  time  at  Greenwich  we  must  take  from  the  Nautical  Almanac  the  suit's  right 
ascension,  the  moon^s  right  ascension,  the  moon''s  declination,  the  moon's  horizontal  parallax, 
and  the  moon''s  seniidiameter,  to  lohich  ive  must  add  the  augmentation  from  Table  XV. 
To  the  observed  altitude  we  must  apply  the  con-ection  of  the  moon's  semidiameter. 
by  adding  it,  if  the  lower  limb  be  observed,  or  subtracting  it,  if  the  upper  limb  be 
observed  ;  from  this  sum  or  difference  we  must  subtract  the  dip  of  the  horizon,  and 
we  shall  obtain  the  moon's  central  altitude.  To  this  we  must  add  the  correction  for 
parallax  and  refraction,  and  we  shall  obtain  the  moon's  correct  altitude,  which  is  to 
be  used  in  the  rest  of  the  calculation.  This  correction  for  parallax  and  refraction 
can  easily  be  found,  as  in  page  171,  by  means  of  Table  XIX.,  by  subtracting  the  tabu- 
lar number,  corresponding  to  the  altitude  and  horizontal  parallax,  from  5U'  42"  ;  the 
remainder  will  be  the  correction  for  parallax  and  refraction,  to  be  used  as  above  ;  and 
tlien  we  find  the  time  by  the  following  rule : — 

RULE. 

Add  together  the  moon's  correct  altitude,  the  ship's  latitude,  and  the  polar  distance ; 
from  the  iialf-sum  subtract  the  moon's  correct  altitude,  and  note  the  remainder ;  then 
add  together  the  log.  secant  of  the  latitude,  the  log.  cosecant  of  the  polar  distance, 
(rejecting  10  from  each  index,)  the  log.  cosine  of  the  half-sum,  and  the  log.  sine  of  the 
remainder;  half  the  sum  of  these  four  logarithms  will  be  the  log.  sine  of  half  the  hoiu- 
angle  ;  take  out  the  corresponding  time  in  the  column  marked  P.  M.,  in  Table  XXVII., 
and  apply  it  to  the  moon's  right  ascension,  by  subtracting  when  the  moon  is  east  of 
the  meridian,  or  adding  when  west  of  the  meridian ;  the  sum  or  difference  will  be  the 
right  ascension  of  the  meridian.  From  the  right  ascension  of  the  meridian  (increased 
by  24  hours  if  necessary)  subtract  the  sun's  right  ascension ;  the  remainder  will  be 
the  apparent  time  at  the  ship,  and  by  ai)plying  to  it  the  equation  of  time  found  in 
the  Nautical  Almanac,  we  shall  get  the  required  mean  solar  time  at  the  meridian  of 
ihe  place  of  observation. 

EXAMPLE. 

"'hen  the  mean  time  at  Greenwich,  by  the  chronometer,  was,  Nov.  29'',  y""  52" 
astronomical  account,  the  altitude  of  the  moon's  upper  limb  was  observed,  when 


214       TO  FIND  THE  TIME  AT   SEA   BY  THE  MOON'S   ALTITUDE. 

west  of  the  meridian,  and  found  to  be  60°  25'  8",  the  latitude  of  the  place  30°  20^  N, 
and  the  dip  4'.    Required  the  mean  time  of  observation. 

We  have  from  the  Nautical  Almanac,  for  the  time,  Nov.  29"^  2^  52"",  the  sun's  right 
ascension,  IG""  22™  45' ;  the  moon's  right  ascension,  9^  26""  23' ;  the  moon's  declina- 
tion, 20°  32'  47"  N.,  or  polar  distance,  09°  27'  13" ;  the  moon's  horizontal  parallax, 
54' 23"  ;  the  moon's  semidiameter,  14'  49"  + Aug.  Table  XV.  14"  =  15'  3". 

J)'s  observed  altitude,  upper  limb 60°  25^  08" 

j)'s  semidiameter sub.        15  03 

60  10  05 
Dip sub.  4  00 

^'s  central  altitude 60  06  05 

Parallax  and  refraction n:  59^  42"  —  Cor.  Tab.  XIX.  33'  08"  ...  .add        26  34 

J)'s  correct  altitude 60  32  39 

3)'s  correct  altitude 60°  32'  39" 

Latitude 30  20   00  Secant 0.06394 

Polar  distance 69  27   13  Cosecant 0.02854 

Sum 2)160  19  52 

Half-sum 80  09   56  Cosine 9.23249 

J) 's  correct  altitude 60  32   39 

Remainder 19  37   17  Sine 9.52608 

Sum 2 )  18.85105 

Half-sum . . .  sine  9.42552 

Corresponding  to  this,  in  the  column  P.  M.,  is 2**  03"'  36' 

2)'s  right  ascension 9   26    23 

Sum  (being  west  of  the  merid.)  gives  right  ascension  of  the  meridian  11    29    59 
Subtract  the  sun's  right  ascension 16   22    45 

Gives  the  apparent  tim^  at  the  ship 19   07    14 

Equation  of  time sub.  11    19 

Mean  time  at  the  ship Nov.  28''.  18    55    55 

Mean  time  at  Greenwich  by  chronometer Nov.  29"^.  02   52    00 

Difference  is  the  longitude  by  the  chronometer 7   56    05  "W. 

It  very  frequently  happens,  that,  a  few  minutes  before  or  after  taking  the  sun's 
meridian  altitude  for  the  determination  of  the  latitude,  we  can  observe  the  moon's 
altitude  for  the  i-egulation  of  the  time ;  and  as  the  latitude  by  observation  is  then 
known  accurately,  without  depending  on  the  ship's  run  for  any  considerable  length 
of  time,  it  will  operate  to  render  the  regulation  of  the  chronometer  by  the  moon's 
altitude  more  accurate.  In  like  manner,  if  we  observe  the  latitude  by  the  moon's 
meridian  altitude,  we  can,  at  nearly  the  same  time,  take  an  observation  of  the  sun's 
altitude  to  regulate  the  chronometer. 


215 


TO    FIND    THE    TIME    AT    SEA    BY  A 
PLANET'S    ALTITUDE. 


We  may  use  either  of  the  large  planets,  Jupitei",  Saturn,  Mars,  or  Venus,  for 
determining  tJie  time  at  sea  ;  and  the  process  is  very  nearly  the  same  as  that  in  the 
preceding  section,  where  tlie  moon's  altitude  is  used.  In  this  case,  we  must  ascertain 
the  time  of  observation,  reduced  to  the  meridian  of  Greenwich,  either  by  a  chronom- 
eter regulated  to  that  meridian,  or  by  knowing  pretty  nearly  the  mean  time  of  obser- 
vation at  the  ship,  and  the  longitude ;  for  by  adding  the  longitude  in  time  to  th 
mean  time  at  the  ship  by  the  watch,  if  the  longitude  be  west,  or  subtracting  the 
longitude  if  it  be  east,  we  shall  obtain  the  corresponding  time  at  Greenwich.  With 
•this  time  at  Greenwich,  we  must  take,  from  the  Nautical  Almanac,  the  suit's  right 
ascension,  the  planeVs  right  ascension,  the  planeVs  declination,  or  polar  distance.  The 
parallax  and  semidiameter  might  also  be  noticed,  but  the  corrections  from  these 
quantities  are  so  small  that  they  may  be  neglected,  as  only  amounting  to  a  few 
seconds.  Then  from  the  observed  central  altitude  of  the  planet  we  must  subtract 
the  dip  and  the  refraction,  and  we  shall  obtain  the  planet's  correct  altitude.*  With 
these  we  may  find  the  time  by  the  following  rule : — 

RULE. 

Add  together  the  planet's  correct  altitude,  the  ship's  latitude,  and  the  polar 
distance ;  from  the  half  sum  subtract  the  planet's  correct  altitude,  and  note  the 
remainder ;  then  add  together  the  log.  secant  of  the  latitude,  the  log.  cosecant  of  the 
polar  distance,  (rejecting  10  from  each  index,)  the  log.  cosine  of  the  half-sum,  and  the 
log.  sine  of  the  remainder  ;  half  the  sum  of  these  four  logarithms  will  be  the  log.  sine 
of  half  the  hour  angle;  take  out  the  corresponding  time  in  the  column  marked  P.  M., 
Table  XXVII.,  and  apply  it  to  the  planet's  right  ascension,  by  subtracting  from  the 
riglit  ascension  when  the  planet  is  east  of  the  meridian,  or  adding  when  west  of  the 
meridian  ;  the  sum  or  difference  will  be  the  right  ascension  of  the  meridian.  From 
the  right  ascension  of  the  meridian  (increased  by  24  hours  if  necessary)  subtract  the 
sun's  right  ascension ;  the  remainder  will  be  the  apparent  time  at  the  ship ;  and  by 
applying  to  it  the  equation  of  time,  found  in  the  Nautical  Almanac,  we  shall  get  the 
required  mean  solar  time,  at  the  meridian  of  the  place  of  observation. 


EXAMPLE    I. 

In  the  latitude  42^  22'  N.,  and  longitude  70°  15'  W.,  on  May  26'>  7"  IS""  35',  astro- 
nomical time,  by  a  watch  which  was  very  nearly  regulated  for  mean  time  at  the  ship, 
observed  the  central  altitude  of  the  planet  Jupiter,  by  a  fore  observation,  and  found 
it  to  be  32°  16'  23"  ;  the  planet  being  to  the  west  of  the  meridian,  and  the  dip  4'  8" 
Required  the  mean  time  of  observation  at  the  ship. 

Adding  the  longitude  4''  41™  to  the  time  by  the  watch,  we  get  the  mean  time  at 
Greenwich,  May  26"  ll''  59™  35';  and  with  this  time  we  get,  from  the  Nautical 
Almanac,  the  sun's  right  ascension  4''  15™  04' ;  Jupiter's  right  ascension  T"  8"  32' ; 
Jupiter's  declination  22°  48'  39"  N.,  or  polar  distance  67°  11'  21". 

*  Wlien  verj'  gjreat  accuracy  is  required,  we  may  notice  Ihe  parallax  in  altitude,  which  is  found  in 
Table  X.  A.,  and  is  to  be  added  to  the  correct  altitude  computed  by  the  above  rule.  We  may  also 
find  the  correction  of  refraction  and  parallax,  by  entering  Table  XVII.  in  the  pa^e  corresponding  to 
the  horizontal  parallax  of  the  planet,  and  taking  out  the  corresponding  number,  which,  being  subtracted 
from  GO',  gives  the  correction  for  parallax  and  refraction,  at  one  operation. 


2U\  VO  FIND   THE   TIME  AT   SEA  BY  A   PLANET'S   ALTITUDE. 

Observed  altitude 32°  16'  23" 

Dip  4'  8",  ref.  1'  30".  .sub.  5  38 

Correct  altitude 32  10  45 

Latitude 42  22  00  Secant 0.13145 

Polar  distance. 07  11  21  Cosecant 0.03537 


Sum 2  )  141  44  06 

Half-sum 70  52  03  Cosine  9.51555 

Altitude 32  10  45 


Remainder 38  41  18  Sine 9.79594 

Sum 2  )  19.47831 

Half-sum sine  9.73916 

Corresponding  to  this,  in  the  column  P.  M.,  is 4''  20""  06' 

Planet's  right  ascension 7   08  32 

Sum  (being  west  of  meridian)  gives  right  ascension  of  meridian 11   34  38 

Subtract  the  sun's  rijjht  ascension 4   15  04 


Gives  the  app^nrent  time  at  tlie  ship 7   19  34 

Equation  of  time  by  Nautical  Almanac sub.  3  14 


Mean  time  at  the  ship 7  16  20 


The  time  by  the  watch  being  7^  18""  35%  it  is  59'  too  slow  for  apparent  time ; 
and  2™  15^  too  fast  for  mean  time. 

EXAMPLE  II. 

January  5''  15'"  40™  24%  1836,  astronomical  mean  time,  at  a  place  in  the  latitude  of 
24°  16'  N.,  longitude  34°  56'  W.  of  Greenwich,  observed  the  central  altitude  of  the 
planet  Saturn,  by  a  fore  observation,  and  found  it  to  be  28°  15';  the  planet  being  east 
of  the  meridian,  and  the  dip  3'  41".    Required  the  mean  time  of  observation  at  the  ship. 

Adding  the  longitude  2''  19™  44'  to  the  time  by  the  watch,  we  get  the  mean  time  at 
Greenwich,  Jan.  5 '  18''  00™  08' ;  and  with  this  time  we  get  from  the  Nautical  Almanac 
the  sun's  right  ascension  19''  5™  13' ;  Saturn's  right  ascension  14''  10™  22' ;  Saturn's 
declination  10°  36'  17"  S.,  or  polar  distance  100°  36'  17". 

Observed  altitude 28°  15'  00" 

Dip  3'  4 1 ",  ref.  1'  46",  sub.  5  27 

Correct  altitude 28   09  33 

Latitude 24   16  00  Secant 0.04018 

Polar  distance 100  36   17  Cosecant 0.00749 


Sum 2)153  01  50 

Half-sum 76*  30   55  Cosine 9.36770 

Altitude 28   09   33 


Remainder 48  21   22  Sine 9.87349 

Sum 2  )  19.28886 

Half-sum sine  9.64443 

Correspondiu;.;  to  this,  in  column  P.  ]M.,  is 3''  29™  20' 

Saturn's  rigiit  ascension 14   10  22 

Difference  (being  east  of  meridian)  gives  right  ascension  of  meridian. .  10  41   02 
Add  24'',  and  subtract  the  sun's  right  ascension 19  05   13 

Gives  the  apparent  time  at  the  ship 15  35   49 

Equation  of  time  by  Nautical  Almanac add  5   46 

Mean  time  at  the  ship 15  41   35 

The  time  by  the  watch  being  15''  40™  24',  it  was  4™  35'  too  fast  for  apparent  time, 
and  1™  11'  too  slow  for  mean  time. 


217 


TO   FIND    THE    APPARENT    TIME    BY    A 
STAR'S    ALTITUDE. 


Correct  the  observed  altitude  for  the  dip  and  refraction,  (the  dip  being  generally  4 
minutes  when  the  obsei-vation  is  taken  on  the  deck  of  a  common-sized  vessel ;)  find 
the  ship's  latitude  at  the  time  of  observation,  and  the  star's  right  ascension  and 
ileclination  in  Table  VIII.*  Add  together  the  star's  correct  altitude,  the  ship's  lati- 
tude, and  the  polar  distance  ;  from  the  half-sum  subtract  the  star's  altitude,  an.d  note 
the  remainder.  Then  add  together  the  log.  secant  of  the  latitude,  the  log.  cosecant 
of  the  polar  distance,  (rejecting  10  in  each  index,)  the  log.  cosine  of  the  half-sum,  and 
the  log.  t^ine  of  the  remainder;  half  the  sum  of  these  four  logarithms  will  he  the  log. 
sine  of  half  the  hour  angle  ;  take  out  the  corresponding  time  in  the  column  marked 
1'.  M.  (Table  XXVII.)  and  apply  it  to  the  star's  right  ascension,  by  subtracting  when 
the  star  is  ea&t  of  the  meridian,  or  adding  when  west  of  the  meridian ;  the  sum  or 
difference  will  be  the  right  ascension  of  the  meridian.  From  the  right  ascension  of 
the  meridian  (increased  by  24  hours  if  necessary)  subtract  the  sun's  right  ascension, 
taken  from  the  Nautical  Almanac  ;  f  the  remainder  will  be  the  apparent  time  at  the 
ship,  and  by  applying  to  it  the  equation  of  time,  we  get  the  mean  time  at  the  ship. 

EXAMPLE    I. 

Suppose  that,  on  September  8'' 14'' 19"  20',1836,  astronomical  time,  as  shown  by  a 
clironometer,  regulated  to  mean  time  at  Greenwich,  when  in  the  latitude  of  7^  45'  S., 
and  longitude  of  29°  12'  E.  from  Greenwich,  the  altitude  of  the  star  Procyon,  being 
then  east  of  the  meridian,  was  observed  by  a  fore  observation,  and  found  to  be  28°  16', 
and  the  dip  4'.     Required  the  mean  time  of  observation  at  the  ship. 

By  inspection  in  the  Nautical  Almanac,  we  find  that,  on  the  above-mentioned  day, 
Procyon's  right  ascension  was  7''  30™  44%  and  the  declination  5°  39'  N.,  or  polar 
distance  95°  39' nearly,  agreeing  nearly  with  the  result  from  Table  VIII.,  corrected 
for  the  annual  variations,  &c. 

Sun's  right  ascension  by  Nautical  Almanac,  Sej)t.  8,  at  mean  noon —  11''  07""  47' 

Correction,  Table  XXXI.,  for  14"  19™  10%  mean  time add  2    09 

Sun's  right  ascension  at  the  time  of  observation 11   09    50 

Star's  observed  altitude 28°  16' 

Dip  4',  ref  2',  Table  XIL,  sub.  0 

Star's  correct  altitude 28   10 

Latitude 7  45         Secant 0.00,399 

Polar  distance 95  39         Cosecant 0.00212 

Sum 2)131   34 

Half-sum 05  47         Cosine 9.01298 

Altitude 28   10 


Remainder 37  37         Sine 9.78560 

Sum 2  )  19.40469 


Ilalf-sum 9.70234 


*  The  right  ascensions  and  declinations  of  the  stars  in  'fable  VIIL  are  the  mean  values  for  January 
1st.  1830,  and  must  be  reduced  to  the  time  of  observation  by  means  of  the  annual  variation  g^iven  in  thp 
sime  table.  When  vcrj'  great  accuracy  is  refjuircd,  the  riglit  ascensions  and  declinations,  thus  obtained, 
must  be  corrected  for  the  aberration  and  nutation,  as  explained  in  the  precepts  of  Tables  XLIl 
XLIII.  j  but  in  general  these  corrections  ma}-  be  neglected.  These  corrections  are,  iiowever,  all 
noticed  in  the  places  of  100  of  the  most  noted  tixed  stars,  given  in  the  Nautical  Almanac  since  the  year 
1834,  for  every  ten  days  in  the  year;  and  when  any  of  these  stars  are  u>c(l,  the  places  must  be  taken 
out,  to  the  nearest  day,  from  the  Nautical  Almanac,  without  any  further  correction,  because  the  varia 
lions  in  ten  days  are  very  small.  Thus,  on  July  29,  183G,  Procyon's  right  ascension  was  1^  30">  43», 
north  polar  distance  84°  21'  29",  or  5°  38'  31"  N.  declination,  corresponding  to  95°  38'  41"  so7i/h  polar 
distance.     This  additional  table  of  the  Nautical  Almanac  simplifies  this  kind  of  calculation  considerably. 

t  The  sun's  right  ascension  and  the  equation  of  time  are  to  be  taken  from  the  Nautical  Almanac,  for 

28 


218        TO   FIND   THE   APPARENT   TIME   BY   A   STAR'S  ALTITUDE. 

Corresponding  to  this  half-sum,  in  Table  XXVIL,  m  column  P.  M.,  is    4''  02™  04' 
Star's  right  ascension 7  30  44 

Right  ascension  of  the  meridian 3  28  40 

Increased  by  24'",  it  is 27  28  40 

Subtract  the  sun's  right  ascension 11   09   56 

Leaves  the  apparent  time  at  the  ship 16   18  44 

Equation  of  time  by  the  Nautical  Almanac sub.  2  43 

Mean  time  at  the  ship 16   16  01 

Now,  the  time  by  the  chronometer  being  14''  19™  20',  it  was  too  slow  for  apparent 
time  by  l*"  59"  24%  or  l**  56™  41'  too  slow  for  mean  time. 

We  have,  in  this  example,  supposed  the  time  at  Greenwich  to  be  given  by  the 
chronometer,  which  is  the  most  simple  way  of  proceeding ;  but  if  you  have  no 
chronometer,  regulated  to  Greenwich  time,  you  must,  in  the  usual  manner,  estimate 
as  nearly  as  you  can  tlie  time  at  Greenwich,  by  adding  the  longitude,  if  west,  to  the 
time  at  the  ship,  or  subtracting  the  longitude,  if  east ;  and  then  use  this  time  in  finding 
the  numbers  from  the  Nautical  Almanac. 


EXAMPLE    II. 

Suppose  that, on Aprill6*  12 ''13"'03%  1836. astronomical  time,  as  shown  by  a  chro- 
nometer, regulated  to  mean  time  at  Greenwich,  when  in  the  latitude  of  48°  57'  N., 
and  longitude  of  67°  25'  W.,  the  altitude  of  Aldebaran,  when  west  of  the  meridian, 
was  22°  25',  and  the  dip  4'.     Required  the  apparent  time  at  the  ship. 

In  the  Nautical  Almanac,  we  find  on  that  day  that  Aldebaran's  right  ascension  was 
4"  26'"  aO%  declination  16°  10'  N.,  or  polar  distance  73°  50'. 

Sun's  right  ascension  by  Nautical  Almanac,  April  16'',  at  mean  noon    . .  1''  38™  20' 
Cor.  Table  XXXI.,  for  12"  13'"  03^ 1   53 

Sun's  right  ascension  at  the  time  of  observation 1   40   13 


Star's  observed  altitude 22°  25' 

Dip  4',  refraction  2',  Tab.  XII. 6_ 

Star's  correct  altitude 22  19 

Latitude 48  57         Secant 0.18262 

Polar  distance 73  50         Cosecant 0.01752 

Sum 2 )  145   08 

Half-sum 72  33         Cosine 9.47694 

Altitude 22  19 


Remainder 50  14         Sine 9.88573 

Sum 2)19.56281 

Half-sum 9.78140 


Corresponding  to  this  half-sum,  in  Table  XXVIL,  column  P.  M.,  is 4"  57"' 33' 

Star's  right  ascension 4   26  30 

Right  ascension  of  the  meridian 9  24  03 

Subtract  the  sun's  right  ascension 1   40   13 

Leaves  the  apparent  time  at  the  ship 7  43  50 

Equation  of  time  by  the  Nautical  Almanac sub.  25 

Mean  time  at  the  ship       7   43  23 

Now,  the  time  by  the  chronometer  being  12''  13™  03%  it  was  too  fast  for  apparent 
time  4''  29™  13%  or  4''  29™  38'  for  mean  time. 


This  method  of  obtaining  the  time  by  the  stars  would  be  accurate,  if  a  good 
horizon  could  be  obtained ;  but  as  that  is  not  always  the  case,  it  is  best  to  regulate 
your  watch  by  the  sun. 

the  time  at  Greenwich  given  by  a  chronometer,  or  by  applying  the  longitude  to  the  estimated  lime  a> 
the  ship,  .n  the  usual  manner. 


219 


TO    REGULATE    A    CHRONOMETER    BY 
EQUAL  ALTITUDES    OF    THE   SUN, 


A  CHRONOMETER  may  be  regulated  on  shore  by  observing  in  the  morning  and 
evening  tlie  times  wlien  the  sun  is  at  the  same  altitude,*  for  the  middle  between 
these  times  would  be  the  apparent  time  of  noon  by  the  chronometer,  if  the  declination 
of  the  sun  remained  the  same  during  the  observation ;  but  if  the  declination  varies, 
as  is  generally  the  case,  the  apparent  time  of  noon,  determined  in  this  manner, 
which,  for  distinction,  we  shall  call  the  middle  /iHie,)must  be  corrected  for  the  change 
of  declination  by  an  equation,  called  the  equation  of  equal  altitudes,  and  the  middle 
time  thus  corrected  will  be  the  correct  time  of  apparent  noon  by  the  chronometer. 
For  greater  accuracy,  several  altitudes  should  be  taken  in  the  morning,  and  corre- 
sponding ones  in  the  afternoon,  and  the  mean  of  the  times  of  the  morning  and 
evening  observations  should  be  res])ectively  taken,  and  the  equation  of  equal  alti- 
tudes, corresponding  to  the  mean  of  all  the  observations,  must  be  calculated  and 
applied  to  the  middle  time,  as  if  a  single  set  of  observations  only  Iiad  been  taken. 

In  noting  the  times  of  observation,  we  must  count  the  hours  in  numeral  succession, 
so  that  if  some  of  the  observations  are  taken  before  IQ"*  by  the  chronometer,  and 
others  after  12'',  the  next  hour  to  12''  must  be  called  IS*",  the  next  14",  &c.  Half  the 
sum  of  the  times  of  observation,  corresponding  to  any  set  of  observations,  (or  the  mean 
of  a  number  of  observations,)  will  be  the  middle  time,  and  the  difference  of  the  times 
of  observation  will  be  the  elapsed  time. 

The  equation  of  equal  altitudes  ponsists  of  two  parts,  which  may  be  calculated  by 
the  following  rule : — 

RULE.' 

1.  To  the  constant  log.  8.8239  add  the  log.  cotangent  of  the  latitude,  the  log.  sine 
corresponding  to  the  elapsed  time  found  in  the  cohunn  1*.  M.  of  Table  XXVII.,  the 
proportional  logarithm  of  the  hours  and  minutes  of  the  elapsed  time,  reckoned  as 
minutes  and  seconds,  and  the  proportional  logarithm  of  the  daily  variation  of  the  sun's 
declination ;  the  sum  (rejecting  .30  in  the  index)  will  be  the  proportional  logaritbm  of 
the  first  part  of  the  equation  of  equal  altitudes,  reckoning  minutes  and  seconds  as 
seconds  and  thirds  respectively. 

2,  To  the  constant  log.  8.8239  add  the  log.  cotangent  of  the  sun's  declination,  the  log. 
tangent  corresponding  to  the  elapsed  time  found  in  the  column  P.M.  of  Table  XXVII., 
the  proportional  logarithm  of  the  hoiu's  and  minutes  of  the  elapsed  time  reckoned  as 
minutes  and  seconds,  and  the  proportional  logarithm  of  the  daily  variation  of  the  sun's 
declination ;  the  sum  (rejecting  30  in  the  index)  will  be  the  proportional  logarithm  of 
the  second  part  of  the  equation  of  equal  altitudes,  reckoning  minutes  and  seconds  as 
seconds  and  thirds  respectively. 

The  first  part  of  tlie  equation  of  equal  altitudes  is  to  be  added  to  the  middle  time 
when  the  sun  is  receding  from  the  elevated  pole,  otherwise  subtracted  ;f  and  the 
second  part  is  to  be  added  when  the  declination  is  increasing,  but  subtracted  when 
decrei^ing  ;  J  these  two  corrections,  being  a])i)Iied  to  the  middle  time,  will  give  the 
apparent  time  of  noon  by  the  chronometer. 

*  Tlie  alliludes  should  be  taken  when  the  sun  rises  or  falls  fast.  The  best  lime  for  observation  is 
when  the  bearing  of  the  sun  is  nearly  east  or  west,  if  the  altitude  exceed  8°  or  10°,  so  as  to  avoid  the 
irregular  refraction  near  the  horizon.  In  general,  two  or  three  hours  from  noon  will  be  suflicient.  An 
artificial  horizon,  formed  by  a  vessel  filled  with  mercury,  may  be  used  in  taking  these  altitudes. 

t  Thus,  in  north  latitudes,  the  first  part  is  to  be  added  from  the  summer  to  tlie  winter  solstice,  when 
the  polar  distance  is  increasing,  and  subtracted  the  rest  of  the  year,  when  the  polar  distance  is 
decreasing. 

i  It  is  here  supposed  that  the  elapsed  time  is  less  than  12  hours,  which  is  generally  the  case ;  but  if 
that  time  exceeds  12  hours,  the  second  part  must  be  applied  in  a  contrary  manner  to  the  above  rule. 


2'iO       TO   REGULATE   A   CHRONOMETER  BY   EQUAL   ALTITUDES. 

EXAMPLE. 

Suppose  that,  on  the  9th  of  May,  1836,  civil  account,  in  the  latitude  of  40°  N.,  and 
longitude  10°  W.,  the  following  observations  were  taken  at  equal  altitudes  of  the  sun; 
required  the  error  of  the  watch. 

t^lt.  0'*  loiver  limh.  Times  per  chron.  Times  per  chron. 

A.  M.  p.  M. 

15°  35'  GhaQ-^Sl'  17"  32"  18* 

15  45  6  31   07  17  31   00 

15  55  6  32  14  17  29  54 

Sum        93  12  93  12 

Mean  6  31   04  17  31   04 
6  31   04 

Difference  is  elapsed  time  11   00  00 

Sum 2)24  02  08 


Middle  time 12  01   04 


Constant  log 8.8239  8.8239 

Latitude  40° cotancent  10.0762  Declination  17°  27'.  cotangent  10.5026 

Elapsed  time  11" sine    9.9963  Tangent 10.8806 

Elapsed  time  IP,  or  IF  P.  L.     1.2139  1.2139 

Variation  declin.  15'  46"*  P.L.     1.0575  1.0575 

1st  part  12"  14"'  P.  L 1.1678  2d  part  0"  36'"  P.  L 2.478[i 

The  first  part  of  tliis  equation,  12"  14'",  is  subtractivc,  because  the  sun  is  proceeding 
towards  the  elevated  pole ;  and  the  second  part,  36"',  is  additive,  because  the  declina- 
tion is  increasing,  so  that  the  whole  equation  is  about  12  seconds  subtractive ;  this, 
being  ajiplied  to  .the  middle  time,  12''  1'"  4',  gives  the  time  of  apj)arent  noon  by  the 
chronometer,  12''  0'"  52%  so  tliat  the  chronometer  is  52  seconds  too  fast  for  ap[iarent 
time. 

*  On  May  9,  at  noon,  by  the  Nautical  Almanac,  the  declination  was  17°  26'  27",  and  on  the  follow- 
ing noon  17°  42' 13",  the  ditTerence  15' 4G",  beinj  the  daily  variation;  the  declination  corresponding 
to  the  longitude  of  10°  W  .  heins  17°  27'  N.  ilearly. 


221 


TO    REGULATE    A    CHRONOMETER     BY 
MEANS  OF  A  TRANSIT  INSTRUMENT. 


This  method  excels  all  others  in  brevity  and  accuracy ;  but  it  can  only  be  used  on 
shore,  and  with  tlie  transit  instrument  that  has  been  adjusted  with  the  greatest  possi- 
ble care,  so  as  to  liave  the  motion  of  the  line  of  collimation  of  the  telescope  perfectly 
in  the  plane  of  the  meridian.  We  have  already  given,  from  pages  145  to  152,  the 
methods  of  making  these  adjustments,  and  of  observing  these  transits ;  we  shall  now 
inseit  several  examples  for  illustration. 

To  determine  the  time  hy  the  sun's  transit  over  the  middle  wire  of  the  telescope. 

In  obsei-vations  of  this  kind,  we  must  note,  by  the  chronometer,  the  times  of  the 
transit  of  the  first  and  second  limbs  of  the  sun  over  the  meridian  wire  ;  the  mean  of 
the  two  observations  will  be  the  time  of  apparent  noon,  by  the  chronometer.  Then 
the  equation  of  time  is  to  be  taken  from  the  Nautical  Almanac  for  the  apj)arent  noon 
at  Greenwich,  and  the  correction  applied  to  it  for  the  longitude  of  the  place  of  obser- 
vation, which  is  easily  obtained  by  the  means  of  the  horary  variation  given  in  the  same 
work.  Applying  this  equation  to  the  apparent  time,  by  adding  or  subtracting, 
according  to  the  directions  in  tlie  Nautical  Almanac,  we  get  the  mean  time  of  ap- 
parent noon.  The  difference  between  this  time  and  the  tune  by  the  chronometer,  will 
be  the  error  of  the  chronometer  in  mean  time;  moreover  the  difference  between  the 
time  by  the  chronometer  and  IS'',  will  be  the  error  of  the  chronometer  for  apparent 
lime. 

EXAMPLE    I. 

Near  noon,  at  the  commencement  of  the  SOt"  of  January,  183G,  according  to  the 
astronomical  computation  of  time,  in  a  place  30°,  or  2'',  west  of  Greenwich,  observed 
the  transits  of  the  limbs  of  the  sun  over  the  meridian  wire  of  the  transit  instrument, 
for  the  purpose  of  regulating  a  chronometer.  It  is  required  to  find,  from  these 
observations,  the  error  of  the  chronometer,  either  for  apparent  or  mean  time. 

Transit  of  the  first  limb  by  the  chronometer ll**  56'"10'.5 

Transit  of  the  second  limb  by  the  chronometer 11   58  27  .0 

Sum 2)        14  37.5 

Half-sum  is  the  time  of  apparent  noon  by  the  chronometer 11   57   18  .7 

Equation  of  time  by  Nautical  Almanac,  at  apparent  noon,  Greenwich         13'"21'.C3 
Correction  for  longitude,  2"  X  0.432 add M 

Equation  of  time  at  the  place  of  observation add         13  22  .5 

Apparent  time  of  observation  at  noon 12   00  00  .0 

Mean  time  of  observation 12   13  22  .5 


Hence  it  appears,  that  the  chronometer  is  too  slow  for  apparent  time  2  41  .3 

Chronometer  too  slow  for  mean  time IG  03  .8 


EXAMPLE  II. 

In  another  observation  of  the  sun's  transit,  similar  to  the  preceding,  made  June  25, 
183G,  in  the  longitude  of  G0°,  or  4'^,  east,  we  shall  suppose  that  the  time  of  the 


222     TO   REGULATE   A   CHRONOMETER   BY   A   TRANSIT   INSTRUMENT. 

Transit  of  the  first  limb,  by  the  chronometer,  was lli*"  02'"  lO'.O 

Transit  of  the  second  limb,  by  the  chronometer 12  04  27.8 

Sum 2 ) 6  37.8 

Half-sum  is  time  of  apparent  noon  by  the  chronometer 12  03  IS  .9 

Equation  of  time  by  Nautical  Almanac  at  apparent  noon  at  Gceenwich  2""  14'.S4 

CoiTection  for  longitude,  4*^  X  0.529 2  .12 

Equation  of  time  at  the  place  of  observation add  2   12  .2 

Ap-parent  time  of  observation  at  noon 12  00  00  .0 

Mean  time  of  observation 12  02   12  .2 

Hence  it  appears,  that  the  chronometer  is  too  fast  for  apparent  time  3™  18'.9 

And  too  fast  for  mean  time 1   06  .7 


To  determine  the  time  hy  the  sun's  transit,  observed  at  the  Jive  wires  of  the 

telescope. 

If  the  telescope  of  the  transit  instrument  be  furnished,  as  usual,  with  five  equidistant 
and  parallel  wires,  two  on  each  side  of  the  meridian  wire,  we  can,  with  very  little 
extra  time  or  trouble,  make  the  observations  of  the  transits  of  the  first  limb  of  the  sun 
at  all  the  Avires,  and  mark  down  the  corresponding  times  by  the  chronometer,  in  five 
sejiarate  columns,  on  tlie  same  horizontal  line,  from  left  to  right.  Immediately  after- 
wards,* make  the  observations  of  the  transits  of  the  second  limb  of  the  sun,  over  the 
same  wires,  and  mark  these  times  below  the  former  numbers  respectively,  taking  them 
in  a  contrary  order,  or  from  right  to  left.  The  sums  of  the  two  numbers  in  each  of 
tho  five  columns  will  be  nearly  the  same,f  and  the  mean  of  the  whole  will  be  the  time 
of  the  transit  of  the  sun's  centre  over  the  meridian,  as  shown  by  the  chronometer. 
Comparing  tliis  with  the  time  of  apparent  noon,  12'',  we  get  the  error  of  the  chronom- 
eter for  a])parent  time  ;  or  by  comparing  it  with  the  mean  time  of  noon,  we  get  the 
error  of  the  chronometer  for  mean  time,  as  in  the  two  preceding  examples. 


EXAMPLE   III. 

July  23,  183C,  in  the  longitude  of  74°,  or  4''  56™,  W.,  the  following  observations  of 
the  times  of  the  transit  of  the  sun's  limbs  over  tlie  wires  of  the  transit  instrument 
were  made.     Required  the  error  of  the  chronometer  for  mean  time. 


First  limb 

Second  limb. . . , 

Sum 


I. 

^  05\0 
09.3 


II. 

5-^  32'.0 
8   42.1 


14    14.3    14    14.1 


Sum 

Mean  of  all  is  transit  by  chronometer. 
Mean  time  of  ai)parent  noon 

Chronometer  too  fust  for  mean  time. . 
Chronometer  too  fast  for  apparent  time 


III. 

12'>  Oo-"  59S5 
12  08    14  .3 


24  14 


13.8 
J4.3 
14.1 
14.2 
14.3 


10)70.7 


12"  07™  07».07 
12  06    07.61 


' 59^46 
'  07».07 


IV. 

6™27» 
7   47 


V. 

6™  54M 
7    20.2 


14    14.2    14    14.3 


Equation  of  Time. 
Noon  at  Greenwich  -}-  6""  07'..32 
Corr.  4"  56™  X 0.059  .29 

Equation  of  time.         6    07.61 

12  h 


12  6   07.6] 


*  We  Iiavc  already  remarked,  in  penoe  150,  (hat  the  wires  are  so  fixed  in  the  telescope,  that  the  first 
limb  of  the  siin  ])asscs  over  all  of  thoni  licfore  the  second  limb  arrives  at  the  first  wire. 

t  This  equality  in  the  sums  renders  it  unnecessary  to  write  down  the  hours  of  the  observation,  except 
in  the  middle  column  ;  and  we  may  also  neglect,  in  the  column  of  minutes,  the  figures  which  stand  for 
lens  of  minutes;  retaining  the  full  expression  of  the  lime  only  in  the  middle  column. 


TO  REGULATE  A   CHRONOMETER  BY   A  TRANSIT   INSTRUMENT.    223 


EXAMPLE   IV. 

May  14, 1836,  in  the  longitude  of  45°,  or  S*",  east,  the  following  transits  of  the  sun's 
limb  over  the  wires  of  the  transit  instrument,  were  obsei'ved.  Required  the  error  of 
the  chronometer  for  mean  time. 


First  limb. ... 
Second  limb. . 

Sum 


I. 

10'.5 
14.0 


IL 

53"^  37'.5 
56    46.5 


50    24  .5    50    24  .0 


Sum 

Mean  of  all  the  transits  by  chronometer 
Mean  time  of  ap|)arent  noon 


Chronometer  too  slow  for  mean  time. 
Chronometer  too  slow  for  app.  time . . 


III. 

1P54'"05'.0 
II  56    19.7 


23  50 


24.7 
24.5 
24.0 
24.6 
24.3 


10 )  122  .1 


11"55"'12^2I 
II   56   03.74 


0'"5P.53 
4"-  47'.79 


IV. 

54"'32».5 
55    52.1 


50   24.6 


V. 

54"'59».3 
55    25.0 


50    24.3 


Equation  of  Time. 
Noon  at  Greenwich  — 3"  56'.30 
Corr.  3"  X  0.014. ._ M 

Equation  of  time  —    3    56  .26 
Apparent  noon    12  00    00.00 

Mean  noon II   56   03  .74 


To  determine  the  time  hy  the  transit  of  a  fixed  star  over  the  meridian. 

In  observations  with  the  transit  instrument,  it  is  most  commonly  the  case,  that  the 
chronometer  wliich  is  used  in  making  the  observations,  will  give  the  viean  time  at 
Greenwich  within  a  few  seconds ;  *  and  for  this  time  we  must  find,  in  the  Nautical 
Almanac,  the  sun's  right  ascension  and  that  of  the  star.  Subtracting  the  former  from 
the  latter,  (increased  by  24''  when  necessary,)  we  get  the  apparent  time  of  the  star's 
transit  over  the  meridian  ;  and  by  applying  to  it  the  equation  of  time,  taken  from  the 
Nautical  Almanac,  for  the  above  time  at  Greenwich,  we  obtain  the  mean  time  at  the 
place  of  observation.  The  difference  between  this  and  the  time  of  tlie  transit,  as 
noted  by  the  chronometer,  will  represent  its  error.  We  may,  as  in  observations  of 
the  sun,  use  the  middle  wire  only,  and  note  the  time  of  the  transit,  when  the  star  is 
bisected  by  that  wire ;  or,  with  greater  chance  of  accuracy,  we  may  take  the  mean  of 
the  observed  times  of  passing  the  five  wires,  as  a  more  correct  time  of  the  actual 
transit.     To  illustrate  this,  we  shall  give  the  following  examples : — 


EXAMPLE    V. 

July  24, 1836,  in  the  longitude  of  44°  39',  or  2''  58"  36%  east,  observed  the  transit  of 
the  star  Arcturus  over  tlie  middle  wire  of  the  telescope,  the  time  by  the  chronometer, 
which  was  snpposred  to  be  regidated  very  nearly  for  mean  time  in  the  meridian  of 
Greenwich,  being  8''  00™  10'.  Required  the  mean  time  of  the  transit  at  the  place  of 
observation. 

S"  15™  05».79 
29  .70 


0's  right  ascension  at  noon,  at  Greenwich,  by  Nautical  Almanac. . 
Correction  for  3''  00'"  10'  X  9'.89I 

©'s  right  ascension  at  the  estimated  time  at  Greenwich 

Star's  right  ascension  at  the  same  time,  by  Nautical  Almanac 


8   15   35.49 
14  08    12.13 


Subtract  ©'s  right  ascension,  gives  the  apparent  time  of  observation    5   52   36  .64 

Equation  of  time  at  noon,  Greenwich -{-Q""  08'.74 

Correction  for  3"  00™  10'  X  0'.035 


Jl 

Corrected  equation  of  time -|"  ^    08  .85 . 

Mean  time  of  observation 

Time  by  the  chronometer 

Error  of  the  chronometer  for  mean  time 

Error  of  the  chronometer  for  apparent  time 


+ 6   08.85 

5  58   45.49 
3  00    10.00 

2  58   35.49 
2  52  26.64 


••When  we  have  no  good  regulation  of  the  chronometer,  from  Greenwich,  we  must  estimate  the 
lime  at  that  place,  from  the  supposed  time  at  the  place  of  observation,  by  applying  to  it  the  longitude  ; 
adding  when  west,  or  subtracting  when  east  ;  repealing  the  operation  if  we  should  find,  after  calculating 
ilw  observations  of  the  transit,  that  any  essential  error  was  made  in  the  time  at  the  place  of  observation. 


224    TO   REGULATE   A   CHRONOMETER  BY  A  TRANSIT  Il^STKljMENT. 

EXAMPLE  VL 

March  10,  1836,  in  the  longitude  of  17°  18',  or  1"  OO""  12%  east,  observed  the  transit 
of  the  star  Siriusover  the  five  wires  of  the  telescope,  at  the  times  by  the  chronometer 
as  given  below  ;  the  chronometer  being  supposed  to  give  very  nearly  the  mean  time 
at  Greenwich.    Required  the  mean  tune  of  this  ti'ansit  at  the  place  of  observation. 

/-  First  wire G"  14'"  01 '.5 

\  Second  wire 14  28  .7 

Time  of  transit  by  the  clu-onometer.  <  Meridian  wire 14  56  .0 

V  Fourth  wire 15  23  .2 

^  Fifth  wire 15  50  .6 

Sum 5124  40--.0 


Mean  of  all  the  times  by  the  chronometer  is G*"  14""  56  .0 


Co 


2)'s  right  ascension  at  noon,  at  Greenwich,  by  the  Nautical  Almanac  23''  23"  10  .85 
;orrection  for  6"  14"'  56=  X  9M86 57  .40 

(v)'s  right  ascension  at  the  estimated  time  at  Greenwich 23  24   08  .25 

Star's  right  ascension  at  the  same  time  by  the  Naut.  Almanac -j- 24''  30   37  55.39 

Subtract  0's  i-ight  ascension,  gives  the  apparent  time  of  obsei-vation     7   13  47  .14 

Equation  of  time  for  noon  at  Greenwich 10"'  25'.  45 

Correction  for  G"  14-"  56^  X  0^G68 4.17 

Corrected  equation  of  time 10  21  .28 add         10  21  .28 

Mean  time  of  observation 7  24   08  .42 

Time  by  the  chronometer G   14  56  .00 

Error  of  the  chronometer  for  mean  time  1   09   12  .42 

Error  of  the  chronometer  for  apparent  time 0  58  51  .14 

We  may  in  the  same  way  find  the  time  by  a  transit  of  the  planet,  either  by  taking 
the  mean  of  the  times  of  the  transits  of  the  two  limbs  of  the  planet  across  the  middle 
wire,  or  the  mean  of  the  times  of  the  limbs  passing  all  the  wires;  then  the  calculation 
is  to  be  made,  as  in  Examples  V.  VI. ;  taking  from  the  Nautical  Almanac,  and  using 
the  right  ascension  of  the  planet,  instead  of  that  of  the  star.  This  method  is  so  plain, 
that  it  will  not  be  necessary  to  give  any  examples.  The  ti-ansit  of  the  moon  might 
also  be  used  ;  but  the  calculation  becomes  so  complex,  on  account  of  the  rapidity  of 
her  motion,  that  it  is  wholly  inexpedient  to  use  such  observations  for  regulating  a 
chi'onometer. 


225 


LUNAR    OBSERVATIONS. 


Almost  all  the  methods  of  determining  the  difference  of  longitude  between  .iny 
two  places,  depend  on  the  general  principle  of  finding  the  dilference  between  the 
times  of  taking  any  observation,  estimated  under  tlie  meridian  of  both  those  places. 
For.  in  any  place,  it  is  the  time  of  apparent  noon  when  the  sun  is  on  the  meridian  ; 
and  as  the  sun,  by  his  diurnal  motion,  appears  on  the  meridian  of  Greenwich  (from 
which  the  longitude  is  reckoned)  one  hour  earlier  than  in  a  place  in  15°  west  longi- 
tude,* and  cue  hour  later  than  in  a  place  in  15°  east  longitude,  and  in  proportion  for 
a  greater  or  less  longitude,  it  follows  that,  if,  at  the  time  of  taking  an  observation,  the 
corresponding  time  at  Greenwich  be  known,  the  longitude  of  the  place  of  observation 
will  be  found  by  alloAving  15°  for  every  hour  of  diffei'ence  betv\een  those  times,  the 
longitude  being  east  when  the  time  at  Greenwich  is  earlier  than  at  the  place  of 
observation,  otherwise  west.  It  is  immaterial  whether  the  times  at  both  places  be 
estimated  for  apparent  or  mean  time,  as  the  interval  is  the  same  when  both  are 
apparent  as  when  both  are  mean ;  it  is,  however,  universally  the  practice,  at  i)resent, 
to  use  mean  time  in  all  these  calculations.  Now,  an  observer,  at  any  place,  may 
determine  the  apparent  or  mean  time  at  any  moment,  by  a  watch  regulated  by  any 
of  the  preceding  methods ;  and  if,  at  the  same  moment,  the  apparent  or  mean  time 
at  Greenwich  could  be  obtained,  nothing  more  would  be  necessary  for  determining 
the  longitude.  One  method  of  determining  the  time  at  Greenwich  is  by  a  watch 
regulated  to  Greenwich  time  ;  for  it  is  evident  that  if  a  watch  could  be  so  constructed 
as  to  go  uniformly  at  all  times,  and  in  all  places,  an  observer,  furnished  with  a  watch 
thus  regulated,  would  only  have  to  compare  the  time  at  the  place  of  observation  with 
the  time  at  Greenwich,  shown  by  the  watch,  and  the  difference  of  the  times  would 
give  the  difference  of  longitude.  This  method  is  useful  in  a  short  run  ;  but  in  a  long 
voyage,  implicit  confidence  cannot  be  placed  in  an  instrument  of  such  a  delicate  con- 
struction, and  liable  to  so  many  accidents.  Another  method  of  determining  the 
longitude,  is  by  observing  the  beginning  or  end  of  an  eclipse  of  the  moon,  or  the 
satellites  of  Jupiter,  and  taking  the  difference  between  the  mean  time  of  observation 
and  the  mean  time  given  in  the  Nautical  Almanac  for  the  meridian  of  Greenwich  ; 
it  being  evident  that  such  an  eclipse  must  be  observed  at  both  places  at  the  same 
moment  of  absolute  time  ;  consequently  the  difference  of  the  times  will  be  the  differ- 
ence of  longitude.  An  observation  of  an  eclipse  of  the  sun,  or  an  occultation,  afler 
making  allowance  for  parallax,  &c.,  as  taught  in  the  Appendix  to  this  work,  may  be 
used  in  like  manner ;  and  this  is  a  very  accurate  method.  However,  observations  of 
eclipses  are  but  of  small  practical  utility  at  sea ;  for  those  of  the  sun  and  moon  happen 
too  seldom,  and  the  difficulty  of  oliserving  the  eclipses  of  Jupiter's  satellites  prevents 
that  method  from  being  made  use  of  In  the  present  improved  state  of  the  Nautical 
Almanac,  we  may  easily  determine  the  longitude  on  shore,  by  means  of  a  transit 
instrument,  by  observing  the  time  of  the  moon's  transit  over  the  meridian,  or  by 
observing  the  difference  between  the  time  of  the  moon's  transit  and  that  of  some 
well-known  and  near  star.  Other  metliods  of  finding  the  longitude  at  sea  have  been 
proposed,  but  among  them  all  there  is  not  one  of  such  practical  utility,  as  that  by 
measuring  the  angular  distance  of  the  moon  from  the  sun,  or  from  certain  fixed  stars 
situated  near  the  ecliptic,  usually  called  a  hmar  ohservation,  or,  more  frequently, 
"a  lunary  For  observations  of  this  kind  may  be  taken,  in  fair  weather,  at  all  times 
(except  near  the  time  of  new  moon)  when  the  objects  are  more  than  8°  or  10°  above 
the  horizon ;  and  as  the  moon  moves  in  her  orbit  about  1'  in  2'"  of  time,  it  follows 
that,  if  her  angular  distance  can  be  ascertained  from  the  sun  or  star  within  1',  the  time 
at  Greenwich  will  be  known  within  2  minutes,  and  the  longitude  within  30  miles. 

*  Because  the  sua,  by  his  apparent  diurnal  molion,  describes  360  degrees  in  21  hours,  wliich  makes 
15  degrees  in  an  hour. 
29 


226  LUiNAR  OBSERVATIONS. 

To  facilitate  tliis  methotl,  there  is  annually  published,  by  the  Commissioners  of  Lon- 
gitude in  England,  a  Nautical  Almanac,  containijig  the  true  angular  distances  of  the 
moon  from  the  sun,  from  the  four  large  planets,  and  from  nine  bright  fixed  stars,  for 
the  beginning  of  every  third  hour  of  mean  time  for  the  meridian  of  Greenwich  ;  and 
the  mean  time  corresponding  to  any  intermediate  liom-  may  be  found  by  proportional 
parts  :  hence,  an  observation  of  these  angular  distances  l)eing  taken  in  any  place,  and 
the  corresponding  mean  time  at  Greenwich  being  found  by  the  Almanac,  and  com- 
pared with  the  mean  tune  at  the  ship,  tlieir  difference  will  be  the  longitude  of  the 
l)lace  of  observation.  But  before  tlie  observed  angular  distance  is  compared  with 
those  in  the  Nautical  Almanac,  the  corrections  for  parallax  and  refraction  must  be 
applied  to  obtain  the  true  distance  ;  for,  the  moon  being  seen  always  lower  than  her 
true  i)lace,  and  the  sun  and  stars  higher,  the  true  distance  is  almost  always  greater  or 
less  than  the  observed  distance. 

The  angular  distances  of  the  moon  from  the  sun  and  proper  fixed  stars  and  planets, 
are  generally  given  in  the  Nautical  Almanac  from  one  object  on  each  side  of  her,  to 
afford  a  greater  number  of  opportunities  of  observation,  and  to  enable  the  observer  to 
correct,  iu  a  great  degree,  the  errors  of  the  instnmient,  the  adjustments,  or  a  faulty 
liabit  of  observing  the  contact  of  the  limbs,  because  these  errors  have  a  natural  ten- 
dency to  correct  each  other,  in  taking  the  mean  of  observations  made  with  objects 
on  different  sides  of  the  moon.  Before  taking  the  observation,  the  Nautical  Almanac 
must  be  examined,  to  see  from  what  objects  the  distances  are  computed,  and  from 
them  only  must  the  distances  be  measured. 

There  are  only  nine  fixed  stars  and  four  planets  from  which  the  angular  distances 
are  computed  in  the  Nautical  Almanac  ;  and  as  it  is  of  the  greatest  importance  to  be 
able  to  discover  them  easily,  we  shall  here  add  a  number  of  remarks  which  will  be 
found  useful  for  that  ])urpose. 

Tlie  best  way  of  discovering  any  star  or  planet,  is  by  means  of  a  celestial  globe  ; 
observing  that,  when  a  planet  is  used,  we  must  estimate  roughly,  by  inspecting  the 
Nautical  Almanac,  the  right  ascension  and  declination  of  the  planet,  and  make  a  mark 
on  the  corresponding  point  of  the  globe  with  a  pencil,  or  by  attaching  a  small  piece 
of  moist  jiaper,  and  this  must  be  considered  as  the  place  of  the  planet.  If  a  globe 
cannot  be  obtained,  the  time  of  passing  the  meridian,  and  the  meridian  altitude  of  the 
object,  may  be  calculated  ;  and  by  observing  at  that  time,  the  object  may  be  easily 
discovered.  The  distances  marked  in  the  Nautical  Almanac  afford  also  to  the 
observer  an  easy  method  of  knowing  the  star  or  planet  from  which  the  moon's  dis- 
tance is  to  be  observed ;  for  he  has  nothing  to  do  but  to  set  the  sextant  or  circle  to 
the  distance  comi)uted  roughly  for  the  apparent  time,  estimated  nearly  for  the 
meridian  of  Greenwich,  and  direct  his  sight  to  the  east  or  west  of  the  moon,  accord- 
ing as  the  object  is  marked  E.  or  W.  in  the  Nautical  Almanac  ;  and,  having  found  the 
reflected  image  of  the  moon  upon  the  horizon  glass,  sweep  the  instrument  to  the 
right  or  left,  and  the  image  will  pass  over  the  sought  star  or  planet,  if  above  the 
horizon,  and  the  weather  clear:  the  star  or  planet  is  always  one  of  the  brightest,  and 
is  situated  nearly  in  the  ai-c  passing  through  the  moon's  centre,  perpendicular  to  the 
line  connecting  the  two  horns. 

The  computed  distance  made  use  of  in  sweeping  for  the  star,  may  be  found  in  this 
manner: — Reckon  the  apparent  time  at  the  ship  in  the  manner  of  astronomers,  (by 
counting  24  hours  from  noon  to  noon,  and  taking  the  day  one  less  than  the  sea 
account;)  to  this  time  apply  the  longitude  turned  into  time,  by  adding  in  west,  or 
subtracting  in  east  longitude;  the  sum  or  difference  will  be  the  apparent  time  at 
Greenwieii  nearly.  Take  tlie  distances  from  the  Nautical  Almanac  for  the  time 
immediately  {)receding  and  following  this  estimated  time,  and  note  the  diffcTcnce  of 
these  distances;  then  say.  As  3*",  or  180"",  is  to  the  dilTerence  of  the  distances,  so  is 
the  difference  between  the  a])parent  time  at  Greenwich  and  the  next  preceding  time, 
set  down  in  the  Nautical  Almanac,  to  a  pro]iortional  part  to  be  added  to  the  next 
preceding  distance  taken  from  the  Nautical  Almanac,  if  the  distance  be  increasing, 
but  subtracted  if  decr(;asing  ;  the  sum  or  difference  will  be  the  distance  at  which  the 
quadrant  or  sextant  is  to  be  fixed. 

In  sweeping  for  the  stars  by  this  method,  it  will  often  happen  that  two  or  more  are 
swept  ujion  at  once  ;  this  might  cause  some  difficulty  to  an  inexperienced  observer, 
who  would  be  at  a  loss  to  know  which  to  jnake  use  of.  To  remove  this,  the  follow- 
ing description  of  these  stars  is  added: — 


LUNAR  OBSERVATIONS. 


227 


«  ARIETIS. 


■VV 


This  star  bears  about  vvest,  distant  22P,  from  the  Pleiades,  or  the 
Seven  Stars ;  it  is  of  the  second  magnitude,  and  may  be  known 
by  means  of  the  star  n,  of  the  third  magnitude,  situated  S.  W. 
from  a  Arietis,  at  the  distance  of  3^  degrees.  South  from  the 
star  n,  at  the  distance  of  1^°,  is  the  star  r,  of  the  fourth  magnitude. 
The  northernmost  of  these  stars  is  a  Arietis. 


ALDEBARAN. 


About  35°  E.  S.  E.  from  «  Arietis,  and  14°  S.  E.  from  the 
Pleiades,  or  Seven  Stars,  is  tiie  bright  star  Aldebaran.  Near  this 
star,  to  the  westward,  are  six  or  seven  stars  of  tlie  third  or  fourth 
magnitude,  forming,  willi  Aldebaran,  a  figure  resembling  tlie  let- 
ter V,  as  is  represented  in  the  adjoined  figure,  where  Aldebaran 
is  marked  a.  At  the  distance  of  23°  from  this  star,  in  a  S.  E. 
direction,  are  three  \ery  bright  stars,  situated  in  a  straight  line, 
near  to  each  other,  forming  the  belt  of  Orion. 


POLLUX.  At  the   distance  of  45^  from    Aldebaran,  in  the  direction  of 

E.  N.  E.,  is  the  star  Pollux,  whicli,is  a  bright  star,  though  not  of 
the  first  magnitude.  N.  W.  from  it,  distant  5°,  is  the  star  Castor, 
of  nearly  the  same  magnitude  ;  and  you  will  almost  always  sweep 
both  at  once  :  the  southernmost  is  the  one  used. 


REGULUS 

4.        "^ 


% 


^   EfigiilxiS. 


^         SPICA. 


E.  by  S.  \  S.  from  Pollux,  at  the  distance  of  37i°,  is  the  star 
Regulus,  of  the  first  magnitude  ;  to  the  northward  of  this  star  (at 
the  distance  of  8°)  is  a  star  of  the  second  magnitude  ;  near  to 
these  are  five  stars  of  the  third  magnitude,  the  whole  forming  a 
cluster  resembling  a  sickle,  represented  in  the  adjoined  figure, 
Regulus  being  in  the  extremity  of  the  handle.  A  line  drawn 
from  the  northern  polar  star,  through  its  pointers,  passes  about 
12°  to  the  eastward  of  Regulus. 


E.  S.  E.  from  Regulus,  at  the  distance  of  54°,  is  the  star  Spica, 
of  the  first  magnitude,  with  no  very  bright  star  near  it;  S.  W. 
from  this  star,  at  the  distance  of  about  16°,  are  five  stars  of  tlie 
third  or  fourth  magnitude,  situated  as  in  the  adjoined  figure ;  the 
two  northernmost  of  these  stars,  ?;,  v,  form  a  straight  line  with 
Spica,  and  by  this  mark  it  may  be  easily  discovered.  A  line 
drawn  from  the  northern  polar  star,  through  the  middle  star  of  the 
tail  of  the  Great  Bear,  will  pass  near  to  Spica. 


ANTARES. 


% 


a  AQUILiE. 


^ 


E.  S.  E.  from  Spica,  at  the  distance  of  4G°,  is  the  star  Afitares, 
in  2G°  of  south  declination  ;  it  is  a  remarkable  star,  of  a  reddish 
color ;  on  each  side  of  it,  to  the  W.  N.  W.  and  S.  S.  E.,  about  2° 
distant,  is  a  star  of  the  third  or  fourth  magnitude,  no  very  bright 
star  bein?  near. 


N.  E.  from  Antares,  at  the  distance  of  G0°,  is  the  very  bright 
star  a  AquiI(B  ;  N.  N.  W.  from  which,  at  2°  distance,  is  a  star  of 
the  third  magnitude,  and,  S.  S.  E.,  at  3°  distance,  another  star  of 
a  less  magnitude.  These  three  stars  appear  nearly  in  a  straight 
line.     The  star  a  Aquilse  is  nearly  of  the  same  color  as  Antares. 


FOMALHAUT. 


a  PEGASI. 


M'r 


•■X- 


S.  E.  from  a  Aquilae,  at  the  distance  of  60°,  is  the  star  Fomalhaut, 
which  is  a  bright  star  of  high  soutliern  declination  its  altitude 
in  northern  latitudes  being  small,  never  exceeding  4U°  m  the  lati- 
tude of  40-^  N.  This  star  bears  nearly  south  from  the  star  a  Peg- 
asi,  distant  45°.  A  line  drawn  from  the  pointers,  through  the 
northern  polar  star,  and  continued  to  the  opposite  meridian,  will 
pass  very  near  to  a  Pegasi  and  Fomalhaut. 

E.  by  N.  from  u  AquiltB,  at  the  distance  of  43°,  and  westward 
from  a  Arielis,  at  the  distance  of  44°,  is  the  star  a  Pegasi,  which 
inay  be  known  by  means  of  four  stars  of  different  magnitudes, 
situated  as  in  the  adjoined  figure  ;  in  which  a  represents  a  Pegasi, 
(i  a  star  of  the  second  magnitude,  bearing  north  of  it,  distant  13°  . 
the  others  are  of  less  magnitudes,  and  two  of  them,  ?;,  u,  form  a 
straight  line  with  the  star  a  Pegasi ;  and  by  this  mark  it  may  be 
easily  discovered 


228  LUNAR   OBSERVATIONS. 


General  Remarks  on  the  taking  of  a  Lunar  Observation. 

The  accuracy  of  a  lunar  obsei*vation  depends  chiefly  on  the  reguhntion  of  the 
cljroiiometer,  and  on  the  exact  measurement  of  the  angular  distance  of  tiie  moon 
from  the  sun  or  star ;  a  small  error  in  the  observed  altitudes  of  those  objects,  will  not 
in  general  much  affect  the  result  of  the  calculation. 

The  best  method  of  regulating  a  clironometei:  at  sea,  is  by  taking  an  altitude  of 
the  sun  when  rising  or  falling  quickly,  or  when  bearing  nearly  east  or  west,  the  alti- 
tude being  sufficiently  great  to  avoid  the  irregular  refraction  near  the  horizon,  and 
noting  the  time  by  the  chronometer.  With  this  altitude,  the  latitude  of  the  ])lace, 
and  the  sun's  declination,  find  the  mean  time  of  observation  by  either  of  the 
I)receding  methods ;  the  difference  between  this  time  and  that  shown  by  the  chro- 
nometer will  show  how  much  it  is  too  fast  or  slow.  A  single  observation,  taken 
with  care,  will  generally  be  exact  enough;  but  if  greater  accuracy  is  required,  the 
:iiean  of  a  number  of  observations  may  be  taken.  If  the  distance  of  the  sun  and 
moon  be  observed  when  the  sun  is  three  or  four  points  distant  from  the  mei-idian, 
the  mean  time  of  observation  may  be  deduced  from  the  altitude  of  the  sun  taken 
at  the  precise  time  of  measuring  the  distance;  this  will  render  the  use  of  a  chronom- 
eter unnecessary,  and  will  prevent  any  irregularity  *  in  its  going  from  affecting  the 
result  of  the  observation.  If  a  night  observation  is  to  be  taken,  the  chronometer 
should  be  regulated  by  an  altitude  of  the  sun  taken  the  preceding  evening,  and  its 
going  examined  by  means  of  another  observation  taken  the  next  morning  ;  for  the 
time  found  by  an  altitude  of  a  star  cannot  be  so  well  depended  upon,  except  in  the 
morning  and  evening  twilight,  as  the  horizon  is  generally  ill-defined ;  but  the  altitude 
may  be  sufficiently  exact  for  finding  the  correction  used  in  determining  the  angular 
distance. 

Although  all  the  instruments  used  in  these  observations  ought  to  be  well  adjusted, 
3'et  particular  care  should  be  taken  of  the  sextant  or  circle  used  in  measuring  the 
angular  distance  of  the  moon  from  the  sun  or  star,  since  an  error  of  1'  in  this  distance 
will  cause  an  error  of  nearly  30'  in  the  longitude  deduced  therefrom.  When  a  great 
angular  distance  is  to  be  measured,  it  is  absolutely  necessary  to  use  a  telescojie,  and 
the  ])arallelism  of  it,  with  respect  to  the  plane  of  the  instrument,  must  be  carefully 
examined ;  but  in  measuring  small  distances,  the  use  of  the  telescope  is  not  of  such 
great  importance,  and  a  sight-tube  may  then  be  used,  taking  care,  however,  that  the 
eye  and  point  of  contact  of  the  objects  on  the  horizon-glass  be  equally  distant  from 
the  plane  of  the  instrument.  But  it  ought  to  be  observed,  that  it  is  always  conducive 
to  accuracy  to  use  a  telescope,  and,  after  a  little  practice,  it  is  easily  done. 

Whilst  one  person  is  observing  the  distance  of  the  objects,  two  others  ought  to  be 
observing  the  altitudes.  The  chronometer  should  be  placed  near  one  of  the 
o!)servers,  or  put  into  the  hands  of  a  fourth  person  appointed  to  note  die  time ;  the 
observer  who  takes  the  angular  distance  giving  previous  notice  to  the  others  to  be 
ready  with  their  altitudes  by  the  time  he  lias  finished  his  observation  ;  which  being 
done,  the  time,  altitudes,  and  distance,f  should  be  carefully  noted,  and  other  sets  of 
observations  taken,  which  must  be  done  within  the  space  of  15  minutes,  and  the 
mean  of  all  these  observations  must  be  taken  and  worked  as  a  single  one. 

When  a  ship  is  close-hauled  to  the  wind,  with  a  large  sea,  or  when  sailing  before  the 
wind,  and  rolling  considerably,  it  is  difficult  to  measure  the  distance  of  the  objects ; 
but  when  the  wind  is  enough  upon  the  quarter  to  keep  tlie  ship  steady,  there  is  no 
difficulty,  especially  in  small  distances,  which  are  much  more  easily  measured  than 
large  ones,  and  are  not  so  liable  to  error  from  an  ill  adjustment  of  the  telesco])e  :  an 
observer  would  therefore  do  well  to  choose  those  times  lor  observation  when  the 
distance  of  the  objects  is  less  than  70"  or  80°.  An  observation  of  the  sun  and  moon 
is  generally  m^.  'i  easier  to  take  when  the  altitude  of  the  moon  is  less  than  that  of 
tlie  sun,  because  the  instrument  will  be  held  in  a  more  natural  and  easy  manner 
When  the  moon  is  near  the  zenith,  the  observation  is  generally  difficult  to  take,  and 
liable  to  be  erroneous,  because  the  observer  is  forced  to  place  himself  in  a  disagreea- 
ble posture.     For  the  same  reason,  an  observation  of  the  moon  and  a  star  or  planet 


*  It  is  not  unromnion  to  find  a  clifTerence  in  l!ic  regulation  of  a  chronnmelcr  in  the  forenoon  and 
afternoon;  tliis  dilferonce  generally  arises  from  the  uncertainty  in  the  estimated  latitude, or  some  sJiglit 
error  in  the  observation,  and  perliaps  partly  from  the  irregularity  in  the  going  of  the  chroiiometer. 

t  If  the  distances  are  measured  liy  a  circular  ins.trumcnt,  it  will  not  be  necessary  to  note  the  several 
distances  measured,  but  only  the  times  and  altitudes,  as  the  sum  of  all  the  distances  measured  by  the 
circle  will  be  given  b)'  the  instrument  at  the  end  of  the  observations ;  and  if  the  aliiiudes  of  the 
objects  are  also  measured  by  circular  instruments,  it  will  not  be  necessary  to  note  the  several  altitudes, 
but  only  the  times  of  observation. 


LUNAR  OBSERVATIONS.  229 

is  generally  much  easier  to  take  when  the  star  or  planet  is  lower  than  the  moon. 
This  situation  of  the  objects  may  in  most  cases  lie  obtained  by  taking  the  observation 
at  a  |)ro|)er  time  of  the  day.  But  it  nnist  be  observed,  that  neither  of  the  objects,  if 
possil)le,  ought  to  be  at  a  less  altitude  than  10^,  upon  account  of  the  uncertainty  of 
the  refraction  near  the  horizon  ;  fir  the  horizontal  refraction  varies  from  133'  to  3G'  40" 
only  by  an  alteration  of  40°  in  the  tliermomcter.  This  alteration  might  cause  an 
error  of  two  degrees  in  the  longitude,  with  an  observer  who  uses  the  mean  refraction. 

In  measming  the  distance  of  the  moon  from  the  sun,  we  must  bring  the  moon's 
round  limb  in  contact  with  the  nearest  limb  of  the  sun.  In  measuring  the  distance 
of  the  moon  from  a  planet  or  fixed  star,  her  round  limb  must  be  brought  in  contact 
with  the  centre  of  the  star  or  planet;  observing  that,  the  scmidiameter  of  the  planet 
being  oidy  a  lew  seconds,  the  centre  of  it  can  be  estimated  sufficiently  near  for  all 
the  purposes  of  this  observation.* 

In  taking  the  altitude  of  the  moon,  the  round  limb,  whether  it  be  the  ujjpcr  or 
lower,  must  be  brought  to  the  iiorizon.  In  damp  weather,  it  is  rather  dirficult  to 
observe  the  altitude  cf  the  stars,  on  account  of  their  dimness,  particularly  a  Pegasi 
and  u  Arietis.  Sometimes  they  are  so  dim  that  they  cannot  be  seen  through  the 
holes  of  the  sight-vane  of  a  quadrant,  particularly  if  the  mirrors  are  not  well 
silvered ;  in  this  case,  the  vane  must  be  turned  aside,  and  the  eye  held  in  nearly  the 
same  place,  or  the  altitude  must  be  taken  by  a  sextant  furnished  with  a  sight-tube. 

We  have  here  sujiposed  that  there  were  obsei-vers  enough  to  measure  the  altitudes 
when  the  distance  was  observed  ;  but  if  that  is  not  the  case,  the  altitudes  may  be 
estimated  by  either  of  the  methods  which  will  be  hereafter  given. 

Preparations  necessary  for  working  a  Lunar  Observation. 

Find  the  mean  time  of  observation  b}'^  astronomical  account,  reckoning  tlie  hours 
from  noon  to  noon  in  numerical  succession  from  1  to  24,  and  taking  the  day  one  less 
than  the  sea  account ;  to  this  time  apply  the  longitude  turned  into  time  by  Table  XXI.f 
by  adding  if  in  west  longitude,  but  subtracting  if  in  east;  the  sum  or  difference  J  will 
be  the  supposed  time  at  Greenwich,  or  reduced  time. 

In  ])age  III.  of  the  month  of  the  Nautical  Almanac,  find  the  moon's  scmidiameter 
and  horizontal  parallax,  for  the  nearest  noon  and  midnight  before  and  after  the 
reduced  time,  and  find  the  difference  of  the  parallaxes  and  the  difference  of  the  semi- 
diameters  ;  then  enter  Table  XI.  with  these  differences  respectively  in  the  side 
column,  and  the  reduced  time  at  the  top;  opposite  the  former,  and  under  the  latter, 
will  stand  the  corrections  §  to  be  ap])lied  respectively  to  the  semidiameter  and  hori- 
zontal parallax  ;iiarked  first  in  the  Nautical  Alman-'c,  additive  if  increasing,  subtractive 
if  decreasing;  the  sum  or  difference  will  be  the  horizontal  semidiameter  and  the 
horizontal  parallax,  res])ectively,  at  the  time  of  observation.  To  this  horizontal  semi- 
diameter must  be  added  the  augmentation  from  Table  XV.  corresponding  to  tlie 
moon's  altitude;  the  sum  will  be  the  true  scmidiameter  of  the  moon. 

The  sun's  true  semidiameter  is  to  be  found  in  pagell.  of  the  month  of  the  Nautical 
Almanac. 

To  the  observed  altitude  of  the  sun's  or  moon's  lower  limb  add  12' ;  but  if  the  up])er 
limbs  were  observed,  subtract  20',  ami  fi-om  the  observed  altitude  of  the  star  or  planet 
subtract  4',  and  you  will  have  nearly  the  apparent  altitudes  of  those  objects  respec- 
tively.ll 

*  If  3113'  °ne  wishes  to  proceed  witli  perfect  accurac}',  he  may  bring'  the  round  lim!)  of  liie  moon  to 
the  nearest  limb  of  the  planet,  and  l!ien  apply  the  planet's  semidiameter,  taken  from  the  Nautical  Alma- 
nac, in  ilie  same  manner  as  in  observations  of  the  sun. 

t  Or  by  multiplying  by  4  se.icageslmall}',  in  the  manner  directed  in  the  note  page  170. 

t  \\nien  the  sum  exceeds  24  hours,  you  must  subtract  24  hours,  and  add  one  to  the  day  of  the  month  ; 
and  when  the  time  to  be  subtracted  is  greater  than  tlie  mean  time,  the  latter  must  be  increased  by  24 
iiours,  and  one  day  taken  from  the  day  of  the  month,  conformably  to  the  usual  rules  of  addition  and 
subtraction.  If  the  chronometer  used  in  taking  the  observation  be  regulated  to  Greenwich  time,  this 
part  of  the  calculation  will  be  unnecessary,  because  the  reduced  time  at  Greenwich  will  be  given 
direcll3'  by  the  chronometer. 

§  These  corrections  may  be  found  easily  without  the  table,  by  saying,  As  12  hours  are  to  the  reduced 
time,  (rejecting  12  hours  when  it  exceeds  12.)  so  is  the  difference  of  semidiameter  or  parallax  for  12 
hours  to  the  corresponding  correction.  If  the  reduced  time  cannot  be  found  accurately  in  the  table, 
you  must  use  the  nearest  numbers,  which  will,  in  general,  be  sufficiently  accurate. 

II  These  altitudes  are  supposed  to  be  taken  at  sea  by  a  fore  observation  ;  and  the  application  cf  the 
above  numbers  will  give  the  apparent  altitudes  corresponding  to  observations  taken  on  the  deck  of  a 
common-sized  vessel  (where  the  dip  is  about  4'  or5')  to  a  sufficient  degree  of  accuracy  ;  if  the  observer 
was 'lO  or  50  feet  above  the  water,  1'  or  2'  might  be  taken  from  these  altitudes.  The  propriety  of 
using  these  numbers  will  appear  by  considering  that  every  wave,  by  raising  the  ship  above  the  level 
of  the  sea,  will  alter  the  dip,  and  that  an  error  of  1'  or  2'  in  the  altitudes  will  in  general  cause  but  a 


230  LUNAR   OBSERVATIONS 

To  the  observed  distance  of  the  moon  from  a  star  or  planet  add  the  moon's  true 
Bemidiameter,  if  her  nearest  limb  was  obsei-ved,  but  subtract  that  semidiameter  if 
her  farthest  limb  was  observed ;  the  sum  or  difterence  will  be  the  apparent  distance. 
But  to  the  observed  distance  of  the  sun  and  moon^s  nearest  limbs,  add  their  true  semidiame- 
ters  ;  (he  sum  will  be  the  apparent  distance. 

These  preparations  are  necessary  in  every  method  of  woi-king  a  lunar  observation 
The  most  noted  methods  are  those  of  Dunthorne,  Borda,  Maskelyne,  Rios,  Witchell, 
L}'ons,  &c.,  and  improvements  thereon  by  various  authors. 

Dunthorne's  and  similar  methods  have  one  great  advantage  in  not  being  liable  to 
a  variety  of  cases;  but  tliese  methods  are  tedious,  when  tables  of  logarithms  to  min- 
utes only  are  used,  by  reason  of  tlie  great  exactness  required  in  proportioning  the 
iogaj-jthms  to  seconds.  This  is  obviated  in  the  excellent  methods  published  by  Rios 
and  Stansbury ;  but  they  require  large  and  expensive  tables,  and  on  that  account  are 
not  in  very  general  use.  Witchell's  and  Lyons's  methods  do  not  labor  under  the; 
inconvenience  of  requiring  large  tables,  nor  do  they  require  any  particular  notice  of 
the  seconds  in  finding  the  log.  sines  and  log.  tangents ;  but  these  methods,  as  they 
were  originally  published,  are  embarrassed  with  a  variety  of  cases ;  sometimes  tlie 
corrections  are  additive,  sometimes  subtractive ;  and  learners  find  a  difficulty  in  rightly 
applying  tliem.  To  remedy  this,  a  method  was  published  in  the  first  edition  of  this 
work,  in  which  two  corrections  were  constantly  additive,  two  subtractive,  and  one 
small  correction  was  additive  when  the  distance  was  less  than  90°,  but  subtractive 
when  above  90°.  This  method  was  further  improved  in  the  Appendix  to  that  edition, 
liy  means  of  four  new  tables,  whicii  are  inserted  in  this  edition,  and  numbered  XVII. 
XV^III.  XIX.  and  XX.,  by  means  of  which  the  work  is  considerably  shortened,  and 
ail  tlie  corrections  rendered  additive.  This  method  will  now  be  given,  after  making 
a  few  lemarks  on  the  manner  of  taking  the  corrections  and  logarithms  from  these 
new  tables. 

Table  XVII.  contains  a  correction  and  logarithm  to  be  used  when  the  moon's  dis- 
tance from  a  star  or  planet  is  observed ;  and  Table  XVIII.  is  a  similar  one,  to  be 
used  when  the  moon's  distance  from  the  sun  is  observed.  Table  XVII.  contains  six 
pages,  corresponding  to  the  horizontal  parallax  of  the  planet,  supposing  it  to  be  either 
0'',  5",  10",  15",  20",  25",  or  30",  as  at  the  top  of  the  pages  respectively  ;  and  tha 
page  is  to  be  used  which  agrees  the  nearest  with  the  horizontal  parallax  of  the  |)lane 
at  tlie  time  of  observation.*  These  tables  are  so  extended,  that  no  proportional  parts 
are  necessary  in  taking  out  the  corrections  and  logarithms,  except  tiie  altitude  of  tlie 
sun  or  star  be  less  than  7°  30',  and  at  such  altitudes  an  observation  is  liable  to  error  on 
account  of  the  uncertainty  of  the  refraction  ;  so  that,  in  using  these  tables,  it  is  suffi- 
ciently accurate  to  find  the  number  nearest  to  the  given  altitude  of  the  sun  or  star, 
and  make  use  of  the  corresponding  correction  and  logarithm.  Thus,  if  the  star's 
altitude  be  12°  25',  the  nearest  number  in  Table  XVII.  is  12°  24',  corresponding  to 
which  are  the  correction  55'  45",  and  the  logarithm  1.31G1. 

Taljle  XIX.  contains  the  corrections  and  logarithms  corresponding  to  the  moon's 
horizontal  parallax  and  altitude,  both  being  found  at  the  same  opening  of  the  book. 
Tlie  corrections  for  seconds  of  parallax  and  minutes  of  altitude  are  easily  taken  out 
by  means  of  Tables  A,  B,  C,  placed  in  the  margin.  The  method  of  finding  these 
corrections  is  given  at  the  bottom  of  the  table  :  they  are  always  additive. 

Besides  the  two  logarithms  taken  from  Table  XVII.  (or  XVIII.)  and  XIX.,  this 
new  rule  requires  only  four  logarithms  to  be  taken  from  Table  XXVII.  to  four  jilaccs 
of  figures,  and  to  the  nearest  minute,  it  being  in  general  unnecessary  to  proportion 
fiir  the  seconds. 

We  shall  now  give  the  rule  for  correcting  the  distance,  and  shall,  for  brevity,  use 
the  words  sine,  secant,  and  cosecant,  instead  of  Zog-.^'ne,  log.  secant,  and  log.  cosecant, 
respectively,  and  the  same  ]iractice  will  be  observed  in  the  second,  third,  and  Iburtii 
iiiLthods  of"  correcting  the  distance. 

small  error  in  the  result  of  the  calculation  of  a  lunar  observation,  so  that  for  all  practical  purposes  the 
above  numbers  may  be  esteemed  as  sufficiently  exact.  It  may  also  be  observed,  that  the  error  arising 
from  this  source  will  not  generally  be  greater  than  that  arising  from  neglecting  the  equations  depending 
on  the  spheroidal  form  of  tlie  earth,  and  on  the  density  and  temperature  of  the  air}  equations  which  are 
almost  alwa^'s  neglected. 

If  any  one  wishes  to  olitain  the  apparent  altitudes  strictly,  he  must,  from  the  observed  altitudes, 
subtract  the  dip  of  the  horizon  taken  from  Table  XIII.,  anu  add  or  subtract  the  semidiameter  of  the 
object,  according  as  the  lower  or  upper  limb  is  observed. 

*  In  strictness,  when  the  horizontal  parallax  diirors  from  those  in  the  table,  we  ouHit  to  take  the 
numbers  for  the  next  greater  and  the  next  less  number,  and  take  a  proportional  part  of  llie  dillcrences 
but  tills  degree  of  accuracy  is  wholly  unnecessary  In  nautical  observations. 


LUNAR  OBSERVATIONS.  231 


FIRST   METHOD 

Of  correcting  the  apparent  distance  of  the  moon  from  the  sun*  in  which  there 
is  no  variety  of  cases,  all  the  corrections  being  additive. 

Add  the  apparent  distance  of  tlie  moon  from  the  sun  to  their  a])parent  altitudes, 
and  note  tlie  half-sum.  Tlie  difference  between  tJie  half-sum  and  the  a])parent  dis- 
tance call  the  first  remainder;  and  the  difference  between  the  half-sum  and  the  sun's 
apparent  altitude  call  the  second  remainder. 

Take  from  Table  XXVII.  the  following  logarithms,  which  mark  beneath  each  other 
in  two  columns,  viz.  the  sine  of  the  apparent  distance,  to  be  marked  in  both  columns, 
the  cosecant  of  the  second  remainder,  to  be  marked  also  in  both  cohnnns,  the  secant 
of  the  first  remainder  to  be  placed  in  tlie  first  column,  and  the  secant  of  the  half-sum 
in  the  second  column.f 

Enter  Table  XVIII.  (or  Table  XVII.  if  a  star  or  planet  be  used),  and  take  out  the 
correction  corresj)onding  to  the  sun's  altitude  (or  star  or  planet's);  take  also  from  the 
same  table  the  corresponding  logarithm,  which  place  in  column  1st. 

Enter  Table  XIX.  with  the  moon's  ai)parent  altitude  and  horizontal  jiarallax ;  find 
the  corresjionding  correction,  which  j)lace  under  the  former  correction,  and  the 
logarithm,  which  place  in  column  2d. 

The  sum  of  the  four  logarithms  f  of  column  first  will  be  the  proportional  logarithm 
of  the  first  correction,  and  the  sum  of  the  logarithms  of  column  second  f  will  be  the 
proportional  logarithm  of  the  second  correction;  these  corrections  being  found  in 
Table  XXII.  are  to  be  ])laced  under  the  former  corrections. 

Enter  Table  XX.,  and  find  tlie  numbers  which  most  nearly  agree  with  the  observed 
distance  and  the  observed  altitudes  of  the  objects,  and  take  out  the  corresponding 
correction  in  seconds,  which  is  to  be  placed  under  those  already  found.  Then,  by 
adding  all  these  corrections  to  the  apparent  distance,  decreased  by  2^,  we  shall  get 
the  true  distance  nearly .| 

To  determine  the  longitude  from  the  true  distance. 

if  the  true  distance  of  the  objects  can  be  found  in  the  Nautical  Almanac,  in  either 
of  the  j)ages  where  the  distances  are  marked,  on  the  day  of  the  observation,  the  time 
Vv'ill  !)e  found  at  the  top  of  the  page.  If  the  tiaie  distance  cannot  be  found  exactly,  in 
the  Nautical  xA.lmanac,  you  must  find  the  two  which  are  nearest  to  it,  the  one  greater 
and  the  other  less  than  the  true  distance ;  and  take  out  that  one  which-  corresponds 
with  tlie  earliest  or  first  of  these  times,  with  the  corresponding  proportional  logarithm. 
Find  the  difference  between  this  first  distance  and  the  true  distance,  and  take  out  its 
proportional  logarithm  from  Table  XXII.  The  difference  between  these  two  pro- 
portional logarithms  will  be  the  proportional  logarithm  of  a  jiortion  of  time,  to  be 
added  to  the  time  standing  over  the  first  distance  in  the  Nautical  Almanac,  and  the 
sum  will  be  the  mean  time  of  the  observation  at  Greenwich.  The  difference  between 
this  time  and  the  mean  time  at  the  ship,  being  turned  into  degrees  and  minutes  by 
Talile  XXI.,  will  be  the  true  longitude  of  the  ship  from  Greenwich,  at  the  time  of 
observation.  This  longitude  will  be  east  if  the  time  at  the  ship  be  greater  than  that 
at  Greenwich,  otherwise  west.§ 

To  exemplify  the  preceding  rules,  we  shall  now  give  several  examples  of  correcting 
the  apjtarent  distance,  including  also  the  preparation  and  the  determination  of  tlie 
longitude  from  the  true  distance. 

*  'I'lils  rule  is  the  same  as  tlial  for  corrcctiiin;  the  distance  of  the  moon  from  a  star  or  a  planet,  except 
in  reading  star  or  planet  for  sun,  and  usin^  Table  XVII.  instead  of  Table  XVIII. 

t  Rejecting'  always  the  tens  in  the  indices. 

i  The  distance  obtained  by  this  rule  is  not  perfectly  correct,  since  several  small  corrections  must  be 
applied  to  obtain  the  true  distance  to  the  nearest  second,  viz.  (1)  The  refraction  taken  from  Talile  XII. 
which  is  made  use  of  in  constructing  Tables  XVII.  XVIII.  and  XIX.,  ought  to  be  corrected  for  the 
dilTercnt  heights  of  the  barometer  and  thermometer,  as  directed  in  page  154.  (2)  A  correction  must  be 
applied  for  the  spheroidal  figure  of  the  earth.  And  (3)  a  very  small  correction  ought  to  be  made  in 
the  numbers  of  Table  XX.  when  the  D's  horizontal  paralla,\  varies  from  57' 30".  But  to  notice  all 
these  corrections  would  increase  the  calculation  very  much,  and  the  result  of  a  single  observation,  in 
which  all  these  things  were  noticed,  would  probably  not  be  so  accurate  as  the  mean  of  two  or  three 
observations,  taken  at  different  times  of  the  day,  in  which  these  corrections  were  neglected  ;  and  the 
time  necessary  to  take  and  work  the  latter  observations  would  not  be  much  greater  than  to  work  a 
single  observation,  in  which  all  the  corrections  were  noticed. 

§  It  may  be  necessary  to  observe  that,  if  the  times  at  the  ship  and  Greenwich  fall  on  different  days, 
the  latest  day  is  to  be  reckoned  the  greatest,  though  the  hour  of  the  day  may  be  the  least ;  thus,  ITiii 
lav  1  hour  is  to  be  esteemed  greater  than  IGth  day  "22  hours. 


232 


LUNAR  OBSERVATIONS. 


EXAMPLE    I. 

Suppose  that,  on  tlie  7th  of  January,  183G,  sea  account,  at  11"  57'  past  midnight, 
mean  time,  in  the  longitude  of  127°  30'  E.,  by  account,  the  observed  distance  of  the 
farthest  iinil)  of  the  moon  from  the  star  Aidebaran,  was  G8°  36'  00",  the  observed 
altitude  of  tlie  8tar  32°  14',  and  tlie  observed  altitude  of  the  moon's  lower  limb  3-1°  43' 
Required  the  true  longitude. 


Preparation. 

Sea  account,  Jan.  7,  is  by  N.  A.  Jan.  6^  ISh  li  >»  67» 

Longitude  127°  30' E 8    30     00 

Reduced  time Jan.  G'i    3^  ilm  67» 


3  scmidiam.  Jan.  G,  noon  15' 05''          Jhor.  par.  Jan.  G,  noon. .  55' £0"        tH^  observed  alt.. . .  32°  14 
midni<rlit  15  09  midnlglit  55  3-1  Subtract 4 


D.flerence 
Table  XL 


15  OG 
9 


Aug.  Table  XV 

J)  semidiameler 15' 15" 


Difference. 
Table  XL. 


14  *  apparent  alt....  32   10 

4 


1)  hor.  par 55' 24"        D  obs.  alt.  L.  L.  .  3^1°  43 

Add 12 

D  apparent  alt.  . .  34°  65' 


App.  dist.  08' 21' 
^app.alt.  32  10 
])  app.  alt.  31  55 


Sum.... 

135  21) 

Half-sum 

C7  43 

App.  dist. 

C8  21 

1  Rem.  . . 

0  38 

Half-sum 

()7  43 

*  app.  alt 

.32  10 

2  Rem.... 

35  33 

Observed  distance  *  D   F.  L 68°  3G' 00" 

])  semidiameler subtract  15   15 

Apparent  distance*  D 68°  20'  45" 


To  find  the  true  distance. 


Col.  1. 

Pine 9.9C82 

2  Hem.  35°  3.T.  Cosec.  0.23.55 
1  Item.    0  38.  Sec. 
TaUeXVir.  ..Log 

1  Corr.  2'  14"  P.  L.. 


0.0000 
1.7018 

1.9055 


Col.  2. 

Same 9.9G82 

Same 0.2355 

Half-sum  07°  43'.  Sec.  0.4212 

Table  XlX.f Leg.  0.2238 

2  Corr.  25'  30"  P.  L.  .  0.8487 


App.  dist.  less  2'  =  66'  20*  45" 


Table  XVII 

Table  XIX.*.... 
1  Corr 

58  29 
15  37 
2  14 

2  Corr 

25  30 

Table  XX 

True  distance... 

25 

..  68°  03'  OC 

To  find  the  longitude. 


True  distance 68"  03'  00" 

Distance  l>v  N.  A.  at  3i> 67    41    43 


DiiTercnco 0    21    17 

Oh  4lr 
Add    3 


Prop,  log 2872 

Prop,  log 9272 

14' Prop.  log.  diff...  6400 


Mean  time  at  Greenwich 3    41    14 

Mean  time  at  the  sliip 12    11    57 


Difference  is  longitude  in  time 8    30   43  ==  127°  40' 45"  E.  from  Greenwich. 


*  This  corr.  =  Corr.  Tab.  XIX.  15'  05"  +  Corr.  Tab.  A.  29"+  Corr.  Tab.  C.  3"  : 
t  Tliis  log.  =  Log.  Tal).  XIX.  2231  +  Log.  Tab.  C.  7  =2238. 


15' 37 


LUNAR  OBSERVATlOlfS. 


233 


EXAMPLE  n. 

Suppose,  1836,  April  2^  2^  03™  50'  A.  M.,  mean  time,  sea  accoimt,  in  the  longitude 
of  172^  E.,  by  account,  the  observed  distance  of  the  moon's  farthest  limb  irorn 
Antares,  Avas  01°  04'  00",  the  observed  altitude  of  the  star  G8°  29',  the  observed  alti- 
tude of  tlie  moon's  lower  limb  45°  23'.     Re(|uired  the  true  longitude. 


Preparation. 

Sea  account,  April  2,  or  by  N.  A.,  April  I'l  l-J-i'  OSmSO" 

Longitude  172°  E 11   28    00 

Reduced  time April  1 J  021'  35m  50^ 


])  semidiam.  April  1,  noon  13' 59" 
midnight  16     4 

DitTercncc 5 

TableXI 1^ 

Sum IG  00 

Aug.  Table  XV H 

D  semidiameter IG' 11'' 


D  horizontal  par.  noon  58'  38" 
niidiilHit  58 -SG 


DlfTcrcnce 

TableXI 

D  horizontal  parallax    58'  42" 


18 
4 


*  observed  a\\ G8°  29' 

Subtract 4 

^  apparent  alt G8°  23' 

D  obs.  alt.  L.  L.  ..  43°  23' 
Add 12 

y>  apparent  alt 45°  35 


Observed  distance  *  K  F.  L 61°  04' 00" 

Subtract  ]>  semidiameter IG   11 

.  Apparent  distance  *  D C0°  47'  49" 


App.  dist.  60°  4B' 
J(f  app.alt.  C8  25 
])  app.  alt.  45  35 
Sum....  174  48 


Half-sum  87  24 
1st  Rem..  26  36 
Sd  Rem. .  18  59 


To  Jind  the  true  distance. 


Col.  1. 

Sine 9.9410 

2dRein.  lS°59'.Cosec.  0.4877 
1st  Rem  .26  36  ...?ec.  0.0186 
Table  XVII Loe.  1.9438 


1st  Corr  0'  41".  P.  L.  2.4-211 


Col.  2. 

Same 9.9410 

Same 

Half-Sinn  87"  24'.  Sec. 

Talile  XIX. t Log. 

2d  Corr.  1'56"..P.L. 


App.  dist.  less  2"  =  58°  47<  49' 

Talile  XVII 59  37 

Table  XIX.* 19  32 

1st  Corr 0  41 

2d  Corr 1  56 

Table  XX 19 

True  distance 60°  09'  6A! 


To  Jind  the  true  longitude. 


True  distance G0°  09'  54" 


Distance  by  N.  A.  at  Oh. 
DifTerence 


Gl   40   13  Prop.Iog. 


1    30   19   Prop,  log 


23^18 

2995 


2h  35m  0G» Prop.  log.  diff.  0G47 

Add    0  no  00 

Mean  time  at  Greenwich 2  35    06 

Mean  time  at  the  ship 14  03    50 

Difference  is  longitude  in  time 11    23    41-=172°  11'  E.  from  Greenwich. 


*  This  corr.  =  Corr.  Tab.  XIX.  19'  17"  +  Corr.  Tab.  A.  12"  + Corr.  Tab.  B.  3"=  19' b'2". 
t  This  log.=  Log.  Tab.  XIX.  1915+  Log.  Tab.  C.  9=  1954 

30 


2.34 


LUNAR  OBSERVATIONS. 


EXAMPLE   III. 

SupjKjse  that,  on  the  30th  of  Oct.  1836,  sea  account,  in  the  forenoon,  in  the  longitude 
of  80°  W.,  by  account,  the  following  observations  of  the  sun  and  moon  were  taken; 
the  times  being  noted  by  a  chronometer  which  was  3™  47'  too  slow  for  mean  time 
at  the  place  of  observation.    Required  the  true  longitude. 

Preparation. 


Time  per  IVatch. 

Observed  Distance 
©  d    N.  L. 

Observed  Altitude 
QL.L. 

Observed  Altitude 
5  L.  L. 

H.  M.    S. 

9  38  01 
9  39  04 
9  40  06 
9  41  00 
9  41  49 

5 ) 200  00 

9  40  00 
Error     +  3  47 
Mean  ^,,3,, 

O          1          II 

111  35  49 
35   13 
34  47 
34  20 
33  66 

174   10 

111  34  50 
D  S.  D.           14  63 
©  S.  D.           16  09 

App.dist.  112  05  52 

O         1 

24  47 
24  51 
24  55 

24  59 

25  03 

124  35 

24  55 
Add 12 

0         1 

26   17 
21 
25 
29 
33 

125 

26  25 
Add 12 

©App.  alt.25  07 

D  App.  alt.  26  37 

Sea   account,  30lh  October,  or  N.  A.,  October  29'!  21i>  43n'47»  or  9>'43m47'  A.M. 
Longitude  80°  \V 5   20   00 

Reduced  time Ootober  30^  3''  03"'47» 


D  semidiameter,  Oct.  30,  noon....  14' 46" 
midnight    14   46 

Difference 0 

TableXI 0 

Sum I'l  46 

Aug.TableXV 7 

J)  semidiameter 14' 53" 


])  horizon,  par.  Oct.  30,  noon. ...  54'  10" 
midnignt    54    10 

0 

0 


Difference 

TableXI 

])  horizontal  parallajc 54'  10 


App.dist.  112°  06' 
©app. alt.  25  07 
D  app.  alt.  26  37 

Sum 1G.3  50 

Half-sum.  81  55 
1st  Rem..  30  11 
SiIRem...    .'J6  48 


To  fond  the  true  distance. 

Col.  1. 

Sine 9.9G69 

2dRem.56°48'.Co3ec.  0.0774 
lstRem.30  11... Sec.  0.0633 
Table  XVllI....Log.  1.6336 
lstCorr.3'16"..P.L 


Col.  2. 

Same 9.9669 

Same 0.0774 

Half-sum  81°  55'.  Sec.  0.8520 
Table  XIX. t Log.  0.2376 


1.7412  2d  Corr.  13'  14".  P.L.  1.1339 


To  find  the  true  longitude 

True  distance 111°33'  53" 

Distance  by  N.  A.  at  Oh 112  54  10 

Difference 


App.  dist.  less  2°  =  110° 05'  59" 


Table  XVlll.... 

Table  XIX.* 

1st  Corr 

2d  Corr 

Table  XX 

True  distance  ... 


58  06 

13  10 

3  16 

13  14 

13 


Prop,  iog 3458 

1    20   17  Prop,  log 3506 


Add 


2h  58m  01» Prop.  log.  diff.  004-8 

0 


Mean  time  at  Greenwich.... Oct.  30''  02h  58'T'01» 
Mean  time  at  the  ship Oct.  29   21   43  47 

Difference  is  longitude  in  time 5   14   14  =  78°  33'  30''  W.  from  Greenwich. 


*  This  corr.  =  Corr.  Tab.  XIX.  12' 23"  + Corr.  Tab.  A.  44"+ Corr.  Tab.  B.  8 ''=13' 10". 
t  This  log.  =  Log.  Tab.  XIX.  2364  + Log.  Tab.  C.  12  =  2376. 


LUNAR  OBSERVATIONS. 


•23,- 


EXAMPLE   IV. 


Suppose  that,  on  the  12th  of  May,  183G,  sea  account,  at  about  l**  P.  M.,  in  the 
latitude  of  30°  S.,  and  in  tlie  longitude  of  4°  00'  E.,  by  account,  the  following  obser- 
vations of  the  sun  and  moon  were  taken  ;  the  sun  being  so  situated  that  the  apparent 
time  could  be  observed  by  her  altitude.     Reciuired  the  true  longitude. 

Preparation. 


Observed  Distance 
m  C  N.L. 

Obsen-ed  Altitude 

m  L.L. 

Observed  Altitude 
D   U.L. 

o         /         // 

46  07  09 
06  01 
04  56 

3 )  18  06 

Mean 46  06  02 

o      / 

39  59 
45 
31 

135 

39  45 

—  2 

39  43 
Add 12 

©  app.  alt.  39  55 

O      1 

28  32 
28  07 

27  42 

84  21 

28  07 
+  01 

28  08 
Subtract. .         20 

D  app.  alt.   27  48 

Index  errors ....         —  03 

Corrected  dist...  46  05  59 
©  semidiameter.         15  51 
D  semidiameter.         15  25 

Apparent  dist...  46  37  15 

Sea  account,  May  12,  or  N.  A.,  May  lid  ih    Qm  00» 

Longitude  4°  00' E 16    00 

Reduced  time May  ll^  Oi>  44m  OO' 


D  semidiameter.  May  11,  noon  ...  15'  17' 
midnight  15  13 


Difference 
Table  XI. 


15  17 


Aug.  Table  XV 8_ 

D  semidiameter 15'  25" 


D  horizontal  p-irallax,  noon 56' 04" 

midniirht ...  55  49 


Difference. 
Table  XI.. 


I)  horizontal  parallax 56' 03" 


App.  dist.  46°  37' 
©app. alt.  39  .55 
Dapp.  alt.    27  48 


Sum 114  20 

Half  sum  57  10 
1st  Rem.  .  10  33 
2(1  Rem...     17  15 


Tojlnd  the  true  distance. 

Col.  1. 

Sine 9.8G14 

2cl  Rem.  17°  lo'.Cosec.  0.5279 
1st  Rem. 10  33... Sec.  0.0074 
Table  XVni....Log.  1.8307 
1st  Corr.  1'  4 


P.  L.  2.2274 


Col.  2. 

Same 9.&314 

Same 0..5279 

Half-sum  57°  10'.  Sec.  0.2C58 
Talile  XIX.t....Log.  0.2215 
2(1  Corr.  23'  5,= 


P.  L.  0.87n(i 


App.  dist.  less  2°  = 

=  44 

'37' 15' 

TaMe  XVIII.... 

58  59 

Table  XIX.* 

U  51 

1  04 

2(1  Corr 

23  55 

Table  XX 

34 

True  distance 4G°  13' 41" 


0  correct  altitude  39°  54' 

Latitude  of  ship..  30  00 

Polar  distance...  107   58 

Sum 177   52 


Tojind  the  mean  time  and  the  true  longitude. 

True  distance 46°  13' 41" 

By  N.  A.  at  Oh  ....  46  34  02  . 

Difference 


Secant..  10.06247 
Cosecant  10.02171 


Half-sum 88  56  . 

.  Cosine 

Half-sum alt.     49   02  . 

.  Sine.. 

Sum.. 

Apparent  time. .  l^  0">  3» . 

..  Sine.. 

Eq.  of  time  .  sub.        3  54 

Mean  time  ....  0    56  09 

8.2G9S8 
9.87800 

18.23206 

9.11603 


20  21   . 

Difference 0°41'32". 

Add 0 


Prop.  log.  3097 
Prop.  log.  9467 
Prop.  log.  6,370 


Mean  time  at  Green.    0  41  32 
Mean  time  at  ship. .    0  56  09 

Longitude  in  time..         14  37= 


:3°39'15"E.froro 
Greenwich. 


*  This  corr.  =  Corr.  Tab.  XIX.  ll'02"  +  Corr.  Tab.  A.49"  +  rorr.  Tab.  B.  3"=n'.'>4". 
'  This  log  =  Log.  Tab.  XIX.  2207  +  Log.  Tab.  C.  8  ==  221 5. 


236 


LUNAR   OBSERVATIONS. 


EXAMPLE  V. 

Supj)Ose  that,  on  the  13th  of  Februaiy,  1830,  sea  account,  at  8''  36™  00',  mean 
time,  A.  M.,  in  the  longitude  of  1G°  W.  from  Greenwich,  by  account,  six  distances  of 
tlie  Sim  and  moon's  nearest  Ihnbs  were  observed,  by  a  circle  of  reflection,  to  bo 
273"  09'  06",  the  corresponding  times  and  altitudes  being  as  in  the  following  table. 
Kenuired  the  true  longitude. 

Preparation. 


Mean    Time  per 
Watch,  A.  M. 


H.  M.  s. 

8  33  24 

W3f. 

35  18 

36  36 

37  04 
39  02 

Sums    6)36  00 


Mean 
time 


8  36  00 


Observed  Distance 
Q  d  N.L. 


Sum  of  the  dis- 
tances taken  from 
the  circle  at  the 
end  of  the  obser- 
vations. 


273°  9'  06' 

45  31  31 
©S.D.  16  13 
5  S.  D.  16  29 

App.dist.  46  ai  13 


Observed  Altitude 
®L.L. 


Add. 


27  42 

27  M 
is  02 

28  12 
28  21 
28  44 

55 

28  09 
12 


0  app.  alt.  28  21 


ObseT^-ed  Altitude 
D  U.  L. 


42  24 
42  42 

42  51 

43  01 
43  11 
43  21 

17  30 

42  55 
Subtract. .       20 


])  app.  alt.  42  35 


February  13,  sea  account,  or  by  N.  A.,  February  12<l  20ti  36™  00» 
Longitude  16°  W 1   04    00 

Reduced  time February  \'2A  21>'  40™  00> 


])  semidiamcter,  Feb.  12,  midnight  16'  17' 

Feb.  13,  noon...  16  18 

Diflerence 1 

TableXl 1 


16  13 
II 


Aug.  Table  XV 

>  semidiametcr 16^29" 


J)  luir.  parallax,  Feb.  12^  midnight  59' 46" 
Feb.l3,nooc...  59  47 

Difference 1 

TableXl 1^ 

D  horizontal  parallax 59' 47" 


To  f.nd  the  true  distance. 


\pp.  dist.  46° 04' 
©app.  alt.  28  21 
D  app.  alt.  42  V.5 
Sum 117  00 


Half-sum  58  30 
1st  Rem.  12  26 
2d  Kern.    30  09 


Col.  1. 

Sine 9.8574 

2d  Kein.  30°  09'  Cosec.  0.2991 
1st  Rem. 12  23  ..Sec.  0.0103 
Table  XVIII.  ..Log.  1.G874 


1st  Corr.  2*  31".  P.  L.  1.8542 


Col.  2. 

9.8574 
0.2991 
0.2819 
0.1878 
0.G2G2 

App.  dist.  less  2° 
Table  XVIII.... 

Table  XIX.* 

li;t  Corr 

=  44 

04' ly 

Same 

58  22 

Half-sum  58°  30'  Sec 
Table  XIX. t  ...Log 
2d  Corr.  42'  34".  P.  L. 

16  41 
2  31 

0(1  Corr 

42  34 

Table  XX 

34 

True  distance... 

..    4(3 

04-65' 

To  find  the  true  longitude. 

True  distance 46°04'  55" 

Distance  by  N.  A.,  Feb.  12^  21^ . . . .  4^)  28  04 Prop,  log 2551 

Difference 23  09  Prop.log 8907 

0h41'n39' Prop.  log,  diff.  6356 

Add  21 

Mean  time  at  Greenwich 21   41    39 

Mean  time  at  the  ship 20  36    00 

Difference  is  lon-itude  in  time 1  05    39  =  16°  24'  45"  W.  from  Greenwich. 


This  corr.  :=  Corr.  Tab.  XIX.  16'  29" -f  Corr.  Tab.  A.  9"-t-Corr.  Tab.  B.  3"=  16'  41". 
This  log.  =  Log.  Tab.  XIX.  1875+  Log.  Tab.  C.  3=  1878. 


LUNAR  OBSERVATIONS. 


2:17 


EXAMPLE   VL 

Suppose  that,  on  the  21st  of  June,  183G,  sea  account,  at  C'  50™  40'  P.  J>L,  mean  time, 
in  the  longitude  of  Gl°  W.,  by  account,  tlie  observed  distance  of  the  nearest  limb  of 
the  moon  from  the  centre  of  tlie  planet  Venus,  was  35°  59'  57",  the  observed  altitude 
of  the  planet  23°  00',  and  the  observed  altitude  of  the  moon's  lower  limb  37°  31' 
Required  the  true  longitude. 


Preparation. 

Sea  account,  June  21st,  is  by  N.  A.  June  SOJ    Gi"  50™  40' 
Long-itude  61°  W.  in  lime 4    04    00 


Reduced  time June  20J  lOi"  51"i  AQ^ 


J>  semidiam.  June  20,  noon  15'  10" 
midniffht  13  15 


Difference 5 

TaJbleXI 5 

Sum 15   15 

Au?.  TableXV 10 


D  semidiaineter 13' 25" 


D  hor.par.  June  20,  noon  55'  38" 
midniglit  55  59 

Difference 21 

Table  XI 19 


D  horizontal  parallax      55'  57" 


5  observed  alt. 
Subtract 


23°  00' 
4 


$  apparent  alt 22°  56' 

D  obs.  a!t.  L.  L...  37°  31 
Add 12 


»  apparent  alt 37°  43' 


Observed  distance  5  ?  N.  L 35°  59' 57' 

I)  scmidiameter add  15  25 

Apparent  distance  5  9 36°  15' 22" 


To  find  the  true  distance. 


App.  tli?t. 

36° 

to' 

?  app. alt 

22 

56 

5  app.  alt 

37 

43 

Sura 

9G 

54 

Half-sum 

48 

27 

1st  Rem.. 

12 

12 

od  Rem.  . 

25 

31 

Col.  1. 

Sine 9.7718 

2dRem.25''31'.Cosec.  0.3658 
1st  Rem. 12  12... Sec.  0.0099 
Table  XVII.  ..  Log.*  1.6348 
lstCorr.2'53"..r.  L.  1.78^3 


Col.  2. 

Same 9.7718 

Same 0.3658 

Ilalf-sum  48°  27'.  Sec.  0.178:3 

Table  XIX Log.  0.2185 

2d  Corr.  52'  35"  .  P  L.  0..53 14 


App.  dist.  less  2°  =  34°  W  22* 

Table  XVII.* 58  05 

Table  XIX 16  40 

1st  Corr 2  58 

2d  Corr 52  35 

Table  XX 41 

True  distance 36°  26'  2J" 


To  find  the  true  longitude. 


True  distance 36°26'21" 

Distance  by  N.  A.  at  911 35  28   17  Prop.  log 2985 

Prop,  log 4913 

Prop.  log.  diff.  1928 


Difference 0  58  04 


Ih  53™  28s 

Add    9 


Mean  time  at  Greenwich 10  55    28 

Mean  time  at  the  ship 6   50    40 

Difference  is  longitude  in  time 4  04    48  =  61°  12'  W.  from  Greenwich. 


*  The  horizontal  parallax  of  Venus  being  20"  by  the  Nautical  Almanac,  we  must,  in  finding  from 
Table  XVIl.  the  correction  and  logarithm,  use  that  table  which  is  marked  at  the  top,  "  Parallax  20," 


being  the  93d  page. 


238 


LUNAR  OBSERVATIONS. 


EXAMPLE   Vn. 

Suppose  that,  on  the  27th  of  August,  1836,  sea  account,  at  0''  50""  08'  A.  M.,  mean 
time,  in  the  longitude  25°  W.,  by  account,  tlie  observed  distance  of  the  farthest  limb 
of  the  moon  from  the  centre  of  the  planet  Mars,  was  114°  05'  17",  tlie  observed  altitude 
of  the  planet  10°  30',  and  the  obsei-ved  altitude  of  the  moon's  upper  limb  22°  51' 
Requii'ed  the  true  longitude 


Preparation. 

Sea  account,  August  27.  is  by  N.  A.  August  20^  12h  SOLOS' 

Longitude  25°  W.  in  time 1    40    00 

Reduced  time August  26(1  14*>  30"  08» 


D  semidiam.  Aug.  26,mid.  16' 10" 
Aug.  27,  noon  16    5 

Difference 5 

TableXI 1 


IG 


Aug.  Table  XV. 


D  semidiamcler IG' 15" 


])  her.  par.  Aug.  26,  mid.  59'  20" 
Aug.  27,  noon  59  00 

Difference 20 

TableXI 4^ 

D  horizontal  paral!a,x. .  59' 16" 


(f  observed  alt 10°  30- 

Subtract. 4 

cf  apparent  alt.  ...   10°  26' 


D  observed  alt.  U.L.  22°  51' 

Subtract 20 

D  apparent  ah 22°  31 


App.  dist.  113M9I 
(f  app.  alt.  10  2G 
D  app.  alt.  22  31 


Sum 14G  46 

Half-sum.  73  23 
1st  Rem. .  40  2fi 
SdRem...    62  57 


Observed  distance  J  5  F.  L 114°  05' 17" 

5  scmidiameler subtract  16    15 

Apparent  distance  d"  ])  113°  49' 02" 


To  find  the  true  distance. 


Col.  1, 

Sine 9.9614 

2aRem.62°57'.Cosec.  0.0503 
lstReni.40  26... Sec.  0.1185 
Table  XVII.... Log.*  1.2525 
lstCorr.7'27"..P.  L.    1.3827 


Col.  2, 

Same 9.9614 

Same 0.0503 

Half-sum  73°  23'.  Sec.  0.5437 
Table  XIX Log.  0.1998 


2d  Corr.  31'  38".  P.L.  0.7552 


To  find  the  true  longitude. 


App.  dist.  less  2'  =  111" 49'  02" 
Table  XVII.*....  55  03 

Table  XIX 7  12 

1st  Corr 7  27 

2dCnrr 31  38 

Table  XX 14 

True  distance 113''30'36' 


True  distance 113°30'  3G" 

Distance  by  N.  A.  at  I2h 1'.4  55   06 Prop,  log 2455 

Difference 1    24  30  Prop,  log 3284 

2i>  28ni  44" Prop.  log.  diff.  08^9 

Add     12 


Jlean  time  at  Greenwich 14  28    44 

Mean  time  at  the  ship 12  50  08 

Difference  is  lona-itude  in  time  . . . 


1   38  36  =  24°  39'  W.  from  Greenwich. 


•  Tiie  horizontal  parallax  of  Mars  being  4".93,  by  the  Nautical  Almanac,  we  ma^'find  the  correction 
and  logarithm  in  Table  XVII.,  page  90,  rorrpsponding  to  the  nearest  parallax  5'' 


LUNAR  OBSERVATIONS.  239 

SECOND   METHOD 

Of  Jinding  the  true  distance  of  the  moon  from  a  ,tar* 

This  method  is  giounded  on  that  which  was  first  published  by  Mr.  Lyons,  and 
afterwards  improved  by  various  persons  by  the  introduction  of  tables  similar  to 
Tables  XLVIL,  XLVIIL,  of  the  present  collection.  In  Lyons's  method  there  are 
four  principal  corrections,  and  several  small  ones,  like  those  which  are  included  in 
Table  XX.;  the  first  and  second  of  these  corrections  depend  on  the  refraction;  the 
third  and  fourth,  on  the  moon's  parallax.  These  two  last  corrections  correspond  very 
nearly  to  the  first  and  second  of  the  present  improved  method.  The  first  and  seconti 
corrections  of  Lyons's  method,  with  all  the  smaller  corrections,  are  given  very  nearly 
by  means  of  Table  XLVIIL,  under  the  name  of  the  tlurd  correction  of  the  present 
method ;  the  numbers  in  this  table  are  liable  to  an  error  of  a  few  seconds  in  conse- 
quence of  using  the  moon's  mean  horizontal  parallax  in  computing  the  numbers. 
Several  of  the  quantities  in  each  page  of  die  table  have  been  compared  by  means  of 
Shcpard's  tables  with  the  correct  results,  for  the  extreme  values  of  the  moon's 
horizontal  parallax  ;  and  it  has  been  found  that  an  error  exceeding  5"  will  rarely  occur 
in  computing  the  distance  from  the  numbers  in  the  table,  if  the  process  of  interpola- 
tion be  carefully  attended  to,  when  the  proposed  distance  and  the  altitudes  are  not 
expressly  given  in  the  table,  as  most  commonly  happens. 

Wheii  this  tabular  form  was  first  adopted  in  finding  this  third  correction,  the  inter- 
vals were  much  longer  than  they  now  are,  and  the  table  contained  only  one  page  ;  the 
process  of  interpolation  was  then  difiicult,  and  liable  to  a  considerable  degree  of 
inaccuracy,  sometimes  amounting  to  more  tlian  half  a  minute.  This  som-ce  of  error 
has  been  successively  diminished  by  increasing  the  number  of  pages  in  the  table  ; 
and  it  was  finally  published  by  ]Mr.  Thompson,  in  nearly  the  same  form  as  in 
Table  XLVIIL  of  the  present  collection,  which  is  so  extended  that  we  can,  without 
much  error,  neglect  wholly  the  process  of  interpolation,  and  take  out,  by  mere 
insi>ection,  the  tabular  correction  for  the  nearest  degrees  in  the  tal)le  corresponding 
to  the  distance  and  altitudes.  Thus,  if  the  a})parent  distance  be  29°  10',  the  moon's 
apparent  altitude  21°  15',  and  the  star's  apparent  altitude  18°  25',  we  must  enter  the 
table  in  jiage  278,  corresponding  to  the  a|)parent  distance  28°,  moon's  altitude  21°, 
star's  altitude  18°,  and  take  out  the  corresponding  correction  1'  19" ;  which  differs 
but  very  little  from  tlie  true  value,  found  by  interpolation. 

This  second  method  has  not  the  same  advantage  as  the  first  method,  of  being 
wholly  free  from  cases,  for  the  second  correction  is  found  at  the  top  of  Table  XLVIL 
when  the  distance  is  greater  than  90°,  and  at  the  hoitom  when  less  tlian  90°  ;  moreover 
the  effect  of  the  parallax  of  the  sun,  or  that  of  a  planet,  is  sometimes  additive,  and  at 
other  times  subtraclive.  In  this,  as  well  as  in  the  third  and  fourth  methods,  the 
preparation  is  the  same  as  in  the  first  method  ;  and  the  process  of  finding  the  longi- 
tude from  the  true  distance  is  also  the  same  :  it  will  therefore  be  unnecessary  to 
repeat  the  rules  for  these  calculations,  which  we  have  given  in  pages  229,  231, 
and  we  shall  restrict  ourselves  to  the  explanation  of  the  process  for  computing  the 
true  distance,  which  is  done  in  the  following  manner : — 

RULE.  ' 

To  the  proportional  logarithm  of  the  moon's  horizontal  parallax,  (Table  XXII.)  arid 
the  log.  cosecant  of  the  star's  apparent  altitude,  (Table  XXVII.)  the  log.  sine  of  the 
star's  apparent  distance,  (Table  XXVII. ;)  the  sum  (rejecting  the  tens  in  the  indices) 
will  be  a  logarithm  which  is  to  be  found  in  Table  XLVIL ;  and  the  corresponding 
number  of  degrees,  minutes,  and  seconds,  taken  at  the  top  of  the  page,  is  the  first 
correction. 

To  the  proportional  logarithm  of  the  moon's  horizontal  parallax,  (Table  XXII.)  add 
the  log.  cosecant  of  the  moon's  apparent  altitude,*  (Table  XXVII.)  and  the  log. 
tangent  of  the  apparent  distance,  (Table  XXVII. ;)  the  sum  (rejecting  the  tens  in  the 
indices)  will  be  a  logarithm  which  is  to  be  found  in  Table  XLVIL;  and  the  corre- 
sponding second  con-ection  is  to  be  found  at  the  top  of  the  table,  if  the  apparent  distance 
exceed  90° ;  but  the  second  correction  is  to  be  found  at  the  bottom  of  the  table,  if 
the  apparent  distance  be  less  than  90°. 

*  1  he  same  rule  may  be  used  for  the  sun  or  a  planet,  correcting  for  the  parallax  by  means  of 
Tables  XLIX.  an!  L.,  as  will  be  shown  hcrcalier, 


240  LUNAR  OBSPmVATIONS. 

Take  the  third  correction,  by  in?^">ection,  from  Table  XL VIII.,  for  tlie  nearest  degrees 
corresponding  to  the  apparent  distances  and  altitudes. 

Add  these  tliree  corrections  to  the  apparent  distance ;  the  sum,  decreased  by  10° 
gives  the  ti'ue  distance  of  the  moon  from  the  star. 

When  the  sun  is  used,  instead  of  a  star,  we  must  take  out  the  correction  for  the 
sun's  parallax,  in  the  part  P,  of  the  same  page  of  Table  XLVIII.  in  which  the  third 
correction  is  found ;  and  this  correction  is  to  be  applied,  by  addition  or  subtraction, 
according  to  its  sign  in  the  table,  to  the  true  distance  above  computed,  as  for  a  star. 

When  a  planet  is  used,  we  can  find  the  correction  of  the  distance  for  the  planet's 
parallax,  by  means  of  Tables  XLIX.,  L.  The  first  of  these  tables,  lieing  entei-ed  with 
the  nearest  degrees  of  the  distance  and  altitudes,  gives  the  correction,  with  its  sign, 
supposing  the  horizontal  parallax  to  be  100".  This  is  reduced  to  the  actual  parallax, 
by  means  of  Table  L.  We  may  also  find  this  correction  very  nearly  by  the  table 
marked  P,  on  the  same  page  of  Table  XLVIII.  where  the  third  correction  is  found ; 
which  gives  tlie  correction  of  the  distance,  with  its  sign,  supposing  the  horizontal 
parallax  to  be  equal  to  the  sun's  mean  parallax,  8".6  ;  if  the  horizontal  parallax  of  the 
planet  be  greater  or  less  than  8".G,  this  correction  must  be  increased  or  decreased  in 
the  same  proportion,  always  retaining  the  same  sign.  The  coiTection  thus  found  is 
to  be  apjjlied  to  the  true  distance,  above  computed  tor  a  star. 


EXAMPLE   VIII. 

[Being  the  same  as  Example  III.,  page  234.] 

Suppose  that,  on  the  30th  of  October,  sea  account,  in  the  forenoon,  in  the  longitude 
of  80°  W.,  by  account,  at  O'^  43™  47%  mean  time,  the  observed  distance  of  the  nearest 
limb  of  the  sun  and  moon  was  111°  34'  50",  the  altitude  of  the  sun's  lower  limb 
24°  55',  and  the  altitude  of  the  moon's  lower  limb  26°  25'.  Requh'ed  the  true 
longitude. 

The  preparation  is  the  same  as  in  page  234,  which,  for  want  of  room  on  this  page, 
we  shall  not  repeat,  but  merely  give  the  results,  namely: — A]iparent  distance 
112°  05'  52" ;  {v)'s  apparent  altitude  25°  07' ;  D's  apparent  altitude  20°  37' ;  ])'s  semi- 
diameter  14'  53" ;   2)'s  horizontal  parallax  54'  10". 


To  find  the  true  distance. 


Dhor.par...     0°5i'10" Prop.  log.    0.5215 

©app.alt...  25  07  00 Cosec 10.3722 

App.clist...ll2  03  52 Sine 9.9GG9 

IstCorr. ...     4  35  ll..Tab.XLVII.  Log.0.8G0G 

SdCorr 4  50  09 

SdCorr 2  45 


Sum— 10°=111   33  57 
©par.Tab.P.  —6 


Same 0.5215 

D  apparent  altilude  26°  37' Cosec. .  10.3487 

Tangent  10.3914 

2d  Corr.  Tab.  XL  VII Los 1.2616 


111   33  51  =  True  distance,  diflering  2"  from  the  first  method  in  page  234. 


To  find  the  true  longitude. 

True  distance 111°33'51" 

Distance  by  M.  A.  at  O'' 112  54  10  Prop,  log 3458 

Difiercnce 1   20  19  Prop,  log 3505 

2i»58n>033 Prop.  log.  diff.  0017 

*  Add    0 


Mean  time  at  Greenwich. . .  .Oct.  SQJ   Sh  58™  03a 
Mean  time  at  the  ship Oct.  29  21   43  47 

DifTerence  is  longitude  in  time 5  14   16  =  78°  34'  V/.  from  GreenHicli. 


LUNAR   OBSERVATIONS. 


241 


EXAMPLE    IX. 
[Same  as  Example  I.,  page  232.] 

Suj»pose  that,  on  the  7th  of  January,  183G,  sea  account,  at  11™  57'  mean  time, 
past  midnight,  in  the  longitude  of  127°  30'  E.,  by  account,  tlie  observed  distance  of 
the  fai-thest  Hmb  of  the  moon  from  the  star  Aldebaran,  was  G8°  3G'  00",  the  observed 
altitude  of  the  star  32°  14',  and  the  observed  altitude  of  the  moon's  lower  limb 
34°  43'.     Required  the  true  longitude. 


Preparation. 

Sea  account,  Jan.  7,  is  by  N.  A.  Jan.  C^  ISh  llm  57» 
Longitude  127°  30' E 8    30     00 


Reduced  time Jan.  C<i    3^  41n>  57' 


Jsemidiam.  Jan.  G,  noon  15' 05''        ])  hor. par.  Jan.  G,  noon..  55' 20"        -5^  observed  alt....  32°  14 


midniarbt  15  09 


Difference 
Table  XI. 


Auff.  Table  XV. 


15  06 
9 


5  semidiameter 15' 15" 


midnight  55  34  Subtract. 


Difference. 
Table  XI.. 


14 

4 


I)  horizontal  parallax....  55' 24" 


B  obs.  alt.  L.  L. 
Add 


4 


*  apparent  alt....  32  10 


34°  45' 
12 

D  apparent  alt.  . .  34°  65' 


Observed  distance  *  D   F.  L 68°  36' 00" 

D  semidiameter subtract         15  15 

Apparent  distance*  D 68°  20' 45" 


To  find  the  true  distance. 


Bhor.par 0°55'24" Prop.  log.  0.5118 

*app.  alt 3210  00 Cosec.  0.2738 

App.  dist G3  20  '15 Sine  9.9632 

IstCorr 4  28  16.  .Tab.  XLVII.  Log.  0.7538 

2dCorr 5  12  35 

SdCorr.Tab.XLVIII.     1  25 

Sum  — 10°=  68°  03' 01"  =  True  distance,  differing  1"  from  the  first  method,  in  page  232. 


Same 0.5113 

B apparent  altitude  34°  55'.. Cosec.  0.2423 

Tangent  0.4012 

2dCorr.  Tab.  XLVII Log.  1.1553 


To  find  the  longitude. 

True  distance 68°  03'  01" 

Distance  by  N.  A.  at  3'> G7   41   43  Prop.  lo"-.  . . .  2872 

Difference 0    21    18  Prop,  log 9269 

0''  41m  16> prop_  lo^  ji(y  5397 

Add    3 

Mean  time  at  Greenwich 3    41    16 

Mean  time  at  the  ship 12    11    57 


Difference  is  longitude  in  time. ...     8    30   41  =  127°  40'  15"  E.  from  Grocnwiclj. 

3 


242  LUNAR  OBSERVATIONS. 

THIRD  METHOD 

Of  finding  the  true  distance  of  the  moon  from  the  sun,  a  planet,  or  a  star. 

RULE. 

From  the  sun's  refraction  (Table  XH.)  take  his  parallax  in  altitude,  (Table  XIV.;) 
the  remainder  call  the  correction  of  the  swi's  altitude.  In  like  manner,  if  a  planet 
be  used,  we  must  find  the  planet's  refraction,  (in  Table  XII.)  and  subtract  from  it  the 
parallax  in  altitude,  (Table  X.  A.;)  the  remainder  will  be  the  correction  of  theplaneVs 
altitude.  Knt  if  a  star  be  used,  we  must  find  the  refraction,  (Table  XII.)  and  that  will 
be  the  correction  of  the  star''s  altitude.* 

From  the  proportional  logarithm  of  the  moon's  horizontal  parallax,  (increasino^  the 
index  by  10,)  take  the  sine  of  the  moon's  apparent  zenith  distance,  (Table  XXVII. ;) 
the  remainder  will  be  the  prop.  log.  of  the  parallax  in  altitude,  which  must  be  found 
in  Table  XXII.,  and  the  moon's  refraction  (Table  XII.)  subtracted  therefrom  ;  the 
remainder  will  be  the  correction  of  the  moon's  altitude.f 

Add  together  the  api)arent  distance  of  the  sun  and  moon,  ([ilanet  and  moon,  or  star 
and  moon,)  and  their  apparent  zenith  distances,  (or  complement  of  their  apparent 
altitudes,)  and  note  the  half-sum  of  these  numbers ;  the  difference  between  the  half- 
sum  and  the  moon's  apparent  zenith  distance  call  t\\e  first  remainder ;  and  the  differ- 
ence between  the  half-sum  and  the  sun's  (planet  or  star's)  apparent  zenith  distance, 
call  the  second  remainder. 

To  the  constant  log.  9.C990  add  the  cosecant  of  the  half-sum,  and  the  sine  of  the 
ap{)arent  distance,  (both  taken  from  Table  XXVII. ;)  the  sum  (rejecting  20  from  the 
index)  will  be  a  reserved  logarithm. 

To  the  reserved  logarithm  add  the  sine  of  the  sun's  (planet  or  star's;  apparent 
zenith  distance,  the  cosecant  of  the  first  remainder,  (both  taken  from  Table  XXVII.) 
and  the  ]M-op.  log.  of  the  correction  of  the  sun's  (planet  or  star's)  altitude,  (Table 
XXII.;)  the  sum  (rejecting  30  from  the  index)  will  be  the  prop.  log.  of  the  fiist  cor- 
rection, to  l)c  found  in  Table  XXII. 

To  the  reserved  logarithm  add  the  sine  of  the  moon's  apparent  zenith  distance,! 
the  cosecant  of  the  second  remainder,  (Table  XXVII.)  and  the  prop.  log.  of  the 
correction  of  the  moon's  altitude,  (Table  XXII. ;)  the  sum  (rejecting  30  fi-om  the  index) 
will  be  the  ])rop.  log.  of  the  second  correction,  to  be  found  in  Table  XXII. 

Then,  to  the  a])parent  distance  add  the  correction  of  the  moon's  altitude,  and  the 
fii'st  correction,  and  subtract  the  smn  of  the  second  correction  and  the  correction  of 
the  sun's  (|)lanet  or  star's)  altitude;  the  remainder  will  be  the  corrected  distance. 

Enter  Table  XX.,  and  find  the  numbers  which  most  nearly  agree  with  the  observed 
distance,  and  the  observed  altitudes  of  the  objects,  and  take  out  the  corresponding 
correction  in  seconds,  which  is  to  be  added  to  the  corrected  distance,  and  then  18' 
subtracted  from  the  sum  ;  *the  remainder  will  be  the  true  distance.J 

We  shall  now  give  an  example  of  this  third  method  of  correcting  the  distance  ;  but 
it  will  be  unnecessary  to  repeat  the  preparation  and  the  process  to  find  the  longittide, 
as  it  is  very  nearly  the  same  as  in  page  232. 


EXAMPLE  X. 

[Same  as  Example  I.,  preceding.] 

Suppose  the  apparent  distance  of  tlio  centre  of  the  moon  from  the  star  Aldebaran 
was  G8°  20'  45",  the  apparent  altitude  of  the  star  32^  10',  the  apparent  altitude  of  tlie 


*  We  may  also  find  this  correction  by  means  of  Table  XVII.,  or  Table  XVIII.j  taking  the 
difTerence  ticlwccn  tlie  taljular  number  and  GO'  for  the  correction ;  using  Table  XVIII,  for  the  sun,  and 
Table  XVII.  for  a  planet,  or  a  fixed  star. 

t  Tliis  correction  may  very  easily  be  found  by  means  of  Table  XIX.,  by  subtracting  tlie  tabular 
number  from  59' 42";  for  tiie  remainder  will  be  the  correction  of  the  moon's  altitude  for  parallax  and 
refraction. 

t  N(!glecting  the  small  corrections  mentioned  in  a  note  marked  i,  in  page  231. 


LUNAR   OBSERVATIONS. 


243 


moon's  centre  34°  55',  and  the  moon's  horizontal  parallax  55'  24".     Required  the  true 
distance  of  the  moon  from  the  star. 


WOO' 
3  app.  alt....  34  55 
])  zenith  dist.  55  05 


90°  00' 
^  app.  alt....  32  10 
^  zenith  dist.  57  50 


Hor.  par.  55'  24" P.  L.  10.5118 

])  zenith  dist.  55°  05' Sine    9.9138 

45' 26"... P. L.     0.5980 

5  refraction  ....     1  21 

Corr.  D  altitude   44   05 


^refraction  l'31f 


App.  dist G8°21' 

5  zenith  dist.  55  05 

^  zenitli  dist.  57  50 

Sum 181    16 

Ifalf-sum 90  38 

I>  zenith  dist.  55  05 

1st  Rem 35  33 

llalf-snm  ....  90  33 

^  zenith  dist.  57  50 

Qd  Rem 32  48 


Constant  log 9.6990 

Half-sum  90°  38' Cosec.  10.0000 

Dist.  C8°  21' Sine  9.9682 

Reserved  log 9.6G72 

*  zenith  dist.  57°  50'  ...Sine  9.9276 
1st  Rom.  35°  33' Cosec.  10.2355 

*  Corr.  1' 31" P.  L.  2.0744 

1st  Corr.  2'  15" P.  L.  1 .9047 


Reserved  log 9.6G72 

5  zenith  dist.  55°  05'... Sine    9.9138 

2d  Rem.  32°  48' Cosec.  10.26G2 

])  Corr.  44'  05" P.  L.    0.6110 

2d  Corr.  1°  2' 40" P.  L.     0.4582 


Apparent  distance G8°  20'  45" 

First  correclioii add  2   15 

Correction  ])  altitude 44   05 

69  07  05 
Second  correction. ..   1°    2' 40" 
Correction  ^  altitude  1   31    sub.     14   11 

Corrected  distance 68  02  54 

Correction  Table  XX.  —  1 8" 7 

68°  03'  01"  agreeing  within  1"  of  the  first  melliod. 

This  method,  as  well  as  the  first,  was  invented  hy  the  autlior  of  this  work,  who 
also  improved  Witchell's  metliod,  and  reduced  considerably  the  number  of  cases. 
These  improvements  were  made  in  consequence  of  a  suggestion  of  the  late  Cliief 
Justice  Parsons,  (a  gentleman  eminently  distinguished  for  his  mathematical  acquire- 
ments,) who  had  somewhat  simplified  Witchell's  process;  and  it  was  found,  upon 
e.xainination,  that  this  improvement  could  be  extended  fartlicr  than  he  had  done  it, 
and  that  the  number  of  cases,  with  the  manner  of  ap|dying  the  corrections,  could  be 
rendered  more  simple  and  symmetrical.  This  improvement  of  Witchell's  process 
we  shall  now  insert  as  the  fourth  method  of  computation. 


FOURTH  METHOD 

Of  finding  the  true  distance  of  the  moon  from  the  sun,  a  planet,  or  a  star. 

RULE. 

From  the  sun's  refraction  (Table  XII.)  take  his  parallax  in  altitude,  (Table  XIV. ;) 
the  remainder  will  be  the  correction  of  the  sun's  allilude.  In  like  manner,  if  a  planet 
be  used,  we  must  find  the  planet's  refraction,  (in  Table  XII.)  and  subtract  from  it  the 
parallax  in  altitude,  (Table  X.  A. ;)  the  remainder  will  he  the  correction  of  the  planet's 
alliludp.  But  if  a  star  be  observed,  we  must  find  the  refraction,  (Table  XII. ;)  and 
that  will  be  the  correction  of  the  starts  altitude* 

From  the  proportional  logarithm  of  the  moon's  horizontal  parallax,  (increasing  the 
index  by  10,)  take  the  cosine  of  the  moon's  apparent  altitude,  (Table  XXVII.;)  the 
remainder  will  be  the  proportional  logarithm  of  the  moon's  parallax  in  altitude ;  from 
wlilch  subtracting  the  moon's  refraction,  (Table  XII.)  the  remainder  will  be  the  cor- 
rection of  the  moon's  allilude.f 

*  This  correction  may  be  found  in  Table  XVII.  or  XVIII.,  as  is  shown  in  a  note  to  the  third 
methofl,  in  page  242. 

t  This  correction  may  be  found  by  Table  XIX.,  as  is  sho'mi  in  a  note  to  the  tliird  method,, 
m  page  242. 


244  LUNAR   OBSERVATIONS 

1.  Add  together  the  apparent  aUitudes  of  the  moon  and  sun,  (planet  or  star,)  and 
take  the  halt-sum ;  subtract  the  least  altitude  from  the  greatest,  and  take  the  half- 
difference  ;  then  add  together 

The  tangent  of  the  half-sum, 

The  cotangent  of  the  half-difference. 

The  tangent  of  half  llie  apparent  distance  ; 

The  sum  (rejecting  20  in  the  index)  will  be  the  tangent  of  the  angle  A,  'which 
must  be  sought  for  in  Table  XXVII.,  and  taken  out  less  than  90°  when  the  sun's 
altitude  is  less  than  the  moon's,  otherwise  greater  than  90°,  *  The  difference  of  the 
angle  A,  and  half  the  apparent  distance,  is  to  be  called  the  first  angle,  and  their  sum 
the  second  angle. 

2.  Add  together  the  tangent  of  the  first  angle, 

The  cotangent  of  the  sun,  planet,  or  star's  apparent  altitude, 

The  prop.  log.  of  the  correction  of  the  sun,  planet,  or  star's  altitude; 

The  sum  (rejecting  20  in  tlie  index)  will  be  the  prop.  log.  of  the  first  correction. 

Or  the  refraction  (Table  XIJ.)  corresponding  to  the  first  angle,  or  its  sup])lement, 
will  be  the  first  correction  nearly;  particularly  if  the  altitude  of  the  sun,  planet,  or 
star,  be  great,  and  the  first  angle  be  near  90°. 

3.  Add  together  the  tangent  of  the  second  angle. 
The  cotangent  of  the  moon's  apparent  altitude. 

The  prop.  log.  of  the  correction  of  the  moon's  altitude ; 

The  sum  (rejecting  20  in  the  index)  will  be  the  prop.  log.  of  the  second  correction. 

4.  The  first  correction  is  to  be  added  to  the  apparent  distance  when  the  first  angle 
is  less  than  90°,  otherwise  subtracted  ;  and  in  the  same  manner  the  second  correction 

.is  to  be  added  when  the  second  angle  is  less  than  90°,  otherwise  subtracted.     By 
applying  these  two  corrections,  we  shall  obtain  the  corrected  distance. 

Enter  Table  XX.,  and  find  the  numbers  which  most  nearly  agree  with  the  observed 
distance  and  the  observed  altitudes  of  the  objects,  and  take  out  the  corresponding 
third  correction  in  seconds,  which  is  to  be  added  to  the  corrected  distance,  and  then 
18"  subtracted  from  the  sum  ;  the  remainder  will  be  the  true  distance. 

We  shall  now  give  an  example  of  this  fourth  method  of  correcting  the  distances 
omitting,  as  before,  the  preparation  and  the  computation  of  the  longitude  from  the  tru» 
distance. 

EXAMPLE   XL 

[The  same  as  Example  I.,  preceding.] 

Suppose  the  apparent  distance  of  the  centre  of  the  moon  from  the  star  Aldebara.! 
was  68°  20'  45",  the  apjiarent  altitude  of  the  star  32°  10',  the  ai)parent  altitude  of  thri 
moon's  centre  34*^55',  and  the  moon's  horizontal  parallax  55'  24".  Required  the  trv.e 
distance  of  the  moon  from  the  star. 

D  app.  alt.  34°  55' 
*  app.  alt.  32   10 

Sum G7  05  Half-sum  ..  33° 33'. .  Tang.    9.82IG1         Ilor.  par.    55' 21"  ...  .P.  L.  10.51 13 

Difference  _2_15  Ilalf-diff.  . .     1°  23'  Cotang.  11.G1711         D  app.  alt.  3t°55' ...  Cosine     9.J138 

Half-di.st.. .  34°  10' .  .Tang.    9.83171  45'  2G"  ....  P.  L.    0.5'JSC 

Angle  A.. .  86°  5G'  .  .Tang.  1 1.27043  1'  21"  5  refraction. 

Difference  is 1st  angle..  52° 4G' .  .Tang.  10.1192  4-1.   05    Corr.  J)  altiHulc. 

*  app.  alt   32°  10'  Cotang.  10.2014 

Corr.  *alt.    1'31"..P.  L.    2.0744  Apparent  distance GS°20'45" 

IstCorr.  ..    0'44"..P.  L.    2.3950  1st  correction add  0  U 

Sumis 2d  angle..  121° OG'.. Tang.  10.2195  '  68  2129 

p  app.  alt.  34°  55'  Cotang.  10.15G1  2d  correction sub.  18  31 

Corr.  D  alt.  44'  05"  . . P.  L.    O.GllO  3d  angle G8  02  55 

2dCorr....  18'  34".. P.  L.    0.98G6  3d  corr.  Table  XX.— 18" 7_ 

Trnedis'anee C8  03  02 

Agreeing  within  2"  of  the  first  method 

*  Every  cotangent  in  Table  XXVII.  corresponds  to  two  angles,  the  one  greater  than  90°,  the  otiiei 
less  than  90°. 


LUNAR  OBSERVATIONS. 


245 


Mdhod  of  correcting  for  the  second  differences  of  the  motions  of  the  bodies  in 
computing  a  lunar  observation. 

In  all  the  preceding  calculation?,  we  have  neglected  the  second  difFerences  of  the 
moon's  motion,  in  the  intervals  of  3  hours,  between  the  times  in  which  the  distances 
are  marked  in  the  Nautical  Almanac.  The  correction  arising  from  this  soiiue  is 
o'cnerally  quite  small,  and  mav,  in  most  cases,  be  neglected,  as  coming  within  the 
Umits  of  the  usual  errors  of  such  observations.  It  is,  however,  very  easy  to  find  tins 
correction  by  means  of  the  following  table,  which  is  similar  to  that  in  page  484  of  the 
Nautical  Almanac  for  183G.  In  using  this  table,  we  must  find  the  difierence  between 
the  two  projjortional  logarithms,  conesjionding  to  the  distances  in  the  Nautical  Alma- 
nac, which  include  the  given  distance.  This  difierence  is  to  be  sought  tor  at  the  top 
of  the  table  ;  and  at  the  side  we  must  find  the  interval  which  is  calculated  m  the  last 
part  of  the  process  of  compiuiiig  the  true  longitude,  being  the  tiiue  between  the  hour 
marked  first  in  the  Nautical  Almanac,  and  the  mean  time  of  observation  atGreenwicJi. 
Tlie  number  of  seconds  in  the  table  corresponding  to  these  two  arguments  is  to  be 
applied,  according  to  the  directions  in  tlie  table,  as  a  correction  to  the  time  at 
(Greenwich,  computed  by  either  of  the  preceding  methods. 

Example  1.  Thus,  in  the  example  page  232,  we  find  that  the  two  proportional 
lo<^arithms  corresponding,  on  January  tith,  to  3''  and  C',  are  2872,  2864,  whose 
difference  is  8;  and  the  interval  past  3",  computed  in  page  232,  is  0"  41"  14'. 
Entering  the  table  with  8  at  the  top,  and  0''  40'"  at  the  side,  (which  is  the  nearest 
mimbor°to  the  interval  0"  41™  U%)  we  get  the  correction  2%  to  be  added  to  die  time 
at  Greenwich,  Q^  41'"  14%  (computed  in  page  232,)  because  the  logarithms  are 
decreasing  ;  hence  the  corrected  time  at  Greenwich  is  3''  41"'  16". 


example  page  237,  we  find  that  the  two  ])roportional  1 
)n  June  20th,  to  9"  and  12",  are  2985  and  29G9,  whose  differ 


loga- 
ence 


Example  2.     In  the 

lithms  corresj)onding,  on ,  , 

is  IG.  Under  this,  and  0])i)osite  the  interval  1''  55'"  28%  computed  in  page  237,  (or  the 
nearest  tabular  nunilier  2''  0'",)  we  find  a  correction  4=  to  be  added  to  the  time  at 
Greenwich  10"  55'"  23%  computed  in  i)age  237,  making  the  corrected  time  at  Green- 
wich 10"  55™  32'. 


Table,  showing  the  Correction  required  on  account  of  the  Second  Differences  of  the 
Distances  in  the  jYautical  Almanac,  in  ivorkhig  a  Lunar  Observation. 


Find  at  the  top  of  the  table  the  diflcrence  between  the  proportional  logarithm  taken  from  the 
iNantical  Almanac,  in  working  a  kuiar  observation,  and  that  which  immediately  follows  it,  and  at  the 
side  the  interval  between  the  hour  marked  in  the  Nautical  Almanac,  and  the  mean  time  of  the 
observation  of  the  meridian  at  Greenwich.  The  corresponding  namber  is  a  correction,  in  seconds, 
which  is  to  be  added  to  the  time  at  Greenwich,  deduced  from  either  of  the  preceding  methods  of 
working  a  lunar  observation  if  the  proportional  logarithms  are  decreasing,  but  stihtracled  if  the  pro- 
portional logarithms  are  increasing ;  the  sum  or  difference  will  be  the  corrected  time  at  (Jreenwich. 


Approxi- 
mate 
Interval. 


n .  M . 
0  0 
0  1(1 

0  20 


n.  M. 
:i  0 
•2  50 
•2  40 


0  3(1-2  ?,() 

0  40]'2  20 

0  502  10 

roo;27)o 

1  lOll  50 
1  201  40 
1  301  30 


Difference  of  tlie  Proportional  Logarithms  in  the  NaiUical  Almanac. 


4  I  8  :i2|l6l20|24|28|32!3()|40|44i48|52|5G|G0!64J()8|72|76!80|84|8S|SJ2|nG 


Correction  of  the  Time  at  Greenwich  for  Second  Differences. 


s.\  s. 
0    0 

11 
12 


2'2 
2    3 

2    3 


2   3 

2  4 

3  4 
3i4 


3  3 
3I4 

4  5 


6    7 

G  7 
G  7 
Gl8 


7   8 

010 

lO'll 


s.  I  s. 
0     " 
3 
6 


91011  12  13 
9;ill2'l314 
9  101111214  15 
9  !lO  11112  14  15 


10  10 
12  13 
1415 


111213 
14  1511G 

1C1718 


14  IG  17  1849  20 

15  17il8  19  20,21 
lG]7!l9  20'21,22 
lG18il9'20>2ll23 


10 


1414 

1718 
20:21 


0 

G 

12 

151G17 
19  2021 
22  2324 


21  22  23124  25127 

22  24  25  2G27  2S 

2r?25  2G,27  28j29 
24'25',2G27,29l30 


Ajypvo.ri- 

mate 
Interval. 


n.  M.  H.M. 

0  03  0 
0  ]0]2  50 
0  20  2  40 

0  30^2  30 
0  40|2  20 
0  50|2  JO 


246 


LUNAR  OBSERVATIONS. 


Mctliod  of  taking  a  lunar  observation  by  one  observer. 

Three  obsei^vers  are  requii-ed  to  make  the  necessary  obsenations  for  determining 
rne  longitude ;  one  to  measure  the  distance  of  the  bodies,  and  the  others  to  take  the 
altitudes.  In  case  of  not  having  a  sufficient  number  of  instruments  or  observers  to 
take  the  altitudes,  it  has  been  customary  to  calculate  them ;  there  being  given  the 
latitude  of  the  place,  the  apparent  time,  the  right  ascensions,  and  the  declinations 
of  the  objects.  These  calculations  are  long,  when  an  altitude  of  a  star  is  to  be  com- 
puted, and  much  more  so  when  that  of  the  moon  is  required ;  and  a  considerable 
degree  of  accuracy  is  required  in  finding,  from  the  Nautical  Almanac,  the  moon's 
right  ascension  and  declination,  which  must  be  liable  to  some  error  on  account  of  the 
uncertainty  of  the  ship's  longitude.  The  following  method  of  obtaining  those  alti-- 
tudes  is  far  more  simple,  and  sufliciently  accurate.  This  method  depends  on  the 
supposition  that  the  altitudes  increase  or  decrease  uniformly. 

Before  you  measure  the  distance  of  the  bodies,  take  their  altitudes,  and  note  the 
times  by  a  chronometer ;  then  measure  the  distance,  and  note  the  time,  (or  you  may 
measure  a  immber  of  distances,  and  note  the  corresponding  times,  and  take  the  mean 
of  all  the  times  and  distances  for  the  time  and  distance  respectively ;)  after  you  have 
measured  the  distances^  again  measure  the  altitudes,  and  note  the  times ;  tiien,  from 
the  two  observed  altitudes  of  either  of  the  objects,  the  sought  altitude  of  that  object 
may  be  fouud  in  the  following  manner: — 

Add  together  the  proportional  logarithm  (Table  XXII.)  of  the  variation  of  altitude* 
of  the  object  between  the  two  times  of  observing  the  altitudes,  and  the  prop.  log.  of 
the  time  elapsed  between  taking  the  first  altitude  and  measuring  the  distance;  from 
the  sum  subtract  the  prop,  log.f  of  the  time  elapsed  between  obsei'vmg  the  two  altitudes 
of  that  object ;  the  remainder  will  be  the  prop.  log.  of  the  correction,  to  be  api)lied  to 
the  first  altitude,  additive  or  subtractive,  according  as  the  altitude  was  increasing  or 
decreasing ;  to  the  altitude,  thus  corrected,  apply  the  correction  for  dip  of  the  horizon 
and  semidiameter,  as  usual. 


EXAMPLE. 

Suppose  the  distances  and  altitudes  of  the  sun  and  moon  were  obseiTcd,  as  in  the 
following  table  ;  it  is  required  to  find  the  altitudes  at  the  time  of  measuring  the  mean 
distance. 

Observations. 


Mcjui. 


Times  by 
chronometer. 

2h3m20' 

2  4    20 
2   5   50 

..2    4    30 


Dist.  0  and 
a  JV.  L. 
40'=  0'  00" 
40  0  30 
40   1   30 


40  0  40 


Times  by 
chrunvmcter, 
2h  2m  0» 

2   6  10 


Difference , 


4  10 


Obs.  alt. 

])'sL.L. 
20°  4G' 
21    20 


Times  by 
chronometer. 

2h2m30' 

2   7  00 
Difference. .      4   30 


Ohs.  alt. 

O^sL.L. 
40°  20' 
39    12 

1      8 


Variation  ])  's  altitude. . 
Time  1st  observation  J) 


34'  Prop.  log.    7238 

oh  2m  Qa 


Mean  time  of  observing  }  a   a  tn 
distance >  ^  *  JU 


Difference. 


2  30  Prop.  log.  1.8573 


Elapsed  time  between 
the  two  observations 

Correction  of  altitude. . . 

First  altitude  of  moon  . . 

Alt.  D  's  L.  L.  at  time  of 
the  mean  obs.  of  dist. 


2.5811 
4m  10'  Prop.  log.  1.G355 

0°  20'  Prop,  lo! 
20  46   add. 


9456 


IzL 


V^ariation  0's  altitude.         1°   8'  Prop.  log.     4223 
Time  1st  observation  0  2ii2ra30» 
Time  mean  observation  2  4    30 

Difference 2    00  Prop.  log.  1.9542 

Sum 2.3770 

Elapsed  time  between 
the  two  observations 


4    30    Prop.  log.  1.6021 


Correction  of  altitude...     0°  30'  Prop.  log.    7749 

Sub.  from  ©'s  1st  altitude  40  20 

Alt.  0's  L.  L.  at  time  of~) 

the  mean  observation  >  39   50 
of  ihe  distajices j 


Thus,  at  the  time  S"*  4™  30%  the  mean  observed  distance  of  the  sini  and  moon's 
nearest  limbs  was  40°  0'  40",  the  altitude  of  the  moon's  lower  limb  21°  &,  and  the 
altitude  of  the  sun's  lower  limb  39°  50' ;  these  altitudes  must  be  corrected  for  dip  and 
semidiameter  as  usual. 


*  Table  XXII  is  only  calculated  as  far  as  3°,  and  if  the  variation  of  altitude  exceed  that  quantiiy, 
you  must  enter  the  table  with  minutes  and  seconds,  instead  of  degrees  and  minutes ;  and  the  correction 
of  altitude  taken  out  in  minutes  and  seconds  must  be  called  degrees  and  minutes  respective!  v 

i  Or  add  its  arithmetical  complement,  neglecting  10  in  the  index  of  the  .sum 


LUiNAR   OBSERVATIONS.  247 

In  tliis  manner  I  have  often  obtained  the  altitudes  in  much  less  time  than  they 
could  have  been  obtained  by  other  calculations. 

The  same  method  may  be  used  for  finding  the  sun's  altitude,  when  taking  an 
azimuth,  by  noting  the  times  of  taking  the  observations  by  a  chronometer,  and  taking 
two  altitudes,  the  one  before,  the  other  after  the  observation,  and  jiroporlioning  the 
altitudes  as  above. 

Any  person  who  wishes  to  calculate  strictly  the  apparent  altitudes,  may  proceed 
according  to  the  following  rules : — 


The  apparent  time,*  the  ship's  latitude  and  longitude,  and  the  sun's  declination 
given,  to  find  the  apparent  altitude  of  his  centre. 

RULE. 

With  the  apparent  time  from  noon,  enter  Table  XXIII.,  and  from  the  column  of 
rising  take  out  the  logarithm  corresponding,  to  which  add  the  log.  cosine  of  the 
latitude,  and  the  log.  cosine  of  the  sun's  declination ;  their  sum  (rejecting  20  in  the 
index)  will  be  the  logarithm  of  a  natural  number,  which  being  suijtracted  lioin  the 
natural  cosine  of  the  sum  of  the  declination  and  latitude,  when  they  are  of  difterent 
names,  or  the  natural  cosine  of  their  difference,  when  of  the  same  name,  will  leave 
the  natural  sine  of  the  sun's  true  altitude  at  the  given  time.  The  refraction,  less 
parallax,  being  added  to  the  true  altitude,  will  give  the  apparent  altitude. 

In  general,  it  will  be  near  enough  to  take  out  the  refraction  only  from  Table  XII., 
and  neglect  the  parallax. 

EXAMPLE   I. 

Ilequired  the  true  altitude  of  the  sun's  centre,  in  latitude  49°  57'  N.,  and  longitude 
75°  W.,  July  20,  183G,  at  C'  50™  30'  in  the  morning,  apparent  time,  sea  account. 

12"  0™0» 
Apparent  time G  56  30 

Apparent  time  from  noon    5    3  30  Its  log.  in  column  of  rising 4.87850 

Latitude 49  57    ON.  Its  log.  cosine 9.80852 

Declination  at  that  time . .  19  24  15  N.  Its  log.  cosine 9.97460 

Natural  number  45880  Its  log.  —4.66162 
Difference  .• 30  32  45  Natural  cosine     86123  

True  altitude 23  44  Natural  sine  . .  40243 

Refraction add  2 

Apparent  altitude 23  4G 


EXAMPLE    II. 

What  will  be  the  true  altitude  of  the  sun's  centre,  in  the  latitude  of  39°  20'  N.,  and 
the  longitude  of  40°  50'  W.,  November  26,  1836,  at  3"  21'"  30%  apparent  time,  in  the 

alternoon,  sea  account  ? 

Apparent  time  from  noon     3''21'^30'  Its  log.  in  column  of  rising 4.5.5900 

Latitude 39   20   00  N.  Its  log.  cosine 9.88844 

Declination  at  that  time  20   53   09  S.  Its  log.  cosine 9.97048 

Natural  number  26177  Its  log.  =  4.41792 
Sum 60   13   09  Natural  cosine    49668  ' 

True  altitude 13  35  Natural  sine. . .  23491 

Refraction add  4  

Apparent  altitude 13   39 


*  If  the  mean  time  be  ^ven,  we  must  dedtice  from  it  the  apparent  time,  by  applying  the  equation 
Table  IV.  A.,  with  a  different  sign  from  that  in  the  table,  as  taught  in  the  introduction  to  the  tables 
remarking,  however,  this  equation  is  found  more  correctly  in  page  II.  of  the  Nautical  Almanac. 


248  LUNAR   OBSERVATIOISS. 

The  apparent  time,  toith  the  latitude  and  longitude  of  the  ship,  given,  to  Jind  thi 
apparent  altitude  of  the  moon's  centre. 

Turn  tlie  longitude  into  time,  (by  Table  XXI.)  and  if  in  west  longitude  add  it  to, 
but  in  east  longitude  subtract  it  from,  tlie  apparent  time  *  at  the  ship  ;  tlie  sum  or 
diflerence  will  be  the  apparent  time  at  Greenwich.  From  this  we  may  deduce  the 
ini-an  time  at  Greenwich,  which  is  wanted  in  finding  the  moon's  right  ascension 
and  declination. 

Tak"  the  sun's  right  ascension  from  tlie  Nautical  Almanac  for  the  preceding  noon 
at  Greenwich,  and  add  thereto  the  correction  taken  from  Table  XXXI.  corresponding 
to  the  hours  and  minutes  of  the  time  at  Greenwich  ;  the  sum  will  be  the  sun's  right 
ascension,  which,  being  added  to  the  ajjparent  time  at  the  ship,  will  give  tlie  right 
ascension  of  the  meridian,  rejecting  24  hours  when  the  sum  exceeds  24  hours. 

Take  Irom  the  Nautical  Almanac  the  moon's  right  ascension  and  declination  ibr 
the  lime  at  (jreenwich  ;  then  the  diflerence  between  the  moon's  right  ascension  and 
the  right  ascension  of  the  meridian,  will  be  the  moon's  distance  f  from  tlie  meridian, 
with  which  enter  Table  XXIIL,  and  take  out  the  corres])onding  logarithm  from  the 
column  of  rising,  and  add  thereto  the  log.  cosine  of  the  latitude  of  the  ship,  and  the 
log.  cosine  of  the  declination  of  the  moon;  the  sum  (rejecting  20  in  the  index)  will 
be  the  logarithm  of  a  natural  number,  (Table  XXVI.)  which,  being  subtracted  from 
the  natural  cosine  (Table  XXIV.)  of  the  sum  of  the  declination  and  latitude  when  of 
different  names,  or  the  natural  cosine  of  their  difference  when  of  the  same  name,  will 
leave  the  natm-al  sine  of  the  moon's  true  altitude  ;  from  which  subtracting  the  correc- 
tion corresponding  to  the  altitude  in  Table  XXIX.|  there  will  remain  the  apparent 
altitude  nearly. 

EXAMPLE. 

What  was  the  moon's  apparent  altitude,  Ajiril  29,  1836,  sea  account,  at  7"  55™  52' 
P.  M.,  in  latitude  42°  34'  S.,  longitude  C5°  07'  30"  W.,  from  Greenwich  ? 

April  29,  sea  account,  or  by  astrononfical  account April  28      7''  55"  52' 

Longitude  05°  07'  30"  W.,  in  time 4    20    30 

Apparent  time  at  Greenwich April  28  12    16    22 

Sun's  right  ascension,  April  28^  12'^  10™  22%  by  Nautical  Almanac. . .     2''  25'"  11 ' 
Api)arent  time  at  the  ship 7    55    52 

Right  ascension  of  the  meridian 10    21    03 

j)'s  right  ascension  in  time 12    33    27 

2)'s  distance  from  the  meridian - 2    19    24 

Corresponding,  to  which,  in  the  column  log.  rising,  \s 4.21027 

Latitude 42°  34'  S Cosine        9.80717 

3)'s  declination 0  10  N Cosine      10.00000 

Natural  number  11952     Log...        4.07744 
Sum 42  50  Natural  cosine  .  73333 

3)'s  true  altitude 37  52  Natural  shie. . .  01381 

Correction  Table  XXIX.  44 

])'s  apparent  altitude 37   08  nearly. 

This  altitude  would  be  decreased  nearly  2',  if  the  true  correction  of  the  altitude, 
corresponding  to  i!:e  3)'s  horizontal  parallax,  59',  were  used,  as  may  be  seen  in 
note  \,  at  the  bo'.iom  of  the  page. 

*  The  apparent  lime  is  counted  from  noon  to  noon,  marking  the  hours  from  1  hour  to  24  hours.  We 
may  remark,  thai  lliis  process  of  findinj^  llic  lime  at  Greenwich  is  unnecessary  when  you  liavc  a 
chronometer  rcaulntcd  for  mean  lime  at  Grcenwicli,  because  we  can  immediately  obtain  tlie  appctrertl 
time,  by  a|>]ilying  the  equalion  of  lime,  taken  from  the  Nautical  Almanac,  or  from  Table  IV.  A.,  using 
a  diflerent  sign  from  ihat  in  tlie  table. 

t  When  the  distance  exceeds  12  hours,  you  must  enter  Table  XXIII.  with  the  diflerence  between 
that  distance  and  2l  hours. 

I  In  slriclness  you  ought,  instead  of  this  correction,  to  use  the  correction  of  the  moon's  altitude, 
corresponding  to  Iier  apparent  altitude  and  horizontal  parallax.  This  is  easily  found  in  Table  XIX., 
using  the  D  's  horizontal  parallax  and  the  apparent  altitude  found  by  the  above  process,  a-id  subtracting 
tlie  tabular  corrocti(m  from  59'  42".  Thus,  if  the  )) 's  horizontal  parallax  is  59',  and  the  )) 's  apparent 
altitude  37°  8',  this  correction  would  lie  59'  42"— 13'  65"=45'  47",  instead  of  44'.  which  is  used  above. 


LUNAR  OBSERVATIONS.  249 


Tkc  apparent  time,  icith  the  latitiidc  and  longitude  qf  the  ship,  being  given,  to 
find  the  apparent  altitude  of  the  centre  of  a  planet. 

Turn  the  longitude  into  time,  (by  Table  XXI. ;)  and  if  west,  add  it  to,  but  if  east 
longitude,  subtract  it  from,  the  apparent  time  at  llie  ship;  the  sum,  or  ditlercnco,  will 
be  the  apparent  time  at  Greenwich.  From  this  we  may  deduce  the  mean  time  at 
Greenwich,  which  is  required  in  finding  the  right  ascension  and  declination  of  the 
planet.* 

Take  the  sun's  right  ascension  from  tlie  Nautical  Almanac,  for  the  preceding  noon 
at  Greenwich,  and  add  thereto  the  correction  taken  from  Table  XXXI.,  corresi)ond- 
ing  to  the  hours  and  minutes  of  the  time  at  Greenwich  ;  the  sum  will  bo  the  sun's 
right  ascension,  which,  beuig  added  to  the  apparent  time  at  the  ship,  will  give  the  right 
ascension  of  the  meridian,  rejecting  24  hours  when  the  sum  exceeds  24  hours. 

Take  from  the  Nautical  Almanac  the  ])lanet's  right  ascension  and  declination  for 
the  time  at  Greenwich  ;  then  the  difference  between  the  jjlanct's  right  ascension  and 
the  right  ascension  of  the  meridian,  will  be  the  ]»lanet's  distance  \  from  the  meridian  ; 
with  wiiich  enter  Table  XXIII.,  and  take  out  the  corresponding  logarithu),  from  the 
column  of  rising,  and  add  thereto  the  log.  cosine  of  the  latitude  of  tiie  ship,  and  the 
log.  cosine  of  the  declination  of  the  j)lanet;  the  sum  (rejecting  20  in  tiie  index)  will 
be  the  logarithm  of  a  natural  Jiumbcr,  (Table  XXVI.)  which,  being  subtracted  from 
the  natural  cosine  (Table  XXIV.)  of  tlie  sum  of  the  declination  ancl  latitude  when  of 
dilierent  names,  or  the  natural  cosine  of  tlieir  difference  when  of  the  same  name,  will 
leave  the  natural  sine  of  the  planet's  true  altitude ;  to  which  add  the  correction  of 
altitude  for  parallax  and  refraction,  and  we  shall  get  the  a])parent  altitude ;  observing 
that  this  correction  is  found  in  Table  XVII.,  in  the  page  corresponding  to  the 
horizontal  parallax  of  the  planet;  the  difference  between  the  tabular  number  and 
GO  being  the  correction  of  the  planet's  altitude  for  refraction  and  parallax. 


EXAMPLE. 

What  was  the  planet  Jui)iter's  apparent  altitude,  April  29,  183G,  sea  account,  at 
1^  55'"  52^  P.  AI.,  in  latitude  42°  34'  S.,  longitude  65°  7'  30''  W.  fronr  Greenwich .' 

April  29,  sea  account,  is  by  astrononfical  account April  28^    7*"  55™  52' 

Longitude  G5°  07'  30"  W.,  in  time 4    20    30 

Apparent  time  at  Greenwich April  28  12    16    22 

©'s  right  ascension,|:April  28'  12''  IG™  22'  by  Nautical  Almanac    2    25    11 

Apparent  time  at  the  ship 7    55    52 

Right  ascension  of  the  meridian 10    21    03 

J/'s  right  ascension,  in  time 6    47    08 

^'s  distance  from  the  meridian 3    33    55 


Corresponding  to  which,  in  the  colinnn  of  log.  rising,  is 4.G0733 

Latitude 42°  34'  S Cosine        9.86717 

Declination  23   IG  N Cosine        9.96316 

Natural  number  27395     Log...        4.43766 
Sum 65   50        Natural  cosine     40939 

^'s  true  altitude 7°  47'       Natural  sine  . . .  13544 

Correction  Taiile  XVII.  add  7  § 

J^'s  apparent  altitude 7  54 

*  TIr.s  is  more  easily  obtained  by  a  cliroiiometer  ref;uhited  to  Greenwich  time,  as  in  tlic  ]irerci!ing 
example  of  finding  llie  altitude  of  the  moon. 

t  When  the  distance  exceeds  12  hours,  you  must  enter  Table  XXIII.  with  the  dilTerence  betweou 
ihat  distance  and  24  liours. 

X  The  sun's  right  ascension  at  noon,  April  28,  is  2i>  23  ">  IS^and  the  horary  motion  9^484,  which, 
for  12h  IG-n  2i^  givcs,by  Table  XXXI.,  1 16"  =:  1'  60"  nearly ;  adding  this  to^  23">  15',  we  get  the 
©'s  right  ascension  2h  25'"  \\».  The  planet's  right  ascension  and  declination  are  found  by  inspection  in 
'he  Nautical  Almanac. 

^  This  correction  is  found  in  page  89,  Jupiter's  parallax  being  only  1".5.  The  tabular  correction 
corresponding  to  the  apparent  altitude  7°  64'  is  53'  26"  ;  subtracting  this  from  CO',  we  get  d'  31".  or 
nearly  7',  for  the  correction  arising  from  the  refraction  and  parallax 


250  LUNAR   OBSERVATIONS. 


The  apparent  time,  the   Iqfitude   and   longitude,  given,  to  find  the  apparent 
altitude  of  a  fixed  star. 

RULE. 

Turn  the  longitude  into  time,  and  add  it  to,  or  subtract  it  from,  the  apparent  time  * 
at  the  ship,  according  as  tlie  longitude  is  west  or  east ;  the  sum  or  difference  Avill  be 
the  time  at  Greenwich.  The  apparent  time  at  Greenwich  may  also  be  found  by 
means  of  a  chronometer,  as  in  the  preceding  example,  page  248. 

Find,  in  the  Nautical  Almanac,  the  sun's  right  ascension  for  the  noon  preceding 
the  time  at  Greenwich,  and  add  thereto  the  correction  corresponding  to  the  hours 
and  minutes  of  the  time  at  Greenwich,  (using  Tables  XXX.  XXXI.  if  necessary  ;) 
the  sum  will  be  the  sun's  right  ascension,  which  being  added  to  the  apparent  time  at 
the  ship,  will  give  the  right  ascension  of  the  meiidian,  rejecting  24  hours  when  the  sum 
exceeds  24  hours. 

Find  the  star's  right  ascension  and  declination  in  the  Nautical  Almanac,  or  by 
means  of  Table  VJII.,  as  taught  in  page  217. 

The  difference  between  the  star's  right  ascension  and  the  right  ascension  of  the 
meridian,  will  be  the  distance  of  the  star  from  the  meridian. 

Find  in  the  cokmni  of  rising  of  Table  XXIIL  the  logarithm  corresponding  to  the 
star's  distance  from  the  meridian,!  and  add  thereto  tlie  log.  cosine  of  the  latitude  of 
the  ship,  and  the  log.  cosine  of  the  declination  of  the  star;  the  sum  (rejecting  20  in  the 
index)  will  be  the  logarithm  of  a  natural  number,  (Table  XXVI.)  which  being 
subtracted  from  the  natural  cosine  (Table  XXIV.)  of  the  sum  of  the  declination  and 
latitude  when  of  different  names,  or  the  natural  cosine  of  their  difference  when  of  the 
same  name,  will  leave  the  natural  sine  of  the  star's  true  altitude. 

The  refraction  being  added  to  the  true  altitude,  will  give  the  apparent  altitude. 

EXAMPLE. 

What  was  the  apparent  altitude  of  Aldebaran,  at  Philadelphia,  April  12,  1836,  sea 
account,  at  5''  57'"  18'  in  the  afternoon,  apparent  time  ? 

The  star's  right  ascension  and  declination  are  found  by  inspection  in  the  Nautical 
Almanac,  as  below ;  this  being  the  shortest  and  most  accurate  method  of  finding 
them. 

App.  time  by  astronomical  account,  April  11''  5''57'"  18' 
Longitude  75^  9'  W 5     0   36 

Time  at  Greenwich April  11   10   57   54 


©'s  right  ascension, April  11,  at  noon,bvN.A.     1    19   54 
Variation  for  10"  57'"  54^  by  Table  XXXI.  1    41 

0's  right  ascension  at  the  time  of  observation     1    21    35 
Jlpparcnt  time  of  observation 5   57    18 

Right  ascension  of  the  meridian 7    18    53 

H<'s  right  ascension  by  Nautical  Almanac. .     4    26    30 

:^^'s  distance  from  the  meridian  f 2   52   23     Its  log.  in  col.  rising  4.43102 

Latitude  of  Philadelphia . .  39°  57'  N 77 Cosine  9.88457 

#'s  declination 16   10  N Cosine  9.98248 

Natural  number  19864 Its  log.  4.29807 

Difference 23   47         Natural  cosine  .  91508  " 


True  altitude 45  46         Natural  sine  .. .  71644 

Refraction add  1  


Apparent  altitude 45  47 


*  The  apparent  time  must  be  taken  (as  usual)  one  day  less  than  the  sea  account,  and  the  hour  must 
be  reckoned  from  noon  to  noon  in  numerical  succession  from  1  to  2-1.  It  may  also  be  observed  that,  if 
tlie  observer  be  fiiriiislicd  willi  a  chronometer,  regulated  to  mean  Greenwich  time,  this  part  of  the 
operation  ma}'  be  saved,  reducing  the  mean  time  to  apparent,  by  applying  the  equation  Table  IV.  A. ^ 
or  that  found  in  the  Nautical  Almanac,  as  in  the  preceding  rules. 

t  If  die  distance  from  the  meridian  exceed  ]2  hours,  vou  must  subtract  it  from  21  hours,  before 
entering  Table  XXIII. 


LUNAR  OBSERVATIONS. 


251 


Method  of  combining  several  lunar  observations  together. 

As  a  lunar  observation  is  liable  to  some  degree  of  uncertainty,  on  account  of  the 
imperfections  of  the  instruments,  the  unavoidable  errors  of  the  observations,  and  the 
imperfections  in  the  reductions,  it  will  generally  be  conducive  to  accuracy  to  combine 
together  several  observations,  taken  on  the  same  day,  or  on  two  or  three  successive 
days;  and  this  may  be  done  in  the  following  manner: — 

After  working  the  lunar  observation,  and  finding  the  mean  time  of  the  observation 
on  the  meridian  of  Greenwich,  by  either  of  the  ])ieceding  methods,  we  must  compare 
this  time  with  the  corresponding  time  of  observation,  as  shown  by  the  chronometer, 
and  the  difference  will  be  the  error  of  the  chronometer  for  mean  time  at  Greenwich, 
as  shown  by  that  lunar  observation.  Other  observations,  being  taken  on  the  same,  or 
on  successive  days,  and  computed  in  the  same  manner,  will  also  give  the  errors  of 
the  chronometer,  corresponding  to  these  observations  respectively.  The  mean  of  all 
these  errors,  being  found,  will  represent  very  nearly  the  error  of  the  chronometer, 
relative  to  the  mean  time  at  Greenwich,  and  corres])onding  to  that  moment  of  time 
which  residts  from  taking  the  mean  of  all  the  times  of  observation  at  Greenwich,  for 
all  the  lunar  observations. 

Having  obtained  in  this  way  the  error  of  the  chronometer  relative  to  Greenwich 
time,  and  knowing  its  daily  rate  of  loss  or  gain,  we  can  determine  at  any  moment 
the  mean  time  at  Greenwich,  by  the  chronometer,  as  it  is  given  by  the  mean  of  all 
these  observations.  Comparing  this  mean  time  widi  the  corresponding  mean  time  at 
the  same  moment  at  the  ship,  as  found  by  taking  the  sim's  altitude,  or  by  any  other 
of  the  methods  explained  in  pages  208 — 218,  the  difference  will  be  the  longitude  of 
the  ship,  resulting  fron^^the  mean  of  all  these  observations. 


Tunes  bij  the  Chronometer. 
April  6 


10" 

30 

40 

20 

16 


6  01 

20 

Simi 6)23   58 

54 

Mean,  April  6'    3'>  59" 

>49s 

EXAMPLE    I. 

Mean  Times  at  Greenwich 
by  Lunar  Observations. 

April  6*1  2"  12™  20- 

2  32  38 

3  42  05 

4  22  25 

5  18  34 

6  03  16 

6)24  11  18 


Errors  of  the  Chronometer 
for  (Jreenwich  Time. 


2" 

00' 

2 

20 

1 

40 

2 

10 

2 

18 

1 

56 

12 

24 

2" 

04= 

April  6''     4"  01  "'53= 

Hence  it  appears,  that,  by  the  mean  of  the  six  lunar  observations,  when  tlie  time  by 
tlie  chronometer  was,  April  6',  3''  59"'  49%  it  was  2™  04=  too  slow  for  mean  time  at 
Greenwich. 

We  shall  now  suppose,  that,  on  April  6'  4''  30™  00%  by  the  chronometer,  an 
altitude  of  the  sun  was  taken,  and  the  mean  time  at  the  ship  deduced  therefrom, 
April  6' 6'' 24'"  56%  and  that  it  v.-as  required  to  find  the  longitude  of  the  ship  ;  the 
chronometer  moving  uniformly  without  gain  or  loss;  we  shall  have 


30"™  00= 
2    04 


Time  by  the  chronometer April  6'' 

Error  of  the  chronometer  by  the  lunar  observations add 

Mean  time  at  Greenwich April  6 

Rlean  time  at  the  ship April  6 

Longitude  east  of  Greenwich 1   52    52  =  28^  13 


4   32 
6   24 


04 

56 


Times  by  the  Chronometer. 

July  61  3^  IS™  06= 
7    4   16    15 


3) 
July 

8 

5 

17 

12 

21 

12 

48 

33 

ean 

7 

4' 

16" 

'11^ 

EXAMPLE  II. 

Mean  Times  at  Greeiiwich  by 
Lunar  Obseri-atioiis. 

July  6-1  3"  17™  16= 
7    4   18    23 


8     5   19 

24 

3)21  12  55 

03 

July  7'  4"  IS" 

'21 

Errors  of  the  Chronometer 
for  Greenwich  Time. 

2™  10= 
2  08 
2    12 


3)6    30 


2"' 10= 


252  TO   FIND   THE   LONGITUDE   BY    ECLIPSES 

The  mean  of  these  three  observations  makes  the  chronometer  too  slow  for  Grcen- 
wicli  time  2'"  10' ;  and  if  we  suppose  the  instrument  to  be  well  regulated  for  mean 
time,  and  on  July  8'^'  4''  10™  15'  by  the  chronometer,  the  mean  time  at  the  ship 
deduced  from  the  sun's  altitude,  was  July  8^  2''  15™  25%  we  shall  have, 

Time  by  chronometer July  8 '  4'^  10"  15' 

Error  by  the  lunar  observations add  2    10 

Mean  time  at  Greenwich July  8    4   12    25 

Mean  time  at  the  ship July  8    2   15    25 

Longitude  west  of  G-reenwich 1   57    00  =:  20^  15' 

This  process  may  be  used  for  regulating  a  chronometer  when  it  has  accidentally 
stopped,  or  h;is  been  allowed  to  run  down.  For,  by  comparing  the  two  above 
examples,  suj)i)osing  them  to  have  been  taken  by  the  same  chronometer, 

The  first  set  gives  the  error  April  6'  3'"  59™  49'  equal  to  -|-  2™  04' 
The  second  set  gives  the  error  July  7   4    IG    11    equal  to -j- 2    10 

Gain  in  92  days -j-  G= 

This  is,  however,  an  imperfect  method  of  determining  the  daily  gain  or  loss  of  the 
chronometer,  on  account  of  the  imperfection  of  the  observations ;  and  is  only  to  be 
used  in  cases  of  absolute  need. 


Tojind  tlic  longitude  hy  the  eclipses  of  Jupiter's  satellites. 

The  eclipses  of  the  satellites  are  given  in  the  Nautical  AlnAnac  for  mean  time  at 
Greenwich,  and  also  for  sideral  time.  There  are  two  kinds  of  these  eclipses — an 
immersion,  denoting  the  instant  of  the  disappearance  of  the  satellite  by  entering  into 
the  shadow  of  Jupiter,  and  an  emersion,  or  the  histant  of  the  appearance  of  the  satellite 
in  coming  from  the  shadow.  The  immersions  and  emersions  generally  happen  when 
the  satellite  is  at  some  distance  from  the  body  of  Jupiter,  excejit.  near  the  opposition 
of  Jupiter  to  the  sun,  when  the  satellite  approaches  to  his  body.  Before  the  opposi- 
tion, they  liai)pen  on  the  west  side  of  Jupiter,  and  after  the  oi)position,  on  the  east 
side.  But  if  an  astronomical  telescope  is  used,  which  reverses  the  objects,  the  appear- 
ance will  be  directly  the  contrary.  The  configurations,  or  the  positions  in  which 
Jupiter's  satellites  appear  at  Greenwich,  are  given,  in  the  Nautical  Almanac,  every 
night,  when  visible. 

As  these  eclipses  hapi)en  almost  dail}',  they  afford  the  most  ready  means  of  deter- 
mining the  longitude  of  jjiaces  on  land,  and  might  also  be  ap|)]ied  at  sea,  if  the  obser- 
vations could  be  taken  with  sufficient  accuracy  in  a  ship  under  sail,  which  can  hardly 
be  done,  since  the  least  motion  of  a  telescope  which  magnifies  sufficiently  to  make 
these  observations,  would  throw  the  object  out  of  the  field  of  view. 

Having  regulated  your  chronometer  for  mean  time  at  the  jilace  of  observation,  you 
must  then  find  nearly  the  mean  time  at  which  the  eclipse  will  begin  at  that  place ; 
this  may  be  done  as  follows  : — Find  from  the  Nautical  Almanac  the  77iean  time  of  an 
immersion,  or  emersion,  and  apply  thereto  the  longitude  turned  into  time,  by  adding 
when  in  east,  Init  subtracting  when  in  west  longitude;  the  sum  or  difference  will  be 
nearly  the  7ncff?i  time  when  the  eclipse  is  to  be  observed  at  the  given  place.  If  there 
be  any  uncertainty  in  the  longitude  of  the  place  of  observation,  you  must  begin  to 
look  out  for  the  eclipse  at  an  earlier  period  ;  and  when  the  eclipse  begins,  you  must 
note  the  time  by  the  chronometer,  and  after  ap])lying  the  correction  for  the  error  of 
the  chronometer,  if  there  be  any,  you  will  have  the  mean  time  of  the  eclipse  at  the 
place  of  observation  ;  the  difference  between  this  and  the  mean  time  in  the  Nautical 
Almanac,  being  turned  into  degrees,  will  be  the  longitude  from  Greenwich. 

EXAMPLE. 

Supj)Ose  that,  on  the  21st  of  August,  183G,  sea  account,  in  the  longitude  of 
127°  5.5'  W.,  I)y  account,  an  innuersion  of  the  first  satellite  of  Jui)iter  was  observed, 
at  10''  24™  47'  P.  M.  mean  time.     Required  the  longitude. 

By  Nautical  Almanac,  the  time  of  innnersion  is,  . .    August  20th  19''    0™    7' 

•By  observation,  August  21,  sea  account,  or  by  N  A August  20th  10   24   47 

Longitude  in  time °   "^5  20 

which,  being  turned  into  degrees,  gives  128°  50'  W.  for  the  longitude  of  the  place  of 
observation. 


TO   FIND   THE   LONGITUDE   BY   A   CHRONOMETER.  253 


To  find  the  longitude  by  an  eclipse  of  the  moon. 

The  determination  of  the  longitude  by  an  eclipse  of  the  moon,  is  performed  by 
comparing  the  times  of  the  beginning  or  ending  of  tlie  eclipse,  as  also  the  times 
when  any  number  of  digits  are  eclipsed,  or  when  the  earth's  shadow  begins  to  touch 
or  leave  any  remarkable  spot  in  the  moon's  face ;  the  dift'erence  of  these  times 
Ijetween  the  like  observations  made  at  different  places,  turned  into  degrees,  will  be 
the  difference  of  longitude  of  those  places. 

When  the  beginning  or  end  of  an  eclijise  of  the  moon  is  observed  at  any  place,  the 
longitude  of  that  place  may  be  easily  found  by  comparing  the  time  of  observation 
with  the  time  given  in  the  Nautical  Almanac  ;  for  the  difference  between  the  observed 
mean  time  of  beginning  or  ending,  and  the  mean  time  given  in  the  Nautical  Almanac, 
will  be  the  shi])'s  longitude  in  time,  which  may  be  turned  into  degrees  by  Table  XXI. 
Thus,  if  the  beginning  of  an  eclipse  of  the  moon  was  observed  October  25,  1836,  sea 
account,  at  5''  21  "\  mean  time  ;  the  mean  time  at  Greenwich  by  the  Nautical  Almanac 
being  October  24,  or  October  25,  sea  account,  at  0''  38'",  their  difference,  4''  43'",  is  the 
longitude  of  the  place  of  observation  =:  70°  45',  which  is  east  from  Greenwich, 
because  the  time  at  the  place  of  oijservation  is  greatest. 


To  find  the  longitude  by  a  perfect  time-keeper  or  chronometer. 

■  It  was  before  observed,  that  if  a  chronometer  could  be  made  in  so  perfect  a  man- 
ner as  to  move  imifbrmly  in  all  places,  and  at  all  seasons,  the  longitude  might  easily 
be  deduced  therefrom,  by  comj»aring  the  mean  time  shown  by  the  chronometer, 
regulated  to  the  meridian  of  Greenwich,  (or  some  other  known  meridian,)  with  the 
mean  time  at  the  place  of  observation ;  for  tlie  difference  of  these  times  would  be 
the  difference  of  longitude  between  that  meridian  and  the  \Aacc  of  observation.  The 
moderate  prices  of  good  chronometers  now,  in  comparison  with  their  values  many 
years  since,  together  with  the  various  im])rovements  in  their  construction,  have 
caused  this  method  of  determining  the  longitude  to  be  very  much  used  within  a  few 
years ;  we  shall  therefore  explain  fully  the  use  of  this  instrument,  the  methods  of 
regulating  and  ascertaining  its  rate  of  going,  and  give  examples  of  the  calculations 
for  finding  the  longitude. 

If  a  chronometer  is  to  be  used  on  a  voyage,  it  must  be  adjusted,  and  its  rate  of 
going  ascertained,  before  sailing.  This  is  most  conveniently  done  on  shore  by  observ- 
ing, with  a  transit  instnnnent,  the  times  of  the  transits  of  the  sim,  or  some  lixed  star, 
over  the  meridian,  as  is  taught  in  pages  221 — 224.  If  you  have  no  instrun)enl  of  this 
kind,  the  regulation  may  be  made  by  taking  altitudes*of  the  sun  or  some  other 
heavenly  body,  and  finding  therefrom  the  mean  time  of  observation,  by  any  of  the 
methods  before  given  in  pages  208 — 218.  The  best  way  of  making  these  last  obser- 
vations on  land,  is  by  an  artificial  horizon  of  quicksilver;  finding  and  correcting  the 
altitudes  in  exactly  the  same  way  as  in  computing  the  latitude  in  page  204.  Comjiaring 
the  mean  time  of  observation,  obtained  in  this  way,  with  the  time  by  the  chronometer, 
shows  how  much  it  is  then  too  fast  or  too  slow  for  the  meridian  of  the  pla-ce  of  obser- 
vation ;  and  by  repeating  the  ojieration  on  a  future  d.ay,  the  rate  of  going  may  be 
ascertained.  If  it  is-  found  to  gain  or  lose  a  few  seconds,  or  parts  of  a  second,  per  day, 
that  allowance  must  be  made  on  all  future  observations  at  sea.  Thus  if,  on  the  1st  of 
June,  1836,  at  5''  10'"  20%  by  the  chronometer,  the  mean  time,  deduced  from  an 
observation  of  the  sun's  altitude,  was  5''  12'"  40%  the  chronometer  would  then  be  too 
slow  by  the  tiifference  of  those  times,  2'"  20';  and  if,  on  the  21st  of  June  following, 
the  time  by  the  chronometer  was  4''  15"  35%  when  the  mean  time  was  4''  18'"  17%  the 
chronometer  would  then  be  too  slov/  by  the  difference  of  those  times,  or  2'"  42'';  and 
the  rate  would  have  varied,  in  20  days,  from  2"'  20%  to  2"'  42%  which  is  a  difference 
of  22'  in  20  daj's,  being  IM  per  day;  and  this  rate  must  be  allowed  on  all  futm-e 
observations  at  sea,  until  a  new  regulation  can  be  obtained,  at  some  place  whose 
longitude  is  known.  It  is  best  to  have  a  considerable  number  of  days'  interval  between 
the  two  observations  for  fixing  the  rate,  since  by  this  means  it  may  be  determined  to 
tenths  of  a  second  ;  the  absolute  error  of  the  observations  being  reduced,  in  finding 
the  daily  rate,  by  dividing  by  the  number  of  days.  Thus,  if  the  above  difference  of 
22'  had  been  erroneous  2%  and  the  true  value  20%  the  daily  rate  would  be  one 
second,  instead  of  I'.l,  varying  only  one  tenth  of  a  second,  notwithstanding  the 
observations  on  which  the  rate  was  established  contained  an  error  of  two  seconds. 

Having  regulated  a  chronometer,  in  the  manner  first  mentioned,  at  a  place  whose 
'ongitude  from  Greenwich  is  known,  it  is  easy  to  find  how  much  it  is  too  fltst  or  too 

*  See  Tab.  LVII. 


254  TO  FIND   THE   LONGITUDE   BY   A   CHRONOMETER. 

slow  for  the  meridian  of  Greenwicli,  by  reducing  the  mean  time  at  the  i)lace  of 
the  observer,  as  found  by  observations,  to  the  meridian  of  Greenwich,  by  adding  the 
longitude  if  west,  subtracting  if  east ;  the  sum  or  difference  will  be  the  mean  time  of 
observation  in  the  meridian  of  Greenwich  ;  the  difference  between  this  and  the  time 
given  by  the  chronometer,  shows  how  much  it  is  too  fast  or  too  slow  for  Greenwich 
meaii  time.  Thus,  by  adding  the  longitude,  which  we  shall  suppose  to  be  4''  5<3"',  to 
the  mean  time  of  the  above  observation,  5''  12'"  40%  we  get  10''  8"^  40'  for  the  mean 
time  at  Greenwich  ;  from  which  subtracting  the  time  by  the  chronometer,  5''  10™  20% 
we  obtain  4"  58"^  20'  for  the  error  of  the  chronometer  relative  to  mean  time  at  Green- 
wich ;  being  too  sloiv  for  that  time.      * 

The  chronometer  having  been  thus  regulated  to  Greenwich  time,  and  the  daily  rate 
of  its  going  ascertained,  if  this  rate  should  remain  unaltered,  the  time  at  Greenwich 
will  be  known  by  it,  at  any  UToment  at  sea ;  and  if  at  tluit  moment,  by  any  observation 
of  the  sun,  moon,  planet,  or  a  fixed  star,  the  mea7i  time  at  the  ship  be  found  by  any  of 
the  methods  explained  in  pages  208,  &c.,  the  difference  between  this  7nea?i  time  at 
the  ship,  and  the  mean  time  at  Greenwich,  shown  by  the  chronometer,  will  be  the 
longitude,  which  may  be  turned  into  degi'ees  and  minutes  by  Table  XXI. 

EXAMPLE   I. 

Wishing  to  regulate  a  cliroiiometer,  in  a  place  whose  latitude  is  51°  30'  N.,  and 
longitude  130°  E.  from  Greenwich,  I  observed,  October  10,  1848,  at  8^  21"'  A.  31.,  sea 
account,  by  a  cln-onometer,  the  altitude  of  the  sun's  lower  limb,  by  a  fore  observation, 
13°  32',  the  correction  for  semidiameter,  parallax,  and  dip,  being  12'.  It  is  required 
to  find  the  error  of  the  chronometer  for  mean  time  at  Greenwich. 

The  mean  time  of  this  observation,  at  the  meridian  of  the  ship,  computed  as  in 
Example  I.,  jiage  209,  is  7^  54'^  18=  A.  M.,  or  October  9'  19"  54'"  18%  astronomical 
account.  From  this  subtract  *  the  longitude  130°,  turned  into  time  8''  40"^,  (by 
Table  XXL)  we  get  the  corresponding  mean  time  at  Greenwich,  Oct.  9%  11''  14"'  18"; 
and  as  the  time  by  the  chronometer  is,  October  9'',  20''  21'"  00%  it  is  too  fast  for  mean 
time  at  Greenwicli  by  the  difference  of  those  two  quantities,  or  9^  6'"  42'. 

EXAMPLE   II. 

May  10,  183G,  at  5^  30""  P.  M.,  sea  account,  by  a  chronometer,  in  latitude  39°  54'  N,, 
in  a  place  whose  longitude  was  known  to  be  35°  45'  E.  from  Greenwich,  the  altitude 
of  the  sun's  lower  limb  by  a  fore  observation  was  15°  45',  the  correction  for  dip, 
pai-allax,  and  semidiameter,  being  12'.  It  is  required  to  find  the  error  of  the  chro- 
nometer for  mean  time  at  Greenwich. 

The  mean  time  of  this  observation,  computed  as  in  Example  II.,  page  210,  is 
May  9>  5''  30™  39%  astronomical  computation.  From  this  subtract  *  the  longitude, 
35°  45',  turned  into  time,  2"  23'",  bv  Table  XXL;  the  remainder.  May  9'  3"  7"'  39', 
is  the  mean  time  at  Greenwich.  The  difference  between  this  and  the  time  by  the 
chronometer,  5''  30'",  is  2''  22'"  21%  which  expresses  how  much  the  chronometer  is 
too  fast  for  Greenwich  mean  time. 


EXAMPLE    III. 

Suppose  tliat,  on  July  27,  1836,  sea  account,  the  mean  time  was  found,  by  an 
altitude  of  the  sun,  to  be  1"  11'"  IG'  P.  M.,  when,  by  a  chronometer  well  regulated  to 
mean  time  at  Greenwich,  it  was  4"  3'"  8'  P.  M.     Reciuired  the  longitude. 

JNIean  lime  at  the  place  of  observation  P  11""  IG' 
Time  at  Greenwich  by  chronometer. .  4     3      8 

Difference  in  the  longitude 2  51    52  —  42°  58'  W.,the  longitude  being 

west,  because  the  time  at  Greenwich  is  the  greatest. 

EXAMPLE   IV. 

Suppose  that,  on  I\Iay  14,  183G,  sea  account,  the  mean  time  was  found,  by  an 
altitude  of  the  sun,  to  be  3''  59"'  09'  P.  M.,  when  the  time  by  the  chronometer  was 

*  This  is  to  be  added,  if  the  ship's  long^ilude  is  west. 


TO   FLND   THE   LOiN'GlTUDE   BY    A    CHRO>fOMETER.  255 

2''  P,  M.,  tlie  chronometer  being  too  slow  for  mean  Gi'ecnwich  time  11'"  9'.    Rciinired 

the  longitude. 

Time  by  chronometer 2^    0""  00" 

Chronometer  too  slow  for  jnea?i  lime  at  Greenwich 11     9 

J\Iea7i  time  at  Greenwich 2   11    09  P.  M. 

Blean  time  at  the  ship 3   59    09 

Difference  is  the  longitude 1   48   00  =  27=  00'  E. 


EXAMPLE  V. 

Suppose  that,  on  June  14,  183G,  sea  account,  in  a  place  whose  longitude  from 
Greenwich  was  known,  a  number  of  observations  were  taken  to  ascertain  the  going 
of  the  chronometer  ;  and  it  was  found,  that,  on  that  day,  it  was  10'  too  slow  for  mean 
Greenwich  time,  and  lost  time  2^  per  day  ;  and  that,  on  July  14, 1836,  sea  account,  the 
time  per  chronometer  was  G""  0™  G'  P.  M.,  when,  by  an  observed  altitude  of  the  sun, 
the  viean  time  was  1''  21'"  32'  P.  31.     Required  the  longitude. 

Error  of  chronometer,  June  14 0''  00™  10'  slow. 

30  days,  at  2' 1      0  slow. 

Error  July  14 1    10  slow. 

Time  per  chronometer G     0      G 

Time  at  Greenwich G     1     IG 

Me-an  time  at  place  of  observation • 1    21    32 

Longitude 4  39    44  =  G9°  5G'  W. 


EXAMPLE   VI. 

Sui)pose  that,  on  June  15,  1836,  in  tlie  afternoon,  astronomical  account,  at  Boston, 
in  the  latitude  of  42°  21'  15"  N.,  and  longitude  71°  04'  09"  VV.,  several  angular  dis- 
tances of  the  sun's  lower  limb,  from  its  reflected  image  in  a  basin  of  quicksilver,  were 
observed,  and  the  times  noted  by  a  chronometer,  which  was  supposed  to  be  very 
nearly  regulat'jd  for  mean  time  at  Greenwich  ;  the  times  and  altitudes  being  as  below  ; 
the  thermometer  standing  at  7G°,  and  the  barometer  at  30°.05.  Required  the  error 
of  the  chronometer  I'elative  to  mean  time  at  Greenwich. 

Times   by  the   chronometer,  June  15' 7'' 55'" 20' Observed  angle  91°  16' 20" 

56  12 .  DO  57  40 

57  01  90  39  48 

57  46 90  22  54 

58  36 90  04  36 

59  24 89  46  42 

Sum  .  .  ..-44    19  Sum. . .  .6)543  08  00 

Mean  of  the  times June  15'  7''  57"  23'.2 Mean  angle   90  31  20 

Half  the  mean  angle  is  equal  to  altitude  ©'s  lower  limb 45  15   IQ 

Refraction,  Table  XFL— 57"— Parallax,  Table  XIV.+  6"=— 51" 

Table  XXX VL,  Thermometer  76°,  correction  — 3"  (      ^ 

Barometer  30°.05,  correction -j- 1"  ^      — 

Correction  for  refraction  and  parallax 53 sub.  53 

45  14  47 

0's  semidiameter  *  by  Nautical  Almanac         15  46 

©'s  true  altitude 45  30  33 

*  111  fiiuling  the  suii's  declir.ation,  semidiamcter,  &lc..  from  the  Nautical  Almnnaf ,  llie  time  at 
Grceiiwicii  is  supiiosed  to  be  tlie  same  as  the  mean  time  of  the  observation  by  the  chronometer 
7ti  57m  238.2,  whivh  is  supnosed  to  be  very  nearly  regulated  to  mean  time  at  Greenwich.  If  you  have 
no  chronometer  reg-ujated  for  that  meridian,  j^bu  must  estimate  the  time  at  Greenwich  in'tlic  usual 
manner,  by  jidding  to  the  mean  lime  at  the  ship  the  longitude  if  west,  or  subtracting  it  if  east 


256  TO   FIND   THE   LONGITUDE   BY  A    CHRONOMETER. 

(v)'s  true  altitude  45°  30'  33" 

Latitude 42  21  15    Secant..  0.13136 

Polar  distance. .  66  38  49     Cosecant  0.03712 

Sum 2 )  1.54  30  37 

Half-sum 77  15  19    Cosine..  9.34362 

Remainder  . . . .  31  44  46    Sine ....   9.72111 

Sum   2 )  19.23321 

Sine  of  half-sum  9.61660  corresponds  to  3''  15"27'.5  ajiip.time. 

Equation  of  time  by  the  Nautical  Almanac -|-  9'.4 

Rlean  time  at  the  place  of  observation 3   15    36'.9 

Add  the  longitude  of  Boston,  in  time 4  44    16'.6 

3Iean  time  at  Greenwich 7  59    53'.5 

Time  by  the  chronometer,  as  above 7  57    23'.2 

Chronometer,  error slow         2""  30^3 

Hence  it  appears  that,  on  the  15th  of  June,  1836,  astronomical  time,  at  7'' 57™  23'.2, 
by  the  chronometer,  it  was  too  slow  for  Greenwich  time  2'"  30'.3.  Suppose,  now, 
that,  a  few  days  afterwards,  as  an  example,  on  June  25,  at  about  the  same  hour  in  the 
afternoon,  a  similar  set  of  altitudes  were  observed,  and  the  times  noted  by  the  same 
chronometer,  the  result  of  the  calculation  making  the  chronometer  too  slow  by 
2"'  45'.6 ;  then  we  shall  find  that,  in  the  interval  of  10  days,  from  June  15  to  June  25, 
it  has  varied  by  the  quantity  2'"  45'.6 — 2™  30'.3  =  15'.3.  Dividing  this  variation  by 
10,  (the  nun)!)er  of  days  in  the  interval,)  we  get  1^53  for  the  daily  rate  of  loss  in 
the  chi-ononieter.  If  other  sets  of  observations  are  made,  which  give  results  differing 
a  little  from  P.53,  we  can  use  the  mean  of  the  different  sets,  as  the  most  probable 
value  of  tije  rate  of  the  chronometer. 

EXAMPLE   VII. 

On  the  15th  of  Jime,  1830,  astronomical  account,  at  about  S""  45™  P.  ]M.,  in  the 
meridian  of  Cape  Cod,  which  bore  south,  distant  about  9  miles,  took  four  altitudes  of 
the  sun,  and  noted  the  times  by  the  chronometer,  as  in  the  table  below ;  the  eye  being 
19  feet  above  the  level  of  the  sea,  the  thermometer  at  65°,  and  the  barometer  at  29 
inches.  It  is  required  to  determine  the  error  of  the  chronometer  for  mean  time  at 
Greenwich. 

Times  by  the  chronometer 8^^  25™  36^ Observed  angle  40°  00'  07" 

26  32 39  50  18 

27  22 3940  14 


Sum  3)     79    30  Sum  3)119  30  39 

Mean  of  the  tlu-ce  observations. . .  8  26    30      ©'s  altitude 39  50  13 

Su{)poscd  error  of  the  chronome- >    i     i     on      t^-     -n  i  i    vtti  i  4   i-r 

t,'  r  .■  , /-I  •  ,   >  +  1    30      Dip,  Table XIll sub.  4  17 

ter  for  mean  tune  at  Greenwich  ^    '  ■ ' 

Estimated  mean  time  at  Greenwich  8  28    00  39  45  56 

Refraction,  Ta!)le  XII —V  8" 

Parallax,  Talile  XIV 4-7" 

Table  XXXVL  Thermometer —2" 

Barometer — 1" sub.  1  04 

39  44  52 
©'s  sen^.idiamcter 15  46 

©'s  true  altitude 40  00  38 

With  the  above  estiniated  time  at  Greenwich,  we  find,  from  the  Nautical  Almanac, 
the  sun's  declination  23°  21'  14''  N.,  the  sim's  semidiainetor  15' 46'',  and  the  equation 
of  time  -|-  9\7.  The  latitude  of  Cape  Cod  being  42°  3'  N.,  and  as  it  is  distant  9',  in  a 
south  direction,  the  latitude  of  the  ship  is*  42°  12'  N. 

*  The  ship  being'  on  the  meridian,  we  must  adil  the  whole  distance  9'  to  the  latitude  of  Cape  Cod, 
to  get  tlie  latitude  of  tiic  ship;  but  if  the  bearing  be  in  any  other  direction,  we  must  calculate  by  mesui'j 


TO  FUND   THE   LONGITUDE   BY   A    CHRONOMETER.  257 

(v)'s  true  altitude  40"  00' 38" 

Latitude 42  12  00  Secant. .  O.L^030 

Polar  distauce .  ■  GG  38  46  Cosecant  0.03712 

Sum 2  )  148  51  24 

[Lalf-sum 74  25  42  Cosine . .  9.42885 

Ileanaiuder 34  25  04  Sine ....  9.75222 


Sum   2)19.34849 

Sine  of  liall-sum  9.G7424  corresponds  to  3''  45'"  29'.    npp.  time 

Equation  of  time  by  tlie  Nautical  Almanac -f-  9'.7 

ftlean  time  at  the  place  of  observation 3  45    38'.7 

Add  the  longitude  of  Cape  Cod,  70°  4' —  4  40    1(^0 

Rlean  time  at  Greenwich 8  25    54°.7 

Time  by  the  chronometer 8   2G    30'.0 

Error  of  the  chronometer  fur  mean  time  at  Green wicli. . .  .fast  35".3 

EXAMPLE  VIII. 

At  New  York,  on  the  5th  of  June,  183G,  by  a  transit  of  the  sun  over  the  meridian,  it 
was  found  that  a  chronometer  was  too  fast  for  mean  time  at  Greenwicli,  by  2'"  8\5; 
and  by  another  transit,  on  the  next  day,  June  Gth,  it  was  too  fast  2'"  lO'.O.  From  these 
observations  it  follows,  that  the  daily  gain  of  the  chronometer  at  that  time  was  1'.5. 
The  instrument  was  then  taken  on  board  a  ship,  which  sailed  immLdiutc'ly  on  a 
voyage  along  tlie  seacoast,  and,  after  a  passage  of  10  days,  arrived  at  a  place  whose 
longitude  from  Greenwich  had  been  well  ascertained.  There,  by  o'jsorvation,  it  was 
(bund,  that  at  noon,  June  16,  183G,  the  chronometer  was  2'"  30'.5  too  fast  for  mean 
time  at  Greenwich  ;  having  gained  22'.0  in  11  days;  or  at  the  mean  daily  rate  of  2^0, 
instead  of  1S5,  which  was  the  rate  at  the  commencement  of  the  voyage.  Now,  the 
chronometer  being  a  new  one,  and  it  being  generally  found  that  the  daily  rate  of  such 
an  instrument  is  constantly  increasing,  it  is  required  to  find  the  error  of  the  chro- 
nometer at  noon  on  every  day  of  the  voyage,  supposing  the  daily  rate  of  gain  to 
increase  uniformly;  the  object  in  thus  finding  the  actual  error  on  each  day,  being 
for  the  purjjose  of  ascertaining  the  longitudes  of  several  capes  and  places  which  were 
observed  during  the  voyage. 

The  calculation  of  this  example  is  made  as  in  the  annexed  table.  Its  first  column 
contains  the  days  of  tlie  month.  The  second  column  contains  the  estimated  error  of 
the  chronometer,  supposing  its  daily  gain  to  be  1\5,  as  at  die  commencement  of  the 
voyage.  The  third  column  contains  the  gain  of  the  chronometer  on  every  successive 
day,  supposing  this  uniform  daily  increment  of  the  rate  to  be  a  fraction  of  a  second, 
which  is  represented  by  /.  The  fourth  column  contains  the  error  of  the  chronometer 
on  each  day,  e.\i)ressed  in  terms  of/;  the  numbers  in  this  column  are  found  by  adding 
succes.-iively  the  daily  gain  in  column  3,  to  the  error  of  the  chronometer  on  the  [)reced- 
ing  noon.  Thus,  on  June  13,  the  error  at  noon  is  2™  20'.5-|-28  t,  and  the  daily  gain 
between  June  13th  and  14th  is  1^5-|-8  t ;  adding  together  these  two  quantities,  we 
obtain  2™  22'.0-j-3G  t,  for  the  error  of  the  chronometer,  June  14,  at  noon  ;  being  the 
same  as  in  coUunn  4.  Proceeding  in  this  way,  by  successive  additions,  we  obtain  the 
error  of  the  chronometer,  June  16,  at  noon,  equal  to  2'"  25^0-|-55<,•  and  as  this  was 
found  by  observation  to  be  2'"  30'..5,  wc  shall  have  2™  25'.0-|-55<=::2"'  30'.5  ;  whence 
we  get  55  <  =  2™  30'.5  —  2"' 25'.0  =r  5'.5.  Dividing  this  by  55,  the  coefiicient  of<, 
we  get  /:^0'.l.  Hence  tlie  daily  gain  in  the  acceleration  is  izr:  O'.l ;  and  by  substi- 
tuting this  value  of  t  in  the  errors  at  noon  on  the  difterent  days,  given  in  column  4, 
we  get  the  corresponding  numbers  in  column  5,  which  represent  how  much  the 
chronometer  is  too  fast  for  mean  time  at  Greenwich  at  each  noon,  from  June  5  to 
June  16 ;  supposing  the  daily  acceleration  of  the  rate  to  be  0^1,  or  yV  of  a  second. 
Taking  the  successive  daily  differences  of  these  errors,  we  get,  as  in  colun^.n  6,  the 
daily  gain  of  the  chronometer,  which  increases  from  1^5  to2\5  during  tlie  voyage. 

of  the  table  of  difference  of  latitude  and  departure,  the  latitude  and  longitude  of  the  ship,  at  tlie  time 
of  observation,  in  the  same  manner  as  when  ta-king  a  departure  from  tlie  land.  Thus,  if  the  true  hear- 
ing of  the  cape,  in  the  above  example,  were  S.  S.  \V.  9',  tlicdiflcreiicc  of  latitude  will  hi  8'..3,  departure 
3'.4.,  dillc-roiicc  of  longitude  4'.()  ;  lience  the  latitude  of  the  ship  will  be  42°  3' +  8'. 3  =  42°  ll'.3  = 
42°  11'  18",  and  the  longitude  70°  4'  —  4'.G  =  Gy°  59'.4  =  G9°  59'  24'' ,;  which  must,  in  this  case,  be 
»6ed  instead  of  the  above  values. 

33 


258 


TO   FIND   THE    LOxXGITUDE   BY  A   CHRONOMETER 


Col.  1. 

Col.  2. 

Col.  3. 

Col.  4. 

Col.  5. 

Col.  6. 

Error    of    the 
chronometer, 

Daily  gain,  sup- 
posing   the    rate 

Error    of   the    chro- 

Error    of    ike 

jDai/)/  ™iH 

Daies. 

siipposing     the 
daily    cra'ui    tu 
be  ls.5. 

to    be    uniformly 
increasing  by  the 
quantity  t. 

day,      expressed      as 
terms  of  t. 

noon   each  day 
in  time. 

in  seconds 

June    5. 

2'"  08».5 

1«.5 

2m 

08'.5 

2™  08^5 

1^5 

"       C. 

2    10.0 

1  .5  +  « 

2 

10.0 

2    10.0 

1  .G 

"       7. 

2    11.5 

1  .5  +  2  t 

2 

11  .5  + < 

2    11  .6 

1  .7 

"       8. 

2    13.0 

1  .5  -f  3  < 

2 

13.0  +  3< 

2    13.3 

1  .8 

"      9. 

2    14.5 

1  .5  +  4  < 

2 

14  .5  +  0  « 

2    15.1 

1  .9 

"     10. 

2    IG.O 

1  .5  +  5< 

2 

IG  .0  +  10« 

2    17.0 

2.0 

"     11. 

2    17.5 

1  .5  +  6  i 

2 

17  .5  4- 15  « 

2    19.0 

2.1 

"     12. 

2    19.0 

1  .5  4-  7  < 

2 

19.0  +  2U 

2    21.1 

2.2 

"     13. 

2    20.5 

1  .5  4-  8  f 

2 

20  .5  +  23  < 

2    23.3 

2.3 

"     14. 

2    22.0 

1  .5  +  9  < 

2 

22  .0  +  30  « 

2    25.6 

2.4 

"     1.5 

2    23.5 

1  .5  +  10  « 

2 

23.5  +  45^ 

2    23.0 

2  .5 

"     1(1. 

2    25.0 

o 

25  .0  -f  55  f 

2    30  .5 

EXAMPLE   IX. 

We  shall  suppose,  as  in  the  preceding  example,  that  at  noon  June  5,  1836,  the 
chronometer  ^vas  too  fast  2™  8".5,  and  at  noon  June  C,  1836,  it  was  too  last  '2'"  lO'.O; 
indicating  a  daily  gain  of  I'.5.  In  proceeding  on  a  voyage,  the  vessel  stopped,  on 
the  10th  of  June,  1830,  at  a  port  whose  longitude  was  unknown ;  and,  witli  a  view 
to  determine  this  longitude  by  the  chronometer,  observations  were  made,  by  wiiich 
it  was  found,  that  between  the  successive  noons  of  June  10th  and  June  11th,  18.3(3, 
the  daily  gain  was  2".0.  It  is  req^uired  to  determine  the  error  of  the  chronometer  on 
the  different  days,  supposing  the  daily  gain  to  ije  uniform.  The  actual  rate  of  the 
chronometer  is  i)articularly  required  on  the  10th  and  11th  of  June,  so  that  we  may 
use  the  rate  of  the  chronometer  in  finding  the  longitude  of  the  place  arrived  at. 

In  the  intervals  of  the  two  days,  commencing  June  5  and  June  10,  the  daily  gams 
were  respectively  1".5  and  2*.0 ;  having  increased  0^.5,  in  tlie  daily  rate,  in  an  interval 
of  5  days;  being  at  the  rate  of  O-.l  per  day.  With  this  daily  increase,  we  can 
comjjute  the  daily  gain,  as  in  column  2  of  the  following  table  ;  and  from  these 
numbers  we  can  deduce  successively  the  errors  of  the  chronometer,  as  in  column  3. 


Col.  1. 

Col.  2. 

Col.  3. 

Dates. 

Daily  rate  of 

Chronometer  too 

gain. 

fast. 

June     5. 

1^5 

2'"  08«.5 

"       6. 

1  .6 

2    10.0 

"       7. 

1  .7 

2    11  .6 

"       8. 

1  .8 

2    13.3 

"       9. 

1  .9 

2    15.1 

"     10. 

2.0 

2    17.0 

"     11. 

2    19.0 

Hence  it  ai)pears,  that  on  June  10,  the  chronometer  was  2™  IV'.O  too  fast  tbr 
Greenwich  mean  time;  and  on  June  11,  it  was  2™  19'.0;  which  can  be  used  in 
determining  the  longitude. 


TO   FIND   THE   LONGITUDE   BY   A   CHRONOMETER.  259 


Precautions  in  using  a  chronometer. 

We  shall  close  this  article  on  chronometers,  by  the  following  directions  relative  ta 
the  manner  of  taking  care  and  using  them,  published  in  a  small  tract  on  this  subject, 
by  Mr.  Stansbury : — In  carrying  a  chronometer  to  and  from  a  ship,  you  must  secure 
the  gimbals  by  the  stay,  to  keep  it  steady ;  and  by  all  means  avoid  giving  the  instru- 
ment a  quick  circular  motion.  A  chronometer  should  be  placed  so  as  to  expose  it  as 
little  as  possible  to  sudden  shocks,  from  tlie  sea  striking  the  ship,  or  from  the  shutting 
of  doors,  &c.  It  ought  not  to  be  exposed  to  a  current  of  air.  Nothing  magnetic 
should  be  allowed  near  it.  When  the  chronometer  is  on  board  a  ship,  free  the  staj', 
let  the  instrument  swing  horizontally,  and  place  it  securely,  and  so  that  it  may  be  dis- 
turbed as  little  as  possible  during  the  voyage ;  using  for  deck-observations  a  common 
watch,  which  must  be  compared  with  the  chronometer  before  and  after  any  obser- 
vation. In  winding  up  a  chronometer,  turn  it  over  gently ;  jnit  the  valve  back,  a{)ply 
the  key,  turn  it  moderately,  and  avoid  sudden  jerks.  A  pocket  chronometer  must  be 
held  inunovable  in  the  one  hand,  whilst  winding  with  the  other,  in  order  to  avoid  a 
circular  motion,  which  may  not  only  alter  the  rate,  but  injure  the  instrument.  If  a 
chronometer  should  happen  to  run  down,  or  stop,  it  nnist,  when  wound  up,  have  a 
quick  circular  motion  in  the  plane  of  the  dial  to  set  it  agoing.  Never  touch  the  hands 
to  set  the  chronometer,  but  wait  till  the  time  arrives  at  which  they  point.  Be  regular 
in  winding.  Get  an  observation  as  soon  as  you  leave  a  port,  to  ascertain  if  you  have 
the  correct  difference  from  Greenwich  time ;  and  in  case  it  should  happen  to  stop,  or 
to  run  down,  during  the  passage,  it  may  be  corrected  by  lunar  observations,  by  the 
method  explained  in  pages  251,  252. 

It  has  been  found  that  chronometers  gain  by  an  increase  of  the  density  of  the 
air,  and  lose  by  a  decrease  of  the  density.  The  firing  of  guns  on  board  a  vessel  will 
sometimes  alter  the  rate  of  going,  unless  the  instriunent  be  well  suspended,  or  held 
in  the  hand  during  the  fii-ing.  Any  sudden  jar  will  sometimes  alter  the  rate.  The 
imperfection  of  the  oil  used  wiii,  after  some  time,  impair  the  instrument.  The 
mechanism  for  correcting  the  changes  in  the  temperature  may  not  do  it  com- 
pletely, and  some  error  may  arise  from  this  source.  Notwithstanding  these  various 
causes  of  error,  it  is  wonderful  to  observe  how  accurately  some  of  these  chronometers 
perform  their  office. 

The  manner  of  using  a  chronometer  in  finding  the  longitude  by  means  of  observa- 
tions of  the  moon's  transits  over  the  meridian,  with  a  transit  instrument,  will  be  given 
in  the  Appendix  to  this  work. 

On  a  variation  chart. 

In  the  year  1700,  Dr.  Ilalley  published  a  chart,  in  which  the  lines  of  the  variation 
of  the  conipass  were  drawn,  for  the  purpose  of  determining  the  longitude  by  means 
of  the  observed  variation  ;  and,  since  that  time,  several  charts  of  this  kind  have  been 
published  for  the  same  purpose  ;  but  the  method  is  not  sufficiently  accurate  to  be  of 
any  practical  use.  A  variation  chart  is,  however,  useful,  as  a  subject  of  scientific 
intiuiry,  and  for  the  purpose  of  correcting  a  ship's  course.  The  latest  and  by  far 
the  best  work  of  this  kind,  is  that  of  the  Admiralty,  published  in  185'J,  and  repub- 
lished by  E.  &,  G.  W.  Blunt,  in  1860.     Every  navigator  should  have  it  for  daily  use. 


2G0 


METHOD  OF  KEEPING 
A    SHIP'S    RECKONING    OR    JOURNAL 

AT   SEA. 


A  ship's  regkoning  is  that  account,  by  which  it  can  be  known  at  any  time  where 
the  ship  is,  and  on  what  course  or  courses  she  must  steer  to  gain  her  port.  Dead 
RECKONING  is  that  account  deduced  from  the  ship's  run  from  tlie  last  observation. 


THE    LOG-BOARD. 


H. 

K. 

F. 

Courses. 

Winds. 

Lce- 
icay. 

Transactions. 

2 

6 

S.  W. 

N.  E. 

4 

5 

5 

G 

5 

N.W.byW. 

8 

5 

Moderate    gales 

10 

4 

5 

E.  N.  E. 

N.  W. 

and  fair  weather. 

12 

4 

5 

At  8  A.  M.,  saw 

2 

4 

5 

a     ship    to    tlie 

4 

4 

5 

northward. 

G 

4 

5 

8 

5 

S.  W. 

W.  N.  W. 

1 

No  observation. 

10 

4 

5 

12 

4 

The  daily  occuiTences  on  board  a  ship  are  marked  on  a  board  or  slate,  called  the 
log-board  or  log-slate,  kept  in  the  steerage  for  that  purpose,  being  usually  divided  into 
seven  columns :  the  first  contaiss  the  hours  from  noon  to  noon,  being  marked  by 
some  for  every  two  hours,  but  usually  for  every  single  hour;  in  the  second  and  third 
columns  are  the  knots  and  fathoms  the  ship  is  found  to  run  per  hour,  set  against  the 
hours  when  the  log  was  hove.  Some  navigators  do  not  divide  the  knot  into  ten 
fathoms,  but  into  half-knots  only,  making  the  tliird  column  H.  K.  The  fourth  colimin 
contains  the  courses  steered  bj^  compass ;  the  fifth,  the  winds ;  the  sixth,  the  lee- way  ;* 
and  the  seventh,  the  alteration  of  the  sails,  the  business  done  aboard,  and  what  other 
remarks  the  oflicer  of  the  watch  thinks  proper  to  insert.  For  it  should  be  obsrrvetl, 
that  it  is  usual  to  divide  a  ship's  company  into  two  parts,  called  the  starboard  and 
larboard  watches,  who  do  the  duty  of  the  ship  for  four  hours  and  four  hours,  alter- 
nately, except  from  4  to  8  P.  M.,  which  is  divided  into  two  watches.  The  i-emarks 
made  on  the  log-board  are  daily  copied  into  a  book,  called  the  Log-Book,  which  is 
ruled  like  the  log-board.  This  book  contains  an  authentic  record  of  the  shi])'s  trans- 
actions ;  and  the  persons  who  keep  a  i-eckoning,  transcribe  them  into  thc\i- journals,  and 
thence  make  the  necessary  deductions  relative  to  the  ship's  ])lace,  every  day  at  noon  ; 
this  o])eration  is  called  working  a  dajfs  loork.  While  a  ship  is  in  port,  the  reinarlv5 
entered  in  the  Log-I>ook  m-e  c-,\\\cAharhor-u'ork,OY  harhor-journal ;  and  the  day  is  then 
estimated  according  to  the  civil  comjuUation,  as  on  shore;  that  is,  from  midnight  to 
midniglit;  but  at  sea,  the  da3's  work  ending  at  noon  is  dated  the  same  as  the  civil 
day,  so  that  the  day's  work  marked  Blonday  begins  on  Sunday  noon,  anrl  ends  on 
Monday  at  noon;  the  day  thus  marked  is  called  a  nautical  day;  the  first  12  hours 
oemg  marked  P.  M.,  the  latter  A.  M.  There  are  various  ways  of  kecjiing  journals  at 
sea,  according  to  the  different  tastes  of  navigators.  Some  keep  only  an  abstract  of 
each  day's  transactions,  specifying  the  weather,  what   sliips   or   lunds   were  seen, 

The  cause  of  the  lee-way,  and  manner  of  allowing  for  it,  are  c.xi)laiiiecl  in  llic  following-  i)age. 


METHOD   OF   KEEPING   A   JOURNAL   AT   SEA.  261 

accidents  on  board,  the  latitude,  longitude,  course,  and  run ;  these  particulars  being 
drawn  from  the  ship's  Log-Book.  Others  keep  a  full  copy  of  the  Log-13ook,  and  the 
deductions  drawn  therefrom,  arranged  in  proper  columns  ;  this  is  the  most  satisfactory 
method  to  tliose  who  may  have  occasion  to  uispect  the  Journal ;  and  we  have  adopted 
it  in  the  following,  but  shall  give  an  abstract,  at  the  end,  conformable  to  the  other 
method. 

When  a  ship  is  about  losing  sight  of  the  land,  the  bearing  of  some  noted  place 
(whose  latitude  and  longitude  are  known)  must  be  observed,  and  its  distance  estima- 
ted and  marked  on  the  Log-Book  ;  this  is  called  takiiig  a  departure.  In  working  tiiis 
first  day's  work,  the  calculation  is  to  be  made  in  the  same  manner  as  if  the  ship  had 
sailed  that  distance  from  that  j)lace  upon  a  course  opposite  to  that  bearing,  and  that 
course  and  distance  are  to  be  entered  accordingly  into  the  traverse  table,  after  allow- 
ing for  the  variation. 

To  allow  for  the  variation. 

We  have  already  taught  the  methods  of  finding  the  variation,  which  must  be 
allowed  on  all  courses  steered,  and  on  all  bearings  taken  with  the  compass;  to  the 
right  hand,  if  the  varialion  be  east,  but  to  the  left  hand,  if  ivest ;  the  observer  being 
supposed  to  be  placed  in  the  centre  of  the  comi)ass,  looking  towards  the  point  from 
which  the  variation  is  to  be  allowed. 


EXAMPLES. 

Courses  bij 

compass. 

Variation,  in  points. 

True  courses. 

N.  E.  by 

E. 

2    W. 

N.  E.  by  N. 

N.  E. 

U  E. 

N.  E.  by  E.  i  E 

N.  W. 

3    W. 

W.  by  IN'. 

S.  K 

3    E. 

S.  by  E. 

s.  s.  w. 

1^  W. 

S.  i  W. 

E.  S.  E. 

u  vv. 

E.  1  S. 

S.  W.  i  w. 

h  w. 

S.  W.  i  S. 

N.  N.  E. 

IE. 

U  E. 

N.  E.  i  E. 

To  find  the  he-icay,  and  alloio  for  it. 

The  courses  must  likewise  be  cori-ected  for  lee-way ;  the  nature  of  which  may  be 
thus  explained : — When  a  sliip  sails  upon  a  wind,  in  a  fresh  gale,  that  part  of  the  wind 
which  acts  upon  the  hull  and  rigging,  together  with  a  considerable  part  of  the  force 
exerted  on  the  sails,  tends  to  drive  her  immediately  from  the  direction  of  the  wind, 
or,  as  it  is  termed,  to  leeward.  But  since  the  !)ow  of  a  ship  exposes  less  surface  to 
tlie  water  than  the  side,  the  resistance  will  be  less  in  the  first  case  than  in  the  second  ; 
the  velocity,  therefore,  in  the  direction  of  her  head,  will,  in  most  cases,  be  greater  than 
the  velocity  in  the  direction  of  her  side,  and  the  ship's  course  will  be  between  the  two 
directions  ;  and  the  angle  contained  between  the  course  towards  which  the  ship's  head 
is  directed,  and  the  course  she  really  describes  through  the  water,  is  termed  her  let- 
way.  The  quantity  of  lee-way  to  be  allowed  will  depend  upon  a  variety  of  circum- 
stances ;  as  the  mould  and  trim  of  the  ship ;  the  quantity  of  sail  she  carries ;  her 
velocity  through  the  water,  &o. :  hence  no  general  rules  can  be  laid  down  with 
accui-acy  that  will  determine  the  quantity  of  lee-way  in  all  cases.  The  following 
have,  however,  been  usually  given  by  most  writers  on  navigation : — 

1.  When  a  ship  is  close-haided,  with  jdl  her  sails  set,  the  water  smooth,  and  a  light 
breeze  of  wind,  she  is  then  supposed  to  make  little  or  no  lee-way. 

2.  When  the  top-gallant  sails  are  handed,  allow  1  point. 

3.  When  under  close-reefed  topsails,  allow  2  points. 

4.  When  one  topsail  is  handed,  allow  2^  points. 

5.  When  both  topsails  are  handed,  allow  3^  points. 

6.  When  tlie  fore-course  is  handed,  allow  4  points. 

7.  When  under  the  mainsail  only,  allow  5  points. 

8.  When  under  a  balanced  mizzen,  allow  6  points. 

9.  When  under  bare  poles,  allow  7  points. 

As  these  allowances  depend  entirely  on  the  quantity  of  sail  set,  without  regard  to 
any  other  circumstance,  it  is  evident  that  they  can  be  considered  only  as  probable 


262 


METHOD   OF   KEEPING   A   JOURNAL   AT   SEA. 


conjectures,  and  may  indeed  sei-ve  to  work  up  the  day's  work  of  a  Journal  that  has 
been  neglected.  But  smee  the  computation  of  a  ship's  way  depends  mucli  upon  the 
accuracy  of  this  allowance,  it  would  be  proper  for  the  officer  of  the  watch  to  mark 
the  lee-way  on  the  log-board,  in  the  column  reserved  for  tllat  purpose.  The  lee-way 
may  be  estimated  by  observing  the  angle  which  the  wake  of  the  ship  makes  with  the 
point  right  astern,  by  means  of  a  semicircle  marked  on  the  tafterel,  and  divided  into 
points  and  quarters ;  by  means  of  which  the  angle  contained  between  the  direction 
of  the  wake  and  the  point  of  the  compass  directly  astern,  may  be  easily  as- 
certained. 

The  lee- way,  thus  determined,  is  to  be  allowed  on  all  courses  steered,  to  the  right  hand 
of  the  course  steered,  when  the  larboard  tacks  ait  aboard,*  but  to  the  left  hand,  ivhen  the 
starboard  tacks  are  aboard ;  the  person  making  the  allowance  being  supposed  to  be 
looking  towards  the  point  of  the  compass  the  ship  is  sailing  upon. 


Courses  steered. 

N.  W. 

E.  N.  E. 
E.  S.  E. 
W.  by  N. 
E.  N.  E.  h  E. 


Winds. 
N.  N.  E. 
North. 
South. 
N.  by  W. 
S.  E. 


EXAMPLES. 


Lee-way. 
1  point. 
2 
1 

h 
3 


True  courses. 
N.  W.  by  W. 
East. 
E.  by  S. 
W.  h  N. 
N.  E.  h  N. 


When  the  variation  and  lee-way  are  both  to  be  allowed  on  a  course,  you  may  do  it 
at  once,  by  allowing  their  sum  when  they  are  both  the  same  way,  or  their  difference 
when  the  allowance  is  to  be  made  in  differentways,  taking  care  to  make  the  allowance 
in  the  same  way  as  the  greater  quantity  ought  to  be,  whether  it  be  the  variation 
or  lee-way. 


EXAMPLE  L 

A  ship  steers  W.  by  N.,  with  her  lar- 
board tacks  aboard,  and  makes  one  point 
lee-way,  there  being  two  points  westerly 
variation.     Required  the  true  course. 

Lee- way  to  the  right-liand 1  point. 

Vai-iation  to  the  left 2  points. 

Difference  allowed  to  the  left . .  1  point. 

Whence  the  course  is  west. 


EXAMPLE  n. 

A  ship  steers  E.  S.  E.,  with  her  star- 
board tacks  aboard,  and  makes  two  ))oint3 
lee-vvay,  there  being  one  point  westerly 
variation.     Required  the  true  course. 

Lee-way  to  the  left 2  points. 

Variation  to  the  left 1  point. 

Sum  allowed  to  the  left 3  points. 

Whence  the  coui-se  is  E.  by  N. 


In  a  violent  gale,  with  a  head  wind  and  heavy  sea,  when  it  woidd  be  dangerous  to 
carry  sail,  it  is  usual  to  lie  to  under  sufficient  sail  to  prevent  the  vessel  from  rolling  so 
much  as  to  endanger  the  masts  and  rigging.  When  a  ship  is  lying  to,  the  tiller  is  put 
over  to  leeward,  and  when  the  ship  has  head-way,  the  rudder  acts  upon  her  to  bring 
her  to  the  wind ;  the  ship  then  loses  her  way  in  the  water,  which  ceasing  to  act  on 
the  rudder,  her  head  falls  off  from  the  wind,  and  the  sail  which  is  set  fills  and  gives 
her  fresh  way  through  the  water,  which  acting  on  the  rudder,  brings  her  head  again 
to  the  wind.  Thus  the  ship  is  kept  continually  falling  off  and  coming  to.  In  tliis 
case,  you  must  observe  the  points  on  which  she  comes  iqi  and  falls  off,  and  take  the 
middle  between  the  two  points  for  the  apparent  course,  from  which  allow  the 
vai'iation  and  lee-way,  and  you  will  obtain  the  true  course. 


EXAMPLE. 

A  ship,  lying  to  under  lier  mainsail,  with  her  starboard  tacks  aboartl,  comes  uj)  E. 
by  S.,  and  Ihlls  off  N.  E.  by  E.,  there  being  one  point  westerly  variation,  and  she 
makes  5  jioints  lee-way.     What  course  does  she  make  good  ? 

The  middle  between  E.  by  S.  and  N.  E.  by  E.  is  E.  by  N. ;  and  by  allowing  ti 
points  to  the  left  hand  (viz.  5  for  lee-way  and  1  for  variation)  tlie  true  com-se  will  be 
obtained,  N.  by  E. 


I 


ec  llie  note,  page  199. 


METHOD    OF    KEEl'IXG    A    JOURNAL    AT   SEA. 


2G3 


To  exercise  the  learner,  we  sliall  add  tlie  examples  ol'  correcting  for  variation  and 
lee-way  contained  in  the  following  table  : — 


THE    TABLE. 


Courses  steered. 


N.W.  A  W. 
W. 

w.  s.  w. 
w. 

W.  by  N. 

s.  w. 


s. 

s.  s.  w. 

s.  w. 

w. 

W.  by  N. 

S. 
E.  by  S. 
E.  N.  E. 

E. 

E. 


S 
E.  S.  E. 
VV.  S.  W. 

W.  by  N. 
N.  W. 


S. 

N.  bv  E. 
N.  W.'by  N. 
N.  W.  by  W. 

W.  by  S. 


Winds. 


N.  N.  E. 

N.  N.  W. 
S. 

s.  s.  w. 

N.  by  W. 
W.  N.  W. 


W.  S.  V/. 

W. 

N.  W.  by  W. 

S.  S.   VV. 

N.  by  W. 

E.  S.  E. 

S.  h  E. 

N. 

N.  b^  E. 


E.  S.  E. 

N.  E. 

S. 

S.  W.  by  S. 

W.  S.   VV. 


w.  s.  w. 

N.  W.  by  W. 
W.  by  S. 

N.  by  E. 
N.  W.  by  N. 


Lee- 

Varia- 

way 

tion 

points. 

points. 

h 

1  W. 

i 

1  W. 

1 

1  W. 

5 

1  W. 

H 

k\v. 

U 

5  W. 

1 

14  w. 

1 

li  w. 

i 

H  w. 

11 

li  w. 

1 

li  w. 

o 

li  w. 

1 

li  w. 

n 

li  w. 

1 

li  VV. 

0 

liW. 

h 

1:1  W. 

h 

15  VV. 

a 

li|  w. 

1 

Yi  W. 

1 

V\  W. 

1 

I  E. 

k 

1    E. 

li 

1    E. 

l.-t 

liE. 

n 

2i  E. 

Courses  corrected. 


N.  r^  W. 
S.  6h  W. 
S.  (\\  W. 

\V. 
S.  7  W. 
S.  11  W. 


S.  S.  E. 

S.  i  E. 
S.  S.  W.  i  W. 

W.  h  N. 
W.  S.  W.  I  W. 

S.  i  W. 

E.  by  N. 
E.  N.  E.  i  E. 

E.  i  N. 
E.  N.  E.  I  E. 


S.  by  E.  i  E. 

E.  I  S. 
S.  W.  by  W. 

W.  i  N. 
N.  W.  I  W. 


S.  i  E. 

N.  N.  E.  5  E. 

N.  s  w. 

N.  W.  by  W.  i  W. 

W.  .i  S." 


If  the  ship  has  been  acted  npon  by  a  current  or  a  heave  of  the  sea,  yon  must  allow 
tlie  sf^t  and  drift  as  a  course  and  distance  in  the  Traverse  Table,  as  directed  in 
l)age  125. 

Having  corrected  the  courses  for  lee- way  and  variation,  and  estimated  the  dititancea 
sailed,  the  latitude  and  longitude  in  at  noon  are  to  be  found  by  either  of  the  preceding 
methods  of  sailing.  Tiie  latitude  and  longitude,  thus  calculated,  are  called  the 
latitiide  and  longitude  by  (/ea(/-?-ecA'oni/?n;-;  and  if  the  real  course  and  distance  made 
good  by  the  ship  could  be  e^<timated  accui-ately  by  the  compass  and  log,  nothing  more 
would  be  necessary  to  determine  tlie  ship's  place  at  any  time;  but  by  reason  of  the 
various  accidents  that  attend  a  ship's  way,  such  as  heave  of  the  s.;a,  unknown 
curri'Uts,  ditJi.rent  rates  of  sailing  between  tlie  times  of  heaving  the  log,  sudden 
S(iu  ills,  imjjroper  allowance  for  lee-way  and  variation,  the  latitude  and  longitude  of 
the  shij),  as  deduced  from  tlie  reckoning,  will  frequently  differ  from  the  latitude  and 
longitude  by  observation.  In  this  case,  it  will  be  jiropcr  to  re-examine  the  calculation, 
to  s;e  whether  a  just  allowance  has  been  made  for  lee-way,  variation,  bad  steerage, 
drift  of  the  sea,  error  of  the  log-line  and  glass,  &.c.,  since  it  will  sometimes  be  found 
that  a  different  and  more  probable  estimate  of  some  of  these  quantities  will  make  the 
dead-reckoning  agree  more  nearly  with  the  observations,  liefbre  the  method  of 
finding  tlie  longitude  by  lunar  observations  was  introduced,  the  mariner  had  no  other 
ob.sei-vation  to  be  depended  on  except  his  latitude^  and  it  was  then  usual  to  make 
allowances  for  supjiosed  errors  in  the  courses  and  distances,  so  as  toanake  tlie  latitude 
by  observation  and  dead-reckoning  agree.  The  method  of  doing  this  consists  ia 
finding,  by  the  difference  of  latitude  by  observation,  and  the  departure  by  account, 
the  corrected  course,  distance,  and  difference  of  latitude,  by  Case  II.  of  31iddle  Lati- 
tude, or  Mercator's  Sailing,  as  in  the  following  example  : — 


EXAMPLE. 

Yesterday  at  noon  we  were  in  the  latitude  of  39"  18'  N.,  and  by  an  observation  ut 
noon  this  day  are  in  the  latitude  of  37°  48'  N. ;  our  dead-reckoning  gives  107   miles 


204  METHOD    OF   KEEPING    A   JOURNAL   AT   SEA. 

southing  and  64  miles  westing.     Required  the  course,  distance,  and  difference  of 
longitude. 

With  the  difference  of  latitude  by  observation,  90  miles,  (the  difference  of  37°  48 
and  39°  18',)  ami  the  departure  by  dead-reckoning,  64  miles,  I  find  by  Cnse  11.  of  JMiddle 
Latitude  Sailing,  the  course  nearly  35°,  and  the  distance  110  miles  ;  and  with  the  middle 
latitude  by  observation,  38°  33',  and  the  departure,  64  miles,  I  f.iid  the  difference  of 
longitude  to  be  82  miles.  If  the  middle  latitude  by  dead-reckoning,  38°  24',  had  been 
taken,  the  residt  would  have  been  nearly  the  same. 

It'  you  have  not  had  an  observation  for  several  days,  and  then  find  an  error  in  the 
iatitiidc  by  account,  you  may  on  these  princii)les  correct  the  latitude  on  the  inter- 
mediate thiys,  by  saying,  Jls  the  sum  of  all  the  distances  sailed,  since  the  first  observation, 
is  to  the  iL'hok  error  in  the  latitude,  so  is  the  sum  of  the  distances  sailed,  from  the  timz  of 
taking  the  frst  observation  to  the  noon  of  any  particular  day,  to  the  correction^  of  the 
latitude  h)  dead-reckoning  on  that  day,  southerly  if  the  last  latitude  by  obscriation  is  soidh 
of  the  latitude  by  dead-reckoning,  otherwise  northerly.  Tluis,  if  the  latitudes  liy  dead- 
reckoning  at  noon,  on  four  successive  days,  were"  41°  0',  41°  30',  42°  0',  43°  0',  the 
latitude  by  observation  on  the  first  day  41°  0',  and  on  the  last  day  43°  15',  differing  15 
miles  from  tlie  latitude  by  account;  the  distances  sailed  by  the  log,  on  the  three  tiays 
respectively,  30,  90,  and  105  miles ;  we  must  say.  As  the  whole  sum  of  the  distances, 
225  miles,  is  to  the  error  of  the  latitude,  15  miles,  so  is  the  first  distance,  30,  to  the 
correction  of  the  second  latitude,  2',  and  so  is  the  sum  of  30  and  90  (  =  120)  to  the 
correction  of  the  third  latitude,  8';  so  that  the  corrected  latitudes  will  be  41°  0',  41° 
30'  -j- 2' =  41°  32',  42°  0'-j-8'  =  42°  8'  and  43°  15',  and  the  corrected  differences  of 
latitude  on  tl  e  successive  days  will  be  32',  36',  and  67',  with  which  and  the  departure 
by  dead-reckc  ning,  the  corrected  courses,  distances,  Sec,  on  each  day,  may  be  found, 
if  diought  necessary  ;  but  as  the  corrected  longitude  is  not  sensibly  altered  l)y  any  of 
these  corrections,  it  appears  to  be  in  general  wholly  unnecessary  to  make  any  altera- 
tion in  the  Journal  on  this  account.  But  if  it  be  thought  proper  to  notice  these 
corrections  in  plotting  off  the  track  of  a  ship,  it  will  be  necessary  first  to  ])lot  off  the 
courses  by  dead-reckoning,  and  then  to  place  the  points  arrived  at,  at  the  end  of  each 
day,  as  much  to  the  north  or  south  of  the  i)laces  by  dead-reckoning  as  will  make  the 
latitudes  of  those  points  agree  with  the  corrected  latitudes  found  by  the. above  rule. 

The  latitude  and  longitude  being  found  by  the  preceding  methods,  we  may  thence 
determine  the  bearing  and  distance  of  the  jjlace  of  destination  ;  but  when  the  mariner 
is  fearful  that  his  longitude  by  account  is  inaccurate,  and  he  has  no  lunar  observations 
or  chronometer  to  correct  it,  he  must  get  into  the  latitude  of  the  place,  and  (if 
possible)  run  east  or  west,  according  to  his  situation  and  the  prevailing  state  of  the 
winds. 

We  have  now  given  all  the  rules  necessary  for  working  a  day's  work,  and,  lor  the 
convenience  of  the  learner,  (to  enable  liim  to  refer  to  them  easily,)  we  have  here 
collected  them  in  the  eight  following  articles: — 


Rules  for  worldng  a  dai/'s  %oork. 

J.  Correct  the  several  courses  sailed*  for  variation  and  lee-way,  and  enter  them  in 
a  traverse  table,  and  opi»osite  to  each  course  place  the  distance  run  on  that  course, 
found  bv  sununing  up  the  knots  and  fathoms  sailed  l)y  the  ship  on  that  coiu-se.  Find 
in  Table  L  or  II.  the  difference  of  latitude  and  dei)arture  corresjionding  to  each  course 
and  distance,  and  set  them  in  their  respective  cohunns;  then  the  difference  between 
the  sums  of  the  northings  and  southings  will  be  the  difii^rciice  of  latitude  made  good, 
of  the  same  name  witii  the  greater;  and  the  difference  between  the  sums  of  the 
eastings  and  westings  will  be  the  departure  made  good,  of  the  same  name  with  the 
greater  quantity. 

2.  Seek  in  Table  L  or  IL  until  the  above  difference  of  latitude  and  the  departure  are 
found  together  in  their  respective  colunms;  opposite  to  these  will  be  the  distance 
made  good,  and  at  the  top  or  bottom  of  the  page,  according  as  the  departure  is  less 
or  greater  than  the  difference  of  latitude,  will  be  found  the  course. 

3.  If  the  latitude  from  which  the  ship's  departure  is  taken,  or  yesterday's  latitude, 
be  of  the  same  name  as  the  difference  of  latittide,  add  them  together;  but  if  of 
different  names,  take  their  difference;  the  sum  or  remainder  will  be  the  present 
latitude,  of  the  same  name  as  the  greater. 

*  The  set  aiul  drift  of  a  current  (if  there  be  any)  is  ic  he  reckoned  as  a  course  and  tlistancc,  and  ou 
tlie  first  day  after  losing  siglil  of  the  land,  the  hearing  and  dislaiice  of  it  are  to  he  taken  into  account. 


METHOD  OF  KEEPING  A  JOURNAL  AT  SEA.  265 

4.  Find  the   middle  latitude  between  the  latitude   of  yesterday  and  this   day 
■v^-hich  take  as  a   course  in  Table  II.,  and  seek  for  the  departure   in  the  column  of 
difference   of  latitude  ;  then  will  the   distance   corresponding  bo  the   difference  of 
longitude,  of  the  same  name  as  the  departure. 

5.  If  the  longitude  in  yesterday  be  of  the  same  name  as  the  difference  of  longitude, 
add  them  together;  but  if  of  different  names,  take  their  difference;  the  sum  or 
remainder  will  bo  the  longitude  in,  of  the  same  name  as  the  greater. 

6.  If  a  lunar  oljservation  were  taken  at  any  time  of  the  day,  you  must  find,  by  the 
above  method,  the  difference  of  longitude  maila  since  taking  the  observation  for 
regulating  the  watch,  and  thence  the  longitude  in  at  uoou  by  that  observation,  and 
enter  it  in  the  Journal  as  the  longitude  by  observation. 

Whenever  it  is  possible  to  obtain  satisfactory  observations,  lunar  distances  should 
be  taken. 

7.  Ifj'ouhavc  a  chronometer,  regulated  for  mean  time  at  Greenwich,  and  you 
can  find,  by  observation,  the  meantime  at  the  ship,  the  difference  between  these  two 
times  will  be  the  longitude  of  the  ship  at  the  time  of  observation,  as  shown  by  the 
chronometer.  This  longitude,  reduced  to  noon,  by  means  of  the  log,  may  also  be 
entered  in  the  Journal. 

1/ navigators  would  reject  the  absurd  mode  of  reckoning  h>j  the  sex-t>xy,  and  adopt 
ASTKO^o:siiCAi.  time,  it  would  lessen  their  labor  and  tend  to  much  greater  accuracy  in 
their  daihj  works.     Why  cannot  this  be  done  ? 

8.  Find  on  a  general  chart  the  spot  corresponding  to  the  latitude  and  longitude  by 
observation,  and  that  place  will  represent  the  situation  of  the  ship,  whence  the 
bearing  and  distance  of  the  intended  port  may  be  found.  The  same  may  be  obtained 
by  middle  latitude  sailing,  by  inspection  of  Table  II.,  thus:  Find  the  middle  latitude 
between  the  place  of  the  ship  and  the  proposed  place,  and  seek  for  that  latitude  as  a 
course  in  Table  11.,  and  find,  in  the  corresponding  page  of  the  table,  the  difference  of 
longitude  (between  the  ship  and  the  proposed  place)  in  the  distance  column,  opposite 
to  which,  in  the  latitude  column,  will  be  the  departure.  Seek  in  Table  II.  for  this 
departure  and  the  difference  of  latitude  (between  the  ship  and  the  proposed  place)  till 
they  are  found  to  agree  ;  corresponding  thereto  will  be  the  bearing  and  distance 
required.  If  the  magnetic  bearing  be  required,  the  variation  must  be  allowed  on  the 
true  Ijearing ;  to  the  right  hand  if  the  variation  is  westerly,  or  to  the  left  hand  if 
easterly. 

9.  When  the  latitude,  by  account,  is  uncertain,  the  known  position  of  the  ship 
"  on  a  line  of  bearing"  may  be  of  very  great  importance.  In  this  case  the  mode 
of  proceeding  is  shown  on  page  205,  or  in  Sumner's  work. 

We  shall  now  proceed  to  exemplify  the  above  rules;  first  by  a  few  examples  of 
separate  days'  works,  and  then  by  a  Journal  from  Boston  to  Madeira,  kept  in  the 
usual  form. 

O-k 


266 


METHOD   OF  KEEPING  A   JOURNAL  AT  SEA. 


EXAMPLE  I. 

Yesterday,  at  noon,  we  were  in  the  latitude  of  48°  21'  N.,  and  the  longitude  of 
36°  28'  W.,  and  have  sailed  till  this  day  at  noon,  as  per  log-board.  Required  the 
course  and  distance  made  good,  with  the  latitude  and  longitude  in. 

LOG-BOARD. 


H. 
2 

K. 

f) 

F. 

Courses. 

Winds. 

N. 

Lee- 
icaij. 

Remarks. 

S.  W.  by  W.  I  W. 

These   24  hours,    moderate    gales 

4 

5 

5 

and  cloudy  weather. 

f) 

5 

N.  W. 

At  4  P.  M.,  spoke  ship  Washing- 

8 

5 

ton,  from  New  York,  bound   to 

10 

3 

G 

S.  W.  1  W. 

Cork. 

12 

3 

4 

2 

3 

4 

4 

4 

5 

6 

4 

6 

At  6  A.  M.,  stowed  the  anchors, 

8 

5 

S.  W.  i  S. 

W.N.W. 

and  unbent  the  cables,  and  coiled 

10 

4 

5 

them  between  decks. 

12 

4 

Variation  2^  points  westerly.* 

Courses. 

Dist. 

N. 

S. 

E. 

W. 

S.  W.  4  S. 
S.  S.  VV.  h  w. 
S.  by  W.'i  W. 

43 

39 
27 

33.2 
34.4 

aj.8 

27.3 

18.4 
7.8 

DifF.  Latitude  93.4 

Dep.     53.5 

TRAVERSE  TABLE.  By  examining  the  log-board,  it 

api)ears  that  the  siiip  goes  large, 
and  makes  no  lee-way ;  there- 
fore, by  allowing  the  variation 
on  each  of  the  courses,  they  will 
stand  as  in  the  adjoined  Traverse 
Table.  Then  the  distances 
marked  on  the  log-board  must 
be  summed  u])  and  doubled, 
because  they  are  marked  only  for  eveiy  two  hours.f  In  allowing  for  the  knots, 
we  must  reckon  10  to  a  mile ;  and  when  the  tenths  are  above  5,  we  must  add  1 
mile  to  the  distance.  Having  found  the  distances,  we  must  find  the  corresponrling 
differences  of  latitude  and  departures,  in  Table  I.  or  II.,  and  then,  with  the  whole 
difference  of  latitude  and  departure,  we  must  find  the  course  and  distance  made 
good,  and  the  difference  of  longitude,  by  Case  II.  of  Middle  Latitude  Sailing. 

In  the  ])resent  e.xample,  the  difference  of  latitude  is 93'  =    1°  33'  S. 

Yesterday's  latitude 48  21  N. 

The  difference  is  the  latitude  in 40   48  N. 

Sinn  of  the  latitudes 95     9 

Middle  latitude 47  34 


Widi  the  difference  of  latitude  made  good,  93.4  S.,  and  the  departure,  53.5  W.,  We 
must  enter  Table  II.,  and  we  shall  find  they  corresjiond  nearly  to  a  course  of 
S.  30°  W.,  and  distance  108  miles.  Then,  with  the  middle  latitude  47°  34',  or  48°, 
we  must  enter  Table  II.,  and  we  shall  find  the  departm'e  53.5  in  the  latitude  column; 
ojiposite  to  which,  in  the  distance  column,  is  die 

Difference  of  longitude 80'  z=z    1°  20'  W. 

Longitude  left 36  28  W. 

Sum  is  the  lon<ntude  in 37  48  W. 


*  As  these  c.vamples  were  given  only  to  illustrate  the  rules,  we  have  not  been  attentive  to  mark  the 
true  x'ariation. 

t  In  long-  vo3-ages,  it  is  customary  to  mark  the  log-board  every  hour ;  in  that  case,  the  distances 
mcU'ked  on  the  log,  being  summed  up,  will  be  the  true  distance  sailed. 


METHOD   OF   KEEPING  A  JOURNAL  AT  SEA. 


267 


EXAMPLE   IL 

Yesterday,  at  noon,  we  were  in'  the  latitude  of  35°  4C'  N.,  and  the  longitude  of 
17°  42'  W.,  and  have  sailed  till  this  noon  as  per  log-board.  Requu-ed  the  latitude  and 
longitude  in,  and  the  bearing  and  distance  of  Cape  St.  Vincent. 


LOG-BOARD. 


H. 
1 

K. 

6 

F. 
G 

Courses. 

Winds. 

Lee- 
way. 

Remarks. 

S.byE.  iE. 

S.  W.  i  w. 

These  24  hours,  moderate  gales 

2 

6 

6 

' 

and  clear  weather. 

3 

5 

8 

4 

5 

8 

5 

5 

8 

6 

5 

8 

7 

5 

8 

8 

5 

8 

9 

5 

S.  S.  E. 

s.  w. 

n 

10 

5 

1] 

5 

2 

At  8  A.  M.,  saw  a  ship  to  wind- 

12 

5 

2 

ward,  steering  east. 

1 

5 

3 

2 

5 

3 

3 

5 

5 

S.  S.  E.  i  E. 

S.  W.  i  s. 

H 

4 

5 

5 

5 

5 

5 

G 

5 

5 

7 

5 

5 

8 

5 

5 

9 

5 

G 

S. E.  by  S. 

S.  W.  by  S. 

H 

10 

5 

G 

11 

5 

4 

12 

5 

4 

Variation,  ^  point  easterly. 

TRAVERSE    TABLE. 


Courses. 

Dist. 

N. 

S. 

E. 

W. 

S.S.E.|E. 
S. E.  ^  S. 
S.  E.  i  S. 
S.  E.  i  E. 

48 
31 
33 
22 

Diff. 

Lat. 

41.2 

24.9 
24.5 

14.8 

105.4 

24.7 
18.5 
22.2 
16.3 

81.7 

Dep. 

The  courses  being  corrected  for  Ice-way 
and  variation,  and  the  distances  summed  uj), 
(but  not  doubled,  since  the  log-board  is 
marked  for  every  hour,)  will  stand  as  in  the 
adjoined  traverse  table.  Hence,  the  difler- 
euce  of  latitude  made  good  is  105.4  S.,  and 
the  departure  81.7  E. ;  consequently  the 
course  is  S.  38°  E.,  and  the  distance  133 
miles  nearly. 


Latitude  left 35°4ry  N. 

Difference  of  latitude 1  45  S. 

Latitude  in 34     1  N. 

Sum  of  the  latitudes 09  47 

Middle  latitude 34  53 


With  tlie  middle  latitude  34°  53',  or  35° 
and  the  departure  81.7,  the  diff.  of  long 
is  found  to  be  100  miles  =     1°  40'  K. 

Longitude  left 17  42  W 

Longitude  in 10     2  W 


To  find  the  hearing  and  distance  of  Cape  St.  Vincent. 

Latitude  in 34°  1'  N.        ]\Ier.  parts  2173       Longitude  in 10°  2'  W. 

Cape  St.  Vincent's  lat.  37    3  N.        Mer.  parts  2390       Cape  St.  Vin.  long.    9  2  W. 

Difference  of  latitude   3°  2'— 182'    3Ier.diff.lat.  223       Diff.  longitude. . .    7°  0'=r  420' 


J3Y  LOGARITILMS. 


To  find  tilt  bearing. 
Ab  mer.  diff.  latitude  223. .  .log.    2.34830 

Is  to  radius 45° 10.00000 

So  is  diff.  longitude.  420. .  .log.    2.02325 


To  lansent  course  02°  02' 


10.27495 


To  find  the  distance. 

As  radius 45°       . . ; . .  10.00000 

Is  to  prop.  diff.  lat       182 2.20007 

So  is  secant  course  02°  02' 10.32887 


To  the  distance. 


388.1 


2.58894 


Hence,  the  bearing  of  Cape  St.  Vhicent  isN.  02°  02'  E.,  and  distance  338.1  miles. 


268 


METHOD   OF  KEEPING  A  JOURNAL  AT  SEA. 


EXAMPLE  in. 

Suppose  that,  at  the  end  of  the  sea-day,  March  10, 1860,  we  were  in  the  latitude  of 
43°  34'  N.,  and  the  longitude  of  50°  E.,  and  have  sailed  till  next  noon,  as  per  log- 
board.  Required  the  latitude  and  longitude  in,  and  the  variation  of  the  compass, 
liaving  taken  for  this  purpose  the  observation  marked  on  the  log-board. 

LOG-BOARD. 


H. 
2 

K. 

4 

F. 
5 

Courses. 

Winds. 
South. 

Lee- 
way. 

Remarks. 

w.  s.  w. 

These  24  liours,  moderate  gales  ;  found  a 

4 

4 

5 

small  current  setting  N.  E.,  at  the  rate 

6 

4 

5 

of  1  mile  in  4  hours. 

8 

4 

10 

4 

12 

4 

2 

3 

5 

4 

3 

5 

S.  W.  by  W. 

S.  by  E. 

6 

3 

At    8  A.   M.,    sun's    magnetic    azimuth 

8 

3 

N.,  125°  19'  E. ;  altitude  of  O's  lower 

10 

3 

limb,  18°  43':  correction   for  dip   and 

12 

3 

5 

semidiameter,  12'  additive. 

In  calculating  the  variation  from  the  above  observation,  it  is  necessary  to  find  the  declina- 
tion and  latitude  at  the  time  of  observation.  The  former,  at  noon  ending  the  sea-daj',  March 
11,  18G0,  was  3^  32'  S.  by  Table  IV. ;  the  correction  for  the  longitude  50°  E.  is  +  3'  13", 
and  fur  the  time  from  noon  4''  is  +  3'  51" ;  therefore  the  whole  correction  is  nearly  1',  which, 
being  added  to  3°  32'  gives  the  declination  at  the  time  of  observation  3*^  39'  S. ;  conse- 
quently the  polar  distance  93°  39'.  To  find  the  latitude,  we  must  see  by  the  log-board 
what  courses  and  distances  the  ship  has  sailed,  from  noon  to  the  time  of  observation,  at  8 
A.  M.,  viz. :  W.S.W.  58  miles,  and  S.W.  by  W.  19  miles  ;  the  current  setting  in  the  same 
time  N.E.  5  miles  ;  these  courses  must  be  corrected  for  one  point  westerlj' variation,  which 
is  found  to  be  nearly  its  value,  by  a  rough  calculation  made  with  the  latitude  in,  the  pre- 
ceding noon ;  and  by  arranging  these  courses  and  distances  in  a  traverse  table,  we  find 
that  the  difference  of  latitude  made  good,  at  8  A.  M.,  is  about  41  miles  ;  consequently  the 
latitude  in,  at  the  time  of  observation,  is  nearly  42°  53'  N.  ;  the  observed  altitude  of  the 
sun's  lower  limb  is  18°  43' ;  the  correction  for  dip  and  semidiameter  +  12',  and  the  refrac- 
tion by  Table  XII. — 3'  nearly  ;  consequently  the  sun's  correct  altitude  is  18°  62'.  With 
these  data,  the  true  azimuth  is  calculated,  as  in  page  160. 

Polar  distance 93°  39' 

Latitude 42   53 Secant   0.13505 

Altitude 18   52 Secant   0.02393 

Sum 155   24 


Half  sura 77   42 Cosine   9.32844 

Polar  distance 93   39 


Remainder 15   57 Cosine   9.98295 

Sum    19.47042 
9.78521 


Half-sum  is  log  cosine 57°   5' 

True  azimuth N.  114   10  E. 

Magnetic  azimuth N.  125  19  E. 

Variation 11     9  W. 


or  nearly  1  point. 


TRAVERSE    TABLE. 


The  variation  being  allowed  on  all  the  courses, 
and  on  the  set  of  the  current,  and  the  distances 
being  summed  up,  the  traverse  table  will  be  as 
adjoined  ;  and  the  difference  of  latitude  made 
good  =  49.8  S.,  departure  =  (57.5  W.  Hence  the 
course  made  good  is  S.  53i°  W.,  and  the  distance 
=  84  miles.  Subtracting  the  difference  of  lati- 
tude 50',  from  latitude  left  43°  34',  there  remaina 
the  latitude  in  42°  44'  N.  Hence  we  have  the 
middle  latitude  43°  9',  with  which,  and  the  de- 
parture 07.5,  we  find  the  difference  cf  loncritude 
to  be  <)2',  or  1°  32'  W.,  nearly ;  and  by  subtract- 
ing it  from  the  longitude  left,  50°  E.,  we  get  the  longitude  in  48°  28'  E. 


Courses. 

Disl. 

N. 

5.0 
5.0 

.  Lat 

S. 

32.2 
22.6 

54.8 
5.0 

49.8 

E. 

3.3 
3.3 

Dep 

W. 

48.2 
22.6 

70.8 
3.3 

67.5 

S.  W.  by  W. 
S.  W. 

N.  E.  by  N. 

53 

32 

6 

Diff 

METHOD    OF   KEEPING  A   JOURNAL   AT   SEA. 


2G9 


EXAMPLE  IV. 

Yesterday,  at  noon,  we  were  in  tlie  latitude  of  40°  19'  N.,  and  in  tlie  longiltide 
of  67°  58'  W.,  and  have  sailed  till  this  noon  as  per  log-book,  llecjuired  the  hearing 
and  distance  of  Cape  Cod. 


LOG-BOARD. 


H. 
1 

K. 
1 

F. 

Courses. 

Winds. 

Lce- 
icaij. 

1 

Remarhs. 

W.  N.  W. 

North. 

First    part    of    tliese   24  hours,  hght 

2 

1 

breezes    and    fine    weather ;    latter 

3 

1 

part,  pleasant  gales  and  cloudy 

4 

1 

5 

2 

5 

G 

3 

7 

1 

5 

8 

1 

f) 

9 

1 

5 

10 

1 

11 

1 

N.  W. 

N.  N.  E. 

1 

Saw  great  quantities  of  Gulf  weed  and 

12 

1 

rock  weed. 

1 

2 

5 

N.  W.  h  N. 

N.  E.  h.  E. 

1 

2 

2 

5 

3 

2 

5 

4 

2 

5 

5 

3 

N.  N.  \V. 

N.  E.  by  E. 

0 

At  7  A.  M.,  water  discolored,  sounded 

6 

3 

no  bottom. 

7 

3 

8 

3 

9 

4 

10 

4 

11 

4 

5 

E.  N.  E. 

Latitude,  by  observation,  40°  50'  N. 

12 

4 

5 

Variation  ^  of  a  point  W. 

TRAVERSE    TABLE. 


Coiirscs. 

Dlst. 

N. 

S. 

E. 

W. 

15.0 
1.8 
8.C 

14.9 

W.  4  N. 

N.W.byW.^W. 

N.W.by  W.,^W. 

N.  N.  W.  %  W. 

15 
2 

10 
29 

0.7 

0.9 

.5.1 

24.9 

Diff.  Lat.  31  .nl 

Dep 

.40.3 

The  distances  are  to  be  siinnned  up, 
and  marked  in  the  traverse  table  without 
doubling,  because  the  log-board  is  mark- 
ed for  every  hour.  IJy  working  tliis 
day's  work  like  tlie  others,  we  find  tlie 
difference  of  latitude  made  good  :=:  31.G 
N.,  and  the  departure  40.3  W. ;  hence 
tlie  course  N.  52°  W.  neai'ly, and  distance 
51  miles. 


Latitude  left 40°  19'  N, 

Difference  of  latitude 32  N. 

Latitude  in  by  dead-reckoning  40  51  N. 

Sum  of  latitudes 81   10 

Middle  latitude 40  35 


With  the  middle  latitude  40^°,  and  the 
departure  40.3,  we  find  tlie  difference 
of  longitude  is 0°  53'  W. 

Longitude  left C7  58  W. 

Longitude  in C8  51  W 


To  find  the  heaiing  and  distance  of  Cape  Cod. 

Latitude  in  by  obser.  40°50'N. 
Latitude  of  Cape  Cod  42     3  N. 

Difference  of  latitude 


1   13;=  73  miles. 


Middle  latitude 41  26i 


Long,  in  by  D.  R.  . 
Long,  of  Cape  Cod 


Diff.  of  longitude 


n8°51'W. 
70     4  W. 

1  13  —  73  miles. 


With  the  difference  of  longitude,  73  miles,  and  tlie  middle  latitude,  41°26i'or 
41i°,  we  find  the  departure,  54.6  nearly;  with  which,  and  the  difference  of  latitude, 
"3  miles,  the  bearing  of  Cape  Cod  is  found  to  be  N.  37°  AV.,  distant  91  miles. 


270 


JOURNAL 

OF   A   VOYAGE  FROM   BOSTON   TO   MADEIRA. 


H. 

K. 

F 

1 

~ 

~ 

2 

3 

4 

5 

6 

7 

8 

9 

6 

5 

10 

6 

5 

11 

6 

5 

12 

6 

5 

1 

6 

2 

6 

3 

6 

4 

C 

5 

6 

5 

6 

6 

5 

7 

G 

8 

C 

9 

G 

10 

G 

11 

7 

12 

7 

Courses. 


E.  by  S. 


Jfinds. 


N.  W. 


North. 


Lee- 
loay. 


Remarks  on  board, Friday,  March25,18Q0 


At  noon,  got  under  way,  with  a  fine  breeze 
from  tJie  N.  W. 

Got  a  good  bearin<T  of  the  sun  at  noon,  and 
found  the  variation  and  local  attraction  of 
the  standard  compass  8o  W.  Ship  head- 
ing West. 

At  8  P.  M.,  Cape  Cod  light-house  bore 
S.  by  E.  ?i  E.,  distant  14  miles ;  from 
which  I  take  iiiy  departure. 


Thermo   at  noon 41o 

do.      at  midnight 35o 

Biirom.  at  noon 29.90 

do.    at  niidnighc 29.78 

Variation  3  oi  a  point  westerly. 


Course. 


N.85°34'E. 


Dist. 


Lat. 


N. 
7 


Dcp. 


E. 

94 


Lat.  by 
D.  R. 


N. 
42°  10' 


Lat.  by 
Ohs. 


Diff. 
LoniT. 


E. 

00  7/ 


Longitude  in,  by 
D.  R.     Liin.Ohs.     Chrmi. 


W. 

G7°  57' 


W. 
G7°  58' 


TRAVERSE    TABLE. 


Courses. 

Diot. 

N. 

S. 

E. 

W. 

N.N.W.  h.  W. 
E.  i  S. 

14 

101 

12.3 

5.0 

1  G.G 
100.9{ 

12.3 
5.0 

5.0 

100.9 
G.G 

G.G 

Diff.  Lat.  7.3 

Dep.  94.3 

that  tlie  (lifTerenre  of  latitude  made  good  is 
N.  85°  34'  E.,  and  distance  95  miles. 
Latitude  sailed  from,  or  Cape  Cod's  lat.... 
Diflerence  of  latitude 


Sum  of  latitudes. 
Middle  latitude  . 


42° 

3'N.  1 

0 

7 

N. 
N. 

42 

10 

84 

13 

42 

7 

Cape  Cod  beaiinj^  from  the  ship  S.  by  E.  J  E.,  distant 
14  miles,  is  the  -same  as  if  the  ship  had  sailed  from  it 
14  miles,  upon  the  opposite,  or  N.  by  W.  |  VV.  point  of 
the  cou!pass  :  and,  allowing  for  the  variation,  it  be- 
comes N.N.  \V.  i  VV.  ;  this,  and  the  distance  14  miles, 
are  to  be  set  in  the  traverse  table,  as  the  first  course 
and  distance. 

The  ship  sailed  all  day  upon  an  E.  by  S.  crurse  by 
compass,  which,  by  allowing  the  variation,  is  E.  \  S. 
The  wlKile  distance  sailed  (or  the  sum  of  all  the  dis- 
tances) is  101  miles.  With  these  courses  and  dis- 
tanres,  we  find  the  corresponding  difterences  of  latitude 
and  departures  ;  and  by  subtracting  the  southing  from 
the  northing,  and  the  westing  from  the  easting,  vi-e  find 
and  the  departure  94.3  E.,  which  correspond  to  a  course 


Then,  with  the  middle  latitude  42°  as  a  course,  we 
mu-^t  enterTable  II., and  against  the  departure  94.3, 
(or  94. 4, which  is  the  nearest  tabular  number,)  found 
ill  the  latitude  column,  is  127=: the  dilference  of 
longitude  in  the  distance  column. 

T.onsitude  from,  or  Cape  Cod's  longitude  70°   4'W. 

DilfercMice  of  longitude 2     7  E. 

Longitude  in 67  57  W. 


To  Jlnd  the  bearing  and  distance  of  Funchal. 


Latitude  in 42°10'N. 

Punchal's  latitude  ..  32  38  N. 

Difference  of  latitude    9  39 
60 

In  miles 572 


Meridional  parts 2795 

Meridional  parts 2073 

Meridional  diff.  latitude    722 


Longitude  in 67*,'i7'W 

Eunchal's  longitude..  IG  54  W 


Difference  of  longitude  51     3 
CO 


In  miles 30C3 


With  the  meridional  difference  of  latitude  722  miles,  and  difference  of  longitude  3003  miles,  the  bearing  is 
found  to  be  S.  76°  44'  E. ;  and  with  this  bearing  fallen  as  a  course,  and  the  proper  difference  of  latitude  573 
miles,  the  distance  is  found  to  be  2193  miles,  by  Case  1.  of  .Mercator's  Sailing. 


JOURNAL  OF  A  VOYAGE  FROM  BOSTON  TO  MADEIRA. 


r.  1 


H. 

K. 

F. 

T 

7 

2 

7 

3 

7 

4 

7 

5 

7 

6 

7 

7 

7 

8 

7 

9 

7 

10 

7 

11 

7 

12 

7 

1 

7 

2 

7 

3 

6 

G 

4 

6 

G 

5 

6 

4 

G 

6 

4 

7 

6 

4 

8 

6 

4 

9 

6 

G 

10 

G 

G 

11 

G 

5 

12 

6 

5 

Courses. 


E.  by  S. 


E.byS.dS. 


E.  S.  E. 


Winds. 


N.  by  E, 


N.  N.  E. 


Lee- 

way. 


Remarks  on  board,  Saturday,  March  20,  1860. 


Fresh  gales  and  pleasant  weather. 

Saw  a  number   of  fishing-vessels   to  the 
southward. 


At  noon  observed  the  altitude  of  the 

sun's  lower  limb,  bearing  south . .  50°  31' 
Add  for  semidiameter,  dip,  &c.. . .     0  12 

[Refractiun,  bc'mg  small,  is  negloclciL]  

Correct  altitude 50  43 

Subtract  from 90  00 

(v)'s  zenith  distance 39  17  N. 

(2)'s  correct  declination 2  26  N. 

Latitude  by  observation 41  43  N. 


Took  a  double  altitude  of  the  sun  at  10'>  and 
ll^  15'",  and  found,  by  the  first  method,  on 
page  180,  the  latitude  at  the  time  of  the 
second  observation  to  be  41°  44'  N. 


Variation  5  of  a  point  westerly. 


Course. 


S.80"15'E. 


Dlst. 


162 


Diff. 
Lat. 


S. 
27 


Dep. 


E. 

IGO 


Lat.  by 
D.  R. 


N. 
41°  43' 


Lat.  by 
Obs. 


N. 
41°  43' 


Diff. 
Lonsr. 


E. 
3°  35' 


Longitude  in,  by 
D.  R.     Lun.Obs.     Chron. 


W. 

64°  22' 


64°  20' 


TRAVERSE    TABLE. 


Courses. 

Dist. 

N. 

S. 

E. 

W. 

E.  iS. 

E.  1  S. 

E.S.E4E. 

42 
42 

79 

2.1 

6.2 

19.2 

41.9 
41.5 
76.G 

DifF.  Lat.  27.5 

160.0  Dep. 

The  variation  being  allowed  on  each 
course,  and  the  distances  summed  up,  they 
will  stand  as  in  the  adjoining  traverse  table ; 
hence,  by  means  of  Table  I.,  we  find  the 
difierence  of  latitude  27.5,  and  the  departure 
IGO.O,  which  correspond  to  the  course  of  nearly 
S.  80°  15'  E.,  and  the  distance  1G2  miles. 


Yesterday's  latitiulc 42°10'N. 

Difference  of  latitude  27  S. 

Latiludein 41   43  N. 

Sum  of  latitudes 83  53 

Middle  latitude 41   6G 


With  the  middle  latitude  41°  5G',  or  42°,  as  a 
course,  we  must  enter  Table  IL,  and  seek  for  the 
dcjiarture  IGO.O  in  the  latitude  column;  the 
nearest  number  to  which  is  159.8,  correspond- 
ing:^ to  'I'e  distance  215,  which  is  therefore  the 
dillercnce  of  longitude,  equal  to. . . .     3°  35'  E. 

Yesterday's  longitude C7  57  W. 

Longitude  in G 4  22  W. 


To  find  Ike  hearing  and  distance  ofFunchal, 


Latitude  in 4I°43'N. 

Funchal's  latitude..  32   38  N. 


Difference  of  latitude    9     5 
60 


meridional  parts 2759 

IMcridional  parts 2073 

Meridional  diff.  latitude    G8G 


Longitude  in 64°22'\V. 

Funchal's  longitude. .  16    54  W. 

Difference  of  longitude  47  28 
60 


In  miles 545 


In  miles 2848 


By  Case  I.  of  Blercator's  Sailing,  we  find  the  bearing  of  Fimchal  to  be  S.  7G°  27'  E.,  and  its  distance 
2326  miles. 

When  the  sun  was  upon  the  meridian,  the  altitude  of  his  lower  limb  was  observed,  and  found  to  be 
50°  31',  to  which  add  12'  for  the  semidiameter,  parallax,  and  the  dip  of  the  horizon  ;  the  refraction 
(given  in  Table  XII.)  for  this  altitude,  being  small,  is  neglected;  hence  the  correct  central  altitude  was 
50°  43',  which,  being  subtracted  from  90°,  leaves  the  zenith  distance  39°  17',  which  must  be  called 
north,  because  the  sun  bore  south  when  on  the  meridian ;  then,  in  Table  IV.,  we  find  the  sun's  declina- 
tion at  noon  at  Greenwich  =2°  22'  N.;  to  this  add  the  correction  4'  taken  from  Table  V.,  correspond- 
ing 1.0  the  ship's  longitude ;  the  sum  is  2°26'N.=:the  correct  declination;  and  since  the  declination 
and  zenith  distance  are  both  north,  we  must  add  them  together,  and  the  sum  will  be  the  latitude  by 
observation  =  41°  4-3'  N.,  which  agrees  with  the  latitude  bv  account. 


372 


JOURNAL   OF   A   VOYAGE 


H. 


Courses. 


E.  S.  E. 


Winds. 


N.  by  E. 


N.  N.  E. 


N.E.byN. 


Lee- 
ivay. 


Remarks  on  board,  Sunday,  March27,  1860. 


All  these  24  hours,  fresh  breezes  and  clear. 

Observed  the  distance  of  the  sun  from  the 
moon.  The  longitude  at  noon,  by  ihejirst 
method,  on  page  231,  was  found  to  be  60° 
16'  W. 

Meridional  alt.  sun's  lower  limb. .  51°  52' 
Add  for  semidiameter,  dip,  &c.. .         12 

Sun's  correct  altitude 52  04 

Subu-act  from 90  00 

Sun's  zenith  distance 37  56  N. 

Sun's  correct  declination 2  50  N. 

Latitude  observed 40  46  N. 


Thermometer  at  noon 47o 

do.  at  midnight , 45o 

Barometer  at  noon 29.70 

do.      at  midnight   29.62 

Variation  |  of  a  point  westerly,  per  amplitude. 


Course. 


E.S.E.IE. 


Dist. 


192 


Dif. 
Lat. 


S. 
47 


Dcp. 


E. 
186 


Lat.  Inj 
D.  R. 


N. 
40°  5fi' 


Lat.  bij 
Ohs. 


N. 
40°  40' 


Diff. 
Lons- 


E. 

4°  8' 


D.  R. 


Longitude  in,  by 


W. 

G0°  14' 


Lun.Obs. 

o  D 
60°  16' 


Chron. 


TRAVERSE    TABLE. 

Course. 

Dist. 
I9'i 

N.          S. 
DiiF.  Lat.  4G7 

E.     1    W. 

E.S.E.5E. 

18G.2   Dep. 

The  ship  sailed  all  day  ujjon  tiie 
same  course,  which,  being  corrected 
for  the  variation,  is  E.  S.  E.  %  E. ;  the 
whole  distance  sailed  is  192  miles, 
and   the    ditierence  of   latitude  is   47 

miles= 0'47'S. 

Yesterday's  latitude 41  43  N. 

Latitude  by  daily  reckoning 40  .56  N. 

So  that  the  latitude  by  account  difTers  10  miles  from  the  latitude  by  observation. 
Latitude  yesterday  by  observation  41°  43'  N. 
Latitude  by  observation  this  day  40   46  N. 


Difference  of  lat.  by  observation..         57 

SSum  of  latitudes 82   29 

Middle  latitude 41    14 


With  the  middle  latitude  41°  14'  as  a  course, 
and  the  departure  1SG.2  as  diiference  of 
latitude,  we  find  the  corresponduig  distance 
218,  wliich  is  equal  to  the  ditference  of 
longitude 4°    8' E. 

Yesterday's  longitude 04    22  W. 

Lontritude  in GO    14  W. 


Tojind  (he  heating  and  distance  of  Funckal. 


Latitude  in 4()°40'N. 

Funchal's  latitude  32  33  N. 

Ditf.  of  latitude..     8     8 
GO 

In  miles 488 


Meridional  parts  2G83 
Meridional  parts  2073 


Mer.  diff.  lat. 


010 


Longitude  in 00°  14' W. 

Funchal's  longitude  10   54  W. 

DifTerpnce  longitude  43  20 
GO 


In  miles 


2G0O 


With  tiie  meridional  difference  oflatitude  and  difference  of  longitude,  the  bearing  is  found 
to  be  S.  7()°  48'  E. ;  witli  that,  and  the  proper  difference  oflatitude,  the  distance  is  found  to 
be  2137  miles,*  by  Case  I.  Mercator. 


*  If  llie  course  was  calculated  to  seconds,  and  the  mcridioiml  parts  taken  to  one  or  two  places  of 
decimals,  it  would  sometimes  make  a  dilTerciice  of  a  few  miles  in  the  calculated  distance.  We  may 
here  remark,  that,  as  this  Journal  is  only  designed  to  exemplify  the  rules  of  navigation,  we  have  not 
endeavored  to  give  the  tnic  variation. 


FROM    BOSTOxN    TO    .MADEIRA. 


273 


H. 

K. 

F. 

1 

~i 

2 

7 

3 

6 

G 

4 

G 

6 

5 

G 

G 

6 

7 

5 

4 

8 

5 

4 

9 

5 

6 

10 

5 

G 

11 

5 

6 

12 

5 

6 

1 

5 

3 

2 

5 

3 

3 

5 

5 

4 

5 

5 

5 

G 

6 

G 

7 

6 

8 

G 

9 

G 

10 

6 

11 

5 

12 

5 

Courses. 


S.E.byE. 


S.  E. 


S.E.byS. 


Winds. 


N.E.byE. 


E.  N.  E. 


E.  by  N. 


Lee- 
way. 

1 


Remarks  on  board,  JVonday,  March  28, 18G0. 

Fresh  gales,  with  rahi. 

At  4  A.  J\I.,  spoke  the  ship  Franklin,  from 

Philadelphia,  boimd  to  Lisbon. 
At  noon,  observed  meridian   alti- 
tude S's  lower  limb 53°  57' 

Add  for  semidiameter,  &c 0   12 

0's  correct  altitude 54     9 

Subtract  from 90  00 

#'s  zenith  distance 35  51  N. 

0's  correct  declination 3  13  N. 

Latitude  observed 39    4  N. 

By  an  altitude  of  the  pole  star  taken  at  9"* 
P.  M.,  the  latitude  was  found,  by  the  rule 
on  page  206,  to  be  38o  32'  N. 

Thermo,  at  noon 48° 

do.       at  midnight 44^ 

Barom.  at  noon 29.60 

do.    at  midnight 29.88 

Variation,  5  of  a  point  westerly. 


Course. 


S.42°29'E 


Dist. 


138 


Diff. 
Lat. 


S. 
102 


Dep. 


E. 

93 


Lat.  by 
D.  R. 


N. 
39°  4' 


Lat.  by 
Obs. 


N. 
39°  4' 


Diff. 
Lon<T. 


E. 

2°  2' 


Longitude  in,  by 
D.  R.     Lun.Ohs.     Chron. 


W. 

58°  12' 


W. 

58^^  14' 


TRAVERSE 

TABLE. 

Courses. 

Dist. 

N. 

s. 

E. 

W. 

S.  E.  |E. 
S.  E.  i  S. 
S.S.E.aE. 

50 
44 
46 

29.8 
32.6 
39.5 

40.2 
29.5 
23.0 

Diff.  Lat.  101.9!  93.3  Dep. 

The  lee-way  and  variation  being  allowed  on 
the  courses,  they  will  stand  as  in  the  adjoined 
traverse  table.  Then,  Avith  the  difference  of 
latitude  and  departure,  the  course  is  found  to 
be  S.  42°  29'  E.,  and  the  distance  138  miles. 


Yesterday's  latitude 40°  4G'  N. 

Difference  of  latitude   102'  =     1  42  S. 

Latitude  in 39    4  N. 

Sum  of  latitudes 79  50 

Middle  latitude 39  55 


AVith  the  middle  latitude  39°  55',  or  40°,  as 
a  course,and  the  departure  93.3,  taken  as 
difference  of  lat.,  the  difference  of  long, 
is  found  to  be  122  miles  ..=.    2°  2^  E. 

Yesterday's  longitude GO  14  W. 

Longitude  in 58  12  W. 


The  course  made  good  each  day  is  marked  in  the  Journal  to  degrees  and  minutes, 
as  it  was  calculated  by  logarithms ;  but  for  practical  purposes,  it  is  sufficiently  exact 
to  find  it  to  the  neai-est  degree,  by  means  of  Table  II. 


To  find  the  hearing  and  distance  of  Funchal. 

[By  Case  I.  Middle  Lalitiide  Sailing.] 

Latitude  in 39°   4'  N.  Longitude  in 58°  12'  W. 

Funchal's  latitude 32  38  N.  Funchal's  longitude IG  54  W. 

Difference  of  latitude.     6  26  =  386  miles.  Difference  of  longitude  41  18 

Sum  of  latitudes 71  42  ■ 

Middle  latitude 35  51  In  miles 2478 

With  the  middle  latitude  35°  51',  or3G°,  as  a  course,  and  the  difference  of  longitude 
2478,  as  a  distance,  we  may  calculate  the  departure ;  with  that  and  the  difference  of 
latitude,  we  can  find  the  distance  and  course,  by  Case  I.  of  Middle  Latitude  Ssilirg. 
35 


274 


JOURNAL   OF   A    VOYAGE 


Courses. 


South. 


S.  iE. 


Winds. 


E.  S.  E. 


E.bySiS, 


Lee- 
way. 

1 


li 


Remarks  on  hoards  Tuesday,  March  29,  I860. 


These  24  liours,  moderate,  pleasant  weatlier. 
Mer.  altitude  sun's' lower  limb. . .   55o37' 
Add  for  semidiameter,  dip,  &c. . .     0   12 

0  's  correct  altitude 55   49 

Subtract  from 90   00 


0's  zenith  distance 34   UN. 

0's  correct  declination 3    37  N. 


Latitude  observed 37   48  N. 


Took  a  lunar  observation  by  observing  the 
distance  of  the  moon  from  the  sun.  The 
longitude  at  noon,  calculated  by  the  first 
method,  on  page  231,  vras  58°  17'  W. 
Noting  the  time  by  chronometer  when  the 
distance  was  taken,  and  calculating  the 
time  by  the  rules  on  page  209  or  210, 
from  the  observed  altitude  of  the  sun,  the 
longitude  bv  chronometer  was  found  to 
be  580  13'  W.  at  noon. 

Variation,  1  point  westerly. 


Course. 


South. 


Dist. 


86 


Diff. 
Lat. 


S. 
86 


Dep, 


Lat.  by 
D.R. 


N. 
37°  38' 


Lat.  hy 
Ohs. 


N. 
37°  48' 


Diff. 
Lons. 


Longitude  in,  by 
D.  R.     Lun.  Obs.     Chron. 

~Yr7~ 

58°  13' 


W. 

58°  12' 


W. 

58°  17' 


TRAVERSE    TABLE. 


Course. 

Dist. 

N. 

S. 

E.  1  W. 

South. 

86 

86.0 

1 

86.0  Diff.  Lat. 

The  lee-way  and  variation  being  allowed  on 
both  courses,  tliey  become  south ;  the  whole 
distance  sailed,  86  miles,  is,  therefore,  the 
difference  of  latitude  by  account,  the  departure 
being  nothing  ;  consequently,  the  ship  is  in  the 
same  longitude  as  yesterday. 


Yesterday's  latitude 39°   4'  N. 

Difference  of  latitude 86  rr:    1  26  S. 

Latitude  in,  by  dead-reckoning 37  38  N. 

The  latitude  by  obseiTation  was  37°  48'  N. ;  differing  10  miles  from  the  account 


Tojind  the  bearing  and  distance  of  Punchal. 


Latitude  in 37°48'N. 

Funchal's  latitude  32  38  N. 


Meri«iional  parts  2453 
Meridional  parts  2073 


Longitude  in 58°12'W. 

Funchal's  longitude  10  54  W. 


Diff.  of  latitude. 


5  10 
60 


Mer.  diff.  lat.. . .    380      Diff.  of  longitude 


41  18 
60 


In  miles 310 


In  miles 2478 


Hence  the  bearing  is  found  to  be  S.  81°  17'  E.,  and  the  distance  2046  miles,  by 
Case  I.  of  3Icrcator's  Sailing;  and  the  same  may  be  found  by  Middle  Latitude,  which 
is  the  most  exact  method  when  tlie  two  latitudes  differ  but  little ;  and  it  is  the  way 
in  which  the  calculation  will  be  made  in  the  rest  of  the  Journal.  If  great  accuracy 
were  required,  we  might  correct  the  middle  latitudes  by  the  numbers  in  page  76 
but  we  have  not  thought  it  to  be  necessary  in  the  present  Journal 


FROM   BOSTON  TO  MADEIRA. 


275 


n. 


K. 


Courses. 


East. 


fflnds. 


N.  N.  E. 


3 
3 
3 
3 
Lav  to ;  up,  S.  E.  by  E. ; 

6fF,S.E.  byS.   Drift,  li 

niiles  per  hour. 

Up,  S. ;  off,  S.  W.     Drift, 
li  miles  per  hour. 


S.  E.  by  S. 


2 

5 

E.  by  N. 

2 

5 

3 

3 

3 

5 

3 

5 

2 

5 

2 

5 

2 

5 

2 

5 

2 

2 

Lee- 
way. 


la 


Rejiarks  on  board,  Wednesday^  March  30,1860. 


These  24  hours,  fresh  galea  and  squally. 
Handed  the  fore  and  main  courses. 


At  midnight,  mere  moderate.    Wore  ship, 
and  set  the  courses. 


At  G  A.  IVI.,  set  the  topsails  close-reefed. 

Thermometer  at  noon. 53o 

do.  at  midnight 50° 

Barometer  at  noon 29.00 

do.      at  midnight 28,22 

Variation,  1  point  westerly. 


Course. 


N.7G°  17'E. 


Dist. 


31 


Diff- 
Lat. 


N. 
7 


Dep. 


E. 

30 


Lat.  bij 
D.R. 


Lat.  by 
Ohs. 


Diff. 
Lon'r. 


E. 

0°33' 


Longitude  in,  by 
D.  R.     Lun.Obs.     Citron. 


W. 

57°  34' 


57°  36' 


TRAVERSE    TABLE. 


Courses. 

Dist. 

N. 

S. 

E. 

W. 

E.  S.  E. 

South. 

W.  S.  W. 

N.  E.  !^  E. 

12 
6 
6 

32 

20.3 

4.G 
6.0 
2.3 

11.1 
24.7 

5.5 

20.3 
12.9 

12.9 

35.8 
5.5 

5.5 

Diff.  Lat.  7.4 

Dep.  30.3 

Yesterday's  latitude 37°48'  N. 

Difference  of  latitude 7  N. 

Latitude  in 37  55  N 

Sum  of  latitudes 75  43 

Middle  latitude 37  51 


Taking  the  middle  points  (viz.  S.  E.  and 
S.  S.  W.)  between  the  points  to  which  the 
ship  comes  to  and  falls  off",  as  taught  in  the 
rules  of  lying  to,  and  then  allowing,  as  be- 
fore, for  the  variation  and  lee-way,  the 
traverse  table  will  stand  as  adjoined. 

With  the  difference  of  latitude  and 
departure,  the  course  is  found  to  be 
N.  76°  17'  E.,  and  the  distance  31  miles. 


With  the  middle  latitude  37°  51'  (or  38°) 
as  a  course,  and  the  departure  30.3  used 
as  difference  of  latitude,  we  find  the 
difference  of  longitude  to  be     0°  38'  E. 

Yesterday's  longitude 58  12  W. 

Longitude  in 57  34  W. 


To  find  the  beating  and  distance  of  Funchal. 


Latitude  in 37°  55'  N. 

Funchal's  latitude . . .  32  38  N. 

Difference  of  latitude     5  17 =317  miles. 

Stun  of  latitudes 70  33 

Middle  latitude 35  IC, 


Longitude  in 57°34'  W 

Funchal's  lonsritude 16  54  W. 


Difference  of  longitude. 


40  40 
60 


In  miles 2440 


Witli  the  middle  latitude  35°  16',  and  the  difference  of  longitude  2440,  the  depart- 
lu-e  is  found  to  be  1992;  with  that,  and  the  difference  of  latitude  317,  the  bearing  of 
Funchal  is  found  to  be  S.  80°  57i'  E.,  and  the  distance  2017  miles. 


276 


JOURNAL   OF   A   VOYAGE 


H. 

K. 

F. 

1 

~5 

2 

5 

3 

5 

4 

5 

5 

5 

6 

G 

5 

6 

7 

5 

4 

8 

5 

4 

9 

5 

5 

10 

5 

5 

11 

6 

12 

6 

1 

7 

2 

7 

3 

7 

4 

7 

5 

7 

6 

7 

7 

7 

8 

7 

9 

7 

10 

7 

11 

8 

12 

8 

Courses, 


E,  S.  E. 


E.byS.iS. 


Winds. 


South. 


S.  iE. 


Lee- 
way. 


Remarks  on  hoard,  Thursday,  March  3J,  1860. 


Pleasant  gales  and  fail-  weather. 


Observed  the  distance  of  the  planet  Venus 
from  the  moon,  and  found  the  longitude  at 
noon  to  be  54°  31'  by  the  first  method, 
page  231. 

Observed  the  magnetic  azimuth  of  the  sun, 
and  found,  by  the  rules  on  pages  160-161, 
the  variation  to  be  1  point  vresterly. 


Thermometer  at  noon 56'^ 

do.  at  midnight 54° 

Barometer  at  noon 29.31 

do.      at  midnight 29.25 

Variation,  1  point  westerly,  per  azimuth. 


Course. 


East. 


Dist. 


151 


Diff. 
Lat. 


Dep. 


E. 

151 


LmI.  by 
D.R. 


N. 
37°  55' 


Lat.  hy 
Ohs. 


Diff. 
Loner. 


E. 

3°  11' 


Longitude  in,  by 
D.  R.         5    2         Chron. 


W. 

54°  23' 


54°  31' 


54°  25' 


The  variation  and  lee-way  being  allowed  on  both  courses,  it  appears  that  the  ship 
lias  ma-de  a  due  east  course  ;  the  distance  sailed,  151  miles,  is  the  departure ;  and  tlie 
difference  of  longitude  is  found  by  Case  II.  of  Parallel  Sailing.  The  latitude  in  is 
the  same  as  yesterday's  latitude,  37°  55'  N.  Taking  this  as  a  coui-se,  and  the  depart- 
ure, 15],  as  difference  of  latitude,  the  distance  which  corresponds  is  the  difference 
of  longitude,  191  miles =    3°  11'  E. 

Yesterday's  longitude 57  34  W 

Longitude  in 54  23  W 


To  find  tJie  bearing  and  distance  of  Funchal. 

Latitude  in 37°  55'  N.  Longitude  in 54°  23'  W. 

Funchal's  latitude . .     32  38  N.  Funchal's  longitude 16  54  W. 

Difference  of  latitude     5  17  rr  317  miles.         Difference  of  longitude 37  29  W. 

Sum  of  latitudes. ...  70  33  — ^ 

Middle  latitude 35  16  In  miles 2249 


Ile^ico,  by  Case  I.  of  Middle  Latitude  Sailing,  the  departure  is  found  to  be  1836 
miles,  the  bearing  of  Funchal  S.  80°  12'  E.,  and  the  distance  1863  miles. 


FROM   BOSTON   TO   MADEIRA. 


277 


H. 

K. 

F. 

1 

8 

2 

8 

3 

8 

4 

8 

5 

8 

4 

G 

8 

4 

7 

8 

6 

8 

8 

6 

9 

8 

5 

10 

8 

5 

11 

8 

5 

12 

8 

5 

1 

9 

2 

9 

3 

9 

4 

9 

5 

8 

G 

G 

8 

G 

7 

8 

4 

8 

8 

4 

9 

8 

5 

10 

8 

5 

11 

9 

12 

9 

Courses. 


E.  S.  E. 


E.byS.iS. 


East. 


Winds. 


s.  s.  w. 


S.byW. 


South. 


S.  by  E, 


Let- 

ivay. 


Remarks  on  hoard,  Friday,  Apiil  1,  1860. 

Fresh  gales  and  pleasant  weathei*. 
Observed   meridian  altitude    O's 

lower  limb 57°  03' 

Correct  for  semidiameter,  dip,  &c. .        12 

0's  correct  altitude 57    15 

Subtract  from 90   00 

0's  zenith  distance 32    45  N. 

^'s  declination 4   45  N. 

Latitude  to  be  observed 37   30  N. 

At  the  sun's  setting,  observed  the  bearing  of 
its  centre,  and  from  the  rules  on  pages  159 
and  161  the  true  amplitude  Avas  calculated 
and  the  variation  found  to  be  1  point  vres- 
terly. 

Thermometer  at  noon 55° 

do.  at  midnight 51° 

Barometer  at  noon 30.10 

do.      at  midnight 30.00 


Course. 


S.85°24'E. 


Dist. 


202 


DIff. 
Lat. 


S. 
IG 


Dep. 


E. 

201 


Lat.  by 
D.  R. 


N. 
37°  39' 


Lat.  bij 
Obs. 


N. 
37°  30' 


Dlff. 

Lonrr. 


E. 
4°  15' 


Longitude  in,  by 
D.  R.     Lun.Obs.     Chron. 


W. 

50°  8' 


W. 

50°  10' 


TRAVERSE 

TABLE. 

Courses. 

Dist. 

N. 

S. 

E. 

W. 

E.  by  S. 

E.  i  S. 

E.N.E.iE. 

100 
70 
35 

10.2 

19.5 
6.9 

98.1 
69.7 
33.5 

10.2 

2G.4 
10.2 

201.3  Dep. 

Diff 

'.  Lat.  16.2 

The  courses  being  corrected  for  leeAvay 
and  variation,  the  traverse  table  will  be  as 
here  given. 


Hence  the  coui-se  is  S.  85°  24'  E.,  distance  202  miles. 


Yesterday's  latitude 37°  55'  N. 

Difference  of  latitude IG  S. 

Latitude  in,  by  account 37  39  N. 


With  the  middle  latitude  37°  42',  and  the 
departure  201.3,  the  difference  of  longi- 
tude is  255 =:  4°  15'  E. 

Yesterday's  longitude 54  23  W. 

Longitude  in  by  account 50    8  W 


The  latitude  by  observalion  differs  9  miles  from  the  latitude  by  dead-reckonmg. 


278 


JOURNAL  OF   A   VOYAGE 


H. 

K. 

F. 

1 

~G 

T 

2 

6 

5 

3 

7 

5 

4 

7 

5 

5 

7 

6 

7 

7 

8 

8 

8 

9 

8 

5 

10 

8 

5 

11 

8 

5 

12 

8 

5 

1 

9 

2 

9 

3 

9 

4 

9 

5 

9 

6 

9 

7 

9 

8 

9 

9 

9 

5 

10 

9 

5 

11 

9 

5 

12 

9 

5 

Courses. 


E.  S.  E. 


E.  S.  E. 


JVinds. 


South. 


S.  W. 


Lee- 
way. 


Remarks  on  hoard,  Saturday,  April  2, 1860. 


Fresh  gales,  with  rain. 


Saw  a  ship  to  the  southward. 

This  da}^,  took  a  lunar  observation,  by 

measuring  the   distance  of  the  moon 

from  the  star  Pollux ;  the  longitude  at 

noon,  deduced  from  this  observation, 

was  450  50' W.,  by  the  rule  on  page  231. 


Thermo,  at  noon 58° 

do.      at  midnight 54o 

Barom.  at  noon 29.76 

do.     at  midnight. 29.72 

Variation,  1  point  westerly. 


Course. 


S.79°56'E. 


Dist. 


202 


Diff- 
Lat. 


D 


ep. 


S. 
35 


E. 

199 


Lat.  by 
D.  R. 


N. 
36°  55' 


Lat.  by 
Obs. 


Diff. 
Lons- 


E. 

409' 


D.R. 


Longitude  in,  by 


W. 

450  59' 


5  * 


W. 

450  50' 


Citron. 


W. 

46°  01' 


TRAVERSE    TABLE. 


Courses. 

Dist. 

N. 

S. 

£. 

W. 

E.  i  S. 
E.  by  S. 

42 

160 

4.1 
31.2 

41.8 
156.9 

DiiF.  Lat.  35.3 

198.7  Dep. 

The  lee-way  and  variation  being  allowec 
on  the  courses,  the  traverse  table  will  be 
as  here  given ;  hence,  the  course  was 
S.  79°  5G'  E.,  and  the  distance  202  miles. 


Yesterday's  latitude 37°  30'  N. 

Difference  of  latitude 35    S. 

Latitude  in 36  55  N. 

Sum  of  latitudes 74  25 

Middle  latitude 37  12 


With  the  middle  latitude  37°  12',  and  the 
departure  198.7,  the  difference  of  long, 
is  found  to  be  249  miles  rr    4°    9'  E. 

Yesterday's  longitude 50     8  W 

Longitude  in 45  59  W. 


Tojind  the  hearing  and  distance  of  Funchal. 

Latitude  in 30°  55'  N.  Longitude  in 45°  59'  W. 

Funchal's  latitude 32  38  N.  Funchal's  longitude 16  54  W. 

Difference  of  latitude.     4  17  =  257  miles.  Difference  of  longitude  29     5 

Sum  of  latitudes 69  33  ^  . 

Middle  latitude 34  46  In  miles 1745 


Hence,  by  Case  I.  of  Middle  Latitude  Sailing,  the  bearing  of  Funchal  is  found  to 
be  S.  79°  50'  E.,  and  its  distance  1456  miles. 


FROxM   BOSTON   TO  MADEIRA. 


279 


H. 

K. 

F. 

Courses. 

1 

9 

6 

E.  S. E. 

2 

9 

6 

3 

9 

4 

4 

9 

4 

5 

9 

6 

9 

7 

9 

8 

9 

9 

9 

5 

10 

9 

5 

11 

9 

5 

12 

9 

5 

1 

9 

2 

9 

3 

9 

4 

9 

5 

9 

G 

9 

7 

9 

8 

9 

9 

9 

10 

9 

11 

9 

12 

9 

iVinds. 


West. 


N.  W 


North. 


Lee- 
way. 


Remarks  on  board,  Sunday,  April  3, 1860. 


Fresh  gales  and  rainy  weather ;  latter  part 

clear. 
A  great  swell  from  the  N.  E.,  for  which  I 

allow  9  miles. 


Observed  altitude  ©'s  lower  liiiih 

at  noon ". 59^02' 

Correct  for  semidiameter,  &c..add     0  12 

©'s  correct  altitude 59  14 

Subtract  from 90  00 

©'s  zenith  distance 30  46  N. 

0's  declination 5  31 N. 

Latitude  observed 3G  17  N. 

Took  a  lunar  observation,  by  observing  the 

distance  of  the  moon  from  the  star  Spica ; 

the   resulting  longitude  at  noon  was  41° 

41'  W. 
Observed  the  magnetic  azimuth  of  the  sun, 

as  on  March  31,  and  found  the  variation  to 

be  \i  point  westerly. 


Course. 


S.79°22'E. 


Dist. 


217 


Diff. 
Lat. 


S. 
40 


Dcp. 


E. 
213 


Lat.  hlj 
D.  R. 


N. 
30°  15' 


Lat.  by 
Ohs. 


N. 
36°  17' 


Diff. 

Lonrr. 


E. 

4°  25' 


D.  R. 


Longitude  in,  bij 


W. 
41°  34' 


p  *     j   Chron. 


W.  W. 

41°  41'  I  41°  36' 


TRAVERSE    TABLE. 


Courses. 

Dist.     N. 

S. 

E. 

W. 

E.  5  S. 
S.S.W.^W. 

220 
9 

32.3 

7.7 

217.G 

4.6 

DifF.  Lat.  40.0 

217.6 
4.6 

4.6 

213.0  Dep. 

With  the  middle  latitude  36°  3G',  and  the 
departure  213  miles,  the  difference  of 
long,  is  found,  2G5  miles  =     4°  25'  E. 

Yesterday's  longitude 45  59  W. 

Longitude  in 41  34  W. 


In  this  day's  work,  the  swell  is  considered 
as  a  current,  setting  the  ship  9  niiles  jier 
day ;  and  since  the  swell  comes  from  the 
N.  E.,  it  must  set  the  ship  S.  W.,  and  allov/- 
ing  the  variation  S.  S.  W.  I  W.  'J  miles, 
these  are  placed  as  a  course  and  distance 
in  the  traverse  table. 

With  the  difference  of  latitude  and  depart- 
ure, the  course  is  found  to  be  S.  79°  22'  E., 
and  the  distance  217  miles. 

Yesterday's  latitude 36°55'  N. 

Difierence  of  latitude 40  S. 

Latitude  in 36  15  N. 


To  find  the  bearing  and  distance  ofFunchal. 


Latitude  in 3G°17'N. 

Funclial's latitude...  32  38  N. 


Longitude  in 41°  34'  W. 

Funchal's  longitude 16  54  W. 


Difference  of  latitude    3  39  =219  miles.         Difference  of  longitude. 


Sum  of  latitudes G8  55 

Middle  latitude 34  27 


24  40  W. 

60 


In  miles 1480 


Hence,  by  Case  L  Middle  Latitude  Sailing,  the  bearing  of  Funchal  is  found  to  be 
S.  79°  50'  E.,  and  its  distance  1240  miles. 

To  find  the  bearing  and  distance  of  Funchal  by  Mercato/s  Chart. 

Havlnjj  pricked  off  llic  place  of  the  ship  at  noon,  Iny  a  ruler  from  that  point  to  Funclial;  take  the 
nearest  distance  between  the  centre  of  tlie  compass  and" the  ruler  ;  then  slide  one  foot  of  the  compasses 
alonff  the  edge  of  the  ruler,  keeping  the  other  foot  at  the  greatest  distance  from  it,  and  it  will  be  found 
to  run  nearl/upon  the  E.  bv  S.  line,  which  is  therefore  the  bearing  ofFunchal :  then  take  m  your  com- 
passes the  extent  from  the  place  of  the  ship  to  Funchal,  and  apply  it  to  the  graduated  meridian,  setlmg 
one  font  as  much  above  one  place  as  the  other  is  below  the  other  place,  and  the  e.vtent  will  be  found  to 
measure  20A  degrees,  or  1230  miles,  which  was  the  distance  of  the  shio  from  Funchal.  nearly 


280 


JOURNAL   OF   A   VOYAGE 


H. 


Courses. 


E.  S.  E. 

S.  E. 

S.  S.  E. 
S.  by  E. 

S.  by  E. 


Winds. 


N.  E. 


E.  N.  E. 


East. 
E.  by  S. 


Lee- 
way. 

] 


n 


n 


Remarks  on  hoard,  Monday,  Jlpril  4,  1860. 


First   part,  fresh   gales;   latter   part,   mc 
moderate  ;  a  heavy  sea  runinng. 

Meridian  altitude  @'s  lower  limb  61o07' 
Correction  for  semidiameter,  &c.,        12 


®'a  correct  altitude 61    19 

Subtract  from 90   00 


's  zenith  distance 28   41  N. 

's  declination   5   54  N. 


Latitude  observed 34  35  N. 


Took  a  lunar  observation,  by  observing  the 
distance  of  tlie  moon  from  the  planet 
Mars,  and  the  longitude  at  noon,  by  rule 
on  page  231,  was  found  to  be  40^  17'  W. 

Thermometer  at  noon COo 

do.  at  midnight 56° 

Barometer  at  noon. 29.61 

do.      at  midnight 29.53 

Variation,  1;^  point  westerly. 


Course. 


S.37°45'E. 


Diet. 


104 


Diff. 
Lat. 


S. 

82 


Dep. 


E. 

64 


Lat.  ly 
D.  R. 


N. 
34°  55' 


Lat.  ly 
Obs. 


N. 
34°  35' 


Diff. 
Lons- 


E. 

]°  ]8' 


Longitude  in,  by 
D.R.        D  J*         Chron. 


W. 

40°  16' 


W. 

40°  i: 


w. 

40°  13' 


TRAVERSE    TABLE. 


The  courses,  being  corrected  for  lee-way 
and  variation,  will  stand  as  in  the  adjoined 
traverse  table. 

Then,  with  the  difference  of  latitude, 
82.4,  and  the  departure,  63.8,  we  find  the 
course  S.  37°  45'  E.  and  the  distance  104  miles. 


Yesterday's  latitude 30°  17'  N. 

Di-fference  of  latitude 1  22   S. 

Latitude  by  account 34  55   N. 


Courses. 

Dist. 

40 
20 
8 
10 
33 

N. 

S. 

13.5 
13.4 

7.2 
15.7 
32.0 

E. 

37.7 
14.8 

3.4 
3.1 

4.8 

W. 

E.S.  E.iE. 

S.  E.  iE. 

S.  S.  E.  i  E. 

S.  by  E. 

S.  1  E. 

Diff.  Lat.  82.4 

03.8  Dep. 

Yesterday's  latitude 30°  17'  N. 

Latitude  in  by  observation ....  34  35  N. 

Sum  of  latitudes 70  52 

Middle  latitude 35  26 


With  the  departure,  03.8  miles,  and  the 
middle  latituile,35°2G',  we  find  the  difl^l 
of  longitude  to  be  78  miles  zz:     1°  18'E. 

Yesterday's  longitude 41   84  W. 

Longitude  in 40  16  W. 


To  find  the  hearing  and  distance  of  Funchal. 


Latitude  in 34°35'N. 

Funchal's latitude...  32  38  N. 


Longitude  in 40°  10'  W. 

Funchal's  loiiiritude 10  54  W 


23  22 

GO 


Difference  of  latitude     1  57  =  117  miles.         DiflTerence  of  longitude. 

Middle  latitude 33  36 

In  miles 1402 

Hence,  by  Case  L  Middle  Latitude  Sailing,  the  bearing  of  Funchal  is  found  to  be 
B  84°17'E.,  and  Its  distance  1174  miles. 


FROM    BOSTOiN    TO   MADEIRA. 


281 


H. 

K. 

F. 

1 

T 

2 

3 

3 

2 

4 

2 

5 

6 

7 

8 

9 

10 

11 

12 

1 

3 

4 

2 

3 

4 

3 

4 

6 

4 

4 

6 

5 

5 

5 

6 

5 

5 

7 

6 

5 

8 

6 

5 

9 

7 

10 

7 

11 

8 

12 

8 

Courses. 


S.  E. 


Calm. 


E.  S.  E. 


Winds. 


E.  N.  E. 


N.  N.  E. 


Z.ee- 
ivay. 

1 


Remarks  on  board,  Tuesday,  April  5,  1860. 


First  part  of  these  24  hours,  small  breezes, 
and  calm  ;  latter  part,  fresh  gales. 

At.4  P.  M.,  got  out  the  boat,  and  tried  the 
current;  found  it  running  E.  1  mile  per 
hour,  and  suppose  the  ship  has  been  in 
this  cuiTcnt  these  24  hours. 


Meridian  altitude  ©'s  lower  limb  Gl°43' 
Correction  for  semidtameter,  &c.  12 

©'s  correct  altitude 61  55 

Subtract  from 90  00 

©'s  zenith  distance 28  05  N. 

©'s  declination G  16  N. 

Observed  latitude 34  21  N. 


Took  a  lunar  observation,  by  observing  the 
distance  of  the  moon  from  the  planet 
Saturn ;  the  resulting  longitude  at  noon 
was  38o  13'  \Y. 

Variation,  \\  point  westerly. 


Course 

Dist. 

Diff. 
Lat. 

Dcp. 

Lat.  by 
D.R. 

Lat.  by 
Obs. 

Diff. 
Long. 

S.83°36'E. 

101 

S. 
11 

E. 

100 

N. 
34°  24' 

N. 
34°  21' 

E. 
2°1' 

D.R. 


Longitude  in,  by 


D  h 


w. 

33°  15' 


W. 

33°  13' 


Chron. 


AV. 

38°  17' 


TRAVERSE 

TABLE. 

Courses. 

Dist. 

N. 

S. 

E. 

W. 

S.  E.  iE. 

E.  1  S. 
E.N.E4E. 

10 
70 
24 

5.8 

0.7 
10.3 

7.4 
G9.2 
23.3 

5.8 

17.0 

5.8 

99.9  Dep. 

Diff 

•.  Lat.  11.2 

In  addition  to  the  courses  sailed,  we  must 
also  allow  24  miles  for  the  set  of  the  cui-rent 
in  the  direction  of  E.,  per  compass,  or 
E.  N.  E.  l  E.,  ti-ue  course. 


With  the  difference  of  latitude  11.2,  and  the  departure  99.9,  the  course  is  found  to 
be  S.  83°  36'  E.,  and  the  distance  nearly  101  miles. 


Yesterday's  latitude 34°  35'  N. 

Difference  of  latitude 0  11  S. 

Latitude  in,  by  account 34  24  N. 


With  the  middle  latitude  34°  28',  and  the 
departure  99.9,  we  find  the  difference  of 
longitude  to  be  121  miles.  .=  2°    1'  E. 

Yesterday's  longitude 40  16  W. 

Longitude  in .      .  38  15  W. 


To  find  the  heanng  and  distance  of  Funchal. 


Longitude  in 38°15'W 

Funchal's  longitude 16  54  W 

Difllercnce  of  longitude. .  21  21 
60 


Latitude  in 34°  21'  N. 

Funchal's  latitude 32  38  N. 

Difference  of  latitude       1  43  =  103  miles. 
Sum  of  latitudes 66  59 

Middle  huitude 33  30  nearly. 

Hence,  by  Case  I.  of  Middle  Latitude  Sailimr,  the  bearing  of  Funchal  is  found  to 
oe  S.  84°  30'  E.,  and  its  distance  1073  miles. 
36 


In  miles 1281 


282 


JOURNAL   OF   A   VOYAGE 


H. 


F.     Courses. 


E.  S.  E. 


JVinds. 


North. 


iee- 
vmj. 


Remarks  on  hoard,  Wednesday,  April  6, 1860. 


Fine  fresli  gales,  and  clear  weather. 

Meridian  altitude  @'s  lower  limb  62°  39' 
Correction  for  dip,  &c 12 

@'s  correct  altitude 62   51 

Subtract  from 90  00 


@'8  zenith  distance 27   09  N. 

%'s  declination C   39  N. 

Observed  latitude iWis  N. 


Observed  the  bearing  of  the  sun's  centre  at 
setting;  the  variation  therefrom  was  found 
to  be  1^  point  westerly,  by  the  rules  on 
pages  159-161. 

Took  a  lunar  observation,  by  observing  the 
distance  of  the  moon  from  the  star  Regu- 
lus;  the  longitude  at  noon,  deduced  there- 
from, was  33°  55'  W. 

Thermometer  at  noon 61° 

do.  at  midnight 58° 

Barometer  at  noon 29.99 

do.       at  midnight 29.91 


Course. 


E.  I  S. 


Dist. 


21G 


Lat. 


S. 
32 


Dep. 


E. 

214 


Lat.  by 
D.R. 


N. 
33°  4y 


LmI.  by 
Obs. 


N. 
33°  48' 


Diff. 
Long-. 


E. 

4°  18' 


Longitude  in,  by 
D.R.        ([  ^         Chron. 


W. 

33°  57' 


W. 

33°  55' 


W. 

33°  59' 


The  course  corrected  for  variation  is  E.  |   S.,  distance  216  miles;   hence  the 
difference  of  latitude  is  31.7,  and  the  depailure  213.7  miles. 


Yesterday's  latitude 34°  21'  N. 

Difference  of  latitude 32  S. 

Latitude  in 33  49  N. 

Sum  of  latitudes,  by  obser.  ...  C8  09 

Middle  latitude 34  05 


With  the  middle  latitude  34°  05',  and  the 
departure  213.7  miles,  we  find  the 
difference  of  longitude  to  be  258 
miles =    4°18'E. 

Yesterday's  longitude 38  15  W 

Longitude  m 33  57  W. 


Tojind  the  hearing  and  distance  of  Funchal. 

Latitude  in 33°  48'  N.  Longitude  in 33°  57'  W. 

Funchal's  latitude 32  38  N.  Funchal's  longitude 16  54  W. 

Difference  of  latitude       1  10  rr:  70  miles.  Difference  of  longitude  17     3  W. 

Sum  of  latitudes C6  26  

Middle  latitude 33  13  In  miles 1023 


Hence  the  bearing  of  Funchal  is  found  to  be  S.  85°  19'  E.,  and  its  distance  859 
miles. 


FROM   BOSTON  TO  MADEIRA. 


283 


H. 

K. 

F. 

1 

lo" 

2 

10 

3 

10 

4 

10 

5 

10 

6 

10 

7 

8 

4 

8 

8 

4 

9 

8 

6 

10 

8 

6 

11 

8 

5 

12 

8 

5 

1 

8 

2 

8 

3 

8 

5 

4 

8 

5 

5 

8 

4 

6 

8 

4 

7 

8 

6 

8 

8 

6 

9 

8 

10 

8 

11 

8 

12 

8 

Courses. 


E.  S.  E. 


S.E.byE-iE. 


IVinds. 


N.  N.W. 


North. 


Lee- 
way. 


Remarks  on  hoard,  Thursday,  April  7, 1860. 


Fresh  gales  and  pleasant  weather,  with 
a  large  sea. 


Observed  the  meridian  altitude  of  the  moon, 
and  the  latitude  at  noon  was  found  to  be, 
by  the  rules  on  pages  170  and  171,  33° 
IS'N. 

Observed  the  meridian  altitude  of  Regulus ; 
the  resulting  latitude  at  noon,  by  the 
rule  on  page  166,  was  33°  IT  N. 

Observed  the  magnetic  azimuth  of  the  sun, 
and  found  the  variation,  per  rules  on 
pages  160  and  161,  to  be  li  point  wes- 
terly. 

Thermo,  at  noon   63° 

do.      at  midnight 61° 

Barom.  at  noon 30.22 

do.      at  midnight 30.18 

Variation,  per  azimuth,  1^  point  westerly. 


Course. 


S.S0°20'E. 


Dist. 


210 


Diff. 
Lat. 


S. 
35 


Dep. 


E. 

207 


Lat.  hy 
D.  R. 


Lat.  by 
Obs. 


N. 
33°  13' 


Diff. 
Lon;T. 


E. 

4°  8' 


Longitude  in,  by 
D.  R.  Chron. 


W. 

29°  49' 


W. 

23°  43' 


TRAVERSE 

TABLE. 

Courses. 

Dist. 

N. 

S. 

E. 

W. 

E.  i  S. 
E.  by  S. 

60 
150 

5.9 
29.3 

59.7 
147.1 

Diff.  Lat.  35.2 

206.8  Dep. 

By  the  adjoined  traverse  table,  the  differ- 
ence of  latitude  is  35.2,  and  tlie  departure 
206.8;  hence,  the  course  is  S.80°  20'  E.,  and 
tiie  distance  209.8,  or  210  miles. 


Yesterday's  latitude 33°48'  N. 

Diffeuence  of  latitude 35  S. 

Latitude  in,  by  account 33  13  N. 

Sum  of  latitudes 07     1 

Middle  latitude 33  30 


With  the  middle  latitude  33°  30',  and  the 
departure  200.8,  we  find  the  didbrence 
of  longitude  248  miles,  or. .     4°    8'  E. 

Yesterday's  longitude 33  57  W. 

Longitude  in 29  49  W. 


To  find  the  bearing  and  distance  of  Funchal. 


Latittido  in 33°  13'  N. 

Funchal's  latitude 32  38  N. 


Longitude  in 29°49' W. 

Funchal's  longitude 10  54  W. 


Difference  of  latitude, 


35 


Difference  of  longitude. 


Sum  of  latitudes 65  51 

Middle  latitude 32  55 


12  55 

GO 


In  miles 775 


Hence  the  bearing  of  Funchal  is  found  to  be  S.  80°  55'  E.,  and  its  distance  6.')2 
miles.  ' 


284 


JOURNAL   OF   A   VOYAGE 


H. 

K. 

F. 

1 

8 

2 

8 

3 

8 

5 

4 

8 

5 

5 

8 

5 

6 

8 

5 

7 

8 

8 

8 

9 

8 

10 

8 

11 

8 

12 

8 

1 

8 

2 

8 

3 

8 

4 

8 

5 

7 

5 

6 

7 

5 

7 

7 

5 

8 

7 

5 

9 

7 

5 

10 

7 

5 

11 

7 

5 

12 

7 

5 

Courses 


E.byS.iS. 


S.  E. 


East. 


Winds. 


N.  N.  E. 


E.  N.  E. 


S.  S.  E. 


Lee- 
way. 


Remarks  on  hoard,  Friday,  April  8, 1860. 


First  part,  fresh   gales,  and  clear;    latter 
part,  rainy  weather. 


At  6  A.  M.,  the  wind  hauled  suddenly  to  the 
S.S.E. 

Observed  the  meridian  altitude  of  the  planet 
Saturn,  and,  from  the  rule  on  page  174, 
the  latitude  at  noon  was  32°  52'  N. 

Took  a  lunar  observation,  by  observing  the 
distarice  of  the  moon  from  the  planet 
Mars;  the  longitude  at  noon,  by  the  method 
on  page  231,  was  found  to  be  26°  17'  W. 

Thermometer  at  noon 63° 

do.  at  midnight 59° 

Barometer  at  noon 29-95 

do.      at  midnight 29.91 


Variation,  IJ  point  westerly. 


Course. 


S.83°45'E 


Dist. 


172 


Diff- 
Lilt. 


S. 
19 


Dep. 


E. 
171 


Lat.  hij 
D.  R. 


N. 
32°  54' 


Lat.  hij 
Obs. 


Diff. 
Loner. 


E. 

3°  24' 


Longitude  in,  by 
D.R.        D    J  Chron. 


W. 

2G=>  25' 


W. 

26°  17' 


W. 

26°  2G' 


TRAVERSE    TABLE. 


Courses. 

Dist. 

N. 

S. 

E. 

W. 

East. 

S.  E.  by  E. 

N.E.byE.^E. 

50 
80 
GO 

25.7 

44.4 

50.0 
6G.5 
54.2 

25.7 

44.4 

25.7 

170.7  Dep. 

Diff 

:  Lat.  18.7 

The  lee-way  and  variation  being  allowed 
on  the  courses,  they  will  stand  as  in  the 
adjoined  traverse  table ;  then,  with  the  dif- 
ference of  latitude  18.7,  and  the  departure 
170.7,  the  course  is  found  to  be  S.83°  45'  E., 
and  the  distance  172  miles. 


Yesterday's  latitude 33°  13'  N. 

Difference  of  latitude 19   S. 

Latitude  in 32  54  N. 

Sum  of  the  latitudes 06     7 

Middle  latitude 33     3 


With  the  middle  latitude  33°  3',  and  tlie  de- 
parture 170.7,  we  find  the  difK  of  long, 
to  be  nearly  204  miles  . .  =    3°  24'  E. 

Yesterday's  longitude 29  49  W. 

Longitude  in 26  25  W. 


To  find  live  beaiing  and  distance  of  Funchal. 


Latitude  in 32°  54'  N. 

Funchal's  latitude 32  38  N. 

Difference  of  latitude IG 

Sum  of  latitudes 65  32 

Middle  latitude 32  46 


Longitude  in 26°25'  W. 

Funchal's  longitude 16  54  W. 

Difference  of  longitude. 


9  31 
GO 


In  miles 571 


Hence  the  bearing  of  Funchal  is  found  to  be  S. 
miles. 


3°  5'  E.,  and  its  distance  480 


FROM   BOSTON  TO   MADEIRA. 


285 


II. 

K. 

F. 

1 

7 

5 

2 

7 

5 

3 

8 

4 

8 

5 

8 

5 

6 

8 

5 

7 

9 

8 

9 

9 

9 

10 

9 

11 

9 

12 

9 

1 

9 

2 

9 

3 

9 

4 

9 

5 

9 

6 

9 

7 

9 

8 

9 

9 

9 

10 

9 

11 

9 

12 

9 

Courses. 


E.byS4S. 


E.  by  S. 


JFinds, 


South. 


Lee- 
way. 


Remarks  on  board,  Saturday,  April  0,  18G0. 


Fine  breezes,  with  variable  weatlxn-. 
Meridian  altitude  gi's  lower  limb  64°  37' 
Correction  for  dip,  &c 12 

0's  correct  altitude 64   49 

Subtract  from 90   GO 

^'s  zenith  distance 25    UN. 

@'s  declination 7   45  N. 

Observed  latitude 35    56  N. 

Observed  the  meridian  altitude  of  the  planet 
Mars ;  and  the  latitude  at  noon,  by  the 
rule  on  page  174,  was  found  to  be  32°  54' 
North. 
Took  a  lunar  observation,  by  observing  the 
distance,of  the  moon  from  Spica  :  the  lon- 
gitude at  noon,  deduced  therefrom,  was 
22°  09'  W. 

Thermometer  at  noon G4° 

do.  at  midnight 61° 

Barometer  at  noon 29.85 

do.      at  midnight 29.80 

Variation,  li  point  westerly. 


Course 


N.89°12'E, 


Dist. 


210 


Diff. 
Lat. 


Dep. 


E. 

209 


Lat.  by 
D.R. 


N. 
32°  57' 


Lat.  by 
Ohs. 


N. 
32°  5G' 


Diff- 

Lonrr. 


E. 

4°  10' 


Longitude  in,  by 
D.  R.  D  *         Chron. 


W. 

22°  15' 


W. 

22°  CD' 


W. 
22°  17' 


TRAVERSE    TABLE. 


Courses. 

Dist. 

N. 

S. 

E. 

W. 

E.  i  S. 
E.  <^N. 

120 

90 

8.8 

5.9 

119.9 
89.C 

Diff.  Lat. 

8.8 
5.9 

2.9 

5.9 

209.5  Dep. 

The  variation  being  allowed  on  the  courses, 
they  will  stand  as  in  the  adjoined  table;  then, 
with  the  difference  of  latitude  2.9,  and  the 
departure  209.5,  the  course  is  found  to  be 
N.  89°  12'  E.,  and  the  distance  210  miles, 
nearly. 


Yesterday's  latitude 32°  54'  N. 

Difterencc  of  latitude 3  N. 

Latitude  by  account 32  57  N. 


With  the  middle  latitude  32°  55',  and  the 
departure  209.5,  the  difference  of  longi- 
tude is  found  250  miles. . .  =r  4°  10'  E. 

Yesterday's  longitude 2G  25  W. 

Longitude  in 22  15  W. 


To  find  the  hearing  and  distance  of  Funchal. 


Latitude  in 32°5G'N. 

Funchal's  latitude 32  38  N. 

Difference  of  latitude    18 

Sum  of  latitudes 65  34 

Middle  latitude 32  47 


Longitude  in 22°  15'  W. 

Funchal's  lonj^itude 10  54  W. 


Difference  of  longitude . 


5  21 

60 


In  miles 321 


Hence  the  bearing  of  Funchal  is  found  to  be  S.  86°  11'  E.,  and  its  distance  270 
miles. 


2S6 


JOURNAL  OF  A  VOYAGE  FROM  BOSTON  TO  MADEIRA. 


H. 

K. 

F. 

1 

~9" 

T 

2 

9 

5 

3 

9 

5 

4 

9 

5 

5 

9 

5 

6 

9 

5 

7 

9 

8 

9 

9 

9 

10 

9 

11 

9 

12 

9 

1 

9 

2 

9 

3 

9 

4 

9 

5 

9 

0 

9 

7 

9 

8 

9 

9 

9 

10 

9 

11 

12 

Courses. 
S.KbyK 


E.byS.|  S. 


Winds. 


s.  s.w. 


Lee- 
way. 


Remarks  on  board,  Sunday,  April  10,  1860. 


All  this  day,  fine  breezes,  with  very  clear 
weather. 


At  10  A.  M.,  made  the  land ;  the  southern 
part  of  Madeira  bearing  per  compass 
E.  by  S.  I  S.,  distant  19  leagues. 

Observed  the  distance  of  the  moon  from  the 
star  Fomalhaut;  the  longitude  at  noon 
was  17°  15'  W. 

Thermometer  at  noon 65° 

do.  at  midnight 63° 

Barometer  at  noon 29.94 

do.      at  midnight 29.90 

Variation,  \%  point  westerly. 


Course. 


S.83°57'E. 


Disl. 


256 


Diff. 
Lat. 


S. 
27 


D> 


ep. 


E. 

255 


Lat.  by 
D.  R. 


N. 
32°  29' 


Lat.  btj 
Obs. 


Diff. 
Loner. 


E. 

5°  3' 


D.R. 


Longitude  in,  bij 


W. 
17°  12'    17°  15' 


D  * 


Chron. 


TRAVERSE    TABLE. 


Courses. 

E.S.  E.|E. 
East. 
East. 

Dist. 

Ill 

90 
57 

N. 

S. 

E. 

W. 

27.0 

107.7 
90.0 
57.0 

DifF.  Lat.  27.0 

254.7  Dep. 

In  the  traverse  table  are  placed  the  bearing 
and  distance  of  the  land  at  10  A.  M.,  (after 
allowing  the  variation.)  Hence  the  whole 
difference  of  latitude  is  27  miles,  the  depart- 
ure 254.7,  the  course  S.  83°  57'  E.,  and  the 
distance  256  miles. 


Yesterday's  latitude 32°  56'  N. 

Difference  of  latitude 27  S. 

Latitude  by  account 32  29  N. 

Sum  of  latitudes 65  25 

Middle  latitude 32  42 


With  the  middle  latitude  32°  42',  and  the 
departure  254.7,  the  dift^  of  longitude 
is  found  to  be  303  miles  =     5°  3'  E. 

Yesterday's  longitude 22  15  W. 

Longitude  of  S.partof  Madeira  17  12  W. 


Therefore  the  latitude  of  the  southern  point  of  Madeira,  by  account,  is  32°  29'  N., 
and  its  longitude  17°  12'  W.  These  values  differ  but  little  from  those  in  the  Table  of 
Latitudes  and  Longitudes ;  we  may,  therefore,  conclude  that  the  Journal  is  nearly 
correct,  and  the  latitude  and  longitude  of  tliat  part  of  IMadeira  well  laid  down. 


Monday,  April  11,  1836. — Pleasant  gales  and  fair  weather.     At  4  P.  M.,  came  to 
off"  Funclial.     At  8  P.  M.,  went  on  shore. 


A«   ABSTRACT   OF   THK   FOREGOING  JOURNAL. 


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288 


In  preparing  an  abstract  of  the  Log,  the  following  symbols  and  abbrevia. 
tions  are  proposed.  Those  denoting  the  force  of  the  wind  and  the  state  of  the 
weather,  were  suggested  by  Captain  Beaufort,  R.  N.,  and  are  used  by  many 
navigators. 

O     Latitude  by  meridian  altitude  of  the  sun. 
D  do.  do.  do.  moon. 

i^        do.  do.  do.  star. 

©0  D  D  ^^     Latitude  by  double  altitudes 
L.  C.     Longitude  by  Ciironometer. 
L.  D.     Longitude  by  Dead  reckoning. 
1.  D.     Latitude  by  Dead  reckoning. 
0  d   ])  :4^     Longitude  by  Lunar  observations. 

™     Temperature  in  depth. 


0. 

Calm. 

1. 

Iiight  air. 

2. 

Light  breeze, 

3. 

Gentle  breeze, 

4. 

Moderate  breeze, 

5. 

Fresh  breeze, 

6. 

Strong  breeze. 

7 

Moderate  gale, 

8. 

Fresh  gale, 

9. 

Strong  gale. 

10. 

Whole  gale. 

IL 

Storm, 

12. 

Hurricane, 

Figures  to  denote  the  force  of  the  wind. 


Ship  has  steerage  way. 
^  or  tha"t  in  which  a  well-conditioned 
>■   man-of-war  would  go  in  smooth  wa- 
5    tcr  and  clean  full. 


1  to  2  knots. 
3  "  4    « 

5  "  6    " 

J  Royals. 
Top-gallant  sails  over  single  reefa 
Double-reefed  topsails. 
I  Triple-reefed  topsails, 
t  Close-reefed  topsails  and  courses, 
under       Close-reefed  main-topsail  and  reefed  foresaiL 
"  Storm  stay-sails. 

"  Bare  pole.s 


Letters  to  denote  the  state  of  the  weather, 

b.  Blue  sky ;  whether  with  clear  or  hazy  atmdsphere. 

c.  ,  Cloudy  ;  but  detached  opening  clouds. 

d.  Drizzling  rain. 

f.  Foggy — F,  thick  fog. 

g.  Gloomy,  dark  weather, 
h.     Hail. 

1.  Lightning. 

m.  Misty,  hazy  atmosphere. 

o.  Overcast ;  the  whole  sky  being  covered  with  an  impervious  cloud. 

p.  Passing  temporary  showers. 

q.  Squally. 

r.  Rain  ;  continued  rain. 

s.  Snow. 

t.  Thunder. 

u.  Ugly,  threatening  appearance  of  the  weather. 

V.  Visibility  of  distant  objects  whether  the  sky  be  cloudy  or  not. 

w.  Wet  dew. 

Under  any  letter,  indicates  an  extraordinary  degree. 

All  tne  ordinary  phenomena  of  the  weather  may  be  easily  recorded  by  the  combination  of 
these  letters.     Thus : 

g^  V.    Gloomy,  dark  weather,  but  distant  objects  remarkably  visible. 


TO  FIND   THE  LONGITUDE  BY  A   CHRONOMETER.  289 

[Continued  from  page  258.1 

EXA]MPLE  X. 

Suppose  a  vessel  in  a  port  in  west  longitude  on  the  Gth  of  June,  1836,  finds  the  daily 
rate  of  the  chronometer  to  be  -{-  5%  and  after  steering  west,  in  18  days  arrives  on  the 
24th  of  June  in  port,  and  finds  by  observation  tliat  the  daily  rate  is  -j-  8".  Required 
the  correction  of  the  observed  longitude  on  the  18th  of  June. 

The  daily  rate  at  tiie  first  port  is 5' 

The  daily  rate  at  the  second  port  is 8' 

Sum 13 

Mean  daily  rate 6^5 

Let  the   difference  of  longitude   between  the  two  places  by  the  first 

daily  rate  be 23°5(y  00" 

And  by  tiie  mean  daily  rate 23  43  15 

DiflTerence  is  the  correction  of  the  longitude  of  the  second  port  for  18 

days,  and  is  easterly 6  45 

Log.  of  the  correction  of  the  second  port,  &  45''  =  405" 2.60746 

Log.  of  18  days  by  Table  A.  ar.  co 7.76700 

Constant  Log 10.37446 

Log.  of  12  days  (from  6th  to  18th  of  June)  by  Table  A 1.89209 

Log.  of  correction,  184".7  =  3'  4"  7 2.26655 

This  correction,  it  is  evident,  gives  the  place  of  observation  on  the  18th  more  east- 
erly, because  the  second  place  of  observation  is  to  tlie  eastward  of  the  position  given 
by  the  daily  rate  at  tlie  port  of  departure. 

Having  the  coiistant  loganthm  as  above,  the  coiTections  for  the  other  days  are  readily 
found,  by  substituting  for  the  log.  of  12  days  the  log.  from  Table  A.  of  the  days  elapsed 
since  the  rate  was  first  ascertained. 

This  method,  by  Rossel,  is  found  in  the  3d  vol.  of  Biot.  Astronomie  Physique.  The 
logarithms  are  used  in  the  same  manner  as  proposed  by  Galbraith. 

TAELE    A. 


Days. 

1 

Log. 
0.00000 

Days. 
21 

Log. 
2.36361 

Days. 
41 

Log. 

Days. 
61 

Log. 
3.27669 

Days. 
81 

Log. 

3.52127 

Days. 
101 

Log. 
3.71189 

2.93500 

2 

0.47712 

22 

2.40312 

42 

2.95569 

62 

3.29070 

82 

3.53186 

102 

3.72041 

.3 

0.77815 

23 

2.44091 

43 

2.97589 

63 

3.30449 

83 

3.54233 

103 

3.72884 

4 

1.00000 

24 

2.47712 

44 

2.99564 

64 

3.31806 

84 

3.55267 

104 

3.73719 

5 

1.17609 

25 

2.51188 

45 

3.01494 

65 

3.33143 

85 

3.56289 

105 

3.74547 

6 

1.32222 

26 

2.54531 

46 

3.03383 

06 

3.34459 

86 

3.57299 

106 

3.75366 

7 

1.44716 

27 

2.57749 

47 

3.05231 

67 

3.35755 

87 

3.58297 

107 

3.76178 

8 

1.55630 

28 

2.60853 

48 

3.07041 

68 

3.37033 

88 

3.59284 

108 

3.76982 

9 

1.05321 

29 

2.63845 

49 

3.08814 

69 

3.38292 

89 

3.60260 

109 

3.77779 

10 

1.74036 

30 

2.66745 

50 

3.10551 

70 

3.39533 

90 

3.61225 

110 

3.78569 

11 

1.81954 

31 

2.69548 

51 

3.12254 

71 

3.40756 

91 

3.62180 

111 

3.79351 

12 

1.89209 

32 

2.72263 

52 

3.13925 

72 

3.41963 

92 

3.63124 

112 

3.80127 

13 

1.95904 

33 

2.74896 

53 

3.15564 

73 

3.43152 

93 

3.64058 

113 

3.80895 

14 

2.02119 

34 

2.77452 

54 

3.17173 

74 

3.44326 

94 

3  64982 

114 

3.81657 

15 

2.07918 

35 

2.79934 

55 

3.18752 

75 

3.45484 

95 

3.65896 

115 

3.82413 

16 

2.13354 

36 

2.82347 

56 

3.20303 

76 

3.46627 

96 

3.66801 

116 

3.83161 

17 

2.18409 

37 

2.84696 

57 

3.21827 

77 

3.47756 

97 

3.67697 

117 

3.83904 

18 

2.23300 

38 

2.86982 

58 

3.23325 

78 

3.48869 

98 

3.68583 

118 

3.84640 

19 

2.27875 

39 

2.89209 

59 

3.24797 

79 

3.49969 

99 

3.69461 

119 

3.85370 

20 

2.32222 

40 

2.91381 

60 

3.26245 

80 

3.51055 

100 

3.70329 

120 

3.86094 

TABLE  1. 

[Pags-1    1 

Difference  of  Latituc 

e  and  Departure  for  ^Paiirt 

, 

N.^E. 

N.  iW. 

S.:5E. 

s.^w 

! 

Dist.    Lat. 

Dcp. 

Dist. 

Lat. 

Dep. 

Dist. 

Lat. 

Dep. 

05.9 

Dist. 

Lat. 

Dep. 

Dist. 

241 

Lat. 

240.7 

Dep. 

II. 8 

I     01  .0 

00.0 

61 

60.9 

o3  0 

121 

120.9 

i8i 

180.8 

08.9 

2  02^ 

3  o3# 

00.  r 

62 

6r  .9 

o3.c 

22 

121  .9 

06.0 

82 

181.8 

08.9 

42 

241.7 

II. 9 

00. 1 

63 

62.9 

o3.] 

23 

122.9 

06.0 

83 

182.8 

09.0 

43 

242.7 

II  .9 

4 

04.0 

00.2 

64 

63.9 

o3.i 

24 

123.9 

06.1 

84 

i83.8 

09.0 

44 

243  7 

12,0 

5 

c5.o 

00.2 

65 

64.9 

o3.2 

2b 

124.8 

06.1 

8b 

i84.8 

09. 1 

45 

244.7 

12.0 

6 

06.0 

00.3 

66 

65.9 

o3.2 

20 

125.8 

06.2 

86 

i85.8 

09.1 

46 

245.7 

12.1 

7 

07.0 

00.3 

67 

66.9 

o3.3 

27 

126.8 

06.2 

87 

186.8 

09.2 

47 

246.7 

12. 1 

8 

08.0 

00.4 

68 

67.9 

o3.3 

28 

127.8 

06.3 

88 

187.8 

09.2 

48 

247-7 

12.2 

9 

09.0 

00.4 

69 

68. 9 

o3.4 

29 

128.8 

06.3 

89 

188.8 

09.3 

49 

248.7 

12.2 

10 

10.0 

00.5 

70 

69.9 

o3.4 

3o 
i3, 

129.8 

06.4 

90 

189.8 

09.3 

bo 

25l 

249.7 
250.7 

12.3 

11 

1.1 .0 

00.5 

71 

70.9 

o3.5 

i3o.8 

ob.4 

191 

190.8 

09.4 

12.3 

12 

12.0 

00.6 

72 

71.9 

o3.5 

32 

i3i  .8 

06.5 

92 

191 .8 

09.4 

52 

25l  .7 

12.4 

i3 

i3.o 

00.6 

73 

72.9 

o3.6 

■33 

i32.8 

06.5 

93 

192.8 

09.5 

53 

252.7 

12.4 

1 4 

i4.o 

00.7 

74 

73.9 

o3.6 

34 

i33.8 

06.6 

94 

193.8 

09.5 

54 

253.7 

12.5 

1 5 

i5.o 

00.7 

75 

74.9 

o3.7 

3  b 

i34.8 

06.6 

95 

194.8 

09.6 

5b 

254.7 

12.5 

i6 

16.0 

00.8 

76 

75.Q 

o3.7 

3(3 

i35.8 

06.7 

96 

195.8 

09.6 

56 

255.7 

12.6 

17 

17.0 

00.8 

77 

76.9 

o3.8 

37 

i36.8 

06.7 

97 

196.8 

09.7 

57 

256.7 

12.6 

i8 

18.0 

00.9 

78 

77-9 

o3.8 

38 

137.8 

06.8 

98 

197.8 

09.7 

58 

257-7 

12.7 

19 

19.0 

00.9 

79 

78.9 

03.9 

39 

i38.8 

06.8 

99 

198.8 

09.8 

b9 

258.7 

12.7 

20 

20.0 

01 .0 

8g 

79-9 

03.9 

40 

139.8 

06.9 

200 

199.8 

09.8 

bo 

259.7 

12.8 

21 

21 .0 

01  .0 

81 

80.9 

04.0 

i4i 

140.8 

06.9 

201 

200.8 

09.9 

261 

260.7 

12.8 

22 

22.0 

01 .1 

82 

81 .9 

o4.o 

42 

i4i.8 

07.0 

02 

201.8 

09.9 

b2 

261 .7 

12.9 

23 

23.0 

01 .1 

83 

82.9 

04. 1 

43 

142.8 

07.0 

o3 

202.8 

10. 0 

63 

262.7 

12.9 

24 

24.0 

01 .2 

84 

83.9 

o4. 1 

44 

143.8 

07.1 

04 

2o3.8 

10. 0 

(!>4 

263.7 

i3.o 

25 

25.0 

01.2 

85 

84.9 

o4.2 

Ab 

144.8 

07.1 

o5 

204.8 

10. 1 

65 

264.7 

l3.G 

26 

26.0 

01.3 

86 

85.9 

04.2 

46 

145.8 

07.2 

06 

2o5.8 

10. 1 

66 

265.7 

i3.i 

27 

27.0 

01.3 

87 

86.9 

04.3 

4i 

146.8 

07.2 

07 

S06.8 

10-2 

67 

266.7 

i3.i 

23 

28-0 

01 .4 

88 

87.9 

04.3 

48 

147-8 

07.3 

08 

207.7 

10.2 

68 

267.7 

l3.2 

29 

29.0 

01 .4 

89 

88.9 

04.4 

49 

i48.8 

07.3 

09 

208.7 

10.3 

69 

268.7 

l3.2 

3o 
3i 

3o.o 
3i.o 

01 .5 
01.5 

90 

89-9 

04.4 

bo 

149.8 

07.4 

10 

209.7 

10.3 

70 

269.7 

l3.2 

91 

90.9 

o4.5 

i5i 

i5o.8 

07.4 

211 

210.7 

10.4 

271 

270.7 

i3.3 

32 

32. 0 

01 .6 

92 

91.9 

04. b 

b2 

i5i.8 

07.5 

12 

211 .7 

70.4 

72 

271.7 

i3.3 

33 

33.0 

01 .6 

93 

92.9 

04.6 

53 

i52.8 

07.5 

i3 

212.7 

10.5 

73 

272.7 

i3.4 

M 

34.0 

01.7 

94 

93.9 

04.6 

54 

i53.8 

07.6 

i4 

213.7 

10.5 

74 

273.7 

i3.4 

3b 

3b. 0 

01.7 

95 

94.9 

04.7 

bb 

i54.8 

07.6 

i5 

214.7 

10.5 

75 

274.7 

i3.5 

36 

36.0 

01.8 

96 

95.9 

04.7 

bb 

i55.8 

07.7 

16 

215.7 

10.6 

76 

275.7 

i3.5 

37 

37.0 

01.8 

97 

96.9 

04.8 

57 

1 56.8 

07.7 

17 

216.7 

10.6 

77 

276.7 

i3.6 

38 

38.0 

01 .9 

98 

97-9 

04.8 

b8 

157.8 

07.8 

18 

217.7 

10.7 

78 

277.7 

i3.6 

39 

39.0 

01 .9 

99 

98.9 

04.9 

b9 

i58.8 

07.8 

19 

218.7 

10.7 

79 

278.7 

i3.7 

40 

4o.o 

02.0 

TOO 

99.9 

04.9 
o5.o 

bo 

159.8 

07.9 

20 

219.7 

10.8 

80 

279.7 

i3.7 

4i 

4 1 .0 

02.0 

lOI 

100.9 

161 

160.8 

07.9 

221 

220.7 

10.8 

281 

280.7 

i3.8 

42 

41.9 

02.1 

02 

mi  .9 

o5.o 

b2 

i6t.8 

07.9 

22 

221 .7 

10.9 

82 

281.7 

i3.8 

43 

42.9 

02 . 1 

o3 

102.9 

o5.i 

63 

162.8 

08.0 

23 

222.7 

10.9 

83 

282.7 

i3.9 

44 

43.9 

02  .2 

o4 

103.9 

o5.i 

64 

i63.8 

08.0 

24 

223.7 

II  .0 

64 

283.7 

i3.9 

45 

44.9 

02.2 

o5 

104.9 

05.2 

65 

164.8 

08.1 

25 

224.7 

II  .0 

85 

284.7 

.'4-0 

46 

45.9 

02.3 

06 

105.9 

o5.2 

66 

i65.8 

08.1 

26 

225.7 

II. I 

86 

285.7 

14.0 

47 

46-9 

02.3 

07 

106.9 

o5.3 

67 

166.8 

08.2 

27 

226.7 

II  .1 

87 

286.7 

14.1 

48 

47.9 

02.4 

08 

107.9 

o5.3 

68 

167.8 

08.2 

28 

227.7 

II  .2 

88 

287.7 

14.1 

49 

48.9 

02.4 

09 

108.9 

o5.3 

69 

168.8 

08.3 

29 

228.7 

11.2 

89 

288.7 

l4.2 

5o 

49.9 

02.5 

10 

109.9 

o5.4 

70 

169.8 

08.3 

3o 

229.7 

II. 3 

90 

289.7 

l4.2 

5i 

50.9 

02.5 

III 

110-9 

o5.4 

171 

170.8 

08.4 

23l 

23o.7 

II. 3 

291 

290.6 

i4.3 

b2 

bi.9 

02.6 

12 

III  .9 

o5.5 

72 

171. 8 

08.4 

32 

23i  .7 

II. 4 

92 

291 .6 

i4.3 

bJ 

52.9 

02.6 

i3 

112. 9 

o5.5 

73 

172.8   08.5 

33 

232.7 

II. 4 

93 

292.6 

14.4 

b4 

53.9 

02.6 

i4 

113.9 

o5.6 

74 

173.8   08.5 

34 

233.7 

II. 5 

94 

293.6 

i4-4 

bb 

54.9 

02.7 

lb 

114. 9 

o5.6 

75 

174.8   08.6 

35 

234.7 

II. 5 

95 

2^4.6 

i4.b 

bb 

bb.9 

02.7 

16 

115.9 

05.7 

76 

175.8    dS.6 

36 

235.7 

II. 6 

96 

295.6 

i4.5 

b7 

b6.9 

02.8 

17 

116. 9 

o5.7 

77 

176.8 

08.7 

37 

236.7 

II. 6 

97 

296.6 

i4.6 

b8 

b7.9 

02.8 

18 

117. 9 

o5.8 

78 

177.8 

08.7 

38 

237.7 

II. 7 

98 

297.6 

14.6 

^9 

b8.9 

02.9 

19 

118.9 

o5.8 

79 

178.8 

08.8 

39 

238.7 

11.7 

99 

298.6 

14.7 

fin 

b9.9 

02.9 

20 

119.9 

05.9 

80 

179.8 

08.8 

40 

239.7 

II. 8 

3oo 

299.6 

14.7 

Oisi. 

\^op. 

Lat. 

Df^t. 

nop. 

Lat. 

Di.l. 

Dnp. 

I,at. 

Dist. 

Dop. 

Lnt. 

Dist. 

Dep. 

Lat. 

E.  :{N. 

E.riS. 

W.  .4N. 

w.  \  S. 

[For  75  Points.,     j 

Fjge  2] 

TABLE  L 

DilTereace  of  Latitude  and  Departure  for  J  Point. 

IV. iE 

JS.iVV 

S.^E. 

S.iW. 

Dist. 

Lat. 

Dep. 
00. 1 

Dist. 

Lat. 

Dep. 

Dist. 
121 

Lat. 

120.4 

Dep. 

Dist. 

Lat. 

Dep. 

Dist. 

Lat. 

Dep. 

I 

01 .0 

61 

60.7 

06.0 

11.9 

181 

180.1 

17-7 

241 

239.8 

23.6 

2 

3 

02.0 
o3.o 

00.2 
00.3 

62 
63 

61 .7 
62.7 

06.1 
06.2 

22 

23 

121  .4 
122.4 

12.0 
12. I 

bo 

iSi.i 
182.1 

17.8 
17.9 

42 
43 

240.8 
24%B 

23.7 
23.8 

4 

o4.o 

00.4 

64 

63.7 

06.3 

24 

123.4 

12.2 

84 

i83.i 

18.0 

AA 

242.8 

23.9 

5 

o5.o 

00.5 

65 

64.7 

06.4 

25 

124.4 

12.3 

85 

184. 1 

18.1 

45 

243.8 

24.0 

6 

06.0 

00.6 

66 

65.7 

06.5 

26 

125.4 

12.4 

86 

■85.1 

18.2 

46 

244.8 

24.1 

7 

07.0 

00.7 

67 

66.7 

06.6 

27 

126.4 

12.4 

87 

1S6. 1 

18.3 

47 

245.8 

24.2 

8 

u8.o 

00.8 

68 

67.7 

06.7 

28 

127.4 

12.5 

88 

1S7.1 

18.4 

48 

246.8 

24.3 

9 

09.0 

0D.9 

69 

68.7 

06.8 

29 

128.4 

12.6 

89 

188.1 

18.5 

49 

247-8 

24.4 

lO 

10. 0 

01  .0 

70 

69.7 

06.9 

3o 

129.4 

12.7 

90 

189. 1 

18.6 

5o 

248.8 

24.5 

II 

10.9 

01  .1 

71 

70.7 

07.0 

i3i 

i3o.4 

12.8 

191 

190.1 

18.7 

25l 

249.8 

24.6 

la 

n.9 

01 .2 

72 

71-7 

07.1 

32 

i3i.4 

12.9 

92 

191 .1 

18.8 

52 

2  5o.8 

24.7 

i3 

12. Q 

01 .3 

73 

72.6 

07.2 

33 

i32.4 

l3.0 

93 

193. 1 

18.9 

53 

251.8 

24.8 

i4 

i3.9 

01 .4 

74 

73.6 

07.3 

34 

i33.4 

i3.i 

94 

193. 1 

19.0 

51 

252.8 

24.9 

i5 

14.9 

01.5 

75 

74.6 

07.4 

35 

i34.3 

l3.2 

95 

194.1 

19. 1 

55 

253.8 

25.0 

i6 

i5.9 

01 .6 

76 

75.6 

07.4 

36 

i35.3 

i3.3 

96 

195. 1 

19.2 

56 

254.8 

25.1 

17 

16.9 

01.7 

77 

76.6 

07.5 

37 

i36.3 

i3.4 

97 

196.1 

19.3 

57 

255.8 

25.2 

i8 

17.9 

01.8 

78 

77.6 

07.6 

38 

137.3 

i3.5 

98 

197.0 

19-4 

58 

256.8 

25.3 

19 

18.9 

01 .9 

79 

78.6 

07.7 

39 

i38.3 

i3.6 

99 

198.0 

.9.5 

59 

257.8 

25.4 

20 

19.9 

02.0 

80 

79.6 

07.8 

4o 

139.3 

13.7 

200 

199.0 

.9.6 

60 

258.7 

25.5 

21 

20.9 

02.1 

81 

80.6 

07.9 

i4i 

i4o.3 

i3.8 

201 

200.0 

19.7 

261 

259.7 

25.6 

22 

21 .9 

02.2 

82 

81.6 

08.0 

42 

i4i.3 

13.9 

02 

201 .0 

19.8 

62 

260.7 

25.7 

23 

22.9 

02.3 

83 

82.6 

oS.i 

43 

142.3 

14.0 

o3 

202.0 

19.9 

63 

261 .7 

25.8 

24 

23.9 

02.4 

84 

83.6 

08.2 

AA 

143.3 

i4-i 

04 

2o3.o 

20.0 

64 

262.7 

25.9 

25 

24.9 

02.5 

85 

84.6 

08.3 

45 

144.3 

l4-2 

o5 

204.0 

20.1 

65 

263.7 

26.0 

26 

25.9 

02.5 

86 

85.6 

08.4 

46 

145.3 

14.3 

06 

2o5.o 

20.2 

66 

264.7 

26.1 

27 

26.9 

02.6 

87 

86.6 

08.5 

47 

146.3 

14.4 

07 

206.0 

20.3 

67 

265.7 

26.2 

28 

27.9 

02 .7 

88 

87.6 

08.6 

48 

147.3 

14.5 

08 

207 . 0 

20.4 

68 

266.7 

26.3 

29 

28.9 

02.8 

89 

88.6 

08.7 

49 

148.3 

14.6 

09 

208 . 0 

20.5 

69 

267.7 

26.4 

3o 

29.9 

02.9 

90 

89.6 

08.8 

5o 

149-3 

14.7 

10 

209.0 

20.6 

70 

268.7 

26.5 

3i 

3o.9 

o3.o 

91 

90.6 

08.9 

i5i 

i5o.3 

14.8 

211 

210.0 

20.7 

271 

269.7 

26.6 

32 

3i.8 

o3.i 

92 

91 .6 

09.0 

52 

i5i.3 

14.9 

12 

21  I  .0 

20.8 

72 

270.7 

26.7 

33 

32.8 

03.2 

93 

02.6 

09.1 

53 

i52.3 

i5.o 

i3 

2  12.0 

20.9 

73 

271.7 

26.8 

34 

33.8 

o3.3 

94 

93.5 

09.2 

54 

i53.3 

i5.i 

i4 

2l3.0 

21 .0 

74 

272.7 

26.9 

35 

34.8 

o3.4 

95 

94.5 

09.3 

55 

i54.3 

l5.2 

i5 

214.0 

21. 1 

7b 

273.7 

27.0 

36 

35.8 

o3.5 

96 

95.5 

09.4 

u6 

i55.2 

i5.3 

16 

2l5.0 

21 .2 

7b 

274.7 

27.1 

3? 

36.8 

o3.6 

97 

96.5 

09.5 

57 

i56.2 

i5.4 

17 

2:6.0 

21.3 

77 

275.7 

27.2 

38 

37.8 

o3.7 

98 

97.5 

09.6 

58 

157.2 

i5.5 

18 

217.0 

21 .4 

78 

276.7 

27.2 

39 

38.8 

o3.8 

99 

98.5 

09.7 

59 

i58.2 

lb. 6 

19 

217.9 

21.5 

79 

277.7 

27.3 

4o 

39.8 

03.9 

100 

99.5 

09.8 

60 

159.2 

i5.7 

20 

218.9 

21.6 

80 

278.7 

27.4 

4i 

40.8 

04.0 

lOI 

100.5 

09.9 

161 

160.2 

i5.8 

221 

219.9 

21.7 

281 

279.6 

27.5 

42 

41.8 

o4.i 

02 

loi  .5 

10. 0 

62 

161 .2 

15.9 

22 

220.9 

21.8 

82 

280.6 

27.6 

43 

42.8 

04.2 

03 

102.5 

10. 1 

63 

162.2 

16.0 

23 

221  .9 

21 .9 

83 

281.6 

27.7 

M 

43.8 

04.3 

04 

io3.5 

10.2 

64 

i63.2 

16.1 

24 

222.9 

22.0 

84 

282.6 

27.8 

45 

44.8 

04.4 

o5 

104.5 

10.3 

65 

164.2 

16.2 

25 

223.9 

22.1 

85 

283.6 

27.9 

46 

45.8 

04.5 

06 

io5.5 

10.4 

66 

i65.2 

16.3 

26 

224.9 

22.2 

86 

284.6 

28.0 

47 

46.8 

04.6 

07 

106.5 

10.5 

67 

166.2 

16.4 

27 

225.9 

22.2 

87 

285.6 

28.1 

48 

47.8 

04.7 

08 

107.5 

10.6 

68 

167.2 

16.5 

28 

226.9 

22.3 

88 

286.6 

28.2 

49 

48.8 

04.8 

09 

108.5 

10.7 

69 

168.2 

16.6 

29 

227.9 

22.4 

89 

287.6    28.3 

5o 

49-8 

'.4.9 

10 

109.5 

10.8 

70 

169.2 

16.7 

JO 

228.9 

22.5 

90 

2S8.6    28.4 

5i 

5o.8 

OD.O 

III 

110.5 

10.9 

171 

170.2 

16.8 

23l 

229.9 

22  6 

291 

289.6 

28.5 

52 

5i  .7 '  o5.i 

12 

III. 5 

1 1 .0 

72 

171 .2 

ib.9 

32 

230.9 

22    ; 

92 

290.6 

28.6 

53 

52.7 

t)5.2 

i3 

112.5 

11 .1 

73 

172.2 

17.0 

33 

23l  .9 

22.8 

93 

291 .6 

28.7 

54 

53.7 

o5.3 

i4 

ii3.5 

II  .2 

74 

1 73 . 2 

17. 1 

M 

'32.9 

22.9 

94 

292.6 

28.8 

55 

54.7 

o5.4 

i5 

114.4 

II. 3 

75 

174.2 

17.2 

35 

233.9 

23. 0 

95 

293.6 

28.9 

56 

55.7 

o5.5 

16 

ii5.4 

II. 4 

76 

175.2 

17.3 

36 

234.9 

23.1 

96 

294.6 

29.0 

57 

56.7 

o5.6 

17 

116.4 

II. 5 

77 

176. 1 

17.3 

37 

235.9 

23.2 

97 

295.6 

29.1 

58 

57.7 

()5.7 

18 

117.4 

II. 6 

78 

177-1 

17-4 

38 

236.9 

23  3 

98 

296.6 

29.2 

59 

58.7 

o5.8 

19 

118.4 

II. 7 

79 

178. 1 

17.5 

39 

237.8 

^i.A 

99 

297.6 

29.3 

60 

59.7 

05.9 

20 

119.4 

II. 8 

80 

179. 1 

17.6 

4o 

238.8 

23.5 

3oo 

298.6 

29.4 

i  Oop.  i   I.nt. 

Dlsl 

Dep.  1  Lat. 

Dist. 

Dep 

Lat. 

Dist. 

Den. 

Lat. 

Disi.l    Dep. 

Lat. 

E.iN. 

E.^S. 

W.  h  N. 

W.  h  S.             [For  7.i  Points.     J 

TABLE  L 

[Page  3 

Difference  of  Latitude  and  Departure  for  f  Point 

N.|E. 

N.|  W 

S.|E. 

S.5W 

• 

Disi. 

Lat. 

Dep. 

Dist. 

Lat. 

Dep. 

Dist. 

Lat. 

Dep. 

Dist. 

Lat. 

Dep. 

Dist. 

Lat. 

Dep. 

35.4 

I 

01 .0 

00. 1 

61 

60.3 

09.0 

121 

119. 7 

17.8 

181 

179.0 

26.6 

241 

238.4 

2 

02.0 

00.3 

62 

61.3 

09.1 

22    120.7 

17.9 

82 

180.0 

26.7 

42 

239.4 

35.5 

3 

o3.o 

00.4 

63 

62.3 

09.2 

23 

121  .7 

18.0 

83    181.0 

26.9 

43 

240.4 

35.7 

4 

o4.o 

00.6 

64 

63.3 

09.4 

24 

122.7 

18.2 

84  1 182.0 

27.0 

44 

241.4 

35.8 

5 

04.9 

00.7 

65 

64.3 

09.5 

25 

123.6 

18.3 

85    i83.o 

27.1 

45 

242.3 

35.9 

6 

05.9 

00.9 

66 

65.3 

09.7 

26 

124.6 

18.5 

86    184.0 

27.3 

46 

243.3 

36.1 

7 

06.9 

01 .0 

67 

66.3 

09.8 

27 

125.6 

18.6 

8-    i85  0 

27.4 

47 

244.3 

36.2 

8 

07.9 

01.2 

68 

67.3 

10.0 

28 

126.6 

18.8 

88 

186  0 

27.6 

48 

245.3 

36.4 

9 

08.9 

01 .3 

69 

68.3 

10. 1 

29 

127.6 

18.9 

89 

187.0 

27-7 

49 

246.3 

36.5 

10 

09.9 

or  .5 

70 

69.2 

10.3 

3o 

128.6 

19.1 

9" 
191 

187.9 
18S.9 

27.9 
28.0 

bo 

25? 

247-3 
248.3 

36.7 
36.8 

1 1 

10.9 

01 .6 

71 

70.2 

10.4 

i3i 

129.6 

19.2 

12 

11.9 

01.8 

72 

71.2 

10.6 

32 

i3o.6 

19.4 

92 

189.9 

28.2 

52 

249.3 

37.0 

i3 

12.9 

01 .9 

73 

72.2 

10.7 

33 

i3i.6 

19.5 

93 

190.9 

28.3 

53 

25o.3 

37.1 

i4 

i3.8 

02.1 

74 

73.2 

10.9 

34 

i32.5 

19.7 

94 

191. 9 

28.5 

54 

25i.3 

37.3 

i5 

14.8 

02.2 

75 

74.2 

11 .0 

35 

i33.5 

19.8 

95 

192.9 

28.6 

55 

252.2 

37.4 

i6 

i5.8 

02.3 

76 

75.2 

11  .2 

36 

i34.5 

20.0 

96 

193.9 

28.8 

5b 

253.2 

37.6 

17 

16.8 

02.5 

77 

76.2 

11.3 

37 

i35.5 

20. 1 

97 

194.9 

28.9 

i)7 

254-2 

37.7 

i8 

17.8 

02.6 

7« 

77.2 

II. 4 

38 

i36.5 

20.2 

98 

193.9 

29.1 

58 

255.2 

37-9 

'9 

18.8 

02.8 

79 

78.1 

11.6 

39 

137.5 

20.4 

99 

196.8 

29.2 

59 

256.2 

38.0 

2C) 

19.8 

02.9 

80 

79.1 

11.7 

4o 

i38.5 

20.5 

200 

197.8 

29.3 

bo 

257.2 

38.1 

21 

20.8 

o3.i 

81 

80.1 

II. 9 

i4i 

139..  5 

20.7 

201 

198.8 

29.5 

261 

258.2 

38.3 

22 

21.8 

03.2 

82 

81. 1 

12.0 

42 

i4o.5 

20.8 

02 

199.8 

29.6 

62 

259.2 

38.4 

23 

22.8 

o3.4 

83 

82.1 

12.2 

43 

i4i.5 

21 .0 

o3 

200.8 

29.8 

63 

260.2 

38.6 

24 

23.7 

o3.5 

84 

83.1 

12.3 

44 

142.4 

21 .1 

04 

201.8 

29.9 

64 

2bi  .1 

38.7 

25 

24.7 

03.7 

85 

84.1 

12.5 

A5 

143.4 

21.3 

o5 

202.8 

3o.i 

65 

262.1 

38.9 

26 

25.7 

o3.8 

86 

85.1 

12.6 

46 

144.4 

21.4 

06 

2o3.8 

3o.2 

bb 

263.1 

39.0 

27 

26.7 

04.0 

87 

86.1 

12.8 

4i 

145.4 

21 .0 

07 

204.8 

3o-4 

67 

264.1 

39.2 

28 

27.7 

04.1 

88 

87.0 

12.9 

48 

146.4 

21.7 

08 

205.7 

3o.5 

b8 

265.1 

39.3 

29 

28.7 

04.3 

89 

88.0 

i3.i 

49 

147-4 

21 .9 

09 

206.7 

3o.7 

69 

266.1 

39.5 

3o 

29.7 

04.4 

90 

89.0 

l3.2 

i3.4 

5o 

148.4 

22.0 

10 

207.7 

3o.8 

70 

267.1 

39.6 
39.8 

3i 

3o.7 

04.5 

91 

90.0 

i5i 

149.4 

22.2 

211 

208.7 

3i  .0 

271 

268.1 

32 

3i.7 

04.7 

92 

91 .0 

i3.5 

52 

i5o.4 

22.3 

12 

209.7 

3i.i 

72 

269.1 

39.9 

33 

32.6 

04.8 

93 

92.0 

i3.6 

53 

i5i.3 

22.4 

i3 

210.7 

3i.3 

73 

270.0 

4o.  I 

34 

33.6 

o5.o 

94 

93.0 

i3.8 

54 

i52.3 

22.6 

i4 

211 .7 

3i.4 

74 

271 .0 

40.2 

35 

34.6 

o5.i 

95 

94.0 

i3.9 

55 

i53.3 

22.7 

i5 

212.7 

3i.5 

7^ 

272.0 

40.4 

36 

35.6 

o5.3 

q6 

95.0 

i4.i 

56 

i54.3 

22.9 

16 

2i3.7 

dr. 7 

76 

273.0 

40.5 

37 

36.6 

o5.4 

97 

96.0 

l4.2 

57 

i55.3 

23. 0 

17 

214.7 

3i.8 

77 

274.0 

40.6 

38 

37.6 

o5.6 

q8 

96.9 

14.4 

58 

i56.3 

23.2 

18 

2i5.6 

32. 0 

78 

275.0 

4o.8 

39 

38.6 

05.7 

QQ 

97-9 

i4.5 

59 

157.3 

23.3 

19 

216.6 

32.1 

79 

276.0 

40.9 

4o 
4i 

39.6 
4o.6 

05.9 
06.0 

100 

98.9 

14.7 

6c) 

i58.3 

23.5 

20 

217.6 

32.3 

80 

277.0 

4i.i 

101 

99.9 

i4.8 

161 

159.3 

23.6 

221 

218.6 

32.4 

281 

278.0 

4l  -2 

42 

4i.5 

06.2 

02 

100.9 

i5.o 

62 

160.2 

23.8 

22 

219.6 

02.6 

8,2 

278.9 

4i.4 

43 

42.5 

06.3 

o3 

101 .9 

i5.i 

63 

161.2 

23.9 

23 

220.6 

32.7 

83 

279.9 

41.5 

44 

43.5 

06.5 

o4 

102.9 

i5.3 

^4 

162.2 

24.1 

24 

221 .6 

32.9 

84 

280.9 

4i  .7 

45 

44.5 

06.6 

o5 

103.9 

i5.4 

65 

i63.2 

24.2 

25 

222.6 

33.0 

85 

281.9 

4i.8 

46 

45.5 

06.7 

06 

104.9 

i5.6 

66 

164.2 

24-4 

26 

223.6 

33.2 

8b 

282.9 

42  .0 

47 

46.5 

06.9 

07 

io5.8 

.5.7 

67 

i65.2 

24.5 

27 

224.5 

33.3 

«7 

283.9 

42.1 

48 

47.5 

07.0 

08 

106.8 

1 5. 8 

68 

166.2 

24.7 

28 

225.5 

33.5 

88 

284.9 

42.3 

49 

48.5 

07.2 

09 

107.8 

16.0 

69 

167.2 

24.8 

29 

226.5 

33.6 

89 

285.9 

42.4 

5o 
"57 

49-5 

07.3 

10 

108.8 

16.1 

70 

168.2 

24.9 

3o 

227.5 

33.7 

90 

286.9 

42.6 

5o.4 

07.5 

1 1 1 

109.8 

16.3 

171 

169. 1 

25.1 

23l 

228.5 

33.9 

291 

287.9 

42.7 

52 

5i.4 

07.6 

12 

IJ0.8 

16.4 

72 

170. 1 

25.2 

32 

229.5 

34.0 

92 

288.8 

42.8 

53 

52.4 

07.8 

i3 

III. 8 

16.6 

73 

171. 1 

25.4 

33 

23o.5 

34.2 

93 

289.8 

43.0 

54 

53.4 

07.9 

i4 

112. 8 

16.7 

74 

172. 1 

25.5 

34 

23i.5 

34.3 

94 

290.8 

43.1 

55 

54.4 

08.1 

i5 

ii3.8 

16.9 

75 

173. 1 

25.7 

35 

232.5134.5 

9'J 

291.8 

43.3 

56 

55.4 

08.2 

16 

114.7 

17.0 

76 

174. 1 

25.8 

36 

233.4; 34.6 

9b 

292.8  i  43.4 

57 

56.4 

08.4 

17 

115.7 

17.2 

11 

175. 1 

26.0 

37 

234.4 

34.8 

97 

295.8  U3. 6 

58 

57.4 

08.5 

18 

116. 7 

17.3 

78 

176. 1 

26.1 

38 

235.4 

34  9 

98 

294.8 

43.7 

59 

58.4 

08.7 

19 

117. 7 

17.5 

79 

177-1 

26.3 

39 

236.4 

35.1 

99 

295.8 

43.9 

60 

59.4 

08.8 

20 

118.7 

17.6 

80 

178.1 

26.4 

40 

237.4 

35.2 

3oo 

296.8 

44.0 

Disl. 

Dop. 

Lat. 

i);.t 

Dep. 

Lat. 

Dist 

Dep. 

Lat. 

Dist 

Dep. 

Lat. 

Dist. 

Dep. 

Lat. 

L. 

E.$N. 

E.|S. 

W.|  N. 

W.3  8. 

[iFor  "li  Points.     | 

Page  4] 

TABLE  L 

Difference  of  Latitude  and  Departure  for  1  Point. 

NbyE. 

N.byW.                       S.byE.                      S 

byW. 

Dist. 

Lat. 

Dep. 
00.2 

Dist. 

Lat. 

Dep. 

Dist. 

Lat. 

Dep. 

Dist. 

Lat. 

Dep. 

Dist. 

Lat. 

Dep. 

47-0 

I 

01 .0 

61 

59.8 

II. 9 

121 

118. 7 

23.6 

181 

177.5 

35.3 

241 

236.4 

2 

02.0 

00.4 

62 

60.8 

12. 1 

22 

1 19. 7 

23.8 

82 

178.5 

35.5 

42 

237.4 

47.2 

3 

02.9 

00.6 

63 

61.8 

12.3 

23 

120.6 

24.0 

83 

179.5 

35.7 

43 

238.3 

47-4 

4 

o3 . 9 

00.8 

64 

62.8 

12.5 

24 

121 .6 

24.2 

84 

180.5 

35.9 

M 

239.3 

47.6 

5 

04.9 

01 .0 

65 

63.8 

12.7 

25 

122.6 

24.4 

85 

181.4 

36.1 

45 

240.3 

47.8 

b 

o5.9 

01 .2 

66 

64.7 

12.9 

26 

123.6 

24.6 

86 

182.4 

36.3 

46 

241.3 

48.0 

7 

06.9 

01 .4 

67 

65.7 

i3.i 

27 

124.6 

24.8 

87 

i83.4 

36.5 

47 

242.3 

48.2 

8 

07.8 

01 .6 

68 

66.7 

i3.3 

28 

125.5 

25. 0 

88 

184.4 

36.7 

48 

243.2 

48.4 

9 

08.8 

01.8 

69 

67.7 

i3.5 

29 

126.5 

25.2 

89 

i85.4 

36.9 

49 

244.2 

48.6 

10 

09.8 

02.0 

70 

68.7 

13.7 

3o 

127.5 

25.4 

90 

186.3 

37.1 

5o 

245.2 

48.8 

II 

10.8 

02.1 

71 

69.6 

i3.9 

i3i 

128.5 

25.6 

191 

187.3 

37.3 

25l 

246.2 

49.0 

12 

II. 8 

02.3 

72 

70.6 

14.0 

32 

129.5 

25.8 

92 

188.3 

37.5 

52 

247.2 

49.2 

iJ 

12.8 

02.5 

73 

71 .6 

14.2 

33 

1 3o .  4 

25.9 

93 

189.3 

37.7 

53 

248.1 

49-4 

i4 

13.7 

02.7 

74 

72.6 

i4.4 

M 

i3i.4 

26.1 

94 

190.3 

37.8 

54 

249.1 

49.6 

li) 

14.7 

02.9 

7i> 

73.6 

14.6 

35 

i32.4 

26.3 

95 

191 .3 

38.0 

55 

25o.i 

49.7 

lb 

i5.7 

o3.i 

76 

74.5 

i4.8 

36 

i33.4 

26.5 

96 

192.2 

38.2 

56 

25l  .1 

49.9 

I? 

lb. 7 

o3.3 

77 

75.5 

i5.o 

37 

i34.4 

26.7 

97 

193.2 

38.4 

57 

252.1 

5o.i 

i8 

17.7 

o3.5 

78 

76.5 

l5.2 

38 

i35.3 

26.9 

98 

194.2 

38  6 

58 

253.0 

5o.3 

19 

18.6 

o3.7 

79 

77.5 

i5.4 

39 

i36.3 

27.1 

99 

195.2 

38.8 

59 

254-0 

5o.5 

20 

19.6 

03.9 

80 

78.5 

i5.6 

4o 

137.3 

27.3 
27.5 

200 

196.2 

39.0 

60 

255. 0 

50.7 

21 

20.6 

04.1 

81 

79-4 

i5.8 

i4i 

i38.3 

201 

197. 1 

39.2 

261 

256. 0 

5o.Q 

22 

21.6 

04.3 

82 

80.4 

16.0 

42 

139.3 

27.7 

02 

198.1 

39.4 

62 

257.0 

5i.i 

2ci 

22.6 

04.5 

83 

81.4 

16.2 

Ai 

i4o.3 

27.9 

OJ 

199. 1 

39.6 

63 

257.9 

5i.3 

24 

23.5 

04.7 

84 

82.4 

16.4 

M 

i4i  .2 

28.1 

04 

200.1 

39.8 

64 

255.9 

5i.5 

2b 

24.5 

04.9 

85 

83.4 

16.6 

45 

142.2 

28.3 

o5 

201 .1 

4o.o 

65 

259.9 

5i.7 

2b 

25.5 

o5. 1 

86 

84.3 

16.8 

46 

143.2 

28.5 

ob 

202.0 

4o.2 

66 

260.9 

5i.9 

27 

26.5 

o5.3 

87 

85.3 

17.0 

47 

144.2 

28.7 

07 

203.0 

40.4 

67 

261 .0 

52.1 

28 

27.5 

o5.5 

88 

86.3 

17.2 

48 

145.2 

28.9 

08 

204.0 

40.6 

68 

262.9 

52.3 

29 

28.4 

05.7 

89 

87.3 

17.4 

49 

i46.i 

29.1 

09 

2o5.0 

40.8 

69 

263.8 

52.5 

Jo 

29.4 

05.9 

90 

88.3 

17.6 

5o 

i47-i 

29.3 

10 

206.0 

4i  .0 

70 

264.8 

52.7 

3i 

3o.4 

06.0 

91 

89.3 

17.8 

i5i 

148.1 

29.5 

211 

206.9 

4i  .2 

271 

265.8 

52.9 

J2 

3i.4 

06.2 

92 

90.2 

17.9 

52 

149.1 

29.7 

12 

207.9 

41.4 

72 

266.8 

53.1 

33 

32.4 

06.4 

93 

91 .2 

18.1 

53 

i5o.  I 

29.8 

i3 

208.9 

41.6 

73 

267.8 

53.3 

M 

33.3 

06.6 

94 

92.2 

18.3 

54 

i5i  .0 

3o.o 

i4 

209.9 

41.7 

74 

268.7 

53.5 

35 

34.3 

06.8 

95 

93.2 

18.5 

55 

l52.0 

3o.2 

i5 

210.9 

41.9 

75 

269.7 

53.6 

36 

35.3 

07.0 

96 

94.2 

18.7 

56 

i53.o 

3o.4 

16 

211.8 

42. 1 

76 

270.7 

53.8 

37 

36.3 

07.2 

97 

95.1 

18.9 

57 

1 54.0 

3o.6 

17 

212.8 

42.3 

77 

271.7 

54.0 

38 

37.3 

07.4 

98 

96.1 

19.1 

58 

i55.o 

3o.8 

18 

2i3.8 

42.5 

78 

272.7 

54.2 

39 

38.3 

07.6 

99 

97.1 

19.3 

59 

155.9 

3i  .0 

19 

214.8 

42.7 

79 

273.6 

54.4 

40 

39.2 

07.8 

100 

98.1 

19.5 

60 

i56.9 

3l.2 

20 

2i5.8 

42.9 

80 

274.6 

54.6 

4i 

40.2 

08.0 

lOI 

99.1 

19.7 

161 

157.9 

3i.4 

221 

216.8 

43.1 

281 

275.6 

54.8 

42 

41.2 

08.2 

02 

100. 0 

19.9 

62 

i58.9 

3i.6 

22 

217.7 

43.3 

82 

276.6 

55.0 

43 

42.2 

08.4 

o3 

loi  .0 

20. 1 

63 

159.9 

3i.8 

23 

2IS.7 

43.5 

83 

277.6 

55.2 

4A 

43.2 

08.6 

04 

102.0 

20.3 

Q>^ 

160.8 

32.0 

24 

219.7 

43.7 

84 

278.5 

55.4 

4b 

44.1 

08.8 

o5 

io3.o 

20.5 

65 

161.8 

32.2 

25 

220.7 

43.9 

85 

279.5 

55.6 

46 

45.1 

09.0 

06 

104.0 

20.7 

66 

162.8 

32.4 

26 

221.7 

44.1 

86 

280.5 

55.8 

47 

46.1 

09.2 

07 

104.9 

20.9 

67 

i63.8 

32.6 

27 

222.6 

44.3 

87 

281.5 

56.0 

48 

47.1 

09.4 

08 

105.9 

21 .1 

68 

164.8 

32.8 

28 

223.6 

44.5 

88 

282.5 

56.2 

49 

48.1 

09.6 

09 

106.9 

21.3 

69 

i65.8 

33.0 

29 

224-6 

44.7 

89 

283.4 

56.4 

bo 

49.0 

09.8 

10 

107.9 

21.5 

70 

166.7 

33.2 

3u 

225.6 

44.9 
45".  I 

90 

284.4 

56.6 

5i 

5o.o 

09.9 

III 

108.9 

21.7 

171 

167.7 

33.4 

23l 

226.6 

291 

285.4 

56.8 

b2 

5i  .0 

lO.I 

12 

109.8 

21 .9 

72 

168.7 

33.6 

32 

227.5 

45.3 

92 

286.4 

57.0 

53 

52.0 

10.3 

i3 

110.8 

22.0 

73 

169.7 

33.8 

33 

228.5 

45.5 

q3 

287.4 

57.2 

54 

53.0 

10.5 

i4 

III. 8 

22.2 

74 

170.7 

33.9 

34 

229.5 

45.7 

94 

288.4 

57.4 

55    53.9 

10.7 

i5 

112.8 

22.4 

73 

171 .6 

34.1 

35 

23o.5 

45.8 

95 

289.3 

57.6 

56 

54.9 

10.9 

16 

ii3.8 

22.6 

76 

172.6 

34.3 

36 

231.5 

46.0 

96 

290.3 

i)7.7 

^7 

55.9 

II. I 

17 

114.8 

22.8 

77 

173.6 

34.5 

37 

232.4 

46.2 

97 

291 .3 

57.9 

58 

5b. 9 

II. 3 

18 

ii5.7 

23.0 

78 

174.6 

34.7 

38 

233.4 

46.4 

g8 

292.3 

58.1 

59 

57  9 

II. 5 

19 

116.7 

23.2 

79 

175.6 

34.9 

39 

234.4 

46.6 

99 

293.3 

58.3 

bo 

58  8 

.11.7 

20 

117.7 

23.4 

80 

176.5 

35.1 

4o 

235.4 

46.8 

3oo 

294.2 

58.5 

IVp. 

Lat. 

Dist. 

Dop. 

Lat. 

Dist. 

Dep. 

Lat. 

Dist. 

Dep. 

Lat. 

Dist.     Dep. 

Lat. 

E.byN. 

E.byS.                 VV.byN. 

W.byS. 

[For  7  Points. 

TABLE  I. 

LPage  5 

Difference  of  Latitude  and  Departure  for  1|  Points. 

N.byE.iE. 

N.byW.^W.                 S.byE.iE.                 S  byW.:iW 

Dist. 

Lat. 

Dep. 

Dist. 
61 

Lat. 

Dep. 

Dist. 

Lat. 

Dep. 
29.4 

Dist. 

Lat.      Dep. 

Dist. 

Lat. 

Dep. 

I 

01  .o 

00.2 

59.2 

14.8 

121 

1 17.4 

i8i 

175.6    44.0 

24 1 

233.8 

58.6 

2 

01 .9 

00.5 

62 

60.1 

i5.i 

22 

118. 3 

29.6 

82 

176.5    44.2 

42 

234.7 

58.8 

3 

02.9 

00.7 

63 

61. 1 

i5.3 

23 

119. 3 

29.9 

83 

177.5 

44.5 

43 

235.7 

59.0 

4 

o3.9 

01 .0 

64 

62.1 

i5.6 

24 

120.3 

3o.i 

84 

178.5 

44.7 

44 

236.7 

59.3 

5 

04.9 

01.2 

65 

63.1 

i5.8 

23 

121 .3 

3o.4 

85 

179.5 

45.0 

45 

237.7 

59.5 

6 

o5.8 

01 .5 

66 

64. 0 

16.0 

26 

122.2 

3o.6 

86 

180.4 

45.2 

46 

238.6 

59.8 

7 

06.8 

01.7 

67 

65.0 

16.3 

27 

123.2 

30.9 

87 

181. 4 

45.4 

4i 

239.6 

60.0 

8 

07.8 

01 .9 

68 

66.0 

16.5 

28 

124.2 

3i.i 

88 

182.4 

45.7 

48 

240.6 

60.3 

9 

08.7 

02.2 

69 

66.9 

16.8 

29 

125. I 

3i.3 

89 

i83.3 

45.9 

49 

241.5 

60.5 

10 

09.7 

02.4 

70 

67.9 

17.0 

3o 

126. 1 

3i.6 

90 

184.3 

46.2 

5o 

242.5 

60.7 

1 1 

10.7 

02.7 

71 

68.9 

17.3 

i3i 

127. 1 

3i.8 

191 

i85.3 

46.4 

231 

243.5 

61 .0 

12 

II. 6 

02.9 

72 

69.8 

17.5 

32 

128.0 

32.1 

92 

186.2 

46.7 

52 

244.4    61.2  1 

i3 

12.6 

03.2 

73 

70.8 

17.7 

33 

129.0 

32.3 

93 

187.2 

46.9 

53 

245.4 

61.5 

i4 

i3.6 

o3.4 

74 

71.8 

18.0 

34 

i3o.o 

32.6 

94 

18S.2 

47.1 

54 

246.4 

61.7 

i5 

14.6 

o3.6 

75 

72.8 

18.2 

35 

i3i.o 

32.8 

95 

189.2 

47-4 

55 

247-4 

62  .0 

if) 

i5.5 

03.9 

76 

73.7 

18.5 

36 

i3i  .9 

33.0 

96 

190. 1 

47-6 

56 

248.3 

62.2 

17 

16.5 

04.1 

77 

74.7 

18.7 

37 

132.9 

33.3 

97 

191 .1 

47.9 

57 

249.3 

62.4 

i8 

17.5 

04.4 

7» 

75.7 

19.0 

38 

133.9 

33.5 

98 

192. 1 

48.1 

58 

25o.3 

62.7 

'9 

1S.4 

04.6 

79 

76.6 

19.2 

39 

i34.8 

33.8 

99 

193.0 

48.4 

59 

25l  .2 

62.9 

2() 

19.4 

04.9 

80 

77.6 

19.4 

40 
i4i 

i35.8 
i36.8 

34.0 
34.3 

200 

194.0 

48.6 

60 

252.2 

63.2 

2  1 

20.4 

o5.i 

81 

78.6 

19.7 

201 

195.0 

48.8 

26! 

253.2 

63.4 

22 

21.3 

o5.3 

82 

79.5 

19.9 

42 

137.7 

34.5 

02 

195.9 

49.1 

62 

254.1 

63.7 

23 

22  3 

o5.6 

83 

80.5 

20.2 

43 

i38.7 

34.7 

o3 

196.9 

49.3 

63 

255.1 

63.9 

24 

23.3 

o5.8 

84 

81.5 

20.4 

44 

139.7 

35.0 

04 

197.9 

49.6 

64 

256.1 

64.1 

25 

24.3 

06.1 

85 

82.5 

20.7 

45 

140.7 

35.2 

o5 

198.9 

49.8 

65 

257. 1 

64.4 

26 

25.2 

06.3 

86 

83.4 

20.9 

46 

i4i.6 

35.5 

06 

199.8 

DO.  I 

66 

258. 0 

64.6 

27 

26.2 

06.6 

87 

84.4 

21 .1 

4i 

142  .6 

35.7 

07 

200.8 

5o.3 

67 

259.0 

64.9 

'      28 

27.2 

06.8 

88 

85.4 

21 .4 

48 

143.6 

36. 0 

08 

201.8 

5o.5 

68 

260.0 

65.1 

1      ^9 

28.1 

07.0 

89 

86.3 

21 .6 

49 

144.5 

36.2 

09 

202.7 

5o.8 

69 

260 . 9 

65.4 

3o 

29.1 

07.3 

90 

87.3 

21 .9 
22.1 

5o 

145.5 

36.4 

10 

2o3.7 

5i.o 

70 
271 

261 .9 
262.9 

65.6 

3i 

3o.  I 

07.5 

91 

88.3 

i5i 

146.5 

36.7 

211 

204.7 

5i.3 

65.8 

32 

3i  .0 

07.8 

92 

89.2 

22.4 

52 

147.4 

36.9 

12 

2o5.6 

5i.5 

72 

2.63.8 

66.1 

33 

32.0 

08.0 

93 

90.2 

22.6 

53 

148.4 

37.2 

i3 

206.6 

5i.8 

73 

264.8 

66.3 

34 

33.0 

08.3 

94 

91 .2 

22.8 

54 

149.4 

37.4 

i4 

207.6 

52.0 

74 

265.8 

66.6 

35 

34.0 

08.5 

95 

92.2 

23.1 

55 

i5o.4 

37.7 

i5 

208.6 

52.2 

7^ 

266.8 

66.8 

36 

34.9 

08.7 

96 

93.1 

23.3 

56 

i5i.3 

37.9 

16 

209.5 

52.5 

76 

267.7 

67.1 

37 

35.9 

09.0 

97 

94.1 

23.6 

57 

i52.3 

38.1 

17 

210.5 

52.7 

77 

268.7 

67.3 

38 

36.9 

09.2 

98 

96.1 

23.8 

58 

i53.3 

38.4 

18 

211.5 

53.0 

78 

269.7 

67.5 

39 

37.8 

09.5 

99 

96.0 

24.1 

59 

i54.2 

38.6 

19 

212.4 

53.2 

79 

270.6 

67.8 

4o 

38.8 

09.7 

100 

97 -o 

24.3 

60 

1 55. 2 

38.9 

20 

2i3.4 

53.5 

80 

271 .6 

68.0 
68.3 

4i 

39.8 

10. 0 

lOI 

98.0 

24.5 

161 

i56.2 

39.1 

221 

214.4 

53.7 

281 

272.6 

42 

40.7 

10.2 

02 

98.9 

24.8 

62 

I57.I 

39.4 

22 

2i5.3 

53.9 

82 

273.5 

68.5 

43 

41.7 

10.4 

o3 

99.9 

25.0 

63 

1 58. 1 

39.6 

23 

216.3 

54.2 

83 

274.5 

68.8 

44 

42.7 

10.7 

04 

100.9 

25.3 

64 

159. 1 

39.8 

24 

2,7.3 

54.4 

84 

275.5 

69.0 

45 

43.7 

10.9 

o5 

lOI  .9 

25.5 

65 

160. 1 

4o.i 

25 

218.3 

54.7 

85 

276.5 

69.2 

46 

44.6 

II  .2 

06 

102.8 

25.8 

66 

161 .0 

40.3 

26 

219.2 

54.9 

86 

277.4 

69.5 

47 

45.6 

II. 4 

07 

io3.8 

26.0 

67 

162.0 

40.6 

27 

220.2 

55.2 

87 

278.4 

69.7 

48 

46.6 

II. 7 

08 

104.8 

26.2 

68 

i63.o 

40.8 

28 

221.2 

55.4 

88 

279.4 

70.0 

49 

47.5 

II. 9 

09 

105.7 

26.5 

69 

163.9 

4t.i 

29 

222. 1 

55.6 

89 

280.3 

70.2 

5o 
'57 

48.5 

12. 1 

10 

106.7 

26.7 

70 

164.9 

4i.3 

3o 
^3? 

223.1 

53.9 

90 

281.3 

70.5 

49-5 

12.4 

1 1 1 

107.7 

27.0 

171 

165.9 

41.5 

224.  1 

56.1 

291 

282.3 

70.7 

52 

5o.4 

12.6 

12 

108.6 

27.2 

72 

166.8 

4i.8 

32 

225.0 

56.4 

92 

283.2 

71.0 

53 

5i.4 

12.9 

i3 

109.6 

27.5 

73 

167.8 

42.0 

33 

226.0 

56.6 

93 

284.2 

71.2 

54 

52.4 

i3.i 

i4 

no. 6 

27.7 

74 

168.8 

42.3 

34 

227.0 

56.9 

94 

285.2 

71.4 

55 

53.4 

i3.4 

i5 

III  .6 

27.9 

75 

169.8 

42.5 

35 

228.0 

57.1 

95 

286.2 

71.7 

56 

54.3 

i3.6 

16 

112. 5 

28.2 

76 

170.7 

42.8 

36 

228.9 

57.3 

96 

287. 1 

71.9 

!>7 

55.3 

i3.8 

17 

1x3.5 

28.4 

77 

171.7 

43.0 

37 

229.9 

57.6 

97 

288.1 

72.2 

58 

56.3 

i4.i 

18 

114.5 

28.7 

78 

172.7 

43.3 

38 

23o.9 

57.8 

98 

289.1 

72.4 

59 

57.2 

i4.3 

19 

ii5.4 

28.9 

79 

173.6 

43.5 

39 

231.8 

58.1 

99 

290.0 

72.7 

()0 

58.2 

i4.6 
I.at. 

20 

116. 4 

29.2 

80 

174.6 

43.7 

4o 
Dist. 

232.8 

Dep. 

58.3 
Lat. 

3oo 

291 .0 

72.9 

l>is(.|  Dop. 

Dist. 

Dop. 

i,a'. 

Dist. 

Dop. 

Lat. 

Dist. 

Dep. 

Lat. 

E.N.E.5E. 

E.S.E.fE. 

W.N.W.sW.           W.S.W.5W. 

[Far  6^  Points. 

Page  6] 

TABLE  L 

Difference  of  Latitude  and  Departure  for  li  Points. 

N.byEiE. 

N.byW.iW.                  S.byEdE                   S.byW.^W. 

Disi. 

Lat. 

Dep. 

Dist. 

Lat. 

Dep. 

Dist. 

Lat. 

Dep. 

Dist. 

Lat. 

Dep. 

Dist. 
241 

Lat. 
23o.6 

Dep. 

I 

01 .0 

00.3 

61 

58.4 

17-7 

121 

ii5.8 

35.1 

181 

173.2 

52.5 

70.0 

2 

01 .9 

00.6 

62 

59.3 

18.0 

22 

-  -^  _ 

35.4 

82 

174.2 

52.8 

42 

231.6 

70.2 

3 

02.9 

00.9 

53 

60.3 

18.3 

23 

117  7 

35.7 

83 

175. 1 

53.1 

4'i 

232.5 

70.5 

4 

o3.8 

01 .2 

64 

61.2 

1?  u 

24 

118. 7 

36.0 

84 

176. 1 

53.4 

44 

233.5 

70.8 

5 

o4.8 

01.5 

65 

62.2 

18.9 

25 

119.6 

36.3 

85 

177.0 

53.7 

45 

234.5 

71. 1 

6 

o5.7 

01.7 

66 

63.2 

19.2 

26 

120.6 

36.6 

86 

178.0 

54.0 

46 

235.4 

71.4 

7 

06.7 

02.0 

67 

64.1 

19.4 

27 

121 .5 

36.9 

87 

178.9 

54.3 

47 

236.4 

71.7 

8 

07.7 

02.3 

68 

65.1 

19.7 

28 

122.5 

37.2 

88 

179.9 

54.6 

48 

237.3 

72.0 

9 

08.6 

02.6 

69 

66.0 

20.0 

29 

123.4 

37-4 

89 

180.9 

54.9 

49 

238.3 

72.3 

lO 

09.6 

02.9 

70 

67.0 

20.3 

3o 
i3i 

124.^ 

37.7 

90 

181.8 

55.2 

5o 
251" 

239.2 

72.6 
72.9 

II 

10.5 

o3.2 

71 

67.9 

20.6 

125.4 

38.0 

191 

182.8 

55.4 

240.2 

12 

II. 5 

o3.5 

72 

68. 9 

20.9 

32 

126.3 

38.3 

92 

183.7 

55.7 

52 

241 .1 

73.2 

i3 

12.4 

o3.8 

73 

69.9 

21.2 

33 

127.3 

38.6 

93 

184.7 

56. 0 

53 

242 . 1 

73.4 

U 

i3.4 

o4.i 

74 

70.8 

21.5 

■M 

128.2 

38.9 

94 

i85.6 

56.3 

54 

243.1 

73.7 

li) 

14.4 

04.4 

75 

71.8 

21.8 

35 

129.2 

39.2 

95 

186.6 

56.6 

55 

244-0 

74.0 

i6 

i5.3 

04.6 

76 

72.7 

22.1 

36 

i3o.i 

39.5 

96 

187.6 

56-9 

56 

245. u 

74.3 

17 

16.3 

04.9 

77 

73.7 

22.4 

37 

i3i  .1 

39.8 

97 

188.5 

57.2 

57 

245.9 

74.6 

i8 

17.2 

o5.2 

78 

74.6 

22.6 

38 

l32.I 

4o.i 

98 

189.5 

57.5 

58 

246.9 

74.9 

19 

18.2 

o5.5 

79 

75.6 

22.9 

39 

i33.o 

4o.3 

99 

IVC.4 

57.8 

59 

247.8 

75.2 

20 

19.1 

o5.8 

80 

76.6 

23.2 

40 

i34.o 

40.6 

200 

191-4 

58.1 

60 
261 

248.8 

75.5 

21 

20.1 

06. 1 

81 

77.5 

23.5 

i4i 

134.9 

40.9 

201 

192.3 

58.3 

249.8 

22 

21 .1 

06.4 

82 

78.5 

23.8 

42 

135.9 

4i  .2 

02 

193.3 

58.6 

62 

250.7 

76.1 

23 

22.0 

06.7 

83 

79-4 

24.1 

43 

i36.8 

4i.5 

o3 

194.3 

58.9 

63 

25l  .7 

76.3 

24 

23.0 

07.0 

84 

80.4 

24.4 

44 

137.8 

4i.8 

04 

195.2 

59.2 

64 

262.6 

76.6 

2b 

23.9 

07.3 

85 

81.3 

24.7 

45 

i38.8 

42.1 

o5 

1 96 . 2 

59-5 

65 

253.6 

76.9 

2b. 

24.9 

07.5 

86 

82.3 

25.0 

46 

139.7 

42.4 

06 

197.1 

59.8 

66 

264.5 

77-2 

27 

25.8 

07.8 

87 

83.3 

25.3 

47 

140.7 

42.7 

07 

198.1 

60.1 

67 

255.5 

77-5 

2S 

26.8 

08.1 

88 

84.2 

25.5 

48 

141.6 

43.0 

08 

199.0 

60.4 

68 

256.5 

77-8 

29 

27.8 

08.4 

89 

85.2 

25.8 

49 

142.6 

43.3 

09 

200.0 

60.7 

69 

267.4 

78.1 

3o 

28.7 

08.7 

90 

86.1 

26.1 

5o 

143.5 

43.5 

10 

201 .0 

61 .0 
61.3 

70 
271 

268.4 

78.4 
78.7 

3i 

29.7 

09.0 

9' 

87.1 

26.4 

i5i 

144.5 

43.8 

211 

201 .9 

269.3 

32    3o.6 

09.3 

92 

88.0 

26.7 

52 

145.5 

44.1 

12 

202.9 

61.5 

72 

260.3 

79.0 

33 

3i.6 

09.6 

93 

89.0 

27.0 

53 

146.4 

44.4 

i3 

2o3.8 

61.8 

73 

261 .2 

79.2 

S4 

32.5 

09.9 

94 

90.0 

27.3 

54 

147.4 

44.7 

i4 

204.8 

62.1 

74 

262.2 

79-5 

6t> 

33.6 

10.2 

95 

90.9 

27.6 

55 

148.3 

45.0 

i5 

205.7 

62.4 

75 

263.2 

79-8 

35 

34.4 

10.5 

96 

91.9 

27.9 

56 

149-3 

45.3 

16 

206.7 

62.7 

76 

264.1 

80.1 

^7 

35.4 

10.7 

97 

92.8 

28.2 

57 

l5o.2 

45.6 

17 

207.7 

63. 0 

77 

265.1 

80.4 

d^ 

36.4 

II. 0 

98 

93.8 

28.4 

58 

i5i  .2 

45.9 

18 

208.6 

63.3 

78 

266.0 

80.7 

39 

37.3 

II. 3 

99 

94.7 

28.7 

59 

1 52. 2 

46.2 

iq 

209.6 

63.6 

7C/ 

267.0 

81.0 

40 

38.3 

II. 6 

100 

95.7, 

29.0 

60 

i53.i 

46.4 

20 
221 

210.5 

63.9 

64.2 

ho 

267.9 

81.3 
Si. 6 

4i 

39.2 

II. 9 

101 

96.7 

29.3 

161 

1 54. 1 

46.7 

211 .5 

281 

268.9 

42 

40.2 

12.2 

02 

97.6 

29.6 

62 

i55.o 

47-0 

22 

212.4 

64-4 

82 

269.9 

81.9 

i4J 

4i.i 

12.5 

o3 

98.6 

29.9 

63 

i56.o 

47-3 

23 

2i3.4 

(XI.7 

83 

270.8 

82.2 

44 

42.1 

12.8 

04 

99.5 

30.2 

64 

i56.9 

47-6 

24 

214.4 

65. 0 

84 

271 .8 

82.4 

4S 

43.1 

i3.i 

o5 

100.5 

3o.5 

65 

157.9 

47-9 

25 

2i5.3 

65.3 

85 

272.7 

82.7 

46 

44.0 

i3.4 

06 

loi  .4 

3o.8 

66 

1 58. 9 

48.2 

26 

216.3 

65.6 

86 

273.7 

83. 0 

47 

45.0 

i3.6 

07 

102.4 

3i.i 

67 

159.8 

48.5 

27 

217.2 

65.9 

87 

274.6 

83.3 

4« 

45.9 

13.9 

08 

io3.3 

3i.4 

68 

160.8 

48.8 

?^S     2,8.2 

66.2 

88 

276.6 

83.6 

49 

46.9 

l4.3 

09 

104.3 

3i.6 

69 

161 .7 

49.1 

21}      -19.1 

66.5 

89 

276.6 

83.9 

bo 

47.8 

14.5 

10 

io5.3 

3i  .9 

70 
171 

162.7 
i63.6 

49.3 
49.6 

3o 

23l 

iA).l 

66.8 

90 

277.5 

84.2 
84.5 

5i 

48.8 

i4.8 

11  ! 

106.2 

32.2 

221  .1 

67.1 

291 

278.5 

52 

49.8 

i5.i 

12 

107.2 

32.5 

72 

164.6 

49-9 

32 

222.0 

67.3 

92 

279.4 

84.8 

53 

5o.7 

i5.4 

l3 

108. 1 

32.8 

73 

165.6 

5o.2 

33 

223.0 

67.6 

93 

280.4 

85.1 

54 

5. .7 

i5.7 

i4 

109. 1 

33.1 

74 

I&6.5 

5o.5 

34 

223.9 

67.9 

94 

281.3 

85.3 

55 

52.6 

16.0 

i5 

IIO.O 

33.4 

75 

167.5 

5o.8 

35 

224.9 

68.2 

q5 

282.3 

85.6 

56 

53.6 

16.3 

16 

11 1 .0 

33.7 

76 

168.4 

5i.i 

36 

225.8 

68.5 

96 

283.3 

86.9 

57 

54.5 

16.5 

17 

112. 0 

34.0 

7^ 

169.4 

5i.4 

37 

226.8 

68.8 

97 

284.2 

86.2 

58 

55.5 

16.8 

18 

112. 9 

34.3 

78 

170.3 

5i.7 

38 

227.8 

69.1 

98 

285.2 

86. 6 

59 

56.5 

17. 1 

19 

113.9 

34.5 

79 

171 .3 

52.0 

39 

228.7 

69.4 

99 

2S6.1 

86.8 

bo 

57.4 

17-4 

20 

114. 8 

34.8 

80 

172.2 

52.3 

40 
Dist. 

229.7 

Dep. 

69.7 

3  00 

287.1 

87.1 

Dis. 

Dep. 

Lat. 

Uisl. 

Dop. 

Lat. 

Dist. 

Dep. 

Lat. 

Lat. 

l^ist.l    Dep. 

Lat. 

E.N.E.AE. 

E.S.E.iE. 

VV.N.W.^W. 

W.S.W..JIW. 

[For  G.i  Points. 

r~ 

1 

TABLE  L 

11 

'age  7    1 

Difference  of  Latitude  and  Departure  for  If  Points. 

N.byE.; 

[E. 

N.byW.^VV.                  S.byE.^E.                  S.byW.^W. 

ir.si. 

Lai. 

Dep. 

Disl. 

Lat. 

Dcp. 

Uist.     Lat. 

Dep. 

40.8 

Dist. 

Lat. 

Dep. 

Disl. 

L.ii. 

Dep. 

I   00.9 

GO.  3 

61 

57.4 

20.6 

121 

113.9 

181 

170.4 

61.0 

241 

226.9 

81.2 

2 101 .9 

00.7 

62 

58.4 

20.9 

22 

4i.i 

82 

I7'.4 

61.3 

42 

227.9 

81.5 

3   02.8 

01 .0 

63 

59.3 

21 .2 

23 

iij.8 

41.4 

83 

172.3 

61.7 

43 

228. s 

81.9 

4|o3.8 

01 .3 

64 

60.3 

21 .6 

24 

116.8 

4i  .S 

84 

173.2 

62.0 

44 

299.7 

82.2 

5 

04.7 

01.7 

65 

61.2 

21.9 

25 

117.7 

42.1 

85 

174.2 

62.3 

45 

23o.7 

82.5 

6 

ob.6 

02.0 

66 

62.1 

22.2 

26 

118.6 

42.4 

86 

175.1 

62.7 

46 

23 1  .6 

82.9 

7 

06.6 

02.4 

67 

63.1 

22.6 

27 

119.6 

42.8 

87 

176.1 

63 .0 

47 

232.6 

83.2 

8 

07.5 

02.7 

68 

64. 0 

22.9 

28 

120.5 

43.1 

88 

177.0 

63.3 

48 

233.5 

83.5 

9 

08.5 

o3.o 

69 

65. 0 

23.2 

29 

121.5 

43.5 

89 

17S.0 

63.7 

49 

234. i 

83.9 

10 

09.4 

o3.4 

70 

65.9 

23.6 

3o 

122.4 

43.8 

90 

178.9 

64.0 

5o 

235.4 

84.2 

1 1 

10.4 

03.7 

71 

66.8 

23.9 

i3i 

123.3 

44.1 

•91 

179-8 

64.3 

25l 

236.3 

84  6 

12 

11.3 

04.0 

72 

67.8 

24.3 

32 

124.3 

44.5 

92 

180.8 

64.7 

52 

237.3 

84.9 

i3 

12.2 

04.4 

73 

68.7 

24.6 

33 

125.2 

44.8 

93 

181. 7 

65.0 

53 

238.2 

85.2 

i4 

(3.2 

04.7 

74 

69.7 

24.9 

34 

126.2 

45.1 

94 

182.7 

65.4 

54 

239.2 

85.6 

lb 

I4.I 

o5.i 

7b 

70.6 

2b. 3 

35 

127.1 

45.5 

95 

i83.6 

65.7 

55 

240. 1 

85.9 

i5 

i5.i 

o5.4 

76 

71.6 

25.6 

36 

128.0 

45.8 

96 

184.5 

66.0 

56 

241 .0 

86.2 

17 

16.0 

o5.7 

77 

72.5 

25.9 

37 

129.0 

46.2 

97 

i85.5 

66.4 

57 

242.0 

86.6 

18 

16.9 

06.1 

78 

73.4 

26.3 

38 

129.9 

46.5 

98 

186.4 

66.7 

58 

242.9 

86.9 
87.3 

19 

17.9 

06.4 

79 

74.4 

26.6 

39 

i3o.9 

46.8 

99 

187.4 

67.0 

59 

243.9 

20 

18.8 

06.7 

80 

75.3 

27.0 

4o 

i3i.8 

47.2 

200 

18S.3 

67.4 

60 

244.8 

87.6 

21 

19.3  J07.1 

81 

76.3 

27.3 

i4i 

i32.8 

47-5 

201 

189.3 

67.7 

261 

245.7 

87.9 

22 

2v0.7 

07.4 

82 

77.2 

27.6 

42 

133.7 

47-8 

02 

1 90 . 2 

68.1 

62 

246.7 

88.3 

23 

21.7 

07.7 

83 

78.1 

28.0 

43 

i34.6 

48.2 

o3 

191 . 1 

68.4 

63 

247.6 

88.6 

24 

22.6 

08.1 

84 

79.1 

28.3 

44 

i35.6 

48.5 

04 

192. 1 

68.7 

64 

248.6 

8S.9 

25 

33.5 

08.4 

85 

80.0 

28.6 

45 

i36.5 

48.8 

o5 

1 93 . 0 

69.1 

65 

249.5 

89.3 

26  24.5 

08.8 

86 

81.0 

29.0 

i3-.5 

49.2 

06 

194.0 

69.4 

66 

250.5 

89.6 

27  25.4 

09.1 

87 

81.9 

29.3 

47 

i38.4 

49-!^ 

07 

194.9 

69.7 

67 

25i.4 

89.9 

28 

26.4 

09.4 

88 

82.9 

29.6 

48 

139.3 

49.9 

08 

195.8 

70.1 

68 

252.3 

90.3 

^9 

27.3 

09.8 

89 

83.8 

3o.o 

49 

i4o.3 

5o.2 

09 

196.8 

70.4 

69 

253.3 

90.6 

3o 

28.2 

10. 1 

90 

84.7 

3o.3 

bo 

l4l  .2 

5o.5 

10 

197.7 

70.7 

70 

254-2 

91.0 

3i 

29.2 

10.4 

91 

85.7 

3o.7 

i5i 

142.2 

50.9 

211 

198.7 

71.1 

271 

255.2 

91.3 

32 

3o.  I 

10.8 

92 

86.6 

3i  .0 

52 

143. 1 

5i  .2 

12 

199.6 

71-4 

72 

256. 1 

91.6 

33 

3i.i 

1 1 . 1 

93 

87.6 

3i.3 

53 

I44.I 

5i.5 

i3 

200.5 

71.8 

73 

257.0 

92.0 

34 

32. 0 

II. 5 

94 

88.5 

3. .7 

54 

145.0 

51.9 

i4 

201.5 

72.1 

74 

258. 0 

92.3 

35 

33.0 

II. 8 

9b 

89.4 

32. 0 

55 

145.9 

52.2 

i5 

202.4 

72.4 

7b 

258.9 

92.6 

36 

33.9 

12. 1 

96 

90.4 

32.3 

56 

146.9 

52.6 

16 

2o3.4 

72.8 

76 

259.9 

93.0 

37 

34.8 

12.5 

97 

91 .3 

3a. 7 

57 

147.8 

52.9 

17 

204.3 

73.. 

77 

260.8 

93.3 

38 

35.8 

12.8 

98 

92.3 

33.0 

58 

148.8 

53.2 

18 

2o5.3 

73.4 

78 

261 .7 

93.7 

39 

36.7 

i3.i 

99 

93.2 

33.4 

59 

i49-7 

53.6 

19 

206.2 

73.8 

79 

262  .7 

94.0 

4o 
4i 

37.7 
38.6 

i3.5 

100 

94.2 

33.7 

60 

i5o.6 

53.9 

20 

207.1 

74.1 

80 

263.6 

94.3 

i3.8 

lOI 

95.1 

34.0 

161 

i5i.6 

54.2 

221 

208.1 

74.5 

281 

264.6 

94.7 

42 

39.5 

i4.i 

02 

96.0 

34.4 

62 

i52.5 

54.6 

22 

209.0 

74.8 

82 

265.5 

95.0 

4i 

40. b 

14.5 

o3 

97.0 

34.7 

63 

i53.5 

54.9 

23 

210.0 

75.1 

83 

266.5 

95.3 

44 

41.4 

i4.8 

04 

97-9 

35.0 

64 

i54.4 

55.2 

24 

210.9 

75.5 

84 

267.4 

95.7 

4^ 

42.4 

l5.2 

ob 

98.9 

35.4 

65 

i55.4 

55.6 

25 

211. 8 

75.8 

85 

268.3 

96.0 

46 

43.3 

i5.5 

06 

99.8 

35.7 

66 

i56.3 

55.9 

26 

212.8 

76.1 

8() 

269 . 3 

96.4 

47 

44.3 

i5.8 

07 

K)0 . 7 

36. 0 

67 

157.2 

56.3 

27 

213.7 

76.5 

87 

270.2 

9*'-7 

48 

45.2 

16.2 

08 

101.7 

56.4 

68 

i58.2 

56.6 

28 

214.7 

76.8 

88 

271 .2 

97.0 

49 

46.1 

16.5 

09 

102.6 

36.7 

69 

159.1 

56.9 

29 

2i5.6 

77.1 

89 

272 . 1 

97.4 

bo 

47.1 

16.8 

10 

io3.6 

37.1 
37.4 

70 

160. 1 

57.3 

3o 

216.6 

77.5 

90 

273.0 

97.7 
98.0 

5( 

48. 0 

17.2 

III 

104.5 

171 

161 .0 

57.6 

23l 

217.5 

77.8 

291 

274.0 

b2 

4'9.o 

17. b 

12 

io5.5 

37.7 

72 

161 .9 

57.9 

3p 

218.4 

78.2 

92 

274.9 

98.4 

b3 

49.9 

17.9 

i3 

106.4 

38.1 

73 

162.9 

58.3 

33 

219.4 

78.5 

93 

275.9 

98.7 

b4 

bo. 8 

18.2 

i4 

107.3 

38.4 

74 

i63.8 

58.6 

34 

220.3 

78.8 

94 

276.8 

99.0 

bb 

bi.8 

18.5 

lb 

108.3 

38.7 

75 

164.8 

59.0 

35 

221 .3 

79.2 

95 

277.8 

99.4 

bb 

52.7 

18.9 

16 

109.2 

39.1 

76 

165.7 

59.3 

36 

222.2 

79.5 

96 

278  7 

99-7 

b7 

b3.7 

19.2 

17 

110.2 

39.4 

77 

166.7 

59.6 

37 

223.1 

79.8 

97 

279.6 

100. 1 

b8 

54.6 

19. b 

i8 

I II  .1 

39.8 

78 

167.6 

60.0 

38 

224.1 

80.2 

98 

280.6 

100.4 

D9 

bb.6 

19.9 

19 

1 12.0 

4o.i 

79 

168.5 

60.3 

39 

225. 0 

80.5 

99 

281.5 

100.7 

bo 

bb.b 

20.2 

20 

it3.o 

40.4 

80 

169.5 

60.6 

40 
Dist. 

226.0 
Drp. 

80.9 

3oo 

282.5 

101. 1 

Dist. 

Dcp. 

Lat. 

Dist. 

Dpp. 

Lat. 

Dist. 

Drp. 

Lat. 

Lni. 

Di<i, 

Dcp. 

Lat. 

E.N.E.^E. 

E.S.E.^E. 

W.N.W.iW. 

W.S.W.iW. 

[For  C)^  Po 

Ills. 

Page  8] 

TABLE  I 

Difference  of  Latitude  and  Departure  for  2  Points. 

iN.N.E. 

N.N.W.                      S.S.E.                      S.S.W. 

Oist. 

Lat. 

Dcp. 

Disl. 

Lat. 

Dep. 
TO" 

Dist. 

Lat. 

Dep. 

Dist. 

Lat. 

Dep. 

Dist. 

Lat. 

Dep. 

I 

00.9 

00.4 

61 

56.4 

121 

III. 8 

46.3 

181 

167.2 

69.3 

241 

222.7 

92.2 

2 

01 .8 

00.8 

62 

57.3 

23.7 

22 

112. 7 

46.7 

82 

168. 1 

69.6 

42 

223.6 

92.6 

3 

02.8 

01 . 1 

63 

58.2 

24.1 

23 

ii3.6 

47.1 

83 

169. 1 

70.0 

Ai 

224.5 

93.0 

/. 

03.7 

01.5 

64 

59.1 

24.5 

24 

114.6 

47.5 

84 

170.0 

70.4 

M 

225.4 

93.4 

5 

<>4.6 

01.9 

65 

60.1 

24.9 

25 

ii5.5 

47-8 

85 

170.9 

70.8 

45 

226.4 

93.8 

6 

u5.5 

02.3 

66 

61.0 

25.3 

26 

116.4 

48.2 

86 

171. 8 

71.2 

46 

227.3 

94-1 

7 

06.5 

02.7 

67 

61.9 

25.6 

27 

117.3 

48.6 

87 

172.8 

71.6 

47 

228.2 

94.5 

8 

07.4 

o3.i 

68 

62.8 

26.0 

28 

118. 3 

49.0 

88 

173.7 

71.9 

48 

229.1 

94.9 

Q 

08.3 

o3.4 

69 

63.7 

26.4 

29 

119. 2 

49-4 

89 

174.6 

72.3 

49 

23o.O 

95.3 

lO 

09.2 

o3.8 

70 

64.7 

26.8 

3o 
i3i 

120. 1 

J21  .0 

49.7 
5o.i 

90 

175.5 

72.7 

5o 

23l  .0 

95.7 

1 1 

10.2 

04.2 

71 

65.6 

27.2 

191 

176.5 

73.1 

25l 

23l  .9 

96.1 

13 

II  .1 

04.6 

72 

66.5 

27.6 

32 

122.0 

5o.5 

92 

177-4 

73.5 

52 

232.8 

96.4 

i3 

12.0 

o5.o 

73 

67.4 

27.9 

33 

122.9 

50.9 

93 

178.3 

73.9 

53 

233.7 

96.8 

i4 

12.0 

o5.4 

74 

68.4 

28.3 

34 

123.8 

5i.3 

94 

179.2 

74-2 

54 

234.7 

97-2 

i5 

I3.q 

05.7 

75 

69.3 

28.7 

35 

124.7 

5i.7 

95 

180.2 

74.6 

55 

235.6 

97.6 

i6 

i4.8 

06. 1 

76 

70.2 

29.1 

36 

125.6 

52. 0 

96 

181. 1 

75.0 

56 

236.5 

98.0 

17 

l5.7 

06.5 

77 

71. 1 

29.5 

37 

126.6 

52.4 

97 

182.0 

75.4 

57 

237.4 

98.3 

iS 

16.6 

06.9 

78 

72.1 

29.8 

38 

127.5 

52.8 

98 

182.9 

75.8 

58 

238.4 

98.7 

19 

17.6 

07.3 

79 

73.0 

3o.2 

39 

128.4 

53.2 

99 

183.9 

76.2 

59 

23q.3 

99.1 

20 

18.5 

07.7 

80 

73.9 

3o.6 
3i  .0 

4o 
i4i 

129.3 

53.6 

200 

184.8 

76.5 

60 

240 . 2 

99.5 

21 

19.4 

08.0 

81 

74.8 

i3o.3 

54.0 

201 

185.7 

76.9 

261 

241 .1 

99.9 

22 

20.3 

08.4 

82 

75.8 

3i.4 

42 

i3i  .2 

54.3 

02 

186.6 

77.3 

62 

242.1 

100.3 

?.3 

21 .2 

08.8 

83 

76.7 

3i.8 

43 

l32.I 

54.7 

o3 

187.5 

77-7 

63 

243.0 

100.6 

24 

22.2 

09.2 

84 

77.6 

32.1 

AA 

i33.o 

55.1 

04 

188.5 

78.1 

64 

243.9 

101.0 

25 

23.1 

09.6 

85 

78.5 

32.5 

45 

i34.o 

55.5 

o5 

189.4 

78.5 

65 

244.8 

101.4 

26 

24.0 

09.9 

86 

79-5 

32.9 

46 

134.9 

55.9 

06 

190.3 

78.8 

66 

245.8 

101.8 

27 

24.9 

10.3 

87 

80.4 

ii.i 

47 

i35.S 

5'j.3 

07 

191 .2 

79.2 

67 

246.7 

102.2 

28 

25.9 

10.7 

88 

81.3 

33.7 

48 

i36.7 

56.6 

08 

192.2 

79.6 

68 

247.6 

102.6 

^9 

26.8 

1 1 . 1 

8q 

82.2 

34.1 

49 

137.7 

57.0 

09 

193.1 

80.0 

69 

248.5 

102.9 

3o    27.7 

II. 5 

90 

83.1 

34.4 

5o 

i38.6 

57  .-4 

10 

194.0 

80.4 

80.7 

70 

249.4 

io3.3 

3 1 

28.6 

II. 9 

91 

84.1 

34.8 

i5i 

139.5 

57.8 

21 1 

194.9 

271 

25o.4 

103.7 

3?. 

29.6 

12.2 

92 

85.0 

35.2 

52 

140.4 

58.2 

12 

195.9 

81. 1 

72 

251.3 

104.1 

33 

3o.5 

12.6 

93 

85.9 

35.6 

53 

141.4 

58.6 

i3 

196.8 

81.5 

73 

252.2 

104.5 

34 

3i.4 

i3.o 

94 

86.8 

36. 0 

54 

142.3 

58.9 

i4 

197-7 

81.0 

74 

253.1 

104.9 

35 

32.3 

i3.4 

95 

87.8 

36.4 

55 

143.2 

59.0 

i5 

19S.6 

82.3 

7^ 

254.1 

105.2 

36 

33.3 

i3.8 

96 

88.7 

36.7 

56 

144. 1 

^9-7 

16 

199.6 

82.7 

76 

255. 0 

io5.6 

37 

34.2 

l4.2 

97 

89.6 

37.1 

57 

145.0 

60.1 

17 

200.5 

83.0 

77 

255. Q 

106.0 

38 

35.1 

14.5 

98 

90.5 

37.5 

58 

i46.o 

60.5 

18 

201 .4 

83.4 

78 

256.8 

106.4 

39 

36.0 

14.9 

99 

91.5 

37.9 

59 

146.9 

60.8 

'9 

202.3 

83.8 

79 

257.8 

106.8 

40 

37.0 

i5.3 

100 

92.4 

38.3 

60 
i6x 

147.8 
148.7 

61 .2 
61.6' 

20 

2o3.3 

84.2 

80 

258.7 

107.2 

41 

37.9 

l5.7 

lOI 

93.3 

38.7 

221 

204  .  2 

84.6 

281 

259.6 

107.5 

42 

38.8 

16. 1 

02 

94.2 

39.0 

62 

149.7 

62.0 

22 

2o5.i 

85.0 

82 

260 . 5 

107.9 

43 

39.7 

16.5 

o3 

95.2 

39.4 

63 

i5o.6 

62.4 

23 

206.0 

85.3 

83 

261.5 

108.3 

AA 

40.7 

16.8 

04 

96. 1 

39.8 

64 

i5i.5 

62.8 

24 

206.9 

85.7 

84 

262.4 

108.7 

45 

41.6 

17.2 

o5 

97.0 

40.2 

65 

i52.4 

63.1 

2  5 

207.9 

86.1 

85 

263.3 

109.1 

46 

42.5 

17.6 

06 

97-9 

40.6 

66 

i53.4 

63.5 

26 

208.8 

86.5 

86 

264.2 

109.4 

47 

43.4 

18.0 

07 

98.9 

40.9 

67 

154.3 

63.9 

27 

209.7 

86.9 

87 

265.2 

109.8 

48 

44.3 

18.4 

08 

9Q.8 

41.3 

68 

155.2 

64.3 

28 

210.6 

87.3 

88 

266. 1 

1 10.2 

4q 

45.3 

18.8 

09 

100.7 

41.7 

69 

i56.i 

64.7 

29 

21  I  .6 

87.6 

89 

267.0 

110.6 

5o 

46.2 

19. 1 

10 

loi  .6 

42.1 

70 

157. 1 

65.1 

3o 

212.5 

88.0 

90 

267.9 

11 1.0 
111.4 

5r 

47.1 

19.5 

III 

102.6 

42.5 

171 

i5&.o 

65.4 

23l 

2i3.4 

88.4 

291 

268.8 

52 

48. 0 

19.9 

r? 

io3.5 

42.9 

72 

i58.9 

65.8 

32 

214.3 

88.8 

92 

269.8 

111.7 

53 

49.0 

20.3 

i3 

104.4 

43.2 

73 

159.8 

66.2 

33 

2i5.3 

89.2 

93 

270.7 

112.1 

54 

49.9 

20.7 

i4 

105.3 

43.6 

74 

160.8 

66.6 

M 

2l6.2 

89.5 

94 

271 .6 

112. 5 

55 

5o.8 

21 .0 

i5 

106.2 

44.0 

7^ 

161 .7 

67.0 

35 

217. 1 

89.9 

9^ 

272.5 

112. 9 

56 

5i.7 

21.4 

16 

107.2 

U.A 

76 

162.6 

67.4 

36 

218.0 

90.3 

96 

273.5 

ii3.3 

57 

52.7 

21.8 

17 

108. 1 

44.8 

77 

i63.5 

67.7 

37 

219.0 

90.7 

97 

274.4 

!i3.7 

58 

53.6 

22.2 

18 

109.0 

45.2 

78 

164.5 

68.1 

38 

219.9 

91.1 

9« 

275.3 

1  i4o 

5q 

54.5 

22.6 

19 

109.9 

45.5 

79 

i65.4 

68.5 

39 

220.8 

91.5 

99 

276.2 

114.4 

60 

55.4 

23. 0 

20 

no. 9 

45.9 

80 

166.3 

68.9 

4o 

221  .7 

91.8 

3  00 

277-2 

11 4-8 

Di^t. 

Dop. 

Lat. 

Dist. 

Drp. 

Lat. 

Dist. 

Dep. 

I>at. 

Dist.i    Dep. 

Lat. 

Dist. 

Dep. 

Lat. 

E.N.E. 

E.S.E. 

W.N.W. 

w.s.w. 

[For  6  Points. 

TABLE  L 

[Page  9 

Diflference  of  Latitude  and  Departure  for  2|-  Points. 

iX.N.E. 

^E.                 N.N.W.iW.                 S.S.E.AE.                 S.S.W.^W. 

Dlst.    Lai. 

Dcp. 

Dist. 

Lat. 

Dep. 

Dist 

Lat. 

Dep. 

Dist. 

Lat. 

Dep. 

Dist. 

Lat. 

Dcp. 

I    00.9 

00.4 

61 

55.1 

26.1 

121 

109.4 

5. .7 

181 

i63.6 

77-4 

241 

217.9 

ic3  c 

2 

01.8 

00. c 

b2 

56. 0 

26.5 

22 

no. 3 

52.2 

82 

164.5 

77.8 

42 

218.8 

io3.5 

3 

02 .7 

01 .3 

63 

57.0 

26.9 

23 

III. 2 

52.6 

83 

i65.4 

78.2 

43 

219.7 

103.9 

4 

o3.6 

01.7 

b4 

57-9 

27.4 

24 

112. 1 

b3.o 

84 

166.3 

78.7 

44 

220.6 

104.3 

5 

o4.5 

02. 1 

6b 

58.8 

27.8 

25 

ii3.o 

53.4 

85 

167.2 

79.1 

45 

221.5 

104.8 

0 

o5.4 

02.6 

6b 

59.7 

28.2 

2b 

113.9 

53.9 

86 

168. 1 

79.5 

46 

222.4 

io5,2 

7 

06.3 

o3  0 

67 

60. G 

28.6 

27 

114.8 

54.3 

87 

169.0 

80.0 

47 

223.3 

105.6 

8 

07.2 

o3.4 

68 

61.5 

29.1 

28 

11D.7 

54.7 

88 

169.9 

80.4 

48 

224.2 

106.0 

9 

08.1 

o3.8 

69 

62.4 

29.5 

29 

116. 6 

bb.2 

89 

170.9 

80.8 

49 

225.1 

106.5 

10 

09.0 

04.3 

70 
71 

63.3 
64.2 

29.9 

3o.4 

3o 

117. 5 

bb.6 

90 

171. 8 

81.2 

5o 

226.0 

106.9 

II 

09.9 

04.7 

i3i 

118. 4 

56.0 

191 

172.7 

81.7 

25l 

226.9 

107.3 

12 

10.8 

ob.i 

72 

65.1 

3o.8 

32 

119. 3 

56.4 

92 

173.6 

82.1 

52 

227.8 

107.7 

i3 

II. 3 

o5.b 

li 

66.0 

3l.2 

■6-^ 

120.2 

56.9 

93 

174.5 

82.5 

53 

228.7 

10S.2 

i4 

12.7 

06.0 

l4 

66.9 

3i.b 

3h 

121 . 1 

57.3 

94 

175.4 

82.9 

54 

229.6  1  108.6  1 

lb 

i3.b 

06.4 

7b 

67.8 

32.1 

3b 

122.0 

57-7 

95 

176.3 

83.4 

55 

230.5 

109.0 

16 

i4.5 

06.8 

76 

68.7 

32.b 

36 

122.9 

58.1 

96 

177.2 

83.8 

56 

23i.4 

109.5 

17 

lb. 4 

07.3 

77 

69.6 

32.9 

37 

123.8 

58.6 

97 

178.1 

84.2 

57 

232.3 

109.9 

18 

lb. 3 

07.7 

7» 

70.5 

ii.>, 

38 

124.8 

59.0 

98 

179.0 

84.7 

58 

233.2 

1 10.3 

19 

17.2 

oS.i 

V 

71.4 

33.8 

39 

125.7 

59.4 

99 

179.9 

85.1 

59 

234.1 

110.7 

20 

18. 1 

08.6 

80 

72.3 

34.2 

4o 

126.6 

59.9 

200 

180.8 

85.5 

60 

235. 0 

III. 2 

21 

19.0 

09.0 

81 

73.2 

34.6 

i4i 

127.5 

60.3 

201 

181. 7 

85.9 

261 

235.9 

111.6 

22 

19.9 

09.4 

bii 

74.1 

3d. I 

42 

128.4 

60.7 

02 

182.6 

86.4 

62 

236.8 

112.0 

2J 

20.8 

09.8 

83 

75.0 

3b.b 

43 

129.3 

61. 1 

o3 

i83.5 

86.8 

63 

237.7 

112.4 

24 

SI. 7 

10.3 

84 

75.9 

3b.9 

•44 

i3o.2 

61.6 

04 

184.4 

87.2 

64 

238.7 

112.9 

2b 

22.6 

10.7 

8b 

76.8 

3b.3 

4':^ 

i3i .  I 

62.0 

o5 

i85.3 

87.6 

65 

239.6 

ii3.3 

2b 

23.5 

II  .1 

8b 

77-7 

3b.8 

46 

l32.0 

62.4 

06 

186.2 

88.1 

66 

240.5 

II3.7 

27 

24.4 

II. 5 

B7 

78.6 

37.2 

47 

132.9 

62.9 

07 

187.1 

88.5 

67 

241.4 

114.2 

28 

25.3 

12.0 

88 

79.6 

37.b 

48 

i33.8 

63.3 

08 

188.0 

88.9 

68 

242.3 

114.6 

29 

26.2 

12.4 

89 

80.5 

38.1 

49 

134.7 

63.7 

09 

188.9 

89.4 

69 

243.2 

ii5.o 

Jo 

27.1 

12.8 

90 

81.4 

38. b 

bo 

i35.6 

64.1 

10 

189.8 

89.8 

70 

244.1 

1 1 5.4 

3i 

28.0 

i3.3 

91 

82.3 

38.9 

i5i 

i36.5 

64.6 

211 

190.7 

90.2 

271 

245.0 

115-9 

32 

28.9 

i3.7 

92 

83.2 

39.3 

52 

137.4 

65.0 

12 

191 .6 

90.6 

72 

245.9 

116.3 

3i 

29.8 

i4.i 

9^ 

84.1 

39.8 

53 

i38.3 

65.4 

i3 

192.5 

91. 1 

73 

246.8 

116.7 

34 

3o.7 

i4.b 

94 

85. 0 

4o.2 

b4 

109.2 

65.8 

i4 

193.5 

91.5 

74 

247.7 

117.2 

3b 

3i.b 

ib.O 

9b 

85.9 

4o.b 

bb 

i4o.i 

66.3 

i5 

194.4 

91.9 

75 

248.6 

II  7.6 

36 

32. b 

lb. 4 

96 

86.8 

4i.o 

bb 

i4i  .0 

66.7 

16 

195.3 

92.4 

76 

249.5 

118.0 

37 

33.4 

lb. 8 

97 

87.7 

4i.b 

b7 

141.9 

67.1 

17 

196.2 

92.8 

77 

25o.4 

118.4 

38 

34.4 

16.2 

98 

88.6 

41.9 

b8 

142.8 

67.6 

18 

197.1 

93.2 

78 

25i.3 

1 18.9 

39 

^0.3 

lb. 7 

99 

89.5 

42  3 

b9 

143.7 

68.0 

19 

198.0 

93.6 

79 

252.2 

119.3 

40 
4i 

3b. 2 

17. 1 

100 

90.4 

42.8 

bo 

144.6 

6S.4 

20 

198.9 

94.1 

80 

253.1 

119-7 

37.1 

17.5 

lOI 

91 .3 

43.2 

161 

145.5 

68.8 

221 

199.8 

94.5 

281 

254.0 

1 20. 1 

42 

38. 0 

18.0 

02 

92.2 

43.b 

62 

146.4 

69.3 

22 

200.7 

94-9 

82 

254.9 

120.6 

43 

38.9 

18.4 

o3 

93.1 

44.0 

63 

147.4 

69.7 

23 

201 .6 

95.3 

•83 

255.8 

1 21.0 

44 

39.8 

18.8 

o4 

94.0 

44.b 

64 

148.3 

70.1 

24 

202.5 

95.8 

84 

256.7 

121.4 

4b 

40.7 

19.2 

Ob 

94.9 

44.9 

bb 

149.2 

70.5 

25 

2o3.4 

96.2 

85 

257.6 

121. 9 

46 

41.6 

19.7 

Ob 

9b.8 

4b.3 

bb 

i5o.i 

71.0 

26 

204.3 

96.6 

86 

258.5 

122.3 

47 

42.  b 

20.1 

07 

96.7 

4b.7 

67 

i5i  .0 

71-4 

27 

205.2 

97.1 

87 

259.4 

122.7 

48 

4i.4 

20.5 

08 

97.6 

46.2 

b8 

i5i  .9 

71.8 

28 

206.1 

97.5 

88 

260.3 

123. 1 

^9 

44.  i 

21.0 

09 

98.5 

4b.b 

69 

i52.8 

72.3 

29 

207.0 

97.9 

89 

261.3 

123.6 

bo 

45.2 

21.4 

10 

99.4 

47-0 

70 

153.7 

72.7 

3o 

207.9 

98.3 

90 

262.2 

124.0 

5i 

46.1 

21.8 

III 

100.3 

47-5 

171 

i54.6 

73.1 

23l 

208.8 

98.8 

291 

263.1 

124.4 

b2 

47 -o 

22.2 

12 

101.2 

47-9 

72 

i55.5 

73.5 

32 

209.7 

99.2 

92 

264.0 

124.8 

b3 

47-9 

22.7 

i3 

102.2 

48.3 

73 

i56.4 

74.0 

33 

210.6 

99.6 

93 

264.9 

125.3 

^4 

48.8 

23.1 

i4 

io3.i 

48.7 

74 

157.3 

74.4 

34 

211  .5 

lOO.O 

94 

265.8 

125.7 

bb 

49.7 

23. b 

lb 

104.0 

49.2 

75 

i58.2 

74.8 

35    212.4 

100.5 

95 

266.7    126.1  1 

Db 

5o.6 

23.9 

lb 

104.9 

105.8 

49.6 

76 

159. 1 

75.2 

36 

2i3.3 

100.9 

q6 

267.6 

1266 

57   5i.5 

24.4 

17 

5o.o 

77 

160.0 

75.7 

37 

214.2 

101.3 

97 

268.5 

127,0 

58 

b2.4 

24.8 

18 

106.7 

bo.b 

78 

160.9 

76.1 

38 

2l5.1 

101.8 

q8 

269.4 

127.4 

b9 

b3  3 

2b. 2 

19 

107.6 

50.9 

79 

161. 8 

76.5 

39 

216. I      102.2 

QQ 

270.3 

127.8 

bo 

54.2    25.7 

20 

108.5 

bi.3 

80 

162.7 

77.0 

4o 

217.0      102.6 

3oo 

271 .2 

128.3 

Dist. 

Dpp.  1  Lat. 

Dist. 

Dep. 

Lat. 

Dist. 

Dep. 

Lat. 

Dist. 

Dcp.      Lat. 

Dist. 

Dnp. 

Lat. 

N.E.byE4E. 

S.E.byE.^E. 

N.W.byW.sW. 

S.W.byW.^W. 

[For  h\  Points. 

Page  10] 

TABLE  L 

Differ 

ence  of  Latitude  and  Departure  for  2^  Points. 

N.N.E.iE. 

N.N.W.^W.                 S.S.E.^E.                 S.S.W.JW. 

D!St.[  Lat. 

Deo. 

Dist. 

Lat. 

D«p. 

Dist. 

Lat. 

Dep. 

Dist. 

Lat. 

Dep. 

Dist. 

Let. 

Dep. 

1 1 3.6 

I  1  00 . 9 

00.5 

61 

53.8 

28.8 

121 

106.7 

57.0 

181 

159.6 

85.3 

241 

212.5 

2 

01.8 

00.9 

62 

54  7 

29.2 

22 

107.6 

57.5 

82 

160.5 

85.8 

42 

2i3.4 

114.1 

3 

02.6 

01 .4 

63 

55.6 

29.7 

23 

108.5 

58.0 

83 

161 .4 

86.3 

43 

214.3 

1 14.5 

4 

o3.5 

01 .9 

64 

56.4 

3o.2 

24 

109.4 

58.5 

84 

162.3 

86.7 

44 

2l5.2 

1 1 5.0 

5 

o4.4 

02.4 

65 

57.3 

3o.6 

25 

no. 2 

58.9 

85 

i63.2 

87.2 

45 

216. I 

1 1 5.5 

6 

o5.3 

02..  8 

66 

58.2 

3i.i 

26 

III  .1 

59.4 

86 

164.0 

87.7 

46 

217.0 

1 16.0 

7 

o6.2 

o3.3 

67 

59.1 

3i.6 

27 

[12.0 

59.9 

87 

164.9 

88.2 

47 

217.3 

116.4 

8 

07.1 

o3.8 

68 

60.0 

32.1 

28 

112. 9 

60.3 

88 

i65.8 

88.6 

48 

218.7 

116.9 

9 

07.9 

04.2 

69 

60.9 

32.5 

29 

ii3.8 

60.8 

89 

166.7 

89.1 

49 

219.6 

117.4 

lO 

08.8 

04.7 

70 

61.7 

33.0 

3o 

114. 6 

61.3 

90 

167.6 

89.6 

5o 

220.5 

117.8 

1 1 

09.7 

05.2 

71 

62.6 

33.5 

i3i- 

ii5.5 

61.8 

191 

168.4 

90.0 

25l 

221  .4 

118.3 

12 

10.6 

o5.7 

72 

63.5 

33.9 

32 

116. 4 

62.2 

92 

169.3 

90.5 

52 

222.2 

118.8 

i3 

II. 5 

06.1 

73 

64.4 

34.4 

33 

117. 3 

62.7 

93 

170.2 

91.0 

53 

223.1 

119.3 

i4 

12.3 

06.6 

74 

65.3 

34.9 

M 

118. 2 

63.2 

94 

171. 1 

91.5 

54 

224.0 

119-7 

i5 

l3.2 

07.1 

75 

66.1 

3b.4 

35 

119. 1 

63.6 

95 

172.0 

91.9 

55 

224.9 

120.2. 

i6 

14. 1 

07.5 

76 

67.0 

35.8 

36 

119. 9 

64.1 

96 

172.9 

92.4 

56 

225.8 

120.7 

17 

i5.o 

08.0 

77 

67.9 

3bJ 

37 

120.8 

64.6 

97 

173.7 

92.9 

57 

226.7 

121. 1 

i8 

l5.9 

08.5 

78 

68.8 

36.8 

38 

121 .7 

65.1 

98 

174.6 

93.3 

58 

227.5 

121.6 

19 

16.8 

09.0 

79 

69.7 

37.2 

39 

122.6 

65.5 

99 

175.5 

93.8 

59 

228.4 

122. 1 

20 

17.6 

09.4 

80 

70.6 

37.7 

4o 

123.5 

66.0 

200 

176.4 

94.3 

60 

229.3 

122.6 

21 

18.5 

09.9 

81 

71.4 

38.2 

i4i 

124.4 

66.5 

201 

177.3 

94.8 

261 

23o.2 

123.0 

22 

19.4 

10.4 

82 

72.3 

38.7 

42 

I2D.2 

66.9 

02 

178. 1 

95.2 

62 

23l  .1 

123.5 

23 

20.3 

10.8 

83 

73.2 

39.1 

43 

126. 1 

07.4 

00 

179.0 

95.7 

63 

23l  .9 

124.0 

24 

21.2 

II. 3 

84 

74.1 

39.6 

44 

127.0 

67.9 

%4 

179.9 

96.2 

64 

232.8 

124.4 

25 

22.0 

II. 8 

85 

75.0 

a'o.i 

45 

127.9 

68.4 

ob 

180.8 

96.6 

65 

233.7 

124.9 

26 

22.9 

12.3 

86 

75.8 

40.5 

46 

128.8 

68.8 

06 

181. 7 

97.1 

66 

234.6 

125.4 

27 

23.8    12.7 

«7 

76.7 

4i.o 

47 

129.6 

69.3 

07 

182.6 

97.6 

671  235.5 

125.9 

28 

24.7 

l3.2 

88 

77.6 

41.5 

48 

i3o.5 

69.8 

08 

i83.4 

98.1 

68 

236.4 

126.3 

29 

25.6 

.3.7 

89 

78.5 

42.0 

49 

i3i.4 

70.2 

09 

184.3 

98.5 

69 

237.2 

126.8 

3o 

26.5 

i4.i 

90 

79-4 

42.4 

bo 

1 32. 3 

70.7 

10 

i85.2 

99.0 

70 

238.1 

127.3 

3i 

27.3 

14.6 

91 

80.3 

42.9 

i5i 

i33.2 

71.2 

21 1 

186.1 

99.5 

271 

239.0 

127.7 

32 

28.2 

i5.i 

92 

81. 1 

4i.4 

b2 

i34.i 

71.7 

12 

187.0 

99-9 

72 

239.9 

128.2 

33 

29.1 

i5.6 

93 

82.0 

43.8 

53 

134.9 

72.1 

i3 

187.8 

100.4 

73 

■2A0.S 

128.7 

34 

3o.o 

16.0 

94 

82.9 

44.3 

54 

i35.8 

72.6 

i4 

188.7 

100.9 

74 

241.6 

129.2 

35 

3o.9 

16.5 

95 

83.8 

44.8 

bb 

1 36. 7 

73.1 

i5 

189.6 

101.4 

75 

242.5 

129.6 

36 

3l.7 

17.0 

96 

84.7 

45.3 

b6 

137.6 

73.5 

16 

190.5 

101.8 

76 

243.4 

i3o.i 

37 

32.6 

17-4 

97 

85.5 

45.7 

57 

i38.5 

74.0 

17 

191 .4 

102.3 

77 

244.3 

i3o.6 

38 

33.5 

17.9 

98 

86.4 

46.2 

b8 

139.3 

74.5 

18 

192.3 

102.8 

78 

245.2 

i3i.o 

39 

34.4 

.8.4 

99 

87.3 

46.7 

59 

l40.2 

75.0 

19 

193. 1 

io3,2 

79 

246.1 

i3i.5 

4o 

35.3 

18.9 

100 

88.2 

47-1 

60 

i4i  .1 

75.4 

20 

194.0 

103.7 

80 

246.9 

l32.0 

4i 

36.2 

.9.3 

lOI 

89.1 

47.6 

161 

142.0 

75.9 

221 

194.9 

104.2 

281 

247.8 

i32.5 

4s 

37.0 

19.8 

02 

90.0 

48.1 

62 

142.9 

76.4 

22 

195.8 

io4;7 

82 

248.7 

132.9 

43 

37.9 

20.3 

o3 

90.8 

48.6 

63 

143.8 

76.8 

.23 

196.7 

io5.i 

83 

249.6 

i33.4 

44 

38.8 

20.7 

04 

91.7 

49.0 

64 

144.6 

77.3 

24 

197.6 

io5.6 

84 

25o.5 

133.9 

45 

J9.7 

21.2 

o5 

92.6 

49.5 

65 

145.5 

77.8 

25 

198.4 

1 06. 1 

85 

25i.3 

1 34.3 

40 

40.6 

21.7 

06 

93.5 

5o.o 

66 

146.4 

78.3 

26 

199.3 

106.5 

86 

252.2 

134.8 

47 

41.5 

22.2 

07 

94.4 

5o.4 

67 

147.3 

78.7 

27 

200.2 

107.0 

87 

253.1 

1 35.3 

48 

42.3 

22.6 

08 

95.2 

50.9 

68 

i48.2 

79.2 

28 

201. 1 

107.5 

88 

254.0 

i3b.8 

49 

43.2 

23.1 

09 

96.1 

5i.4 

69 

149.0 

79-7 

29 

202.0 

107.9 

89 

254.9 

i36.2 

bo 

44.1 

23.6 

10 

97.0 

51.9 

70 

149.9 

80.1 

3o 

202.8 

108.4 

90 

255.8 

1 36.7 

5i 

45  0 

24.0 

III 

97-9 

52.3 

171 

i5o.8 

80.6 

23l 

203.7 

108.9 

291 

256.6 

137.2 

52 

45  9    24.5 

12 

98.8 

52.8 

72 

i5i  .7 

81. 1 

32 

204.6 

109.4 

92 

257.5 

137.6 

53 

46.7 

25.0 

i3 

99-7 

53.3 

73 

i52.6 

81.6 

33 

2o5.5 

109.8 

93 

258.4 

i38.i 

54 

47.6 

25.5 

i4 

100.5 

53.7 

74 

i53.5 

82.0 

34 

206.4 

110.3 

94 

259.3 

i38.6 

55 

48.5 

25.9 

i5 

101.4 

54.2 

75 

i54.3 

82.5 

35 

207.3 

1 10.8 

95 

260.2 

139.1 

56 

49-4 

26.4 

16 

102.3 

54.7 

76 

i55.2 

83. 0 

36 

208.1 

III. 2 

96 

261 .0 

139.5 

57 

50.3 

26.9 

17 

I03.2 

55.2 

77 

i56.i 

83.4 

37 

209.0 

1 1 1.7 

97 

261 .9 

i4o.o 

58 

5l.2 

27.3 

18 

104. 1 

55.6 

78 

157.0 

83.9 

38 

209.9 

112. 2 

98 

262.8 

i4o5 

59 

52  .0 

27.8 

19 

104.9 

56.1 

79 

157.9 

84.4 

39 

210.8 

112.7 

99 

263.7 

140.9 

6o 
Dist 

52.9 

28.3 

20 

io5.8 

56.6 

80 

i58.7 

84.9 

40 

21 1 .7 

ii3.i 

3oo 

264.6 

i4i4 

Dep. 

l.iit. 

Dist. 

Pep. 

I.ai. 

Dlsl. 

Dop. 

Lat. 

Dist. 

Dep. 

Lat. 

Dist. 

Dep. 

Lai. 

N.E.byE.AE.        S 

E.byE.^E. 

N.W.byW.^W. 

S.W.byW.^W. 

[For  5^  Points. 

TABLE  L 

[Page  U 

Difference  of  Latitude  and  Departure  for  2|  Points. 

N.N.E4E.                N.N.W.|W.                 S.S.E.|E.                S.S.W.|W. 

Disl. 

Lat. 

Dop. 

Dist. 

Lat. 

Dep. 

Dist. 

Lat. 

Dep. 

Dist. 

Lat.    !  Dep. 

Dist. 
241 

Lat. 
206.7 

Dep. 
123.9 

I 

00.9 

00.5 

61 

52.3 

3i.4 

121 

io3.8 

62.2 

181 

i55.2 

93.1 

2 

01.7 

01. 0 

62 

53.2 

J  1. 9 

22 

104.6 

62.7 

82 

i56.i 

93.6 

42 

207.6 

124.4 

3 

02.6 

01 .5 

63 

54.0 

32.4 

23 

io5.5 

63.2 

83 

157.0 

94.1 

43 

208.4 

124.9 

4 

o3.4 

02.1 

64 

54.9 

32.9 

24 

106.4 

63.7 

^4 

157.8 

94.6 

44 

209.3 

125.4 

5 

04.3 

02.6 

65 

55.8 

33.4 

25 

107.2 

64.3 

85 

i58.7 

95.1 

45 

210. 1 

126.0 

6 

o5.i 

o3.i 

66 

56.6 

33.9 

26 

108. 1 

64.8 

86 

159.5 

95.6 

46 

21 1 .0 

126.5 

7 

06.0 

o3.6 

67 

57.5 

34.4 

27 

108.9 

65.3 

87 

160.4 

96.1 

47 

211 .9 

127.0 

8 

06.9 

04.1 

68 

58.3 

35.0 

28 

109.8 

65.8 

88 

161. 3 

96.7 

48 

212.7 

127.5 

9 

07.7 

04.6 

69 

59.2 

35.5 

29 

no. 6 

66.3 

89 

162.1 

97.2 

49 

2i3.6 

128.0 

lO 

08.6 

o5.i 

70 

60.0 

36.0 

3o 

III  .5 

66.8 

90 

i63.o 

97-7 

5o 

214.4 

128.5 

II 

09.4 

o5.7 

71 

60.9 

36.5 

i3i 

112.4 

67.3 

191 

i63.8 

98.2 

25l 

2i5.3 

1 29.0 

12 

10.3 

06.2 

72 

61.8 

37.0 

32 

Il3.2 

67.9 

92 

164.7 

98.7 

52 

216. 1 

129.6 

i3 

II. 2 

06.7 

73 

62.6 

37.5 

33 

114.1 

68.4 

93 

i65.5 

99.2 

53 

217.0 

i3o.i 

iS 

12.0 

07.2 

74 

63.5 

38.0 

34 

114.9 

68.9 

94 

166.4 

99-7 

54 

217.9 

i3o.6 

i5 

12.9 

07.7 

75 

64.3 

38.6 

35 

ii5.8 

69.4 

95 

167.3 

100.3 

55 

218.7 

i3i.i 

i6 

i3.7 

08.2 

76 

65.2 

39.1 

36 

116. 7 

69.9 

96 

168.1 

100.8 

56 

219.6 

i3i.6 

17 

14.6 

08.7 

77 

66.0 

39.6 

37 

117.5 

70.4 

97 

169.0 

101.3 

57 

220.4 

l32.I 

i8 

i5.4 

09.3 

7« 

66.9 

4o.i 

38 

118.4 

70.9 

98 

169.8 

101.8 

58 

221 .3 

1 32.6 

19 

16.3 

09.8 

79 

67.8 

40.6 

39 

119. 2 

71.5 

99 

170.7 

102.3 

59 

222.2 

1 33.2 

20 

17.2 

10.3 

80 

68.6 

4i.i 

4o 
i4i 

120.1 

72.0 

200 

171 .5 

102.8 

60 

223.0 

i33.7 

21 

18.0 

10.8 

81 

69.5 

4t.6 

120.9 

72.5 

201 

172.4 

io3.3 

261 

223.9 

i34.2 

22 

18.9 

11.3 

82 

70.3 

42.2 

42 

121.8 

73.0 

02 

173.3 

io3.8 

62 

224.7 

134-7 

23 

19.7 

II. 8 

83 

71.2 

42.7 

43 

122.7 

73.5 

o3 

174.1 

104.4 

63 

225.6 

i35.2 

24 

20.6 

12.3 

84 

72.0 

43.2 

44 

123.5 

74.0 

04 

175.0 

104.9 

64 

226.4 

135.7 

25 

21 .4 

12.9 

85 

72.9 

43.7 

45 

124.4 

74.5 

o5 

175.8 

io5.4 

65 

227.3 

1 36.2 

26 

22.3 

i3.4 

86 

73.8 

44.2 

46 

125.2 

75.1 

06 

176.7 

105.9 

66 

228.2 

1 36.8 

27 

23.2 

13.9 

87 

74.6 

44.7 

47 

126. X 

75.6 

07 

177.5 

106.4 

67 

229.0 

137.3 

26 

24.0 

14.4 

88 

75.5 

45.2 

48 

126.9 

76.1 

q8 

178.4 

106.9 

68 

229.9 

137.8 

29 

24.9 

14.9 

89 

76.3 

45.8 

49 

127.8 

76.6 

09 

179.3 

107.4 

69 

230.7 

1 38.3 

3o 

25.7 

i5.4 

90 

77.2 

46.3 

5o 

128.7 

77-1 

10 

180. 1 

108.0 

70 

231.6 

i38.8 

3i 

26.6 

i5.9 

91 

78.1 

46.8 

i5i 

129.5 

77.6 

21 1 

181. 0 

108.5 

271 

232.4 

139.3 

32 

27.4 

16.5 

92 

78.9 

47.3 

52 

i3o.4 

78.1 

12 

181. 8 

109.0 

72 

233.3 

139.8 

33 

28.3 

17.0 

93 

79.8 

47.8 

53 

i3i  .2 

78.7 

i3 

182.7 

109.5 

73 

234.2 

140.4 

M 

29.2 

17.5 

94 

80.6 

48.3 

54 

l32.1 

79.2 

i4 

i83.6 

IIO.O 

74 

235.0 

140.9 

35 

3o.o 

18.0 

95 

81.5 

48.8 

55 

132.9 

79-7 

i5 

184.4 

110.5 

75 

235.9 

141.4 

36 

3o.9 

18.5 

96 

82.3 

49-4 

56 

i33.8 

80.2 

16 

i85.3 

III.O 

76 

236.7 

141.9 

37 

3i.7 

19.0 

97 

83.2 

49.9 

57 

134.7 

80.7 

17 

186.1 

11 1.6 

77 

237.6 

142.4 

38 

32.6 

19.5 

9S 

84.1 

5o.4 

58 

i35.5 

81.2 

18 

187.0 

112. 1 

78 

238.4 

142.9 

39 

33.5 

20.1 

99 

84.9 

50.9 

59 

i36.4 

81.7 

19 

187.8 

112.6 

79 

239.3 

143.4 

4o 

34.3 

20.6 

100 

85.8 

5i.4 

60 

137.2 

82.3 

20 

188.7 

ii3.i 

80 

240 . 2 

143.9 

4i 

35.2 

21 .1 

lOI 

86.6 

51.9 

161 

i38.i 

82.8 

221 

189.6 

ii3.6 

281 

241 .0 

144.5 

42 

36.0 

21 .6 

02 

87.5 

52.4 

62 

139.0 

83.3 

22 

190.4 

114.1 

82 

241.9 

145.0 

43 

36.9 

22.1 

o3 

88.3 

53.0 

63 

139.8 

83.8 

23 

191 .3 

114.6 

83 

242.7 

145.5 

44 

37.7 

22  .6 

o4 

89.2 

53.5 

64 

i4o.7 

84.3 

24 

192. 1 

ll5.2 

84 

243.6 

1 46.0 

45 

38.6 

23.1 

o5 

90.1 

54.0 

65 

i4i.5 

84.8 

25 

193.0 

II5.7 

85 

244.5 

146.5 

46 

39.5 

23.6 

06 

90.9 

54.5 

66 

142.4 

85.3 

26 

193.8 

116.2 

86 

245.3 

147-0 

47 

40.3 

24.2 

07 

91.8 

55.0 

67 

143.2 

85.9 

27 

194.7 

116.7 

87 

246.2 

147.5 

48 

4t  .2 

24.7 

08 

92.6 

55.5 

68 

144.1 

86.4 

28 

195.6 

117.2 

88 

247.0 

I48.I 

49 

42.0 

25.2 

09 

93.5 

56.0 

69 

i45.o 

86.9 

29 

196.4 

117-7 

89 

24-7.9 

148.6 

be 

42.9 

25.7 

10 

94.4 

56.6 

70 

145.8 

87.4 

3o 

197.3 

118. 2 

90 

248.7 

1 49. 1 

5i 

43.7 

26.2 

III 

95.2 

57.1 

171 

146.7 

87.9 

23l 

198.1 

118.8 

291 

249.6 

149.6 

52 

44.  b 

26.7 

12 

96.1 

57.6 

72 

147.5 

88.4 

32 

199.0 

1 19.3 

92 

250.5 

i5o.i 

53 

45.5 

27.2 

i3 

96.9 

58.1 

73 

148.4 

88.9 

33 

199.9 

1 19.8 

93 

251.3 

i5o.6 

54 

46.3 

27.8 

i4 

97.8 

58.6 

74 

149.2 

89.5 

M 

200.7 

120.3 

94 

252.2 

i5i.i 

55 

47.2 

28.3 

i5 

98.6 

59.1 

75 

i5o.i 

90.0 

35 

201 .6 

120.8 

q5 

253. 0 

i5i.7 

56 

48.0 

28.8 

16 

99.5 

59.6 

76 

i5i.o 

90.5 

36 

202.4 

121.3 

96 

253.9 

l52.2 

i)7 

48.9 

29.3 

17 

100.4 

60.2 

77 

i5i.8 

91.0 

37 

2o3.3 

121.8 

97 

254-7 

i52.7 

58 

49.7 

29.8 

18 

lOI  .2 

60.7 

78 

i52.7 

91.5 

38 

204.1 

122.4 

98 

255.6 

i53.2 

59 

5o.6 

3o.3 

19 

102. 1 

61.2 

79 

i53.5 

92.0 

39 

2o5.o 

122.9 

99 

256.5 

i53.7 

bo 

5i.5 

3o.8 

20 

102.9 

61.7 

80 

154.4 

92.5 

40 

205.9 

123.4 

3  00 

257.3 

154.2 

Dist-i  Dcp. 

I. at. 

Dist. 

Dep. 

Lat. 

Dist. 

Dep. 

Lat. 

Dist. 

Dep. 

Lat. 

Dist. 

Dep. 

Lat. 

N.E.byE.iE.        S.E.byE.iE. 

N.W.byW.^W. 

S. W.by W.^W .       [For  5^  Points. 

rage  12]                                            TABLE  I. 

DilTerenca  of  Latitude  and  Departure  for  3  Points. 

N.E.byiN.                    N.W.byN.                   S.E.byS.                   S.W.byS. 

Dist.    Lai. 

Dep. 

Disi. 

Lat. 

Dep. 

Dist. 

Lat. 

Dep. 

Dist. 

181 
82 
83 
84 
85 
86 

87 
88 
89 
90 

Lat. 

Dep. 

Dist. 

Lat. 

Dep. 

I 

2 

3 

4 
5 
6 

7 
8 

9 

lO 

00. 8 
01.7 
02.5 
o3.3 
o4.2 
o5.o 
o5.S 
06.7 
07.5 
08.3 

00.6 
01 .1 
01.7 
02.2 
02.8 
o3.3 
03.9 
04.4 
o5.o 
o5.6 

61 
62 
63 
64 
65 
66 

67 
68 
69 

70 

5o.7 
5i.6 
52.4 
53.2 
54.0 
54.9 
55.7 
56.5 
57.4 
58.2 

33.9 
34.4 
35.0 
35.6 
36.1 
36.7 
37.2 
37.8 
38.3 
38.9 

121 
22 

23 

24 

25 

26 
27 
28 

3? 

100.6 
101.4 

102.3 

io3.i 
103.9 

104.8 
io5.6 
106.4 
107.3 
108. 1 

67.2 
67.8 
68.3 
68.9 
69.4 
70.0 
70.6 
71. 1 

71-7 
72.2 

i5o.5 
i5i.3 

l52.2 

I53.0 
i53.8 

154.7 
i55.5 
i56.3 
157. 1 
i58.o 

100.6 

lOI.I 

101.7 
102.2 
102.8 
io3.3 
103.9 
104.4 
io5.o 
io5.6 

241 
42 
43 

44 
45 
46 
47 
48 

ii 

200.4 
201 .2 
202.0 
202.9 
203.7 
204.5 
2o5.4 
206 . 2 
207.0 
207.9 

133.9 

134.4 
i35.o 
i356 
1 36  I 
1 36.7 
137.2 
137.8 
i38.3 
i38.9 

1 1 

12 

i3 

i4 
i5 
i6 

17 
iS 

19 
20 

09. 1 
10. 0 
10.8 
II. 6 
12.5 
i3.3 
i4.i 
i5.o 
1 5. 8 
16.6 

06. 1 
06.7 
07.2 
07.8 
08.3 
08.9 
09.4 

1 0.0 
10.6 

11. 1 

7' 
72 
73 
74 
75 
76 
77 
78 

79 
80 

59.0 
59.9 
60.7 
61.5 
62.4 
63.2 
64.0 
64.9 
65.7 
66.5 

39.4 
4o.o 
4o.6 
41.1 
41.7 
42.2 
42.8 
43.3 
43.9 
44.4 

i3i 

32 

33 
34 
35 
36 

37 
38 

39 
4o 

108.9 
109.8 
1 10.6 
III  .4 
112. 2 
ii3.i 
113.9 

114.7 
ii5.6 
116. 4 

72.8 
73.3 
73.9 
74.4 
75.0 
75.6 
76.1 
76.7 
77.2 
77.8 

191 

92 
93 

94 

95 
96 

97 
98 

99 
200 

i58.8 
159.6 
160.5 
161. 3 
1 62 . 1 
i63.o 
i63.8 
164.6 
i65.5 
166.3 

106. 1 
106.7 
107.2 
107.8 
108.3 
108.9 
109.4 

1 1 0.0 
110.6 

111. 1 

25l 
52 

53 
54 
55 
56 

57 
58 

60 

208.7 
209.5 
210.4 
211 .2 
212.0 
212.9 
213.7 
214.5 
2i5.4 
216.2 

139.4 
i4oo 
1 40.6 
i4i.i 
141.7 
142.2 
142.8 
143.3 
143.9 
144.4 

21 
22 

23 

24 

25 

26 
27 
28 

19 
3o 

17.5 
18.3 
19. 1 
20.0 

20.8 
21  .6 

22.4 
23.3 
24.1 
24.9 

II. 7 
12.2 
12.8 
i3.3 
13.9 
14.4 
i5.o 
i5.6 
16. 1 
16.7 

81 
82 
83 
84 
85 
86 
87 
88 
89 
90 

67.3 
68.2 
69.0 
69.8 
70.7 
71.5 
72.3 
73.2 
74.0 
74.8 

45.0 
45.6 
46.1 
46.7 
47-2 
47.8 
48.3 
48.9 
49.4 
5o.o 

i4i 
42 
43 
44 
45 

47 
48 

49 

5o 

117. 2 
118. 1 
118. 9 
119.7 
120.6 

121. 4 
122.2 
123.  I 
123.9 

124.7 

7S.3 
78.9 

79-4 
80.0 
80.6 
81. 1 
81.7 
82.2 
82.8 
83.3 

201 
02 
o3 
o4 
o5 
(}6 
07 
08 
09 
10 

167. 1 
168.0 
168.8 
169.6 
170.5 
171 .3 
172. 1 
172.9 
173.8 
174.6 

1 1 1.7 

112. 2 
112.8 

11 3.3 
1 13.9 
114.4 
1 1 5.0 

1 1 5.6 
116.1 

1 16.7 

261 
62 
63 
64 
65 
66 
67 
68 
69 
70 

217.0 
217.8 
218.7 
219.5 
220.3 
221 .2 
222.0 
222.8 
223.7 
224.5 

145.0 
i45.6 
146.1 
146.7 
147.2 
147-8 
148.3 
148.9 
149.4 
i5o.o 

3i 

32 

33 
34 
35 
36 
37 
38 
39 
40 

25.8 
26.6 
27.4 
28.3 
29.1 

^9-9 
3o.8 
3i.6 
32.4 
33.3 

17.2 
17.8 
18.3 
18.9 
19.4 
20.0 
20.6 
21. 1 
21.7 
22.2 

91 

93 
94 
95 
96 

97 
98 

99 

TOO 

75.7 
76.5 
77.3 
78.2 
79.0 
79.8 
80.7 
81.5 
82.3 
83.1 

5o.6 
5i.i 
51.7 
52.2 
52.8 
53.3 
53.9 
54.4 
55.0 
55.6 

i5i 

52 

53 
54 
55 
56 

57 
58 
59 
60 

125.6 
126.4 
127.2 
128. c 

128.9 
129.7 

i3o.5 
i3i.4 

l32.2 

133.0 

83.9 
84.4 
85.0 
85.6 
86.1 
86.7 
87.2 
87.8 
88.3 
88.9 

211 
12 
i3 
i4 
i5 
16 
17 
18 

'9 

20 

175.4 
176.3 
177-1 
177-9 
178.8 
179.6 
180.4 
181. 3 
182.1 
182.9 

117.2 
1 17.8 
118.3 
118.9 
119.4 
120.0 
120.6 
121. 1 
121.7 
122.2 

271 
72 
73 

74 
75 
76 
77 
78 

79 
80 

225.3 

226.2 
227.0 

227.8 
228.7 

229.5 
23o.3 
23l  .1 
232. 0 
232.8 

i5o.6 
i5i.i 
1 5 1. 7 

l52.2 

i52.8 
i53.3 
153.9 
154.4 
i55.o 

1 55.6 
1 56.1 

1 56.7 
157.2 
1578 
i58.3 
1 58.9 
1 59.4 
160.0 
160,6 
161. 1 

4i 
42 
43 
44 
45 
46 

47 
48 

49 
5o 

34.1 

34.9 
35.8 
36.6 
37.4 
38.2 
39. 1 
39.9 
40.7 
4r.6 

22.8 
23.3 
23.9 
24.4 
25.0 
25.6 
26.1 
26.7 
27.2 
27.8 

lOI 
02 

o3 
04 
o5 
06 

07 
08 
09 
10 

84.0 
84.8 
85.6 
86.5 
87.3 
88.1 
89.0 
89.8 
90.6 
91.5 

56.1 
56.7 
57.2 
57.8 
58.3 
58.9 
59.4 
60.0 
60.6 
61. 1 

161 
62 
63 
64 
65 
66 
67 
68 
69 
70 

171 

72 
73 
74 
75 
7fi 
77 
78 

79 
80 

133.9 

134.7 
i35.5 
i36.4 
137.2 
i38.o 
i38.9 
139.7 
i4o.5 
i4i.3 

89.4 
90.0 
90.6 
91. 1 
91.7 
92.2 
92.8 
93.3 
93.9 
94.4 
95.0 
95.6 
96.1 
96.7 
97.2 
97.8 
98.3 
98.9 
99.4 
100. 0 

221 
22 

2  3 

24 

25 

26 

27 
28 

3o 

i83.8 
184.6 
i85.4 
186.2 

187. 1 
187.9 
18S.7 
189.6 
190.4 

191 .2 

122.8 
123.3 
123.9 
124.4 

I25.0 

125.6 
126. 1 

126.7 
127.2 
127.8 

281 
82 
83 
84 
85 
86 
87 
88 

89 
90 

233.6 
234.5 
235.3 
236.1 
237.0 
237.8 
238.6 
239.5 
240.3 
241 .1 

5i 

52 

53 
54 
55 

56 

57 
58 

60 
Disi. 

42.4 
43.2 
44.1 
44.9 
45.7 
46.6 
47.4 
48.2 
49.1 
.49.9 

I)C|). 

28.3 
28.9 
29.4 
3o.o 
3o.6 
3i.i 
3i.7 

32.2 

32.8 
33.3 

Lat. 

II I 
12 
i3 
i4 
i5 
16 
17 
18 

'9 

20 

92.3 
93.1 
94.0 
94.8 
95.6 
96.5 
97.3 
98.1 
98.9 
99.8 

61.7 
62.2 
62.8 
63.3 
63.9 
64.4 
65.0 
65.6 
66.1 
66.7 

142.2 
143.0 
143.8 

144.7 
145.5 
i46.3 

l47-2 

i48.o 
i48.8 
149.7 

23l 
32 

33 

34 
35 
36 
37 
38 
39 
4o 

Dist. 

192. 1 
192.9 
193.7 
194.6 
195.4 
196.2 

197-1 
197.9 
198.7 
199.6 

128.3 
128.9 
129.4 
i3o.o 
i3o.6 
i3i.i 
i3i.7 

l32.2 

1 32.8 
i33.3 

291 

9^ 
93 

94 

96 

97 
98 

3oo 

242.0 
242.8 
243.6 
244.5 
245.3 
246.1 
246.9 
247.8 
24s.  6 
249.4 

161.7 
162.2 
162.8 
i63.3 
163,9 
164.4 
i65.o 
i65.6 
166.1 
166.7 

Hist. 

Dep. 

Lat. 

Disi. 

Dep. 

Lat. 

Dep. 

Lat. 

Dist. 

Dep. 

Lat. 

N.E.byE.               S.E.byE.               N.W.byW.               S.W.by W.           [For  .5  Points. 

;  *" 

^ 

TABLE  L 

frase  13 

Difference  of  Latitude  and  Dep 

irture  for  3^  Points. 

N.E.^N.                    N.W4N. 

S.E.IS.                    S.W.5S. 

Dist 

Lat. 

Dep. 

Disl. 
61 

Lat. 

49.0 

Dep. 

Disl 
121 

Lat. 
97.2 

Dep. 

Disl 

Lat. 

Dep. 

Disl. 

Lat. 

Dep. 

I 

GO .  8 

00.6 

36.3 

72.1 

181 

145.4 

107.8 

241 

193.6 

143.6 

2 

01 .6 

01 .2 

62 

49-8 

30.9 

22 

98.0 

72.7 

82 

i46.2 

10S.4 

42 

194.4 

i44-2 

3 

02.4 

or.t 

63 

5o.6 

37.5 

23 

98.8 

73.3 

83 

i47-o 

109.0 

43 

195.2 

144.S 

4 

()3.2 

02.4 

64 

5i.4 

38.1 

24 

99.6 

73.9 

84 

147-8 

109.6 

44 

1 9b .  0 

145.4 

5 

o4.o 

o3.o 

63 

52.2 

38.7 

2  5 

100.4 

74.5 

85 

148.6 

110.2 

45 

196.8 

145.9 

6 

o4.8 

o3.6 

66 

53.0 

39.3 

26 

lOI  .2 

75.1 

86 

149.4 

1 10.8 

46 

197.6 

146.5 

7 

o5.6 

04.2 

67 

53.8 

39.9 

27 

102.0 

75.7 

87 

i5o.2 

1 1 1.4 

4i 

198.4 

i47-i 

S,o6.4 

04.8 

68 

54.6 

40.5 

28 

102.8 

76.2 

88 

i5i  .0 

1 1 2.0 

48 

199.2 

147-7 

9 

07.2 

o5.4 

69 

55.4 

4i.i 

29 

io3.6 

76.8 

89 

i5i.8 

112.6 

49 

200 . 0 

1 48. 3 

10 

08.0 

06.0 

70 

56.2 

41.7 

3o 

104.4 

77.4 

90 

i52.6 

Il3.2 

5o 

200.8 

148.9 

1 1 

0S.8 

06.6 

71 

57.0 

42.3 

i3i 

I05.2 

78.0 

191 

153.4 

ii3.8 

25l 

201 .6 

149-5 

12 

09.6 

07.1 

72 

57.8 

42.9 

32 

106.0 

78.6 

92 

i54.2 

1 14.4 

52 

202.4 

i5o.i 

i3 

10.4 

07.7 

73 

58.6 

43.5 

33 

106.8 

79.2 

93 

i55.o 

1 1 5.0 

53 

2o3.2 

i5o.7 

i4 

1 1.2 

08.3 

74 

59.4 

44.1 

M 

107.6 

79.8 

94 

155.8 

1 1 5.6 

54 

204.0 

i5i.3 

/5 

12.0 

08.9 

75 

60.2 

44.7 

35 

108.4 

80.4 

9b 

156.6 

116. 2 

55 

204.8 

i5i.9 

i6 

12.9 

.09.5 

76 

61 .0 

45.3 

36 

109.2 

81.0 

96 

i5-  4 

116.8 

56 

2o5.6 

i52.5 

I? 

■  3.7 

10. 1 

77 

61.8 

45.9 

37 

IIO.O 

81.6 

97 

i58.2 

117.4 

57 

206.4 

i53.i 

i8 

i4.5 

10.7 

78 

62.7 

46.5 

38 

no. 8 

82.2 

98 

159.0 

1 17.9 

58 

207.2 

153.7 

IP 

i5.3 

H.3 

79 

63.5 

47-1 

39 

III  .6 

82.8 

99 

159.8 

11S.5 

59 

20S.0 

154.3 

20 

16. 1 

II. 9 

80 
81 

64.3 

47.7 

4o 

112. 4 

83.4 
~847o~ 

200 

1 60 . 6 

1 19.1 

6(, 

J08.8 

1 54  9 

21 

16.9 

12.5 

65.1 

48.3 

i4i 

ii3.3 

201 

161. 4 

119.7 

261 

209.6 

i55.5 

22 

17-7 

i3.i 

82 

65.9 

48.8 

42 

114. 1 

84.6 

02 

162.2 

120.3 

62 

210.4 

1 56. 1 

23 

18.5 

l3.7 

83 

66.7 

49.4 

43 

114. 9 

85.2 

OJ 

i63.i 

120.9 

63 

211. 2 

1 56.7 

24 

19.3 

i4.3 

84 

67.5 

5().o 

44 

115.7 

85.8 

04 

163.9 

121. 5 

64 

212.0 

1.57.3 

25 

20. 1 

r4.9 

85 

68.3 

5o.6 

45 

116. 5 

86.4 

o5 

164.7 

122. 1 

65 

212.8 

157.9 

26 

20.9 

i5.5 

86 

69.1 

5l.2 

46 

117. 3 

87.0 

06 

1 65. 5 

122.7 

66 

2.3.7 

I5S.5 

27 

21 .7 

16. 1 

87 

69.9 

5i.8 

4i 

118. 1 

87.6 

07 

166.3 

123.3 

67 

214.5 

1 59. 1 

28 

22.5 

16.7 

88 

70.7 

52.4 

48 

118. 9 

88.2 

08 

167.1 

123.9 

68 

2i5.3 

159.6 

29 

23.3 

17.3 

89 

71.5 

53.0 

49 

119.7 

88.8 

09 

167.9 

124.5 

69 

216.1 

160.2 

3o 

24.1 

17.9 

90 

72.3 

53.6 

5o 

120.5 

89.4 

10 

168.7 

125. 1 

70 

216.9 

160.8 

3r 

24.9 

18.5 

91 

73.1 

54.2 

i5i 

121 .3 

90.0 

21 1 

169.5 

125.7 

271 

217.7 

161.4 

32 

23.7 

19. 1 

92 

73.9 

54.8 

52 

122. 1 

90.5 

12 

170.3 

126.3 

72 

218.5 

162.0 

33 

26.5 

19.7 

93 

74.7 

55.4 

53 

122.9 

91. 1 

i3 

171. 1 

126.9 

73 

219.3 

163.6 

34 

27.3 

20.3 

94 

75.5 

56.0 

54 

123.7 

91.7 

i4 

171. 9 

127.5 

74 

220. 1 

i63.2 

35 

28.1 

20.8 

95 

76.3 

56.6 

55 

124.5 

92.3 

i5 

172.7 

128. 1 

75 

220.9 

163.8 

36 

28.9 

21.4 

96 

77.1 

57.2 

56 

125.3 

92.9 

16 

173.5 

128.7 

76 

221 .7 

164.4 

3? 

29.7 

22.0 

97 

77-9 

57.8 

b7 

126. 1 

93.5 

17 

174.3 

129.3 

77 

222.5 

i65.o 

38 

3o.5 

22.6 

98 

78.7 

58.4 

58 

126.9 

94.1 

18 

175. 1 

129.9 

78 

223.3 

i65.6 

39 

3t.3 

23.2 

99 

79-5 

59.0 

59 

127.7 

94.7 

19 

175.9 

i3o.5 

79 

224. 1 

16b.  2 

4o 
4i' 

32.1 

23.8 

100 

80.3 

5o.b 

bo 

128.5 

95.3 

20 

176.7 

i3i.i 

80 

224.9 

ibb.S 

32.9    24.4 

lOI 

81. 1 

60.2 

161 

129.3 

959 

221 

i77-b 

i3r.6 

281 

225.7 

167.4 

42 

33.7    25.0 

02 

81.9 

60.8 

62 

i3o.i 

96.5 

22 

178.3 

l32.2 

82 

226.5 

168.0 

43 

34.5    25.6 

o3 

82.7 

61.4 

63 

1 3o .  9 

97.1 

23 

179.1 

i32.8 

83 

227.3 

168.6 

44 

35.3    26.2 

04 

83.5 

62.0 

64 

i3i.7 

97-7 

24 

179.9 

i33.4 

84 

228.1 

169.2 

45 

3b  I    26.8 

o5 

84.3 

62.5 

65 

i32.5 

98.3 

25 

180.7 

1 34.0 

85 

228.9 

169.8 

46 

3(3  9 

27.4 

06 

85.1 

63.1 

()b 

i33.3 

Q8.9 

26 

181. 5 

1 34.6 

86 

229.7 

170.4 

47 

37.8 

28.0 

07 

85.9 

63.7 

67 

i34.i 

99.5 

27 

182.3 

i35.2 

87 

23o.5 

1 71.0 

48 

38.6 

28.6 

n8 

86.7 

64.3 

()8 

134.9 

100. 1 

28 

i83.i 

i35.8 

88 

231.3 

1 7 1 .6 

49 

39.4 

29.2 

09 

87.5 

64.9 

69 

.35.7 

100.7 

29 

183.9 

1 36.4 

89 

232.  I 

172.2 

5o 

4o.2 

29.8 

10 

88.4 

65.5 

70 

i36.5 

IOI.3 
IUI.9 

3u 

184.7 

137.0 

90 
291 

232.9 

233.7 

172.8 
173.3 

5i 

4i  .0 

3o.4 

I II 

89.2 

66.1 

171 

137.3 

23  I 

i85.5 

137.6 

52 

4i.8 

3i.o 

12 

90.0 

66.7 

72 

i38.2 

102.5 

32 

186.3 

i38.2 

92 

234.5 

173.9 

53 

42.6 

3i.8 

i3 

90.8 

67.3 

73 

139.0 

io3.i 

33 

187. 1 

1 38.8 

93 

235.3 

174.5 

M 

4'i.4 

32.2 

i4 

91 .6 

67.9 

74 

139.8 

io3.7 

34 

188.0 

139.4 

94 

236.1 

175.1 

bb 

44.2 

32.8 

i5 

92.4 

68.5 

7b 

140.6 

104.2 

35 

18S.8 

i4o.o 

95 

236-9 

175.7 

5b 

45.0 

33.4 

16 

93.2 

69.1 

76 

i4i.4 

104.8 

36 

189.6 

i4o.6 

96 

237.7 

176.3 

b7 

45.8 

34.0 

17 

94.0 

69.7 

77 

142.2 

io5.4 

37 

190.4 

l4l.2 

97 

238.6 

176.9 

58 

46.6 

34.6 

18 

94.8 

70.3 

78 

143.0 

106.0 

38 

191 .2 

i4i.8 

98 

239.4 

177-5 

59 

47.4 

35.1 

19 

95.6 

70.9 

79 

143.8 

1 06.6 

39 

192.0 

142.4 

99 

240  2 

178.1 

bo 

48.2 

35.7 

20 

96.4 

71.5 

80 

144.6 

107.2 

40 

192.8 

143.0 

3oo 

241 .0 

178.7 

DIst. 

Dop. 

Lat. 

Dist. 

Dcp> 

Lat. 

Dsi.     Dep.  1 

Lai. 

l>i.st. 

Dop.      Lat.   1 

Disl. 

Dop. 

Lat. 

N.E.3E.               S.E3E. 

N.W.s 

W. 

S.W.^V/.               [For  43  Points,     j 

Page  141 

TABLE  L 

Difference  of  Latitude  and 

Departure  for  3^  Points. 

n.e; 

^N.                     N.W.^N. 

S.E.iS.                 S.W.iS.                     1 

Dist.l  Lat. 

Dep. 

Dist. 

Lat. 

Dep. 

Dist. 

Lat. 

Dep. 

Dist. 

Lat. 

Dep. 

Dist. 

Lat. 

Up. 

I 

00.8 

00.6 

61 

47-2 

38.7 

121 

93.5 

76.8 

181 

139.9 

1 14.8 

241 

186.3 

if-wi.9 

2 

01. b 

01 .3 

62 

47-9 

39.3 

22 

94.3 

77-4 

82 

140.7 

1 1 5.5 

42 

187. 1 

ib3.5 

6 

02.3 

01 .9 

63 

48.7 

4o.o 

23 

95.1 

78.0 

83 

i4i.5 

1 1 6. 1 

43 

187.8 

154.2 

4 

o3.i 

02.5 

64 

49-5 

40.6 

24 

95.9 

78.7 

84 

142.2 

1 16.7 

AA 

188.6 

1 54.8 

b 

03.9 

03.2 

65 

5o.2 

4l.2 

2b 

96.6 

79.3 

85 

143.0 

117.4 

45 

189.4 

i55.4 

b 

04. b 

o3.8 

66 

5i.o 

41.9 

2b 

97-4 

79-9 

86 

143.8 

118.0 

46 

190.2 

i56.i 

1 

o5.4 

04.4 

67 

5i.8 

42.5 

27 

98.2 

80.6 

87 

144.6 

118.6 

47 

190.9 

1 56.7 

a 

06.2 

o5.i 

68 

52.6 

43.1 

28 

98.9 

81.2 

88 

145.3 

1 19.3 

48 

191-7 

157.3 

9 

07.0 

Ob. 7 

69 

53.3 

43.8 

29 

99-7 

81.8 

89 

146.1 

1 19.9 

49 

192.5 

1 58.0 

10 

07.7 

06.3 

70 

54.1 

AA-A 

3o 

100.5 

82.5 

90 

146.9 

120.5 

5o 

193.3 

1 58.6 

II 

08.5 

07.0 

71 

54.9 

45.0 

i3i 

loi  .3 

83.1 

191 

147.6 

121.2 

25l 

194.0 

159.2 

12 

09.3 

07.6 

72 

55.7 

4b.7 

32 

102.0 

83.7 

92 

148.4 

121.8 

52 

194.8 

.59.9 

i3 

10. 0 

08.2 

73 

56.4 

46.3 

33 

102.8 

84.4 

93 

149.2 

122.4 

53 

195.6 

160.5 

i4 

10.8 

08.9 

74 

57.2 

46.9 

34 

io3.6 

85.0 

94 

i5o.o 

123. 1 

54 

T96.3 

161.1 

lb 

II. b 

09.5 

75 

58. 0 

47 -b 

35 

104.4 

85.6 

95 

i5o.7 

123.7 

55 

197.1 

161.8 

lb 

12.4 

10.2 

76 

58.7 

48.2 

36 

io5.i 

86.3 

96 

i5i.5 

124.3 

56 

197-9 

162.4 

17 

i3.i 

10.8 

77 

59.5 

48.8 

37 

105.9 

86.9 

97 

i52.3 

125. 0 

57 

198.7 

i63.o 

i8 

13.9 

II. 4 

7B 

60.3 

49-i) 

38 

106.7 

87.5 

98 

i53.i 

125.6 

58 

199.4 

163.7 

19 

14.7 

12. 1 

79 

bi.i 

bo.  I 

39 

107.4 

88.2 

99 

i53.8 

126.2 

59 

200.2 

164.3 

20 

ib.b 

12.7 

80 

61.8 

5o.8 

40 

108.2 

88.8 

200 

i54.6 

126.9 

60 

201 .0 

164.9 

21 

16.2 

i3.3 

81 

62.6 

5i.4 

i4i 

109.0 

89.4 

201 

i55.4 

127.5 

261 

201 .8 

1 65.6 

22 

17.0 

i4.o 

82 

63.4 

52.0 

42 

109.8 

90.1 

02 

i56.i 

128.1 

62 

202.5 

166.2 

2j 

17.8 

14.6 

83 

64.2 

b2.7 

A^ 

no. 5 

90.7 

o3 

i56.9 

128.8 

63 

2o3.3 

166.8 

24 

18. b 

l5.2 

84 

64.9 

b3.3 

AA 

III  .3 

91.4 

04 

157.7 

129.4 

H 

204.1 

167.5 

25 

19.3 

lb. 9 

85 

65.7 

b3.9 

45 

112. 1 

92.0 

o5 

i58.5 

i3o.i 

65 

204.8 

168.1 

2b 

20.1 

16.5 

86 

66.5 

54.6 

46 

112. 9 

92.6 

06 

159.2 

1 30.7 

66 

2o5.6 

168.7 

27 

20.9 

17. 1 

87 

67.3 

bb.2 

47 

ii3.6 

93.3 

07 

160.0 

i3i.3 

67 

206.4 

169.4 

28 

21  .b 

17.8 

88 

68.0 

bb.8 

48 

114.4 

93.9 

08 

160. S 

l32.0 

68 

207.2 

170.0 

29 

22.4 

18.4 

89 

68.8 

bb.b 

49 

ll5.2 

94.5 

09 

161. 6 

132.6 

69 

207.9 

170.7 

Jo 

23.2 

19.0 

90 

69.6 

b7.i 

bo 

116.0 

95.2 

10 

162.3 

i33.2 

70 

208.7 

171.3 

3i 

24.0 

19.7 

91 

70.3 

D7.7 

i5i 

116.7 

95.8 

21 1 

i63.i 

133.9 

271 

209.5 

171.9 

32 

24.7 

20.3 

Q2 

71. 1 

58.4 

b2 

117.5 

96.4 

12 

163.9 

i34.5 

72 

210.3 

172.6 

33 

2b. b 

20.9 

93 

71.9 

59.0 

53 

118. 3 

97.1 

i3 

164.7 

i35.i 

73 

21 1 .0 

173.2 

34 

26.3 

21 .6 

94 

72.7 

b9.b 

M 

119. 0 

97-7 

i4 

i65.4 

i35.8 

74 

211.8 

173.8 

3b 

27.1 

22.2 

95 

73.4 

bo.3 

bb 

119. 8 

98.3 

i5 

166.2 

i36.4 

75 

212.6 

174.5 

3o 

27.8 

22.8 

96 

74.2 

60.9 

5b 

120.6 

99.0 

16 

167.0 

137.0 

76 

2i3.4 

175. 1 

^7 

28  .b 

23.5 

97 

75.0 

bi.b 

!)7 

121 .4 

99.6 

17 

167.7 

137.7 

77 

214. 1 

175.7 

38 

29.4 

24.1 

98 

75.8 

62.2 

58 

122. 1 

100.2 

18 

168.5 

i38.3 

78 

214.9 

176.4 

39 

3o.  I 

24.7 

99 

76.5 

62.8 

59 

122  .9 

100.9 

19 

169.3 

i38.9 

79 

2i5.7 

177.0 

40 

30.9 

25.4 

100 

77.3 

63.4 

60 

123.7 

IOI.5 

20 

170. 1 

139.6 

80 

216.4 

177.6 

4i 

3i.7 

26.0 

lOI 

78.1 

64.1 

161 

124.5 

102. 1 

221 

170.8 

i4o.2 

281 

217.2 

178.3 

42 

32. b 

26.6 

02 

78.8 

64.7 

62 

125.2 

102.8 

22 

171 .6 

i4o.8 

82 

218.0 

178.9 

Ai 

33.2 

27.3 

o3 

79.6 

65.3 

63 

126.0 

io3.4 

23 

172.4 

i4i.5 

83 

218. 8 

179.5 

AA 

34.0 

27.9 

o4 

80.4 

66.0 

64 

126.8 

104.0 

24 

173.2 

142.1 

84 

219.5 

180.2 

4b 

34.8 

28.5 

o5 

81.2 

66.6 

65 

127.5 

104.7 

25 

173.9 

142.7 

85 

220.3 

180.8 

4b 

35.6 

29.2 

06 

81.9 

67.2 

66 

128.3 

io5.3 

26 

174.7 

143.4 

86 

221 .1 

181.4 

47 

3b. 3 

29.8 

07 

82.7 

67.9 
68.5 

67 

129.1 

105.9 

27 

175.5 

144.0 

87 

221 .9 

1S2.1 

48 

37.1 

3o.5 

08 

83.5 

68 

129.9 

106.6 

28 

176.2 

144.6 

88 

222.6 

182.7 

f9 

37.9 

3i.i 

09 

84.3 

69.1 

69 

i3o.6 

107.2 

29, 177.0 

145.3 

89 

223.4 

183.3 

bo 

38.7 

3. .7 

10 

85.0 

69.8 

70 

i3i.4 

107.8 

3o 

177.8 

145.9 

90 

224.2 

184.0 

bi 

39.4 

32.4 

1 II 

85.8 

70.4 

•71 

I  32.  2 

108.5 

23  I 

178.6,  146.5 

291 

224.9 

184.6 

b2 

4o.2 

33.0 

12 

86.6 

71.1 

72 

i33.o 

109.1 

32 

179.3 

147.2 

92 

225.7 

i85.2 

b3 

4i  .0 

33.6 

i3 

87.4 

71-7 

73 

133.7 

109.8 

33 

180.1 

147.8 

93 

226.5 

185.9 

^4 

41.7 

34.3 

i4 

88.1 

72.3 

74 

134.5 

110.4 

34 

1 80 . 9 

148.4 

94 

227.3 

186.5 

bb 

42.5 

34.9 

i5 

88.9 

73.0 

7^) 

i35.3 

1 1  I.O 

35 

181  7 

149.1 

95 

228.0 

187.1 

bb 

43.3 

35.5 

16 

89.7 

73.6 

7b 

i36.o 

1 1 1.7 

36 

182.4 

149.7 

96 

228.8 

187.8 

^7 

44.1 

36.2 

17 

90.4 

74.2 

77 

i36.8 

I  12.3 

37 

i83.2 

i5o.4 

97 

229.6 

18S.4 

b8 

AA-^ 

36.8 

18 

91 .2 

74.9 

78 

137.6 

1 1 2.9 

38 

184.0 

1 5 1.0 

98 

23o.4 

189.0 

b9 

45.6 

J7.4 

19 

92.0 

75.b 

79 

i38.4 

11 3.6 

39 

184.7 

i5i.6 

99 

23 1. 1 

189.7 

bo 

A^'^.A 

38.1 

20 

93.8 

7b.  I 

80 

139. 1 

Il4.2 

40 
I)i.;t. 

i85.5 
Dop. 

i52.3 

3oo 

231.9 

190.3 

Dist. 

I)('|). 

l.iU. 

Dist. 

Dep. 

Lnl. 

Dist. 

Dpp. 

Lat. 

Lnt. 

Dist. 

Dep. 

Lat. 

N.EAE. 

S.E.AE. 

N.W..i^ 

V. 

S.W.AW.               [For  4.i  Points. 

1 

TABLE  L 

1 
fP.ige  15 

Difference  of  Latitude  and  Departure  for  3f  Points. 

N.E-iN.                    N.W4N.                    S.E.^S.                    S.W.iS. 

Dist 

Lai.  '  Dep. 

Dist. 

Lat. 

Dep. 

Dist. 

Lat. 

Dep. 

Dist. 

Lat. 

Dep. 

Dist. 

Lat. 

Dep. 

I 

00 . 7    00 . 7 

61 

45.2 

4i.o 

121 

89.7 

81.3 

181 

i34.i 

I  21.6 

241 

178.6 

161.8 

2 

oi.5'oi.3 

62 

4'i.9 

4i.6 

22 

90.4 

81.9 

82 

134.9 

122.2 

42 

179.3 

162.5 

.1 

02.2 

02.0 

63 

46.7 

42.3 

23 

91. 1 

82.6 

83 

i35.6 

122.9 

Ai 

180. 1 

i63.2 

^ 

()3 .0 

02.7 

64 

47.4 

43.0 

24 

91.9 

83.3 

84 

i36.3 

123.6 

AA 

180.8 

J  63.0 

^. 

o3.7 

o3.4 

65 

48.2 

43.7 

25 

92.6 

83.9 

85 

137. 1 

124.2 

45 

181. 5 

164.6 

f) 

o4.4 

04.0 

66 

48.9 

A^:^ 

26 

93.4 

84.6 

86 

137.8 

124.9 

46 

182.3 

i65.2 

7 

o5.2 

04.7 

07 

49-6 

45.0 

27 

94.1 

85.3 

«7 

i38.6 

125.6 

47 

i83.o 

166.0 

« 

05.9 

o5.4 

68 

5o.4 

45.7 

28 

94.8 

86.0 

88 

139.3 

126.3 

48 

i83.8 

166.5 

9 

06 . 7 

06.0 

69 

5i.i 

46.3 

29 

95.6 

86.6 

89 

i4o.o 

126.9 

49 

184.5 

167.2 

lO 

07.4 

06.7 

70 

51.9 

47-0 

3o 

96.3 

87.3 

90 

140.8 

127.6 

5o 

iS5.2 

167.9 
168.6 

II 

08.2 

07.4 

71 

52.6 

47-7 

i3i 

97.1 

88.0 

191 

141.5 

128.3 

25l 

186.0 

12 

08.9 

08.1 

72 

53.3 

48.4 

32 

97.8 

88.6 

92 

142.3 

128.9 

62 

186.7 

169.2 

i3 

09.6 

08.7 

73 

54.1 

49.0 

33 

98.5 

89.5 

93 

143.0 

129.6 

53 

187.5 

169.9 

(4 

10.4 

09.4 

74 

54.8 

49.7 

M 

99.3 

90.0 

94 

143.7 

i3o.3 

64 

188.2 

170.6 

i5 

1 1 .1 

10. 1 

75 

55.6 

5o.4 

35 

100. 0 

90.7 

95 

144.5 

i3i.o 

66 

188.9 

171-2 

if) 

II. 9 

10.7 

76 

56.3 

5i.o 

36 

100.8 

91.3 

96 

145.2 

i3i.6 

66 

189.7 

171-9 

'7 

12  .6 

II. 4 

77 

57.1 

5i.7 

37 

loi  .5 

92.0 

97 

146.0 

i32.3 

67 

190.4 

172.6 

i8 

i3.3 

12. 1 

78 

57.8 

52.4 

38 

102.3 

92.7 

98 

146.7 

i33.o 

68 

191 .2 

173.3 

■9 

14.1 

12.8 

79 

58.5 

53.1 

39 

io3.o 

93.3 

99 

147-4 

i33.6 

69 

191. 9 

173.9 

JO 

14.8 

i3.4 

80 

59.3 

53.7 

4o 

io3.7 

94.0 

200 

148.2 

134.3 

60 

192.6 

174.6 

21 

i5.6 

14.1 

81 

60.0 

54.4 

i4i 

104.5 

94-7 

201 

148.9 

i35.o 

261  1 193.4 

176.3 

22 

16.3 

j4.8 

82 

60.8 

55.1 

42 

io5.2 

9U 

02 

149.7 

i35.7 

62 

194. 1 

176.9 

2  3 

17.0 

i5.4 

83 

61.5 

55.7 

43 

106.0 

96.0 

o3 

i,5o.4 

i36.3 

63 

194-9 

176.6 

24 

17.8 

16. 1 

84 

62.2 

56.4 

AA 

106.7 

96.7 

04 

i5i  .2 

137.0 

64 

196.6 

177.3 

25 

18.5 

16.8 

85 

63.0 

57.1 

45 

107.4 

97-4 

OD 

i5i  .9 

137-7 

65 

iy6.4 

178.0 

26 

19.3 

17.5 

86 

63.7 

57.8 

46 

108.2 

98.0 

06 

i52.6 

i38.3 

66 

197-1 

178.6 

27 

20.0 

18.1 

87 

64.5 

58.4 

47 

108.9 

98.7 

07 

1 53. 4 

139.0 

67 

197.8 

179.3 

28 

20.7 

18.8 

88 

65.2 

59.1 

48 

109.7 

99.4 

08 

I54.I 

139.7 

68 

198.6 

180.0 

29 

21.5 

19.5 

89 

65.9 

59.8 

49 

1 10.4 

100. 1 

09 

154.9 

i4o.4 

69 

199.3 

180.6 

3o 

22.2 

20.1 

90 

66.7 

60.4 

5o 

III .  I 

100.7 

10 

i55.6 

i4i-o 

70 
271 

200 . 1 

181.3 

3i 

23.0 

20.8 

91 

67.4 

61. 1 

i5i 

1 1 1 .9 

10 1. 4 

211 

i56.3 

141.7 

182.0 

32 

23.7 

21.5 

92 

68.2 

61.8 

52 

112. 6 

102. 1 

12 

157.1 

142.4 

72 

201 .5 

182.7 

33 

24.5 

22.2 

93 

68.9 

62.5 

53 

ii3.4 

102.7 

i3 

157.8 

143.0 

73 

202.3 

i83.3 

34 

25.2 

22.8 

94 

69.6 

63.1 

M 

1 14. 1 

io3.4 

i4 

i58.6 

143.7 

74 

2o3.0 

184.0 

35 

25.9 

23.5 

95 

70.4 

63.8 

55 

ii4.8 

104. 1 

i5 

159.3 

144.4 

76 

2o3.8 

184.7 

36 

26.7 

24.2 

96 

71. 1 

64.5 

56 

ii5.6 

104.8 

16 

160.0 

145.1 

76 

204.5 

186.4 

37 

27.4 

24.8 

97 

71 .9 

65.1 

57 

116. 3 

105.4 

17 

160.8 

145.7 

77 

2o5  .2 

186.0 

38 

28.2 

2  5.5 

98 

72.6 

65.8 

58 

117. 1 

1 06. 1 

18 

161.5 

146.4 

78 

206  0 

186.7 

39 

28.9 

26.2 

99 

73.4 

66.5 

59 

117. 8 

106.8 

19 

162.3 

i47-i 

79 

206  7 

187.4 

4- 
4i 

29.6 

26.9 

TOO 

74.1 

67.2 

6g 

118. 6 

107.4 

20 

i63.o 

i47-7 

80 
281 

207.5 
208,2 

188.0 

3o.4 

27.5 

lOI 

74.8 

67.8 

161 

1 19.3 

108.1 

221 

i63.8 

148.4 

188.7 

42 

3i.i 

28.2 

02 

75.6 

68.5 

62 

120.0 

108.8 

22 

164.5 

149. 1 

82 

208.9 

189.4 

43 

3, .9 

28.9 

o3 

76.3 

69.2 

63 

120.8 

109.5 

23 

i65.2 

149.8 

83 

209.7 

1 90. 1 

4i 

32.6 

29.5 

o4 

77.1 

69.8 

64 

121. 5 

1 10. 1 

24 

166.0 

i5o.4 

84 

210.4 

190.7 

4'') 

33.3 

3o.2 

o5 

77.8 

70.5 

65 

122.3 

1 10.8 

25 

166.7 

i5i.i 

86 

211. 2 

191.4 

46 

34.1 

3o.9 

06 

78.5 

71.2 

66 

123. 0 

111.5 

26 

167.5 

i5i.8 

86 

21 1 .9 

192.1 

i- 

34.8 

3i.6 

07 

79.3 

71.9 
72.5 

67 

123.7 

112.2 

27 

168.2 

i52.4 

87 

212.7 

192.7 

48 

35.6 

32.2 

08 

80.0 

68 

124.5 

1X2.8 

28 

168.9 

i53.i 

88 

2i3.4 

193.4 

49 

36.3 

32.9 

09 

80.8 

73.2 

69 

125.2 

ii3.5 

29 

169.7 

i53.8 

89 

214.1 

194. 1 

5o 
5i 

37.0 
37.8 

33.6 
34.2 

10 

81.5 

73.9 

70 

126.0 

114.2 

3o 

170.4 

i54.5 

90 
291 

214.9 

2i5.6 

_^i8 
196.4 

III 

82.2 

74.5 

171 

126.7 

114.8 

23l 

171 .2 

i55.i 

52 

38.5 

34.9 

12 

83.0 

75.2 

72 

127.4 

ii5.5 

32 

171-9 

i55.8 

92 

216.4 

1 96. 1 

53 

39.3 

35.6 

i3 

83.7 

75.9 

73 

128.2 

1 16.2 

33 

172.6 

i56.5 

93 

2 1 7 . 1 

196.8 

64 

4o.o 

36.3 

i4 

84.5 

76.6 

74 

128.9 

1 16.9 

34 

173.4 

157. 1 

94 

217.8 

197.4 

55 

40.8 

36.9 

i5 

85.2 

77.2 

75 

129.7 

II7-5 

35 

I74-I 

157.8 

96 

218.6 

.98., 

56 

4i.5 

37.6 

16 

86.0 

77-9 

76 

i3o.4 

118. 2 

36 

174.9 

1 58.5 

96    219.3 

198.8 

67 

42.2 

38.3 

17 

86.7 

78.6 

77 

i3i.i 

118.9 

37 

175.6 

169.2 

97 

220.1 

199.5 

58 

43.0 

39.0 

18 

87.4 

79.2 

78 

i3i  .9 

119.5 

38 

176.3 

169.8 

98 

220.8 

200.1 

59 

43.7 

39.6 

19 

88.2 

79-9 

79 

i32.6 

120.2 

39 

177. 1 

160.5 

,99 

221 .5 

200.8 

60 

44.5 

40.3 

20 

88.9 

80.6 

80 

133.4 

120.9 

40 

177.8 

161. 2 

3oo    222. d 

20 1. £ 

DUi. 

I),-P. 

l.at. 

Dist. 

Dep. 

Lat. 

Dist. 

Dep. 

Lat. 

Dist.     Dep.  1 

Lat. 

Dist.     Dep.  1 

Lnt. 

N.E.iE. 

S.E.iE. 

N.W.^W. 

S.W.AW.              [For  4i  Po 

nts. 

Page  16] 

TABLE  I. 

Difference  of  Latitude  and 

Departure  for  4  Points. 

N.E.                        N.W. 

S.E.                        S.W. 

Dist. 

Lai.  1 

Dep. 

Dist. 

Lai. 

Dep. 

Dist. 

Lai. 

Dep. 
85.6 

Dist. 

Lat. 

Dep. 

Disi. 

Lat. 

Dep. 

I    no .  7 

00.7 

61 

43.1 

43.1 

121 

85.6 

181 

128.0 

128.0 

241 

170.4 

170.4 

2     OI.4 

01 .4 

62 

43.8 

43.8 

22 

86.3 

86.3 

82 

128.7 

128.7 

42 

171. 1 

171. 1 

3 

02. 1 

02. 1 

63 

44.5 

44.5 

23 

87.0 

87.0 

83 

129.4 

129.4 

A'i 

171.8 

171.8 

4 

02.8 

02.8 

64 

45.3 

45.3 

24 

87.7 

87.7 

84 

i3o.i 

i3o.i 

AA 

172.5 

172.5 

5 

o3.5 

o3.5 

65 

46.0 

46.0 

25 

88.4 

88.4 

85 

i3o.8 

i3o.8 

45 

178.2 

178.2 

6 

04.2 

04.2 

66 

46.7 

46.7 

26 

89.1 

89.1 

86 

i3i.5 

i3i.5 

46 

178.9 

178.9 

7 

04.9 

04.9 

67 

47.4 

47-4 

27 

89.8 

89.8 

87 

l32.2 

l32.2 

47 

174.7 

174.7 

8 

03.7 

o5.7 

68 

48.1 

48.1 

28 

90.5 

90.5 

88 

132.9 

182.9 

48 

175.4 

175.4 

9 

06.4 

06.4 

69 

48.8 

48.8 

29 

91 .2 

91.2 

89 

i33.6 

i33.6 

49 

176.1 

176.1 

lO 

07.1 

07.1 

70 

49.5 

49.5 

3o 

91.9 

91.9 

90 

i34.4 

134-4 

5o 

176.8 

176.8 

II 

07.8 

07.8 

71 

50.2 

5o.2 

i3i 

92.6 

92.6 

191 

i35.i 

i35.i 

25l 

177.5 

177-5 

12 

08.5 

08.5 

72 

50.9 

50.9 

J2 

93.3 

93.3 

92 

i35.8 

i35.8 

52 

178.2 

178.2 

i3 

09.2 

09.2 

73 

5i.6 

5i.6 

33 

94.0 

94.0 

93 

i36.5 

i36.5 

53 

178.9 

178.9 

i4 

09.9 

09.9 

74 

52.3 

52.3 

M 

94.8 

94.8 

94 

187.2 

187.2 

54 

179.6 

179.6 

i5 

TO. 6 

10.6 

75 

53.0 

53.0 

35 

95.5 

95.5 

95 

187.9 

187.9 

55 

180.8 

180.8 

i6 

II. 3 

II. 3 

76 

53.7 

5J.7 

36 

96.2 

96.2 

96 

i38.6 

1 38.6 

56 

181. 0 

181.0 

17 

12.0 

12.0 

77 

54.4 

54.4 

37 

96.9 

96.9 

97 

139.3 

189.3 

57 

181. 7 

181.7 

i8 

12.7 

12.7 

78 

55.2 

55.2 

38 

97.6 

97.6 

98 

i4o.o 

i4o.o 

58 

182.4 

182.4 

^9 

i3.4 

i3.4 

79 

55.9 

55.9 

39 

98.3 

98.3 

99 

140.7 

140.7 

59 

i83.i 

i83.i 

20 

i4.i 

14.1 

80 
81 

56.6 

56.6 

4o 

99.0 

99.0 

200 

i4i.4 

i4i-4 

60 

i83.8 

iS3.8 

21 

i4.8 

14.8 

57.3 

57.3 

i4i 

99-7 

99-7 

201 

142. 1 

142. 1 

261 

184.6 

184.6 

22 

1 5. 6 

i5.6 

82 

58. 0 

58 .0 

42 

100.4 

100.4 

02 

142.8 

142.8 

62 

i85.3 

i85.3 

9  3 

16.3 

16.3 

S3 

58.7 

58.7 

43 

lOI  .  I 

101. 1 

o3 

143.5 

143.5 

63 

186.0 

186.0 

24 

17.0 

17.0 

84 

59.4 

59.4 

A^ 

101.8 

101.8 

04 

144.2 

144.2 

64 

186.7 

186.7 

25 

17.7 

17.7 

85 

60. 1 

60.1 

45 

102.5 

102.5 

o5 

145.0 

145.0 

65 

187.4 

187.4 

26 1  18.4 

18.4 

86 

60.8 

60-8 

46 

103.2 

io3.2 

06 

145.7 

145.7 

66 

188. 1 

1S8.1 

2"      IQ.I 

19. 1 

87 

61.5 

61.5 

47 

103.9 

103.9 

07 

146.4 

146.4 

67 

188.8 

188.8 

28 

iq.8 

19.8 

88 

62.2 

62.2 

48 

104.7 

104.7 

08 

i47-i 

147-1 

68 

189.5 

189.5 

29 

20.5 

20.5 

8q 

62.9 

62.9 

49 

io5.4 

105.4 

09 

147-8 

i47-« 

69 

190.2 

190.2 

3o 

21  .2 

21 .2 

90 
qi 

63.6 

63.6 

5o 

106. 1 

1 06. 1 

10 

148.5 

148.5 

70 

190.9 

190.9 

3i 

21.9 

21 .9 

64.3 

64.3 

i5i 

106.8 

106.8 

211 

149-2 

149.2 

271 

191 .6 

191.6 

32 

22.6 

22.6 

92 

65.1 

65.1 

52 

107.5 

107.5 

12 

149-9 

149-9 

72 

192.3 

192.8 

33 

23.3 

23.3 

q3 

65.8 

65.8 

53 

T08.2 

108.2 

li 

i5o.6 

i5o.6 

73 

198.0 

198.0 

34 

24.0 

24.0 

94 

66.5 

66.5 

54 

108.9 

I0S.9 

i4 

i5i.3 

i5i.3 

74 

198.7 

198.7 

35 

24.7 

24.7 

95 

67.2 

67.2 

55 

109.6 

109.6 

i5 

1D2.0 

l52.0 

1^ 

194.5 

194.5 

36 

25.5 

25.5 

96 

67.9 

67.9 

56 

no. 3 

1 10.3 

16 

i52.7 

152.7 

76 

195.2 

195.2 

37 

26.2 

26.2 

97 

68.6 

68.6 

57 

III.O 

III.O 

17 

i53.4 

i53.4 

77 

195.9 

193.9 

38 

26.9 

26.9 

9S 

69.3 

69.3 

58 

III. 7 

1 1 1.7 

18 

i54.i 

i54.i 

78 

196.6 

196.6 

39 

27.6 

27.6 

99 

70.0 

70.0 

59 

112. 4 

112.4 

'9 

154.9 

154.9 

79 

197.3 

197.3 

40 

28.3 

28.3 

1 00 

70.7 

70.7 

60 

ii3.i 

ii3.i 

20 

i55.6 

1 55.6 

80 

198.0 

198.0 

4t 

29.0 

29.0 

lOI 

71.4 

71-4 

161 

ii3.8 

ii3.8 

221 

i56.3 

1 56.3 

281 

198.7 

198.7 

42 

29.7 

29.7 

02 

72.1 

72.1 

62 

114.6 

114.6 

22 

157.0 

157.0 

82 

199-4 

199-4 

43 

3o.4 

3o.4 

o3 

72.8 

72.8 

63 

ii5.3 

ii5.3 

23 

157.7 

157.7 

83 

200.1 

200.1 

A/\ 

3l.T 

3i.i 

o4 

73.5 

73.5 

64 

1 16.0 

1 16.0 

24 

i58.4 

i58.4 

84 

200.8 

200.8 

45 

3i.8 

3T.8 

o5 

74.2 

74.2 

65 

116. 7 

1 16.7 

25 

159. 1 

159. 1 

85 

201 .5 

201.5 

46 

32.5 

32.5 

06 

75.0 

75.Q 

66 

117-4 

117.4 

26 

159.8 

159.8 

86 

202.2 

202.2 

47 

33.2 

33.2 

07 

75.7 

75.7 

67 

118. 1 

118. 1 

27 

160.5 

160.5 

87 

202.9 

202.9 

48 

33. Q 

33. p 

08 

76.4 

76.4 

68 

118.8 

1 18.8 

28 

161 .2 

161.2 

88 

2o3.6 

208.6 

49 

34.6 

34.6 

09 

77.1 

77.1 

69 

119.5 

1 19.5 

29 

161.9 

161.9 

89 

204.4 

204.4 

5o 
5t 

35.4 
36.1 

35.4 

36.1 

10 

77.8 

77.8 

70 

120.2 

120.2 

3o 

162.6 

162.6 

90 

205.1 

205.1 

II I 

78.5 

78.5 

171 

120.9 

120.9 

23l 

i63.3 

i63.3 

291 

2o5.8 

2o5.8 

52 

36.8 

36.8 

12 

79.2 

79.2 

72 

121 .6 

1 2 1. 6 

32 

164.0 

164.0 

92 

206.5 

206.5 

53 

37.5 

37.5 

i3 

79-9 

79-9 

73 

122.3 

122.3 

ii 

164.8 

164.8 

93 

207.2 

207.2 

54 

38.2 

38 . 2 

i4 

80.6 

80.6 

74 

123.0 

123.0 

34 

i65.5 

i65.5 

94 

207.9 

207-9 

55 

38.  Q 

38.  Q 

lb 

81.3 

81.3 

75 

123.7 

123.7 

35 

166.2 

166.2 

95 

208.6 

208.6 

56 

39.6 

39.6 

16 

82.0 

82.0 

76 

124.5 

124.5 

3b 

166.9 

166.9 

96 

209.3 

209.3 

57 

40.3 

40.3 

17 

82.7 

82.7 

77 

125.2 

125.2 

37 

167.6 

167.6 

97 

210.0 

210.0 

58 

4i.o 

/r.o 

18 

83.4 

83.4 

78 

125.9 

125.9 

38 

16S.3 

i68.3- 

98 

210.7 

210.7 

59 

41.7 

4l.7 

19 

84.1 

84.1 

79 

126.6 

126.6 

39 

169.0 

169.0 

99 

211 .4 

2 1 1. 4 

60 

42.4 

42.4 

20 

84-9 

84.9 

80 

127.3 

127.3 

40 

169.7  1  169.7 

3oo 

212. 1 

212.1 

Dist. 

Dep. 

Lr,t. 

Dist 

Dep. 

Lat. 

Dist. 

Dep. 

Lat. 

Dist. 

Dep.  1    Lat. 

Dist. 

Dep. 

Lat. 

N.E. 

N.W. 

S.l 

-■ 

S.W. 

[For  4  Points. 

TABLE  IL 

1 
[Page  17 

Difference  of  Latitude  and  Departure  for  1  Degree. 

DIst. 

Lat. 

Dcp. 

Dist. 

Lat. 

Dep. 

Dist. 

Lat. 

Dep. 

Dist. 

Lat. 

Dep. 

Dist. 

Lat. 

Dep. 

I 

01  .0 

00.0 

61 

61 .0 

01 .1 

121 

121 .0 

02.1 

181 

181. 0 

o3.2 

241 

241 .0 

04.2 

2 

02.0 

00.0 

62 

62.0 

01 .1 

22 

122.0 

02.  I 

82 

182.0 

o3.2 

42 

242.0 

04.2 

3 

o3.o 

00. 1 

63 

63.0 

01 . 1 

23 

123.0 

02.1 

83 

i83.o 

03.2 

43 

243.0 

o4.2 

4 

o4.o 

00  I 

64 

64.0 

01 .1 

24 

124.0 

02.2 

84 

184.0 

o3.2 

M 

244.0 

o4.3 

5 

o5.o 

00. 1 

65 

65.0 

01 .1 

25 

125.0 

02.2 

85 

i85.o 

o3.2 

45 

245.0 

04.3 

6 

06.0 

00. 1 

66 

66.0 

01 .2 

26 

126.0 

02.2 

86 

186.0 

o3.2 

46 

246.0 

o4.3 

7 

07.0 

00. 1 

67 

67.0 

01  .2 

27 

127.0 

02.2 

87 

187.0 

o3.3 

47 

247.0 

04.3 

8 

08.0 

00.  I 

68 

68.0 

01 .2 

28 

128.0 

02.2 

88 

188.0 

o3.3 

48 

248.0 

04.3 

9 

09.0 

00.2 

69 

69.0 

01  .2 

29 

129.0 

02.3 

89 

189.0 

o3.3 

49 

249.0 

04.3 

10 

10. 0 

00.2 

70 

70.0 

01 .2 

3o 

i3o.o 

02.3 

90 

190.0 

o3.3 

5o 

25o.O 

04.4 

1 1 

II.O 

00.2 

71 

71.0 

01 .2 

i3i 

i3i  .0 

02.3 

191 

191 .0 

o3.3 

25l 

25l  .0 

04.4 

12 

12.0 

00.2 

72 

72.0 

01    i 

32 

l32.0 

02.3 

92 

192.0 

o3.4 

52 

252.0 

04.4 

i3 

i3.o 

00.2 

73 

73.0 

01    i 

33 

i33.o 

02.3 

93 

193.0 

o3.4 

53 

253.0 

04.4 

i4 

i4-o 

00.2 

74 

74.0 

01 .3 

34 

i34.o 

02.3 

94 

194.0 

o3.4 

54 

254.0 

04.4 

ID 

i5.o 

00.3 

75 

75.0 

01 .3 

35 

i35.o 

02.4 

95 

195.0 

o3.4 

55 

255.0 

04.5 

i6 

16.0 

00.3 

76 

76.0 

01 .3 

36 

i36.o 

02.4 

96 

196.0 

o3.4 

56 

256.0 

04.5 

17 

17.0 

00.3 

77 

77-0 

01 .3 

37 

137.0 

02.4 

97 

197.0 

o3.4 

57 

257.0 

04.5 

i8 

18.0 

00.3 

7S 

78.0 

01 .4 

38 

i38.o 

02.4 

98 

198.0 

o3.5 

58 

258.0 

04.5 

19 

19.0 

00.3 

79 

79.0 

01 .4 

39 

139.0 

02.4 

99 

199.0 

o3.5 

59 

259.0 

o4.5 

20 

20.0 

00.3 

80 

80.0 

01 .4 

40 

i4o.o 

02.4 

200 

200.0 

o3.5 

60 

260.0 

04.5 

21 

21 .0 

00.4 

81 

81.0 

01 .4 

i4i 

i4i.fr 

02.5 

201 

201 .0 

o3.5 

261 

261 .0 

04.6 

22 

22.0 

00.4 

82 

82.0 

01 .4 

42 

142.0 

02.5 

02 

202.0 

o3.5 

62 

262.0 

04.6 

23 

23. 0 

00.4 

83 

83.0 

01 .4 

43 

143.0 

02.5 

03 

203.0 

o3.5 

63 

263. 0 

04.6 

24 

24.0 

00.4 

84 

84. 0 

01 .5 

U 

i44.o 

02.5 

04 

204.0 

o3.6 

64 

264.0 

o4.6 

25 

25.0 

00.4 

85 

85. 0 

01.5 

45 

145.0 

02.5 

o5 

205.0 

o3.6 

65 

265.0 

04.6 

26 

26.0 

00.5 

86 

86.0 

01.5 

46 

146.0 

02.5 

06 

206.0 

o3.6 

66 

266.0 

04.6 

27 

27.0 

00.5 

87 

87.0 

01 .5 

47 

i47-o 

02.6 

07 

207.0 

o3.6 

67 

267.0 

04.7 

28 

28.0 

00.5 

88 

88.0 

01 .5 

48 

148.0 

02.6 

08 

208.0 

o3.6 

68 

268.0 

04.7 

29 

29.0 

00.5 

89 

89.0 

01 .6 

49 

i49-o 

02.6 

09 

209.0 

o3.6 

69 

269.0 

04.7 

Jo 

3o.o 

00.5 

90 

90.0 

01 .6 

5o 

i5o.o 

02.6 

10 

210.0 

o3.7 

70 

270.0 

04.7 

3i 

3i.o 

00.5 

91 

91 .0 

01 .6 

i5i 

i5i  .0 

02.6 

211 

211  .0 

o3.7 

271 

271 .0 

04.7 

32 

32.0 

00.6 

92 

92.0 

01 .6 

52 

l52.0 

02.7 

12 

212.0 

o3.7 

72 

272.0 

04.7 

33 

33.0 

00.6 

93 

93.0 

01 .6 

53 

i53.o 

02.7 

i3 

2l3.0 

o3.7 

73 

273.0 

o4.8 

34 

34.0 

00.6 

94 

94.0 

01.6 

54 

1 54.0 

02.7 

i4 

214.0 

o3.7 

74 

274.0 

04.8 

35 

35.0 

00.6 

95 

95.0 

01.7 

55 

155.0 

02.7 

i5 

2l5.0 

o3.8 

75 

275.0 

04.8 

36 

36.0 

00.6 

96 

96.0 

01.7 

56 

1 56.0 

02.7 

16 

216.0 

o3.8 

76 

276.0 

04.8 

37 

37.0 

00.6 

97 

97.0 

01.7 

57 

157.0 

02.7 

17 

217.0 

o3.8 

77 

277.0 

04.8 

38 

38. 0 

00.7 

98 

98.0 

01.7 

58 

i58.o 

02.8 

18 

218.0 

o3.8 

78 

278.0 

04.9 

39 

39.0 

00.7 

99 

99.0 

01.7 

59 

159.0 

02.8 

19 

219.0 

o3.8 

79 

279.0 

04.9 

40 

4o.o 

00.7 

roo 

lOO.O 

01.7 

60 

160.0 

02.8 

20 

220.0 

o3.8 

80 

280.0 

04.9 

4i 

4i  .0 

00.7 

lOI 

lOI  .0 

01.8 

161 

i6i  .0 

02.8 

221 

221  .0 

03.9 

281 

281 .0 

04.9 

■42 

42.0 

00.7 

02 

102.0 

01.8 

62 

162.0 

02.8 

22 

222.0 

03.9 

82 

282.0 

04.9 

43 

43.0 

00.8 

o3 

io3.o 

01.8 

63 

1 63.0 

02.8 

23 

223.0 

03.9 

83 

283. 0 

04.9 

^■^ 

44.0 

00.8 

04 

104.0 

01.8 

64 

164.0 

02.9 

24 

224.0 

03.9 

84 

284.0 

o5.o 

45 

45.0 

00.8 

o5 

io5.o 

01.8 

65 

i65.o 

02.9 

25 

225.0 

03.9 

85 

285.0 

o5.o 

46 

46. 0 

00.8 

06 

106.0 

01.8 

66 

1 66 . 0 

02.9 

26 

226.0 

03.9 

86 

286.0 

o5.o 

47 

47.0 

00.8 

07 

107.0 

01 .9 

67 

167.0 

02.9 

27 

227.0 

04.0 

87 

287.0 

o5.o 

48 

48.0 

00.8 

08 

108.0 

01.9 

68 

168.0 

02.9 

28 

228.0 

04.0 

88 

288.0 

o5.o 

49 

49.0 

00.9 

09 

109.0 

01 .9 

69 

169.0 

02.9 

29 

229.0 

04.0 

89 

289.0 

o5.o 

5o 

5o.o 

00.9 

10 

IIO.O 

01 .9 

70 

170.0 

o3.o 
o3.o 

Jo 

2  3o.O 

04.0 

90 

290.0 

o5.i 

5i 

5i.o 

00.9 

II I 

II  I  .0 

01.9 

171 

171. 0 

23l 

23l.O 

04.0 

291 

291 .0 

o5.i 

52 

52.0 

00.9 

12 

112. 0 

02.0 

72 

172.0 

o3.o 

32 

232.0 

04.0 

92 

292.0 

o5.i 

53 

53.0 

00.9 

i3 

ii3.o 

02.0 

73 

173.0 

o3.o 

33 

233. 0 

04. 1 

93 

293.0 

o5.i 

H 

54.0 

00.9 

i4 

ii4.o 

02.0 

74 

174.0 

o3.o 

34 

234.0 

04.1 

94 

294.0 

o5.i 

55 

55.0 

01. 0 

i5 

ii5.o 

02.0 

75 

175.0 

o3.i 

35 

235.0 

04.1 

95 

295.0 

o5.i 

56 

56.0 

01 .0 

16 

116.0 

02.0 

76 

176.0 

o3.i 

36 

236. 0 

04.1 

96 

296.0 

o5.2 

^7 

57.0 

01 .0 

17 

117. 0 

02.0 

77 

177.0 

o3.i 

37 

237.0 

04.1 

97 

297.0 

o5.2 

58 

58. 0 

01 .0 

18 

118.0 

02.1 

78 

178.0 

o3.i 

38 

238.0 

04.2 

98 

298.0 

05.2 

59 

59.0 

01 .0 

19 

/19.0 

02.1 

79 

179.0 

o3.i 

39 

239.0 

04.2 

99 

299.0 

o5.2 

bo 

60.0 

01  .0 

20 

120.0 

02.1 

80 

180.0 

o3.i 

4o 

240.0 

04.2 

3(K) 

3()0.o 

o5.2 

I»ist 

Dnp. 

Lai. 

Dist. 

Dop. 

Lat. 

Dist. 

Dep. 

Lat. 

Disi.|    Dep.  1  Lni. 

Dist. 

Dcp. 

Lat. 

[ 

•'or  89  Degrees. 

Page  18] 

TABLE  n. 

Difference  of  Latitude  and  Departure  for  2  Degrees. 

Dist 

Lat. 

Dep. 

Dist. 

Lat. 

Dep. 

Dist. 

Lat. 

Dep. 

Dist. 

Lat. 

Dep. 

Dist.[    Lit. 

Dep. 

08.4 

I 

01 .0 

00.0 

61 

61 .0 

02.1 

121 

120.9 

04.2 

181 

180.9 

06.3 

241 

240.9 

2 

02.0 

00. 1 

62 

62.0 

02.2 

22 

12'. 9 

04.3 

82 

181.9 

06.4 

42 

241.9 

08.4 

3 

o3.o 

00. 1 

63 

63. 0 

02.2 

23 

12... 9 

04.3 

83 

182.9 

06.4 

43 

242.9 

08.5 

4 

04.0 

00. 1 

64 

64.0 

02.2 

24 

123.9 

04.3 

84 

183.9 

06.4 

44 

243.9 

08.5 

5 

o5.o 

00.2 

65 

65.0 

02.3 

25 

124.9 

04.4 

85 

184.9 

06.5 

45 

244.9 

08.6 

6 

06.0 

00.2 

66 

66.0 

02.3 

26 

125.9 

o4-4 

86 

185.9 

06.5 

46 

245.9 

08. e 

7 

07.0 

00.2 

67 

67.0 

02.3 

27 

126.9 

04.4 

87 

186.9 

06.5 

47 

246.8 

08.6 

8 

08.0 

00.3 

68 

68.0 

02.4 

28 

127.9 

04.5 

88 

187.9 

06.6 

48 

247.8 

08.7 

9 

09.0 

00.3 

69 

69.0 

02.4 

29 

128.9 

04.5 

89 

188.9 

06.6 

49 

248.8 

08.7 

10 

10. 0 

00.3 

70 

70.0 

02.4 

3o 

129.9 

04.5 
04.6 

90 

189.9 

06.6 

5o 

249.8 

08.7 

II 

II. 0 

00.4 

71 

71.0 

02.5 

i3i 

1 30.9 

191 

190.9 

06.7 

25l 

250.8 

08.8 

12 

12.0 

00.4 

72 

72.0 

02.5 

32 

i3i  .9 

04.6 

92 

191.9 

06.7 

52 

251.8 

08.8 

i3 

i3.o 

00.5 

73 

73.0 

02.5 

33 

132.9 

04.6 

93 

192.9 

06.7 

53 

252.8 

08.8 

i4 

i4-o 

00.5 

74 

74.0 

02.6 

34 

133.9 

04.7 

94 

193.9 

06.8 

54 

253.8 

08.9 

i5 

i5.o 

00.5 

7^ 

7b. 0 

02.6 

35 

134.9 

04.7 

95 

194.9 

06.8 

55 

254.8 

08.9 

i6 

16.0 

00.6 

76 

76.0 

02.7 

36 

135.9 

04.7 

96 

195.9 

06.8 

56 

255.8 

08.9 

t? 

17.0 

00.6 

77 

77.0 

02.7 

37 

i36.9 

o4.8 

97 

196.9 

06.9 

57 

256.8 

09.0 

i8 

18.0 

00.6 

78 

78.0 

02.7 

38 

137.9 

04.8 

98 

197.9 

06.9 

58 

257.8 

09.0 

19 

19. c 

00.7 

79 

79.0 

02.8 

39 

i38.9 

04.9 

99 

198.9 

06.9 

59 

258.8 

09.0 

20 

20.0 

00.7 

80 

80.0 

02.8 

40 

139.9 

04.9 
04.9 

200 

199.9 

07.0 

60 

259.8 

09.1 

21 

21 .0 

00.7 

81 

8i.o 

02.8 

i4i 

140.9 

201 

200.9 

07.0 

261 

260.8 

09.1 

22 

22.0 

00.8 

82 

82.0 

02.9 

42 

i4i.9 

o5.o 

02 

201 .9 

07.0 

62 

261.8 

09.1 

23 

23. 0 

00.8 

83 

82.9 

02.9 

43 

142.9 

o5.o 

00 

202.9 

07.1 

63 

262.8 

09.2 

24 

24.0 

00.8 

84 

83.9 

02.9 

44 

143.9 

o5.o 

o4 

2o3 . 9 

07.1 

64 

263.8 

09.2 

25 

23. 0 

00.9 

85 

84.9 

o3.o 

45 

144.9 

o5.i 

o5 

204.9 

07.2 

65 

264.8 

09.2 

26 

26.0 

00.9 

86 

85.9 

o3.o 

46 

145.9 

o5. 1 

06 

205.9 

07.2 

66 

265.8 

09.3 

27 

27.0 

00.9 

87 

86.9 

o3.o 

47 

146.9 

o5.i 

07 

206.9 

07.2 

(^7 

266.8 

09.3 

28 

20.0 

01. 0 

88 

87.9 

o3.i 

48 

i47-9 

o5.2 

08 

207.9 

07.3 

68 

267.8 

09.4 

29 

29.0 

01 .0 

89 

88.9 

o3.i 

49 

148.9 

05.2 

09 

208.9 

07.3 

69 

268.8 

09.4 

3o 
Ji 

3o.o 

01 .0 

90 

89.9 

o3.i 

bo 
i5i 

149-9 

o5.2 

10 

209.9 

07.3 

70 

269.8 

09.4 

3i  .0 

01. 1 

9f 

90.9 

o3.2 

i5o.9 

o5.3 

211 

210.9 

07.4 

271 

270.8 

09.5 

32 

3i.o 

01. 1 

92 

91.9 

o3.2 

62 

ibi.9 

o5.3 

12 

21 1 .9 

07.4 

72 

271.8 

09.5 

33 

33.0 

01 .2 

93 

92.9 

o3.2 

53 

Ib2.9 

o5.3 

i3 

212.9 

07.4 

73 

272.8 

09.5 

34 

3'(.o 

01 .2 

94 

93.9 

o3.3 

54 

ib3.9 

o5.4 

i4 

213.9 

07.5 

74 

273.8 

09.6 

35 

35.0 

01 .2 

95 

94.9 

o3.3 

b") 

154.9 

o5.4 

i5 

214.9 

07.5 

7^ 

274.8 

09.6 

36 

36.0 

01.3 

96 

95.9 

o3.4 

56 

ibb.9 

o5.4 

16 

215.9 

07.5 

76 

275.8 

09.6 

37 

J7.0 

01.3 

97 

96.9 

o3.4 

^7 

i56.9 

o5.5 

17 

216.9 

07.6 

77 

276.8 

09.7 

38 

38. 0 

01.3 

98 

97-9 

o3.4 

58 

157.9 

o5.5 

18 

217.9 

07.6 

78 

277.8 

09.7 

39 

39.0 

01 .4 

99 

98.9 

o3.5 

b9 

i58.9 

o5.5 

'9 

218.9 

07.6 

79 

278.8 

09.7 

40 

4o.o 

01 .4 

100 

99.9 

o3.5 

60 

159.9 

ob.6 

20 

219.9 

07.7 

80 

279.8 

09.8 

4i 

4i  .0 

01 .4 

lOI 

100.9 

o3.5 

161 

160.9 

o5.6 

221 

220.9 

07.7 

281 

280.8 

09.8 

45 

42.0 

01.5 

02 

lOI  .9 

o3.6 

62 

161 .9 

o5.7 

22 

221 .9 

07.7 

82 

2S1.8 

09 . 8 

43 

43.0 

01 .5 

o3 

102.9 

o3.6 

63 

162.9 

o5.7 

23 

222.9 

07.8 

Hi 

282.8 

09.9 

44 

44.0 

01.5 

o4 

103.9 

o3.6 

64 

163.9 

o5.7 

24 

223.9 

07.8 

84 

983.8 

09.9 

45 

45.0 

01 .6 

o5 

104.9 

o3.7 

65 

164.9 

o5.8 

25 

224.9 

07.9 

8b 

284.8 

09.9 

46 

46. 0 

01 .6 

06 

105.9 

o3.7 

66 

165.9 

o5.8 

26 

225.9 

07.9 

86 

285.8  1  10. 0  1 

47 

47-0 

01.6 

07 

106.9 

o3.7 

67 

166.9 

o5.8 

27 

226.9 

07.9 

87 

286.8  1  10. 0 

■48 

48. 0 

01.7 

08 

107.9 

o3.8 

68 

167.9 

05.9 

28 

227.9 

08.0 

88    287.8  i  10. 1 

49 

49.0 

01.7 

09 

108.9 

o3.8 

69 

168.9 

o5.9 

29 

228.9 

08.0 

89    288.8  ■  10.1 

5o 

5o.o 

01 .7 

10 

109.9 

o3.8 

70 

169.9 

05.9 
06.0 

3u 

229.9 

08.0 

90    289.8 

10.  1    ! 

5i 

5i.o 

01.8 

III 

no. 9 

03.9 

171 

170.9 

23l 

230.9 

08.1 

291 

290.8 

10.2 

52 

52.0 

01.8 

12 

III  .9 

03.9 

72 

171-9 

06.0 

32 

23l  .9 

08.1 

92 

291.8 

10.2    1 

53 

53.0 

01.8 

i3 

II  2  . 9 

03.9 

73 

172.9 

06.0 

33 

232.9 

08.1 

93 

292.8 

10.2    1 

54 

54.0 

01.9 

i4 

II3.9 

04.0 

74 

173.9 

06.1 

34 

233.9 

08.2 

94    293.8 

10.3    i 

5? 

55.0 

01 .9 

i5 

ii4-9 

04.0 

75 

174.9 

06.1 

35 

234.9 

08.2 

951294.8 

10.3 

56 

56. 0 

02.0 

16 

115.9 

04.0 

76 

175.9 

06.1 

36 

235.9 

08.2 

96^295.8 

10  3 

57 

57.0 

02.0 

17 

no. 9 

04.1 

77 

176.9 

06.2 

37 

236.9 

08.3 

97 

296.8 

10.4 

58 

58. 0 

02.0 

18 

117. 9 

04.1 

78 

177-9 

06.2 

38 

237-9 

08.3 

98 

^9Z-" 

10.4 

5q 

59.0 

02.1 

19 

118.9 

04.2 

79 

178.9 

06.2 

39 

238.9 

08.3 

.99 

298.8 

10.4 

Go 
Dist. 

60.0 
Hep. 

02.1 
Lat. 

20 

119. 9 

04.2 
Lat. 

80 

179.9 

06.3 

40 

239.9 

08.4 

3oo 

299.8 

10.5 
L.it. 

Dist. 

Dep. 

Dist. 

Dep. 

Lat. 

Dist. 

])ep. 

Lril. 

Dist. 

Dep. 

[For  83  Degrees. 

TABLE  IL 

[Page  19 

Difference  of  Latitude  and  Departure  for  3  Degrees. 

Dist. 

Lat. 

Dcp. 

Dist. 

Lat. 

Dcp. 
o3.2 

Dist. 

Lat. 

Dep. 

Dist. 

Lat. 

Dep. 

Dist. 

Lat 

Dep. 

{ 

01 .0 

00. 1 

61 

60.9 

121 

120.8 

06.3 

181 

180.8 

09.5 

24 1 

240.7 

12.6 

2 

02.0 

00. 1 

62 

61 .9 

o3.2 

22 

121. 8 

06.4 

82 

181.8 

09.5 

42 

241.7 

12.7 

,3 

o3.o 

00.2 

C3 

62.9 

o3.3 

23 

122. 8 

06.4 

83 

182.7 

09.6 

43 

242.7 

12.7 

4 

o4.o 

00  2 

64 

63.9 

o3.3 

24 

123.8 

06.5 

84 

183.7 

09.6 

44 

243.7 

12.8 

") 

o5.o 

00.3 

65 

64.9 

o3.4 

25 

124.8 

06.5 

85 

184.7 

09.7 

45 

244.7 

12.8 

6 

06.0 

00.3 

66 

65.9 

o3.5 

26 

125.8 

06.6 

86 

185.7 

09.7 

46 

245.7 

12.9 

7 

07.0 

00.4 

67 

66.9 

o3.5 

27 

126.8 

06.6 

87 

186.7 

09.8 

47 

246.7 

12.9 

8 

08.0 

00.4 

68 

67.9 

o3.6 

28 

127.8 

06.7 

88 

187.7 

09.8 

48 

247.7 

i3.o 

9 

09.0 

00.5 

69 

68.9 

o3.6 

29 

128.8 

06.8 

89 

188.7 

09.9 

49 

248.7 

i3.o 

10 

10. 0 

00.5 

7" 

69.9 

o3.7 

3o 

129.8 

Ob. 8 

90 

189.7 

09.9 

5o 

249.7 

i3.i 

1 1 

1 1 .0 

00.6 

71 

70.9 

03.7 

i3i 

i3o.8 

06.9 

191 

190.7 

10.0 

25l 

250.7 

i3.i 

12 

12.0 

CK).6 

72 

71.9 

o3.8 

32 

i3i.8 

06.9 

92 

191.7 

10. 0 

62 

25l  .7 

l3.2 

i3 

i3.o 

00.7 

73 

72.9 

o3.8 

33 

i32.8 

07.0 

93 

192.7 

10. 1 

53 

252.7 

l3.2 

1 4 

i4.o 

00.7 

74 

73.9 

03.9 

34 

i33.8 

07.0 

94 

193.7 

10.2 

54 

253.7 

i3.3 

i5 

i5.o 

00.8 

75 

74.9 

03.9 

35 

i34.8 

07.1 

95 

194.7 

10.2 

55 

254.7 

i3.3 

i6 

16.0 

00.8 

76 

75.9 

04.0 

36 

i35.8 

07.1 

96 

195.7 

10.3 

5b 

255.6 

i3.4 

17 

17.0 

00.9 

77 

76.9 

04.0 

37 

i36.8 

07.2 

97 

196.7 

10.3 

J)7 

256.6 

i3.5 

i8 

18.0 

00.9 

78 

77-9 

04.1 

38 

137.8 

07.2 

08 

197-7 

10.4 

58 

257.6 

i3.5 

19 

ly.o 

01. 0 

79 

78.9 

04.1 

39 

i38.8 

07.3 

99 

198.7 

10.4 

59 

258.6 

i3.6 

20 

20.0 

01 .0 

80 

79-9 

04.2 

40 

139.8 

07.3 

200 

199.7 

10.5 

bo 

259.6 

i3.6 

21 

21  .0 

01 .1 

81 

80.9 

04.2 

i4i 

i4o.8 

07.4 

201 

200.7 

10.5 

261 

260.6 

i3.7 

22 

22.0 

01 .2 

82 

81 .9 

04.3 

42 

i4i.8 

07.4 

02 

201 .7 

10.6 

b2 

261 .6 

i3.7 

23 

23.0 

01 .2 

83 

82.9 

04.3 

43 

142.8 

07.5 

o3 

202.7 

10.6 

b3 

262.6 

i3.8 

24 

24.0 

01 .3 

84 

83.9 

04.4 

44 

143.8 

07.5 

04 

203.7 

10.7 

64 

263.6 

i3.8 

2  5 

25.0 

01 .3 

85 

84.9 

04.4 

45 

144.8 

07.6 

o5 

204.7 

10.7 

b5 

264.6 

13.9 

26 

26.0 

01 .4 

86 

85.9 

04.5 

46 

145.8 

07.6 

06 

205.7 

10.8 

bb 

265.6 

13.9 

27 

27.0 

01 .4 

87 

86.9 

04.6 

47 

i46.8 

07.7 

07 

206.7 

10.8 

67 

266.6 

14.0 

28 

28.0 

01.5 

88 

87.9 

04.6 

48 

147.8 

07.7 

08 

207.7 

10.9 

b8 

267.6 

14.0 

29 

29.0 

01.5 

89 

88.9 

04.7 

49 

148.8 

07.8 

09 

208.7 

10.9 

69 

268.6 

14.1 

3o 
3i 

3o.o 
3i  .0 

01 .6 
01 .6 

90 

89.9 

04.7 

5o 

149-8 

07.9 

10 

209.7 

1 1 .0 

70 
271 

269.6 
270.6 

i4.i 

91 

90.9 

04.8 

i5i 

i5o.8 

07.9 

211 

210.7 

11. 0 

14.2 

32 

32.0 

01.7 

92 

91.9 

04.8 

52 

i5i.8 

08.0 

12 

211 .7 

II. I 

72 

271.6 

14.2 

33 

33.0 

01.7 

g3 

92.9 

04.9 

53 

i52.8 

08.0 

i3 

212.7 

11 .1 

73 

272.6 

i4.3 

34 

34.0 

01.8 

94 

93.9 

04.9 

54 

i53.8 

08.1 

i4 

213.7 

11 .2 

74 

273.6 

14.3 

35 

35.0 

01.8 

95 

94-9 

o5.o 

55 

i54.8 

08.1 

i5 

214.7 

11.3 

7^ 

274.6 

i4.4 

3G 

36.0 

01 .9 

96 

95.9 

o5.o 

56 

i55.8 

08.2 

16 

215.7 

II. 3 

7b 

275.6 

i4-4 

37 

36.9 

01.9 

97 

96.9 

o5.i 

57 

i56.8 

08.2 

17 

216.7 

11.4 

77 

276.6 

14.5 

38 

37.9 

02.0 

98 

97-9 

o5.i 

58 

157.8 

08.3 

18 

217.7 

11.4 

78 

277.6 

14.5 

39 

38.9 

02  .0 

99 

98.9 

o5.2 

59 

i58.8 

08.3 

19 

218.7 

11.5 

Z9 

278.6 

i4.6 

4o 

39.9 

02. 1 

100 

99.9 

o5.2 

o5.3 

60 

159.8 

08.4 
08.4 

20 

219.7 

11.5 

80 

279.6 

14.7 

4i 

40.9 

02.1 

lOI 

100.9 

161 

160.8 

221 

220.7 

11.6 

281 

280.6 

14.7 

42 

4!  .9 

02.2 

02 

lOI  .9 

o5.3 

62 

161.8 

08.5 

22 

221  .7 

11.6 

82 

281.6 

14.8 

43 

42.9 

02.3 

o3 

102.9 

o5.4 

63 

162.8 

08.5 

23 

222.7 

II. 7 

83 

282.6 

i4.8 

44 

43.9 

02.3 

o4 

103.9 

o5.4 

64 

i63.8 

08.6 

24 

223.7 

11.7 

84 

283.6 

14.9 

45 

44.9 

02.4 

o5 

104.9 

o5.5 

65 

164.8 

08.6 

25 

224.7 

II. 8 

85 

284.6 

14.9 

46 

45.9 

02.4 

06 

105.9 

o5.5 

66 

i65.8 

08.7 

26 

225.7 

II. 8 

8b 

285.6 

i5.o 

47 

46.9 

02.5 

07 

106.9 

o5.6 

67 

166.8 

08.7 

27 

226.7 

11.9 

87 

286.6 

i5.o 

48 

4-7.9 

02.5 

08 

107.9 

05.7 

68 

167.8 

08.8 

28 

227.7 

11.9 

88 

287.6 

i5.i 

49 

48.9 

02.6 

09 

108.9 

o5.7 

69 

168.8 

08.8 

29    228.7 

12.0 

89 

288.6 

i5.i 

5o 

49.9 

02.6 

10 

109.8 

o5.8 

70 

169.8 

08.9 

3o 

229.7 

12.0 

90 

289.6 

l5.2 

5i 

50.9 

02.7 

III 

no. 8 

o5.8 

171 

170.8 

08.9 

23l 

2  3o.7 

12. 1 

291 

290.6 

l5.2 

52 

5. .9 

02.7 

12 

III. 8 

05.9 

72 

171. 8 

09.0 

32 

23l  .7 

12.1 

92 

291 .6 

i5.3 

53 

52.9 

02.8 

i3 

112. 8 

05.9 

73 

172.8 

09.1 

33 

232.7 

12.2 

93 

=9'-6 

i5.3 

54 

53.9 

02.8 

i4 

ii3.8 

06.0 

74 

173.8 

09.1 

34 

233.7 

12.2 

94 

293.6 

i5.4 

55 

54.9 

02.9 

i5 

114.8 

06.0 

75 

174.8 

09.2 

35 

234.7 

12.3 

95 

294.6 

i5.4 

56 

55.9 

02.9 

16 

ii5.8 

06.1 

76 

175.8 

09.2 

36 

235.7 

12.4 

9b 

295-6 

i5.5 

57 

56.9 

o3.o 

17 

116.8 

06.1 

77 

176.8 

09.3 

37 

236.7 

12.4 

97 

296.6 

i5.5 

58 

57.9 

o3.o 

18 

117. 8 

06.2 

78 

177.8 

09.3 

38 

237.7 

12.5 

98 

297.6 

i5.6 

59 

58.9 

o3.i 

19 

118. 8 

06.2 

79 

178.8 

09.4 

39 

238.7 

12.5 

99 

298.6 

i5.6 

bo 

59.9 

o3.i 

20 

119. 8 

06.3 

80 

179-8 

09.4 

4o 

239.7 

12.6 

3oo 

299.6 

i5.7 

nist. 

Dcp. 

Lat. 

|l)is, 

Dcp. 

Lat. 

Dist 

Dop. 

Lat. 

Dist.j    Dep. 

Lat. 

Dist 

Dep. 

Lat. 

[For  87  Degrees. 

Page  20] 

TABLE  IL 

Difference  of  Latitude  and  Departure  for  4  Degrees. 

Dist 

Lat. 

Dep. 

Dist. 

Lat. 

Dep. 
04.3 

Dist. 

Lat. 

Dep. 

Dist 

Lat. 

Dep. 
12.6 

Dist. 

Lat. 

Dep. 

I 

01 .0 

00. 1 

61 

60.9 

121 

120.7 

08.4 

181 

180.6 

241 

240.4 

16.8 

2 

02.0 

00. 1 

62 

61.8 

04.3 

22 

121. 7 

08.5 

82 

181. 6 

12.7 

42 

241.4 

16.9 

3 

o3.o 

00.2 

63 

62.8 

04.4 

23 

122.7 

08.6 

83 

182.6 

12.8 

43 

242.4 

17.0 

4 

o4.o 

00.3 

64 

63.8 

o4.5 

24 

123.7 

08.6 

84 

i83.6 

12.8 

M 

243.4 

17.0 

5 

o5.o 

00.3 

65 

64.8 

04.5 

25 

124.7 

08.7 

85 

184.5 

12.9 

45 

244.4 

17. 1 

6 

06.0 

00.4 

66 

65.8 

o4.6 

26 

125.7 

08.8 

86 

i85.5 

i3.o 

46 

245.4 

17.2 

7 

07.0 

00.5 

67 

66.8 

04.7 

27 

126.7 

08.9 

87 

186.5 

i3.o 

47 

246.4 

17.2 

8 

08.0 

00.6 

68 

67.8 

04.7 

28 

127.7 

08.9 

88 

187.5 

i3.i 

48 

247.4 

17.3 

9 

09.0 

00.6 

69 

68.8 

04.8 

29 

128.7 

09.0 

89 

188.5 

l3.2 

49 

2.48.4 

17.4 

lO 

10. 0 

00.7 

70 

69.8 

04.9 

3o 

129.7 

09.1 

90 

189.5 

i3.3 

5o 

249.4 

17.4 

II 

II. 0 

00.8 

71 

70.8 

o5.o 

i3i 

i3o.7 

09.1 

191 

190.5 

i3.3 

25l 

25o.4 

17.5 

12 

12.0 

00.8 

72 

71.8 

o5.o 

32 

i3i.7 

09.2 

92 

191. 5 

i3.4 

b2 

251.4 

I7.b 

i3 

i3.o 

00.9 

73 

72.8 

o5.i 

■6-6 

132.7 

09.3 

93 

192.5 

i3.5 

53 

262.4 

17.6 

i4 

lA.o 

01 .0 

74 

73.8 

05.2 

M 

133.7 

09.3 

94 

193.5 

i3.5 

54 

253.4 

17.7 

i5 

i5.o 

01 .0 

75 

74.8 

o5.2 

35 

134.7 

09.4 

95 

194.5 

i3.6 

55 

254.4 

17.8 

i6 

16.0 

01. 1 

76 

75.8 

o5.3 

■i6 

1J5.7 

09.5 

96 

195.5 

13.7 

56 

255.4 

17.9 

17 

17.0 

01  .2 

77 

76.8 

o5.4 

37 

136.7 

09.6 

97 

196.5 

i3.7 

57 

256.4 

17.9 

i8 

18.0 

01.3 

78 

77.8 

o5.4 

38 

137.7 

09.6 

98 

197.5 

i3.8 

68 

267.4 

18.0 

19 

19.0 

01.3 

79 

78.8 

o5.5 

39 

i38.7 

09.7 

99 

198.5 

i3.9 

59 

258.4 

18. 1 

20 

20.0 

01.4 

80 

79.8 

o5.6 

40 

139.7 

09.8 

200 

199.5 

14.0 

bo 

269.4 

18. 1 

2-1 

20.9 

01.5 

81 

80.8 

o5.7 

i4i 

140.7 

09.8 

201 

200.5 

14.0 

261 

260.4 

18.2 

22 

21.9 

01.5 

82 

81.8 

05.7 

42 

141.7 

09.9 

02 

201.5 

14.1 

62 

261.4 

18.3 

23 

11. C) 

01 .6 

83 

82.8 

o5.8 

43 

142.7 

10. 0 

OJ 

202.5 

14.2 

63 

262.4 

18.3 

24 

23.9 

01.7 

84 

83.8 

05.9 

M 

143.6 

10. 0 

o4 

2o3.5 

14.2 

64 

263.4 

18.4 

25 

24.9 

01.7 

85 

84.8 

05.9 

45 

144.6 

10. 1 

o5 

204.5 

14.3 

65 

264.4 

18.5 

26 

25.9 

01.8 

86 

85.8 

06.0 

46 

145.6 

10.2 

06 

2o5.5 

14.4 

66 

265.4 

18.6 

27 

26.9 

01 .9 

87 

86.8 

06.1 

47 

i46.6 

10.3 

07 

206.5 

14.4 

67 

266.3 

18.6 

28 

27.9 

02.0 

88 

87.8 

06.1 

48 

147-6 

10.3 

08 

207.5 

14.5 

68 

267.3 

18.7 

29 

28.9 

02.0 

89 

88.8 

06.2 

49 

i48.6 

10.4 

09 

208.5 

14.6 

69 

268.3 

18.8 

3o 

29.9 

02. 1 

90 

89.8 

06.3 

5o 

149.6 

10.5 

10 

209.5 

14.6 

70 

269.3 

18.8 

3 1    3o.9 

02.2 

91 

90.8 

06.3 

i5i 

i5o.6 

10.5 

an 

210.5 

14.7 

271 

270.3 

18.9 

32 

3i.9 

02.2 

92 

91.8 

06.4 

52 

i5i.6 

10.6 

12 

211 .5 

i4.8 

72 

271.3 

19.0 

33 

32. 9 

02.3 

93 

92.8 

06.5 

53 

i52.6 

10.7 

i3 

212.5 

14.9 

73 

272.3 

19.0 

34 

33.9 

02.4 

94 

93.8 

06.6 

54 

i53.6 

10.7 

i4 

2i3.5 

14.9 

74 

273.3 

19. 1 

35 

34.9 

02.4 

95 

94.8 

06.6 

65 

i54.6 

10.8 

i5 

214.5 

i5.o 

75 

274.3 

19.2 

36 

35.9 

02.5 

96 

95.8 

06.7 

56 

i55.6 

10.9 

16 

2i5.5 

i5.i 

76 

275.3 

19.3 

37 

36.9 

02.6 

07 

96.8 

06.8 

57 

i56.6 

II  .0 

17 

216.5 

i5.i 

77 

276.3 

19.3 

38 

37.9 

02.7 

98 

97.8 

06.8 

58 

157.6 

II. 0 

18 

217.5 

l5.2 

78 

277.3 

19-4 

39 

38.9 

02.7 

99 

98.8 

06.9 

59 

i58.6 

II. I 

19 

218.5 

i5.3 

79 

278.3 

19.6 

4o 
4i 

39.9 

02.8 

100 

99.8 

07.0 

bo 

159.6 

II. 2 

30 

219.5 

i5.3 

80 
281 

279.3 
280.3 

19. D 
19.6 

40.9 

02.9 

lOI 

100.8 

07.0 

161 

160.6 

II. 2 

221 

220.5 

i5.4 

42 

41.9 

02.9 

02 

101.8 

07.1 

62 

161. 6 

II. 3 

22 

221 .5 

i5.5 

82 

281.3 

19.7 

43 

42.9 

o3.o 

o3 

102.7 

07.2 

63 

162.6 

1 1. 4 

23 

222.5 

i5.6 

83 

282.3 

19.7 

Ai 

43.9 

o3.i 

o4 

103.7 

07.3 

64 

i63.6 

II. 4 

24 

223.5 

i5.6 

84 

283.3 

19.8 

45 

44.9 

o3.i 

o5 

104.7 

07.3 

65 

164.6 

II. 5 

25 

224.5 

i5." 

85 

284.3 

19.9 

46 

45.9 

o3.2 

06 

105.7 

07.4 

66 

i65.6 

II. 6 

26 

225.4 

i5.& 

86 

285.3 

20.0 

47 

46.9 

o3.3 

07 

106.7 

07.5 

67 

166.6 

II. 6 

27 

226.4 

i5.8 

87 

2S6.3 

20.0 

48 

47-9 

o3.3 

08 

107.7 

07.5 

68 

167.6 

II. 7 

28 

227.4 

i5.9 

88 

287.3 

20.1 

49 

48.9 

o3.4 

09 

108.7 

07.6 

69 

168.6 

II. 8 

29 

228.4 

16.0 

89 

2S8.3 

20.2 

5o 
5i 

49.9 

o3.5 

10 

109.7 

07.7 

70 

169.6 

II. 9 

3o 

229.4 

16.0 

90 

289  3 

20.2 

50.9 

o3.6 

III 

110.7 

07.7 

171 

170.6 

II. 9 

23  1 

23o.4 

16.1 

291 

290.3 

20.3 

52 

5i  .9 

o3.6 

12 

III. 7 

07.8 

72 

171. 6 

12.0 

32 

23i.4 

16.2 

92 

291 .3 

30.4 

53 

52.9 

o3.7 

i3 

iii.7 

07.9 

73 

172.6 

12. 1 

33 

232.4 

16.3 

93 

292.3 

20.4 

54 

53.9 

o3.8 

i4 

113.7 

08.0 

74 

173.6 

12.1 

34 

233.4 

16.3 

94 

293.3 

20.5 

55 

54.9 

o3.8 

i5 

114.7 

08.0 

75 

174.6 

12.2 

35 

234.4 

16.4 

95 

204.3 

20. D 

56 

55.9 

03.9 

16 

115.7 

08.1 

76 

175.6 

12.3 

36 

235.4 

16.5 

96 

295.3 

20.6 

57 

56.9 

04.0 

17 

116. 7 

08.2 

77 

176.6 

12.3 

37 

236.4 

16.5 

97 

296.3 

20.7 

58 

57.9 

04.0 

18 

117. 7 

08.2 

78 

177.6 

12.4 

38 

237.4 

16.6 

98 

297.3 

20.8 

59 

58.9 

04.1 

19 

118. 7 

08.3 

79 

178.6 

12.5 

39 

238.4 

16.7 

99 

298.3 

20.9 

60 

59.9 

04.2 

20 

119. 7 

08.4 

80 

179.6 

[2.6 

4o 

239.4 

16.7 

3  00 

299.3 

20.9 

Dist. 

Dep. 

Lat. 

Dist. 

Dep. 

Lat. 

Dist.     Dep.  1 

Lat. 

Dist. 

Dep. 

Lat. 

Dist. 

Dep. 

Lat. 

[For  86  Degrees. 

TABLE  II.                                           f!'H='e2i 
Difference  of  Latitude  and  Departure  for  5  Degrees. 

Dist. 

Lat. 

Dep. 

Dist. 

6i 
62 
63 
64 
65 
66 
67 
68 
69 
70 

Lat. 

60.8 

61.8 

62.8 

63.8 

64.8 

65.7 

66.7 

67.7 
68.7 
69.7 

Dep. 
o5.3 
o5.4 
o5.5 
o5.6 
05.7 
o5.8 
o5.8 
05.9 
06.0 
06.1 

Dist. 

Lat.   1  Dep. 

Dist. 

Lat.    1  Dep. 

Dist.l    Lat. 

Dep. 

I 

2 

3 
4 
5 
6 

7 
8 

9 

10 

01 .0 
02.0 
o3.o 
04.0 
o5.o 
06.0 
07.0 
08.0 
09.0 
10. 0 

00. 1 
00.2 
00.3 
00.3 
00.4 
00.5 
00.6 
00.7 
00.8 
00.9 

121 
22 

23 

24 

25 

26 
27 
28 

=9 

3o 

120.5 
121 .5 

122.5 

123.5 
124.5 
125.5 
126.5 
127.5 
128.5 
129.5 

10.5 
10.6 
10.7 
10.8 
10.9 

1 1 .0 

1 1 .1 
II  .2 
II  .2 
II. 3 

181 
82 
83 
84 
85 
86 

87 
88 
89 
90 

180.3 
181. 3 
182.3 
i83.3 
184.3 
i85.3 
186.3 
187.3 
188.3 
189.3 

i5.8 
15.9 
15.9 
16.0 
16. 1 
16.2 
16.3 
16.4 
16.5 
16.6 

241 
42 
43 
A^ 

45 
46 
47 
4^ 

240.1 
241 .1 
242. 1 
243.1 

244.1 
245.1 
246.1 

247-1 

248.1 
249.0 

21 .0 

21 .1 
21.2 

21.3 
21  .4 
21.4 
21.5 
21  .6 

21.7 

21.8 

II 

I  2 

i3 

i4 
i5 
i6 

17 
i8 

19 

20 

II  .0 
12.0 
i3.o 
13.9 
14.9 
15.9 
16.9 

17.9 
18.9 
19.9 

01 .0 

01 .0 

01. 1 
01.2 

01 .3 

01 .4 
01.5 
01 .6 
01.7 
01.7 

01.8 
01 .9 
02.0 
02.1 
02.2 
02.3 
02.4 
02.4 
02.5 
02.6 

71 
72 
73 
74 
75 
76 
77 
78 

79 
80 

70.7 

71-7 
72.7 
73.7 

74.7 
75.7 
76.7 
77-7 
78.7 
79-7 

06.2 
06.3 
06.4 
06.4 
06.5 
06.6 
06.7 
06.8 
06.9 
07.0 

i3i 

32 

33 
M 
35 
36 

37 
38 
39 
4o 

i3o.5 
i3i.5 
i32.5 
i33.5 
i34.5 
i35.5 
i36.5 
137.5 
i38.5 
139.5 

11. 4 

11. 5 

11. 6 

11. 7 

11. 8 

11. 9 
II. 9 
12.0 
12. 1 
12.2 

191 

92 
93 
94 

96 

97 
98 

99 
200 

190.3 
191 .3 
192.3 
193.3 
194.3 
195.3 
196.3 
197.2 
19S.2 
199.2 

16.6 
16.7 
16.8 
16.9 
17.0 
17. 1 
17.2 
17.3 
17.3 
17.4 

25l 

52 

53 

54 
55 
56 
57 
58 

60 

25o.O 
25l  .0 
252.0 

253. 0 
254.0 
255. 0 
256. 0 
257.0 
258. 0 
259.0 

21  .9 
22.0 
22.1 
22.1 
22.2 
22.3 
22.4 
22.5 
22.6 
22.7 

21 
22 
23 
24 
25 

26 

27 
28 

3o 

20.9 
21.9 
22.9 
23.9 
24.9 
25.9 
26.9 
27.9 
28.9 
29.9 

81 
82 
83 

84 
85 
86 
87 
88 
89 
90 

80.7 
81.7 
82.7 
83.7 
84.7 
85.7 
86.7 
87.7 
88.7 
89.7 

07.1 
07.1 
07.2 
07.3 
07.4 
07.5 
07.6 
07.7 
07.8 
07.8 

i4i 
42 
A'i 

45 
46 
47 
48 

49 
5o 

i4o.5 
i4r.5 
142.5 
143.5 
144.4 
145.4 
146.4 
147-4 
148.4 
149.4 

12.3 
12.4 
12.5 
12.6 
12.6 
12.7 
12.8 
12.9 
i3.o 
i3.i 

201 
02 
o3 
o4 
o5 
06 
07 
08 
09 
10 

200.2 
201 .2 
202.2 

2o3.2 
204.2 
2o5.2 
206.2 
207.2 
208.2 
209.2 

17.5 
17.6 

'7.7 
17.8 
17.9 
18.0 
18.0 
18.1 
18.2 
18.3 

261 
62 
63 
64 
65 
66 
67 
68 
69 
70 

260.0 
261 .0 
262.0 
263.0 
264.0 
265.0 
266.0 
267.0 
268.0 
269.0 

22.7 
22.8 
22.9 
23. 0 
23.1 
23.2 

23.3 

23.4 
23.4 

23.5 

3i 

32 

33 
34 
35 
36 
37 
38 
39 
4o 

30.9 
3i  .9 
32.9 
33.9 
34.9 
35.9 
36.9 
37.9 
38.9 
39.8 

02.7 
02.8 
02.9 
o3.o 
o3.i 
o3.i 
o3.2 
o3.3 
o3.4 
o3.5 

91 

92 
93 

94 

95 
96 

97 
98 

99 

100 

90.7 
91 .6 
92.6 
93.6 
94.6 
95.6 
96.6 
97.6 
98.6 
99.6 

07.9 
08.0 
08.1 
08.2 
08.3 
08.4 
08.5 
08.5 
08.6 
08.7 

i5i 

52 

53 

54 
55 
56 
57 
58 

60 

i5o.4 
i5i.4 
i52.4 
i53.4 
i54.4 
i55.4 
i56.4 
157.4 
i58.4 
159.4 

l3.2 
l3.2 

i3.3 
i3.4 
i3.5 
i3.6 
i3.7 
i3.8 
13.9 
13.9 

211 

12 

i3 
i4 
i5 
16 

17 
18 

19 
20 

210.2 
211  .2 
2  12.2 
2l3.2 
2l4.2 
2l5.2 
216.2 
217.2 
218.2 
219.2 

18.4 
18.5 
18.6 
18.7 
18.7 
18.8 
18.9 
19.0 
19. 1 
19.2 

271 

72 
73 
74 
75 
76 
77 
78 

Z9 
80 

270.0 
271 .0 
272.0 
273.0 
274.0 
274.9 
275.9 
276.9 
277.9 
278.9 

23.6 

23.7 

23.8 

23.9. 

24.0 

24.1 

24.1 

24.2 

24.3 

24.4 

4i 
42 
43 
4^ 
45 
46 
47 
48 
49 
5o 

40.8 
4i.8 
42.8 
43.8 
44.8 
45.8 
46.8 
47.8 
48.8 
49-8 

o3.6 
03.7 
o3.7 
o3.8 
03.9 
04.0 

04. 1 

04 . 2 
04.3 
04.4 

lOI 

02 
o3 
04 
o5 
06 
07 
08 
09 
10 

100.6 
loi  .6 
102.6 
io3.6 
104.6 
105.6 
106.6 
107.6 
108.6 
109.6 

08.8 
08.9 
09.0 
09. 1 
09.2 
09.2 
09.3 
09.4 
09.5 
09.6 

161 
62 
63 
64 
65 
66 

67 
68 
69 
70 

160.4 
161. 4 
162.4 
i63.4 
164.4 
t65.4 
166.4 
167.4 
168.4 
169.4 

14.0 
i4.i 
14.2 
i4.3 
i4.4 
i4.5 
i4.6 
14.6 
14.7 
i4.8 

221 
22 

23 

24 

25 

26 

27 
28 

3? 

220.2 
221  .2 
222.2 
223.1 
224.1 
225.1 
226.1 
227.1 
228.1 
229.1 

19.3 
19.3 
19-4 
19.5 
19.6 

'9-7 
19.8 

19.9 

20.0 

20.0 

281 
82 
83 
84 
85 
86 
87 
88 
89 
90 

279.9 
280.9 
281.9 
282.9 
283.9 
284.9 
285.9 
286.9 
287.9 
288.9 

24.5 
24.6 
24.7 
24.8 
24.8 
24.9 
25.0 

25.1 
25.2 

25.3 

5i 

52 

53 
54 
55 
56 
57 
58 
59 
60 

5o.8 
5i.8 
5s. 8 
53.8 
54.8 
55.8 
56.8 
57.8 
58.8 
59.8 

04.4 
04.5 
04.6 

04.7 
04.8 
04.9 
o5.o 
o5.i 
o5.i 
o5.2 

III 
12 
i3 
i4 
i5 
16 

17 
18 

19 

20 

1 10.6 
II 1 .6 
112. 6 
ii3.6 
ii4.6 
ii5.6 
116. 6 
117. 6 
118. 5 
119. 5 

09.7 
09.8 
09.8 
09.9 

10. 0 

10. 1 
10.2 
10.3 
10.4 
10.5 

171 

72 
73 
74 
75 
76 
77 
78 

79 
80 

170.3 
171. 3 

172.3 
173.3 
174.3 
175.3 
176.3 
177.3 
178.3 
179.3 

r4.9 
i5.o 
i5.i 

l5.2 

i5.3 
i5.3 
i5.4 
i5.5 
i5.6 
i5.7 

23l 
32 

33 
M 
35 
36 

37 
38 
39 
4o 

23o.I 
23l  .1 
232.1 

233.1 

234.1 

235.1 
236.1 
237.1 
258.1 
239.1 

20. 1 
20.2 

20.3 

20.4 
20.5 

20.6 

20.7 
20.7 

20    8 

20.9 

291 

92 
93 

94 

9j 
96 

97 
98 

99 

JdO 

289.9 
290.9 
291.9 
292  .9 
293.9 

294.9 
295.9 
296.9 
297.9 
298.9 

25.4 
25.4 
25.5 
25.6 
25.7 
25.8 
25.9 
26.0 
26.1 
26.1 

Disi. 

nop. 

Lat. 

Dist. 

Dep. 

Lat. 

Dist. 

Dep. 

Lat. 

Dist. 

Dep. 

Lat. 

Dist. 

Dep.      Lat.  1 

[ForK 

Degre 

es. 

Page  S22] 

TABLE  IL 

Difference  of  Latitude  and  Departure  for  6  Degrees. 

Dist. 

Lat. 

Dep. 

Dist. 

Lat. 

Dep. 

Dist. 

Lat. 

Dep. 

Dist. 

Lat. 

Dep. 

Dist. 

Lat. 

Dep. 

I 

01  .0 

00. 1 

61 

60.7 

06.4 

121 

120.3 

12.6 

181 

180.0 

18.9 

241 

239.7 

25.2 

2 

02.0 

00.2 

62 

61.7 

06.5 

22 

121 .3 

12.8 

82 

181. 0 

19.0 

42 

240.7 

25.3 

3 

o3.o 

00.3 

63 

62.7 

06.6 

23 

122.3 

12.9 

83 

182.0 

19. 1 

43 

241.7 

25.4 

4 

o4.o 

00.4 

64 

63.6 

06.7 

24 

123.3 

i3.o 

84 

i83.o 

19.2 

AA 

242.7 

I'^.'J 

5 

o5.o 

00.5 

6b 

64.6 

06.8 

25 

124.3 

i3.i 

85 

184.0 

19.3 

45 

243.7 

-5.6 

6 

06.0 

00.6 

66 

65.6 

06.9 

26 

125.3 

l3.2 

86 

i85.o 

19.4 

46 

244.7 

25.7 

7 

07.0 

00.7 

67 

66.6 

07.0 

27 

126.3 

i3.3 

87 

186.0 

19.5 

47 

245.6 

25.8 

8 

08.0 

00.8 

68 

67.6 

07.1 

28 

127.3 

i3.4 

88 

187.0 

19.7 

48 

246.6 

25.9 

9 

09.0 

00.9 

69 

68.6 

07.2 

29 

128.3 

i3.5 

89 

188.0 

19.8 

49 

247-6 

20.0 

10 

11 

09.9 
10.9 

01 .0 

01 .1 

_i2_ 
71 

69.6 

07.3 

3o 

129.3 

i3.6 

90 

189.0 

19-9. 

5o 

25l 

248.6 
249.6 

26.1 
26.2 

70.6 

07.4 

i3i 

i3o.3 

l3.7 

191 

190.0 

20.0 

12 

II. 9 

01 .3 

72 

71.6 

07.5 

32 

i3i.3 

i3.8 

92 

190.9 

20.1 

52 

25o.6 

26.3 

Ij 

12.9 

01 .4 

73 

72.6 

07.6 

33 

i32.3 

i3.9 

93 

191. 9 

20.2 

53 

25i.6 

26.4 

i4 

.3.9 

CI  .5 

74 

73.6 

07.7 

34 

i33.3 

14.0 

94 

192.9 

20.3 

54 

252.6 

26.6 

i6 

14.9 

01 .6 

7!) 

74.6 

07.8 

35 

i34.3 

14. 1 

95 

19J.9 

20.4 

55 

253.6 

26.7 

lb 

,b.9 

01.7 

7b 

75.6 

07.9 

36 

i35.3 

14.2 

96 

194.9 

20.5 

56 

254.6 

26.8 

17 

16.9 

01.8 

77 

76.6 

08.0 

37 

i38.2 

i4.3 

97 

195.9 

20.6 

57 

255.6 

26.9 

i8 

17.9 

01 .9 

78 

77.6 

08.2 

38 

137.2 

14.4 

98 

196.9 

20.7 

58 

256.6 

27.0 

19 

18.9 

02.0 

79 

78.6 

08.3 

39 

i38.2 

i4.5 

99 

197.9 

20.8 

59 

257.6 

27.1 

20 

19.9 

02.1 

80 

79.6 

08.4 

40 

139.2 

14.6 

200 

198.9 

20.9 

60 

258.6 

I'!  .1 
27.3 

21 

20.9 

02.2 

81 

80.6 

08.5 

i4i 

i4o.2 

14.7 

201 

199.9 

21  .0 

261 

259.6 

22 

21 .9 

02.3 

82 

81.6 

08.6 

42 

l4l  -2 

i4.8 

02 

200.9 

21  .1 

62 

260.6 

27.4 

23 

22.9 

02.4 

83 

82.5 

08.7 

43 

142.2 

14.9 

o3 

201 .9 

21  .2 

63 

261 .6 

27.5 

24 

23.9 

02.5 

84 

83.5 

08.8 

AA 

143.2 

i5.i 

04 

202.9 

21.3 

■  H 

262.6 

27.6 

2b 

24.9 

02.6 

8b 

84. b 

08.9 

45 

144.2 

l5.2 

o5 

203.9 

21  .4 

65 

263.5 

27.7 

2b 

2b. 9 

02.7 

86 

85.5 

09.0 

46 

145.2 

i5.3 

06 

204.9 

21.5 

66 

264.5 

27.8 

27 

26.9 

02.8 

87 

86.5 

09.1 

47 

146.2 

i5.4 

07 

205.9 

21  .6 

67 

265.5 

27.9 

28 

27.8 

02 .9 

88 

87.5 

09.2 

48 

147.2 

ib.b 

08 

206.9 

21.7 

68 

266.5 

28.0 

29 

28.8 

o3.o 

89 

88.5 

09.3 

49 

148.2 

i5.6 

09 

207.9 

21.8 

69 

267.5 

28.1 

do 

29.8 

o3.i 

90 

89.5 

09.4 

5o 

149.2 

lb. 7 

10 

208.8 

22.0 

70 

268.5 

28.2 

3i 

3o.8 

o3.2 

91 

90.5 

09.5 

i5i 

i5o.2 

i5.8 

211 

209.8 

22.  1 

271 

269.5 

28.3 

02 

3i.8 

o3.3 

92 

91. b 

09.6 

52 

i5i  .2 

ib.9 

12 

210.8 

22.2 

72 

270.5 

28.4 

33 

32.8 

o3.4 

93 

92.5 

09.7 

53 

l52.2 

16.0 

i3 

211. 8 

22.3 

73 

271.5 

28.5 

M 

33.8 

o3.b 

94 

93.5 

09.8 

54 

i53.2 

16. 1 

i4 

212.8 

22.4 

74 

272.5 

28.6 

'3b 

34.8 

03. 7 

95 

94.5 

09.9 

55 

i54.2 

16.2 

i5 

2i3.8 

22.5 

75 

273.5 

28.7 

3b 

3b. 8 

o3.8 

96 

95.5 

lO.O 

56 

i55.i 

16.3 

16 

214.8 

22.6 

76 

274.5 

28.8 

^1 

3b. 8 

03.9 

97 

96.5 

10. 1 

57 

i56.i 

16.4 

17 

2i5.8 

22.7 

77 

275.5 

29.0 

38 

37.8 

04.0 

^/» 

97.5 

10.2 

58 

157. 1 

16.5 

18 

216.8 

22.8 

78 

276.5 

29.1 

39 

38.8 

04.1 

99 

98.5 

10.3 

59 

i58.i 

16.6 

19 

217.8 

22.9 

79 

277.5 

29.2 

40 
4i 

39.8 

oA-1 

100 

99.5 

10.5 

60 

159. 1 

16.7 

20 

218.8 

23.0 
23.1 

80 

278.5 

29.3 

40.8 

04.3 

lOI 

100.4 

10.6 

161 

160. 1 

16.8 

221 

219.8 

281 

279.5 

29.4 

42 

4i.8 

04.4 

02 

101.4 

10.7 

62 

161 .1 

16.9 

22 

220.8 

23.2 

hi 

280.5 

29.5 

Ai 

42.8 

04. b 

o3 

10.2.4 

10.8 

63 

162. 1 

17.0 

23 

221.8 

23.3 

83 

281.4 

29.6 

A^ 

43.8 

04.6 

o4 

io3.4 

10.9 

64 

i63.i 

17. 1 

24 

222.8 

23.4 

84 

282.4 

29.7 

45 

44.8 

04.7 

ob 

104.4 

II  .0 

65 

164. 1 

17.2 

25 

223.8 

23.5 

85 

283.4 

29.8 

46 

45.7 

04.8 

06 

io5.4 

II  .1 

66 

i65.i 

17-4 

26 

224.8 

23.6 

86 

284.4 

29.9 

47 

4b. 7 

04.9 

07 

106.4 

II  .2 

67 

166. 1 

17.5 

27 

225.8 

23.7 

87 

285.4 

3o.o 

48 

47.7 

o5.o 

08 

107.4 

11.3 

68 

167. 1 

17.6 

28 

226.8 

23.8 

88 

286.4 

3o.i 

49 

48. 7 

Ob. I 

09 

108.4 

II. 4 

69 

168. 1 

17.7 

29 

227.7 

23.9 

89 

287.4 

3o.2 

bo 
5i 

49.7 
5o.7 

05.2 

o5.3 

10 

109.4 

II. 5 

70 

169. 1 

17.8 
17.9 

3o 

228.7 

24.0 

90 

288.4 

3o.3 

III 

110.4 

11.6 

171 

170. 1 

23  I 

229.7 

24. 1 

291 

289.4 

3o.4 

b2 

bi.7 

o5.4 

12 

III  .4 

11.7 

72 

171. 1 

18.0 

32 

230.7 

24.3 

92 

290.4 

3o.5 

b3 

52.7 

ob.b 

i3 

112. 4 

11.8 

73 

172. 1 

18. 1 

33 

23l  .7 

24.4 

93 

291 .4 

3o.6 

54 

b3.7 

ob.fc 

i4 

ii3.4 

II. 9 

74 

173.0 

18.2 

34 

232.7 

24.5 

94 

292.4 

3o.7 

bb 

^4.7 

ob.7 

lb 

114.4 

12.0 

75 

174.0 

18.3 

35 

233.7 

24.6 

95 

293.4 

3o.8 

bb 

bb.7 

05.9 

lb 

ii5.4 

12. 1 

76 

175.0 

18.4 

36 

234.7 

24.7 

96 

294.4 

30.9 

37 

bb.7 

06.0 

17 

116. 4 

12.2 

77 

176.0 

18.5 

37 

235.7 

24.8 

97 

295.4 

3i.o 

b8 

b7.7 

06. 1 

18 

117.4 

12.3 

78 

177.0 

18.6 

38 

236.7 

24.9 

98 

296.4 

3i.i 

b9 

b8.7 

06.2 

19 

118. 3 

12.4 

79 

178.0 

18.7 

39 

1*37.7 

25.0 

Q9 

297.4 

3i.3 

bD 

59.7 

Ob. 3 

20 

119.3 

12.5 

80 

179.0 

18.8 

40 
Disi. 

238.7 
Dep. 

25.1 

3oo 

298.4 

3i.4 

Dist. 

l)c.p. 

Lat. 

Dist 

Dep. 

Lat. 

Dist. 

Dep. 

Lat. 

L;il. 

Dist. 

Dep- 

Lat. 

[For  84  Degrees. 

TABLE  IL 

[Page  23 

Difierence  of  Latitude  and  Departure  for  7  Degrees. 

Disl 

Lat. 

Dep. 

Disl. 

Lai. 

Dep. 

Dist. 

Lat. 

Dep. 

Dist. 

Lat. 

Dep. 

Dist. 

Lai. 

Dep. 
29.4 

I 

01 .0 

00. 1 

61 

60.5 

07.4 

121 

120.1 

14.7 

181 

1-79.7 

22.1 

241 

239.2 

2 

02.0 

00.2 

62 

61.5 

07.6 

22 

121 .1 

14.9 

82 

180.6 

22.2 

42 

240.2 

29.5 

3 

o3.o 

00.4 

63 

62.5 

07.7 

23 

122.1 

i5.o 

83 

181.6 

22.3 

43 

241 .2 

29.6 

4 

o4.o 

00.5 

64 

63.5 

07.8 

24 

123.1 

iS.i 

84 

182.6 

22.4 

AA 

242.2 

29.7 

5 

()5.o 

00.6 

65 

64.5 

07.9 

25 

124.1 

l5.2 

85 

i83.6 

22.5 

45 

243.2 

29.9 

6 

06.0 

00.7 

66 

65.5 

08.0 

26 

125.1 

i5.4 

86 

184.6 

22.7 

46 

244.2 

3o.o 

7 

06.9 

00.9 

67 

66.5 

08.2 

27 

126.1 

i5.5 

87 

i85.6 

22.8 

47 

245.2 

3o.  I 

8 

07.9 

0£  .0 

68 

67.5 

08.3 

28 

127.0 

i5.6 

88 

186.6 

22.9 

48 

246.2 

3o.2 

9 

08.9 

01  .1 

69 

68.5 

08.4 

29 

128.0 

i5.7 

89 

187.6 

23.0 

49 

247.1 

3o.3 

10 

09.9 

01  .2 

70 

69.5 

08.5 

3o 

129.0 

i5.8 

90 

188.6 

23.2 

5o 

248.1 

3o.5 

1 1 

10.9 

01  .3 

71 

70.5 

08.7 

i3i 

1 3o .  0 

16.0 

191 

189.6 

23.3 

25l 

249.1 

3o.6 

12 

II. 9 

01.5 

72 

71.5 

08.8 

32 

i3i  .0 

16. 1 

92 

190.6 

23.4 

52 

25o.i 

3o.7 

i3 

12.9 

01 .6 

73 

72.5 

08.9 

■d-^ 

l32.0 

16.2 

93 

191 .6 

23.5 

53 

25l  .1 

3o.8 

i4 

■  3.9 

01.7 

74 

73.4 

09.0 

34 

i33.o 

16.3 

94 

192.6 

20.6 

54 

252.  I 

3i  .0 

i5 

14.9 

01.8 

75 

74.4 

09.1 

35 

i34.o 

16.5 

95 

193.5 

23.8 

55 

253.1 

3i.i 

i6 

.5.9 

ot  .9 

76 

75.4 

09.3 

36 

i35.o 

16.6 

96 

194.5 

23.9 

56 

254.1 

3l.2 

17 

16.9 

02.1 

77 

76.4 

09.4 

37 

i36.o 

16.7 

97 

195.5 

24.0 

57 

255.1 

3i.3 

i8 

17.9 

02.2 

78 

77.4 

09.5 

38 

137.0 

16.8 

98 

196.5 

24.1 

58 

256.1 

3i.4 

19 

18.9 

02.3 

79 

78.4 

09.6 

39 

i38.o 

16.9 

99 

197.5 

24.3 

59 

257.1 

3i.6 

20 

.9.9 

02.4 

80 

79-4 

09.7 

4o 

139.0 

17. 1 

200 

198.5 

24.4 

60 

258.1 

3. .7 
3i.8 

21 

20.8 

02.6 

8i 

80.4 

09.9 

i4i 

139.9 

17.2 

201 

199.5 

24.5 

261 

259. 1 

22 

21.8 

02.7 

82 

81.4 

10. 0 

42 

140.9 

17.3 

02 

200.5 

24.6 

62 

260.0 

3i.9 

23 

22.8 

02.8 

83 

82.4 

10. 1 

43 

141.9 

17.4 

00 

201 .5 

24.7 

63 

261 .0 

32.1 

24 

23.8 

02.9 

84 

83.4 

10.2 

Ai 

142.9 

17.5 

o4 

202.5 

24.9 

64 

262.0 

32.2 

25 

24.8 

o3.o 

85 

84.4 

10.4 

45 

143.0 

17.7 

o5 

2o3 . 5 

25.0 

65 

263.0 

32.3 

26 

25.8 

03.2 

86 

85.4 

10.5 

46 

144.9 

17.8 

06 

204.5 

25.1 

66 

264.0 

32.4 

27 

26.8 

o3.3 

87 

86.4 

10.6 

47 

145.9 

17.9 

07 

205.5 

25.2 

67 

265.0 

32.5 

28 

27.8 

o3.4 

88 

87.3 

10.7 

48 

146.9 

18.0 

08 

206.4 

25.3 

68 

266.0 

32.7 

29 

28.8 

o3.5 

89 

88.3 

10.8 

49 

i47-9 

18.2 

09 

207.4 

25.5 

69 

267.0 

32.8 

3o 
3i 

29.8 

o3.7 

90 

69.3 

1 1 .0 

5o 

148.9 

18.3 

10 

208.4 

25.6 

70 
271 

268.0 
269.0 

32.9 

33 .0 

3<).8 

o3.8 

91 

90.3 

1 1 .1 

i5i 

149.9 

18.4 

211 

209.4 

25.7 

32 

3i.8 

03.9 

92 

91.3 

II  .2 

52 

1 50.9 

18.5 

12 

210.4 

25.8 

72 

270.0 

33.1 

33 

32.8 

04.0 

93 

92.3 

II. 3 

53 

i5i  .9 

18.6 

i3 

211 .4 

26.0 

7-J 

271  .c 

33.3 

34 

33.7 

o4.i 

94 

93.3 

II. 5 

54 

152.9 

18.8 

i4 

212.4 

26.1 

74    272. c 

33.4 

35 

31.7 

04.3 

95 

94.3 

II. 6 

55 

i53.8 

18.9 

i5 

2i3.4 

26.2 

75 

273.0 

33.5 

36 

35.7 

04.4 

90 

95.3 

II. 7 

56 

154.8 

19.0 

i6 

214.4 

26.3 

76 

273.9 

33.6 

37 

36.7 

04.5 

97 

96.3 

II. 8 

57 

i55.8 

19.1 

17 

2.5.4 

26.4 

77 

274.9 

33.8 

38 

37.7 

04.6 

98 

97.3 

II. 9 

58 

i56.8 

.9.3 

18 

216.4 

26.6 

78 

275.9 

33.9 

39 

38.7 

04.8 

99 

98.3 

12. 1 

59 

157.8 

19-4 

19 

217.4 

26.7 

79 

276.9 

34.0 

4(. 

3y.7 

04.9 

100 

99.3 

12.2 

bo 

i58.8 

19.5 

20 

218.4 

26.8 
26.9 

80 

277.9 

34. 1 

4i 

40.7 

o5.o 

lOI 

100.2 

12.3 

161 

159.8 

19.6 

221 

219.4 

281 

278.9 

34.2 

42 

41.7 

o5.i 

02 

lOI  .2 

12.4 

62 

160.8 

19.7 

22 

220.3 

27.1 

82 

279.9 

34.4 

43 

42.7 

05.2 

o3 

102.2 

12.6 

63 

161. 8 

19.9 

23 

221 .3 

27.2 

83 

280.9 

34.5 

44 

43.7 

o5.4 

04 

I03.2 

12.7 

64 

162.8 

20.0 

24 

222.3 

27.3 

84 

281.9 

34.6 

45 

44.7 

o5.5 

o5 

104.2 

12.8 

65 

i63.8 

20. 1 

25 

223.3 

27.4 

85 

282.9 

34.7 

46 

45.7 

o5.6 

06 

I05.2 

12.9 

66 

164.8 

20.2 

26 

224.3 

27.5 

86 

283.9 

•^4.9 

Si 

46.6 

05.7 

07 

106.2 

i3.o 

C7 

i65.8 

20.4 

27 

225.3 

27.7 

87 

284.9 

35.0 

48 

47-6 

o5.8 

08 

107.2 

l3.2 

68 

166.7 

20.5 

28 

226.3 

27.8 

88 

285.9 

35.1 

49 

48.6 

06.0 

09 

108.2 

i3.3 

69 

167.7 

20.6 

29 

227.3 

27.9 

89 

286.8 

35.2 

5o 

49.6 

06.1 

10 

109.2 

i3.4 

70 

168.7 

20.7 

3u 

228.3 

28.0 

90 

287.8 

35.3 

35.5 

5f 

5o.6 

06.2 

I II 

I  10.2 

i3.5 

171 

169.7 

20.8 

23  1 

229.3 

28.2 

291 

288.8 

52 

5i.6 

06.3 

12 

I  I  I  .2 

i3.6 

72 

170.7 

21  .0 

32 

23o.3 

28.3 

92 

289.8 

35.6 

53 

52.6 

06.5 

i3 

112. 2 

i3.8 

73 

171. 7 

21  .1 

33 

23i.3 

28.4 

93 

290.8 

35.7 

54 

53.6 

06.6 

i4 

1X3.2 

i3.9 

74 

172.7 

21  .2 

34 

232   3 

28.5 

94 

291.8 

35.8 

55 

54.6 

06.7 

i5 

Il4.I 

i4.o 

75 

173.7 

21.3 

35 

233.2 

28.6 

95 

292.8 

36. 0 

56 

55.6 

06.8 

16 

iiS.i 

i4.i 

76 

174.7 

21.4 

36 

234.2 

28.8 

96 

293.8 

36.1 

''7 

56.6 

06 . 9 

17 

116. 1 

14.3 

77 

175.7 

21  .6 

37     235.2 

28.9 

97 

294. 8 

36.2 

58 

57.6 

07.1 

18 

117. 1 

14.4 

78 

176.7 

21.7 

38  1 

236.2 

29.0 

98 

295.8 

36.3 

59 

58.6 

07.2 

19 

118. 1 

i4.5 

79 

177-7 

21.8 

39 

237.2 

29. 1 

99 

296.8 

36.4 

b(. 

59.6 

07.3 

20 

1 19. 1 

i4.6 

80 

178.7 

21  .9 

40 

238.2 

29.2 

3oo    297.8 

36.6 

Dist. 

Dep. 

Lat. 

Disl. 

Dep. 

Lat. 

Dist. 

Dep. 

Lat. 

Disl. 

Dep. 

Lat. 

Dist.     Dep. 

Lat. 

[For  83  Degrees. 

Page  24] 

TABLE  IL 

Difference  of  Latitude  and  Departure  for  8  Degrees. 

Dist. 

1 
Lat. 

Dep. 

Dist. 

Lat. 

Dep. 

Dist. 

Lat. 

Dep. 

Dist. 

Lat. 

Dep. 

Dist. 

Lat. 

Dep. 

I 

01 .0 

00. 1 

61 

60.4 

08.5 

121 

1 19.8 

16.8 

181 

179.2 

25.2 

241 

238.7 

33.5 

2 

02.0 

00.3 

62 

61.4 

08.6 

22 

120.8 

17.0 

82 

180.2 

25.3 

42 

239.6 

33.7 

3 

o3.o 

00.4 

63 

62.4 

08.8 

23 

121. 8 

17. 1 

83 

181.2 

25.5 

43 

240.6 

33.8 

4 

04.0 

00.6 

64 

63.4 

08.9 

24 

122.8 

17.3 

84 

182.2 

25.6 

44 

241 .6 

34.0 

5 

o5.o 

00.7 

65 

64.4 

09.0 

25 

123.8 

17-4 

85 

i83.2 

25.7 

45 

242.6 

34.1 

6 

05.9 
06.9 
07.9 

00.8 

66 

65.4 

09.2 

26 

124.8 

17.5 

86 

184.2 

2D. 9 

46 

243.6 

34.2 

7 

01 .0 

67 

66.3 

09.3 

27 

125.8 

17-7 

87 

i85.2 

26.0 

4i 

244.6 

34.4 

8 

01. 1 

68 

67.3 

09.5 

28 

126.8 

17.8 

88 

186.2 

26.2 

48 

245.6 

34.5 

9 

08.9 

01 .3 

69 

68.3 

09.6 

29 

127.7 

18.0 

89 

187.2 

2b. 3 

49 

246.6 

34.7 

10 

09.9 

01 .4 

70 

69.3 

09.7 

3o 

128.7 

18.1 

90 

188.2 

26.4 

5o 

247.6 

34.8 

II 

10.9 

01.5 

71 

70.3 

09.9 

i3i 

129.7 

18.2 

191 

189.1 

26.6 

25 1 

248.6 

34.9 

12 

II. 9 

01 .7 

72 

71.3 

10. 0 

32 

1 30.7 

18.4 

92 

190.1 

26.7 

52 

249.5 

35.! 

1 3 

12.9 

01.8 

73 

72.3 

10.2 

33 

i3i  .7 

18.5 

93 

191 .1 

26.9 

53 

25o.5 

35.2 

1/1 

I3.Q 

01 .9 

74 

73.3 

10.3 

34 

i32.7 

18.6 

94 

192. 1 

27.0 

54 

25i.5 

35.3 

1 5 

i4.q 

02.1 

75 

74.3 

10.4 

35 

133.7 

18.8 

95 

193.1 

27.1 

55 

252.5 

35.5 

ifi 

i5.8 

02.2 

76 

75.3 

10.6 

36 

134.7 

18.9 

96 

194. 1 

27.3 

5b 

253.5 

35.6 

17 

16.8 

02.4 

77 

76.3 

10.7 

37 

135.7 

19. 1 

97 

195. 1 

27.4 

57 

254.5 

35.8 

18 

17.8 

02.5 

78      77-2 

10.9 

38 

i36.7 

19.2 

98 

196. 1 

27.6 

58 

255.5 

35.9 

19 

18.8 

02  .6 

79      78.2 

II  .0 

39 

137.7 

19.3 

99 

197.1 

27.7 

59 

256.5 

36.0 

20 

19.8 

02.8 

80 

79.2 

II  .1 

4o 

i38.6 

19.5 
19.6 

200 

198. 1 

27.8 

bo 

257.5 

.36.2 

21 

20.8 

02.9 

81 

80.2 

II. 3 

i4i 

139.6 

201 

199.0 

28.0 

261 

258.5 

36.3 

22 

2T,8 

0-3.1 

82 

81.2 

II. 4 

42 

i4o.6 

19.8 

02 

200.0 

28    1 

62 

259.5 

36.5 

23 

22.8 

o3.2 

83 

82.2 

II. 6 

43 

i4i.6 

19.9 

o3 

201 .0 

28.3 

63 

260.4 

36.6 

•?4 

23,8 

o3.3 

84 

83.2 

II. 7 

44 

142.6 

20.0 

04 

202.0 

28.4 

64 

261.4 

36.7 

9.5 

24.8 

o3.5 

85 

84.2 

n.8 

45 

143.6 

20.2 

o5 

203.0 

28.5 

65 

262.4 

36.9 

26 

25.7 

o3.6 

86 

85.2 

12.0 

46 

144.6 

20.3 

06 

204.0 

28.7 

66 

263.4 

37.0 

27 

26.7 

o3.8 

87 

86.2 

12. 1 

47 

145.6 

20.5 

07 

205.0 

28.8 

67 

264.4 

37.2 

28 

27.7 

03.9 

88 

87.1 

12.2 

48 

146.6 

20.6 

08 

206.0 

28.9 

68 

265.4 

37.3 

29 

28.7 

04.0 

8q 

88.1 

12.4 

49 

i47-5 

20.7 

09 

207.0 

29.  I 

69 

266.4 

37.4 

JO 

29.7 

04.2 

90 

89.1 

12.5 

5u 
i5i 

i48.5 
149.5 

20.9 

10 

208.0 

29.2 

70 

267.4 

37.6 

3i 

3o.7 

04.3 

91 

90.1 

12.7 

21  .0 

211 

208.9 

29.4 

271 

268.4 

37.7 

32 

3l.7 

04.5 

92 

91. 1 

12.8 

52 

i5o.5 

21  .2 

12 1 209.9 

29.5 

72 

269.4 

379 

33 

32.7 

04.6 

93 

92.1 

12.9 

53 

i5i.5 

21.3 

i3 

210.9 

29.6 

73 

270.3 

38.-. 

34 

33.7 

04.7 

94 

93.1 

i3.i 

54 

i52.5 

21  .4 

i4 

2  1 1  . 9 

29.8 

74 

271 .3 

38.1 

35 

34.7 

04.9 

95 

94.1 

l3.2 

55 

i53.5 

21  .6 

i5 

212.9 

29.9 

7^ 

272.3 

•38.3 

36 

35.6 

o5.o 

96 

95.1 

i3.4 

5b 

i54.5 

21.7 

lb 

213.9 

3o.i 

76 

273.3 

38.4 

37 

36.6 

o5.i 

97 

96.1 

i3.5 

57 

i55.5 

21  .9 

17 

214.9 

3o.2 

77 

274.3 

38  6 

38 

37.6 

o5.3 

98,  97.0 

i3.6 

58 

i56.5 

22.0 

18 

215.9 

3o.3 

78 

275.3 

38.7 

39 

38.6 

o5.4 

Q9 

98.0 

i3.8 

59 

157.5 

22.1 

19 

210.9 

3o.5 

79 

276.3 

38.8 

4o 

39.6 

o5.6 

100 

99.0 

13.9 

bo 

i58.4 

22.3 

20 

217.9 

3o.6 

80 

277.3 

39.0 

4i 

40.6 

o5.7 

lOl 

1 00.0 

14.1. 

161 

159.4 

22.4 

221 

218.8 

3o.8 

281 

278.3 

39.1 

4? 

41.6 

o5.8 

02 

101 .0 

14.2 

62 

160.4 

22.5 

22 

219.8 

30.9 

82 

279.3 

39.2 

43 

42.6 

06.0 

o3 

102.0 

i4.3 

63 

161.4 

22.7 

23 

220.8 

3i  .0 

83 

280.2 

39.4 

44 

43.6 

06.1 

04 

io3.o 

14.5 

64 

162.4 

22.8 

24 

221.8 

3l.2 

84 

281.2 

39.5 

45 

44.6 

06.3 

o5 

104.0 

14.6 

65 

i63.4 

23. 0 

25 

222.8 

3i.3 

85 

282.2 

39.7 

46 

45.6 

06.4 

06 

io5.o 

14.8 

66 

164.4 

23.1 

2b 

223.8 

3i.5 

8b 

283.2 

39.8 

47 

46.5 

06.5 

07 

106.0 

14.9 

67 

i65.4 

23.2 

27 

224.8 

3i.6 

87 

284.2 

39.9 

48 

47.5 

06.7 

08 

106.9 

i5.o 

68 

166.4 

23.4 

28 

225.8 

31.7 

88 

285.2 

4o.i 

49 

48.5 

06.8 

oq 

107.9 

l5.2 

69 

167.4 

23.5 

29 

226.8 

3i  .9 

89 

286.2 

40.2 

5o 

49-5 

07.0 

10 

108.9 

lb. 3 

70 

168.3 

23.7 

Jo 

227.8 

32.0 

90 

287.2 

40.4 

5i 

5o.5 

07.1 

II I 

109.9 

i5.4 

171 

169.3 

23.8 

23l 

228.8 

32.1 

291 

288.2 

40.5 

5?. 

5t.5 

07.2 

12 

110.9 

i5.6 

72 

170.3 

23.9 

32 

229.7 

32.3 

92 

289.2 

40.6 

03 

52.5 

07.4 

i3 

III  .9 

i5.7 

73 

171 .3 

24.1 

33 

23o.7 

32.4 

93 

290.1 

40.8 

54 

53.5 

07.5 

i4 

112. 9 

15.9 

74 

172.3 

24.2 

34 

23l  .7 

32.6 

94 

291 .1 

40.9 

55 

54.5 

07.7 

i5 

113.9 

16.0 

7^ 

173.3 

24.4 

35 

232.7 

32.7 

95 

292.1 

4i.i 

56 

55.5 

07.8 

16 

114.9 

16. 1 

7b 

174.3 

24.5 

3b 

233.7 

32.8 

9b 

293.1 !4i.2  1 

57 

56.4 

07.9 

17 

115.9 

16.3 

77 

175.3 

24.6 

37 

234.7 

33.0 

97 

294.1 

4i.3 

58 

57.4 

08.1 

18 

116. 9 

16.4 

78 

176.3 

24.8 

38 

235.7 

33.1 

98 

295.1 

41.5 

59 

58.4 

08.2 

19 

117. 8 

16. 6 

79 

177.3 

24.9 

39 

236.7 

33.3 

99 

296.1 

4i  6 

60 

59.4 
j  Dep. 

08.4 
Lat. 

20 

118. 8 

lb. 7 

80 
dTsI. 

178.2 

25.1 

Lat. 

40 
Dist. 

237.7 

33.4 

3oo 
Dist. 

297. 1 
Dep. 

4i.6 

Dist. 

Dep. 

Lat. 

Dep. 

Dep. 

Lat. 

[For  82  Degrees. 

TABLE  II 

[Page  25 

DifFerence  of  Latitude  and  Departure  for  9  Degrees. 

Dist. 

Lat. 

Dep. 

Dist.     Lat. 

Dep. 
09.5 

Dist.     Lat. 

Dep. 

Dist.| 

Lat. 

Dep. 

Dist. 

Lai. 

Dep. 

I 

01 .0 

00.2 

61 

60.2 

121     119. 5 

18.9 

181 

178.8 

28.3 

24 1 

238.0 

37.7 

2 

02.0 

00.3 

62 

61.2 

09.7 

22      120.5 

19. 1 

82 

179.8 

28.5 

42 

239.0 

37.9 

3 

o3.o 

00.5 

63 

62.2 

09.9 

23 

121  .5 

19.2 

83 

180.7 

28.6 

43 

240 . 0 

38. 0 

4 

o4.o 

00.6 

64 

63.2 

lO.O 

24 

122.5 

'9-4 

84 

181 .7 

28.8 

44 

241  .0 

38.2 

'■> 

04.9 

00.8 

65 

64.2 

10.2 

25 

123.5 

19.6 

85 

182.7 

28.9 

45 

242.0 

38.3 

5 

o5.9 

00.9 

66 

65.2 

10.3 

26 

124.4 

19.7 

86 

183.7 

29.1 

4b 

243.0 

38.5 

7 

06.9 

01  .1 

67 

66.2 

10.5 

27 

125.4 

19.9 

87 

184.7 

29.3 

47 

244.0 

38.6 

.S 

07.9 

01 .3 

68 

67.2 

10.6 

28 

126.4 

20.0 

88 

185.7 

29.4 

48 

244.9 

38.8 

9 

08.9 

01 .4 

69 

68.2 

10.8 

29 

127.4 

20.2 

89 

186.7 

29.6 

f9 

245.9 

39.0 

10 

09.9 

01 .6 

70 

69. 1 

1 1 .0 

3o 

128.4 

20.3 

90 

187.7 

29.7 

5o 

246.9 

39.1 

II 

10.9 

01.7 

71 

70.1 

II  .1 

i3i 

129.4 

20.5 

191 

188.6 

29.9 

25l 

247-9 

39.3' 

12 

II. 9 

01 .9 

72 

71. 1 

II. 3 

32 

i3o.4 

20.6 

92 

189.6 

3o.o 

b2 

248.9 

39.4 

i3 

12.8 

02.0 

73 

72.1 

II. 4 

33 

i3i.4 

20.8 

93 

190.6 

3o.2 

53 

249-9 

39.6 

1 4 

i3.8 

02.2 

74 

73.1 

II. 6 

34 

i32.4 

21  .0 

94 

191 .6 

3o.3 

54 

250.9 

39.7 

i5 

i4.8 

02.3 

75 

74.1 

II. 7 

35 

i33.3 

21 .1 

95 

192.6 

3o.5 

55 

25l  .9 

39.9 

1 6 

i5.8 

02.5 

76 

75.1 

II. 9 

36 

i34.3 

21.3 

96 

193.6 

3o.7 

5b 

252.8 

4o.o 

'7 

16.8 

02.7 

77 

76.1 

12.0 

37 

i35.3 

21.4 

97 

194.6 

3o.8 

i>7 

253.8 

40.2 

i8 

17.8 

02.8 

78 

77.0 

12.2 

38 

i36.3 

21.6 

98 

195.6 

3i.o 

58 

254.8 

40.4 

'9 

18.8 

o3.o 

79 

78.0 

12.4 

39 

137.3 

21.7 

99 

196.5 

3i.i 

59 

255.8 

40.5 

20 

19. 8 

o3.i 

80 

79.0 

12.5 

4o 

i38.3 

21.9 

200 

197.5 

3i.3 

bo 

256.8 

40.7 

21 

20.7 

o3.3 

81 

80.0 

12.7 

i4i 

139.3 

22.1 

201 

19S.5 

3i.4 

261 

257.8 

40.8 

22 

21.7 

o3.4 

82 

81.0 

12.8 

42 

i4o.3 

22.2 

02 

199.5 

3i.6 

b2 

258.8 

41 .0 

23 

22.7 

o3.6 

83 

82.0 

i3.o 

43 

i4i  .2 

22.4 

o3 

200.5 

3i.8 

b3 

259.8 

4i.i 

24 

23.7 

o3.8 

84 

83. 0 

i3.i 

44 

142.2 

22.5 

04 

201 .5 

31.9 

64 

260.7 

41.3 

25 

24.7 

03.9 

85 

84.0 

i3.3 

45 

143.2 

22.7 

o5 

202.5 

32.1 

65 

261.7 

41.5 

2fi 

25.7 

04.1 

86 

84.9 

i3.5 

46 

144.2 

22.8 

06 

2o3.5 

32.2 

bb 

262.7 

4i.6 

27 

26.7 

04.2 

87 

85.9    i3.6 

47 

145.2 

23. 0 

07 

204.5 

32.4 

67 

263.7 

4i.8 

28 

27-7 

o4.4 

88 

86.9    i3.8 

48 

146.2 

23.2 

08 

2o5.4 

32.5 

68 

264.7 

4i  .9 

29 

28.6 

04.5 

89 

87.9    i3.9 

49 

l47-2 

23.3 

09 

206.4 

32.7 

69 

265.7 

42.1 

3o 

29.6 

04.7 

90 

88.9 

i4.i 

5o 

148.2 

23.5 

10 

207.4 

32.9 

70 
271 

266.7 

42.2 
42.4 

3i 

3o.6 

04.8 

91 

89.9 

l4.2 

i5i 

149.1 

23.6 

211 

208.4 

33.0 

267.7 

32 

3i.6 

o5.o 

92 

90.9 

14.4 

52 

i5o.i 

23.8 

12 

209.4 

33.2 

72 

268.7 

42.6 

33 

32.6 

o5.2 

93 

91.9 

14.5 

53 

i5i  .1 

23.9 

i3 

210.4 

33.3 

73 

269.6   42.7 

34 

33.6 

o5.3 

94 

92.8 

14.7 

54 

ID2.I 

24.1 

i4 

211 .4 

33.5 

74 

270.6   42.9 

35 

34.6 

o5.5 

95 

93.8 

14.9 

55 

i53.i 

24.2 

i5 

212.4 

33.6 

7^ 

271 .6 

43.0 

36 

35.6 

o5.6 

96 

94.8 

i5.o 

56 

i54.i 

24.4 

16 

2i3.3 

33.8 

76 

272.6 

43.2 

37 

36.5 

o5.8 

97 

95.8 

l5.2 

57 

i55.i 

24.6 

17 

214.3 

33.9 

77 

273.6 

43.3 

38 

37.5 

05.9 

98 

96.8 

i5.3 

58 

i56.i 

24.7 

18 

2i5.3 

34.1 

78 

274.6 

43.5 

3g 

38.5 

06. 1 

99 

97.8 

i5.5 

§9 

157.0 

24.9 

19 

216.3 

34.^ 

79 

275.6 

43.6 

4" 

39.5 

06.3 

100 

98.8 

i5.6 

60 

i58.o 

25.0 

20 
221 

217.3 
218.3 

34.4 

80 

276.6 

43.8 

4i 

40.5 

06.4 

lOI 

99.8 

i5.8 

161 

159.0 

25.2 

34.6 

281 

277.5 

44.0 

42 

41.5 

06.6 

02 

100.7 

16.0 

62 

160.0 

25.3 

22 

219.3 

34.7 

82 

278.5 

44.1 

43 

42.5 

06.7 

o3 

lOI  .7 

16. 1 

63 

161 .0 

25.5 

23 

220.3 

34.9 

83 

279.5 

44.3 

44 

43.5 

06.9 

o4 

102.7 

16.3 

64 

162.0 

25.7 

24 

221 .2 

35.0 

84 

280.5 

44.4 

45 

44.4 

07.0 

o5 

103.7 

16.4 

65 

i63.o 

25.8 

25 

222.2 

35.2 

85 

281.5 

44.6 

46 

45.4 

07.2 

06 

104.7 

16.6 

66 

164.0 

26.0 

26 

223.2 

35.4 

8b 

282.5 

44.7 

47 

46.4 

07.4 

07 

105.7 

16.7 

67 

164.9 

26.1 

27 

224.2 

35.5 

87 

283.5 

44.9 

48 

47.4 

07.5 

08 

106.7 

16.9 

68 

165.9 

26.3 

28 

225.2 

35.7 

88 

284.5 

45.1 

49 

48.4 

07.7 

09 

107.7 

17. 1 

69 

166.9 

26.4 

29 

226.2 

35.8 

89 

285.4 

45.2 

5o 

49.4 

07.8 

ID 

108.6 

17.2 

70 

167.9 

26.6 

3o 

227.2 

36.0 

90 

286.4 

45.4 

5i 

5o.4 

08.0 

1  I  I 

IC9.6 

17-4 

171 

168.9 

26.8 

23l 

228.2 

36.1 

291 

287.4 

45.5 

52 

5i.4 

08.1 

12 

110.6 

17.5 

72 

169.9 

26.9 

32 

229.1 

36.3 

92 

288.4 

45.7 

53 

52.3 

08.3 

l3 

III  .6 

17-7 

73 

170.9 

27.1 

33 

23o.i 

36.4 

93 

289.4145.8  1 

54 

53.3 

08.4 

i4 

112. 6 

17.8 

74 

171-9 

27.2 

34 

23l  .1 

36.6 

94 

290.4 

46.0 

55 

54.3 

08.6 

i5 

ii3.6 

18.0 

75 

172.8 

27.4 

35 

232.1 

36.8 

95 

291 .4 

46.1 

56 

55.3 

08.8 

16 

114.6 

18.1 

76    173.8 

27.5 

36 

233.1 

36.9 

96 

292  .4 

46.3 

57 

56.3 

08.9 

17 

ii5.6 

18.3 

77    174.8 

27.7 

37 

234.1 

37.1 

97 

293.3 

46.5 

58 

57.3 

09.1 

18 

116.5 

18.5 

78    175.8 

27.8 

38 

235.1 

37.2 

98 

294.3 

46.6 

59 

58.3 

09.2 

19 

117. 5 

18.6 

79    176.8 

28.0 

39 

236.1 

37.4 

99 

295.3 

46.8 

bo 
Dist. 

59.3  1  09.4 
Dep.l  Lat. 

20 

118. 5 

18.8 

80    177.8 

28.2 

40 

237.0 

37.5 

3oo 

J96.3 

46.9 

Dist.'    Dep.  1  Lat. 

Dist.     Dep. 

Lat. 

Di^f 

Dep. 

Lat. 

Dist 

Dep. 

Lat. 

[For  81  Degrees. 

Page  26] 

TABLE  IL 

Difference  of  Latitude  and  Departure  for  10  Degrees. 

Disl. 

Lat. 

Dep. 

Dist. 

Lat. 

Dep. 

Dist. 

Lai. 

Dep. 
21 .0 

Disl. 

181 

Lat. 
178.3 

Dep. 

3i.4 

Disl. 

Lat. 

Dep. 

I 

01  .0 

00.2 

61 

60.1 

10.6 

121 

1 19.2 

24 1 

237.3 

4i.8 

2 

02.0 

00.3 

62 

61. 1 

10.8 

22 

120.1     21.2 

82 

179.2 

3i.6 

42 

238.3 

42.0 

3 

o3.o 

00.5 

63 

62.0 

10.9 

23 

121 .1 

21.4 

83 

180.2 

3i.8 

43 

289.3 

42.2 

4 

o3.9 

00.7 

64 

63. 0 

1 1 . 1 

24 

122. 1 

21.5 

84 

181.2 

32.0 

44 

240.3 

42.4 

5 

04.9 

00.9 

65 

64.0 

II. 3 

25 

123. I 

21.7 

85 

182.2 

32.1 

45 

241.3 

42.5 

6 

05.9 

01 .0 

66 

65.0 

II  .0 

26 

124. 1 

21 .9 

86 

i83.2 

32.3 

46 

242.3 

42.7 

7 

06.9 

01 .2 

67 

66.0 

II. 6 

27 

125.  I 

22.1 

87 

184.2 

32.5 

47 

243.2 

42.9 

8 

07. Q 

DI.4 

68 

67.0 

II. 8 

28 

126. 1 

22.2 

88 

i85.i 

32.6 

48 

244.2 

43.1 

9 

08.9 

01 .6 

69 

68.0 

12.0 

29 

127.0 

22.4 

89 

186.1 

32.8 

49 

245.2 

43.2 

lO 

09.8 

01.7 

70 

68.9 

12.2 

3o 

128.0 

22.6 

22.7 

90 

187.1 

33.0 

5o 

246.2 

43.4 

II 

10.8 

01 .9 

71 

69.9 

12.3 

i3i 

129.0 

191 

188. 1 

33.2 

25l 

247.2 

43.6 

12 

II. 8 

02.1 

72 

70.9 

12.5 

32 

i3o.o 

22.9 

92 

189. 1 

33.3 

52 

248.2 

43.8 

i3 

12.8 

02.3 

73 

71.9 

12.7 

33 

i3i  .0 

23.1 

93 

190. 1 

33.5 

53 

249.2 

43.9 

i4 

i3.8 

02.4 

74 

72.9 

12.8 

34 

l32.0 

23.3 

94 

191 .1 

33.7 

54 

25o.i 

44.1 

i5 

14.8 

02.6 

75 

73.9 

i3.o 

35 

132.9 

23.4 

95 

192.0 

33.9 

55 

25l  .  1 

44.3 

i6 

i5.8 

02.8 

76 

74.8 

l3.2 

36 

133.9 

23.6 

96 

193.0 

34.0 

56 

252.1 

44.5 

17 

16.7 

o3.o 

77 

75.8 

i3.4 

37 

134.9 

23.8 

97 

194.0 

34.2 

57 

253.1 

44.6 

i8 

17-7 

o3. 1 

78 

76.8 

i3.5 

38 

135.9 

24.0 

98 

195.0 

34.4 

58 

254.1 

44.8 

19 

18.7 

o3.3 

79 

77.8 

.3.7 

39 

186.9 

24.1 

99 

196.0 

34.6 

59 

255.1 

45.0 

20 

19.7 

o3.5 

80 

78.8 

13.9 

4o 

137.9 

24.3 

200 

197.0 

34.7 

60 

256.1 

45.1 

21 

20.7 

o3.6 

81 

79.8 

i4.i 

i4i 

i38.9 

24.5 

201 

197.9 

34.9 

261 

257.0 

45.3 

22 

21.7 

o3.8 

82 

80.8 

14.2 

42 

139.8 

24.7 

02 

198.9 

35.1 

62 

258.0 

45.5 

23 

22.7 

04.0 

83 

81.7 

14.4 

Ai 

140.8 

24.8 

OJ 

199.9 

35.3 

63 

259.0 

45.7 

24 

23.6 

04.2 

84 

82.7 

i4.6 

44 

i4i.8 

25.0 

o4 

200.9 

35.4 

64 

260 . 0 

45.8 

25 

24.6 

o4.3 

85 

83.7 

i4.8 

45 

142.8 

25.2 

o5 

201 .9 

35.6 

65 

261 .0 

46.0 

26 

25.6 

04.5 

86 

84.7 

14.9 

46 

143.8  j  25.4 

06 

202.9 

35.8 

66 

262.0 

46.2 

27 

q6.6 

04.7 

87 

85.7 

i5.i 

47 

144.8 

25.5 

07 

203.9 

35.9 

67 

262.  G 

46.4 

28 

27.6 

04.9 

88 

86.7 

i5.3 

48 

145.8 

25.7 

08 

204.8 

35. 1 

68 

263.9 

46.5 

29 

28.6 

o5.o 

89 

87. 6 

i5.5 

49 

146.7 

25.9 

09 

2o5.8 

36.3 

69 

264.9 

46.7 

3o 

29.5 

o5.2 

90 

88.6 

i5.6 

5o 

147-7 

26.0 

10 

206.8 

36.5 

70 

265.9 

46.9 

3i 

3o.5 

o5.4 

91 

89.6 

i5.8 

i5i 

148.7 

26.2 

211 

207.8 

36.6 

271 

266.9 

47.1 

32 

3i.5 

o5.6 

92 

90.6 

16.0 

52 

149.7 

26.4 

12 

208.8 

36.8 

72 

267.9 

47-2 

33 

32.5 

o5.7 

93 

91 .6 

16. 1 

53 

i5o.7 

26.6 

i3 

209.8 

37.0 

73 

268.9 

47  4 

34 

33.5 

05.9 

94 

92.6 

16.3 

54 

i5i  .7 

26.7 

i4 

210.7 

37.2 

74 

269.8 

47  G 

35 

34.5 

06.1 

95 

93.6 

16.5 

55 

i52.6 

26.9 

i5 

21 1. 7 

37.3 

75 

270.8 

47.8 

36 

35.5 

06.3 

96 

94.5 

16.7 

56 

i53.6 

27.1 

16 

212.7 

37.5 

76 

271.8 

47.9 

37 

36.4 

06.4 

97 

95.5 

16.8 

57 

i54.6 

27.3 

17 

213.7 

37.7 

77 

272.8 

48.1 

38 

37.4 

06.6 

98 

96.5 

17.0 

58 

i55.6 

27.4 

18 

214.7 

37.9 

7a 

273.8 

48.3 

39 

38.4 

06.8 

99 

97.5 

17.2 

59 

i56.6 

27.6 

19 

2j5.7 

38. 0 

79 

274.8 

48.4 

4o 

39.4 

06.9 

100 

98.5 

17-4 

60 

157.6 

27.8 

PC 

216.7 

38.2 

80 

275.7 

48.6 

4i 

40.4 

07.1 

ipi 

99.5 

17.5 

161 

i58.6 

28.0 

221 

217.6 

38.4 

2Sr 

276.7 

48.8 

42 

41.4 

07.3 

02 

100.5 

17-7 

62 

159.5 

28.1 

22 

218.6 

38.5 

82 

277.7 

49.0 

43 

42.3 

07.5 

o3 

101 .4 

17.9 

63 

160.5 

28.3 

23 

219.6 

38.7 

83 

278.7 

49.1 

M 

43.3 

07.6 

04 

102.4 

18.1 

64 

161. 5 

28.5 

24 

220.6 

38.9 

84 

279.7 

49.3 

45 

A^.6 

07.8 

o5 

io3.4 

18.2 

65 

162.5 

28.7 

25 

221 .6 

39.1 

85 

2S0.7 

49.5 

46 

45.3 

08.0 

06 

104.4 

18.4 

66 

i63.5 

28.8 

26 

222.6 

39.2 

86 

281.7 

49.7 

47 

A^.i 

08.2 

07 

io5  4 

18.6 

67 

164.5 

29.0 

27 

223.6 

39.4 

87 

282.6 

49.8 

48 

47-3 

08.3 

08 

106.4 

18.8 

68 

i65.4 

29.2 

28 

224.5 

39.6 

88 

283.6 

5o.o 

49 

48.3 

08.5 

■  09 

107.3 

18.9 

69 

166.4 

29.3 

29 

225.5 

39.8 

89 

284.6 

5o.2 

5o 
5i 

49.2 

08.7 

10 

108.3 

19.1 

70 

167.4 

29.5 

3u 

226.5 

J9.9 

291 

285.6 

So. 4 

5o.2 

08.9 

III 

109.3 

19.3 

171 

168.4 

29.7 

23l 

227.5 

4o.  1 

286.6 

5o.5 

52 

5l.2 

09.0 

12 

1 10.3 

19-4 

72 

169.4 

29.9 

32 

228.5 

40.3 

92 

287.6 

50.7 

53 

52.2 

09.2 

i3 

III. 3 

ig.6 

73 

170.4 

3o.o 

33 

229.5 

40.5 

93 

288.5 

50.9 

54 

53.2 

09.4 

i4 

1 12.3 

19.8 

74 

171 .4 

3o.2 

34 

23o.4 

40.6 

94 

289.5 

5i.i 

55 

54.2 

09.6 

i5 

ii3.3 

20.0 

75 

172.3 

3o.4 

35 

23i.4 

40.8 

9i 

290.5 

5l.2 

56 

55.1    09.7 

16 

Il4-2 

20. 1 

7^ 

173.3 

3o.6 

36 

232.4 

4i  .0 

96 

291 .5 

bi.4 

57 

56.1    09 . 9 

17 

Il5.2 

20.3 

77 

174.3 

3o.7 

37 

233.4 

41.2 

97,292.5 

5i.6 

58 

57.1    10. 1 

18 

116. 2 

20.5 

78 

175.3 

3o.9 

38 

234.4 

4i.3 

98 

293.5 

5. .7 

59 

58.1    10.2 

19 

117. 2 

20.7 

79 

176.3 

3i.i 

39 

235.4 

4i.5 

99 

294.5 

5.. 9 

6o 

59.1    10.4 

20 

118. 2 

20.8 

Lat. 

80 

177.3 

3i.3 

40 

236.4 

41.7 

3  00 

295.4 

52.1 

Disl. 

Di.p. 

Lat. 

Dist. 

Dep. 

Dist. 

Dep. 

Lat. 

Dist.     Dep. 

Lat. 

Disl. 

Dep.  i 

Lat. 

[I 

^orSC 

)  Degrees. 

TABLE  XL 

[Paf.f  27 

Difference  of  Latitude  and  Departure  for  11  Degrees. 

Dist.    Lat. 

Dep. 

Dist. 

Lat. 

Dcp. 

Dist. 

Lat. 

Dpp. 

Dist. 

Lat. 

^ep. 

Dist. 

Lat 

46.0 

I      01. 0 

00.2 

61 

59.9 

11  .6 

121 

118. 8 

23.1 

181 

177-7 

34.5 

241 

236.6 

2 

02.0 

00.4 

62 

60.9 

II. 8 

22 

119. 8 

23.3 

82 

178.7 

34.7 

42 

237.6 

46.2 

3 

02.9 

00.6 

63 

61.8 

12.0 

23 

120.7 

23.5" 

83 

179.6 

34.9 

43 

238.5 

46.4 

4 

03.9 

00.8 

(^A 

62.8 

12.2 

24 

121.7 

23.7 

«4 

180.6 

35.1 

AA 

239.5 

46.6 

5 

04.9 

01  .0 

65 

63.8 

12.4 

25 

122.7 

23.9 

*5 

181.6 

35.3 

45 

240.5 

46.7 

6 

05.9 

01  .1 

66 

64.8 

12.6 

26 

123.7 

24.0 

86 

182.6 

35.5 

46 

241.5 

46.9 

7 

06.9 

01 .3 

67 

65.8 

12.8 

27 

124.7 

24.2 

87 

i83.6 

35.7 

47 

242.5 

47.1 

8 

07.9 

01 .5 

68 

66.8 

i3.o 

28 

125.6 

24.4 

88 

184.5 

35.9 

48 

243.4 

47.3 

9 

08.8 

01.7 

69 

67.7 

l3.2 

29 

126.6 

24.6 

89 

i85.5 

36.1 

49 

244.4 

47.5 

10 

09.8 

01 .9 

70 

68.7 

i3.4 

3o 

127.6 

24.8 

90 

186.5 

36.3 

Soj  245.4 

47.7 

11 

10.8 

02,1 

71 

69.7 

i3.5 

i3i 

128.6 

25.0 

191 

187.5 

36.4 

25i 

246.4 

47-9 

la 

II. 8  |o2.3 

72 

70.7 

i3.7 

32 

129.6 

25.2 

92 

188.5 

36.6 

52 

247.4 

48.1 

i3 

12.8 

02.5 

73 

71-7 

13.9 

6i 

i3o.6 

25.4 

93 

189.5 

36.8 

53 

248.4 

48.3 

14 

i3.7 

02.7 

74 

72.6 

i4.i 

M 

i3i.5 

25.6 

94 

190.4 

37.0 

54 

249.3 

48.5 

i5 

I4.-7 

02.9 

75 

73.6 

14.3 

35 

i32.5 

25.8 

9^ 

191. 4 

37.2 

55 

250.3 

48.7 

iG 

i5.7 

o3.i 

76 

74.6 

14.5 

36 

i33.5 

26.0 

96 

192.4 

37.4 

56 

251.3 

48.8 

17 

16.7 

o3.2 

77 

75.6 

14.7 

37 

1 34. 5 

26.1 

97 

193.4 

37.6 

57 

252.3 

49.0 

i8 

17.7 

o3.4 

78 

76.6 

14.9 

38 

i35.5 

26.3 

98 

194.4 

37.8 

58 

253.3 

49.2 

19 

18.7 

o3.6 

79 

77.5 

i5. 1 

39 

i36.4 

26.5 

99 

195.3 

38.0 

59 

254.2 

49-4 

20 

19.6 

o3.8 

80 

78.5 

i5.3 

4o 

137.4 

26.7 

200 

196.3 

38.2 

60 

255.2 

49-6 

21 

20.6 

04.0 

81 

79-5 

i5.5 

i4i 

i38.4 

26.9 

201 

197.3 

38.4 

261 

256.2 

49.8 

C2 

21.6 

04.2 

82 

80.5 

1 5. 6 

42 

139.4 

27.1 

02 

198.3 

38.5 

62 

257.2 

5o.o 

23 

22.6 

04.4 

83 

81.5 

i5.8 

A'i 

140.4 

27.3 

o3 

199.3 

38.7 

63 

258.2 

5o.2 

24 

23.6 

04.6 

84 

82.5 

16.0 

AA 

141.4 

27.5 

04 

200.3 

38.9 

64 

259.1 

5o.4 

25 

24.5 

04.8 

85 

83.4 

16.2 

45 

142.3 

27-7 

o5 

201.2 

39.1 

65 

260. 1 

5c.6 

26 

25.5 

o5.o 

86 

84.4 

16.4 

46 

143.3 

27.9 

06 

202.2 

3q.3 

66 

261 .1 

5o.8 

27 

26.5 

o5.2 

87 

85.4 

16.6 

47 

144.3 

28.0 

07 

2o3.2 

39.5 

67 

2rj2 . 1 

5o.9 

28 

27.5 

o5.3 

88 

86.4 

16.8 

48 

145.3 

28.2 

08 

204.2 

39.7 

68 

263.1 

5.    I 

29 

28.5 

o5.5 

89 

87.4 

17.0 

49 

i46.3 

28.4 

09 

205.2 

39.9 

69 

264.1 

5i.3 

3o 
3i 

29.4 

o5.7 

90 

88.3 

17.2 

5o 

l47-2 

28.6 
28.8 

10 

206  .  1 

4<5. 1 

70 

265.0 

3i.5 

2,0. A 

05.9 

91 

89.3 

17-4 

i5i 

I4S.2 

211 

207.1 

4o.3 

271 

266.0 

5i.7 

32 

3i.4 

06.1 

92 

90.3 

17.6 

52 

149.2 

29.0 

12 

208  .  I 

40.5 

72 

267.0 

5i  .9 

33 

32.4 

06.3 

93 

91.3 

17-7 

53 

l5o.2 

29.2 

i3 

209.1 

40.6 

73 

268.0 

52.1 

34 

33.4 

06.5 

94 

92.3 

17.9 

54 

i5i  .2 

29.4 

i4 

210.1 

40.8 

74 

269.0 

52.3 

35 

34.4 

06.7 

95 

93.3 

18. 1 

55 

l52.2 

29.6 

i5 

21  1  .0 

4i  .0 

75 

269.9 

52.5 

36 

35.3 

06.9 

96 

94.2 

18.3 

56 

I53.I 

29.8 

lb 

2  12.0 

4i  .2 

76 

270.9 

52.7 

37 

36.3 

07.1 

97 

95.2 

18.5 

57 

154.1 

3o.o 

17 

2l3.0 

4i.4 

77 

271.9 

52.9 

38   37.3 

07.3 

98 

96.2 

18.7 

58 

i55.i 

3o.  I 

18 

214.0 

4i.6 

78 

272.9 

53.0 

39 

38.3 

07.4 

99 

97.2 

18.9 

59 

i56.i 

3o.3 

19 

2l5.0 

4i.8 

79 

273.9 

53,2 

4<) 

39.3 

07.6 

100 

98.2 

19. 1 

6f. 

157.1 

3o.5 

20 

216.0 

42.0 

80 
281 

274.9 

53.4 

4i 

40.2 

07.8 

lOI 

99.1 

19.3 

161 

i58.o 

3o.7 

221 

216.9 

42.2 

275.8 

53.6 

42 

4i  .2 

08.0 

02 

100. 1 

.9.5 

62 

159.0 

30.9 

22 

217.9 

42.4 

82 

276.8 

53.8 

43 

42.2 

08.2 

o3 

lOI  .  I 

19.7 

63 

160.0 

3i.i 

23 

218.9 

42.6 

83 

277.8 

54.0 

AA 

43.2 

08.4 

04 

102. 1 

19.8 

64 

161 .0 

3i.3 

24 

219.9 

42.7 

84 

278.8 

54.2 

45 

44.2 

08. 6 

o5 

io3.i 

20.0 

65 

162.0 

3i.5 

25 

220.9 

42.9 

85 

279.8 

54.4 

^{6 

45.2 

08.8 

06 

104. 1 

20.2 

66 

i63.o 

3i.7 

26 

221.8 

43.1 

86 

280.7 

54.6 

47 

46.1 

09.0 

07 

io5.o 

20.4 

67 

163.9 

3i  .9 

27 

222.8 

43.3 

87 

281.7 

54.8 

48 

47.1 

09.2 

08 

1 06 . 0 

20.6 

68 

164.9 

32.1 

28 

223.8 

43.5 

88 

282.7 

55.0 

49 

48.1 

09.3 

09 

107.0 

20.8 

69 

165.9 

32.2 

29 

224.8 

43.7 

89 

283.7 

55.1 

5o 

49.1 

09.5 

10 

108.0 

21 .0 

70 

166.9 

32.4 

3o 

225.8 

43.9 

90 

284.7 

55.3 

5i 

30.I 

09.7 

HI 

1 09 . 0 

21.2 

171 

167.9 

32.6 

23l 

226.8 

44.1 

291 

285.7 

55.5 

52 

5i.o 

09.9 

12 

109.9 

21 .4 

72 

168.8 

32.8 

32 

227.7 

44.3 

92 

286.6 

55.7 

53 

52.  0 

10. 1 

i3 

1 10.9 

21.6 

73 

169.8 

33.0 

Si 

228.7 

44.5 

93 

287.6 

55.9 

54 

53.0 

10.3 

lA 

III. 9 

21.8 

74 

170.8 

33.2 

34 

229.7 

44.6 

94 

288.6 

56.1 

^^ 

54.0 

10.5 

i5 

112. 9 

21.9 

75 

171. 8 

33.4 

35 

230.7 

44.8 

95 

289.6 

56.3 

56    55.0 

10.7 

lb 

113.9 

22.1 

76 

172.8 

33.1. 

36 

23l  .7 

45.0 

9b 

290.6 

56.5 

57 

56.0 

10.9 

17 

114.9 

22.3 

77 

173.7 

33.  K 

37 

232.6 

45.2 

97 

291  .5 

56.7 

5b 

56.9 !  1 1 . 1 

18 

ii5.8 

22.5 

78 

174.7 

34.0 

38 

233.6 

45.4 

98 

292  .5 

56.9 

^9 

57.9    II. 3 

19 

116. 8 

22.7 

79 

175.7 

34.2 

39 

234.6 

45.6 

99    293.5 

37.1 

bo 
Disi. 

58.9    11.4 
l>.|..  I   Lat. 

20 

117. 8 

22.9 

8c. 

176.7 

34.3 

4o 

235.6 

45.8 

3t.o    294.5 

57.2 

Dist. 

Dep. 

Lat. 

Dist.     Drp.   1   Lat. 

Dist. 

Dcp. 

Lat. 

Di.st.j    Dep.      Lat. 

[T 

"or  79  Deirrees. 

Page  28] 

TABLE  IL 

Difference  of  Latitude  and  Departure  for  12  Degrees. 

Disl.    Lat. 

Dep. 

Dist. 

Xat. 

Dep. 

Dist. 
121 

Lat. 
118. 4 

Dep. 

25.2 

Dist. 

Lat. 

Dep. 

Dist. 

Lat. 

Dep. 

I     01  .o 

00.2 

61 

5o.7 

12.7 

181 

177.0 

37.6 

241 

235.7 

5o.i 

2     02.0 

00.4 

62 

60.6 

12.9 

22 

119. 3 

25.4 

82 

178.0 

37.8 

42 

236.7 

5o.3 

3    02.9 

00.6 

63 

61.6 

i3.i 

23 

120.3 

25.6 

83 

179.0 

38.0 

43 

237.7 

5o.5 

4 

03.9 

00.8 

64 

62.6 

i3.3. 

►  24 

121 .3 

25.8 

84 

180.0 

38.3 

44 

238.7 

50.7 

b 

04.9 

01 .0 

65 

63.6 

i3.5 

25 

122.3 

26.0 

85 

181. 0 

38.5 

45 

239.6 

5o.Q 

b 

05.9 

01.2 

6b 

64.6 

i3.7 

26 

123.2 

26.2 

86 

181. 9 

38.7 

46 

240.6 

5..i 

7 

Ob. 8 

01.5 

67 

65.5 

13.9 

27 

124.2 

26.4 

87 

182.9 

38.9 

47 

24i  .6 

5i.4 

8 

07.8 

01.7 

68 

66.5 

14. 1 

28 

125.2 

26.6 

88 

183.9 

39.1 

48 

242.6 

5i.6 

9 

08.8 

01 .9 

69 

67.5 

14.3 

29 

126.2 

26.8 

89 

184.9 

39.3 

49 

243.6 

5i.8 

10 

09. S 

02.1 

70 

68.5 

14.6 

3o 

127.2 

27.0 

90 

i85.8 

39.5 

5o 

244.5 

52.0 

II 

10.8 

02.3 

71 

69.4 

i4.8 

i3i 

128.1 

27.2 

191 

186.8 

39.7 

25l 

245.5 

52.2 

12 

II. 7 

02.5 

72 

70.4 

i5.o 

32 

129. 1 

27.4 

92 

187.8 

39.9 

52 

246.5 

52.4 

iJ 

12.7 

02.7 

7^ 

71-4 

l5.2 

33 

iSo.i 

27.7 

93 

188.8 

4o.i 

53 

247.5 

52.6 

i4 

i3.7 

02.9 

74 

72.4 

i5.4 

34 

i3i .  I 

.27-9 

94 

189.8 

4o.3 

54 

248.4 

52.8 

lb 

14.7 

o3.i 

75 

7-:! -4 

i5.6 

35 

l32.0 

28.1 

95 

190.7 

4o.5 

55 

249.4 

53.0 

lb 

lb. 7 

o3.3 

76 

74.3 

i5.8 

36 

i33.o 

28.3 

96 

191. 7 

4o.8 

56 

25o.4 

53.2 

17 

16.6 

o3.5 

77 

75.3 

16.0 

37 

i34.o 

28.5 

97 

192.7 

4i  .0 

57 

25i.4 

53.4 

18 

17. b 

03.7 

78 

76.3 

16.2 

38 

i35.o 

28.7 

98 

193.7 

4i  .2 

58 

252.4 

53.6 

^9 

18. b 

04.0 

79 

77.3 

ib.4 

39 

i36.o 

28.9 

99 

194.7 

41.4 

59 

253.3 

53.8 

20 

19.6 

04.2 

80 
81 

78.3 

16.6 

4o 

1 36. 9 

29. 1 

200 

195.6 

4i.6 

60 

254.3 

54.1 

21 

20.5 

04.4 

79.2 

16.8 

i4i 

137.9 

29.3 

201 

196.6 

4i.8 

261 

255.3 

54.3 

22 

21. b 

04.6 

82 

80.2 

17.0 

42 

i38.9 

29.5 

02 

197.6 

42 .0 

62 

256.3 

54.5 

23 

22.5 

04.8 

83 

81.2 

17.3 

43 

139.9 

29.7 

o3 

198.6 

42.2 

63 

257.3 

54.7 

24 

23. b 

o5.o 

84 

82.2 

17.5 

44 

140.9 

29.9 

04 

199.5 

42.4 

64 

258.2 

54.9 

2b 

24.5 

o5.2 

8b 

83.1 

17.7 

45 

i4i.8 

3o.i 

o5 

200.5 

42.6 

65 

259.2 

55.1 

26 

25.4 

o5.4 

86 

84.1 

17.9 

46 

142.8 

3o.4 

06 

201 .5 

42.8 

66 

260.2 

55.3 

27 

26.4 

o5.6 

87 

85.1 

18.1 

47 

143.8 

3o.6 

07 

202.5 

43.0 

67 

261 .2 

55.5 

28 

27.4 

o5.8 

88 

86.1 

18.3 

48 

144.8 

3o.8 

08 

2o3.5 

43.2 

68 

262.1 

55.7 

29 

28.4 

06.0 

89 

87.1 

18.5 

49 

145.7 

3i  .0 

09 

204.4 

43.5 

69 

263.1 

55.9 

60 

29.3 

06.2 

90 

88.0 

18.7 

5o 

146.7 

3l.2 

10 

2o5.4 

43.7 

70 
271 

264.1 
265.1 

56.1 
56.3 

3i 

3o.3 

06.4 

91 

89.0 

18.9 

i5i 

147-7 

3i.4 

211 

206.4 

43.9 

32 

3i.3 

06.7 

92 

90.0 

19. 1 

52 

148.7 

3i.6 

12 

207.4 

44.1 

72 

266.1 

56.6 

33 

32.3 

06.9 

93 

91 .0 

19.3 

53 

149.7 

3i.8 

i3 

208.3 

44.3 

73 

267.0 

56.8 

M 

6.i.6 

07.1 

94 

91.9 

19.5 

54 

i5o.6 

32. 0 

i4 

209.3 

44.5 

74 

268.0 

57.0 

3b 

34.2 

07.3 

9b 

92.9 

19.8 

55 

i5i.6 

32.2 

i5 

210.3 

44.7 

75 

269.0 

57.2 

3b 

3b. 2 

07.5 

96 

93.9 

20.0 

56 

i52.6 

32.4 

16 

21 1 .3 

44.9 

76 

270.0 

b7.4 

^7 

3b. 2 

07.7 

97 

94.9 

20.2 

57 

i53.6 

32.6 

17 

212.3 

45.1 

77 

270.9 

57.6 

38 

37.2 

07.9 

98 

95.9 

20.4 

58 

i54.5 

32.9 

18 

2l3.2 

45.3 

78 

271.9 

57.8 

39 

38.1 

08.1 

99 

96.8 

20.6 

59 

155.5 

33.1 

19 

214.2 

45.5 

79 

272.9 

58.0 

40 

39.1 

08.3 

100 

lOI 

97.8 
98.8 

20.8 

60 

i56.5 

33.3 
33.5 

20 

2l5.2 

45.7 

80 

273.9 

58.2 
58.4 

4i 

40. 1 

08.5 

21 .0 

161 

157.5 

221 

216.2 

45.9 

281 

274.9 

42 

4i.i 

08.7 

02 

99.8 

21 .2 

62 

i58.5 

33.7 

22 

217. 1 

46.2 

82 

275.8 

58.6 

43 

42.1 

08.9 

o3 

100.7 

21 .4 

63 

159.4 

33.9 

23 

218. I 

46.4 

83 

276.8 

58.8 

44 

43.0 

09.1 

o4 

ior.7 

21 .6 

64 

1UO.4 

34.1 

24 

219.1 

46.6 

84 

277.8 

59.0 

4b 

44.0 

09.4 

o5 

102.7 

21.8 

65 

161.4 

34.3 

25 

220.1 

46.8 

8b 

278.8 

59.3 

4b 

4b. 0 

09.6 

06 

io3.7 

22.0 

66 

162.4 

34.5 

26 

221  .1 

47-0 

86 

279.8 

59.5 

47 

46.0 

09.8 

07 

104.7 

22.2 

67 

163.4 

34.7 

27 

222.0 

47.2 

87 

280.7 

59.7 

48 

47-0 

10. 0 

08 

105.7 

22.5 

68 

164.3 

34.9 

28 

223.0 

47-4 

88 

281.7 

59.9 

49 

47-9 

10.2 

09 

106.6 

22.7 

69 

i65.3 

35.1 

29 

224.0 

47-6 

89 

282.7 

60.1 

bo 

48.9 

10.4 

10 

107.6 

22.9 

70 

166.3 

35.3 

3o 

225.0 

47-8 

90 

283.7 

60.3 

5i 

49.9 

10.6 

III 

108.6 

23.1 

171 

167.3 

35.6 

23  1 

226.0 

48.0 

291 

284.6 

60.5 

b2 

bo. 9 

10.8 

12 

109.6 

23.3 

72 

168.2 

35.8 

32 

226.9 

48.2 

92 

285.6 

60.7 

b3 

bi.8 

II  .0 

i3 

110.5 

23.5 

73 

169.2 

36.0 

33 

227.9 

48.4 

93 

286.6 

60.9 

b4 

b2..8 

II. 2 

i4 

III. 5 

23.7 

74 

170.2 

36.2 

34 

228.9 

48.7 

94 

287.6 

61.1 

bb 

b3.8 

II. 4 

lb 

112. 5 

23.9 

75 

171 .2 

36.4 

35 

229.9 

48.9 

95 

288.6 

61.3 

bb 

b4.8 

II. b 

16 

ii3.5 

24.1 

76 

172.2 

36.6 

36 

230.8 

49.1 

96 

289.5 

61.5 

^7 

bb.8 

II. 9 

17 

114. 4 

24.3 

77 

173. 1 

36.8 

37 

231.8 

49-3 

97 

290.5 

6r  .7 

58    56.7 

12. 1 

18 

ii5.4 

24.5 

78 

174. 1 

37.0 

38 

232.8 

49.5 

98 

291 .5 

62.0 

59    57.7 

12. J 

19 

116.4 

24.7 

79 

175. 1 

37.2 

39 

233.8 

49-7 

99 

292.5 

62.2 

60    58.7 

12.5 

20 

117.4 

24.9 
Lat. 

80 

176.1 

37.4 

_4o 
Dist. 

234.8 
Dep. 

49.9 
Lat. 

3oo 

293.4 

62.4 
Lat. 

Dist.    Dep. 

Lat. 

DIst. 

Dep. 

Disl. 

Dep. 

Lat. 

Dist. 

Deu. 

0 

'"or  78  Degrees 

TABLE  IL 

[Page  29 

Difference  of  Latitude  and  Departure  for  13  Degrees. 

Disl. 

Lat. 

Dep. 

Dis.. 

Lat.    i  Dcp. 

Dist. 

Lat. 

Dep. 

Dist. 

Lat. 

Dep. 

Dist. 

241 

Lat.      Dep.  | 

I 

ot  .0 

00.2 

61 

59.4' !3, 7 

121 

117. 9 

27.2 

181 

176.4 

40.7 

234.8 

54.2 

2 

CI. 9 

CO. 4 

62 

5o.4  1  i3.9 

22 

118. 9 

27.4 

82 

177.3 

40.9 

42 

235.8 

54-4 

3 

02.9 

00.7 

63 

5i.4'i4.2 

23 

119. 8 

27.7 

83 

178.3 

41.2 

43 

236.8 

54.7 

4 

08.9 

00.9 

64 

62.4 

i4.4 

24 

120.8 

27.9 

84 

179-3 

41.4 

44 

237.7 

54.9 

5 

04.9 

01 .1 

65 

63.3 

14.6 

25 

121. 8 

28.1 

85 

180.3 

4i.6 

45 

238.7 

55.1 

6 

o5.8 

01.3 

66 

64.3 

i4.8 

26 

122.8 

28.3 

86 

181.2 

4i.8 

46 

239.7 

55.3 

} 

06.8 

01 .6 

67 

65.3 

i5.i 

27 

123.7 

28.6 

87 

182.2 

42.1 

47 

240.7 

55.6 

s 

07.8 

01.8 

68 

66.3 

i5.3 

28 

124.7 

28.8 

88 

i83.2 

42.3 

48 

241 .6 

55.8 

9 

08.8 

02.0 

69 

67.2 

i5.5 

29 

125.7 

29.0 

89 

184.2 

42.5 

49 

242.6 

56.0 

10 

09.7 

02.2 

70 

68.2 

l5.7 

3o 

126.7 

29.2 

90 

i85.i 

42.7 

5o 

243.6 

56.2 
56.5 

1 1 

10.7 

02.5 

71 

69.2 

16.0 

i3i 

127.6 

29.5 

191 

186.1 

43.0 

25l 

244.6 

12 

II. 7 

f.2.7 

72 

70.2 

16.2 

32 

■128.6 

29.7 

92 

187.1 

43.2 

52 

245.5 

56.7 

i3 

12.7 

02.9 

73 

71. 1 

16.4 

33 

129.6 

29.9 

93 

188.1 

43.4 

53 

246.5 

56.9 

i4 

i3.6 

o3.i 

74 

72.1 

16.6 

34 

i3o.6 

3o.i 

94 

189.0 

43.6 

54 

247-5 

57.1 

i5 

14. b 

o3.4 

75 

73.1 

16.9 

35 

i3i.5 

3o.4 

95 

190.0 

43.9 

55 

248.5 

57.4 

i6 

i5.6 

o3.6 

76 

74.1 

17. 1 

36 

i32.5 

3o.6 

96 

191 .0 

44.1 

56 

249.4 

57.6 

17 

16. b 

o3.8 

77 

73. 0 

17.3 

37 

i33.5 

3o.8 

97 

192.0 

44.3 

57 

25o.4 

57.8 

i8 

17.5 

04.0 

7» 

76.0 

17.5 

38 

i34.5 

3i  .0 

98 

192.9 

44.5 

58 

25i.4 

58.0 

19 

18.5 

04.3 

79 

77.0 

17.8 

39 

i35.4 

3i.3 

99 

193.9 

44.8 

59 

252.4 

58.3 

20 

19.5 

04.5 

80 

77-9 

18.0 

40 

i36.4 

3i.5 

200 

194.9 

45.0 

60 

253.3 

58.5 

21 

20.5 

04.7 

81 

78.9 

18.2 

i4i 

137.4 

3i.7 

201 

195.8 

45.2 

261 

254.3 

58.7 

22 

21.4 

04.9 

82 

79-9 

18.4 

42 

i38.4 

3i  .9 

02 

196.8 

45.4 

62 

255.3 

58.9 

23 

22.4 

o5.2 

83 

80.9 

18.7 

43 

139.3 

32.2 

o3 

197.8 

45.7 

63 

256.3 

59.2 

24 

23.4 

o5.4 

84 

81.8 

18. Q 

44 

i4o.3 

32.4 

04 

198.8 

45.9 

64 

257.2 

59.4 

25 

24.4 

o5.6 

85 

82.8 

19. I 

45 

i4i.3 

32.6 

o5 

199.7 

46.1 

65 

258.2 

59.6 

26 

25.3 

o5.8 

86 

83.8 

19.3 

46 

142.3 

32.8 

06 

200.7 

46.3- 

66 

259.2 

59.8 

27 

26.3 

06.1 

87 

84.8 

19.6 

47 

143.2 

33.1 

07 

201 .7 

46.0 

67 

260.2 

60.1 

28 

27.3 

06.3 

88 

85.7 

19.8 

48 

144.2 

33.3 

08 

202.7 

46.8 

68 

261 .1 

60.3 

29 

28.3 

06.5 

89 

86.7 

20.0 

49 

145.2 

33.5 

09 

2o3.6 

47-0 

69 

262,1 

60.5 

60 

29.2 

06.7 

90 

«7.7. 

20.2 

5o 

146.2 

33.7 

10 

204.6 

47-2 

70 

263.1 

60.7 

3i 

3o.2 

07.0 

91 

88.7 

20.5 

1.5 1 

i47-i 

34.0 

211 

2o5.6 

47.5 

271 

264.1 

61.0 

32 

3l.2 

07.2 

92 

89.6 

20.7 

52 

I48.I 

34.2 

12 

206.6 

47-7 

72 

265.0 

61 .2 

33 

32.2 

07.4 

93 

90.6 

20.9 

53 

149. 1 

34.4 

i3 

207.5 

47-9 

73 

266.0  1 61  4 

34 

33.1 

07.6 

94 

91.6 

21  .1 

54 

i5o.i 

34.6 

i4 

208.5 

48.1 

74 

267,0  ; 61 .6 

35 

34.1 

07.9 

95 

92.6 

21.4 

55 

i5i.o 

34.9 

i5 

209.5 

48.4 

75 

268.0 

61.9 

36 

35.1 

08.1 

96 

93.5 

21  .6 

56 

l52.0 

35.1 

16 

210.5 

48.6 

76 

268.9 

62.1 

37 

36.1 

08.3 

97 

94.5 

21.8 

57 

i53.o 

35.3 

17 

211 .4 

48.8 

77 

269.9 

62.3 

38 

37.0 

08.5 

98 

95.5 

22.0 

58 

i54.o 

35.5 

18 

212.4 

49.0 

78 

270.9 

62.5 

39 

33.0   08.8 

99 

96.5 

22.3 

59 

154.9 

35.8 

19 

2i3.4 

49-3 

79 

271.8 

62.8 

4o 

39.0   09.0 

100 

97-4 

22.5 

60 

155.9 

36. 0 

20 

214.4 

49.5 

80 

272.8 

63.0 

4i 

39.9   09 . 2 

lOI 

98.4 

22.7 

161 

i56.9 

36.2 

221 

2i5.3 

49-7 

281 

273.8 

63.2 

43 

40.9 

09.4 

02 

99.4 

22.9 

62 

157.8 

36.4 

22 

216.3 

49.9 

82 

274.8 

63.4 

43 

41.9 

09.7 

o3 

100.4 

23.2 

63 

i58.8 

36.7 

23 

217.3 

5o.2 

83 

275.7 

63.7 

44 

42.9 

09.9 

04 

loi  .3 

23.4 

64 

159.8 

36.9 

24 

218.3 

5o.4 

84 

276.7 

63.9 

45 

4i.8 

10. 1 

o5 

102.3 

23.6 

65 

160.8 

37.1 

25 

219.2 

5o.6 

85 

277-7 

64.1 

46 

44.8 

10.3 

06 

io3.3 

23.8 

66 

161 .7 

37.3 

26 

220.2 

5o.8 

86 

278.7 

64.3 

47 

45.8 

10.6 

07 

104.3 

24.1 

67 

162.7 

37.6 

27 

221 .2 

5i.i 

87 

279.6 164.6 

48 

46.8 

10.8 

08 

io5.2 

2'i.3 

68 

163.7 

37.8 

28 

222.2 

5i.3 

88 

280.6 ,64.8 

49 

47.7 

II  .0 

09 

106.2 

24.5 

69 

164.7 

38.0 

29 

223.  I 

5i.5 

89 

281.6 

65.0 

5o 

48.7 

II. 2 

10 

107.2 

24.7 

7" 
171 

i65.6 
166.6 

38.2 

3o 

224.1 

51.7 

90 

282.6 

65.2 
65.5 

5i 

49-7 

11.5 

III 

I03.2 

25.0 

38.5 

23l 

225.1 

52.0 

291 

283.5 

32 

50.7 

II. 7 

12 

109.1 

25.2 

72 

167.6 

38.7 

32 

226.1 

52.2 

92 

284.5 

65.7 

53 

5i.6 

II. 9 

i3 

no. I 

25.4 

73 

168.6 

38.9 

33 

227.0 

52.4 

93 

285.5 

65.9 

54 

52.6 

12     I 

i4 

III. I 

25.6 

74 

169.5 

39.1 

34 

228.0 

52.6 

94 

286.5 

66.1 

55 

53.6 

12.4 

i5 

112. 1 

25.9 

75 

170.5 

3q.4 

35 

229.0 

52.9 

95 

287.4 

66.4 

56 

54.6 

12.5 

16 

ii3.o 

26.1 

76 

171. 5 

39.6 

36 

23o.o 

53.1 

96 

288.4 

66.6 

57 

55.5 

12.8 

37 

114.0 

26.3 

77 

172.5 

39.8 

37 

230.9 

53.3 

97 

289.4 

66.8 

58 

56.5 

i3.o 

18 

i:5.c    26.5 

78 

173.4 

4o.o 

38 

23l  .9 

53.5 

98 

290.4 

67.0 

59 

57.5 

i3.3 

19 

:i6.o 

25.5 

79 

174.4 

4o.3 

39 

232.9 

53.8 

99 

291 .3 

67.3 

bu 

5S.5,i3.5 

20 

116.9 

27.0 

80 

175.4 

4o.5 

4:)    233.8 

54.0 

3oo 

292.3 

67.5 

Dist. 

Dpp.  '  Lat. 

Dist, 

Dop 

Lai. 

Dist. 

Dep. 

Lnt. 

Disl.     D.'p. 

Lat. 

Disl. 

Dep. 

Lat. 

V 

For  77  Degrees. 

Page  30] 

TABLE  n. 

Difference  of  Latitude  and  Departure  for  14  Degrees. 

Dist. 

Lat. 

Dep. 

00.2 

Dist. 

Lat. 

Dep. 

Dist. 

Lat. 

Dep. 

Dist 

Lat. 

Dep. 

Dist. 

Lat. 

Dep. 

.58.3 

I 

01. 0 

61 

59.2 

14.8 

121 

117. 4 

29.3 

181 

175.6 

43.8 

241 

233.8 

2 

01.9 

00.5 

62 

60.2 

i5.o 

22 

118.4 

29.5 

82 

176.6 

44-0 

42 

234.8 

58.5 

3 

02.9 

00.7 

63 

61.1 

l5.2 

23 

119. 3 

29.8 

83 

177-6 

44-3 

43 

235.8 

58.8 

4 

03.9 

01  .0 

64 

62.1 

i5.5 

24 

120.3 

3o.o 

84 

178.5 

44.5 

A^ 

236.8 

59.0 

5 

04.9 

01.2 

65 

63.1 

i5.7 

25 

121 .3 

3o.2 

85 

179.5 

44.8 

45 

237-7 

59.3 

6 

o5.8 

01 .5 

66 

64.0 

16.0 

26 

122.3 

3o.5 

86 

180.5 

45.0 

46 

238.7 

59.5 

7 

06.8 

01.7 

67 

65. 0 

16.2 

27 

123.2 

3o.7 

87 

181. 4 

45.2 

47 

239.7 

59.8 

8 

07.8 

01.9 

68 

66.0 

16.5 

28 

124.2 

3i  .0 

88 

182.4 

45.5 

48 

240.6 

60.0 

9 

08.7 

02.2 

69 

67.0 

,6.7 

29 

125.2 

3l.2 

89 

i83.4 

45.7 

49 

241 .6 

60.2 

lO 

09.7 

02.4 

70 

67.9 

16.9 

3o 

126.  I 

3i.4 
3i.7 

90 

184.4 

46. 0 

5o 

242.6 

60.5 

II 

10.7 

02.7 

71 

68.9 

17.2 

i3i 

127. I 

191 

i85.3 

46.2 

25l 

243.5 

60.7 

12 

II. 6 

02.0 

72 

69.9 

17-4 

32 

128.  I 

3i  .9 

92 

186.3 

46.4 

52 

244.5 

61.0 

i3 

12.6 

o3.i 

73 

70.8 

17-7 

33 

129.0 

32.2 

93 

187.3 

46.7 

53 

245.5 

61 .2 

i4 

i3.6 

o3.4 

74 

71.8 

17.9 

34 

i3o.o 

32.4 

94 

188.2 

46.9 

54 

246.5 

61.4 

i5 

i4.6 

o3.6 

75 

72.8 

18. 1 

35 

i3i.o 

32.7 

95 

189.2 

47-2 

55 

247.4 

61.7 

i6 

i5.5 

03.9 

76 

73.7 

18.4 

36 

l32.0 

32.9 

96 

190.2 

47-4 

56 

248.4 

61.9 

17 

16.5 

04. 1 

77 

74-7 

18.6 

37 

132.9 

33.1 

97 

191 .1 

47-7 

57 

249.4 

62.2 

i8 

17.5 

04.4 

78 

75.7 

18.9 

38 

i33.9 

33.4 

98 

192.1 

47-9 

58 

250.3 

62.4 

19 

18.4 

04.6 

79 

76.7 

19. 1 

39 

134.9 

33.6 

99 

193.1 

48.1 

59 

251.3 

62.7 

20 

19.4 

04.8 

80 

77.6 

19.4 

40 

i35.8 

33.9 

200 

194. 1 

48.4 

60 

252.3 

62.9 

21 

20.4 

o5.i 

81 

78.6 

19.6 

i4i 

i36.8 

34.1 

201 

195.0 

48.  G 

261 

253.2 

63.1 

22 

21.3 

o5.3 

82 

79.6 

19.8 

42 

137.8 

34.4 

02 

196.0 

48.9 

62 

254.2 

63.4 

23 

22.3 

o5.6 

83 

80.5 

20. 1 

43 

i38.8 

34.6 

OJ 

197.0 

49.1 

63 

255.2 

63.6 

24 

23.3 

o5.8 

84 

81.5 

20.3 

M 

139.7 

34.8 

o4 

197.9 

49-4 

64 

256.2 

63.9 

25 

24.3 

06.0 

85 

82.5 

20.6 

45 

140.7 

35.1 

o5 

198.9 

49-6 

65 

257.1 

64.1 

26 

25.2 

06.3 

86 

83.4 

20.8 

46 

141.7 

35.3 

06 

199.9 

49-8 

66 

258.1 

64.4 

27 

26.2 

06.5 

87 

84.4 

21 .0 

47 

142.6 

35.6 

07 

200.9 

5o.i 

^7 

2:j9.  I 

64.6 

28 

27.2 

06.8 

88 

85.4 

21.3 

48 

143.6 

35.8 

08 

201.8 

5o.3 

68 

260.0 

64.8 

29 

28.1 

07.0 

89 

86.4 

21.5 

49 

144.6 

36.0 

09 

202.8 

5o.6 

69 

261 .0 

65.1 

3o 

29.1 

07.' 

90 

87.3 

21.8 

5o 

145.5 

36.3 

10 

2o3.8 

5o.8 

70 

262.0 

65.3 

3i 

3o.  I 

07.5 

9' 

88.3 

22.0 

i5i 

146.5 

36.5 

211 

204.7 

5i.o 

271 

263.0 

65.6 

32 

3i  .0 

07,7 

92 

69.3 

22.3 

52 

147-5 

36.8 

12 

205.7 

5i.3 

72 

263.9 

65.8 

33 

32.0 

08.0 

93 

90.2 

22.5 

53 

148.5 

37.0 

i3 

206.7 

5i.5 

73 

264.9 

66.0 

34 

33.0 

08.2 

94 

91 .2 

22.7 

54 

149.4 

37.3 

i4 

207.6 

5i.8 

74 

365.9 

66.3 

35 

34.0 

08.5 

95 

92.2 

23.0 

55 

i5o.4 

37.5 

i5 

208.6 

52.0 

7^ 

266.8 

66.5 

36 

34.9 

08.7 

96 

93.1 

23.2 

56 

i5i.4 

37.7 

16 

209.6 

52.3 

76 

267.8 

66.8 

37 

35.9 

09.0 

97 

94.1 

23.5 

57 

i52.3 

38. 0 

17 

210.6 

52.5 

77 

268.8 

67.0 

38 

36.9 

09.2 

98 

95.1 

23.7 

58 

i53.3 

38.2 

18 

211 .5 

52.7 

78 

269.7 

67.3 

39 

37.8 

09.4 

99 

96. 1 

24.0 

59 

i54.3 

38.5 

19 

212.5 

53.0 

79 

270.7 

67.5 

40 

38.8 

09.7 

100 

97.0 

24.2 

60 

i55.2 

38.7 

20 

2i3.5 

53.2 

80 

271.7 

67-7 

4i 

39.8 

09.9 

lOI 

98.0 

24.4 

161 

i56.2 

38.9 

221 

214.4 

53.5 

281 

272.7 

68.0 

42 

40.8 

10.2 

02 

99.0 

24.7 

62 

157.2 

39.2 

22 

2i5.4 

53.7 

82 

273.6 

68.2 

43 

41.7 

10.4 

o3 

99.9 

24.9 

63 

i58.2 

39.4 

23 

216.4 

53.9 

83 

274.6 

68.5 

U 

42.7 

10.6 

04 

100.9 

25.2 

64 

159. 1 

39.7 

24 

217.3 

54.2 

84 

275.6 

68.7 

45 

43.7 

10.9 

o5 

lOI  .9 

25.4 

65 

160. 1 

39.9 

25 

218.3 

54.4 

85 

276.5 

68.9 

46 

44.6 

II  .1 

Otj 

102.9 

25.6 

66 

161 .1 

40.2 

26 

219.3 

54.7 

86 

277.5 

69.2 

47 

45.6 

II. 4 

07 

io3.8 

25.9 

67 

162.0 

40.4 

27 

220.3 

54-9 

87 

278.5 

69.4 

48 

46.6 

II. 6 

08 

104.8 

26.1 

68 

i63.o 

40.6 

28 

221 .2 

55.2 

88 

279.4 

69.7 

49 

47.5 

II. 9 

09 

io5.8 

26.4 

69 

164.0 

40.9 

29 

222.2 

55.4 

89 

280.4 

69.9 

5o 

48.5 

12. 1 

10 

106.7 

26.6 

70 

i65.o 

4i.i 

3o 

223.2 

55.6 

90 

281.4 

70.2 

5i 

49. "i 

12.3 

III 

107.7 

26.9 

171 

165.9 

41.4 

23  I 

224.1 

55.9 

291 

282.4 

70.4 

52 

5o.5 

12.6 

12 

108.7 

27.1 

72 

166.9 

41.6 

32 

225.1 

56.1 

92 

283.3 

70.6 

53 

5i.4 

12.8 

i3 

109.6 

27.3 

73 

167.9 

41.9 

33 

226.1 

56.4 

93 

284.3 

70.9 

54 

52.4 

i3.i 

i4 

no. 6 

27.6 

74 

168.8 

42.1 

34 

227.0 

56.6 

94 

285.3 

71. 1 

55 

53.4 

i3.3 

1 5 

III  .6 

27.8 

75 

169.8 

42.3 

35 

228.0 

56.9 

95 

286.2 

71.4 

56 

54.3 

i3.5 

16 

112. 6 

28.1 

76 

170.8 

42.6 

■66 

229.0 

57-1 

96 

287.2 

71.6 

57 

55.3 

i3.8 

17 

ii3.5 

28.3 

77 

171-7 

42.8 

37 

23o.o 

57.3 

97 

288.2 

71.9 

58 

56.3 

i4-o 

18 

114.5 

28.5 

78 

172.7 

43.1 

38 

230.9 

57.6 

98 

289.1 

72.1 

59, 57.2 

i4.3 

19 

ii5.5 

28.8 

79 

173.7 

43.3 

39 

231.9 

57.8 

99 

290. 1 

72.3 

60:  58.2 
Dist.l  Dcp. 

i4.5 
Lat. 

20 

116. 4 

29.0 
Lat. 

80 

174.7 

43.5 

4o 

232.9 

58.1 

000 

29 1 . 1 

72.6 

Dist. 

Dcp. 

Dist. 

Dep. 

Lai. 

Dist. 

Dep. 

Lat. 

Dist. 

Dep. 

Lat. 

[For  7G  Degre 

es. 

TABLE  n 

[Page  31 

Dinference  of  Latitude  and  Departure  for  15  Degrees. 

Disl. 

Lai.  1'  Dop. 

Uisl. 

Lat. 

Dep. 

Dist.j    Lat. 

Dep. 

Dist. 

Lat. 

Dep. 

Dist. 

Lat. 

D*.-p. 

I 

ul  .0 

00.3 

61 

58.9 

i5.8 

121 

116. 9 

3i.3 

181 

174-8 

46.8 

241 

232.8 

62.4 

2 

01.9 

00.5 

b2 

59..  9 

16.0 

22 

117. 8 

3i.6 

82 

175.8 

47.1 

42 

233.8 

62.6 

3 

02.9 

00.8 

63 

60.9 

16.3 

23 

118.8 

3i.8 

83 

176.8 

47.4 

43 

234.7 

62.9 

4 

o3.9 

01 .0 

b4 

61.8 

16.6 

24 

119. 8 

32.1 

84 

177-7 

47.6 

44 

235.7 

63.2 

5 

04.8 

01 .3 

bb 

62.8 

16.8 

2b 

120.7 

32.4 

8b 

178.7 

47-9 

45 

236.7 

63.4 

6 

o5.8 

01 .6 

6b 

63.8 

17. 1 

26 

121 .7 

32.6 

86 

'79-7 

48.1 

46 

237.6 

63.7 

7 

Ob. 8 

01.8 

67 

64.7 

17.3 

27 

122.7 

32.9 

87 

180.6 

48.4 

47 

238.6 

63.9 
64.2 

8 

07.7 

02. 1 

68 

65.7 

17.6 

28 

123.6 

33.1 

88 

181.6 

48.7 

48 

239.5 

9 

08.7 

02.3 

69 

66.6 

17.9 

29 

124.6 

33.4 

89 

182.6 

48.9 

49 

240.  L 

64.4 

lO 

09.7 

02.6 

70 

67.6 

18. 1 

3o 

125.6 

33.6 

90 

i83.5 

49.2 

5o 

241.5 

64.7 

II 

10.6 

02.8 

71 

68.6 

18.4 

i3i 

126.5 

33.9 

191 

184.5 

49.4 

25l 

242.4 

65.0 

12 

11. b 

o3.i 

72 

69.5 

18.6 

32 

127.5 

34.2 

92 

185.5 

49.7 

52 

243.4 

65.2 

i3 

12.6 

o3.4 

73 

70.5 

18.9 

^i 

128.5 

M.4 

93 

186.4 

5o.o 

53 

244.4 

65.5 

i4 

i3.b 

o3.6 

74 

71.5 

19.2 

M 

129.4 

34.7 

94 

187.4 

5o.2 

54 

245.3 

65.7 

lb 

i4.b 

03.9 

7!) 

72.4 

19.4 

3b 

i3o.4 

34.9 

95 

188.4 

5o.5 

55 

246.3 

66.0 

i6 

ib.b 

o4.i 

76 

73.4 

19.7 

36 

i3i.4 

35.2 

96 

189.3 

5o.7 

56 

247.3 

66.3 

J7 

lb. 4 

04.4 

77 

74.4 

19.9 

37 

i32.3 

35.5 

97 

190.3 

5i.o 

57 

248.2 

66.5 

i8 

17-4 

04.7 

7» 

7b. 3 

20.2 

38 

133.3 

35.7 

98 

.91.3 

5l.2 

58 

249.2 

66.8 

'9 

18.4 

04.9 

79 

76.3 

20.4 

39 

i34.3 

36. 0 

99 

192.2 

5i.5 

59 

25o.2 

67.0 

20 

.9.3 

05.2 

80 

77.3 

20.7 

4o 

i35.2 

36.2 

200 

193.2 

5i.8 

60 

25l  .1 

67.3 

2! 

20.3 

o5.4 

81 

78.2 

21 .0 

.i4i 

i36.2 

36.5 

201 

194.2 

52.0 

261 

252.1 

67.6 

22 

21.3 

o5.7 

82 

79.2 

21.2 

42 

137.2 

36.8 

02 

195.1 

52.3 

62 

253.1 

67.8 

23 

22.2 

06.0 

83 

80.2 

21.5 

43 

i38.i 

37.0 

o3 

196. 1 

52.5 

63 

254.0 

68.1 

24 

23.2 

06 . 2 

84 

81. 1 

21.7 

44 

139. 1 

37.3 

04 

197.0 

52.8 

64 

255.0 

68.3 

25 

24.1 

06.5 

8b 

82.1 

22.0 

4b 

1 40 . 1 

37.5 

o5 

198.0 

53.1 

65 

256. 0 

68.6 

2b 

2b. I 

06.7 

8b 

83.1 

22.3 

46 

i4i  .0 

37.8 

06 

199.0 

53.3 

66 

256.9 

68.8 

27 

2b. I 

07.0 

87 

84.0 

22.5 

47 

142.0 

38. 0 

07 

199.9 

53.6 

67 

257.9 

69.1 

2« 

27.0 

07.2 

88 

85. 0 

22.8 

48 

143.0 

38.3 

08 

200.9 

53.8 

68 

258.9 

69.4 

29 

28.0 

07.  D 

89 

86.0 

23.0 

49 

143.9 

38.6 

09 

201 .9 

54.1 

69 

259.8 

69.6 

Jo 

29.0 

07.8 

90 

80.9 

23.3 

bo 

144.9 

38.8 

10 

202.8 

54.4 

70 

260.8 

69.9 

3i 

29.9 

08.0 

9' 

87.9 

23.6 

i5i 

145.9 

39.. 

21 1 

2o3.8 

54.6 

271 

261.8 

70.1 

32 

30.9 

08.3 

92 

88.9 

23.8 

52 

146.8 

39.3 

12 

204.8 

54.9 

72 

262.7 

70.4 

iS 

31.9 

08.5 

9i 

89.8 

24.1 

b3 

147.8 

39.6 

i3 

2()5.7 

55.1 

73 

263.7 

70.7 

34 

32.8 

0S.8 

94 

90.8 

24.3 

b4 

148.8 

39.9 

i4 

206 . 7 

55.4 

74 

264.7 

70.9 

jt) 

33.8 

09.  I 

9-^ 

91.8 

24.6 

bb 

149.7 

4o.  1 

i5 

207.7 

55.6 

75 

265.6 

71.2 

3b 

34.8 

09.3 

9b 

92.7 

24.8 

b6 

1 5o .  7 

40.4 

16 

208.6 

55.9 

7t) 

266.6 

71-4 

^7 

3^.7 

09.6 

97 

93.7 

25.1 

i)7 

i5i  .7 

4o.6 

17 

209.6 

56.2 

77 

267.6 

71-7 

38 

3b. 7 

09.8 

98 

94.7 

2b. 4 

b8 

1 52 .6 

40.9 

18 

210.6 

56.4 

78 

268.5 

72.0 

39 

37.7 

10.  I 

99 

95.6 

25.6 

b9 

i53.6 

4i  .2 

19 

211.5 

56.7 

79 

269.5 

72.2 

40 

38.  b 

10.4 

i(i() 

96.6 

25.9 

60 

154.5 

41.4 
41.7 

20 

212.5 

56.9 

80 

270.  D 

72.5 

4i 

39.6 

10.6 

101 

97.6 

26. 1 

161 

i55.5 

221 

213.5 

57.2 

281 

271.4 

72.7 

42 

40.6 

10.9 

02 

98.5 

26.4 

62 

i56.5 

4i  .9 

22 

214.4 

57-5 

82 

272.4 

73.0 

43 

4i.b 

I  I  .  I 

o3 

99.5 

26.7 

63 

157.4 

42.2 

23 

2i5.4 

57-7 

83 

273.4 

73.2 

44 

42. b 

II. 4 

04 

100.5 

26.9 

64 

158.4 

42.4 

24 

216.4 

58. 0 

84 

274.3 

73.5 

4b 

43.5 

1 1. 6 

ob 

loi  .4 

27.2 

65 

159.4 

42.7 

25 

217.3 

58.2 

85 

275.3 

73.8 

4f) 

4^.4 

11.9 

Ob 

102.4 

27.4 

66 

160.3 

43.0 

26 

218.3 

58.5 

86 

276.3 

74.0 

47 

4':). 4 

12.2 

07 

io3.4 

27.7 

67 

161. 3 

43.2 

27 

219.3 

58.8 

87 

277.2 

74.3 

48 

4b. 4 

12.4 

08 

104.3 

28.0 

68 

162.3 

43.5 

28 

220.2 

59.0 

88 

278.2 

74.5 

i:'9 

47.3 

12.7 

09 

io5.3 

28.2 

69 

i63.2 

43.7 

29 

221 .2 

59.3 

89 

279.2 

74.8 

bo 

48.3 

12.9 

10 

106.3 

2S.5 

70 

164.2 

44.0 

3o 

222.2 

59.5 

90 

280.1 

75.1 

bi 

49.3    1 3. 2  1 

1 1 1 

1 07 . 2 

28.7 

171 

i65.2 

44.3 

23l 

223.1 

59.8 

291 

281. 1 

75.3 

b2 

bf  > .  2 

i3.b 

12 

108.2 

29.0 

72 

166. 1 

44.5 

32 

224.1 

60.0 

92 

282.1 

75.6 

33 

bi.2 

13.7 

i3 

109. I    29.2 

73 

167.1 

44.8 

33 

225.1 

60.3 

93 

283. 0 

75.8 

b4 

b2.2 

14.0 

i4 

1 10. 1 

29.5 

74 

168.1 

45.0 

34 

226.0 

60.6 

94    284.0 

76.1 

bb 

b3.r 

l4.2 

lb 

III  .1 

29.8 

75 

-169.0 

45.3 

35 

227.0 

60.8 

95 

284.9 

76.4 

bb 

b4.i 

i4.b 

lb 

112.0 

Bo.o 

76    170.0 

45.6 

36 

228.0 

61.1 

96 

285.9 

76.6 

b7 

bb.i 

i4.8 

17 

ii3.o 

3o.3 

77 

171 .0 

45.8 

37 

228.9 

61.3 

97 

286.9 

76.9 

b8 

bb.o 

i5.o 

18 

114.0 

3o.b 

78 

171.9146.1  1 

38 

229.9 

61.6 

98 

287.8 

11  ■' 

b9 

b7.o 

i5.3 

'9 

114.9 

3o.8 

79 

172.9 

46.3 

39 

230.9 

61 .9 

99 

288.8 

11-4 

fx) 

b8.o 

i5.5 

l.at. 

20 

1 15.9 

3i.i 

80 

173.9 

46.6 

4o 

231.8 

62.1 

3()o 

289.8 

77-6 

Dist.  nop. 

I)ist.|    Dep. 

Lat. 

Dist. 

Dep. 

Lai. 

Dist. 

Dep. 

Lat. 

Dist. 

Dep. 

Lat. 

I 

[> 

"or  75  Defrre 

es. 

Page  32j 

TABLE  IL 

Difference  of  Latitude  and  Departure  for  16  Degrees 

Dist 

Lat. 

Dep. 

Dist. 

Lat. 

Dcp. 

Dist. 

Lat. 

Dep. 

Dist 

Lat. 

Dep. 

Dist. 

Lat. 

Dep. 

I 

01 .0 

00.3 

61 

58.6 

16.8 

121 

116.3 

33.4 

181 

174.0 

49.9 

241 

23l  .7 

66.4 

2 

01 .9 

00.6 

62 

59.6 

17. 1 

22 

117.3 

33.6 

82 

174.9 

5o.2 

42 

232.6 

66.7 

3 

C  2  . 9  1  DO  .  8 

63 

60.6 

17-4 

23 

Ij8.2 

33.9 

83 

175.9 

5o.4 

43 

233.6 

67.0 

4 

o3.8 

01 .1 

64 

61.5 

17.6 

24 

119. 2 

34.2 

84 

176.9 

50.7 

44 

234.5 

67.3 

5 

04.8 

01 .4 

65 

63.5 

17.9 

25 

120.2 

34.5 

85 

177.8 

5i.o 

45 

235.5 

67.5 

6 

o5.8 

01.7 

66 

63.4 

18.2 

26 

121 .1 

34.7 

86 

178.8 

5i.3 

46 

236.5 

67.8 

7 

06.7 

01 .9 

67 

64.4 

18.5 

27 

122.1 

35.0 

87 

179.8 

5i.5 

47 

237.4 

68.1 

8 

07.7 

02.2 

68 

65.4 

.8.7 

28 

123.0 

35.3 

88 

180.7 

5i.8 

48 

238.4 

68.4 

9 

08.7 

02.5 

69 

66.3 

19.0 

29 

124.0 

35.6 

89 

181.7 

52.1 

49 

239.4 

68.6 

10 

09.6 

02.8 

70 

67.3 

19.3 

3o 

125. 0 

35.8 

90 

182.6 

52.4 
52.6 

5o 

240.3 

68.9 

II 

10.6 

o3.o 

71 

68.2 

19.6 

i3i 

125.9 

36.1 

191 

i83.6 

25l 

241.3 

69.2 

12 

II. 5 

o3.3 

72 

69.2 

19.8 

32 

126.9 

36.4 

92 

184.6 

52.9 

52 

242.2 

69.5 

i3 

12.5 

o3.6 

73 

70.2 

20.1 

33 

127.8 

36  7 

93 

185.5 

53.2 

53 

243.2 

69.7 

i4 

i3.5 

03.9 

74 

71. 1 

20.4 

34 

128.8 

36.9 

94 

186.5 

53.5 

54 

244.2 

70.0 

i5 

14.4 

04.1 

75 

.72.1 

20.7 

35 

129.8 

37.2 

95 

187.4 

53.7 

55 

245.1 

70.3 

i6 

1 5. 4 

04.4 

76 

73.1 

20.9 

36 

1 30.7 

37.5 

96 

188.4 

54.0 

56 

246.1 

70.6 

17 

i6.3 

04.7 

77 

74.0 

21  .2 

37 

i3i.7 

37.8 

97 

189.4 

54.3 

57 

247.0 

70.8 

i8 

17.3 

o5.o 

78 

75.0 

21.5 

38 

l32.7 

38. 0 

98 

190.3 

54.6 

58 

248.0 

71.1 

19 

18.3 

o5.2 

79 

75.9 

21.8 

39 

i33.6 

38.3 

99 

191 .3 

54.9 

59 

249.0 

71.4 

20 

19.2 

o5.5 

80 

76.9 

22.1 

4o 

i34.6 

38.6 

200 

192.3 

55.1 

60 

249.9 

71.7 

21 

20.2 

o5.8 

81 

77-9 

22.3 

i4i 

i35.5 

38.9 

201 

193.2 

55.4 

261 

250.9 

71.9 

22 

21. 1 

06.1 

82 

78.8 

22.6 

42 

i36.5 

39.1 

02 

194.2 

55.7 

62 

25l  .9 

72.2 

23 

22.1 

06.3 

83 

79.8 

22.9 

43 

137.5 

3q.4 

00 

195.1 

56.0 

63 

252.8 

72.5 

24 

23.1 

06.6 

84 

80.7 

23.2 

44 

i38.4 

39.7 

o4 

196. 1 

56.2 

64 

253.8 

72.8 

25  I  24.0 

06.9 

85 

81.7 

23.4 

45 

139.4 

4o.o 

o5 

197.1 

56.5 

65 

254.7 

73.0 

26)25.0 

07.2 

86 

82.7 

23.7 

46 

140.3 

40.2 

06 

198.0 

56.8 

66 

255.7 

73.3 

27  •■  26.0 

07.4 

87 

83.6 

24.0 

47 

i4i.3 

40.5 

07 

199.0 

57.1 

67 

256.7 

73.6 

28  1  26.9 

07.7 

88 

84.6 

24.3 

48 

142.3 

4o.8 

08 

199.9 

57.3 

68 

257.6 

73.9 

29127.9 

08.0 

89 

85.6 

24.5 

49 

143.2 

4i.i 

09 

200.9 

57.6 

69 

258.6 

74.1 

30J28.8 

08.3 

90 

86.5 

24.8 

5o 

144.2 

4i.3 

10 

201 .9 

57.9 

70 

259.5 

74.4 

3r 129.8 

08.5 

91 

87.5 

25.1 

i5i 

145.2 

4i.6 

211 

202.8 

58.2 

271 

260.5 

74.7 

32  130.8 

08.8 

92 

88.4 

25.4 

52 

146.1 

41.9 

12 

2o3.8 

58.4 

72 

261 .5 

75.0 

33i3i.7 

09.1 

93 

89.4 

25.6 

53 

i47-i 

42.2 

i3 

204.7 

58.7 

73 

262.4 

75.2 

34'32.7 

09.4 

94 

90.4 

25.9 

54 

148.0 

42.4 

i4 

205.7 

59.0 

74 

363.4 

75.5 

35.33.6 

09.6 

95 

91.3 

26.2 

55 

149.0 

42.7 

i5 

206.7 

59.3 

75 

264.3 

75.8 

36  134.6 

09.9 

96 

92.3 

26.5 

56 

i5o.o 

43.0 

16 

207.6 

59.5 

76 

265.3 

76.1 

37 

35.6 

10.2 

97 

93.2 

26.7 

57 

i5o.9 

43.3 

17 

208.6 

59.8 

77 

266.3 

76.4 

38 

36.5 

10.5 

98 

94.2 

27.0 

58 

i5i  .9 

43.6 

iS 

209.6 

60.1 

78 

267.2 

76.6 

3q 

37.5 

10.7 

99 

95.2 

27.3 

59 

i52.8 

43.8 

19 

210.5 

60.4 

79 

268.2 

76.9 

40 

38.5 

II  .0 

100 

96.1 

27.6 

60 

i53.8 

44.1 

20 

211 .5 

60.6 

80 

269.2 

77.2 

4i 

39.4 

II. 3 

lOI 

97.1 

27.8 

161 

154.8 

44.4 

221 

212.4 

60.9 

281 

270. 1 

77.5 

42 

40.4 

II. 6 

02 

98.0 

28.1 

62 

155.7 

44.7 

22 

2i3.4 

61 .2 

82 

271 .1 

77-7 

43 

4i.3 

II. 9 

o3 

99.0 

28.4 

63 

i56.7 

44.9 

23 

214.4 

61.5 

83 

272.0 

78.0 

44 

42.3 

12. 1 

04 

100. 0 

28.7 

64 

157.6 

45.2 

24 

2i5.3 

61.7 

84 

273.0 

78.3 

45 

43.3 

12.4 

o5 

100.9 

28.9 

65 

i58.6 

45.5 

25 

216.3 

62.0 

85 

274.0 

78.6 

46 

44.2 

12.7 

06 

lOI  .9 

29.2 

66 

159.6 

45.8 

26 

217.2 

62.3 

86 

274.9 

78.8 

47 

45.2 

i3.o 

07 

102.9 

29.5 

67 

160.5 

46. 0 

27 

218.2 

62.6 

87 

275.9 

79.1 

48 

46.1 

l3.2 

08 

io3.8 

29.8 

68 

161. 5 

46.3 

28 

219.2 

62.8 

88 

276.8 

79-4 

49 

47.1 

i3.5 

09 

104.8 

3o.o 

69 

162.5 

46.6 

29 

220.1 

63.1 

89 

277.8 

79-7 

5o 

48.1 

i3.8 

10 

io5.7 

3o.3 

70 

i63.4 

46.9 

3o 

221 .1 

63.4 

90 

278.8 

79-9 

5i 

49.0 

i4.i 

III 

106.7 

3o.6 

171 

164.4 

47-1 

23  I 

222.1 

63.7 

291 

279.7 

80.2 

52 

5o.o 

i4.3 

12 

107.7 

3o.9 

72 

i65.3 

47.4 

32 

223.0 

63.9 

92 

280.7 

80.5 

53 

50.9 

14.6 

i3 

108.6 

3i.i 

73 

166.3 

47.7 

33 

224.0 

64.2 

93 

2S1.6 

80.8 

54 

31.9 

14.9 

i4 

109.6 

3i.4 

74 

167.3 

48. 0 

34 

224.9 

64.5 

94 

282.6 

81.0 

55 

52.9 

l5.2 

i5 

110.5 

31.7 

75 

168.2 

48.2 

35 

225.9 

64.8 

95 

283.6 

81.3 

56 

53.8 

i5.4 

16 

III. 5 

32.0 

76 

169.2 

48.5 

36 

226.9 

65.1 

96 

284.5 

81.6 

57 

54.8 

i5.7 

17 

112.5 

32.2 

77 

170.1 

48.8 

37 

227.8 

65.3 

97 

285.5 

81.9 

58 

55.8 

16.0 

18 

ii3.4 

32.5 

78 

171. 1 

49.1 

38 

228.8 

65.6 

98 

286.5 

82.1 

5q 

56.7 

16.3 

19 

114.4 

32.8 

79 

172. 1 

49-3 

39 

229.7 

65.9 

99 

287.4 

82.4 

60 

67.7 

16.5 

20 

ii5.4 

33.1 
Lat. 

80 

173.0 

49.6 

40 

23o.7 

66.2 

3oo 

288.4 

82.7 

Uist. 

Dop. 

I,at. 

Dist. 

Dep. 

Dist. 

Dep.      Lat.  | 

Dist. 

Dep. 

Lat. 

Dist.j 

Dep. 

Lat. 

0 

For  74  Degrees. 

TABLE  IL 

1  Page  33 

Difference  of  Latitude  and  Depart 

ure  for  17  Degrees. 

Dist. 

Lat. 

Dep. 

Dist. 

Lat. 

Dep. 

Dist. 

Lat. 

Dep. 

Dist. 

Lat. 

Dep. 

Dist. 

Lat. 

Dep. 

I 

01 .0 

00.3 

61 

58.3 

17.8 

121 

115.7 

35.4 

181 

173. 1 

52.9 

241 

23o.5 

70.5 

2 

01 .9 

00.6 

62 

59.3 

18.1 

22 

116.7 

35.7 

82 

174.0 

53.2 

42 

231.4 

70.8 

3 

02.9 

00.9 

63 

60.2 

18.4 

23 

117. 6 

3b. 0 

83 

175.0 

53.5 

4-^ 

232.4 

71.0 

4 

o3.8 

01 .2 

64 

61 .2 

18.7 

24 

118.6 

36.3 

84 

176.0 

53.8 

44 

233.3 

71.3 

5 

04.8 

01.5 

65 

62.2 

19.0 

25 

119.5 

36.5 

85 

176.9 

54.1 

45 

23^1.3 

71.6 

6 

00.7 

01.8 

66 

63.1 

19.3 

26 

120.5 

36.8 

86 

177-9 

54.4 

46 

235.3 

71.9 

7 

06.7 

02.0 

67 

64.1 

19.6 

27 

121 .5 

37.1 

87 

178.8 

54.7 

47 

236.2 

72.2 

8 

07.7 

02.3 

b8 

65.0 

19.9 

28 

122.4 

37.4 

88 

179.8 

55.0 

48 

237.2 

72.5 

9 

08.6 

02.6 

69 

66.0 

20.2 

29 

123.4 

37.7 

89 

180.7 

55.3 

49 

238.1 

72.8 

10 

09.6 

02.9 

70 

66.9 

20.5 

3o 

124.3 

38. 0 

90 

181.7 

55.6 

5o 

239.1 

73.1 

II 

10.5 

o3.2 

71 

67.9 

20.8 

i3i 

125.3 

38.3 

191 

182.7 

55.8 

25l 

240.0 

73.4 

12 

II. 5 

o3.5 

72 

68.9 

21 .1 

32 

126.2 

38.6 

92 

i83.6 

56.1 

52 

241 .0 

73.7 

i3 

12.4 

o3.8 

73 

69.8 

21.3 

33 

127.2 

38.9 

93 

184.6 

56.4 

53 

241.9 

74.0 

i4 

i3.4 

04.1 

74 

70.8 

21 .6 

34 

128. 1 

39.2 

94 

iS5.5 

56.7 

54 

242.9 

74.3 

i5 

i4.3 

04.4 

76 

71-7 

21.9 

35 

129.1 

39.5 

95 

186.5 

57.0 

55 

243.9 

74.6 

i6 

i5.3 

04.7 

76 

72.7 

22.2 

36 

i3o.i 

39.8 

96 

187.4 

57.3 

56 

244.8 

74.8 

17 

16.3 

o5.o 

77 

73.6 

22.5 

37 

i3i  .0 

4o.i 

97 

188.4 

57.6 

57 

245.8 

75.1 

i8 

17.2 

o5.3 

7» 

74.6 

22.8 

38 

l32.0 

40.3 

98 

189.3 

57.9 

58 

246.7 

75.4 

19 

18.2 

o5.6 

79 

75.5 

23.1 

39 

132.9 

40.6 

99 

190.3 

58.2 

59 

247-7 

75.7 

20 

19.1 

o5.8 

80 

76.5 

23.4 

4o 

133.9 

40.9 
4i  .2 

200 
201 

191 .3 
192.2 

58.5 
58.8 

60 

248.6 

76.0 

21 

20. 1 

06.1 

81 

77.5 

23.7 

i4i 

i34.8 

261 

249.6 

76.3 

22 

21 .0 

06.4 

82 

78.4 

24.0 

42 

i35.8 

4i.5 

02 

193.2 

59.1 

62 

25o.6 

76.6 

23 

22.0 

06.7 

83 

79-4 

24.3 

43 

i36.8 

4i.8 

o3 

194.1 

59.4 

63 

251.5 

76.9 

24 

23.0 

07.0 

84 

80.3 

24.6 

44 

137.7 

42.1 

04 

195. 1 

59.6 

64 

252.5 

77.2 

25 

23.9 

07.3 

85 

81.3 

24.9 

45 

i38.7 

42.4 

o5 

196.0 

59.9 

65 

253.4 

77.5 

26 

24.9 

07.6 

86 

82.2 

25.1 

46 

139.6 

42.7 

06 

197.0 

60.2 

66 

254.4 

77.8 

27 

25.8 

07.9 

87 

83.2 

25.4 

47 

i4o.6 

43.0 

07 

198.0 

60.5 

67 

255.3 

78. 1 

28 

26.8 

08.2 

88 

84.2 

25.7 

48 

i4i.5 

43.3 

08 

198.9 

60.8 

68 

256.3 

78.4 

29 

27-7 

08.5 

89 

85.1 

26.0 

49 

142.5 

43.6 

09 

199.9 

61.1 

69 

257.2 

78.6 

3o 

28.7 

08.8 

90 

86.1 

26.3 

5o 

143.4 

43.9 

10 

200.8 

61.4 

70 

258.2 

78.9 

3i 

29.6 

09.1 

91 

87.0 

26.6 

i5i 

144.4 

44.1 

211 

201.8 

61.7 

271 

259.2 

79.2 

32 

3o.6 

09.4 

92 

88.0 

26.9 

52 

145.4 

44.4 

12 

202.7 

62.0 

72 

260.1 

79.5 

33 

3i.6 

09.6 

93 

88.9 

27.2 

53 

i46.3 

44.1 

i3 

2o3.7 

62.3 

73 

261. 1 

79.8 

34 

32.5 

09.9 

94 

89.9 

27.5 

54 

147.3 

45.0 

i4 

2o4.6 

62.6 

74 

262.0 

80.1 

35 

33.5 

10.2 

95 

90.8 

27.8 

55 

148.2 

45.3 

i5 

2o5.6 

62.9 

75 

263.0 

80.4 

3G 

34.4 

10.5 

96 

91.8 

28.1 

56 

149.2 

46.6 

16 

206.6 

63.2 

76 

263.9 

80.7 

37 

35.4 

10.8 

97 

92.8 

28.4 

57 

1 5o .  I 

45.9 

17 

207.5 

63.4 

77 

264.9 

81.0 

38 

36.3 

II  .1 

98 

93.7 

28.7 

58 

i5i.i 

46.2 

18 

208.5 

63.7 

78 

265.9 

81.3 

39 

37.3 

II. 4 

99 

94-7 

28.9 

59 

l52.1 

46.5 

19 

209.4 

64.0 

79 

266.8 

81.6 

4o 

38.3 

II. 7 

100 

95.6 

29.2 

60 

i53.o 

4b. 8 

20 

210.4 

64.3 

80 

267.8 

81.9 

4i 

39.2 

12.0 

101 

96.6 

29.5 

161 

1 54.0 

47.1 

221 

211 .3 

64.6 

281 

268.7 

82.2 

42   40.2 

12.3 

02 

97.5 

29.8 

62 

154.9 

47.4 

22 

212.3 

64.9 

82 

269.7 

82.4 

43  4i.i 

12.6 

o3 

98.5 

3o.i 

63 

155.9 

47.7 

23 

2i3.3 

65.2 

83 

270.6 

82.7 

44 

42.1 

12.9 

04 

99.5 

3o.4 

64 

i56.8 

47-9 

24 

214.2 

65.5 

84 

271 .6 

83.0 

45 

43.0 

l3.2 

o5 

100.4 

30.7 

65 

157.8 

48.2 

25 

2l5.2 

65.8 

85 

272.5 

83.3 

46 

44.0 

i3.4 

06 

101.4 

3i  .0 

6-6 

i58.7 

48.5 

26 

216. I 

66.1 

86 

273.5 

83.6 

47 

44.9 

i3.7 

07 

102.3 

3i.3 

67 

159.7 

43.8 

27 

217. I 

66.4 

87 

274.5 

83.9 

48 

45.9 

14.0 

08 

io3.3 

3i.6 

68 

160.7 

49.1 

28 

218.0 

66.7 

88 

275.4 

84.2 

49 

46.9 

14.3 

09 

104.2 

3i  .9 

69 

161.6 

49-4 

29 

219.0 

67.0 

89 

276.4 

84.5 

5o 

47.B 

i4.b 

10 

io5  2 

32.2 

70 

162.6 

49.7 

3o 

220.0 

67.2 

90 

277.3 

84.8 

5i 

48.8 

14.9 

III 

ic6. 1 

32.5 

171 

163.5 

5o.o 

23l 

220.9 

67.5 

291 

278.3 

85.1 

52 

49-7 

l5.2 

12 

107. 1 

32.7 

72 

164.5 

5o.3 

32 

221  .9 

67.8 

92 

279.2    85.4 

53 

5o.7 

i5.5 

i3 

108.1 

33.0 

73 

i65.4 

5o.6 

33 

222.8 

68.1 

93 

280.2    85.7 

54 

5i.6 

[5.8 

i4 

109.0 

33.3 

74 

166.4 

50.9 

34 

223.8 

68.4 

94 

281.2 

86.0 

53 

52. f) 

16.1 

i5 

IIO.O 

33.6 

75 

167.4 

5l.2 

35 

224.7 

68.7 

95 

282.1 

86.2 

56 

53.6 

16.4 

16 

110.9 

33.9 

76 

168.3 

5i.5 

36 

225.7 

69.0 

g6 

283.1 

86.5 

b7 

54.5 

16.7 

17 

III  .9 

34.2 

77 

169.3 

5i.7 

37    226.6 

69.3 

97 

284.0 

86.8 

d8 

t)5.3 

17.0 

18 

112.8 

34.5 

78 

170.2 

52.0 

38 

227.6 

69.6 

98 

285.0 

87.1 

59 

5b. 4 

17.2 

19 

ii3.8 

34.8 

79 

171 .2 

52.3 

39 

228.6 

69.9 

99 

285.9 

87.4 

bo 
I)ist. 

57.4 

17.5 

20 

114.8 

35.1 

80 
Disl. 

172. 1 

52.6 

40 

2?q.5 

70.2 

3oo 

286.9 

87.7 

Dep. 

Lr.t. 

Disl. 

Dep. 

Lat. 

Dop. 

Lat. 

Disl. 

■Dep. 

Lat. 

Dist. 

Dep. 

Lat. 

[1 

''or  73  Degrees. 

I";i;,'e  :U 

TABLE  IL 

Dilference  of  Latitude  and  Departure  for  13  Degrees. 

Dist 

Lai. 

Oep. 
00.3 

Dlsl. 

Lat. 

D-.p. 

Dist.]    Lat. 

Dep. 

Disl. 

Lat. 

Dep. 

Dist. 

Lat.    1  D,.p. 

I 

01  .0 

61 

58. 0 

18.9 

121  1  1 15. 1 

37.4 

181 

172. 1 

55.9 

241 

229.2 

74.5 

2 

01.9 

00.6 

b2 

59.0 

19.2 

22 

116.0 

37.7 

82 

173. 1 

56.2 

42 

23o.2 

74.8 

d 

02.9 

00.9 

M 

59.9 

i9.b 

23 

117. 0 

38. 0 

83 

174.0 

56.6 

43 

23l  .1 

75.1 

4 

o3.8 

01  .2 

b4 

60.9 

19.8 

24 

117. 9 

38.3 

84 

175.0 

56.9 

44 

232.1 

75.4 

i) 

o4.8 

01. b 

bb 

61.8 

20.1 

2b 

118. 9 

38.6 

85 

175.9 

57.2 

45 

233.0 

75.7 

b 

Ob.  7 

01 .9 

bb 

62.8 

20.4 

26 

119.8 

38.9 

86 

176.9 

57.5 

46 

234.0 

76.0 

7 

06.7 

02.2 

b7 

63.7 

20.7 

27 

120.4 

39.2 

87 

177.8 

57.8 

47 

234.9 

76.3 

8 

07.6 

02. b 

b8 

64.7 

21 .0 

28 

121  .7 

39.6 

88 

178.8 

58.1 

48 

235.9 

76.6 

9 

08.6 

02.8 

b9 

65.6 

21.3 

29 

122.7 

39.9 

89 

179-7 

58.4 

49 

236.8 

76.9 

lO 

09.5 

o3. 1 

70 

66.6 

21. b 
21.9 

3o 

123.6 

4o.2 

90 
191 

180.7 
181. 7 

58. 7 
59.0 

5o 

237.8 

77-3 

II 

10.5 

o3.4 

71 

67.5 

i3i 

124.6 

40.5 

25l 

238.7 

77-6 

12 

II. 4 

o3.7 

72 

68.5 

22.2 

32 

125.5 

4o.8 

92 

182.6 

59.3 

52 

239.7 

77-9 

i3 

12.4 

04.0 

7'i 

69.4 

22.6 

33 

126.5 

4i.i 

93 

i83.6 

59.6 

53 

240.6 

78.;. 

i4 

i3.3 

04.3 

74 

70.4 

22.9 

34 

127.4 

41.4 

94 

184.5 

59-? 

54 

241.6 

78.5 

15 

14.3 

04. b 

7b 

71.3 

23.2 

35 

128.4 

41.7 

95 

185.5 

60.3 

55 

242.5 

78.8 

i6 

lb. 2 

04.9 

76 

72.3 

23.5 

36 

129.3 

42.0 

96 

186.4 

60.6 

56 

243.5 

79.1 

17 

16.2 

ob.3 

77 

73.2 

23.8 

37 

i3o.3 

42.3 

97 

187.4 

60.9 

57 

244.4 

79-4 

i8 

17. 1 

ob.b 

78 

74.2 

24.1 

38 

i3i  .2 

42.6 

98 

188.3 

61 .2 

58 

245.4 

79-7 

19 

18. 1 

05.9 

79 

7b. I 

24.4 

39 

\3i.i 

43.0 

99 

189.3 

61.5 

59 

246.3 

80.0 

20 

19.0 

06.2 
06.5 

80 

7b.  I 

24.7 

40 

i33.i 

43.3 

200 

190.2 

61.8 

60 

247-3 

80.3 

21 

20.0 

81 

77.0 

25.0 

i4i 

i34.i 

43.6 

201 

191 .2 

62.1 

261 

248.2 

80.7 

22 

20.9 

06.8 

82 

78.0 

2b. 3 

42 

i35.i 

43.9 

02 

192. 1 

62.4 

62 

249.2 

81.0 

23 

21 .9 

07.1 

83 

78.9 

23.6 

43 

1 36.0 

44.2 

o3 

193.1 

62.7 

63 

25o.i 

81.3 

24 

22.6 

07.4 

84 

79-9 

26.0 

44 

137.0 

44.5 

04 

194.0 

63. 0 

H 

25l  .  I 

81.6 

2b 

23.8 

07.7 

8b 

80.8 

26.3 

45 

137.9 

44.8 

o5 

195.0 

63.3 

65 

252.0 

81.9 

2b 

24.7 

08.0 

86 

81.8 

26.6 

46 

13S.9 

45.1 

06 

195.9 

63.7 

66 

253.0 

82.2 

27 

2b. 7 

08.3 

87 

82.7 

26.9 

4i 

139.8 

45.4 

07 

196.9 

64.0 

67 

253.9 

82.5 

28 

26.6 

08.7 

88 

83.7 

27.2 

48 

140.8 

4b. 7 

08 

197-8 

64.3 

68 

254.9 

82.8 

29 

27.6 

09.0 

89 

84.6 

27.5 

49 

141.7 

46.0 

09 

198.8 

64.5 

69 

255.8 

83.1 

Jo 

28. b 

09.3 

90 

85.6 

27.8 

5o 

142.7 

46.4 

10 

199.7 

64-9 

70 

256.8 

83.4 

3i 

29.5 

09.6 

91 

86.5 

28.1 

i5i 

143.6 

46.7 

211 

200.7 

65.2 

271 

257.7 

83.7 

32 

3o.4 

09.9 

92 

87.5 

28.4 

52 

144.6 

47-0 

12 

201 .6 

65.5 

72 

258.7 

84.1 

:i6 

3i.4 

10.2 

93 

88.4 

28.7 

53 

145.5 

47.3 

i3 

202.6 

65.8 

73 

259.6 

84.4 

34 

32.3 

10.5 

94 

89.4 

29.0 

54 

146.5 

47-6 

i4 

2o3.5 

66.1 

74 

260.6 

84.7 

3!) 

dd.d 

10.8 

9b 

90.4 

29-4 

55 

147-4 

47-9 

i5 

204.5 

66.4 

75 

261.5 

85.0 

3b 

34.2 

II  .1 

9b 

91.3 

29.7 

56 

148.4 

48.2 

16 

2o5.4 

66.7 

76 

262.5 

S5.3 

^7 

3b. 2 

II. 4 

97 

92.3 

3o.o 

57 

149-3 

48.5 

17 

206.4 

67.1 

77 

263.4 

85.6 

38 

3b. I 

II. 7 

98 

93.2 

3o.3 

58 

i5o.3 

48.8 

18 

207.3 

67-4 

78 

264.4 

85.9 

39 

37.1 

12. 1 

99 

94.2 

3o.6 

59 

i5i  .2 

49. 1 

19 

208.3 

67.7 

79 

265.3 

86.2 

40 

38. 0 

12.4 

100 

9b.; 

30.9 

3l.2 

6(j 

l52.2 

49-4 

20 

209.2 

68.0 
68.3 

80 

266.3 

86.5 
86.3 

4i 

39.0 

12.7 

lOI 

96.1 

161 

153.1 

49.8 

221 

210.2 

281 

267  .2 

42 

39.9 

i3.o 

02 

97.0 

3i.5 

62 

i54.i 

5o.i 

22 

211.1 

68.6 

82 

268.2 

87.1 

4i 

40.9 

i3.3 

o3 

98.0 

3i.8 

63 

i55.o 

5o.4 

23 

212.1 

68. 9 

83 

269.1 

87.5 

44 

4i.8 

i3.6 

04 

98.9 

32.1 

64 

i56.o 

50.7 

24 

2l3.0 

69.2 

84 

270. 1 

87.8 

4b 

42.8 

i3.9 

ob 

99.9 

32.4 

65 

i56.9 

5i.o 

25 

214.0 

69.5 

85 

271 .1 

88. 1 

4b 

43.7 

14.2 

06 

100.8 

32.8 

66 

.57.Q 

5i.3 

26 

214.9 

69.8 

86 

272.0 

88.4 

47 

44.7 

14.5 

07 

101.8 

33.1 

67 

i5S.8 

5i.6 

27 

215.9 

70.1 

87 

273.0 

88.7 

48 

4b. 7 

14.8 

08 

102.7 

33.4 

68 

159.8 

5i.9 

28 

216.8 

70.5 

88 

273.9 

89.0 

49 

4b.  b 

16. 1 

09 

103.7 

33.7 

69 

160.7 

52.2 

29 

217.8 

70.8 

89 

274.9 

89.3 

bo 
5i 

47.b 

ib.b 

10 

104.6 

34.0 

70 

161 .7 

52.5 
52.8 

3o 

23  I 

218.7 

71. 1 

90 
291 

275.8 
276  8 

89.6 

48.5 

i5.8 

II I 

io5.6 

34.3 

171 

162.6 

219.7 

71.4 

89.9 

ba 

49-i) 

lb. I 

12 

106.5 

34.6 

72 

i63.6 

53.2 

32 

220.6 

71.7 

93 

277-7 

90.2 

bd 

5o.4 

lb. 4 

i3 

107.5 

34.9 

73 

164.5 

53.5 

33 

221  .6 

72.0 

93 

278.7 

90.5 

64 

bi.4 

lb. 7 

i4 

108.4 

35.2 

74 

i65.5 

53.8 

34 

222.5 

72.3 

94 

279.6 

90.9 

bb 

b2.3 

17.0 

lb 

109.4 

35.5 

75 

166.4 

-M-i 

35 

223.5 

72.6 

95 

280.6 

91.2 

bb 

b3.3 

.7.3 

lb 

no. 3 

35.8 

76 

167.4 

54.4 

36 

224.4 

72.9 

96 

281.5 

91.5 

b7 

b4.2 

.7.b 

17 

III  .3 

36.2 

77 

168.3 

54.7 

37 

225.4 

73.2 

97 

282.5 

91.8 

b8 

bb.2 

17.9 

18 

1 12.2 

36.5 

78 

169.3 

55.0 

38 

226.4 

73.5 

98 

283.4 

92.1 

b9 

b6.i 

18.2 

•9 

I  l3.2 

36.8 

79 

170.2 

55.3 

39 

227.3 

73.9 

99 

284.4 

92.4 

bo 

b7.i 

18. b 

20 

114.1 

37.1 

80 

171 .2 

55.6 

4o 
Dist. 

228.3 

Dep. 

74-2 
Lat. 

3  00 

285.3 

92.7 

Dist. 

De,,. 

Lat. 

Dist 

Do  p. 

Lai. 

Dist. 

Dop. 

Lat. 

Dist. 

Dep. 

Lat. 

[1 

^or  72  Degrees. 

TABLE  II. 

[Page  35 

Difference  of  Lat 

tude  and  Departure  for  19  Degrees. 

Dist 

Lat. 

Dep. 

00.3 

Dist. 

Lat. 

Do  p. 

Dist. 

Lat. 

Dcp. 

39.4 

Dist. 

Lat. 

Dep. 

Dist. 

Lat. 

Dep 

I 

00.9 

61 

57-7 

19.9 

121 

114.4 

181 

171. 1 

58.9 

241 

227.9 

78.5 

2 

01.9 

00.7 

62 

58.6 

20.2 

22 

ii5.4 

39.7 

82 

172. 1 

59.3 

42 

228.8 

78.8 

3 

02.8 

01 .0 

63 

59.6 

20. D 

23 

116. 3 

4o.o 

83 

173.0 

59.6 

43 

229.8 

79.1 

4 

o3.8 

01 .3 

64 

60.5 

20.8 

24 

117. 2 

40.4 

84 

174.0 

59.9 

AA 

230.7 

79-4 

5 

04.7 

01 .6 

65 

61.5 

21  .2 

25 

118.2 

40.7 

85 

174.9 

60.2 

45 

23l  .7 

79.8 

6 

o5.7 

02.0 

66 

62.4 

21.5 

26 

119. 1 

4i  .0 

86 

175.9 

60.6 

46 

232.6 

80.1 

7 

06.6 

02.3 

67 

63.3 

21.8 

27 

120. 1 

4i.3 

87 

176.8 

60.9 

47 

233.5 

80.4 

8 

07.6 

02.6 

68 

64.3 

22.1 

28 

121 .0 

41.7 

88 

177.8 

61 .2 

48 

234.5 

80.7 

9 

08.5 

02.9 

69 

65.2 

22.5 

29 

122.0 

42.0 

89 

178.7 

61.5 

49 

235.4 

81. 1 

10 

09.5 

o3.3 

70 

66.2 

22.8 

3o 

122.9 

42.3 

90 

179.6 

61 .9 

5o 

236.4 

81.4 
81.7 

II 

10.4 

o3.6 

71 

67.1 

23.1 

i3i 

123.9 

42.6 

191 

180.6 

62.2 

25l 

237.3 

12 

II. 3 

03.9 

72 

68.1 

23.4 

32 

124.8 

43.0 

92 

181.5 

62.5 

52 

238.3 

82.0 

i3 

12.3 

o4.2 

73 

69.0 

23.8 

33 

125.8 

43.3 

93 

182.5 

62.8 

53 

239.2 

82.4 

i4 

l3.2 

04.6 

74 

70.0 

24.1 

M 

126.7 

43.6 

94 

i83.4 

63.2 

54 

240.2 

82.7 

i5 

l4.2 

04.9 

75 

70.9 

24.4 

35 

127.6 

44.0 

95 

184.4 

63.5 

55 

241 .1 

53.0 

i6 

i5.i 

05.2 

76 

71.9 

24.7 

36 

128.6 

AA.'i 

96 

i85.3 

63.8 

56 

242.1 

83.3 

17 

16. 1 

o5.5 

77 

72.8 

25.1 

37 

129.5 

44.6 

97 

186.3 

64.1 

57 

243.0 

83.7 

i8 

17.0 

05.9 

78 

73.8 

25.4 

38 

i3o.5 

44-9 

98 

187.2 

64.5 

58 

243.9 

84.0 

'9 

18.0 

06.2 

79 

74.7 

25.7 

39 

i3i.4 

45.3 

99 

188.2 

64.8 

59 

244.9 

84.3 

20 

18.9 

06.5 

80 

75.6 

26.0 

40 

i32.4 

45.6 

200 

189. 1 

65.1 

60 

245.8 

84.6 
85. 0 

21 

19.9 

06.8 

81 

76.6 

26.4 

i4i 

i33.3 

45.9 

201 

190.0 

65.4 

261 

246.8 

22 

20.8 

07.2 

82 

77.5 

26.7 

42 

134.3 

46.2 

02 

191 .0 

65.8 

62 

247.7 

85.3 

23 

21.7 

07.5 

83 

78.5 

27.0 

A'i 

i35.2 

46.6 

OJ 

191. 9 

66.1 

63 

248.7 

85.6 

24 

22.7 

07.8 

84 

79-4 

27.3 

AA 

i36.2 

46.9 

o4 

192.9 

66.4 

64 

249.6 

86.0 

25 

23.6 

08.1 

85 

80.4 

27.7 

45 

137. 1 

47-2 

o5 

193.8 

66.7 

65 

25o.6 

86.3 

26 

24.6 

08.5 

86 

81.3 

28.0 

46 

i38.o 

47.5 

06 

194.8 

67.1 

66 

251.5 

86.6 

27 

25.5 

08.8 

87 

82.3 

28.3 

47 

139.0 

47-9 

07 

195.7 

67.4 

67 

252.5 

86.9 

38 

26.5 

09.: 

88 

83.2 

28.7 

48 

139.9 

48.2 

08 

196.7 

67.7 

68 

253.4 

87.3 

29 

27.4 

09.4 

89 

84.2 

29.0 

49 

140.9 

48.5 

09 

197.6 

68.0 

69 

254.3 

87.6 

3o 

28.4 

09.8 

90 

85.1 

29.3 

5o 

i4i.8 

48.8 

10 

198.6 

68.4 

70 

255.3 

87.9 
88.2 

3i 

29.3 

10. 1 

9' 

86.0 

29.6 

i5i 

142.8 

49.2 

211 

199.5 

68.7 

271 

256.2 

32 

3o.3 

10.4 

92 

87.0 

3o.o 

52 

143.7 

49.5 

12 

200.4 

69.0 

72 

257.2 

88.6 

33 

3l.2 

10.7 

93 

87.9 

3o.3 

53 

144.7 

49-8 

i3 

201 .4 

69.3 

73 

258.1 

88.9 

34 

32.1 

II  .1 

94 

88.9 

3o.6 

54 

145.6 

5o.i 

i4 

202.3 

69.7 

74 

259.1 

89.2 

35 

33.1 

II. 4 

95 

89. 8 

3o.9 

55 

1.46.6 

5o.5 

i5 

2o3.3 

70.0 

75 

260.0 

89.5 

36 

34.0 

II. 7 

96 

90.8 

3i.3 

56 

147-5 

5o.8 

16 

204.2 

70.3 

76 

261.0 

89.9 

3? 

35.0 

12.0 

97 

91.7 

3i.6 

57 

148.4 

5i.i 

17 

205.2 

70.6 

77 

261.9 

90.2 

38 

35.9 

12.4 

98 

92.7 

3i  .9 

58 

149.4 

5i.4 

18 

206. 1 

71.0 

78 

262.9 

90.5 

39 

36  9 

12.7 

99 

93.6 

32.2 

59 

i5o.3 

5i.8 

19 

207.1 

71.3 

79 

263.8 

90.8 

40 
4i 

37.8 

i3.o 

100 

94.6 

32.6 

60 

i5i.3 

52.1 

20 

208.0 

71.6 

80 

264.7 

91.2 

38.8 

i3.3 

lOI 

95.5 

32.9 

161 

l52.2 

52.4 

221 

209.0 

72.0 

281 

265.7 

91.5 

42 

39.7 

i3.7 

02 

96.4 

33.2 

62 

i53.2 

52.7 

22 

209.9 

72.3 

82 

266.6 

91.8 

43 

40.7 

i4.o 

o3 

97-4 

33.5 

63 

i54.i 

53.1 

23 

210.9 

72.6 

83 

267.6 

92.1 

^i 

4i.6 

14.3 

o4 

98.3 

33.9 

64 

i55.i 

53.4 

24 

211. 8 

72.9 

84 

268.5 

92.5 

45 

42.5 

14.7 

o5 

99.3 

34.2 

65 

i56.o 

53.7 

25 

212.7 

73.3 

85 

269.5 

92.8 

46 

43.5 

i5.o 

06 

100.2 

34.5 

66 

157.0 

54.0 

26 

213.7 

73.6 

86 

270.4 

93.1 

^1 

44.4 

i5.3 

07 

101 .2 

34.8 

67 

157.9 

54.4 

27 

214.6 

73.9 

87 

271 .4 

93.4 

48 

45.4 

i5.6 

08 

102. 1 

35.2 

68 

i58.8 

54.7 

28 

2i5.6 

74.2 

88 

272.3 

93.8 

49 

46.3 

16.0 

09 

io3.i 

35.5 

69 

159.8 

55.0 

29 

216.5 

74.6 

89 

273.3 

94.1 

5o 

47.3 

16.3 

10 

104.0 

35.8 

70 

160.7 

55.3 

3o 

217.5 

74.9 

90 
291 

274.2 
275.1 

94.4 
94-7 

5i 

48.2 

16.6 

III 

loS.o 

36.1 

171 

161 .7 

55.7 

23l 

218.4 

75.2 

52 

49.2 

16.9 

12 

105.9 

36.5 

72 

162.6 

56.0 

32 

219.4 

75.5 

92 

276.1 

95.1 

53 

5o.i 

17.3 

i3 

106.8 

36.8 

73 

i63.6 

56.3 

33 

220.3 

75.9 

93 

277.0 

95.4 

54 

5i.i 

17.6 

i4 

107.8 

37.1 

74 

164.5 

56.6 

34 

221  .3 

76.2 

94 

278.0 

95.7 

55 

52.0 

17.9 

i5 

108.7 

37.4 

75 

i65.5    57.0  1 

35 

222.2 

76.5 

95 

278.9 

96.0 

56 

52.9 

I«.2 

16 

109.7 

37.8 

76 

166.4 

57.3 

36 

223.1 

76.8 

96 

279.9 

96.4 

■^1 

5J.9 

18.6 

17 

no. 6 

38.1 

77 

167.4 

57.6 

37 

224.1 

77.2 

97 

280.8 

96.7 

58 

54.8 

18.9 

18 

II  1. 6 

38.4 

78 

168.3 

58.0 

38 

225.0 

77.5 

98 

281.8 

97.0 

59 

55.8 

19.2 

•9 

112. 5 

38.7 

79 

169.2 

58.3 

39 

226.0 

77.8 

99 

282.7 

97.3 

ho 

56.7 

19.5 

20 

ii3.5 

39.1 

80 

170.2 

58.6 

40 

226.9 

78.1 

3oo 

283.7 

97-7 

Disi. 

])e|).      I.at. 

nisi. 

Dcp. 

Lat. 

Dist. 

Dcp. 

Lat. 

Dist. 

Dcp. 

Lat. 

Dist. 

Dep. 

Lat. 

[For  71  Degrees. 

I'age  3G] 

TABLE  IL 

Difference  of  Latitude  and  Departure  for  20  Degrees. 

Dist. 

Lat. 

Dep. 

Dist. 

Lat. 

Dep. 

Dist. 

Lat. 

Dep. 

D.st. 

Lat. 

Dep. 

Dist.     Lat. 

Dep. 

I 

00.9 

00.3 

61 

57.3 

20.9 

121 

ii3.7 

4i.4 

181 

170.1  1 

61.9 

241     226.5 

82.4 

2 

01 .9 

00.7 

62 

58.3 

21 .2 

22 

114.6 

41.7 

82 

171. 0  1  62.2 

42    227.4 

82.8 

3 

02.8 

01 .0 

63 

59.2 

21.5 

23 

ii5.6 

42.1 

83 

172.0  1 62  6 

43 '228.3 

83.1 

4 

o3.8 

01 .4 

64 

60.1 

21  .9 

24 

116. 5 

42.4 

84 

172.9 

62.9 

AA   229.3 

83.5 

5 

04.7 

01.7 

65 

61. 1 

22.2 

25 

117. 5 

42.8 

85 

173.8 

63  3 

45     23o.2 

83.8 

6 

o5.6 

02.1 

66 

62.0 

22.6 

26 

118. 4 

43.1 

86 

174.8 

63.6 

46 

23l  .2 

84.1 

7 

06.6 

02.4 

67 

63.0 

22.9 

27 

1 19.3 

43.4 

87 

175.7 

64.0 

47 

232.1 

84.5 

8,  f.7.5 

02.7 

68 

63.9 

23.3 

28 

120.3 

43.8 

88 

176.7 

64.3 

48 

233.0 

84.8 

P 

08.5 

o3.i 

69 

64.8 

23.6 

29 

121 .2 

44.1 

89 

177.6 

64.6 

49 

234.0 

85.2 

10 

09.4 

o3.4 

70 

65.8 

23.9 

3o 

122.2 

U.b 

90 

178.5 

65.0 
65.3 

5o 

234.9 

85.5 

II 

10.3 

o3.8 

71 

66.7 

24.3 

i3i 

123. 1 

44.8 

191 

179.5 

25l 

235.9 

85.8 

12 

II. 3 

04.1 

72 

67.7 

24.6 

32 

124.0 

45.1 

92 

180.4 

65.7 

52 

236.8 

86.2 

i3 

12.2 

04.4 

73 

68.6 

25.0 

33 

125.0 

45.5 

93 

181.4 

66.0 

53 

237.7 

86.5 

14 

l3.2 

o4.8 

74 

69.5 

25.3 

34 

125.9 

45.8 

94 

182.3 

66.4 

54 

238.7 

86.9 

i5 

i4.i 

o5.i 

75 

70.  D 

25.7 

35 

126.9 

46.2 

95 

i83.2 

66.7 

55 

239.6 

87.2 

16 

i5.o 

o5.5 

76 

71.4 

26.0 

36 

127.8 

46.5 

96 

184.2 

67.0 

56 

240.6 

87.6 

17 

16. G 

o5.8 

.77 

72.4 

26.3 

37 

128.7 

46.9 

97 

i85.i 

67.4 

57 

241.5 

87.9 

18 

16.9 

06.2 

78 

73.3 

26.7 

38 

129.7 

47.2 

98 

186.1 

67.7 

58 

242.4 

88.2 

19 

17.9 

06.5 

79 

74.2 

27.0 

39 

i3o.6 

47-5 

99 

187.0 

68.1 

59 

243.4 

88.6 

20 

18.8 

06.8 

80 

75.2 

27.4 

4o 

i3i.6 

47-9 

200 

187.9 

68.4 

60 

244.3 

88.9 

21 

19.7 

07.2 

81 

76.1 

27.7 

i4i 

i32.5 

48.2 

201 

188.9 

68.7 

261 

245.3 

89.3 

22 

20.7 

07.5 

82 

77.1 

28.0 

42 

i33.4 

48.6 

02 

189.8 

69.1 

62 

246.2 

89.6 

23 

21.6 

07.9 

83 

78.0 

28.4 

43 

134.4 

48.9 

o3 

190.8 

69.4 

63 

247-1 

90.0 

24 

22.6 

08.2 

84 

78.9 

28.7 

^A 

i35.3 

49-3 

04 

191.7 

69.8 

64 

24s.  I 

90.3 

25 

23.5 

08.6 

85 

79-9 

29. 1 

45 

i36.3 

49.b 

o5 

192.6 

70.1 

65 

249.0 

90.6 

26 

24.4 

08.9 

86 

80.8 

29.4 

46 

137.2 

49.9 

06 

193.6 

70.5 

66 

25o.o 

91.0 

27 

25.4 

09.2 

87 

81.8 

29.8 

47 

i38.i 

5o.3 

07 

194.5 

70.8 

67 

250.9 

91.3 

28 

26.3 

09.6 

88 

82.7 

3o.i 

48 

139. 1 

5o.6 

08 

195.5 

71.1 

68 

25i.8 

91.7 

29 

27.3 

09.9 

89 

83.6 

3o.4 

49 

i4o.o 

5i.o 

09 

196.4 

71.5 

69 

252.8 

92.0 

3o 

28.2 

10.3 

90 

84.6 

3o.8 

5o 

i4i  .0 

5i.3 

10 

197.3 

71.8 

70 

253.7 

92.3 

3x 

29.1 

10.6 

91 

85.5 

3i.i 

i5i 

i4i.9 

5i.6 

211 

198.3 

72.2 

271 

254.7 

92.7 

32 

3o.i 

10.9 

92 

86.5 

3i.5 

52 

142.8 

52.0 

12 

199.2 

72.5 

72 

255.6 

93.0 

33 

3i  .0 

II. 3 

93 

87.4 

3i.8 

53 

143.8 

52.3 

i3 

200.2 

72.9 

73 

256.5 

93.4 

34 

3i  .9 

II. 6 

94 

88.3 

32.1 

54 

144.7 

52.7 

i4 

201 . 1 

73.2 

74 

257.5 

93.7 

35 

32.9 

12.0 

95 

89.3 

32.5 

55 

145.7 

53.0 

i5 

202.0 

73.5 

75 

258.4 

94.1 

36 

33.8 

12.3 

96 

90.2 

32.8 

56 

146.6 

53.4 

16 

203.0 

73.9 

76 

259.4 

94.4 

37 

34.8 

12.7 

97 

91 .2 

33.2 

57 

147.5 

53.7 

17 

203.9 

74.2 

77 

260.3 

94-7 

38 

35.7 

i3.o 

98 

92.1 

33.5 

58 

148.5 

54.0 

18 

204.9 

74.6 

78 

261.2 

95.1 

39 

36.6 

i3.3 

99 

93.0 

33.9 

59 

149.4 

54.4 

19 

2o5.8 

74.9 

79 

262.2 

95.4 

40 

37.6 

i3.7 

100 

94.0 

34.2 

60 

i5o.4 

54.7 
55.1 

20 

206.7 

75.2 

80 

263.1 

95.8 

4i 

38.5 

i4.o 

lOI 

94.9 

34.5 

161 

i5i.3 

221 

207.7 

75.6 

281 

264.1 

96.1 

42 

39.5 

14.4 

02 

95.8 

34.9 

62 

l52.2 

55.4 

22 

208.6 

75.9 

82 

265.0 

96.4 

43 

40.4 

14.7 

o3 

96.8 

35.2 

63 

153.2 

55.7 

23 

209.6 

76.3 

83 

265.9 

96.8 

M 

4r.3 

i5.o 

o4 

97-7 

35.6 

64 

I54.I 

56.1 

24 

210.5 

76.6 

84 

266.9 

97.1 

45 

42.3 

i5.4 

o5 

98.7 

35.9 

65 

i55.o 

56.4 

25 

211 .4 

77.0 

8b 

267.8 

97.b 

46 

43.2 

i5.7 

06 

99.6 

36.3 

66 

i56.o 

56.8 

26 

212.4 

77.3 

86 

268.8 

97.8 

47 

AA.'i 

16. 1 

07 

100.5 

36.6 

67 

i56.9 

57.1 

27 

2i3.3 

77.6 

87 

269.7 

98.2 

48 

45.1 

16.4 

08 

loi  .5 

36.9 

68 

157.9 

57.5 

28 

214.2 

78.0 

88 

270.6 

98.5 

49 

46. 0 

16.8 

09 

102.4 

37.3 

69 

i58.8 

57.8 

29 

2l5.2 

78.3 

89 

271 .6 

98.8 

5o 

47-0 

17.1 

10 

io3.4 

37.6 

70 

159.7 

58.1 

3o 

216.1 

78.7 

90 

272.5 

99.2 

5i 

47-9 

17.4 

III 

104.3 

38. 0 

171 

160.7 

58.5 

23l 

217.1 

79.0 

291 

273.5 

99.5 

52 

48.9 

17.8 

12 

io5.2 

38.3 

72 

161. 6 

58.8 

32 

218.0 

79.3 

92 

274.4 

99-9 

53 

49-8 

18. 1 

i3 

106.2 

38.6 

73 

162.6 

59.2 

■Si 

21S.9 

79-7 

93 

275.3 

100.2 

54 

50.7 

1S.5 

i4 

107. 1 

39.0 

74 

i63.5 

59.5 

34 

219.9 

80.0 

94 

276.3 

100.6 

55 

5i.7 

18.8 

i5 

108. 1 

39.3 

75 

164.4 

59.9 

35 

220.8 

80.4 

95 

277.2 

100.9 

56 

52.6 

19.2 

16 

109.0 

39.7 

76 

i65.4 

60.2 

36 

221.8 

80.7 

96 

278.1 

101.2 

57 

53.6 

iq.5 

17 

109.9 

4o.o 

77 

166.3 

60.5 

37 

222.7 

81.1 

97 

279.1 

101.6 

58 

54.5 

19.8 

18 

1 10.9 

40.4 

78 

167.3 

60.9 

38 

223.6 

81.4 

98 

280.0 

101.9 

59 

55.4 

20.2 

19 

III. 8 

40.7 

79 

168.2 

61.2 

39 

224.6 

81.7 

99 

281.0 

102.3 

60 

56.4 

20.5 

20 

112. 8 

4i  .0 

80 

169. 1 

61.6 

40 
Dist. 

225.5 

Dep. 

82.1 
Lat. 

3oo 

281.9 

102.6 

Dist 

Dop. 

Lai. 

Dist. 

Dep. 

Lat. 

Dist. 

Dep. 

Lat. 

Dist. 

Dep. 

Lat. 

For  70  Degress. 

TABLE  IL 

[Page  37 

DilTercnce  of  Latitude  and  Departure  for  21  Degrees. 

Uist. 

Lat.     Dep. 

Dist. 

Lat. 

Dep. 

Dist. 

Lat. 

Dep. 

Dist. 

Lat. 

Dep. 

Dist. 

Lat. 

Dep. 

I 

00.9 

00.4 

61 

56.9 

21 .9 

121 

ii3.o 

43.4 

181 

169.0 

64.9 

24 1 

225.0 

86.4 

2 

01  .9 

00.7 

62 

57.9 

22.2 

22 

113.9 

43.7 

82 

169.9 

65.2 

42 

225.9 

86.7 

3 

02.8 

01. 1 

63 

58.8 

22.6 

23 

114.8 

44.1 

83 

170.8 

65.6 

4^ 

226.9 

87.1 

4 

03.7 

or. 4 

64 

59.7 

22.9 

24 

ii5.8 

44.4 

84 

171.8 

65.9 

44 

227.8 

87-4 

5 

04.7 

01.8 

65 

60.7 

23.3 

25 

116. 7 

44.8 

85 

172.7 

bb.3 

45 

228.7 

87.8 

f) 

o5.6 

02.2 

66 

6i.6 

23.7 

26 

117. 6 

45.2 

86 

173.6 

66.7 

46 

229.7 

88.2 

7 

06.5 

02.5 

67 

62.5 

24.0 

27 

118.6 

45.5 

87 

174.6 

67.0 

47 

23o.6 

88.5 

8 

07.5 

02.9 

68 

63.5 

24.4 

28 

119. 5 

45.9 

88 

175.5 

67.4 

48 

23i.5 

88.9 

9 

08.4 

03.2 

69 

64.4 

24.7 

29 

120.4 

46.2 

89 

176.4 

67.7 

49 

232.5 

89.2 

lO 

1 1 

09.3 

o3.6 

70 

65.4 

25.1 

25.4 

3o 

121 .4 

46.6 
46.9 

90 

177.4 

68.1 

5o 

233.4 

89.6 

10.3 

03.9 

71 

66.3 

i3i 

122.3 

191 

178.3 

68.4 

25l 

234.3 

90.0 

12 

tr  .2 

04.3 

72 

67.2 

25.8 

32 

123.2 

47.3 

92 

179.2 

68.8 

52 

235.3 

90.3 

1 3 

12.1 

04.7 

73 

68.2 

26.2 

33 

124.2 

47-7 

93 

180.2 

69.2 

53 

236  2 

90.7 

i4 

1 3. 1 

o5.o 

74 

69.1 

26.5 

34 

125.  I 

48.0 

94 

181.1 

69.5 

54 

237. 1 

91.0 

i5 

i4.o 

o5.4 

75 

70.0 

26.9 

35 

126.0 

48.4 

95 

182.0 

69.9 

55 

238.1 

91.4 

ifj 

14.9 

o5.7 

76 

71 .0 

27.2 

36 

127.0 

48.7 

96 

i83.o 

70.2 

5b 

239.0 

91.7 

17 

.5.9 

06.1 

77 

71.9 

27.6 

37 

127.9 

49.1 

97 

183.9 

70.6 

57 

239.9 

92.1 

i8 

16.8 

06.5 

78 

72.8 

28.0 

38 

128.8 

49.5 

98 

184.8 

71.0 

58 

240.9 

92.5 

"9 

17-7 

06.8 

79 

73.8 

28.3 

39 

129.8 

49-8 

99 

i85.8 

71.3 

59 

241.8 

92.8 

20 

18.7 

07.2 

80 

74.7 

28.7 

4o 

i3o.7 

5o.2 

200 

186.7 

71.7 

60 

242.7 

93.2 

21 

19.6 

07.5 

81 

75.6 

29.0 

i4i 

i3i.6 

5o.5 

201 

187.6 

72.0 

261 

243.7 

93.5 

22 

20.5 

07.9 

82 

76.6 

29.4 

42 

i32.6 

30.9 

02 

188.6 

72.4 

62 

244.6 

93.9 

23 

21  .5 

08.2 

83 

77.5 

29.7 

43 

i33.5 

5i  .2 

o3 

189.5 

72.7 

63 

24X5 

94.3 

24 

22.4 

08.6 

84 

78.4 

3o.  I 

44 

134.4 

5i.6 

04 

190.5 

73.1 

64 

246.5 

94.6 

25 

23.3 

09.0 

85 

79-4 

3o.5 

45 

i35.4 

52.0 

o5 

191 .4 

73.5 

65 

247-4 

q5.o 

26 

24.3 

09.3 

86 

80.3 

3o.8 

46 

i36.3 

52.3 

o5 

192.3 

73.8 

66 

248.3 

95.3 

27 

25.2 

09.7 

87 

81.2 

3l.2 

47 

137.2 

52.7 

07 

193.3 

74.2 

67 

249.3 

93.7 

28 

26.1 

10. 0 

88 

82.2 

3i.5 

48 

i38.2 

53.0 

08 

194.2 

74.5 

68 

25o.2 

96.0 

29 

27.1 

10.4 

89 

83.1 

3i.9 

49 

139. 1 

53.4 

09 

195.1 

74-9 

69    25 1.  I 

96.4 

JO 

28.0 

10.8 

90 

84.0 

32.3 

5o 

i4o.o 

53.8 

10 

196.1 

75.3 

70      252.1 

96.8 
97.1 

3i 

28.9 

II. I 

91 

85.0 

32.6 

i5i 

i4:  .0 

54.1 

211 

197.0 

75.6 

271 

253.0 

32 

29.9 

11.5 

92 

85.9 

33.0 

52 

141.9 

54.5 

12 

197.9 

76.0 

72 

253.9 

97.5 

33 

3o.8 

IT. 8 

93 

86.8 

33.3 

53 

142.8 

54.8 

i3 

198.9 

70.3 

73 

254.9 

97.8 

34 

3i.7 

12.2 

94 

87.8 

33.7 

54 

143.8 

55.2 

i4 

199.8 

76.7 

74 

255.8 

98. 2 

35 

32.7 

12.5 

95 

88.7 

34.0 

55 

144.7 

55.5 

i5 

200.7 

77.0 

75 

256.7 

98.6 

36 

33.6 

12.9 
i3.3 

96 

89.6 

34.4 

56 

145.6 

55.9 

16 

201 .7 

77-4 

70 

257.7 

98.9 

37, 

34.5 

07 

00.6 

34.8 

57 

146.6 

56.3 

17 

202.6 

77.8 

77 

258.6 

99.3 

38 

35.5 

i3.6 

98 

91.5 

35.1 

58 

147-5 

56.6 

18 

2o3.5 

78.1 

78 

259.5 

99.6 

39 

36.4 

i4.o 

9Q 

92.4 

35.5 

59 

148.4 

57.0 

19 

204.5 

78.5 

79 

260.5 

100.0 

4o 

37.3 

i4.3 

100 

93.4 

35.8 

60 

149.4 

57.3 

20 

2o5.4 

78.8 

80 

261 .4 

100.3 

4 1 

38.3 

14.7 

lOI 

94.3 

36.2 

161 

i5o.3 

57.7 

221 

206.3 

79.2 

281 

262.3 

100.7 

42 

39.2 

i5.i 

02 

95.2 

36.6 

62 

l5l.2 

58.1 

22 

207.3 

79.6 

82 

263.3 

lOI.I 

43 

4o.i 

i5.4 

o3 

96.2 

36.9 

63 

l52.2 

58.4 

23 

208.2 

80.3 

83 

264.2 

101.4 

.  44 

4i.i 

i5.8 

04 

97.1 

37.3 

64 

i53.i 

58.8 

24 

209.1 

84 

265.1 

101.8 

45 

42.0 

16. 1 

o5 

98.0 

37.6 

65 

i54.o 

59.1 

25 

210.1 

bo.b 

85 

266.1 

102.1 

46 

42.9 

16.5 

06 

99.0 

38.0 

66 

i55.o 

59.5 

26 

211 .0 

81.0 

8b 

267.0 

102.5 

47 

43.9 

16.8 

07 

99.9 

38.3 

67 

155.9 

59.8 

27 

211 .0 

81.3 

87 

267.9 

102.9 

48 

44.8 

17.2 

08 

100.8 

38.7 

68 

i56.8 

60.2 

28 

212.9 

81.7 

88 

268.9 

103.2 

49 

45.7 

17.6 

09 

101.8 

39.1 

69 

157.8 

60.6 

29 

2i3.8 

82.1 

89 

269.8 

io3.6 

5o 

4(5.7 

17.9 

10 

1.02.7 

39.4 

70 

i58.7 

60.9 

3o 

214.7 

82.4 

90 

270  .'7 

103.9 

5i 

47.6 

18.3 

III 

io3.6 

39.8 

171 

159.6 

61.3 

23l 

215.7 

82.8 

291 

271.7 

104.3 

52 

48.5 

18.6 

12 

104.6 

4o.i 

72 

160.6 

61.6 

32 

216.6 

83.1 

92 

272.6 

104.6 

53 

49.5 

19.0 

i3 

io5.5 

4o.5 

73 

161.5 

62.0 

33 

217.5 

83.5 

93 

273.5 

io5.o 

54 

5o.4 

19.4 

i4 

106.4 

40.9 

74 

162.4 

62.4 

34 

218.5 

83.9 

94 

274.5  1  io5.4  1 

55 

5i.3 

19.7 

i5 

107.4 

4l.2 

75 

i63.4 

62.7 

35 

219.4 

84.2 

95 

275.4 

105.7 

56 

52.3 

20. 1 

16 

108.3 

4i.6 

76 

164.3 

63.1 

36 

220.3 

84.6 

96 

276.3 

106. 1 

57 

53.2 

20.4 

17 

109.2 

41.9 

77 

i65.2 

63.4 

37 

221 .3 

84.9 

97 

277.3 

106.4 

58 

54.1 

20.8 

18 

no. 2 

42.3 

78 

166.2 

63.8 

38 

222.2 

85.3 

98 

278.2 

106.8 

59 

55.1 

21 .1 

19 

III. I 

42.6 

79 

167.1 

64.1 

3q 

223.1 

85.6 

99 

279.1 

107.2 

60 

56.0 

21.5 

20 

112. 0 

43.0 

80 

168.0 

64.5 

40 

224.1 

86.0 

3oo 

280.1 

107.5 

Disl. 

Dep. 

Lat. 

Dist. 

Dep. 

Lat. 

Disl 

Dep. 

Lat. 

Dist 

Dep. 

Lat. 

Dist 

Dep. 

Lat. 

For  G9  Degi 

ees. 

Page  38J 

TABLL  II. 

1 

DifTerence  of  Latitude  and  Departure  for  22  Degrees. 

Dist. 

Lat. 

Dep. 

Dist. 

Lat. 

Dep. 

Dist. 

Lat. 

Dep. 

Dist. 

Lat. 

Dep. 

Dist. 

Lat. 

Dep. 

I 

00.9 

00.4 

61 

56.6 

22.9 

121 

112. 2 

45.3 

181 

167.8 

67.8 

241 

223.5 

90.3 

2 

01 .9 

00.7 

62 

57.5 

23.2 

22 

Ii3.i 

45.7 

82 

168.7 

68.2 

42 

224.4 

90.7 

3 

02.8 

01. 1 

63 

58.4 

23.6 

23 

ii4.o 

46.1 

83 

169.7 

68.6 

43 

225.3 

91.0 

4 

o3.7 

01.5 

64 

59.3 

24.0 

24 

ii5.o 

46.5 

84 

170.6 

68.9 

AA 

220.2 

91.4 

5 

04.6 

01 .9 

65 

60.3 

24.3 

25 

115.9 

46.8 

85 

171. 5 

69.3 

45 

227.2 

91.8 

6 

o5.6 

02.2 

66 

61 .2 

24.7 

26 

116.8 

47-2 

86 

172.5 

69.7 

46 

228.1 

92.2 

7 

06.5 

02.6 

67 

62.1 

25.1 

27 

117. 8 

47-6 

87 

173.4 

70.1 

47 

229.0 

92.5 

8 

07.4 

o3.o 

68 

63. 0 

25.5 

28 

118. 7 

47-9 

88 

174.3 

70.4 

48 

229.9 

92.9 

9 

08.3 

o3.4 

69 

64. 0 

25.8 

29 

119. 6 

48.3 

89 

175.2 

70.8 

49 

230.9 

93.3 

10 

09.3 

03.7 

70 

64-9 

26.2 

3o 

120.5 

48.7 

90 

176.2 

71.2 

5o 

23i.8 

93.7 

II 

10.2 

04.1 

71 

65.8 

26.6 

i3i 

121 .5 

49.1 

191 

177. 1 

71.5 

25l 

232.7 

94.0 

12 

II . X    o4 . 5 

72 

66.8 

27.0 

32 

122  4 

49.4 

92 

178.0 

71.9 

52 

233.7 

94-4 

i3 

12. 1 

04.9 

73 

67.7 

27.3 

33 

123.3 

49-8 

93 

178.9 

72.3 

53 

234.6 

94-8 

r4 

i3.o 

05.2 

74 

68.6 

27.7 

34 

124.2 

5o.2 

94 

179-9 

72.7 

54 

235.5 

95.2 

1 5 

13.9 

o5.6 

75 

69.5 

28.1 

35 

125.2 

5o.6 

95 

180.8 

73.0 

55 

236.4 

95.5 

i6 

i4.8 

06.0 

76 

70.5 

28.5 

36 

126.  I 

50.9 

96 

181. 7 

73.4 

56 

237.4 

95-9 

I? 

i5.8 

06.4 

77 

71.4 

28.8 

37 

127.0 

5i.3 

97 

182.7 

73.8 

i)7 

238.3 

96.3 

i8 

16.7 

06.7 

78 

72.3 

29.2 

38 

128.0 

5. .7 

98 

i83.6 

74.2 

58 

239.2 

96.6 

19 

17.6 

07.1 

79 

73.2 

29.6 

39 

128.0 

52.1 

99 

184.5 

74-5 

59 

240.1 

97-0 

20 

18.5 

07.5 

80 
81 

74.2 

3o.o 

4o 

129.8 

52.4 

200 

i85.4 

74-9 

60 

241 .1 

97-4 

21 

19.5 

07.9 

75.1 

3o.3 

i4i 

i3o.7 

52.8 

201 

186.4 

75.3 

261 

242.0 

97-8 

22 

20.4 

08.2 

82 

76.0 

3o.7 

42 

i3i  .7 

53.2 

02 

187.3 

75.7 

62 

242.9 

98.1 

23 

21.3 

08.6 

83 

77.0 

3i.i 

43 

i32.6 

53.6 

OJ 

188.2 

76.0 

63 

243.8 

98.5 

24 

22.3 

09.0 

84 

77-9 

3i.5 

M 

i33.5 

53.9 

o4 

189. 1 

76-4 

64 

244.8 

98.9 

25 

23.2 

09.4 

85 

78.8 

3i.8 

45 

i34.4 

54.3 

o5 

190. 1 

76.8 

65 

245.7 

99.3 

26 

24.1 

09.7 

86 

79-7 

32.2 

46 

i35.4 

54.7 

06 

191.0 

77.2 

66 

246.6 

99.6 

27 

25.0 

10. 1 

87 

80.7 

32.6 

47 

i36.3 

55.1 

07 

191.9 

77-b 

67 

247-6 

1 00.0 

28 

26.0 

10.5 

88 

81.6 

33.0 

48 

137.2 

55.4 

08 

192.9 

77-9 

68 

248.5 

100.4 

29 

26.9 

10.9 

89 

82.5 

33.3 

49 

i38.2 

55.8 

09 

193.8 

78.3 

69 

249.4 

100.8 

3o 

27.8 

II  .2 

90 

83.4 

33.7 

5o 

139. 1 

56.2 

10 

194.7 

78.7 

70 

25o.3 

lOI.I 

3i 

28.7 

II. 6 

Qi 

84.4 

34.1 

i5i 

i4o.o 

56.6 

211 

195.6 

79.0 

271 

25i.3 

101.5 

32 

29.7 

12.0 

92 

85.3 

34.5 

52 

140.9 

56.9 

12 

196.6 

79-4 

72 

252.2 

101.9 

33 

3o.6 

12.4 

93 

86.2 

34.8 

53 

141.9 

57.3 

i3 

197.5 

79.8 

73 

253.1 

102.3 

34 

3[.5 

12.7 

94'    87.2 

35.2 

54 

142.8 

57.7 

i4 

198.4 

80.2 

74 

254.0 

102.6 

35 

32.5 

i3.i 

95,    88.1 

35.6 

55 

143.7 

58.1 

i5 

199.3 

80.5- 

75 

255. 0 

io3.o 

36 

33.4 

i3.5 

g6 

89.0 

36.0 

56 

144.6 

58.4 

16 

200.3 

80.9 

76 

255.9 

io3.4 

37 

34.3 

i3.9 

97 

89.9 

36.3 

57 

145.6 

58.8 

17 

201 .2 

81.3 

77 

256.8 

io3.8 

38 

35.2 

14.2 

98 

90.9 

36.7 

58 

146.5 

59.2 

18 

202.1 

81.7 

78 

257.8 

104. 1 

39 

36.2 

i4.6 

99 

91.8 

37.1 

59 

147-4 

59.6 

19 

203.1 

82.0 

79 

258.7 

104.5 

40 

37.1 

i5.o 

100 

92.7 

37.5 

60 

148.3 

59.9 
60.3 

20 

204.0 

82.4 

80 

259.6 

104.9 

4i 

38. 0 

i5.4 

lOI 

93.6 

37.8 

161 

149.3 

221 

204.9 

82.8 

281 

260.5 

io5.3 

42 

38.9 

l5.7 

02 

Q4.6 

38.2 

62 

i5o.2 

60.7 

22 

205.8 

83.2 

82 

261 .5 

io5.6 

43 

39.9 

16. 1 

o3 

95.5 

38.6 

63 

i5i.i 

61. 1 

23 

206.8 

83.5 

83 

262.4 

106.0 

U 

4o.8 

16.5 

04 

96.4 

39.0 

64 

l52.I 

61.4 

24 

207.7 

83-9 

84 

263.3 

106.4 

45 

41.7 

16.9 

o5 

97.4 

39.3 

65 

I53.0 

61.8 

25 

208.6 

84.3 

85 

264.2 

106.8 

46 

42.7 

17.2 

06 

98.3 

39.7 

66 

153.9 

62.2 

26 

209.5 

84-7 

86 

265.2 

107.1 

47 

43.6 

17.6 

07 

99.2 

4o.i 

67 

i54.8 

62.6 

27 

210.5 

85.0 

87 

266.1 

107.5 

48 

44.5 

18.0 

08 

1 00. 1 

40.5 

68 

i55.8 

62.9 

28 

211. 4 

85.4 

88 

267.0 

107.0 

49 

45.4 

18.4 

09 

lOI  .  I 

4o.8 

69 

i56.7 

63.3 

29 

212.3 

85.8 

89 

268.0 

108.3 

5o 

^^.^ 

IS. 7 

10 

102.0 

41.2 

70 

157.6 

63.7 

3o 

2i3.3 

86.2 

90 

268.9 

108.6 

5i 

47.3 

19.1 

III 

102.9 

4i.6 

171 

i58.5 

64.1 

23  I 

214.2 

86.5 

291 

269.8 

109.0 

52 

48.2 

19.5 

12 

io3.8 

42.0 

72 

159.5 

64.4 

32 

2 1 5 . 1 

86.9 

92 

270.7 

109.4 

53 

49.1 

19.9 

i3 

104.8 

42.3 

73 

160.4 

64.8 

33 

216.0 

87.3 

93 

271.7 

109.8 

54 

5o.i 

20.2 

i4 

105.7 

42.7 

74 

161. 3 

65.2 

U 

217.0 

87-7 

94 

272.6 

1 10. 1 

55 

5i.o 

20.6 

i5 

ic6.6 

43.1 

73 

162.3 

65.6 

35 

217.9 

88.0 

95 

273.5 

1 10.5 

56 

5i.9 

21 .0 

16 

107.6 

43.5 

76 

i63.2 

65.9 

36 

218.8 

S8.4 

96 

274.4 

110.9 

57 

52.8 

21 .4 

17 

108.5 

43.8 

77 

164.1 

66.3 

37 

219.7 

88.8 

97 

275.4 

1 1 1.3 

58 

53.8 

21.7 

18 

109.4 

44.2 

78 

i65.o 

66.7 

38 

220.7 

89.2 

98 

276.3 

1 1 1.6 

59 

54.7 

22.1 

19 

no. 3 

44.6 

79 

166.0 

67.1 

39 

221 .6 

89.5 

99 

277.2 

112.0 

60 

55.6 

22.5 
TaT 

20 

II 1 .3 

45.0 

80 

166.9 

67.4 

40 

222.5 

89.9 

3oo 

278.2 

112.4 

Dist. 

De). 

Dist 

Dep. 

Lat. 

Dist. 

Dep. 

Lai. 

Dist. 

Dep. 

Lat. 

Dist. 

Dep. 

Lat. 

[For  G8  Deg 

rees. 

TABLE  IL 

[Page  33 

Difference  of  Latitude  and  Departure  for  23  Degrees. 

Dist. 

Lat. 

Dfp. 

Dist. 

Lai. 

Di-p. 

Dist. 

Lat. 

Dep. 

Dist. 

Lat. 

Dep. 

Dist. 

Lat. 

Dep. 

I 

00.9 

00.4 

61 

56.2 

23.8 

1 21 

III. 4 

47.3. 

iSi 

166.6 

70.7 

241 

221 .8 

94.2 

2 

01 .8 

GO. 8 

62 

b7.. 

24.2 

22 

112.3 

47.7 

82 

167.5 

71.1 

42 

222.8 

94.6 

3 

02.8 

01  .2 

63 

58. 0 

24.6 

23 

Il3.2 

48.1 

83 

168.5 

71.5 

43 

223.7 

94-9 

4 

o3.7 

01 .6 

64 

58.9 

25. 0 

24 

114.1 

48.5 

84 

169.4 

71.9 

44 

224.6 

95.3 

5 

04.6 

02.0 

bb 

59.8 

2b. 4 

25 

ii5.i 

48.8 

85 

170.3 

72.3 

45 

225.5 

9D.7 

6 

ob.5 

02.3 

66 

60.8 

25.8 

26 

116.0 

49.2 

86 

171 .2 

72.7 

46 

226.4 

96.1 

7 

06.4 

02.7 

67 

6,. 7 

26.2 

27 

116. 9 

49.6 

87 

172.1 

73.1 

47 

227.4 

96.5 

8 

07.4 

o3.i 

68 

62.6 

26.6 

28 

117.8 

5o.o 

88 

173. 1 

73.5 

48 

228.3 

96.9 

9 

08.3 

o3.5 

69 

63.5 

27.0 

29 

118.7 

5o.4 

89 

174.0 

73.8 

49 

229.2 

97.3 

10 

09.2 

03.9 

70 

64.4 

27.4 

3o 

119.7 

bo. 8 

90 

174.9 

74.2 

bo 

23o.i 

97-7 

II 

10. 1 

04.3 

71 

65.4 

27.7 

i3i 

120.6 

5l.2 

191 

175.8 

74.6 

25l 

23l  .0 

98.1 

12 

II  .0 

04.7 

72 

66.3 

28.1 

32 

121 .5 

5i.6 

92 

176.7 

75.0 

52 

232.0 

98.5 

i3 

12.0 

oS.i 

73 

67.2 

28.5 

33 

122.4 

52.0 

93 

177.7 

75.4 

53 

232.9 

98.9 

i4 

12.9 

o5.5 

74 

68.1 

28.9 

34 

123.3 

b2.4 

94 

178.6 

75.8 

54 

233.8 

99.2 

lb 

i3.8 

o5.9 

7b 

69.0 

29.3 

35 

124.3 

52.7 

95 

179.5 

76.2 

55 

234.7 

99.6 

lb 

14.7 

06.3 

76 

70.0 

29.7 

36 

125.2 

53.1 

96 

180.4 

76.6 

56 

235.6 

100.0 

17 

ib.b 

06.6 

77 

70.9 

3o.i 

37 

126. I 

b3.5 

97 

181.3 

77.0 

57 

236.6 

100.4 

i8 

ib.b 

07.0 

7S 

71.8 

3o.b 

38 

127.0 

53.9 

98 

182.3 

77.4 

58 

23-7.5 

100.8 

19 

17. b 

07.4 

79 

72.7 

30.9 

39 

128.0 

b4.3 

99 

i83.2 

77.8 

59 

238.4 

101.2 

20 

18.4 

07.8 

80 

73.6 

3i.3 

40 

128.9 

54.7 

200 

184.1 

78.1 

60 

239.3 

101.6 

21 

19.3 

08.2 

81 

74.6 

3. .6 

i4i 

129.8 

55.1 

201 

i85.o 

78.5 

261 

240.3 

102.0 

22 

20.3 

08.6 

82 

75.5 

32. 0 

42 

i3o.7 

55.5 

02 

185.9 

78.9 

62 

241 .2 

102.4 

23 

21.2 

09.0 

83 

76.4 

32.4 

43 

i3i.6 

55.9 

o3 

186.9 

79.3 

63 

242.1 

102.8 

24 

22.1 

09.4 

84 

77.3 

32.8 

44 

i32.6 

56.3 

04 

187.8 

79-7 

64 

243.0 

103.2 

2b 

23.0 

09.8 

85 

78.2 

33.2 

45 

i33.5 

56.7 

o5 

188.7 

80.1 

65 

243.9 

103.5 

26 

23.9 

10.2 

86 

79.2 

33.6 

46 

i34.4 

57.0 

«5 

189.6 

80.5 

66 

244.9 

103.9 

27 

24.9 

10.5 

87 

80.1 

34.0 

47 

i35.3 

57.4 

07 

190.5 

80.9 

67 

245.8 

104.3 

28 

2b. 8 

10.9 

88 

81.0 

'M.4 

48 

i36.2 

b7.8 

08 

191 .5 

81.3 

68 

246.7 

104.7 

29 

26.7 

II  .3 

89 

81.9 

34.8 

49 

137.2 

58.2 

09 

192.4 

81.7 

69 

24-'.  6 

io5.i 

3o 

27. b 

II. 7 

90 

82.8 

3b. 2 

5o 

i3S.i 

58.6 

10 

193.3 

82.1 

70 

248.5 

105.5 

3i 

28.5 

12. 1 

91 

83.8 

35.6 

i5i 

139.0 

59.0 

211 

194.2 

82.4 

271 

249.5 

105.9 

32 

29.5 

12.5 

92 

84.7 

35.9 

52 

139.9 

59.4 

12 

19D.1 

82.8 

72 

25o.4 

106.3 

6:i 

3o.4 

12.9 

93 

85.6 

36.3 

53 

140.8 

59.8 

i3 

196.1 

83.2 

73 

251.3 

106.7 

M 

3i.3 

i3.3 

94 

86.5 

36.7 

54 

i4i.8 

60.2 

i4 

197.0 

83.6 

74 

252.2 

107.1 

6b 

32.2 

i3.7 

95 

87.4 

37.1 

55 

142.7 

60.6 

i5 

197.9 

84.0 

75 

253.1 

107.5 

3b 

33.1 

i4.i 

96 

88.4 

37.5 

56 

143.6 

61 .0 

16 

198.8 

84.4 

76 

254.1 

107.8 

^7 

34.1 

i4.5 

97 

89.3 

37.9 

57 

144.5 

61.3 

17 

199.7 

84.8 

77 

255. 0 

108.2 

38 

3b. 0 

i4.8 

98 

90.2 

38.3 

58 

145.4 

61.7 

18 

200.7 

85.2 

78 

255.9 

108.6 

39 

3b. 9 

l5.2 

99 

91. 1 

38.7 

59 

146.4 

62. 1 

19 

201 .6 

85.6 

79 

256.8 

109.0 

40 

36.8 

i5.6 

100 

92.1 

39.1 

6(3 

147-3 

62.5 

20 

202.5 

86.0 

80 

257.7 

109.4 

4 1 

37.7 

16.0 

lOI 

93.0 

39.5 

161 

148.2 

62.9 

221 

2o3.4 

86.4 

281 

258.7 

109.8 

42 

38.7 

16.4 

02 

93.9 

39.9 

62 

149. 1 

63.3 

22 

204.4 

86.7 

82 

259.6 

110.2 

4d 

39.6 

16.8 

o3 

94.8 

40.2 

63 

i5o.o 

63.7 

23 

2o5.3 

87.1 

83 

260.5 

1 1 0.6 

44 

40.5 

17.2 

04 

95.7 

40.6 

64 

i5i.o 

64.1 

24 

206.2 

87.5 

84 

261 .4 

III.O 

4b 

41.4 

17.6 

o5 

96.7 

4i  .0 

65 

.51.9 

64.5 

25 

207.1 

87.9 

85 

262.3 

1 1 1.4 

4b 

42.3 

18.0 

06 

97.6 

41.4 

66 

152.8 

64.9 

26 

208.0 

88.3 

86 

263.3 

1 11.7 

47 

43.3 

18.4 

07 

98.5 

4i.8 

67 

153.7 

65.3 

27 

209.0 

88.7 

87 

264.2 

1 12. 1 

48 

44.2 

18.8 

08 

99.4 

42.2 

68 

154.6 

65.6 

28 

209.9 

89.1 

88 

265.1 

112. 5 

49 

4b. I 

19. 1 

09 

100.3 

42.6 

69 

i55.6 

66.0 

29 

210.8 

89.5 

89 

266.0 

1 12.9 

bo 

46. 0 

19.5 

10 

loi  .3 

43.0 

7" 

156.5 

66.4 

3o 

21 1 .7 

89.9 

90 
291 

266.9 

ii3.3 

5i 

46.9 

19.9 

III 

102.2 

43.4 

171 

157.4 

66.8 

23l 

212.6 

90.3 

267.9 

1 13.7 

b2 

47.9 

20.3 

12 

io3. 1 

43.8 

72 

i58.3 

67.2 

32 

2i3.6 

90.6 

92 

268.8 

114.1 

bi 

48.8 

20.7 

i3 

104.0 

44.2 

73 

159.2 

67.6 

33 

214.5 

91.0 

93 

269.7 

114.5 

b4 

49.7 

21. 1 

i4 

104.9 

44.5 

74 

160.2 

68.0 

34 

2i5.4 

91.4 

94 

270.6 

1 14.9 

bb 

5o.6    21.5 

lb 

105.9 

44.9 

75 

161. 1 

68.4 

35 

216.3 

91.8 

95 

271 .5 

ii5.3 

bb 

5i.5 

21 .9 
22.3 

16 

106.8 

45.3 

76 

162.0 

68.8 

36 

217.2 

92.2 

96 

272.5 

nb.7 

b7 

b2.b 

17 

107.7 

45.7 

77 

162.9 

69.2 

37 

218.2 

92.6 

97 

273.4 

'  16.0 

b8 

b3.4 

22.7 

18 

108.6 

46.1 

78 

i63.8 

69.6 

38 

219.1 

93.0 

98 

274.3,116.4  1 

b9 

b4.3 

23.1 

19 

109.5 

46.5 

79 

164.8 

69.9 

39 

220.0 

93.4 

99 

275.2 

1 16.8 

bo 

bb.2 

23.4 

20 

no. 5 

46.9 

80 

ibb.7 

70.3 

40 
Dist. 

220.9 

Dep. 

93.8 
Lat. 

3oo 

276.2 

117.2 

Dist. 

Dep. 

Lat. 

Dist. 

Dcp. 

Lat. 

Dist. 

Dep. 

Lat. 

Dist. 

Den. 

Lat. 

[For  C7  Degrees. 

Page  40] 

TABLE  IL 

Difference  of  Latitude  and  Departure  for  24  Degrees. 

Dist. 

I 

Lat. 
00.9 

Dep. 
00.4 

Dist. 

Lat. 

Dep. 

Dist. 

Lat. 

Dep. 

Dist. 

Lat. 

Dep. 

Dist. 

Lat. 

Dep. 

61 

55.7 

24.8 

121 

no. 5 

49.2 

181 

i65.4 

73.6 

241 

220.2 

98.0 

2 

01.8 

00.8 

62 

56.6 

25.2 

22 

III  .5 

49 -(3 

82 

166.3 

74.0 

42 

221 .1 

98.4 

3 

02.7 

01 .2 

63 

57.6 

25.6 

23 

112. 4 

5o.o 

83 

167.2 

74.4 

43 

222.0 

98.8 

4 

o3.7 

01 .6 

64 

58.5 

26.0 

24 

ii3.3 

5o.4 

84 

168.1 

74.8 

M 

222.9 

99.2 

5 

04.6 

02.0 

65 

59.4 

26.4 

25 

Il4-2 

5o.8 

85 

169.0 

75.2 

45 

223.8 

99-7 

6 

o5.5 

02.4 

66 

60.3 

26.8 

26 

ii5.i 

5i.2r 

86 

169.9 

75.7 

46 

224.7 

100. 1 

7 

06.4 

02.8 

67 

61.2 

27.3 

27 

116. 0 

5i.7 

87 

170.8 

76.1 

47 

225.6 

100.5 

8 

07.3 

o3.3 

68 

62.1 

27.7 

28 

116.9 

52.1 

88 

171.7 

76.5 

48 

226.6 

100.9 

9 

08.2 

03.7 

69 

63.0 

28.1 

29 

117. 8 

52.5 

89 

172.7 

76.9 

49 

227.5 

101.3 

10 

09. 1 

04.1 

70 

63.9 

28.5 

3o 

118.8 

52.9 

90 

173.6 

77.3 

5o 

228.4 

101.7 

1 1 

lO.O 

04.5 

71 

64.9 

28.9 

i3i 

119-7 

53.3 

191 

174.5 

77-7 

25l 

229.3 

102.1 

12 

II  .0 

04.9 

72 

65.8 

29.3 

32 

120.6 

53.7 

92 

n^.A 

78.1 

52 

23o.2 

102.5 

i3 

II. 9 

o5.3 

73 

66.7 

29.7 

33 

121 .5 

54.1 

93 

176.3 

78.5 

53 

23l.I 

102.9 

i4 

12.8 

o5.7 

74 

67.6 

3o.  I 

M 

122.4 

54.5 

94 

177.2 

78.9 

54 

232.0 

io3.3 

i5 

i3.7 

06.1 

75 

68.5 

3o.5 

35 

123.3 

54.9 

95 

178. 1 

79-3 

55 

233.0 

103.7 

i6 

14.6 

06.5 

76 

69.4 

30.9 

36 

124.2 

55.3 

96 

179-1 

79-7 

56 

233.9 

104.1 

17 

i5.5 

06.9 

77 

70.3 

3i.3 

37 

125.2 

55.7 

97 

180.0 

80.1 

57 

234.8 

104.5 

18 

16.4 

07.3 

78 

71.3 

3i.7 

38 

126.  I 

56.1 

98 

180.9 

80.5 

58 

235.7 

104.9 

19 

17.4 

07.7 

79 

72.2 

32.1 

39 

127.0 

56.5 

99 

181. 8 

80.9 

59 

236.6 

105.3 

20 

18.3 

oS.i 

80 

73.1 

32.5 

40 

127.9 

56.9 

200 

182.7 

81.3 

60 

237.5 

io5.8 

21 

19.2 

08.5 

81 

74.0 

32.9 

i4i 

128.8 

57.3 

201 

i83.6 

81.8 

261 

238.4 

106.2 

22 

20. 1 

08.9 

82 

74.9 

33.4 

42 

129.7 

57.8 

02 

184.5 

82.2 

62 

239.3 

106.6 

23 

21.0 

09.4 

83 

75.8 

33.8 

43 

i3o.6 

58.2 

o3 

i85.4 

82.6 

63 

240.3 

107.0 

24 

21.9 

09.8 

84 

76.7 

34.2 

A-i 

i3i.6 

58.6 

o4 

186.4 

83.0 

64 

241.2 

107.4 

25 

22.8 

10.2 

85 

77-7 

34.6 

45 

i32.5 

59.0 

o5 

187.3 

83.4 

65 

242.1 

107.8 

26 

23.8 

10.6 

86 

78.6 

35.0 

46 

i33.4 

59.4 

06 

188.2 

83.8 

66 

243.0 

108.2 

27 

24.7 

II. 0 

«7 

79.5 

35.4 

47 

i34.3 

59.8 

07 

189.1 

84.2 

67 

243.9 

108.6 

28 

25.6 

II. 4 

88 

80.4 

35.8 

48 

i35.2 

60.2 

08 

190.0 

84.6 

68 

244.8 

109.0 

29 

26.5 

II. 8 

89 

81.3 

36.2 

49 

i36.i 

60.6 

09 

190.9 

85.0 

69 

245.7 

109.4 

3o 
3i 

27-4 
28.3 

12.2 

12.6 

90 

82.2 

36.6 

5o 

i37.o 

61 .0 

10 

191.8 

85.4 

70 

246.7 

109.8 

91 

83.1 

37.0 

i5i 

137.9 

61.4 

211 

192.8 

85.8 

271 

247.6 

110.2 

32 

29.2 

i3.o 

92 

84.0 

37.4 

52 

i38.9 

6r.8 

12 

193.7 

86.2 

72 

248.5 

1 10.6 

33 

3o.i 

i3.4 

93 

85. 0 

37.8 

53 

139.8 

62.2 

i3 

194.6 

86.6 

73 

249.4 

in.o 

34 

3i.i 

i3.8 

94 

85.9 

38.2 

54 

140.7 

62.6 

i4 

195.5 

87.0 

74 

250.3 

1 1 1.4 

35 

32.0 

14.2 

95 

86.8 

38.6 

55 

i4i.6 

63. 0 

i5 

196.4 

87.4 

75 

25l.2 

III. 9 

36 

32.9 

14.6 

96 

87.7 

39.0 

56 

142.5 

63.5 

16 

197.3 

87.9 

76 

252.1 

112. 3 

37 

33.8 

i5.o 

97 

88.6 

39.5 

57 

143.4 

63.9 

17 

198.2 

88.3 

77 

253.1 

112.7 

38 

34.7 

i5.5 

98 

89.5 

39.9 

58 

144.3 

64.3 

18 

199.2 

88.7 

78    254.0 

ii3.i 

39 

35.6 

15.9 

99 

90.4 

40.3 

59 

145.3 

64.7 

19 

200.1 

89.1 

79 

254.9 

ii3.5 

40 
4i 

36.5 

16.3 

100 

91.4 

40.7 

60 

i46.2 

65.1 

20 

201 .0 

89.5 

80 

255.8 

113.9 

37.5 

16.7 

lOI 

92.3 

4i.i 

161 

i47-i 

65.5 

221 

201 .9 

89.9 

281 

256.7 

114.3 

•42 

38.4 

17. 1 

02 

93.2 

4i.5 

62 

148.0 

65.9 

22 

202.8 

90.3 

82 

257.6 

114-7 

43 

39.3 

17.5 

o3 

94.1 

41.9 

63 

148.9 

66.3 

23 

203.7 

90.7 

83 

258.5 

ii5.i 

Ai 

4o.2 

17.9 

04 

95.0 

42.3 

64 

149-8 

66.7 

24 

204.6 

91. 1 

84 

259.4 

ii5.5 

45 

4i.i 

18.3 

o5 

95.9 

42.7 

65 

1 50.7 

67.1 

25 

2o5.5 

91.5 

85 

260.4 

1 1 5.9 

46 

42.0 

18.7 

06 

96.8 

43.1 

66 

i5i.6 

67.5 

26 

206.5 

91.9 

86 

261.3 

116.3 

47 

42.9 

19. 1 

07 

97-7 

43.5 

67 

i52.6 

67.9 

27 

207.4 

92.3 

87 

202.2 

116.7 

48 

43.9 

19.5 

08 

98.7 

43.9 

68 

153.5 

68.3 

28 

208.3 

92.7 

88 

263.1 

117. 1 

49 

4'i.8 

19.9 

09 

99.6 

44.3 

69 

1 54. 4 

68.7 

29 

209.2 

93.1 

89 

264.0 

1 1 7.5 

5o 

45.7 

20.3 

10 

1 00 . 5 

44.7 

70 

i55.3 

69.1 

3o 

210. 1 

93.5 

90 

264.9 

1 18.0 

5i 

46.6 

20.7 

III 

loi  .4 

45.1 

171 

1 56. 2 

69.6 

23l 

21 1 .0 

94.0 

291 

265.8 

118.4 

52 

47  ^ 

21.2 

12 

102.3 

45.6 

72 

1 57. 1 

70.0 

32 

211 .9 

94.4 

92 

266.8 

118.8 

53 

48.4 

21.6 

1  3 

I03.2 

46. 0 

73 

i58.o 

70.4 

33 

212.9 

94.8 

93 

267.7 

1 19.2 

54 

49.3 

22.0 

i4 

104.1 

A6.A 

74 

159^0 

70.8 

■M 

2i3.8 

95.2 

94 

268.6 

119.6 

55 

5o.2 

22.4 

i5 

io5.i 

46.8 

75 

159.9 

71.2 

35 

214.7 

95.6 

95 

269.5 

120.0 

5b 

5l.2 

22.8 

16 

1 06 . 0 

47.2 

76 

160.8 

71.6 

36 

2i5.6 

96.0 

96 

270.4 

120.4 

!)7 

52.  I 

23.2 

17 

106.9 

47-6 

77 

161 .7 

72.0 

37 

216.5 

96.4 

97 

271 .3 

120.8 

5b 

53.0 

23.6 

18 

107.8 

48. 0 

78 

162.6 

72.4 

38 

217.4 

96.8 

98 

272.2 

1 2 1. 2 

59 

53.9 

24.0 

19 

108.7 

48.4 

79 

i63.5 

72.8 

39 

218.3 

97.2 

99 

273.2 

1 2 1. 6 

bo 

54.8 

24.4 

20 

109.6 

48.8 

80 

164.4 

73.2 

4o 

219.3 

97.6 

3oo 

274.1 

122.0 

Disi. 

De'j. 

Lat. 

Dist. 

Dc.p. 

Lnt. 

Dist. 

Dtp. 

Lat. 

Dist. 

Dep. 

Lat. 

Dist.l    Dep.  1 

Lat. 

[For  GO  Degrees. 

TABLE  IL 

[Page  41 

Difference  of  Latitude  and  Departure  for  25 

Degre 

!es. 

Uist. 

Lat. 

Dcp. 

Dist. 

Lat. 

Dep. 
25.8 

Dist. 

Lat. 

Dep. 

Dist. 

Lat. 

Dcp. 

Dist. 

Lat. 

Dep. 

I 

00.9 

00.4 

61 

55.3 

121 

109.7 

5i.i 

181 

164.0 

76.5 

241 

218.4 

101.9 

2 

01 .8 

00.8 

62 

56.2 

26.2 

22 

1 10.6 

5i.6 

82 

164.9 

76.9 

42 

219.3 

102.3 

3 

02.7 

01 .3 

63 

57.1 

26.6 

23 

III. 5 

52. 0 

83 

165.9 

77.3 

43 

220.2 

102.7 

4 

o3.6 

01.7 

64 

58.0 

27.0 

24 

112.4 

52.4 

84 

166.8 

77-8 

AA 

221.1 

io3.i 

5 

04.5 

02.1 

65 

58.9 

27.5 

25 

ii3.3 

52.8 

85 

167.7 

78.2 

45 

222.0 

103.5 

6 

o5.4 

02.5 

66 

59.8 

27.9 

26 

Il4.2 

53.2 

86 

168.6 

78.6 

46 

223.0 

104.0 

7 

06.3 

o3.o 

67 

60.7 

28.3 

27 

ii5.i 

53.7 

87 

169.5 

79.0 

47 

223.9 

104.4 

8 

07.3 

o3.4 

68 

61.6 

28.7 

28 

116.0 

54.1 

88 

,170.4 

79-^ 

48 

224.S 

104.8 

9 

0S.2 

o3.8 

69 

62.5 

29.2 

29 

1 16.9 

54.5 

89 

171.3 

79-9 

49 

225.7 

105.2 

10 

09.1 

04.2 

7" 

63.4 

29.6 
3o.o 

3o 

117. 8 

54.9 

90 

172.2 

80.3 

5o 

226.6 

105.7 

II 

10. 0 

04.6 

71 

64.3 

i3i 

118.7 

55.4 

191 

173.1 

80.7 

25l 

227.5 

106. 1 

12 

10.9 

o5.j 

72 

65.3 

3o.4 

32 

119.6 

55.8 

92 

174.0 

81. 1 

52 

228.4 

106.5 

i3 

II. 8 

o5.5 

73 

66.2 

30.9 

33 

120.5 

56.2 

93 

174.9 

81.6 

53 

229.3 

106.9 

]4 

12.7 

05.9 

74 

67.1 

3i.3 

34 

121 .4 

56.6 

94 

175.8 

82.0 

54 

23o.2 

107.3 

i5 

i3.6 

06.3 

75 

68.0 

3i.7 

35 

122.4 

57.1 

95 

176.7 

82.4 

55 

23l.I 

107.8 

i6 

i4.5 

06.8 

76 

68.9 

32.1 

36 

123.3 

57.5 

96 

177.6 

82.8 

56 

232.0 

108.2 

17 

i5.4 

07.2 

77 

69.8 

32.5 

37 

124.2 

57.9 

97 

178.5 

83.3 

57 

232.9 

108.6 

i8 

16.3 

07.6 

7S 

70.7 

33.0 

38 

125.1 

58.3 

98 

179.4 

83.7 

58 

233.8 

109.0 

19 

17.2 

08.0 

79 

71.6 

■6i.A 

39 

126.0 

58.7 

99 

180.4 

84.1 

59 

234.7 

109.5 

20 

18.1 

08.5 

80 

72.5 

33.8 

40 

126.9 

59.2 
59.6 

200 

181.3 

84.5 

60 

235.6 

109.9 

21 

19.0 

08.9 

81 

73.4 

34.2 

i4i 

127.8 

201 

182.2 

84.9 

261 

236.5 

1 10.3 

22 

19.9 

09.3 

82 

74.3 

34.7 

42 

128.7 

60.0 

02 

i83.i 

85.4 

62 

237.5 

1 10.7 

23 

20.8 

09.7 

83 

75.2 

35.1 

43 

129.6 

60.4 

o3 

184.0 

85.8 

63 

2  38.4 

III. I 

24 

21.8 

lO.I 

84 

76.1 

35.5 

AA 

i3o.5 

60.9 

04 

184.9 

86.2 

64 

239.3 

111.6 

25 

22.7 

10.6 

85 

77.0 

35.9 

45 

i3i.4 

61.3 

o5 

i85.8 

86.6 

65 

240.2 

1 12.0 

26 

23.6 

II. 0 

86 

77-9 

36.3 

46 

i32.3 

61.7 

06 

186.7 

87.1 

66 

241.1 

1 1 2.4 

27 

24.5 

II. 4 

87 

78.8 

36.8 

47 

133.2 

62.1 

07 

187.6 

87.5 

67 

242.0 

1 1 2.8 

28 

25.4 

II. 8 

88 

79.8 

37.2 

48 

i34.i 

62.5 

08 

188.5 

87-9 

68 

242.9 

ii3.3 

29 

26.3 

12.3 

89 

80.7 

37.6 

49 

i35.o 

63. 0 

09 

189.4 

88.3 

69 

243.8 

113.7 

3o 

27.2 

12.7 

90 

81.6 

38. 0 

5o 

135.9 

63.4 

10 

190.3 

88.7 

70 

244.7 

114.1 

3i 

28.1 

i3.i 

Qi 

82.5 

38.5 

i5i 

i36.9 

63.8 

211 

191.2 

89.2 

271 

245.6 

114.5 

32 

29.0 

i3.5 

92 

83.4 

38.9 

52 

137.8 

64.2 

12 

192.1 

89.6 

72 

246.5 

1 1 5.0 

33 

29.9 

i3.9 

93 

84.3 

39.3 

53 

i38.7 

64.7 

i3 

193.0 

90.0 

73 

247-4 

n5.4 

34 

3o.8 

14.4 

94 

85.2 

39.7 

54 

139.6 

65.1 

i4 

193.9 

90.4 

74 

248.3 

ii5.8 

35 

3i.7 

14.8 

95 

86,1 

4o.  I 

55 

140.5 

65.5 

i5 

194.9 

90.9 

75 

249.2 

1 16.2 

36 

32.6 

l5.2 

96 

87.0 

40.6 

56 

141.4 

65.9 

16 

195.8 

91.3 

76 

25o.i 

116.6 

37 

33.5 

i5.6 

97 

87.9 

4i  .0 

57 

142.3 

66.4 

17 

196.7 

91.7 

77 

25l.O 

117.1 

38 

34.4 

16. 1 

98 

88.8 

41.4 

58 

143.2 

66.8 

18 

197.6 

92.1 

78 

252.0 

117.5 

39 

35.3 

16.5 

99 

89.7 

4i.8 

59 

144. 1 

67.2 

19 

198.5 

92.6 

79 

252.9 

117.9 

40 

36.3 

16.9 

100 

90.6 

42.3 

60 

145.0 

67.6 

20 

199-4 

93.0 

80 

253.8 

118.3 

4i 

37.2 

17.3 

lOI 

91.5 

42.7 

161 

145.9 

68.0 

221 

200.3 

93-4 

281 

254.7 

118.8 

42 

38.1 

17.7 

02 

92.4 

43.1 

62 

146.8 

68.5 

22 

201.2 

93.8 

82 

255.6 

119.2 

43 

39.0 

18.2 

o3 

93.3 

43.5 

63 

147-7 

68.9 

23 

202.1 

94.2 

83 

256.5 

119.6 

A/\ 

39.9 

18.6 

04 

94.3 

44.0 

QA 

148.6 

69.3 

24 

2o3.o 

94-7 

84 

257.4 

120.0 

45 

40.8 

19.0 

o5 

95.2 

aA-A 

65 

149.5 

69.7 

25 

203.9 

95.1 

85 

258.3 

120.4 

46 

41.7 

19.4 

06 

96.1 

44.8 

66 

i5o.4 

70.2 

26 

2o4-8 

95.5 

86 

259.2 

120.9 

47 

42.6 

19.9 

07 

97.0 

45.2 

67 

i5i.4 

70.6 

27 

205.7 

95.9 

87 

260.1 

12 1. 3 

A^ 

43.5 

20.3 

08 

97-9 

45.6 

68 

i52.3 

71.0 

28 

206.6 

96.4 

88 

261.0 

1 2 1. 7 

49 

u.^ 

20.7 

09 

98. 8 

46.1 

69 

i53.2 

71.4 

29 

207.5 

96.8 

89 

261.9 

122. 1 

bo 
5i 

45.3 

21 .1 

10 

99-7 

46.5 

70 

154.1 

71.8 

3o 

208.5 

97-2 

90 

262.8 

122.6 

46.2 

21 .6 

III 

100.6 

46.9 

171 

i55.o 

72.3 

23l 

209.4 

97.6 

291 

263.7 

123.0 

52 

47.1 

22.0 

12 

Id  .5 

47.3 

72 

155.9 

72.7 

32 

210.3 

98.0 

92 

264.6 

123.4 

53 

48.0 

22.4 

i3 

102.4 

47-8 

73 

i56.8 

73.1 

33 

211. 2 

98.5 

93 

265.5 

123.8 

54 

48.9 

22.8 

i4 

io3.3 

48.2 

74 

157.7 

73.5 

M 

212. 1 

98.9 

94 

266.5 

124.2 

55 

49.8 

23.2 

i5 

104.2 

48.6 

75 

i58.6 

74.0 

35 

2l3.0 

99.3 

95 

267.4 

124.7 

56 

5o.8 

23.7 

lb 

io5.i 

49.0 

76 

159.5 

74.4 

36 

2l3.9 

99-7 

96 

268.3 

125. 1 

57 

51.7 

24.1 

17 

106.0 

49.4 

77 

160.4 

74.8 

37 

214.8 

100.2 

97 

269.2 

125.5 

58 

52.6 

24.5 

18 

106.9 

49.9 

78 

161. 3 

75.2 

38 

215.7 

100.6 

98 

270.1 

125.9 

59 

53.5 

24.9 

19 

107.9 

5o.3 

79 

162.2 

75.6 

39 

216.6 

lOI.O 

99 

271.0 

126.4 

bo 

54.4 

25.4 

20 

108.8 

50.7 

8g 

i63.i 

76.1 

4o 

217.5 

101.4 

3oo 

271.9 

126.8 

Dist. 

Pep. 

Lat. 

Dist. 

Dcp. 

Lat. 

Dist.      Dep. 

Lat. 

Dist. 

Dep. 

Lat. 

Dist. 

Dep. 

Lat. 

[1 

^-or  G.^  Degr 

ees. 

Page  421 

TABLE  IL 

1 

Difference  of  Latitude  and  Departure  for  26 

Degrees. 

Disi. 

Lat. 

Uep. 

Disl. 

Lat. 

Dep. 

Dist. 

Lat. 

Dep. 

Dist. 

Lat. 

Dep. 

Dist. 

Lat. 

Dep. 

I 

00.9 

00.4 

61 

54.8 

26.7 

121 

108.8 

53.0 

181 

162.7 

79.3 

241 

216.6 

io5.6 

2 

01.8 

00.9 

62 

55.7 

27.2 

22 

109.7 

53.5 

82 

i63.6 

79.8 

42 

217.5 

106.1 

3 

02.7 

01 .3 

63 

56.6 

27.6 

23 

no. 6 

53.9 

83 

164.5 

80.2 

43 

21S.4 

106.5 

4 

o3.6 

01.8 

64 

57.5 

28.1 

24 

III. 5 

54.4 

84 

i65.4 

80.7 

44 

219.3 

107.0 

5 

04.5 

02.2 

65 

58.4 

28.5 

25 

112. 3 

54.8 

85 

166.3 

81.1 

45 

220.2 

107.4 

6 

o5.4 

02.6 

66 

59.3 

28.9 

26 

Il3.2 

55.2 

86 

167.2 

81.5 

46 

221.1 

107.8 

7 

06.3 

o3.i 

67 

60.2 

i!9.4 

27 

114.1 

55.7 

87 

168.1 

82.0 

47 

222.0 

108.3 

8 

07.2 

o3.5 

68 

61. 1 

29.8 

28 

ii5.o 

56.1 

88 

169.0 

82.4 

48 

222.9 

108.7 

9 

08.1 

03.9 

69 

62.0 

3o.2 

29 

115.9 

56.5 

89 

169.9 

82.9 

49 

223.8 

109.2 

10  1  09.0 

04.4 

70 

62.9 

3o.7 

3o 

116. 8 

57.0 

90 

170.8 

83.3 

5o 

224.7 

109.6 

II . 09 . 9 

04.8 

71 

63.8 

3i.i 

i3i 

117. 7 

57.4 

191 

171.7 

83.7 

25l 

225.6 

1 1 0.0 

12 

10.8 

o5.3 

72 

64.7 

3i.6 

32 

118. 6 

^7.9 

92 

172.6 

84.2 

52 

226.5 

110.5 

i3 

II. 7 

05.7 

73 

65.6 

32. 0 

33 

119. 5 

58.3 

93 

173.5 

84.6 

53 

227.4 

IIO.O 

i4 

12.6 

06.1 

74 

66.5 

32.4 

U 

120.4 

58.7 

94 

174.4 

85.0 

54 

228.3 

111.3 

i5 

i3.5 

06.6 

75 

67.4 

32.9 

35 

121.3 

59.2 

95 

175.3 

85.5 

55 

229.2 

11 1.8 

16 

14.4 

07.0 

76 

68.3 

33.3 

36 

122.2 

59.6 

96 

176.2 

85.9 

56 

23o.l 

112.2 

17 

i5.3 

07.5 

77 

69.2 

33.8 

37 

123. I 

60. 1 

97 

177.1 

86.4 

57 

23l.O 

112.7 

18 

16.2 

07.9 

7S 

70.1 

34.2 

38 

124.0 

60.5 

98 

178.0 

86.8 

58 

231.9 

ii3.i 

19 

17. 1 

08.3 

79 

71.0 

34.6 

39 

124.9 

60.9 

99 

'78.9 

87.2 

59 

232.8 

ii3.5 

20 

18.0 

08.8 

80 

71.9 

35.1 

4o 

125.8 

61.4 

200 

I'/ 9-8 

87.7 

60 

233.7 

114.0 

21 

18.9 

09.2 

81 

72.8 

35.5 

i4i 

126.7 

61.8 

201 

180.7 

88.1 

261 

234.6 

1 14.4 

22 

19.8 

09.6 

82 

73.7 

35. q 

42 

127.6 

62.2 

02 

181.6 

88.6 

62 

235.5 

114.9 

23 

20.7 

10. 1 

83 

74.6 

36.4 

43 

128.5 

62.7 

o3 

182.5 

89.0 

63 

236.4 

115.3 

24 

21.6 

10.5 

84 

75.5 

36.8 

44 

129.4 

63.1 

04 

i83.4 

89.4 

H 

237.3 

115.7 

25  ,  22.5 

II  .0 

85 

76.4 

37.3 

45 

i3o.3 

63.6 

o5 

184.3 

89.9 

65 

238.2 

116. 2 

26 

23.4 

II. 4 

86 

77.3 

37.7 

46 

l3l.2 

64.0 

06 

185.2 

90.3 

66 

239.1 

116.6 

27 

24.3 

II. 8 

87 

78.2 

38.1 

47 

l32.I 

^4.4 

07 

186.1 

90.7 

67 

240.0 

117.0 

28 

25.2 

12.3 

88 

79.1 

38.6 

48 

i33.o 

64.9 

08 

186.9 

91.2 

68 

240.9 

1 17.5 

29 

26.1 

12.7 

89 

80.0 

39.0 

49 

133.9 

65.3 

09 

187.8 

91.6 

69 

241.8 

117.9 

3o 

27.0 

l3.2 

90 

80.9 

39.5 

39.9 

5o 

i34.8 

65.8 

10 

188.7 

92.1 

70 

242.7 

118.4 

3i 

27.9 

i3.6 

Qi 

81.8 

i5i 

135.7 

66.2 

211 

189.6 

92.5 

271 

243.6 

118.8 

32 

28.8 

i4.o 

92 

82.7 

4o.3 

52 

i36.6 

66.6 

12 

190.5 

92.9 

72 

244.5 

119. 2 

33 

29.7 

i4.5 

93 

83.6 

40.8 

53 

137.5 

67.1 

i3 

191.4 

93.4 

73 

245.4 

119.7 

34 

3o.6 

14.9 

94 

84.5 

41.2 

54 

i38.4 

67.5 

i4 

192.3 

93.8 

74 

246.3 

120.1 

35 

3i.5 

i5.3 

95 

85.4 

4i.6 

55 

139.3 

67.9 

i5 

193.2 

94.2 

75 

247.2 

120.6 

36 

32.4 

i5.8 

96 

86.3 

4^.1 

56 

140.2 

68.4 

16 

194.1 

94.7 

76 

248.1 

1 21.0 

37 

33.3 

16.2 

97 

87.2 

42.5 

^7 

i4i.i 

68.8 

17 

195.0 

95.1 

77 

249.0 

121.4 

38 

34.2 

16.7 

98 

88. 1 

43.  c 

58 

142.0 

69.3 

18 

195.9 

95.6 

78 

249.9 

121.9 

39 

35.1 

17. 1 

99 

89.0 

43.4 

59 

142.9 

69.7 

19 

196.8 

96.0 

79 

25o.8 

122.3 

4o 

36. 0 

17.5 
18.0 

100 

89.9   43.8 

bo 

143.8 

70.1 

20 

197.7 

96.4 

80 

251.7 

122.7 

4. 

36.9 

lOI 

90.8 

44. i 

161 

144.7 

70.6 

221 

198.6 

96.9 

281 

252.6 

123.2 

42 

37.7 

18.4 

02 

91.7 

44.1 

62 

145.6 

71.0 

22 

199.5 

97.3 

82 

253.5 

123.6 

43 

38.6 

18.8 

o3 

92.6 

45.2 

63 

i46  5 

71.5 

23 

200.4 

97.8 

83 

254.4 

I24.I 

44 

39.5 

19.3 

04 

93.5 

45.6 

^4 

147-4    71-9 

24 

201.3 

98.2 

84 

255.3 

124.5 

45 

40.4 

19.7 

o5 

94.4 

46.0 

65 

148.3 

72.3 

25 

202.2 

98.6 

83 

256.2 

124.9 

46 

41.3 

20.2 

06 

95.3 

46.5 

66 

149.2 

72.8 

26 

203.1 

99.1 

86 

257.1 

125.4 

47 

42.2 

20.6 

07 

96.2 

46.9 

67 

i5o.i 

73.2 

27 

204-0 

99.5 

87 

358.0 

125.8 

48 

43.1 

21 .0 

08 

97.1 

47.3 

68 

i5i.o 

73.6 

28 

204.9 

99.9 

88 

258.9 

126.3 

49 

44.0 

21.5 

09 

98.0 

47-8 

69 

i5i.9 

74.1 

29 

2o5.8 

10U.4 

89 

259.8 

126.7 

bo 

44.9 

21.9 

10 

98.9 

48.2 
48.7 

70 

152.8 

74.5 

3o 

206.7 

100.8 

90 

260.7 

127.1 

5. 

45.8 

22.4 

III 

99.8 

171 

153.7 

75.0 

23l 

207.6 

101.3 

291 

261.5 

127.6 

52 

46.7 

22.8 

12 

100.7 

49.1 

72 

i54.6 

75.4 

32 

208.5 

101.7 

92 

262.4 

128.0 

53 

47.6 

23.2 

i3 

loi  .6 

49.5 

73 

155.5 

75.8 

33 

209.4 

102.1 

93 

263.3 

128.4 

54 

43.5 

23.7 

.4 

102.5 

5o.o 

74 

i56.4 

76.3 

34 

210.3 

102.6 

94 

264.2 

128.9 

55 

49-4 

24.1 

i5 

io3.4 

5o.4 

75 

157.3 

76.7 

35 

21  1.2 

io3.o 

95 

265.1 

129.3 

56 

5o.3 

24.5 

16 

104.3 

50.9 

76 

i58.2 

77.2 

36 

212.1 

io3.5 

96 

266.0 

129.8 

57 

5l.2 

25.0 

17 

105.2 

5i.3 

77 

159.1 

77.6 

37 

2l3.0 

103.9 

97 

266.9 

l3o.2 

58 

52.1 

25.4 

18 

106. 1 

5i.7 

78 

160.0 

78.0 

38 

213.9 

io4.3 

98 

267.8 

i3o.6 

59 

53.0 

25.9 

19 

107.0 

52.2 

79 

160.9 

78.5 

39 

214.8 

104.8 

99 

268.7 

i3i.i 

60 

53.9 

26.3 

20 

107.9 

52.6 

80 

161.8 

78.9 

4o 

215.7 

io5.2 

3  00 

269.6 

i3i.5 

Dist. 

IK.p. 

Lat. 

Dist. 

Dcp. 

Lat. 

Dist. 

Dep. 

Lat. 

Dist. 

Dep. 

Lat. 

Dist. 

Dcp. 

Lat. 

[ 

■"or  64  Degrees. 

"1 

TABLE  II. 

[ Pago  43 

Differe 

nee  of  Lati 

tude  and  Departure  for  27  Degre 

es. 

Dist. 

Lai. 

Dep. 

Dist. 

Lat. 

Dep. 

Dist. 

Lat. 

Dep. 

Dist. 

Lat. 

Dep. 

Dist. 

Lat. 

Dep. 

I 

00.9 

00.5 

61 

54.4 

27-7 

121 

107.8 

54.9 

181 

161.3 

82.2 

241 

214.7 

109.4 

3 

01.8 

00.9 

62 

55.2 

28.1 

22 

108.7 

55.4 

82 

162.2 

82.6 

42 

21 5.6 

109.9 

3 

02.7 

01 .4 

63 

56.1 

28.6 

23 

109.6 

55.8 

83 

i63.i 

83.1 

43 

216.5 

110.3 

4 

o3.6 

01 .8 

64 

57.0 

29.1 

24 

1 10.5 

56.3 

84 

163.9 

83.5 

AA 

217.4 

110.8 

5 

o4.5 

02.3 

65 

57.9 

29.5 

25 

III. 4 

56.7 

85 

164.8 

84.0 

45 

218.3 

III. 2 

6io5.3 

02.7 

66 

58.8 

3o.o 

26 

112.3 

57.2 

86 

i65.7 

84.4 

46 

219.2 

111.7 

7 

06.2 

o3.2 

67 

59-7 

3o.4 

27 

Il3.2 

57.7 

87 

166.6 

84.9 

47 

220.1 

112.1 

8 

07.1 

o3.6 

68 

60.6 

30.9 

28 

114.0 

58.1 

88 

167.5 

85.4 

48 

221.0 

112.6 

9 

08.0 

04.1 

69 

61.5 

3i.3 

29 

114.9 

58.6 

89 

168.4 

85.8 

49 

221.9 

I  i3.o 

10 

II 

08.9 
09.8 

04.5 
o5.o 

70 

62.4 

3i.8 

3o 

ii5.8 

59.0 

90 

169.3 

86.3 

5o 

222,8 

ii3.5 

71 

63.3 

32.2 

i3i 

116.7 

59.5 

191 

170.2 

86.7 

25l 

2236 

114.0 

12 

10.7 

o5.4 

72 

64.2 

32.7 

32 

117.6 

59.9 

92 

171. 1 

87.2 

52 

224.5 

1 14.4 

i3 

11.6 

05.9 

73 

65.0 

33.1 

33 

118. 5 

60.4 

93 

172.0 

87.6 

53 

225.4 

1 14.9 

i4 

12.5 

06.4 

74 

65.9 

33.6 

34 

119. 4 

60.8 

94 

172.9 

88.1 

54 

226.3 

ii5.3 

i5 

i3.4 

06.8 

75 

66.8 

34.0 

35 

120.3 

61.3 

95 

173.7 

88.5 

55 

227.2 

ii5.8 

i6 

i4.3 

07.3 

76 

67.7 

34.5 

36 

121 .2 

61.7 

96 

174-6 

89.0 

56 

228.1 

116.2 

17 

i5.i 

07.7 

77 

68.6 

35.0 

37 

122. 1 

62.2 

97 

175.5 

89.4 

57 

229.0 

1 16.7 

i8 

16.0 

08.2 

78 

69.5 

35.4 

38 

123.0 

62.7 

98 

176.4 

89.9 

58 

229.9 

117.1 

19 

16.9 

08.6 

79 

70.4 

35.9 

39 

123.8 

63.1 

99 

177-3 

90.3 

59 

23o.8 

117.6 

20 

17.8 

09.1 

80 

71.3 

ib.6 

4o 

124.7 

63.6 
64.0 

200 

178.2 

90.8 

60 

231.7 

118.0 

21 

18.7 

09.5 

81 

72.2 

36.8 

i4i 

125.6 

201 

179.1 

91.3 

261 

232.6 

118.5 

22 

19.6 

10. 0 

82 

73.1 

37.2 

42 

126.5 

64.5 

02 

180.0 

91.7 

62 

233.4 

118.9 

23 

20.5 

10.4 

83 

74.0 

37.7 

43 

127.4 

64.9 

o3 

180.9 

92.2 

63 

234.3 

1 19.4 

24 

21.4 

10.9 

84 

74.8 

38.1 

^^ 

128.3 

65.4 

04 

181.8 

92.6 

64 

235.2 

1 19.9 

25 

22.3 

II. 3 

85 

75.7 

38.6 

45 

129.2 

65.8 

o5 

182.7 

93.1 

65 

236.1 

120.3 

26 

23.2 

II. 8 

86 

76.6 

39.0 

46 

i3o.i 

66.3 

06 

i83.5 

93.5 

66 

237.0 

120.8 

27 

24.1 

12.3 

87 

77.5 

39.5 

47 

i3i  .0 

66.7 

07 

184.4 

94.0 

67 

237.9 

121.2 

28 

24.9 

12.7 

88 

78.4 

4o.o 

48 

i3i.9 

67.2 

08 

i85.3 

94.4 

68 

238.8 

121.7 

29 

25.8 

l3.2 

89 

79.3 

40.4 

49 

i32.8 

67.6 

09 

186.2 

94-9 

69 

239.7 

122. 1 

3o 

26.7 

i3.6 

90 

80.2 

40.9 

5o 

i33.7 

68.1 

10 

187.1 

95.3 

70 
271 

240.6 

122.6 

3i 

27.6 

14.1 

91 

81. 1 

41.3 

i5i 

i34.5 

68.6 

211 

188.0 

95.8 

241.5 

I23.0 

32 

28.5 

i4.5 

92 

82.0 

4i.8 

52 

i35.4 

69.0 

12 

188.9 

96.2 

72 

242.4 

123.5 

33 

29.4 

i5.o 

93 

82.9 

42.2 

53 

i36.3 

69.5 

i3 

189.8 

96.7 

73 

243.2 

123.9 

34 

3o.3 

i5.4 

94 

83.8 

42.7 

54 

137.2 

69.9 

i4 

190.7 

97.2 

74 

244.1 

124.4 

35 

3l.2 

i5.9 

95 

84.6 

43.1 

55 

i38.i 

70.4 

i5 

191.6 

97.6 

75 

245.0 

124.8 

36 

32.1 

16.3 

96 

85.5 

43.6 

56 

139.0 

70.8 

16 

192.5 

98.1 

76 

245.9 

125.3 

37 

33.0 

16.8 

97 

86.4 

44.0 

57 

139.9 

71.3 

17 

193.3 

98.5 

77 

246.8 

125.8 

38 

33.9 

17.3 

98 

87.3 

44.5 

58 

140.8 

71.7 

18 

194.2 

99.0 

78 

247-7 

126.2 

39 

34.7 

'7-7 

99 

88.2 

44.9 

59 

141.7 

72.2 

19 

195.1 

99-4 

79 

248.6 

126.7 

40 
4i 

35.6 

~3'6:y 

18.2 
18.6 

100 

89.1 

45.4 

bo 

142.6 

72.6 

20 

196.0 

99.9 

80 

249.5 

127.1 

lOI 

90.0 

45.9 

161 

143.5 

73.1 

221 

196.9 

100.3 

281 

2  5o.4 

127.6 

42 

37.4 

19. 1 

02 

90.9 

46.3 

62 

144.3 

73.5 

22 

197.8 

100.8 

82 

251.3 

128.0 

43 

38.3 

19.5 

o3 

91.8 

46.8 

63 

145.2 

74.0 

23 

198.7 

101.2 

83 

252.2 

128.5 

U 

39.2 

20.0 

04 

92.7 

47-2 

64 

I46.I 

74.5 

24 

199.6 

101.7 

84 

253.0 

128.9 

45 

4o.  I 

20.4 

o5 

93.6 

47-7 

65 

i47-o 

74.9 

25 

2CJ0.5 

102. 1 

85 

253.9 

129.4 

46   4i.o 

20.9 

06 

94.4 

48.1 

66 

147.9 

7^.4 

26 

201.4 

102.6 

86 

254.8 

129.8 

47 

4i  .91 21.3 

07 

95.3 

48.6 

67 

i48.8 

75.8 

27 

202.3 

io3.i 

87 

255.7 

i3o.3 

48 

42.8 

21.8 

08 

96.2 

49.0 

68 

149.7 

76.3 

28 

203.1 

io3.5 

88 

256.6 

i3o.7 

49 

43.7 

22.2 

09 

97.1 

49.5 

69 

i5o.6 

76.7 

29 

204.0 

104.0 

89 

257.5 

l3l.2 

5o 

44.6 

22.7 

10 

98.0 

49-9 

70 

i5i.5 

77.2 

3o 

204.9 

104.4 

90 

2  58.4 

i3i.7 

5i 

45.4 

23.2 

III 

98.9 

5o.4 

171 

i52.4 

77.6 

23l 

2o5.8 

104.9 

291 

259.3 

i3i.i 

52 

46.3 

23.6 

12 

99.8 

5o.8 

72 

i53.3 

78.1 

32 

206.7 

io5.3 

92 

260.2 

i32.6 

53 

47-2 

24.1 

i3 

100.7 

5i.3 

73 

i54.i 

78.5 

33 

207.6 

io5.8 

93 

261. 1 

i33.o 

54 

48.1 

24.5 

i4 

loi  .6 

5i.8 

74 

i55.o 

79.0 

34 

208.5 

106.2 

94 

262.0 

i33.5 

55 

49.0 

25. 0 

i5 

102.5 

52.2 

75 

155.9 

79-4 

35 

209.4 

106.7 

95 

262.8 

133.9 

56 

49.9 1 25.4 

16 

io3.4 

52.7 

76 

i56.8 

79-9 

36 

210.3 

107.1 

96 

263.7 

134.4 

i)7 

5o.8    25.9 

17 

104.2 

53.1 

77 

157.7 

80.4 

37 

211.2 

107.6 

97 

264.6 

134.8 

58 

5i.7    26.3 

18 

io5.i 

53.6 

78 

i58.6 

80.8 

38 

212. 1 

108.0 

98 

265.5 

i35.3 

59 

52.6    26.8 

19 

106.0 

54.0 

79 

159.5 

81.3 

39 

2l3.0 

108.5 

99 

266.4 

135.7 

bo 

53.5I27.2 

20 

106.9 

54.5 

80 

160.4 

81.7 

40 

2i3.8 

109.0 

3oo 

267.3 

i36.2 

Disi. 

Dop.  i  Lat 

Dist. 

Dep. 

Lat. 

Dist. 

Dep. 

Lat. 

Dist. 

Dep. 

Lat. 

Dist.J  Dep. 

Lat. 

[For  C3  Degrees. 

Piige  4-1] 

TABLE  IL 

Difference  of  Latitude  and  Departu 

re  for  23  Degrees. 

Dist. 

Lat. 

Dep. 

Dist. 

Lat. 

Dep. 

Dist. 

Lat. 

Dep. 

Dist. 

Lat. 

Dep. 

Dist.J   Lat.  1 

Dep. 

I 

00.9 

00.5 

61 

53.9 

28.6 

121 

]o6.8 

56.8 

181 

159.8 

85.0 

241 

212.8 

ii3,i 

2 

01.8 

00.9 

62 

54.7 

29. 1 

22 

107.7 

57.3 

82 

160.7 

85.4 

42 

213.7 

ii3.6 

3 

02.6 

01.4 

63 

55.6 

29.6 

23 

108.6 

57.7 

83 

161.6 

85.9 

43 

214.6 

ii4.i 

4 

o3.5 

01 .9 

64 

56.5 

3o.o 

24 

109.5 

58.2 

84 

162.5 

86.4 

44 

2i5.4 

114-6 

5 

04.4 

02.3 

65 

57.4 

3o.5 

25 

no. 4 

58.7 

85 

i63.3 

86.9 

45 

216.3 

ii5.o 

fi 

o5.3 

02.8 

66 

58.3 

3i  .0 

26 

III  .3 

59.2 

86 

i64-2 

87.3 

46 

217.2 

ii5.5 

7 

06.2 

o3.3 

67 

59.2 

3i.5 

27 

112. 1 

59.6 

87 

i65.i 

87.8 

47 

218.1 

1 16.0 

8 

07.1 

o3.8 

68 

60.0 

3i.9 

28 

ii3.o 

60.1 

88 

166.0 

88.3 

48 

219.0 

1 16.4 

9 

07.9 

04.2 

69 

60.9 

32.4 

29 

113.9 

60.6 

89 

166.9 

88.7 

49 

219.9 

116.9 

10 

08.8 

04.7 

70 

61.8 

32.9 
33.3 

3o 

114. 8 

61.0 

90 

167.8 

89.2 

5o 

220.7 

1 17.4 
1 17.8 

II 

09.7 

o5.2 

71 

62.7 

i3i 

115.7 

61.5 

191 

168.6 

89.7 

25l 

221.6 

12 

10.6 

o5.6 

72 

63.6 

33.8 

32 

116. 5 

62.0 

92 

169.5 

90.1 

52 

222.5 

118.3 

t3 

II. 5 

06.1 

73 

64.5 

34.3 

33 

117. 4 

62.4 

93 

170.4 

90.6 

53 

223.4 

118.8 

i4 

12.4 

06.6 

74 

65.3 

34.7 

34 

118. 3 

62.9 

94 

171.3 

91.1 

54 

224.3 

119.2 

i5 

l3.2 

07,0 

75 

66.2 

35.2 

35 

119. 2 

63.4 

95 

172.2 

91.5 

55 

225.2 

1 19.7 

i6 

i4.i 

07.5 

76 

67.1 

35.7 

36 

120.1 

63.8 

96 

173.1 

92.0 

5b 

226.0 

120.2 

17 

i5.o 

08.0 

77 

68.0 

36.1 

37 

121 .0 

64.3 

97 

173.9 

92.5 

57 

226.9 

120.7 

i8 

i5.9 

08.5 

78 

68.9 

36.6 

38 

121. 8 

64.8 

98 

174.8 

93.0 

58 

227.8 

121. 1 

19 

16.8 

08.9 

79 

69.8 

37.1 

39 

122.7 

65.3 

99 

175.7 

93.4 

59 

228.7 

1 2 1. 6 

20 

17.7 

09.4 

80 

70.6 

37.6 

40 

123.6 

65.7 

200 

176.6 

93.9 

60 
261 

229.6 

23o.4 

122.1 

21 

18.5 

09.9 

81 

71.5 

38. 0 

i4i 

124.5 

66.2 

201 

177.5 

94.4 

122.5 

22 

19.4 

10.3 

82 

72.4 

38.5 

42 

125.4 

66.7 

02 

178.4 

94.8 

62 

231.3 

123.0 

23 

20.3 

10.8 

83 

73.3 

39.0 

43 

126.3 

67.1 

OJ 

179.2 

95.3 

63 

232.2 

123.5 

24 

21  .2 

II. 3 

84 

74.2 

39.4 

44 

127. 1 

67.6 

o4 

180.1 

95.8 

64 

233.1 

123.9 

25 

22.1 

II. 7 

85 

75.1 

39.9 

45 

128.0 

68.1 

o5 

181.0 

96.2 

65 

234.0 

124.4 

26 

23. 0 

12.2 

86 

75.9 

40.4 

46 

128.9 

68.5 

06 

181.9 

96.7 

66 

234.9 

124-9 

27 

23.8 

12.7 

87 

76.8 

40.8 

47 

129.8 

69.0 

07 

182.8 

97.2 

67 

235.7 

125,3 

28 

24.7 

i3.i 

88 

77-7 

4i.3 

48 

i3o.7 

69.5 

08 

183.7 

97-7 

68 

236.6 

125.8 

29 

25.6 

i3.6 

89 

78.6 

4i.8 

49 

i3i.6 

70.0 

09 

184.5 

98.1 

69 

237.5 

126.3 

3o 

26.5 

i4.i 

00 

79-i> 

42.3 

5o 

i32.4 

70.4 

10 

i85.4 

98.6 

70 

238.4 

126.8 

3i 

27.4 

i4.6 

91 

80.3 

42.7 

i5i 

i33.3 

70.9 

211 

186.3 

99.1 

271 

239.3 

127.2 

32 

28.3 

i5.o 

92 

81.2 

43.2 

52 

i34.2 

71-4 

12 

187.2 

99.5 

72 

240.2 

127.7 

33 

29.1 

i5.5 

93 

82.1 

43.7 

53 

i35.i 

71.8 

i3 

188.1 

100. 0 

73 

241.0 

128.2 

34 

3o.o 

16.0 

94 

83. 0 

44.1 

54 

1 36.0 

72.3 

i4 

189.0 

100.5 

74 

241.9 

128.6 

35 

30.9 

16.4 

95 

83.9 

44.6 

55 

1 36. 9 

72.8 

i5 

189.8 

100.9 

75 

242.8 

129.1 

36 

3i.8 

16.9 

96 

84.8 

45.1 

56 

137.7 

73.2 

16 

190.7 

101.4 

lb 

243.7 

129.6 

37 

32.7 

17.4 

97 

85.6 

45.5 

57 

i38.6 

73.7 

17 

191.6 

101.9 

77 

244.6 

i3o.o 

38 

33.6 

17.8 

98 

86.5 

46.0 

58 

139.5 

74.2 

18 

192.5 

102.3 

78 

245.5 

i3o.5 

3q 

34.4 

18.3 

99 

87.4 

46.5 

59 

i4o.4 

74.6 

19 

iy3.4 

102.8 

79 

246.3 

i3i.o 

40 

35.3 

18.8 

100 

88.3 

46.9 

60 

i4i.3 

75.1 

20 

194.2 

io3.3 

80 

247.2 

i3i.5 

4i 

36.2 

19.2 

lOI 

89.2 

47-4 

161 

142.2 

75.6 

221 

195. 1 

io3.8 

2S1 

248.1 

1 3 1. 9 

42 

37.1 

19.7 

02 

90.1 

47-9 

62 

143.0 

76.1 

22 

196.0 

104.2 

82 

249.0 

i32.4 

43 

38.0 

20.2 

o3 

90.9 

48.4 

63 

143.9 

76.5 

23 

196.9 

104.7 

83 

249.9 

132.9 

44 

38.8 

20.7 

o4 

91.8 

48.8 

64 

144.8 

77.0 

24 

197.8 

io5.2 

84 

25o.8 

i33.3 

45 

39.7 

21 . 1 

o5 

92.7 

49-3 

65 

145.7 

77.5 

25 

198.7 

105.6 

85 

25l.b 

133.8 

46 

4o.6 

21 .6 

06 

93.6 

49-8 

66 

146.6 

77-9 

26 

199.5 

106. 1 

8b 

252.5 

1 34-3 

47 

4i.5 

22.1 

07 

94.5 

5o.2 

67 

147-5 

78.4 

27 

200.4 

106.6 

87 

253.4 

134.7 

48 

42.4 

22.5 

08 

95.4 

50.7 

68 

148.3 

78.9 

28 

201.3 

107.0 

88 

254.3 

i35.2 

49 

43.3 

23.0 

09 

96.2 

5l.2 

69 

149.2 

79.3 

29 

202.2 

107.5 

89 

255.2 

i35.7 

5o 
5 1 

44.1 
45.0 

23.5 
23.9 

10 

97.1 

5i.6 

70 

i5o.i 

79.8 

3o 

203.1 

108.0 

90 

256.1 

1 36. 1 

III 

98.0 

52.1 

171 

iSi.o 

80.3 

23l 

204.0 

108.4 

291 

256.9 

1 36.6 

52 

45.9 

24.4 

12 

98.9 

52.6 

72 

i5i  .9 

80.7 

32 

204.8 

108.9 

92 

257.8 

137.1 

53 

46.8 

24.9 

i3 

99.8 

53.1 

73 

i52.7 

81.2 

33 

205.7 

109.4 

93 

258.7 

137.6 

54 

47.7 

25.4 

i4 

100.7 

53.5 

74 

i53.6 

81.7 

34 

206.6 

109.9 

94 

259.6 

)38.o 

55 

48.6 

25.8 

i5 

loi  .5 

54.0 

75 

i54.5 

82.2 

35 

207.5 

no  3 

9b 

260.5 

i38.5 

56 

49-4 

26.3 

16 

102.4 

54.5 

76 

i55.4 

82.6 

36 

208.4 

1 10.8 

96 

261.4 

139.0 

57 

5o.3 

26.8 

17 

io3.3 

54.9 

77 

i56.3 

83.1 

37 

209.3 

111.3 

97 

262.2 

139.4 

58 

5i  .2 

27.2 

18 

104.2 

55.4 

78 

157.2 

83.6 

38 

210. 1 

1 1 1.7 

98 

263.1 

139.9 

5q 

52.1 

27.7 

19 

io5.i 

55.9 

"9 

i58.o 

84.0 

39 

211.0 

112.2 

99 

264.0 

140.4 

6o_ 

Dist" 

53.0 

28.2 

20 

106.0 

56.3 

80 

i58.9 

84.5 

40 

21  1.9 

112.7 

3oo 

264.9 

i4o.8 

Dep. 

1   Lat. 

Dist 

Dep. 

Lat. 

Dist. 

Dep. 

Lat. 

Dist. 

Dep. 

Lat. 

Dist. 

Dep. 

Lat. 

Tor  C2  Deg 

rees. 

TABLE  IL 

rr;,;.;.-    I.", 

Differe 

nee  of  Latitude  and  Departure  for  29  Degrees. 

Disl. 

Lat. 

Dcp. 

Dist. 

Lat. 

Dep. 

Dist. 

Lat. 

Dcp. 

58.7 

Dist. 

Lat. 

Dep. 

Dist. 

Lai. 

Dep. 

I 

00.9 

00.5 

61 

53.4 

29.6 

121 

io5.8 

iSi 

i58.3 

S7.8 

241 

210.8 

1 16.8 

2 

01.7 

01 .0 

62 

54.2 

3o.i 

22 

106.7 

59.1 

82 

159.2 

88.2 

42 

211.7 

117.3 

3 

02.6 

01 .5 

63 

55.1 

3o.5 

23 

107.6 

59.6 

83 

1 60. 1 

88.7 

43 

212.5 

117-8 

4 

o3.5 

01 .9 

64 

56.0 

3i  .0 

24 

108.5 

60.1 

84 

160.9 

89.2 

44 

2 1 3.4 

118.3 

5 

o4.4 

02.4 

65 

56.9 

3i.5 

25 

109.3 

60.6 

85 

161.8 

89-7 

45 

214.3 

118.8 

6 

o5.2 

02.9 

66 

67.7 

32.0 

26 

no. 2 

61.1 

86 

162.7 

90.2 

46 

2l5.2 

119.3 

7 

06.1 

o3.4 

67 

58.6 

32.5 

27 

III. I 

61.6 

87 

i63.6 

90.7 

47 

2l6.0 

119.7 

8 

07.0 

03.9 

68 

59.5 

33.0 

28 

112. 0 

62.1 

88 

164.4 

91. 1 

48 

216.9 

120.2 

9 

07.9 

04.4 

69 

60.3 

33.5 

29 

112. 8 

62.5 

89 

i65.3 

91.6 

49 

217.8 

120.7 

10 

08.7 

04.8 

70 

61 .2 

33.9 

3o 

II3.7 

63.0 

90 

166.2 

92.1 

5o 

218.7 

121.2 

II 

09.6 

o5.3 

71 

62. 1 

34.4 

i3i 

114.6 

63.5 

191 

167.1 

92.6 

25l 

219.5 

121.7 

12 

10. b 

o5.8 

72 

63.0 

34.9 

32 

II5.4 

64.0 

92 

167.9 

93.1 

52 

220.4 

122.2 

i3 

U.4 

06.3 

73 

63.8 

35.4 

33 

116.3 

64.5 

93 

168.8 

93.6 

53 

22  1.3 

122.7 

i4 

12.2 

06.8 

74 

64.7 

35.9 

34 

117. 2 

65. 0 

94 

169.7 

94.1 

54 

222.2 

123.1 

ID 

i3.i 

07.3 

75 

65.6 

36.4 

35 

118. 1 

65.4 

95 

170.6 

94-5 

55 

223.0 

123.6 

i6 

i4-o 

07.8 

76 

66.5 

36.8 

36 

118. 9 

65.9 

96 

I7I-4 

95.0 

56 

223.9 

1 24. 1 

17 

14.9 

q8.2 

77 

67.3 

37.3 

37 

119.8 

66.4 

97 

172.3 

95.5 

57 

224.8 

124.6 

i8 

lb. 7 

08.7 

7S 

68.2 

37.8 

38 

120.7 

66.9 

98 

173.2 

96.0 

58 

225.7 

125.1 

'9 

lb. 6 

09.2 

79 

69.1 

38.3 

39 

121.6 

67.4 

99 

174-0 

96.5 

59 

226.5 

125.6 

20 

17. b 

09.7 

80 

70.0 

.-.8.8 
39.3 

4o 
T41 

122.4 

67.9 

200 

174-9 

97.0 

60 
261 

227.4 
228.3 

126.1 

21 

18.4 

10.2 

81 

70.8 

123.3 

68.4 

201 

175.8 

97-4 

126.5 

22 

19.2 

10.7 

82 

71.7 

39.8 

42 

124.2 

68.8 

02 

176.7 

97-9 

62 

229.2 

127.0 

23 

20.1 

II  .2 

83 

72.6 

40.2 

43 

125. 1 

69.3 

o3 

177.5 

98.4 

63 

23o.O 

127  5 

24 

21 .0 

II. 6 

84 

73.  b 

40.7 

44 

125.9 

69.8 

04 

178-4 

98.9 

64 

230.9 

128.0 

2b 

21  .9 

12. I 

8b 

74.3 

4l.2 

4b 

126.8 

70.3 

o5 

179.3 

99.4 

65   231.8 

128.5 

26 

22.7 

12.6 

86 

75.2 

41.7 

46 

127.7 

70.8 

06 

180.2 

99.9 

66 

232.6 

129.0 

27 

23.6 

i3.i 

^7 

7b. I 

42.2 

47 

128.6 

71.3 

07 

181.0 

100.4 

67 

233.5 

129.4 

28 

24. b 

i3.6 

88 

77.0 

42.7 

48 

129.4 

71.8 

08 

181.9 

100.8 

68 

234.4 

129.9 

29 

2b. 4 

i4.i 

89 

77.8 

43.1 

49 

i3o.3 

72.2 

09 

182.8 

101.3 

69 

235.3 

i3o.4 

3o 

26.2 

i4.5 

90 

78.7 

43.fi 

bo 

l3l.2 

72.7 

10 

183.7 

101.8 

70 

236.1 

i3o.9 

3i 

27.1 

i5.o 

91 

79.6 

44.1 

:5i 

I32.I 

73.2 

211 

i84-5 

102.3 

271 

237.0 

i3i.4 

32 

28.0 

i5.5 

92 

80.5 

44.6 

52 

132.9 

73.7 

12 

i85.4 

102.8 

72 

237-9 

i3i.9 

33 

28.9 

16.0 

93 

81.3 

45.1 

53 

i33.8 

74.2 

i3 

186.3 

io3.3 

73 

238.8 

i32.4 

34 

29.7 

16.5 

94 

82.2 

45.6 

54 

134.7 

74.7 

i4 

187.2 

io3.7 

74 

239.6 

i32.8 

3i) 

3o.b 

17.0 

95 

83.1 

46.1 

55 

i35.6 

75.1 

i5 

188.0 

104.2 

75 

240.5 

i33.3 

36 

3i.5 

17.5 

96 

84.0 

46.5 

56 

i36.4 

75.6 

16 

188.9 

104.7 

76 

241.4 

1 33.8 

37 

32.4 

17.9 

97 

84.8 

47-0 

57 

i37.3 

76.1 

17 

189.8 

10D.2 

77 

242.3 

1 34.3 

38 

33.2 

iS.4 

98 

8b. 7 

47-5 

58 

i38.2 

76.6 

18 

190.7 

105.7 

78 

243.1 

i34.8 

39 

34.1 

.8.9 

99 

86.6 

48.0 

b9 

139.1 

77-1- 

19 

191. 5 

106.2 

79 

244.0 

i35.3 

4o 
4i 

3b.  0 

19.4 

100 

87. b 

48. b 

bo 

139.9 

77.6 

20 

192.4 

106.7 

80 

244-9 

i35.7 

35.9 

19.9 

lOI 

88.3 

49.0 

161 

140.8 

78.1 

221 

193.3 

107.1 

281 

245.8 

i36.2 

42 

36.7 

20.4 

02 

89. 2 

49.5 

62 

i4i.7 

78.5 

22 

194.2 

107.6 

82 

246.6 

i36.7 

43 

37.6 

20.8 

o3 

90.1 

49.9 

63 

142.6 

79.0 

23 

195.0 

108.1 

83 

247.5 

137.2 

44 

38. b 

21.3 

04 

91  .0 

5o.4 

64 

143.4 

79.5 

24 

195.9 

108.6 

84 

248.4 

1 37.7 

4b 

39.4 

21.8 

OD 

91.8 

50.9 

65 

144.3 

80.0 

25 

196.8 

109.1 

85 

249.3 

i38.2 

4(> 

4o .  2 

22.3 

06 

92.7 

bi.4 

66 

145.2 

80.5 

26 

197-7 

109.6 

86 

25o.I 

i38.7 

47 

4i.i 

22.8 

07 

93.6 

5i  ;9 

67 

I46.I 

81.0 

27 

198.5 

no. I 

87 

25l.O 

139.1 

48 

42.0 

23.3 

08 

94.5 

52.4 

68 

i46.9 

81.4 

28 

199-4 

no. 5 

88 

251.9 

139.fi 

49 

42.9 

23.8 

09 

9b. 3 

52.8 

69 

i47-8 

81.9 

29 

200.3 

in.o 

89 

252.8 

i4o.i 

bo 

43.7 

24.2 

10 

96.2 

b3.3 
53.8 

70 

148.7 

82.4 
82.9 

3o 

201.2 

in. 5 

90 

253.6 

i4o.6 

5i 

44.6 

24.7 

III 

97.1 

171 

149-6 

23l 

202.0 

112.0 

291 

254.5 

i4i.i 

b2 

4b. b 

25.2 

12 

98.0 

54.3 

72 

i5o.4 

83.4 

32 

202.9 

112. 5 

92 

255.4 

i4i-6 

b3 

46.4 

2b. 7 

i3 

98.8 

54.8 

73 

i5i.3 

83. q 

33 

2o3.8 

it3.o 

93 

256.3 

142.0 

b4 

47.2  t  26.2 

1 4 

99-7 

55.3 

74 

l52.2 

84.4 

34 

204.7 

n3.4 

94 

257.1 

142.5 

bb 

48.1 

2D. 7 

lb 

100.6 

bb.8 

75 

i53.i 

84.8 

35 

2o5.5 

1 13.9 

95 

258.0 

143.0 

bb 

49.0 

27.1 

16 

loi  .5 

56.2 

76 

1 53. 9 

85.3 

36 

206.4 

n4.4 

96 

2  58.9 

143.5 

t)7 

49.9 

27.6 

17 

102.3 

56.7 

77 

i54.8 

85.8 

37 

207.3 

n4-9 

97 

259.8 

144.C 

b8 

5o.7 

28.1 

18 

io3.2 

57.2 

78 

155.7 

86.3 

38 

208.2 

n5.4 

98 

260.6 

144.5 

b9 

bi.6 

28.6 

19 

104. 1 

37-7 

79 

i56.6 

86.8 

39 

209.0 

1 15.9 

99 

261.5 

145.0 

bo 

b2.b 

29.1 

20 

io5.o 

58.2 

8n 

157-4 

87-3 

4o 

209.9 

116.4 

3  00 

262,4    145.4  1 

Oist. 

Dop. 

I.nl. 

Dist. 

Dop. 

Lat. 

Dist. 

])cp. 

Lat. 

Disl.l   Dcp. 

Lnt. 

Dist. 

Dep.      Lat.    ! 

[ 

^or  Gl  Degrees. 

Page  40] 

TABLE   II 

Difference  of  Lati 

tude  and  Departure  for  30  Degrees. 

Dist. 

Lat. 

Dep. 

Dist. 

Lat. 

Dep. 

Dist. 

Lat. 

Dep. 

Dist.j 

Lat. 

Dep. 

Dist.     Lat.  ( 

Dep. 

I 

00.9 

00.3 

bi 

52.8 

3o.5 

121 

104.8 

60.5 

18, 

1 56.8 

90.5 

241 

208.7 

120.5 

2 

01.7 

01 .0 

62 

53.7 

3i  .0 

22 

105.7 

61 .0 

82 

157.6 

91.0 

42 

209.6 

I  21.0 

3 

02.6 

01 .5 

63 

54.6 

3i.5 

23 

106.5 

61.5 

83 

i58.5 

91.5 

43 

210.4 

121.5 

4 

o3.5 

02.0 

64 

55.4 

32.0 

24 

107.4 

62.0 

84 

159.3 

92.0 

44 

211. 3 

122.0 

5 

04.3 

02.5 

65 

56.3 

32.5 

25 

108.3 

62.5 

85 

160.2 

92.5 

45 

212.2 

122.5 

6 

05.2 

o3.o 

66 

57.2 

33.0 

26 

109. 1 

63.0 

86 

161. 1 

93.0 

46 

2l3.0 

123.0 

7 

06.1 

o3.5 

67 

58. 0 

33.5 

27 

IIO.O 

63.5 

87 

161.9 

93.5 

47 

213.9 

123.5 

8 

06.9 

04.0 

68 

58.9 

34.0 

28 

no. 9 

64.0 

88 

162.8 

94.0 

48 

214.8 

124.0 

9 

07.8 

04.5 

69 

59.8 

34.5 

29 

III. 7. 

64.5 

89 

163.7 

94.5 

49 

215.6 

124.5 

lO 

08.7 

o5.o 

70 

60.6 

35.0 
35.5 

3o 

112. 6 

65. 0 
65.5 

90 

164.5 

95.0 

5o 

216.5 

125.0 

II 

09.5 

o5.5 

71 

61.5 

i3i 

ii3.4 

191 

i65.4 

95.5 

25l 

217-4 

125.5 

12 

10.4 

ob.o 

72 

62.4 

36.0 

32 

114.3 

66.0 

92 

166.3 

96.0 

52 

218.2 

126.0 

i3 

II. 3 

ob.5 

73 

63.2 

36.5 

33 

Il5.2 

66.5 

93 

167. 1 

96.5 

53 

2  1 9. 1 

12fi.5 

i4 

12. 1 

07.0 

74 

64.1 

37.0 

34 

116. 0 

67.0 

94 

168.0 

97.0 

54 

220.0 

1 27.0 

lb 

i3.o 

07.5 

75 

65.0 

37.5 

35 

lib. 9 

67.5 

95 

168.9 

97.b 

55 

220.8 

127.5 

lb 

i3.9 

08.0 

76 

65.8 

38. 0 

36 

117. 8 

68.0 

96 

169.7 

98.0 

56 

221.7 

128.0 

17 

14.7 

08. b 

77 

66.7 

38.5 

37 

118. 6 

68.5 

97 

170.6 

98.5 

57 

222.6 

128.5 

i8 

ib.b 

09.0 

78 

67.5 

39.0 

38 

119. 5 

69.0 

98 

171.5 

99.0 

58 

223.4 

129.0 

19 

ib.b 

09.5 

■79 

68.4 

39.5 

39 

120.4 

69.5 

99 

172.3 

99.5 

59 

224.3 

129.5 

20 

17.3 

10. 0 

80 

69.3 

4o.o 

4o 

121 .2 

70.0 

200 

173.2 

100. 0 

60 

225.2 

i3o.o 

21 

18.2 

10.5 

81 

70.1 

40.5 

i4i 

122. 1 

70.5 

201 

1 74. 1 

100.5 

261 

226.0 

i3o.5 

22 

19. 1 

II  .0 

82 

71.0 

4i  .0 

42 

123.0 

71.0 

02 

174.9 

lOI.O 

62 

226.9 

i3i.o 

23 

19.9 

II. 5 

83 

71.9 

4i.5 

43 

123.8 

71.5 

o3 

175.8 

101.5 

63 

227.8 

i3i.5 

24 

20.8 

12.0 

84 

72.7 

42.0 

44 

124.7 

72.0 

04 

176.7 

102.0 

64 

228.6 

i32.o 

2b 

21.7 

12. b 

85 

73.6 

42.5 

45 

125.6 

72.5 

o5 

177.5 

102.5 

65 

229.5 

i32.5 

2b 

22.5 

i3.o 

86 

74.5 

43.0 

46 

126.4 

73.0 

06 

178.4 

io3.o 

66 

23o.4 

i33.o 

27 

23.4 

i3.5 

87 

75.3 

43.5 

47 

127.3 

73.5 

07 

179.3 

io3.5 

67 

23l.2 

1 33.5 

28 

24.2 

14.0 

88 

76.2 

44.0 

48 

128.2 

74-0 

08 

180.1 

104.0 

68 

232.1 

i34.o 

29 

2b. I 

14.5 

89 

77-1 

44.5 

49 

129.0 

74.5 

09 

181.0 

104.5 

69 

233.0 

i34.5 

So 
3i 

26.0 

ib.o 

90 
91 

77.9 
78.8 

45.0 
45.5 

5o 

129.9 

75.0 

10 

181.9 

io5.o 

70 

233.8 

1 35.0 

i5.5 

i5i 

i3o.8 

75.5 

211 

182.7 

io5.5 

271 

234.7 

i35.5 

32 

27-7 

16.0 

92 

79-7 

46.0 

52 

i3i.6 

76.0 

12 

i83.6 

106.0 

72 

235.6 

1 36.0 

33 

28. b 

16.5 

93 

80.5 

46.5 

53 

i32.5 

76.5 

i3 

184.5 

106.5 

73 

236.4 

136.5 

34 

29.4 

17.0 

94 

81.4 

47-0 

54 

i33.4 

77.0 

i4 

i85.3 

107.0 

74 

237.3 

137.0 

3b 

3c. 3 

.7.5 

95 

82.3 

47. b 

55 

i34.2 

77.5 

i5 

186.2 

107.5 

75 

238.2 

137.5 

3b 

3l.2 

18.0 

96 

83.1 

48. 0 

56 

i35.i 

78.0 

16 

187. 1 

108.0 

76 

239.0 

i38.o 

J7 

32.0 

18.5 

97 

84.0 

48. b 

57 

1 36.0 

78.5 

17 

187.9 

10S.5 

77 

239.9 

138.5 

38 

32  .9 

19.0 

98 

84.9 

49.0 

58 

i36.8 

79.0 

18 

188.8 

109.0 

78 

240.8 

139.0 

39 

33.8 

.9.5 

99 

85.7 

49-^ 

59 

137.7 

79.5 

19 

189.7 

109.5 

79 

241.6 

139.5 

40 

34. b 

20.0 

100 

86.6 

5o.o 

60 

i38.6 

80.0 

20 

190. D 

IIO.O 

80 

242.5 

i4o.o 

4i 

35.5 

20.5 

lOI 

87.5 

5o.5 

161 

139.4 

80.5 

221 

191. 4 

1 10.5 

281 

243.4 

i4o.5 

42 

3b. 4 

21 .0 

02 

88.3 

bi.o 

62 

i4o.3 

81.0 

22 

192.3 

II  1. 0 

82 

244.2 

i4i-o 

4S 

37.2 

21 .5 

o3 

89.2 

5i.5 

63 

i4i  .2 

81.5 

2  3 

193. 1 

1 1 1.5 

83 

245.1 

i4i.5 

44 

38.1 

22.0 

04 

90.1 

52.0 

64 

142.0 

82.0 

24 

194.0 

1 1 2.0 

84 

246.0 

142.0 

4b 

39.0 

22.5 

o5 

90.9 

52. b 

65 

142.9 

82.5 

25 

194.9 

112. 5 

85 

246.8 

142.5 

4b 

39.8 

23.0 

06 

91.8 

b3.o 

66 

143.8 

83.0 

26 

195.7 

ii3.o 

86 

247-7 

143.0 

47 

4o .  7 

23.5 

07 

92.7 

b'3.b 

67 

144.6 

83.5 

27 

196.6 

ii3.5 

87 

248.5 

143.5 

48 

4i.6 

24.0 

08 

93.5 

54.0 

68 

145.5 

84.0 

28 

197.5 

ii4-o 

88 

249.4 

i44-o 

^9 

42.4 

24.5 

09 

94.4 

b4.b 

69 

146.4 

84.5 

29 

198.3 

114.5 

89 

25o.3 

144.5 

bo 

43.3 

25.0 

10 

95.3 

bb.o 

70 

i47-2 

85. 0 

3o 

199.2 

ii5.o 

90 

25l.I 

145.0 

bi 

44.2 

25.5 

1 1 1 

9b.  I 

55.5 

171 

I48.I 

85.5 

23l 

200.1 

ii5.5 

291 

252.0 

145.5 

b2 

45.0 

26.0 

12 

97 -o 

bb.o 

72 

149.0 

86.0 

32 

200.9 

1 16.0 

92 

252.9 

1 46.0 

b3 

4b. 9 

2b. 5 

i3 

97.9   56.5 

73 

149-8 

86.5 

33 

201.8 

1 16.5 

93 

253.7 

146.5 

b4 

4b. 8 

27.0 

i4 

98.7 

b7.o 

74 

1 5o .  7 

87.0 

34 

202.6 

1 17.0 

94 

254-6 

147.0 

bb 

47. b 

27.5 

i5 

99.6 

bv.b 

75 

i5i.6 

87.5 

35 

2o3.5 

117.5- 

95 

255.5 

147-5 

bb 

48. b 

28.0 

lb 

100.5 

b8.o 

76 

152.4 

88.0 

36 

204.4 

118.0 

96 

2  56.3 

i48.o 

b7 

49.4 

28.5 

17 

loi  .3 

58.5 

77 

i53.3 

88.5 

37 

205.2 

118.5 

97 

257.2 

148.5 

b8 

5().2 

29.0 

18 

102.2 

59.0 

78 

154.2 

89.0 

38 

206.1 

1 19.0 

98 

258.1 

149-0 

b9 

bi.i 

29.5 

>9 

io3.i 

59.5 

79 

i55.o 

89.5 

39 

207.0 

119. 5 

99 

258.9 

149.5 

60 

52.0 

3o .  f> 

20 

103.9 

60 .  0 

80 
Dist. 

155.9 

90.0 

40 

207.8 

120.0 

3oo 

259.8 

i5o.o 

Dist. 

n.-p. 

l.nt. 

Dist. 

Dcp. 

I^at, 

Dcp. 

Lat. 

Dist.j  Dep. 

Lat. 

Dist. 

Dep. 

Lat. 

[ 

For  GO  Degrees. 

TABLE   IL 

[!• 

lye  47 

Difference  of  Latitude  and  Departure  for  31  Degrees. 

Dist. 

Lai. 

Dep. 

Dist. 

Lat. 

Dep. 

Dist. 

Lat. 

Dep. 

Dist. 

Lat. 

Dep. 

Dist. 

Lat.    [ 

Dep. 

I 

00.9 

00.5 

61 

52.3 

3i.4 

121 

io3.7 

62.3 

181 

i55.i 

93.2 

241 

206.6 

124. 1 

2 

01 .7 

01 .0 

62 

53.1 

3i  .9 

22 

104.6 

62.8 

82 

1 56.0 

93-7 

42 

207.4 ; 

12-4  6 

3 

02.6 

01.5 

63 

54.0 

32.4 

23 

105.4 

63.3 

83 

1 56.9 

94.3 

43 

208.3  ■ 

125.2 

4 

o3.4 

02.1 

64 

54.9 

33.0 

24 

106.3 

63.9 

84 

157.7 

94.8 

AA 

209.1 1 

125.7 

S 

04.3 

02.6 

65 

55.7 

33.5 

25 

107. 1 

bA.A 

85 

1 58.6 

95.3 

45 

210.0 

126.2 

6 

o5.i 

o3.i 

66 

56.6 

34.0 

26 

108.0 

64-9 

86 

159.4 

95.8 

46 

210.9 

126.7 

06.0 

o3.6 

67 

57.4 

34.5 

27 

108.9 

65.4 

87 

160.3 

96.3 

47 

21 1.7 

127.2 

8 

06.9 

04.1 

68 

58.3 

35.0 

28 

109.7 

65.9 

88 

161. 1 

96.8 

48 

212.6 

127.7 

9 

07.7 

04.6 

69 

59.1 

35.5 

29 

no. 6 

66.4 

89 

162.0 

97-3 

49 

2  13.4 

128.2 

10 

08.6 

o5 .2 

70 

60.0 

3b. I 

3o 

III  .4 

67.0 

90 
191 

162.9 

763".7 

97-9 

5o 

214.3 

128.8 

1 1 

09.4 

05.7 

71 

60.9 

36.6 

i3i 

112. 3 

67.5 

98.4 

25l 

2l5.I 

129.3 

12 

10.3 

06.2 

72 

61.7 

37.1 

32 

u3.i 

68.0 

92 

164.6 

98.9 

52 

216.0 

129.8 

i3 

II  .1 

06.7 

73 

62.6 

37.6 

33 

ii4-o 

68.5 

93 

1 65. 4 

99.4 

53 

216.9 

i3o.3 

i4 

12.0 

07.2 

74 

63.4 

38.1 

M 

114-9 

69.0 

94 

166.3 

99.9 

54 

217.7 

i3o.8 

i5 

12.9 

07.7 

75 

64.3 

38. b 

35 

115.7 

69.5 

95 

167. 1 

100.4 

55 

218.6 

i3i.3 

i6 

i3.7 

08.2 

76 

65.1 

39.1 

36 

116. 6 

70.0 

96 

168.0 

100.9 

56 

219.4 

i3i.8 

17 

i4.6 

08.8 

77 

66.0 

39.7 

37 

117.4 

70.6 

97 

168.9 

101.5 

57 

220.3 

i32.4 

i8 

i5.4 

09.3 

78 

66.9 

4o.2 

38 

118. 3 

71. 1 

98 

169.7 

102.0 

58 

221.1 

i32.9 

19 

16.3 

09.8 

79 

67.7 

40.7 

39 

119. 1 

71.6 

99 

170.6 

102.5 

59 

222.0 

1 33.4 

20 

17. 1 

10.3 

80 

68.6 

4i  .2 

4o 

120.0 

72.1 

200 

171.4 

io3.o 

60 
261 

222.9 
223.7 

133.9 

21 

18.0 

10.8 

81 

69.4 

41.7 

i4i 

120.9 

72.6 

201 

172.3 

io3.5 

134.4 

22 

18. 9 

II. 3 

82 

70.3 

42.2 

42 

121 .7 

73.1 

02 

173.1 

104.0 

62 

224.6 

1 34.9 

23 

19.7 

II. 8 

83 

71. 1 

42.7 

43 

122.6 

73.7 

o3 

174.0 

104.6 

63 

225.4 

i35.5 

24 

20.6 

12.4 

84 

72.0 

Ai.i 

AA 

123.4 

74.2 

04 

174-9 

io5.i 

64 

226.3 

i36.o 

25 

21  .4 

12.9 

85 

72.9 

43.8 

45 

124.3 

74.7 

o5 

175-7 

io5.6 

65 

227.1 

i36.5 

i6 

22.3 

i3.4 

86 

73.7 

AA.'^ 

46 

125. 1 

75.2 

06 

176-6 

106.1 

66 

228.0 

137.0 

27 

23.1 

r3.9 

87 

74.6 

44.8 

47 

126.0 

75.7 

07 

177.4 

106.6 

67 

228.9 

137.5 

28 

24.0 

14.4 

88 

7b. 4 

45.3 

48 

126.9 

7b. 2 

08 

178.3 

107.1 

68 

229.7 

i38.o 

99 

24.9 

[4.9 
i5.5 

89 

76.3 

45.8 

49 

127.7 

7b. 7 

09 

1 79. 1 

107.6 

69 

2  3o.6 

i38.5 

3o 
3i 

25.7 

90 

77-1 

4b. 4 

5o 

128.6 

77-3 

10 

180.0 

108.2 

70 

23i.4 

139. 1 

56.6 

16.0 

91 

78.0 

46.9 

i5i 

129.4 

77.8 

211 

180.9 

108.7 

271 

232.3 

139.6 

32 

27.4 

16.5 

92 

78.9 

47-4 

52 

i3o.3 

78.3 

12 

181.7 

109.2 

72 

233.1 

1 40. 1 

33 

28.3 

17.0 

93 

79-7 

47-9 

53 

i3i.i 

78.8 

i3 

182.6 

109.7 

73 

234.0 

i4o.6 

34 

29. 1 

17.5 

94 

80.6 

48.4 

54 

l32.0 

79-3 

i4 

i83.4 

110.2 

74 

234.9 

i4i-i 

35 

3o.o 

18.0 

95 

81.4 

48.9 

55 

i32.9 

79.8 

i5 

184.3 

110.7 

75 

235.7 

i4i-6 

3G 

3o.Q 

18.5 

96 

82.3 

49-4 

56 

i33.7 

80.3 

16 

i85.i 

111.2 

76    236.6 

142.2 

3? 

3. .7 

19. 1 

97 

83.1 

5o.Q 

57 

i34.6 

80.9 

17 

1S6.0 

111.8 

77  '  237.4 

142.7 

38 

32.6 

19.6 

08 

84.0 

5o.5 

58 

i35.4 

81.4 

18 

1S6.9 

112.3 

78 

238.3 

143.2 

39 

33.4 

20.1 

99 

84.9 

5i.o 

59 

i36.3 

81.9 

'9 

187.7 

112.8 

79 

239.1 

143.7 

4o 

34.3 

20.6 

100 

85.7 

5i.5 

60 

137. 1 

82.4 

20 

188.6 

113.3 

80 

240.0 

144.2 

4i 

35.1 

21 .1 

101 

86.6 

52.0 

161 

i38.o 

82.9 

221 

189.4 

ii3.8 

281 

240.9 

144.7 

42 

36. 0 

21.61      03 

87.4 

52.5 

62 

i38.9 

83.4 

22 

190.3 

114.3 

82 

241.7 

145.2 

43 

36.9 

22.  I 

o3 

88.3 

53.0 

63 

139.7 

84.0 

23 

191. 1 

1 14.9 

83 

242.6 

145.8 

44 

37.7 

22.7 

04 

89.1 

53. b 

64 

1 40.6 

84.5 

24 

192.0 

1.5.4 

84 

243.4 

146.3 

45 

38. 6 

23.2 

o5 

90.0 

54.1 

65 

141.4 

85. 0 

25 

192.9 

115.9 

85 

944.3 

146.8 

46 

39.4 

23.7 

c6 

90.9 

54. b 

66 

142.3 

85.5 

26 

193.7 

116.4 

86 

245.1 

i47-3 

47 

4o:3 

24.2 

07 

91.7 

55.1 

67 

143. 1 

86.0 

27 

194.6 

116.9 

87 

246.0 

147-8 

48 

4i.i 

24.7 

08 

92.6 

55. b 

68 

144.0 

8t3.5 

28 

195.4 

1 17-4 

88 

246.9 

i48.3 

49 

42.0 

25.2 

09 

93.4 

5b.  I 

69 

144.9 

87.0 

29 

196.3 

"7-9 

89 

247-7 

i48.8 

bo 
5i 

42.9 

25.8 

10 

94.3 

5b. 7 

70 

145.7 

87.6 

3o 

197-1 

1 18.5 

90 

248.6 

149.4 

43.7 

26.3 

II I 

95.1 

57.2 

171 

i46.6 

88.1 

23l 

198.0 

1 19  0 

291 

249.4 

149.9 

52 

44.6 

26.8 

12 

96.0 

b7.7 

72 

147-4 

88.6 

32 

198.9 

1 1 9-5 

92 

250.3 

i5o.4 

53  145.4 

27.3 

i3 

96.9 

58.2 

73 

i48.3 

89.1 

33 

199.7 

120.0 

93 

25l.2 

1 50.9 

54146.3 

27.8 

i4 

97-7 

58.7 

74 

149. 1 

89.6 

34 

200.6 

120.5 

94 

252.0 

.51.4 

55 

47.1 

28.3 

i5 

98.6 

59.2 

75 

i5o.o 

90. 1 

35 

201.4 

121. 0 

95 

252.9 

.51.9 

56 

48.0 

28.8 

16 

99.4 

59.7 

76 

1 50.9 

90.6 

36 

202.3 

121.5 

96 

253.7 

i52.5 

^7 

48.9 

29.4 

17 

100.3 

bo.  3 

77 

i5i.7 

91 .2 

37 

2o3.i 

122. 1 

97 

254.6 

1 53.0 

58 

49.7 

99.9 

18 

lOI  .1 

bo. 8 

78 

i52.6 

91.7 

38 

204.0 

122.6 

98 

255.4 

i53.5 

59 

5n.6 

3o.4 

19 

102.0 

bi.3 

79 

1 53.4 

92.2 

39 

204-9 

123.1 

99 

256.3 

i54.c 

bo 

5i.4 

30.9 

90 

102.9 

bi.8 

80 

i54-3   92.7 

40 

2o5.7 

123.6 

3oo 

2D7.1 

.54.5 

1  ).■,.. 

l.nt. 

nist. 

Dop. 

Lat. 

Disi. 

Dop.      Lat. 

Dist. 

Dep. 

Lat. 

Dist. 

Dep. 

Lat. 

[For  59  Degr 

ees. 

Page  4tj] 

TABLE  11. 

'1 

1 
1 

Differe 

nee  of  Latitude  and  Departure  for  32 

Degrees. 

Dist. 

Lat. 

Dep. 

00.5 

Dist. 

Lat. 

Dep. 

Dist. 

Lat. 

Dep. 

Dist. 

Lat. 

Dep. 

Dist. 

Lat. 

Dep. 

1 27.7 

I 

GO. 8 

61 

5i.7 

32.3 

121 

102.6 

64.1 

181 

i53.5 

95.9 

241 

204.4 

2 

01.7 

01 . 1 

62 

52.6 

32.9 

22 

io3.5 

64-7 

82 

i54.3 

96.4 

42 

205.2 

12S.2 

3 

02.5 

01 .6 

63 

53.4 

33.4 

23 

104.3 

65.2 

83 

i55.2 

97.0 

43 

206.1 

128.8 

4 

o3.4 

02.1 

64 

54.3 

33.9 

24 

io5.2 

65.7 

84 

i56.o 

97.5 

44 

206.9 

129.3 

5 

04.2 

02.6 

65 

55.1 

34.4 

25 

106.0 

66.2 

85 

156.9 

98.0 

45 

207.8 

129.8 

6 

o5.i 

o3.2 

66 

56. 0 

35.0 

26 

106.9 

66.8 

86 

157.7 

98.6 

46 

208.6 

i3o.4 

7 

05.9 

o3.7 

67 

56.8 

35.5 

27 

107.7 

67.3 

87 

i58.6 

99.1 

47 

209.5 

iSo.i; 

8 

06.8 

04.2 

68 

57.7 

36.0 

28 

108.6 

67.8 

88 

159.4 

99.6 

48 

210.3 

i3i.4 

9 

07.6 

04.8 

69 

58.5 

36.6 

29 

109.4 

68.4 

89 

160.3 

100.2 

49 

211.2 

1 3 1. 9 

10 

08.5 

o5.3 

70 

59.4 

37.1 

3o 

no. 2 

68.9 

90 

161. 1 

100.7 

5o 

212.0 

i32.5 

II 

09.3 

o5.8 

7' 

60.2 

37.6 

i3i 

III  .1 

69.4 

191 

162.0 

101.2 

25l 

212.9 

i33.o 

12 

10.2 

06.4 

72 

61. 1 

38.2 

32 

III  .9 

69.9 

Q2 

162.8 

101.7 

52 

213.7 

i33.5 

i3 

II  .0 

06.9 

73 

61 .9 

38.7 

33 

112. 8 

70.5 

93 

163.7 

102.3 

53 

214.6 

i34.i 

i4 

11. 9 

07.4 

74 

62.8 

39.2 

34 

ii3.6 

71.0 

94 

164.5 

102.8 

54 

215.4 

i34.6 

i5 

12.7 

07.9 

75 

63.6 

39.7 

35 

114.5 

71.5 

■  95 

i65.4 

io3.3 

55 

216.3 

i35.i 

i6 

i3.6 

08.5 

76 

64.5 

40.3 

36 

ii5.3 

72.1 

q6 

166.2 

103.9 

56 

217.1 

i35.7 

17 

14.4 

09.0 

77 

65.3 

4o.8 

37 

116.2 

72.6 

97 

167. 1 

104.4 

^7 

217.9 

i36.2 

i8 

i5.3 

09.5 

78 

66.1 

4i.3 

38 

117. 0 

73.1 

98 

167.9 

104.9 

58 

218.8 

i36.7 

I    19 

16. 1 

10. 1 

79 

67.0 

41.9 

39 

117.9 

73.7 

99 

168.8 

io5.5 

59 

219.6 

137.2 

20 

17.0 

10.6 

80 

67.8 

42.4 
42.9 

4o 
i4i 

118. 7 

74.2 

200 

169.6 

106.0 

60 

220.5 

137.8 

21 

17.8 

II  .1 

81 

68. 7 

119. 6 

74.7 

201 

170.5 

106.5 

261 

221.3 

i38.3 

22 

18.7 

II. 7 

82 

69.5 

43.5 

■     42 

120.4 

75.2 

02 

171.3 

107.0 

62 

222.2 

i38.8 

23 

19.5 

12.2 

83 

70.4 

44.0 

43 

121.3 

75.8 

OJ 

172.2 

107.6 

63 

223.0 

139.4 

24 

20.4 

12.7 

84 

71.2 

44.5 

44 

122. 1 

76.3 

o4 

173.0 

108. 1 

64 

223.9 

1399 

25 

21 .2 

l3.2 

85 

72.1 

45.0 

45 

123.0 

76.8 

o5 

173.8 

108.6 

65 

224.7 

i4o.4 

26 

22 .0 

i3.8 

86 

72.9 

45.6 

46 

123.8 

77.4 

06 

174.7 

109.2 

66 

225.6 

i4i.o 

27 

22.0 

i4.3 

87 

73.8 

46.1 

47 

124.7 

77-9 

07 

175.5 

109.7 

67 

226.4 

i4i.5 

28 

23.7 

i4.8 

88 

74.6 

46.6 

48 

125.5 

78.4 

08 

ilb.4 

no. 2 

68 

227.3 

142.0 

29 

24.6 

i5.4 

89 

75.5 

47.2 

49 

126.4 

79.0 

09 

177.2 

no.8 

69 

228.1 

142.5 

3o 

25.4 

i5.9 

90 

76.3 

47-7 

5o 

127.2 

79.5 

10 
211 

178. 1 
178.9 

111.3 

70 

229.0 

i43.i 

3i 

26.3 

16.4 

91 

77.2 

48.2 

i5i 

128. 1 

80.0 

111.8 

271 

229.8 

143.6 

32 

27.1 

17.0 

92 

78.0 

48.8 

52 

128.9 

80.5 

12 

179.8 

112.3 

72 

230.7 

144.1 

33 

28.0 

17.5 

93 

78.9 

49.3 

53 

129.8 

81. 1 

i3 

180.6 

112. 9 

73 

23i.5 

144-7 

34 

28.8 

18.0 

94 

79-7 

49-8 

i.4 

i3o.6 

81.6 

i4 

181.5 

113.4 

74 

232.4 

145.2 

35 

29.7 

18.5 

95 

80.6 

5o.3 

55 

i3i.4 

82.1 

i5 

182.3 

113.9 

75 

233.2 

145.7 

36 

3o.5 

19. 1 

96 

81.4 

50.9 

56 

i32.3 

82.7 

16 

i83.2 

1 14.5 

76 

234.1 

146.3 

37 

3i.4 

19.6 

97 

82.3 

5i.4 

57 

i33.i 

83.2 

17 

184.0 

n5.o 

77 

234.9 

146.8 

38 

32.2 

20. 1 

98 

83.1 

51.9 

58 

i34.o 

83.7 

18 

184.9 

n5.5 

78 

235.8 

147-3 

3q 

33.1 

20.7 

9Q 

84. 0 

52.5 

59 

i34.8 

84.3 

19 

185.7 

116.1 

79 

236.6 

147-8 

4o 

33.9 

21 .2 

100 

84.8 

53.0 

60 

i35.7 

84.8 
85.3 

20 

186.6 

116.6 

80 
281 

237.5 
238.3 

i48.4 
148.9 

4! 

34.8 

21.7 

lOI 

85.7 

53.5 

161 

i36.5 

221 

187.4 

117. 1 

4a 

35.6 

22.3 

02 

86.5 

54.1 

62 

137.4 

85.8 

22 

188.3 

117.6 

82 

239.1 

i49-4 

43 

36.5 

22.8 

o3 

87.3 

54.6 

63 

i38.2 

86.4 

23 

189.1 

118.2 

83 

240.0 

i5o.o 

44 

37.3 

23.3 

04 

88.2 

55.1 

64 

139. 1 

86.9 

24 

190.0 

118.7 

84 

240.8 

i5o.5 

45 

38.2 

23.8 

o5 

89.0 

55.6 

65 

139.9 

87.4 

25 

190.8 

119.2 

85 

241.7 

i5i.o 

46 

39.0 

24.4 

06 

89.9 

56.2 

66 

i4o.8 

88.0 

26 

191.7 

119.8 

86 

242.5 

i5i.6 

^1 

39.9 

24.9 

07 

90.7 

56.7 

67 

i4i.6   88.5 

27 

192.5 

120.3 

87 

243.4 

l52.I 

48 

40.7 

25.4 

08 

91 .6 

57.2 

68 

i42.5 

89.0 

28 

193.4 

120.8 

88 

244.2 

i52.6 

49 

4i.6 

26.0 

09 

92.4 

57.8 

69 

143.3 

89.6 

29 

194.2 

1 2 1. 4 

89 

245.1 

i53.i 

5o 

42.4 

26.5 

10 

93.3 

58.3 
58.8 

70 
171 

i44.2 

90. 1 
90.6 

3o 

1 95. 1 

121. 9 

90 

245.9 

i53.7 

5i 

43.3 

27.0 

1 1 1 

94.1 

145.0 

23l 

195.9 

122.4 

291 

246.8 

i54.2 

52 

44.1 

27.6 

12 

95.0 

59.4 

72 

145.9 

91. 1 

32 

196.7 

122.9 

92 

247.6 

154.7 

53 

44. q 

28.1 

i3 

95.8 

59.9 

73 

146.7 

91.7 

33 

197.6 

123.5 

93 

248.5 

i55.3 

54 

45.8 

28.6 

i4 

96.7 

60.4 

74 

i47-C 

92.2 

34 

198.4 

124.0 

94 

249.3 

i55.8 

55 

46.6 

29.1 

i5 

97. b 

60.9 

7^ 

148.4 

92.7 

35 

199.3 

124.5 

95 

250.2 

i56.3 

56 

47.5 

29.7 

16 

98.4 

61.5 

7^3 

149.3 

93.3 

36 

200.1 

I25.I 

96 

2DI.0 

1 56.9 

5? 

48.3 

3o.2 

17 

99.2 

62  .0 

77 

1 5o .  I 

93.8 

37 

201.0 

125.6 

97 

251.9 

157.4 

58 

49-2 

3o.7 

18 

100. 1 

62.5 

78 

1 5 1 .0 

94.3 

38 

201.8 

1 16. 1 

98 

252.7 

157.9 

59 

5o.o 

3[.3 

19 

100.9 

63.1 

79 

i5i.8 

94.9 

39 

202.7 

126.7 

99 

253.6 

i5S.4 

6o 

IV-sl. 

50.9 

3i.8 

20 

1 0 1 . 8 

63.6 

80 

i52.6 

9^-4 

40 

2o3.5 

127.2 

3oo 

254.4 

159.0 

Dep. 

Lat. 

Oisl 

Dep.  1  Lat. 

Dist. 

Dep. 

Lat. 

Dist. 

Dep. 

Lat. 

Dist. 

Dep. 

Lat. 

[For  58  Degrees. 

TABLE  IL 

[I'ugp  '19 

Difterence  of  Latitude  and  Departure  for  33  Degrees. 

i)isl. 

Lat. 

Dep. 

Disl. 

Lat. 

Dep. 

Dist. 

Lat. 

Qcp. 

Dlst. 

Lat. 

Dep. 

Disl. 

Lat. 

Dep. 

I 

00.8 

00.5 

61 

5i  .2 

33.2 

121 

ioi.5 

65.9 

iSi 

i5i.8 

98.6 

24 1 

202.1 

i3i.3 

2 

01.7 

01 .1 

62 

52.0 

33.8 

22 

102.3 

66.4 

82 

1 52.6 

99.1 

42 

2o3.o 

i3i.8 

3 

02.5 

01 .6 

63 

52.8 

34.3 

23 

I03.2 

67.0 

83 

i53.5 

99-7 

43 

2o3.8 

i32.3 

4 

o3.4 

02.2 

64 

53.7 

34.9 

24 

104.0 

67.5 

84 

154.3 

100.2 

A4 

2o4-6 

132.9 

5 

o4.2 

02.7 

65 

54.5 

35.4 

25 

104.8 

68.1 

85 

i55.2 

100.8 

45 

2o5.5 

i33.4 

6 

o5.o 

o3.3 

66 

55.4 

35. Q 

26 

105.7 

68.6 

86 

i56.o 

101.3 

46 

206.3 

1 34.0 

7 

05.9 

o3.S 

67 

56.2 

36.5 

27 

106.5 

69.2 

87 

i56.8 

101.8 

47 

207.2 

134-5 

8 

06.7 

04.4 

68 

57.0 

37:0 

28 

1 07 . 3 

69.7 

88 

i57.7 

102.4 

48 

208.0 

i35.i 

9 

07.5 

04.9 

69 

57.9 

37.6 

29 

108.2 

70.3 

89 

i58.5 

102.9 

49 

208.8 

135.6 

10 

08.4 

o5.4 

70 

58.7 

38.1 

3o 

109.0 

70.8 

90 

159.3 

io3.5 

60 

209.7 

i36.2 

u 

09.2 

06.0 

71 

59.5 

38.7 

i3i 

109.9 

71.3 

191 

160.2 

io4-o 

25l 

210.5 

136.7 

12 

10. 1 

06.5 

72 

60.4 

39.2 

32 

1 10.7 

71-9 

92 

161.0 

104.6 

52 

211.3 

137.2 

i3 

10.9 

07.1 

73 

61 .2 

39.8 

33 

III. 5 

72.4 

93 

161.9 

io5.i 

53 

212.2 

137.8 

i4 

II. 7 

07.6 

74 

62.1 

4o.3 

34 

112. 4 

73.0 

94 

162.7 

105.7 

54 

21 3.0 

i38.3 

13 

12.6 

0S.2 

75 

62.9 

4o.8 

35 

I  l3.2 

73.5 

95 

i63.5 

106.2 

55 

213.9 

138.9 

i6 

i3.4 

08.7 

76 

63.7 

41.4 

36 

ii4.i 

74-1 

96 

164.4 

106.7 

56 

214.7 

139.4 

17 

i4.3 

09.3 

77 

64.6 

41.9 

37 

114.9 

74.6 

97 

i65.2 

107.3 

b7 

2i5.5 

i4o.o 

i8 

I3.I 

09.8 

78 

65.4 

42.5 

38 

u5.7 

73. 2 

98 

166. 1 

107.8 

58 

216.4 

140.5 

19 

i5.9 

10.3 

79 

66.3 

43.0 

39 

116. 6 

75.7 

99 

166.9 

108.4 

59 

217.2 

i4i-i 

20 

16.8 

10.9 

80 

67.1 

43.6 

40 

117. 4 

76.2 

200 

167.7 

108.9 

60 

21S.1 

i4i.6 

21 

.7.6 

II. 4 

81 

67.9 

44.1 

i4i 

118.3 

76.8 

201 

168.6 

109.D 

261 

218.9 

142.2 

22 

18.5 

12.0 

82 

68.8 

44.7 

42 

119. 1 

77.3 

02 

169.4 

IIO.O 

62 

219.7 

142.7 

23 

19.3 

12.5 

83 

69.6 

45.2 

43 

119. 9 

77-9 

o3 

170.3 

110.6 

63 

220.6 

143.2 

24 

20.1 

i3.i 

84 

70.4 

45.7 

4^ 

120.8 

78.4 

04 

171. 1 

III. I 

64 

221.4 

143.8 

23 

2!  .0 

i3.6 

85 

71.3 

46.3 

45 

121. 6 

79-0 

o5 

171.9 

111.7 

65 

222.2 

144.3 

26 

21.8 

14.2 

86 

72.1 

46.8 

46 

122.4 

79.5 

06 

172.8 

112.2 

66 

223.1 

144.9 

27 

22.6 

14.7 

87 

73.0 

47.4 

47 

123.3 

80.1 

07 

173.6 

112. 7 

67 

223.9 

145.4 

28 

23.5 

l5,2 

88 

73.8 

47-9 

48 

124. 1 

80.6 

08 

174.4 

ii3.3 

68 

224.8 

146  0 

29 

24.3 

i5.8 

89 

74.6 

48.5 

49 

125.0 

81.2 

09 

175.3 

ii3.8 

69 

225.6 

146.5 

3o 

25.2 

16.3 

90 

75.5 

49.0 
49- (3 

5o 

125.8 

81.7 

10 

1 76. 1 

1 14.4 

70 

226.4 

i47-i 

3i 

26.0 

16.9 

91 

76.3 

i5i 

126.6 

82.2 

211 

177.0 

114.9 

271 

227.3 

147.6 

32 

26.8 

17.4 

92 

77.2 

.50.1 

52 

127.5 

82.8 

12 

177-8 

ii5.5 

72 

22S.I 

1 48. 1 

33 

27.7 

18.0 

93 

78.0 

50.7 

53 

128.3 

83.3 

i3 

178.6 

1 16.0 

73 

229.0 

148.7 

34 

28.5 

18.5 

94 

78.8 

5l.2 

54 

129.2 

83.9 

i4 

179.5 

116.6 

74 

229.8 

149-2 

35 

29.4 

19. 1 

95 

79-7 

5l.7 

55 

i3o;o 

84-4 

i5 

180.3 

117.1 

7b 

23o.6 

149.8 

36 

3o.2 

19.6 

96 

80.5 

52.3 

56 

i3o.8 

85.0 

16 

181.2 

1 1 7.6 

76 

23i.5 

i5o.3 

37 

3i.o 

20.2 

97 

61.4 

52.8 

57 

i3i  .7 

85.5 

17 

182.0 

118.2 

77 

232.3 

i5o.9 

38 

3, .9 

20.7 

98 

82.2 

53.4 

58 

i32.5 

86.1 

18 

182.8 

1 18.7 

78 

233.2 

i5i.4 

09 

32.7 

21.2 

99 

83. 0 

53.9 

59 

i33.3 

86.6 

19 

183.7 

119.3 

79 

234.0 

l52.0 

40 

33.5 

21.8 

100 

83.9 

54.5 

60 

i34.2 

87.1 

20 

184.5 

119.8 

80 

234.8 

i52.5 

4i 

34.4 

22.3 

lOI 

84.7 

55.0 

161 

i35.o 

87.7 

221 

i85.3 

120.4 

281 

235.7 

i53.o 

42 

35.2 

22.9 

02 

85.5 

55.6 

62 

135.9 

88.2 

22 

186.2 

120.9 

82 

236.5 

1 53.6 

43 

36.1 

23.4 

o3 

86.4 

56.1 

63 

i36.7 

88.8 

23 

187.0 

121. 5 

83 

237.3 

i54-i 

44 

36.9 

24.0 

o4 

87.2 

56.6 

64 

137.5 

89.3 

24 

187.9 

122.0 

84 

238.2 

1 54-7 

45 

37.7 

24.5 

o5 

88.1 

57.2 

65 

i38.4 

89.9 

25 

188.7 

122.5 

85 

239.0 

i55.2 

46 

38.6 

25.1 

06 

88.9 

57.7 

66 

139.2 

90.4 

26 

189.5 

123. 1 

86 

239.9 

i55.8 

47 

3q.4 

25.6 

07 

89.7 

58.3 

67 

i4o.  I 

91 .0 

27 

190.4 

123.6 

87 

240.7 

1 56.3 

48 

40.3 

26.1 

08 

90.6 

58.8 

68 

140.9 

91.5 

28 

191. 2 

124.2 

88 

241-5 

i56.9 

49 

4i.i 

26.7 

09 

91.4 

59.4 

69 

i4i  .7 

92.0 

29 

192. 1 

124.7 

89 

242.4 

157.4 

5o 
5. 

4i  .9 

27.2 

10 

92.3 

59.9 

70 

142.6 

92.6 

3o 

192.9 

125.3 

90 

243.2 

157.9 

42.8 

27.8 

I II 

93.1 

60.5 

171 

143.4 

93. 1 

23l 

193.7 

125.8 

291 

244.1 

i58.5 

52 

43.6 

28.3 

12 

93.9 

61 .0 

72 

144.3 

93-7 

32 

194.6 

126.4 

92 

244.9 

159.0 

53 

4i.4 

28.9 

i3 

94.8 

61.5 

73 

145.1 

94 . 2 

33 

19^-4 

126.9 

93 

245.7 

159.6 

54 

45.3 

29.4 

1 4 

95.6 

62. 1 

74 

145.9 

94.8 

■34 

196.2 

127.4 

9^ 

246.6 

160.1 

55 

46.1 

3o  .0 

i5 

96.4 

62.6 

75 

i46.8 

95.3 

35 

197-1 

128.0 

95 

247-4 

160.7 

56 

47-0 

3o.5 

16 

97.3 

63.2 

76 

147-6 

95.9 

36 

197.9 

128.5 

96 

248.2 

161.2 

!)7 

47.8 

3i  .0 

'7 

98.1 

63.7 

77 

148.4 

96.4 

37 

198.8 

129.1 

97 

249-1 

i6i.8 

58 

48.6 

3i.6 

18 

99.0 

64.3 

78 

149-3 

96.9 

38 

199.6 

129.6 

98 

249.9 

162.3 

59 

49.5 

32.1 

19 

99.8 

64.8 

79 

1 5o .  1    97 . 5 

39 

200.4 

l3o.2 

99 

25o.8 

162.8 

bo 

5o.3 

32.7 

20 

100.6 

65.4 

80 

1 5 1 . 0    98 . 0 

4o 

201.3 

1 30.7 

3  00 

25i.6 

1 63.4 

r-ist.i  Dcp.i  Lai. 

DIst. 

Hop. 

Lat. 

Dist-l    Dop.  1  Lat. 

Disl. 

Dop. 

Lai. 

Disl 

Dep. 

Lat. 

[For  57  Degrees. 

Page  50J 

TABLE   IL 

Difference  of  Latitude  and  Departure  for  34  Degrees. 

Disl. 

I 

Lai. 

Dcp. 
00.6 

Disl. 
~67 

Lat. 

Dep. 
34.1 

Disl. 

Lat. 

Dep. 

Disl. 

Lat. 

Dcp. 

Dist. 

Lat. 

Dep. 

00.8 

5o.6 

121 

100.3 

67.7 

181 

i5o.i 

101.2 

241 

199.8 

134.8 

2 

01.7 

01 .1 

62 

5i.4 

34.7 

22 

lOI  .1 

68.2 

82 

i5o.9 

101.8 

42 

200.6 

i35.3 

3 

02.5 

01.7 

63 

52.2 

35.2 

23 

102.0 

68.8 

83 

i5i.7 

102.3 

43 

201.5 

135.9 

4 

o3.3 

02.2 

64 

53.1 

35.8 

24 

102.8 

69.3 

84 

i52.5 

102.9 

AA 

202.3 

1 36.4 

5 

04. 1 

02.8 

65 

53. q 

36.3 

2  5 

io3.6' 

69.9 

85 

i53.4 

io3.5 

45 

203.1 

137.0 

6 

o5.o 

o3.4 

66 

54.7 

36.9 

26 

IC4.5 

70. b 

86 

i54.2 

104.0 

46 

203.9 

137.6 

7 

o5.8 

o3.9 

67 

55.5 

37.5 

27 

io5.3 

71.0 

87 

1 55.0 

104.6 

47 

204.8 

i38.i 

8 

06.6 

o4.5 

68 

56.4 

38.0 

28 

!06.I 

71.6 

88 

i55.9 

io5.i 

48 

20b.6 

i38.7 

9 

07.5 

o5.o 

69 

57.2 

38.6 

29 

106.9 

72.1 

89 

i56.7 

105.7 

49 

206.4 

139.2 

lO 

08.3 

ob.6 

70 

58. 0 

39.1 

3o 

107.8 

72.7 

90 

ib7.b 

106.2 

bo 

207.3 

139.8 

1 1 

09.1 

06.2 

71 

58.9 

39.7 

i3i 

108.6 

73.3 

191 

i58.3 

106.8 

25l 

208. 1 

140.4 

12 

09.9 

06.7 

72 

59.7 

4o.3 

32 

109.4 

73.8 

92 

159.2 

107.4 

52 

208.9. 

140.9 

IJ 

10.8 

07.3 

73 

60.5 

4o.8 

33 

no. 3 

74.4 

93 

160.0 

107.9 

53 

209.7 

i4i.5 

i4 

II. 6 

07.8 

74 

61.3 

4i.4 

34 

III. I 

74.9 

94 

160.8 

108.5 

54 

210.6 

142.0 

lb 

12.4 

08.4 

75 

62.2 

41.9 

35 

III  .9 

7b. b 

95 

161.7 

109.0 

55 

211.4 

142.6 

lb 

i3.3 

08.9 

76 

63.0 

42.5 

36 

112.7 

7b.  I 

96 

Ib2.b 

109.6 

56 

212.2 

143.2 

17 

14.1 

09.5 

77 

63.8 

43.1 

37 

ii3.6 

76.6 

97 

i63.3 

110.2 

57 

2l3.I 

143.7 

i8 

14.9 

10. 1 

78 

64.7 

43.6 

38 

114. 4 

77.2 

98 

164.1 

1 10.7 

58 

213.9 

144-3 

19 

lb. 8 

10.6 

79 

65.5 

44.2 

39 

Il5.2 

77.7 

99 

i65.o 

1 1 1.3 

59 

2i4-7 

1 44 -8 

20 

lb. 6 

II  .2 

80 

66.3 

44.7 
45.3 

40 

116. 1 

78.3 

200 

i65.8 

1 1 1.8 

60 

2i5.5 

145.4 

21 

17.4 

II. 7 

81 

67.2 

i4i 

116. 9 

78.8 

201 

166.6 

112.4 

261 

216.4 

145.9 

22 

18.2 

12.3 

82 

68.0 

45.9 

42 

117. 7 

79-4 

02 

167.5 

ii3.o 

62 

217.2 

146.5 

23 

19. 1 

12.9 

83 

68.8 

46.4 

43 

118. 6 

80.0 

o3 

168.3 

ii3.5 

63 

218.0 

i47-i 

24 

19.9 

i3.4 

84 

69.6 

47-0 

M 

119. 4 

80.5 

04 

169.1 

ii4.i 

64 

218.9 

147-6 

2!) 

20.7 

i4.o 

85 

70.5 

47.5 

45 

120.2 

81.1 

o5 

170.0 

114.6 

65 

219.7 

148.2 

2b 

21 .6 

14.5 

86 

71.3 

48.1 

46 

121 .0 

81.6 

06 

170.8 

Il5.2 

66 

220.5 

148.7 

27 

22.4 

i5.i 

87 

72.1 

48.6 

47 

121 .9 

82.2 

07 

171.6 

ii5.8 

67 

221.4 

149.3 

28 

23.2 

l5.7 

88 

73.0 

49.2 

48 

122 .7 

82.8 

08 

172.4 

116.3 

68 

222.2 

149-9 

29 

24.0 

16.2 

89 

73.8 

49-8 

49 

123.5 

83.3 

09 

173.3 

116.9 

69 

223.0 

i5o.4 

3o 

24.9 

16.8 

90 

74.6 

5o.3 

5o 

124.4 

83.9 

10 

174. 1 

1 17-4 

70 

223.8 

i5i.o 

3i 

25.7 

.7.3 

91 

75.4 

5o.9 

i5i 

125.2 

84.4 

211 

174.9 

1 18.0 

271 

224.7 

i5i.5 

32 

26.5 

17.9 

92 

76.3 

5i.4 

52 

126.0 

85.0 

12 

175.8 

118.5 

72 

225.5 

l52.I 

33 

27.4 

18.5 

93 

77-1 

52.0 

53 

126.8 

85.6 

i3 

176.6 

119. 1 

73 

226.3 

i52.7 

M 

28.2 

19.0 

94 

77-9 

52.6 

54 

127.7 

86.1 

14 

177-4 

119.7 

74 

227.2 

i53.2 

3b 

29.0 

19.6 

q5 

78.8 

53.1 

55 

128.5 

86.7 

i5 

178.2 

120.2 

75 

228.0 

i53.8 

36 

29.8 

20. 1 

96 

79.6 

53.7 

56 

129.3 

87.2 

16 

179-1 

120.S 

76 

228.8 

i54.3 

37 

3o.7 

20.7 

97 

80.4 

54.2 

57 

i3o.2 

87.8 

17 

179.9 

1 2 1. 3 

77 

229.6 

1 54-9 

38 

3i.5 

21.2 

98 

81.2 

54.8 

58 

i3i  .0 

88.4 

18 

180.7 

1 2 1. 9 

78 

230.5 

i55.5 

39 

32.3 

21.8 

99 

82.1 

55.4 

59 

i3i.8 

88.9 

19 

181.6 

122.5 

79 

231.3 

1 56.0 

4o 
4i 

33.2 
34.0 

22.4 
22 .9 

IOC) 

82.9 

55.9 
56.5 

60 

i32.6 

89.5 

20 

182.4 

123.0 

80 

232.1 

i56.6 

lOI 

83.7 

161 

i33.5 

90.0 

221 

i83.2 

123.6 

281 

233.0 

I57-I 

42 

34.8 

23.5 

02 

84.6 

57.0 

62 

i34.3 

90.6 

22 

184.0 

124. 1 

82 

233.8 

157.7 

43 

35.6 

24.0 

o3 

85.4 

57.6 

63 

i35.i 

91. 1 

23 

i84-9 

124.7 

83 

234.6 

i58.3 

44 

36.5 

24.6 

04 

86.2 

58.2 

64 

1 36.0 

91.7 

24 

185.7 

125.3 

84 

235.4 

1 58.8 

45 

37.3 

25.2 

o5 

87.0 

58.7 

65 

i36.8 

92.3 

25 

186.5 

125.8 

85 

236.3 

159.4 

46 

38.1 

25.7 

06 

87.9 

59.3 

66 

137.6 

92.8 

26 

187.4 

126.4 

86 

237.1 

159.9 

47 

39.0 

26.3 

07 

88.7 

59.8 

67  ,  1 38. 4 

93.4 

27 

188.2 

126.0 

87 

237.9 

160.5 

48 

39.8 

26.8 

08 

89.5 

60.4 

68 

139.3 

93.9 

28 

189.0 

127.5 

88 

238.8 

1 61.0 

f9 

40.6 

27.4 

09 

90.4 

61 .0 

69 

i4o.i 

94.5 

29 

189.8 

128.! 

89 

239.6 

i6r.6 

bo 

TT 

4i.b 
42.3 

28.0 
28.5 

10 
1 1 1 

91.2 

61. b 

70 

140.9 

9b.. 

3o 

190.7 

128.6 

90 

240.4 

162.2 

92.0 

62.1 

171 

i4i.8 

95.6 

23l 

i9'-5 

129.2 

291 

241.2 

162.7 

b2 

43.1 

29.  T 

12 

92.9 

62.6 

72 

142.6 

96.2 

32 

,92.3 

129.7 

92 

242.1 

i63.3 

b3 

43.9 

29.6 

i3 

93.7 

63.2 

73 

143.4 

96.7 

33 

193.2 

i3o.3 

93 

242.9 

i63.8 

b4 

44.8 

3<).  2 

i4 

94.5 

63.7 

74 

144.3 

97.3 

34 

194.0 

i3o.9 

94 

243.7 

164.4 

bb 

45.6 

3o.8 

i5 

95.3 

64.3 

75 

145.1 

97-9 

35 

194.8 

i3i.4 

95 

244.6 

i65.o 

bb 

46.4 

3i.3 

16 

96.2 

64.9 

76 

145.9 

98.4 

36 

195.7 

l32.0 

96 

245.4 

i65.5 

b7 

47.3    31.9 

17 

97.0 

bb.4 

77 

146.7 

99.0 

37 

196.5 

i32.5 

97 

246.2 

166.1 

b8 

48.1    32.4 

18 

97.8 

66.0 

78 

147-6 

99.5 

38 

197.3 

i33.i 

98 

247-1 

166.6 

b9 

48.9   33.0 

'9 

98.7 

bb.b 

79 

148.4 

1 00 . 1 

39 

198.1 

i33.6 

99 

247-9 

167.2 

bo 
Disl. 

49.7    33.6 
Dep.     I, at. 

20 

99.5 

b7.i 

80 

149.2 

1 00 . 7 

40 

199-0 

i34.2 

3oo 

248.7 

167.8 

Disl. 

Dcp. 

Lai. 

Disl. 

Dop. 

Lat. 

Disl. 

Dep. 

Lat. 

Disl. 

Dep. 

Lat. 

1 

[ 

'^cr  5G  Degrees. 

TABLE  IL 

[Page  5]     1 

Difference  of  Latitude  and  Departure  for  35 

Degrees. 

Disi. 

Lat. 

Dep. 

Dist. 

Lat. 

Dep. 

Dist. 

Lat. 

Dep. 

Dist. 

Lat. 

Dep. 

Dist. 

Lat. 

Dep. 

I 

00.8 

00.6 

61 

5o.o 

35.0 

121 

99.1 

69.4 

181 

148.3 

io3.8 

241 

197.4 

i38.2 

2 

01 .6 

01. 1 

62 

5o.8 

35.6 

22 

99.9 

70.0 

82 

i49-i 

104.4 

42 

198.2 

i38.8 

3 

02.5 

01.7 

63 

5i.6 

36.1 

23 

100. 8 

70.5 

83 

i49-9 

io5.o 

43 

199.1 

139.4 

4 

o3.3 

02.3 

64 

52.4 

36.7 

24 

loi  .6 

71.1 

84 

i5o.7 

io5.5 

M 

199.9 

1 40.0 

5 

o4.i 

02.9 

65 

53.2 

37.3 

25 

102.4 

71.7 

85 

i5i.5 

106. 1 

45 

200.7 

i4o.5 

6 

04.9 

o3.4 

66 

54.1 

37.9 

26 

I03.2 

72.3 

86 

152.4 

106.7 

46 

201.5 

i4i.i 

7 

o5.7 

o4.o 

67 

54.9 

38.4 

27 

104.0 

72.8 

87 

i53.2 

107.3 

47 

202.3 

141.7 

8 

06.6 

o4.6 

68 

55.7 

39.0 

28 

104.9 

73.4 

88 

i54.o 

107.8 

48 

2o3.I 

142.2 

9 

07.4 

05.2 

69 

56.5 

39.6 

29 

105.7 

74.0 

89 

1 54.8 

108.4 

49 

2040 

142.8 

10 

08.2 

o5.7 

70 

57.3 

4o.2 

3o 

106.5 

74.6 

90 

i55.6 

109.0 

5o 

204.8 

143.4 

II 

09.0 

06.3 

71 

58.2 

40.7 

i3i 

107.3 

75.1 

191 

i56.5 

109.6 

25l 

2o5.6 

144.0 

12  log. 8 

06.9 

72 

59.0 

4i.3 

32 

108. 1 

75.7 

92 

157.3 

1 10. 1 

52 

206.4 

144.5 

i3 

10.6 

07.5 

73 

59.8 

4i  .9 

33 

108.9 

76.3 

93 

i58.i 

110.7 

53 

207.2 

145.1 

i4 

II. 5 

08.0 

74 

60.6 

42.4 

M 

109.8 

76.9 

94 

i58.9 

111.3 

54 

208.1 

1457 

i5 

12.3 

08.6 

75 

61.4 

43.0 

35 

110.6 

77-4 

95 

159.7 

111.8 

55 

208.9 

i46.3 

if) 

i3.i 

09.2 

76 

62.3 

43.6 

36 

III. 4 

78.0 

96 

160.6 

112.4 

56 

2097 

i46.8 

17 

.3.9 

09.8 

77 

63.1 

44.2 

37 

112. 2 

78.6 

97 

161.4 

1  i3.o 

57 

210.5 

147.4 

i8 

14.7 

10.3 

78 

63.9 

44.7 

38 

ii3.o 

79.2 

98 

162.2 

ii3.6 

58 

211.3 

i48.o 

'9 

i5.6 

10.9 

79 

64.7 

45.3 

39 

113.9 

79-7 

99 

1 63.0 

114.1 

59 

212.2 

1 48 .6 

20 

16.4 

11.5 

80 
81 

65.5 
66.4 

45.9 

46.5 

40 

114.7 

80.3 

200 

1 63.8 

114.7 

60 

2l3.0 

i49-i 

21 

17.2 

12  .0 

i4i 

ii5.5 

80.9 

201 

164.6 

n5.3 

261 

2i3.8 

149.7 

22 

18.0 

12.6 

82 

67.2 

47 -o 

42 

116. 3 

81.4 

02 

i65.5 

1.5.9 

62 

214.6 

i5o.3 

23 

18.8 

l3.2 

83 

68.0 

47.6 

43 

117. 1 

82.0 

o3 

166.3 

116.4 

63 

21 5.4 

i5o.9 

24 

19.7 

i3.8 

84 

68.8 

48.2 

M 

118. 0 

82.6 

04 

167.1 

1 17.0 

64 

216.3 

i5i.4 

25 

20.5 

i4.3 

85 

69.6 

48.8 

i5 

118. 8 

83.2 

o5 

167.9 

117.6 

65 

217.1 

1 52.0 

26 

21.3 

14.9 

86 

70.4 

49-3 

46 

119.6 

83.7 

06 

168.7 

118. 2 

66 

217.9 

i526 

27 

22.1 

i5.S 

87 

71.3 

49.9 

47 

120.4 

84.3 

07 

169.6 

118.7 

67 

218.7 

i53.i 

28 

22.0 

16,1 

88 

72.1 

5o.5 

48 

121 .2 

84.9 

08 

170.4 

1 19.3 

68 

219.5 

153.7 

29 

23.8 

16.6 

89 

72.9 

5i  .0 

49 

122. 1 

85.5 

09 

171.2 

119.9 

69 

220.4 

1 54.3 

3o 

24.6 

17.2 

90 

73.7 

5i.6 

5o 

122.9 

86.0 

10 

172.0 

120.5 

70 

221.2 

154.9 

3i 

25.4 

17.8 

91 

74.5 

52.2 

i5i 

123.7 

86.6 

211 

172.8 

121.0 

271 

222.0 

i55.4 

32 

26.2 

18.4 

92 

75.4 

52.8 

52 

124.5 

87.2 

12 

173.7 

1 2 1. 6 

72 

222.8 

i56.o 

33 

27.0 

18.9 

93 

76.2 

53.3 

53 

125.3 

87.8 

i3 

174.5 

122.2 

73 

223.6 

i56.6 

34 

27.9 

19.5 

94 

77.0 

53.9 

54 

126.1 

88.3 

14 

175.3 

122.7 

74 

224.4 

157.2 

35 

28.7 

20.1 

95 

77.8 

54.5 

55 

127.0 

88.9 

i5 

176.1 

123.3 

75 

225.3 

j57.7 

36 

29.5 

20.6 

96 

78.6 

55.1 

56 

127.8 

89.5 

16 

176.9 

123.9 
124.5 

76 

226.1 

i58.3 

37 

3o.3 

2!  .2 

97 

79.5 

55.6 

57 

128.6 

90.1 

17 

177.8 

77 

226.9 

1 58.9 

38 

3i.i 

21.8 

98 

80.3 

56.2 

58 

129.4 

90.6 

18 

178.6 

125.0 

78 

227.7 

159.5 

39 

3, .9 

22.4 

99 

81. 1 

56.8 

59 

i3o.2 

91.2 

19 

179.4 

125.6 

79 

228.5 

160.0 

40 
4i 

32.8 

22.9 

100 

81.9 

57.4 

60 

i3i.i 

91 .8 

20 

180.2 

126.2 

80 

229.4 

160.6 

33.6 

23.5 

lOI 

82.7 

57.9 

161 

i3i  .9 

92.3 

221 

181.0 

126.8 

281 

23o.2 

161.2 

42 

34.4 

24.1 

02 

83.6 

58.5 

62 

132.7 

92.9 

22 

181.9 

127.3 

82 

23l.O 

161.7 

43 

35.2 

24.7 

o3 

84.4 

59.1 

63 

i33.5 

93.5 

23 

182.7 

127.9 

83 

23i.8 

162.3 

U 

36.0 

25.2 

o4 

85.2 

59.7 

64 

i34.3 

94.1 

24 

i83.5 

128.5 

84 

232.6 

162.9 
i63.5 

45 

36.9 

25.8 

o5 

86.0 

60.2 

65 

i35.2 

94.6 

25 

184.3 

129.1 

85 

233.5 

46 

37.7 

26.4 

06 

86.8 

60.8 

66 

i36.o 

95.2 

26 

i85.i 

129.6 

86 

234.3 

164.0 

47 

38.5 

27.0 

07 

87.6 

61.4 

67 

i36.8 

95.8 

27 

165.9 

l3o.2 

87 

235.1 

164.6 

48 

39.3 

27.5 

08 

88.5 

61.9 

68 

i37.6 

96.4 

28 

186.8 

i3o.8 

88 

235.9 

i65.2 

49 

4o.i 

28.1 

09 

89.3 

62.5 

69 

i3S.4 

96.9 

29 

187.6 

i3i.3 

89 

236.7 

i65.8 

5o 

4i.o 

28.7 

10 

90.1 

63.1 

70 

139.3 

97.5 

3o 

188.4 

i3i.9 

90 

237.6 

166.3 

5i 

41.8 

29.3 

III 

90.9 

63.7 

171 

i4o.i 

98.1 

23  I 

189.2 

i32.5 

291 

238.4 

166.9 

52 

42.6 

29.8 

12 

91.7 

64.2 

72 

140.9 

98.7 

32 

190.0 

i33.i 

92 

239.2 

167.5 

53 

Ai.A 

3o.4 

i3 

92.6 

64.8 

73 

141.7 

99.2 

33 

190.9 

i33.6 

93 

240.0 

168. 1 

54 

44.2 

3i.o 

i4 

93.4 

65.4 

74 

142.5 

99.8 

34 

191.7 

1 34.2 

94 

240.8 

168.6 

55 

45.1 

3i.5 

i5 

94.2 

66.0 

75 

143.4 

100.4 

35 

192.5 

i34.8 

95 

241.6 

169.2 

56 

45.9 

32.1 

16 

95.0 

66.5 

76 

144.2 

100.9 

36 

193.3 

1 35.4 

96 

242.5 

169.8 

67 

46.7 

32.7 

17 

95.8 

67.1 

77 

145.0 

loi  .5 

37 

194. 1 

135.9 

97 

243.3 

170.4 

58 

47. b 

ii.i 

18 

96.7 

67.7 

78 

145.8 

102.1 

38 

195.0 

i36.5 

98 

244.1 

170.9 

D9 

48.3 

33.8 

19 

97.5 

68.3 

79 

i46.6 

102.7 

39 

195.8 

i37.i 

99 

244.9 

171.5 

Oo 

49.1 

M.A 

20 

98.3 

68.8 

80 

147-4 

I03.2 

4o 

196.6 

137.7 

3oo 

245.7 

172.1 

Dist. 

Dep. 

Lat. 

Din. 

Dep. 

Lat. 

Dist. 

Dep. 

Lat. 

Dist. 

Dep. 

Lat. 

Dist. 

Dep. 

i   Lat. 

[ 

For  55  Degrees. 

Page  52] 

TABLE  IL 

Difference  of  Latitude  and  Departure  for  36  Degrees. 

Dibl. 

Lai. 

Dep. 
00.6 

Dist. 

Lat. 

Dep. 

Dist. 

Lat. 

Dep. 

Dist.|    Lat.  i   Dep. 

Dist. 

241 

Lat.      D?p. 

I 

00.8 

61 

49.4 

35.9 

121 

97-9 

71. 1 

181 

146.4 

106.4 

195.0    141.7 

2 

01.6 

01 .2 

62 

50.2 

36.4 

22 

98.7 

71.7 

82 

147.2 

107.0 

42    ^95. 8  1  :42.2  1 

3 

02.4 

01.8 

63 

5i.o 

37.0 

23 

99.5 

72.3 

83 

148.1 

107.6 

43    196.6 

142.8 

4 

03.2 

02.4 

64 

5i.8 

37.6 

24 

100.3 

72.9 

84 

14S.9 

108.2 

44   1974 

143.4 

5 

o4.o 

02.9 

65 

52.6 

38.2 

25 

lOI  .1 

73.5 

85 

i49-7 

108.7 

45    198.2 

144.0 

6 

04.9 

o3.6 

66 

53.4 

38.8 

26 

loi  .9 

74.1 

86 

i5o.5 

109.3 

46 ,  199.0 

144.6 

7 

o5.7 

o4.i 

67 

54.2 

39.4 

27 

102.7 

74.6 

87 

i5i.3 

109.9 

47 

199.8 

145.2 

B 

06.5 

04.7 

68 

55.0 

4o.o 

28 

io3.6 

75.2 

88 

l52.I 

110.5 

48 

200.6 

145.8 

9 

07.3 

o5.3 

69 

55.8 

40.6 

29 

io4.4 

75.8 

89 

152.9 

III. I 

49 

201.4 

146.4 

10 

08.1 

o5.9 

70 

56.6 

4i.i 

3o 

io5.2 

76.4 

90 
1.9 1 

153.7 
154.5 

111.7 

5o 

202.3 

146.9 

II 

08.9 

06.5 

71 

57.4 

4i.7 

i3i 

106.0 

77.0 

1 12.3 

25l 

2o3.I 

147-5 

12 

09.7 

07.1 

72 

58.2 

42.3 

32 

106.8 

77.6 

92 

i55.3 

112. 9 

52 

203.9 

I48.I 

i3 

10.5 

07.6 

73 

59. 1 

42.9 

33 

107.6 

78.2 

93 

i56.i 

ii3.4 

53 

204.7 

148.7 

i4 

11. 3 

08.2 

74 

59.9 

43.5 

34 

108.4 

■   78.8 

94 

i56.9 

ii4.o 

54 

2o5.5 

149-3 

lb 

12. 1 

08.8 

7^ 

60.7 

44.1 

35 

109.2 

79-4 

95 

157.8 

1 14.6 

55 

206.3 

149.9 

i6 

12.9 

09.4 

76 

61.5 

44.7 

36 

IIO.O 

79-9 

96 

1 58.6 

Il5.2 

56 

207.1 

i5o.5 

17 

i3.8 

lO.O 

77 

62.3 

45.3 

37 

no. 8 

80.5 

97 

159.4 

II5.8 

57 

207.9 

i5i.i 

la 

14.6 

10.6 

78 

63.1 

45.8 

38 

III  .6 

81. 1 

98 

160.2 

116.4 

58 

208.7 

i5i.6 

19 

i5.4 

II  .2 

79 

63.9 

46.4 

39 

112. 5 

81.7 

99 

161.0 

1 17.0 

59 

309.:) 

l52.2 

20 

16.2 

II. 8 

80 

64.7 

47-0 

40 

ii3.3 

82.3 

200 

161.8 

117.6 

60 

210.3 

i52.8 

21 

17.0 

12.3 

81 

65.5 

47-6 

i4i 

1 14. 1 

82.9 

201 

162.6 

118.1 

261 

211.2 

i53.4 

22 

17.8 

12.9 

82 

66.3 

48.2 

42 

114.9 

83.5 

02 

163.4 

118.7 

62 

212.0 

i54-o 

23 

18.6 

i3.5 

83 

67.1 

48.8 

43 

ii5.7 

84.1 

OJ 

164.2 

1 19.3 

63 

212.8 

i54.6 

24 

19.4 

i4.i 

84 

68.0 

49-4 

44 

116. 5 

84.6 

o4 

i65.o 

119.9 

64 

2i3.6 

i55.2 

25 

20.2 

14.7 

85 

68.8 

5o.o 

45 

117. 3 

85.2 

o5 

i65.8 

120.5 

65 

214.4 

1 55.8 

26 

21 .0 

i5.3 

86 

69.6 

5o.5 

46 

118. 1 

85.8 

06 

166.7 

121. 1 

66 

2l5.2 

1 56.4 

27 

21.8 

i5.9 

87 

70.4 

5t.i 

47 

118. 9 

86.4 

07 

167.5 

121. 7 

67 

216,0 

1 56.9 

2» 

22.7 

16.5 

88 

71.2 

5i.7 

48 

119-7 

87.0 

08 

168.3 

122.3 

68 

216.8 

157.5 

29 

23.5 

17.0 

89 

72.0 

52.3 

49 

120.5 

87.6 

09 

169.1 

122.8 

69 ,  217.6 

i58.i 

3o 

24.3 

17.6 

90 
91 

72.8 

52.9 

5o 

121 .4 

88.2 

10 

169.9 

123.4 

70  '  218.4 

1 58.7 

3i 

25.1 

18.2 

73.6 

53.5 

i5i 

122.2 

88.8 

211 

170.7 

124.0 

271 

219.2 

159.3 

32 

25.9 

18.8 

92 

74.4 

54.1 

52 

123. 0 

89.3 

12 

171.5 

124.6 

72 

220.1 

159-9 

33 

26.7 

19.4 

93 

75.2 

54.7 

53 

123.8 

89.9 

i3 

172.3 

125.2 

73 

220.9 

160.5 

34 

27.5 

20.0 

94 

76.0 

55.3 

54 

124.6 

90.5 

i4 

173. 1 

125.8 

74 

221.7 

161. 1 

3b 

28.3 

20.6 

95 

76.9 

55.8 

55 

125.4 

91. 1 

i5 

173.9 

126.4 

75 

222.5 

161.6 

36 

29.1 

21 .2 

96 

77-7 

56.4 

56 

126.2 

91.7 

16 

174.7 

127.0 

76 

223.3 

162.2 

37 

29.9 

21.7 

97 

78.5 

57.0 

67 

127.0 

92.3 

17 

175.6 

127.5 

77 

224.1 

162.8 

38 

3o.7 

22.3 

98 

79-3 

57.6 

58 

127.8 

92.9 

18 

176.4 

128. 1 

78 

224.9 

i63.4 

39 

3i.6 

22.9 

99 

80.1 

58.2 

59 

128.6 

93. b 

19 

177.2 

128.7 

79 

225.7 

164.0 

40 

32.4 

23.5 

100 

80.9 

58.8 

6c 

129.4 

94.0 

20 

178.0 

129.3 

80 

226.5 

164.6 

4i 

33.2 

24. 1 

lOI 

81.7 

59.4 

161 

i3o.3 

94.6 

221 

178.8 

129.0 

281 

227.3 

i65.2 

42 

34.0 

24.7 

02 

82.5 

60.0 

62 

i3i.i 

95.2 

22 

179.6 

i3o.5 

82 

228.1 

i65.8 

43 

34.8 

25.3 

o3 

83.3 

60.5 

63 

i3i.9 

95.8 

23 

180.4 

i3i.i 

83 

229.0 

166.3 

44 

35.6 

25.9 

o4 

84.1 

61. 1 

64 

132.7 

96.4 

24 

181.2 

i3i.7 

84 

229.8 

166.9 

45 

36.4 

26.5 

o5 

84.9 

61.7 

65 

i33.5 

97.0 

25 

182.0 

i32.3 

85 

23o.6 

167.5 

46 

37.2 

27.0 

06 

85.8 

62.3 

66 

i34.3 

97.6 

26 

182.8 

i32.8 

86 

23 1. 4 

168.1 

47 

38. 0 

27.6 

07 

86.6 

62.9 

67 

i35.i 

98.2 

27 

i83.6 

i33.4 

87 

232.2 

168.7 

48 

38.8 

28.2 

08 

87.4 

63.5 

68 

135.9 

98.7 

28 

184.5 

i34.o 

88 

233.0 

169.3 

49 

39.6 

28.8 

09 

88.2 

64.1 

69 

i36.7 

99.3 

29 

i85.3 

i34.6 

89 

233.8 

169.9 

bo 

40.5 

29.4 

10 

89.0 

64.7 

70 

137.5 

99.9 

3o 

186.1 

i35.2 

90 

234.6 

170.5 

5i 

4i.3 

3o.o 

III 

89.8 

65.2 

171 

i38.3 

100.5 

23l 

186.9 

i35.8 

291 

235.4 

171.0 

b2 

42.1 

3o.6 

12 

90.6 

65.8 

72 

139.2 

101 .1 

32 

187.7 

i36.4 

92 

236.2 

171.6 

53 

42.9 

3l.2 

i3 

91.4 

66.4 

73 

i4o.o 

lOI  .7 

33 

188.5 

137.0 

93 

237.0 

172.2 

54 

43.7 

3l.7 

i4 

92.2 

67.0 

74 

i4o.8 

102.3 

34 

189.3 

137.5 

94 

237.9 

172.8 

55 

44.5 

32.3 

i5 

93.0 

67.6 

7i> 

i4i.6 

102.9 

35 

190. 1 

i38.i 

95 

238.7 

173-4 

56 

45.3 

32.9 

16 

93.8 

68.2 

76 

142.4 

io3.5 

36 

190.9 

i38.7 

96 

239.5 

174.0 

57 

46.1 

33.5 

17 

94-7 

68.8 

77 

143.2 

104.0 

37 

191.7 

139.3 

97 

240.3 

174.6 

58 

46.9 

34.1 

18 

95.5 

69.4 

78 

i44.o 

104.6 

38 

192.5 

139.9 

98 

241 .1 

175.2 

59 

47.7 

34.7 

19 

9().3 

69.9 

79 

144.8 

I05.2 

39 

193.4 

i4o.5 

99 

241.9 

i7b-7 

60 

48.5 

35.3 

20 

97.1 

70.5 

80 

145.6 

io5.8 

4o 

194.2 

i4i.i 

3oo 

242.7 

176.3 

Hop. 

Lnt. 

Dist. 

Dop, 

Lat. 

Dist. 

Dep. 

Lat. 

Dist.    Dep.  i 

Lat. 

Dist.    Dep. 

Lat. 

[For  54  Degrees. 

—                                                                       \ 

TABLE  11.                                            [i'''s«s3 

.   Difference  of  Latitude  and  Departure  for  37  Degrees. 

l)is>t. 

Lai. 

Dep. 

Disi. 

Lai. 

Dep. 

36.7 
37.3 

37.9 
38.5 
39.. 
39.7 
40.3 
40.9 
4i.5 
42.1 

42.7 
43.3 
43.9 
44.5 
45.1 
45.7 
46.3 
46.9 
47.5 
48.1 

Dist. 

Lat. 

Dep. 

Disi. 

Lat. 

Dep. 

Dist. 

Lat. 

Dep. 

145.6 

146.2 
146.8 
i47-4 
i48.o 
148.6 
149-3 

,49.9 
i5o.5 

i5i.i 
i5i.7 
i52.3 
152.9 
i53.5 
i54.i 
i54.7 
i55.3 
155.9 
i56.5 

I 

2 
3 

4 
5 
6 

7 
8 

9 

10 
12 

i3 
i4 
i5 
iG 
17 
iS 

'9 

20 

00.8 

oi  .6 

02.4 
o3.2 

o4.o 
o4.8 
o5.6 
o6.4 
07.2 
08.0 
08.8 
09.6 
10.4 
11 .2 
12.0 
12.8 
i3.6 
14.4 

l5.2 

16.0 

00.6 
01  .2 
01.8 
02.4 
o3.o 
o3.6 
04.2 
04.8 
o5.4 
06.0 

06.6 
07.2 
07.8 
08.4 
09.0 
09.6 
10.2 
10.8 
II. 4 
12.0 

61 
62 
63 
64 
65 
66 
67 
68 
69 
70 

48.7 
49-5 
5o.3 
5i.i 
5i  .9 
52.7 
53.5 
54.3 
55.1 
55.9 

121 

22 
23 
24 
25 
26 
27 
28 

=9 
3o 

96.6 
97.4 
98.2 
99.0 
99.8 

lOO.D 

loi  .4 

102.2 

io3.o 
I03.8 

72.8 
73.4. 
74.0 
74.6 
75.2 
75.8 
76.4 
77.0 
77.6 
78.2 

181 
82 
83 
84 
85 
86 

87 
88 
89 
90 

1 44 -6 
145.4 
i46.2 
146.9 
i47-7 
i48.5 
i49-3 
i5o.i 
i5o.9 
i5i.7 

108.9 
109.5 
no. I 
110.7 
11 1.3 
11 1.9 
112.5 
ii3.i 
113.7 
114.3 

114.9 
ii5.5 
116.2 
116.8 
II7-4 
118.0 
118.6 
119.2 
119.8 
120.4 

241 
42 
43 
44 
45 
46 
47 
48 

it 

25l 
52 

53 
54 
55 
56 

57 
58 

59 
60 

192.5 
193.3 
194.1 
194.9 
195.7 
196.5 
197.3 
198.1 
198.9 
199.7 
200.5 

201.3 
202.1 
202.9 

2o3.7 
204.5 

205.2 

206.0 

206.8 

207.6 

71 
72 
73 

74 
73 
76 
77 
78 

79 
80 

56.7 
57.5 
58.3 
59.1 
59.9 
60.7 
61.5 
62.3 
63.1 
63.9 

i3i 

32 

33 
34 
35 
36 

37 
38 
39 
4o 

104.6 
105.4 
106.2 
107.0 
107.8 
108.6 
109.4 
110.2 
II 1 .0 
III. 8 

78.8 

79-4 
80.0 
80.6 
81.2 
81.8 
82.4 
83.1 
83.7 
84.3 

191 

93 
94 

95 
96 

97 
98 

99 
200 

i52.5 
i53.3 
1 54.1 
154.9 
155.7 
i56.5 
157.3 
i58.i 
158-9 
159.7 

2t 
2  2 
23 
24 
25 
26 
27 
28 

=9 

3o 

16.8 
17.6 
18.4 
19.2 
20.0 
20.8 
21.6 
22.4 

23.2 

24.0 

12.6 
l3.2 

i3.8 
14.4 
i5.o 
i5.6 
16.2 
16.9 
17.5 
18. 1 

81 
82 
83 
84 
85 
86 

87 
88 
89 
90 

64-7 
65.5 
66.3 
67.1 
67.9 
68.7 
69.5 
70.3 
71. 1 
71.9 

48.7 
49-3 
5o.o 
5o.6 

5l.2 

5i.8 
52.4 
53.0 
53.6 

54.2 

i4i 
42 
43 
44 
45 
46 
47 
48 
49 
5o 

112. 6 
ii3.4 

Il4-2 

ii5.o 
ii5.8 
116.6 
II7-4 
118. 2 
119. 0 
119.8 

84.9 
85.5 
86.1 
86.7 
87.3 
87.9 
83.5 
89.1 
89.7 
90.3 

201 
02 
o3 
o4 
o5 
06 
07 
08 
09 
10 

160.5 
161.3 
162.1 
162.9 
163.7 
164.5 
i65.3 
166.1 
166.5 
167.7 

121.0 
1 2 1. 6 
122.2 
122.8 
123.4 
124.0 
124.6 

125.2 

125.8 
126.4 

261 
62 
63 
64 
65 
66 

67 
68 
69 
70 

208.4 

209.2 
210.0 

210.8 

211.6 

212.4 

2l3.2 
2l4.0 
214.8 

2i5.6 

157.1 
157.7 
i58-3 
1 58-9 
,59.5 
1 60. 1 
160.7 
161.3 
161.9 
162.5 

3i 

32 

33 
34 
35 
36 

37 
3S 
39 
4o 

24.8 

25.6 
26.4 
27.2 
28.0 
28.8 
29.5 
3o.3 
3i.i 
3i  .9 

i8.7 
19.3 
19.9 
20.5 
21. 1 
21.7 
22.3 
22.9 
23.5 
24.1 

9' 

92, 
93 

94 

9? 
96 

97 
98 

99 

ICO 

72.7 
73.5 
74.3 
75.1 
75.9 
76.7 
77.5 
78.3 
79.1 
79-9 

54.8 
55.4 
56. 0 
56.6 
57.2 
57.8 
58.4 
59.0 
59.6 
60.2 

i5i 

52 

53 

54 
55 
56 

57 
58 
59 
6g 

120.6 
121 .4 
122.2 

123.0 

123.8 
124.6 
125.4 
126.2 
127.0 
127.8 

90.0 
91.5 
92.1 
92.7 
93.3 
93.9 
94.5 
95.1 
95.7 
96.3 

211 
12 
i3 

i4 
i5 
16 

17 
18 

19 

20 

168.5 
169.3 
170. 1 
170.9 
171.7 
172.5 
173.3 
174.1 
174.9 
175.7 

127.0 
127.6 
128.2 
128.8 
129.4 
i3o.o 
i3o.6 

l3l.2 

i3i.8 
i32.4 

271 
72 
73 
74 
75 
76 
77 
78 

79 
80 

216.4 

217.2 

218.0 
218.8 

219.6 
220.4 

221.2 
222.0 
222.8 
223.6 

i63.i 
163.7 
164.3 
164.9 
i65.5 
166.1 
166.7 
167.3 
167.9 
168.5 

4i 

42 

43 
44 
45 
46 

47 
-■H 

49 
So 

32.7 
33.5 
3i.3 
35.1 
35.9 
36.7 
37.5 
38.3 
39. 1 
39.9 

24.7 
25.3 
25.9 
26.5 
27.1 
27.7 
28.3 
28.9 
29.^ 
3o.  I 

101 
02 
o3 
04 
o5 
06 
07 
08 
09 

ID 

III 
12 
l3 

■i4 
i5 
16 

17 
18 

19 
20 

Dist. 

80.7 
81.5 
82.3 
83.1 
83.9 

84.7 
85.5 
86.3 
87.1 
87.8 

88.6 
89.4 
90.2 
91 .0 
91.8 
92.6 
93.4 
94.2 
95.0 
95.8 

6u.8 
6r.4 
62.0 
62.6 
63.2 
63.8 
64.4 
65. 0 
65.6 
66.2 

66.8 
67.4 
68.0 
68.6 
69.2 
69.8 
70.4 
71.0 
71.6 
72.2 

161 
62 
63 
64 
65 
66 
67 
68 
69 
70 

128.6 
129.4 
i3o.2 
i3i  .0 
i3i.8 
i32.6 
i33.4 
i34.2 
i35.o 
i35.8 

96.9 
97.5 
98.1 
98.7 
99.3 
99.9 
100.5 
roi  .1 
101 .7 
102.3 

221 
22 

23 

24 

25 

26 

27 
28 

It 

176.5 
1-77.3 
178.1 
178.9 
179-7 
180.5 
181.3 
182.1 
182.9 
183.7 

i33.o 
i33.6 
i34.2 
i34.8 
i35.4 
i36.o 
1 36.6 
137.2 
i37.8 
i38.4 

281 
82 
83 
84 
85 
86 

87 
88 
89 
90 

224.4 
225.2 
226.0 
226.8 
227.6 
228.4 
229.2 

23o.o 
23o.8 
23i.6 

169.1 
169.7 
170.3 
170.9 
171.5 
172.1 
172.7 
173.3 
173.9 
174.5 

5 1 

52 

53 
54 
55 
56 
57 
58 

6() 

40.7 
41.5 
42.3 
43.1 
43.9 
44.7 
45.5 
46.3 
47-1 
47.9 

3o.7 
3i.3 
3.-9 
32.5 
33.1 
33.7 
34.3 

34.9 
35.5 
36.1 

171 
72 
73 
74 
75 
76 
77 
78 

Z9 
80 

i36.6 
137.4 
i38.2 
139.0 
139. S 
i4o.6 
i4i.4 
142.2 
143.0 
143.8 

102.9 
io3.5 
io4.i 
104.7 
io5.3 
i()5.9 
106.5 
107. 1 
107.7 
108.3 

23l 
32 

33 
34 
35 
36 

37 
38 
39 
40 

184.5 
i85.3 
186.1 
186.9 
187.7 
188.5 
189.3 
190.1 
190.9 
191.7 

139.0 
139.6 

l40.3 

140.8 

i4i.4 
142.0 
142.6 
143.2 
143.8 
144.4 

291 

92 
93 
94 

95 
96 

97 
98 

99 
3oo 

232.4 
233.2 

234-0 
234-8 
235-6 
236-4 
237.2 
238.0 
238.8 
239.6 

175.1 

175.7 

1763- 

176.9 

177.5 

178. 1 

178.7 

179.3 

179-9 
180.5 

Dist. 

D,.p. 

Lnt. 

Dep. 

Lat. 

Dist. 

Dep. 

Lat. 

Disi. 

Dep. 

Lat. 

Dist. 

Dep- 

I    Lat. 

[For  .53  Degrees. 

Pago  5-1] 

TABLE  IL 

Difference  of  Latitude  and 

Departure  for  38  Degrees. 

Dist. 

Lat. 

Dep. 

Dist. 

Lat. 

Dep. 
37.6 

Dist. 

Lat. 

Dep. 

Dist. 
181 

Lat. 

142.6 

Dep. 

Dist. 

Lat. 

Dep. 

I 

00.8 

00.6 

6i 

48.1 

121 

95.3 

74.5 

ni.4 

241 

189.9 

148.4 

2 

01 .6 

01 .2 

62 

48.9 

38.2 

22 

96.1 

7b. I 

82 

143.4 

112. 1 

42 

190.7 

149.0 

3 

02.4 

01.8 

63 

49.6 

38.8 

23 

96.9 

7b.7 

83 

144.2 

112.7 

43 

191. 5 

i49-6 

4 

o3.2 

02.5 

64 

bo. 4 

39.4 

24 

97-7 

76.3 

84 

145.0 

n3.3 

A4 

192.3 

i5o.2 

6 

03.9 

o3.i 

65 

bl.2 

4o.o 

25 

98.5 

77.0 

85 

145.8 

113.9 

45 

193.1 

i5o.8 

b 

o4-7 

03.7 

66 

52.0 

4o.6 

26 

99.3 

77-6 

86 

i46.6 

ii4-b 

46 

193.9 

i5i.5 

7 

ob.b 

04.3 

67 

52.8 

4i  .2 

27 

100. 1 

78.2 

87 

\4iA 

Il5.! 

47 

194.6 

l52.I 

« 

Ob. 3 

04.9 

68 

b3.6 

41.9 

28 

100.9 

78.8 

88 

i48.i 

115.7 

48 

195.4 

152.7 

9 

07.1 

ob.b 

69 

b4.4 

42.5 

29 

lOI  .7 

79-4 

89 

148.9 

116.4 

49 

196.2 

i53.3 

10 

07.9 

06.2 

70 

bb.2 

43.1 

3o 

102.4 

80.0 

90 

149.7 

1 17.0 

5o 

197.0 

153.9 

II 

0S.7 

06.8 

71 

55.9 

43.7 

i3i 

io3.2 

80.7 

191 

i5o.5 

117.0 

25l 

197-8 

154.5 

12 

09.5 

07.4 

72 

bb.7 

44.3 

32 

104.0 

81.3 

92 

i5i.3 

118.2 

52 

198.6 

i55.i 

i3 

10.2 

08.0 

73 

b7.b 

44.9 

33 

104.8 

81.9 

93 

l52.1 

118.8 

53 

199.4 

i55.8 

i4 

II  .0 

08.6 

74 

58.3 

45.6 

34 

io5.6 

82.5 

94 

152.9 

119.4 

54 

200.2 

i56.4 

lb 

II. 8 

09.2 

75 

b9.i 

46.2 

35 

106.4 

83.1 

9b 

i53.7 

120. 1 

55 

200.9 

.57.0 

i6 

12.6 

09.9 

76 

^9-9 

46.8 

36 

107.2 

83.7 

96 

ib4.b 

120.7 

56 

201.7 

157.6 

17 

i3.4 

10.5 

77 

60.7 

47-4 

37 

1 08 . 0 

84.3 

97 

i55.2 

121.3 

57 

202. D 

1 58.2 

i8 

l4.2 

11. 1 

78 

61.5 

48.0 

38 

108.7 

85. 0 

98 

i56.o 

1 2 1. 9 

58 

2o3.3 

i58.8 

19 

ib.o 

II. 7 

79 

62.3 

48.6 

39 

109.5 

85.6 

99 

1 56.8 

122.5 

59 

204.1 

159.5 

20 

lb. 8 

12.3 

80 

63. 0 

49-3 
49.9 

4o 

no. 3 

86.2 

200 

1 57.6 

123.1 

60 

204.9 

1 60. 1 

21 

16.5 

12.9 

81 

63.8 

i4i 

III  .1 

86.8 

201 

i58.4 

123.7 

261 

2o5.7 

160.7 

22 

17.3 

i3.5 

82 

64.6 

5o.5 

42 

1 1 1 .9 

87.4 

02 

159.2 

124-4 

62 

206.5 

161.3 

23 

18. 1 

14.2 

83 

65.4 

5i.i 

43 

112. 7 

88.0 

o3 

160.0 

125.0 

63 

207.2 

161.9 

24 

18.9 

i4.8 

84 

66.2 

5l.7 

44 

ii3.5 

88.7 

04 

160.8 

125.6 

H 

208.0 

162.5 

25 

19.7 

i5.4 

85 

67.0 

52.3 

45 

114.3 

89.3 

o5 

161.5 

126.2 

65 

208.8 

i63.2 

26 

20.5 

16.0 

86 

67.8 

52.9 

46 

ii5.o 

89.9 

06 

162.3 

126.8 

66 

209.6 

1 63 .8 

27 

21.3 

16.6 

87 

68.6 

53.6 

47 

ii5.8 

90.5 

07 

i63.i 

127.4 

67 

210.4 

164.4 

28 

22.1 

17.2 

88 

69.3 

54.2 

48 

116. 6 

91.1 

08 

163.9 

128.1 

68 

211.2 

i65.o 

29 

22.9 

17.9 

89 

70.1 

54.8 

49 

117-4 

91.7 

09 

164.7 

128.7 

69 

212.0 

i65.6 

Jo 

23.6 

18. b 

90 

70.9 

55.4 

5o 

118. 2 

92.3 

10 

ibb.5 

129.3 

70 

212.8 

166.2 

3i 

24.4 

19.1 

91 

71-7 

56.0 

i5i 

119. 0 

93.0 

211 

166.3 

129.9 

271 

2i3.6 

166.8 

32 

2b. 2 

19.7 

92 

72.5 

56.6 

52 

119. 8 

93.6 

12 

167.1 

i3o.5 

72 

214.3 

167,5 

33 

26.0 

20.3 

93 

73.3 

57.3 

53 

120.6 

94.2 

i3 

167.8 

i3i.i 

73 

2l5.1 

168.1 

34 

2b. 8 

20.9 

94 

74.1 

57.9 

54 

121 .4 

94.8 

i4 

168.6 

i3i.8 

74 

215.9 

168.7 

35 

27.6 

21.5 

95 

74.9 

58.5 

55 

122. 1 

95.4 

i5 

169.4 

i32.4 

75 

216.7 

169.3 

36 

28.4 

22.2 

96 

75.6 

59.1 

56 

122.9 

96.0 

16 

170.2 

i33.o 

76 

217.5 

169.9 

37 

29.2 

22.8 

97 

76.4 

59.7 

57 

123.7 

96.7 

17 

171.0 

J33.6 

77 

218.3 

170.5 

38 

29.9 

23.4 

98 

77.2 

60.3 

58 

124.5 

97.3 

18 

171.8 

i34.2 

■    78 

219.1 

171. 2 

39 

30.7 

24.0 

99 

78.0 

Ol  .0 

59 

125.3 

97-9 

19 

172.6 

i34.8 

79 

219.9 

171.8 

4o 

Ji.b 

24.6 

100 

78.8 

61.6 

60 

126. 1 

98. b 

20 

173.4 

i3b.4 

80 

220  6 

172.4 

4i 

32.3 

25.2 

lOI 

79.6 

62.2 

161 

126.9 

99.1 

221 

174.2 

1 36.1 

281 

221.4 

173.0 

42 

33.1 

25.9 

02 

80.4 

62.8 

62 

127.7 

99-7 

22 

174.9 

i36.7 

82 

222.2 

173.6 

43 

33.9 

26.5 

o3 

81.2 

63.4 

63 

128.4 

100.4 

23 

175.7 

137.3 

83 

223.0 

174.2 

44 

34.7 

27.1 

04 

82.0 

64.0 

64 

129.2 

101 .0 

24 

176.5 

137.9 

84 

223.8 

174.8 

45 

3b. b 

27.7 

o5 

82.7 

64.6 

65 

1 3o .  0 

101 .6 

25 

177.3 

i38.5 

85 

224.6 

i7b.b 

46 

36.2 

28.3 

06 

83.5 

65.3 

66 

i3o.8 

102.2 

26 

178.1 

139.1 

86 

225-4 

1 76. 1 

47 

37.0 

28.9 

07 

84.3 

65.9 

67 

i3i.6 

102.8 

27 

178.9 

139.8 

87 

226.2 

176.7 

48 

37.8 

29.6 

08 

8b. I 

66.5 

68 

i32.4 

io3.4 

28 

179-7 

140.4 

88 

226.9 

'77-3 

49 

38.6 

3o.2 

09 

85.9 

67.1 

69 

i33.2 

104.0 

29 

180.5 

i4i.o 

89 

227.7 

1779 

bo 

39.4 

3o.8 

10 

86.7 

67.7 

70 

i34.o 

104.7 

3o 

181. 2 

141.6 

90 

228.5 

178.5 

5i 

40.2 

3i.4 

II I 

87.5 

68.3 

171 

134.7 

io5.3 

23  I 

182.0 

142.2 

291 

229.3 

179.2 

52 

4i  .0 

32.0 

12 

88.3 

69.0 

72 

i35.5 

105.9 

32 

182.8 

142.8 

92 

23o.l 

179-8 

53 

4i.8 

32.6 

i3 

89.0 

69.6 

73 

i36.3 

106.5 

33 

i83.6 

143.4 

93 

230.9 

180.4 

54 

42.6 

33.2 

i4 

89.8 

70.2 

74 

137. 1 

1 07 . 1 

34 

184.4 

144.1 

94 

23i.7 

181.0 

bb 

43.3 

33.9 

i5 

90.6 

70.8 

75 

137.9 

107.7 

35 

i85.2 

144-7 

95 

232.5 

181.6 

b6 

44.1 

34.5 

16 

91 .4 

71.4 

76 

i38.7 

108.4 

36 

186.0 

145.3 

96 

233.3 

182.2 

57 

44.9 

35.1 

17 

92.2 

72  .0 

77 

139.5 

1 09 . 0 

37 

186.8 

145.9 

97 

234-0 

182.9 

b8 

4b. 7 

35.7 

18 

93.0 

72.6 

78 

i4o.3 

109.6 

38 

187.5 

146.5 

98 

234.8 

i83.5 

59 

46.5 

36.3 

19 

9J.8 

73.3 

79 

i4i .  I 

1 10.2 

39 

188.3 

147.1 

99 

235.6 

184.1 

bo 

47.3 

36.9 

20 

94.6 

73.9 

80 

i4i.8 

no. 8 

4o 

189.1 

147-8 

3  00 

236.4 

184.7 

Dist. 

Dep. 

Lat. 

Disl. 

Dep. 

Lat. 

Dist. 

Dep. 

Lat. 

Dist. 

Dep. 

Lat. 

Dist.l   Dep. 

Lat. 

[1 

Por  52  Degrees. 

TABLE   IL 

JPuge  5i) 

Difference  of  La 

itude  and  Departure  for  39  Degrees. 

Dist. 

Lai. 

Dep. 

Dist. 

Lat. 

Dep. 

Dist. 

Lat. 

Dep. 

Dist. 

Lat. 

Dep. 

Dist. 

Lat. 

Dep. 

I 

00.8 

00.6 

61 

47-4 

38.4 

121 

94.0 

76.1 

181 

140.7 

113.9 

24 1    18-7.3 

i5i.7 

2 

01 .6 

01 .3 

62 

48.2 

39.0 

22 

94.8 

76.8 

82 

141.-^ 

114.5 

42 

188. 1 

i52.3 

3 

02.3 

01 .9 

63 

49.0 

39.6 

23 

95.6 

77.4 

83 

142.2 

Il5.2 

43 

188.8 

i52.9 

4 

oJ.i 

02.5 

64 

49-7 

4o.3 

24 

96.4 

78.0 

84 

143.0 

11 5.8 

44 

189.6 

i53.6 

b 

o3.9 

o3.i 

6b 

5o.5 

40.9 

25 

97.1 

78.7 

85 

143.8 

116.4 

45 

190.4 

i54.2 

6 

04.7 

o3.8 

66 

bi.3 

4i.b 

26 

97-9 

79.3 

86 

144.5 

117.1 

46 

191.2 

1 54.8 

7 

o5.4 

04.4 

67 

b2.i 

42.2 

27 

9».7 

79-9 

«7 

145.3 

117.7 

47 

192.0 

i55.4 

8 

06.2 

o5.o 

68 

b2.8 

42.8 

28 

99.5 

80.6 

88 

I46.I 

118.3 

48 

192.-' 

1 56. 1 

9 

07.0 

Ob. 7 

69 

b3.6 

4^.4 

29 

100.3 

81.2 

89 

146.9 

118.9 

49 

193.5 

i56.7 

lO 

07.8 

06.3 

70 

54.4 

44.1 

3o 

lOI  .0 

81.8 

90 

•47-7 

1 19.6 

5o 

194.3 

157.3 

1 1 

08.5 

06.9 

71 

55.2 

44.7 

i3i 

I0I.8 

82.4 

191 

148.4 

120.2 

25l 

195.1 

i58.o 

12 

09.3 

07.6 

72 

bb.o 

4b. 3 

32 

102.6 

83.1 

92 

i49-2 

120.8 

52 

195.8 

i58.6 

i6 

10. 1 

08.2 

73 

b6.7 

45.9 

33 

io3.4 

83.7 

93 

i5o.o 

121.5 

53 

196.6 

159.2 

i4 

10.9 

08.8 

74 

b7.b 

46.6 

34 

io4. 1 

84.3 

94 

i5o.8 

122.1 

54 

197.4 

159.8 

lb 

II. 7 

09.4 

75 

b8.3 

47-2 

35 

104.9 

85.0 

95 

i5i.5 

122.7 

55 

J98.2 

160.5 

lb 

12.4 

10. 1 

76 

59. 1 

47.8 

36 

io5.7 

85.6 

96 

i52.3 

123.3 

56 

198.9 

161.1 

17 

l3.2 

10.7 

77 

b9.8 

48. b 

37 

106.5 

86.2 

97 

i53.i 

124.0 

57 

199.7 

161.7 

i8 

i4-o 

II. 3 

78 

bo.b 

49.1 

38 

107.2 

86.8 

98 

153.9 

124.6 

58 

200.5 

162.4 

19 

i4.8 

12.0 

79 

61.4 

49.7 

39 

108.0 

87.5 

99 

154.7 

125.2 

59 

201.3 

i63.o 

20 
21 

ib.b 
16.3 

12.6 

l3.2 

80 

62.2 

5o .  3 

4o 

108.8 

88.1 

200 

1 55.4 

125.9 

60 

202.1 

1 63.6 

81 

62.9 

5i.o 

i4i 

109.6 

88.7 

201 

i56.2 

126.5 

261 

202.8 

164.3 

22 

17.1 

i3.8 

82 

b3.7 

bi.b 

42 

110.4 

89.4 

02 

157.0 

127.1 

62 

2o3.6 

164.9 

2J 

17.9 

14.5 

83 

b4.b 

52.2 

4'^ 

III  .1 

90.0 

o3 

157.8 

127.8 

63 

204.4 

,65.5 

24 

18.7 

lb. I 

84 

bb.3 

52.9 

44 

HI  .9 

90.6 

04 

1 58.5 

128.4 

64 

2o5.2 

166.1 

2!) 

.9-4 

lb. 7 

85 

bb.i 

53.5 

45 

112.7 

91 .3 

o5 

159.3 

129.0 

65 

205.9 

166.8 

2b 

20.2 

16.4 

86 

66.8 

b4.i 

46 

ii3.5 

91.9 

06 

160.1 

129.6 

66 

206.7 

167.4 

27 

21 .0 

17.0 

H7 

67.6 

b4.8 

47 

Il4-2 

92.5 

07 

160.9 

i3o.3 

67 

207.5 

168.0 

28 

21.8 

17.6 

88 

b8.4 

bb.4 

48 

ii5.o 

93.1 

08 

161.6 

1 30.9 

68 

208.3 

168.7 

29 

22  .5 

18.3 

89 

69.2 

56. 0 

49 

ii5.8 

93.8 

09 

162.4 

i3i.5 

69 

209.1 

169.3 

Jo 

23.3 

18.9 

90 

69-9 

bb.b 

5o 

116.6 

94.4 

10 

i63.2 

l32.2 

70 

209.8 

169.9 

3i 

24.1 

19.5 

91 

70.7 

57.3 

i5i 

117. 3 

95.0 

211 

164.0 

i32.8 

271 

210.6 

170.5 

32 

24.9 

20.1 

92 

71.5 

57.9 

D2 

118. 1 

95.7 

12 

164.8 

i33.4 

72 

2  1  1.4 

171.2 

33 

25.6 

20.8 

93 

72.3 

58.5 

53 

118.9 

96.3 

i3 

i65.5 

1 34.0 

73 

212.2 

171.8 

34 

26.4 

21.4 

94 

73.1 

59.2 

54 

119.7 

96.9 

i4 

166.3 

i34.7 

74 

212.9 

172.4 

3  b 

27.2 

22.0 

95 

73.8 

59.8 

55 

120.5 

97.5 

i5 

167.1 

i35.3 

75 

2l3.7 

173.1 

db 

28.0 

22.7 

96 

74.6 

60.4 

56 

121 .2 

98.2 

16 

167.9 

135.9 

76 

214.5 

173.7 

37 

28.8 

23.3 

97 

75.4 

61 .0 

57 

122.0 

98.8 

17 

168.6 

1 36.6 

77 

2i5.3 

174.3 

38 

29.5 

23.9 

98 

76.2 

bi.7 

58 

122.8 

99.4 

18 

169.4 

137.2 

78 

216.0 

175.0 

39 

3o.3 

24. b 

99 

76.9 

62.3 

59 

123.6 

100. 1 

19 

170.2 

137.8 

79 

216.8 

175.6 

4o 

3i.i 

25.2 

100 

77-7 

6:2.9 

60 

124.3 

100.7 

20 

1 71.0 

i38.5 

80 

217.6 

176.2 

4i 

31.9 

25.8 

lOI 

78.5 

63.6 

161 

125.1 

101 .3 

221 

171.7 

139.1 

281 

218.4 

176.8 

42 

32.6 

2G.4 

02 

79.3 

64.2 

62 

125.9 

lOI  .9 

22 

172.5 

139.7 

82 

219.2 

177-5 

43 

i^.4 

27.1 

o3 

80.0 

b4.8 

63 

126.7 

102.6 

23 

173.3 

i4o.3 

83 

219.9 

178.1 

44 

34.2 

27.7 

04 

80.8 

bb.4 

64 

127.5 

103.2 

24 

1 74. 1 

i4i.o 

84 

220.7 

178.7 

4^) 

35.0 

28.3 

o5 

81.6 

6b.  I 

65 

128.2 

io3.8 

25 

174.9 

141.6 

85 

221.5 

179.4 

46 

3b. 7 

28.9 

06 

82.4 

bb.7 

66 

129.0 

104.5 

26 

175.6 

142.2 

86 

222.3 

180.0 

47 

36.5 

29.6 

07 

83.2 

67.3 

67 

129.8 

io5.i 

27 

176.4 

142.9 

87 

223.0 

180.6 

48 

37.3 

3o.2 

08 

83.9 

68.0 

68 

1 3o .  6 

io5.7 

28 

177.2 

143.5 

88 

223.8 

181.2 

^9 

38.1 

3o.8 

09 

84.7 

68.6 

69 

i3i.3 

106.4 

29 

178.0 

1 44. 1 

89 

224.6 

181.9 

bo 

38.9 

3i.b 

10 

8b. b 

69.2 

70 

l32.I 

1 07 . 0 

3o 

178.7 

144.7 

90 

225.4 

182.5 

5, 

39.6 

32.1 

II I 

86.3 

69.9 

171 

132.9 

107.6 

23l 

179.5 

145.4 

291 

226.1 

i83.i 

b2 

40.4 

32.7 

12 

87.0 

70.5 

72 

i33.7 

108.2 

32 

180.3 

i46.o 

92 

226.9 

i83.8 

b3 

41.2 

33.4 

i3 

87.8 

71. 1 

73 

i34.4 

^108.9 

33 

181. 1 

i46.6 

93 

227.7 

184.4 

t)4 

42.0 

34.0 

i4 

88.6 

71-7 

74 

1 35. 2 

109.5 

34 

181.9 

147.3 

94 

228.5 

i85.o 

bb 

42.7 

34.6 

lb 

89.4 

72.4 

75 

i36.o 

no.  1 

35 

182.6 

147.9 

95 

229.3 

i85.6 

bb 

43. b 

35.2 

lb 

90.1 

73.0 

76 

i36.8 

110.8 

36 

i83.4 

148.5 

96 

23o.O 

186.3 

i)7 

44.3 

3b. 9 

17 

90.9 

73. b 

77 

137.6 

III. 4 

37 

184.2 

149.1 

97 

230.8 

166.9 

58 

4b. I 

3b. b 

18 

91.7 

74.3 

78 

i38.3 

112.0 

38 

i85.o 

149.8 

98 

23].6 

187.5 

b9 

45.9 

37.1 

19 

92.5 

74.9 

79 

139. 1 

I  12.6 

3q 

1^.5.7 

i5o.4 

99 

232.4 

188.2 

bo 

45.6 

37.8 

20 

93.3 

75.5 

80 

139.9 

II3.3 

40 

i86.5 

i5i.o 

3  00 

233.1  i  ib8.» 

i)ist. 

Do  p. 

I.nt. 

Dlsi. 

Dep. 

Lat. 

Disi. 

Dep. 

Lat. 

Dist. 

Dep. 

Lat. 

Dist. 

Dep.  1    Lat. 

[1 

^or  51  Decrees 

Page  5G] 

TABLE  IL 

1 

Difference  of  Latitude  and  Departure  for  40  Degrees. 

Dist. 

Lat. 

Dep. 

Dist.  I  Lai. 

Dep. 

Dist. 

Lat. 

Dep. 

Dist. 

181 

Lat. 

i38.7 

Dep. 
116.3 

Dist. 

Lat. 

Dep. 

I 

01J.8 

00.6 

61 

46.7 

39.2 

121 

92.7 

77.8 

241 

184.6 

154.9 

2 

01.5 

01.3 

62 

47. b 

39.9 

22 

93.5 

78. 4 

82 

139.4 

117. 0 

42 

185.4 

i55.6 

3 

02  .J 

01 .9 

63 

48.3 

40.5 

23 

94.2 

79.1 

83 

i4o.2 

1 17.6 

43 

1 86. 1 

i56.2 

4 

o3.r 

02.6 

64 

49.0 

41.1 

24 

95.0 

79-7 

84 

i4i.o 

1 18.3 

AA 

186.9 

1 56.8 

3 

o3.8 

03.2 

65 

49.8 

4i.8 

25 

95.8 

80.3 

85 

141.7 

118.9 

45 

187.7 

1575 

6 

()4.b 

03.9 

66 

5o.6 

42.4 

26 

96.5 

81.0 

86 

142.5 

1 19.6 

46 

188.4 

i58.i 

7 

o5.4 

o4.5 

67 

5i.3 

43.1 

27 

97.3 

81.6 

87 

143.3 

120.2 

47 

189.2 

i58.8 

8 

06.1 

o5.i 

68 

52.1 

43.7 

28 

98.1 

82.3 

88 

144.0 

120.8 

48 

190.0 

159.4 

9 

06.9 

o5.8 

69 

52.9 

44.4 

29 

98.8 

82.9 

89 

144.8 

121. 5 

49 

190.7 

1 60. 1 

10 

11 

07.7 

06.4 

70 

53.6 

45.0 

3o 

99.6 

83.6 
84.2 

90 

145.5 

122. 1 

5o 

25l 

191. 5 
192.3 

160.7 

08.4 

07.1 

71 

54.4 

45.6 

i3i 

100.4 

IQI 

146.3 

122.8 

161.3 

12 

09.2 

07.7 

72 

55.2 

46.3 

32 

lOI  .  I 

84.8 

92 

147-1 

123.4 

52 

193.0 

162.0 

i3 

10. 0 

08.4 

73 

55.9 

46.9 

3:^ 

101.9 

85.5 

q3 

147-8 

124. 1 

53 

193.8 

162.6 

i4 

10.7 

09.0 

74 

56.7 

47.6 

34 

102.6 

86.1 

94 

148.6 

124.7 

54 

194.6 

i63.3 

i5 

II. 5 

09.6 

75 

07.5 

48.2 

35 

io3.4 

86.8 

g5 

149.4 

125.3 

55 

195.3 

163.9 

i6 

12.3 

10.3 

76 

58.2 

48.9 

36 

io4.2 

87.4 

96 

i5o.i 

126.0 

56 

1 96. 1 

164.6 

17 

i3.o 

10.9 

77 

59.0 

49.^ 

37 

io4-9 

88.1 

97 

i5o.9 

126.6 

57 

196.9 

i65.2 

i8 

i3.8 

II. 6 

78 

59.8 

5o.i 

38 

io5.7 

88. 7 

98 

i5i.7 

127.3 

58 

197.6 

i65.8 

19 

14.6 

12.2 

79 

60.5 

5o.8 

39 

106.5 

89.3 

99 

i52.4 

127.9 

59 

198.4 

166.5 

20 

i5.3 

12.9 

80 

61.3 

5i.4 

52.1 

4o 

107.2 

90.0 

200 

i53.2 

128.6 

60 
261 

_i99^ 
199.9 

167.1 

21 

16.1 

i3.5 

81 

62.0 

i4i 

■  108.0 

90.6 

201 

1 54.0 

129.2 

167.8 

22 

16.9 

14.1 

82 

62.8 

52.7 

42 

108.8 

91.3 

02 

154.7 

129.8 

62 

200.7 

168.4 

23 

17.6 

14.8 

83 

63.6 

53.4 

43 

109.5 

91.9 

o3 

i55.5 

i3o.5 

63 

201.5 

169.1 

24 

18.4 

i5.4 

84 

64.3 

54.0 

AA 

no. 3 

92.6 

o4 

1 56.3 

i3i.i 

64 

202.2 

169.7 

25 

19.2 

16. 1 

85 

65.1 

54.6 

45 

III. I 

93.2 

o5 

157.0 

i3i.8 

65 

2o3.o 

170.3 

26 

19.9 

16.7 

86 

65.9 

55.3 

46 

III. 8 

93.8 

06 

157.8 

i32.4 

66 

2o3.8 

171. 0 

27 

20.7 

17.4 

87 

66.6 

55.9 

47 

1 12.6 

94.5 

07 

1 58.6 

1 33. 1 

67 

204.5 

171.6 

28 

21.4 

18.0 

88 

67.4 

56.6 

48 

ii3.4 

Q5.I 

08 

159.3 

133.7 

68 

2o5.3 

172.3 

29 

22.2 

18.6 

89 

68.2 

57.2 

49 

ii4-i 

95.8 

09 

160. 1 

i34.3 

69 

206.1 

172.9 

■50 

23.0 

19.3 

90 

68.9 

57.9 

5o 

114. 9 

96.4 

10 

160.9 

i35.o 

70 

206.8 

173.6 

3i 

23.7 

19.9 

91 

69.7 

58.5 

i5i 

ii5.7 

97.1 

211 

161.6 

i35.6 

271 

207.6 

174.2 

32 

24.5 

20.6 

92 

70.5 

59.1 

52 

116. 4 

97-7 

12 

162.4 

i36.3 

72 

208.4 

174.8 

33 

23.3 

21 .2 

93 

71.2 

59.8 

53 

117. 2 

98.3 

i3 

i63.2 

i36.9 

73 

209.1 

175.5 

34 

26.0 

21 .9 

94 

72.0 

60.4 

64 

118. 0 

99.0 

i4 

163.9 

137.6 

74 

209.9 

176. 1 

35 

26.8 

22.5 

95 

72.8 

61. 1 

55 

118. 7 

99.6 

i5 

164.7 

i38.2 

75 

210.7 

176.8 

36 

27.6 

23.1 

96 

73.5 

61.7 

56 

119. 5 

100.3 

16 

i65.5 

i38.8 

76 

2 1 1.4 

177-4 

37 

28. 3 

23.8 

97 

74.3 

62.4 

57 

120.3 

100.9 

17 

166.2 

139.5 

77 

212.2 

178. 1 

38 

29. 1 

24.4 

98 

75.1 

63. 0 

58 

121. 0 

101.6 

18 

167.0 

i4o.i 

78 

2l3.0 

178.7 

39 

29.9 

25.1 

99 

75.8 

63.6 

59 

121. 8 

102.2 

19 

167.8 

i4o.8 

79 

2i3.7 

179.3 

4o 

3o.6 

25.7 

100 

76.6 

64.3 

60 

122.6 

102.8 

20 

168.5 

i4i.4 
142. 1 

80 

214.5 

180.0 

Ai 

3i.4 

26.4 

lOI 

77-4 

64.9 

161 

123.3 

io3.5 

221 

169.3 

281 

2 1 5.3 

180.6 

42 

32.2 

27.0 

02 

78.1 

65.6 

62 

1 24. 1 

104. 1 

22 

170.1 

142.7 

82 

216.0 

181.3 

43 

32.9 

27.6 

o3 

78.9 

66.2 

63 

124.9 

104.8 

23 

170.8 

143.3 

83 

216.8 

1 8 1. 9 

M 

3J.7 

28.3 

04 

79-7 

66.8 

64 

125.6 

io5.4 

24 

171.6 

i44-o 

84 

217.6 

182.6 

45 

34.5 

28.9 

o5 

80.4 

67.5 

65 

126.4 

106. 1 

25 

172.4 

144.6 

85 

218.3 

i83.2 

46 

35.2 

29.6 

06 

81.2 

68.1 

66 

127.2 

106.7 

26 

173. 1 

145.3 

86 

219. 1 

1 83.8 

47 

3b.  0 

3o.2 

07 

82.0 

68.8 

67 

127.9 

107.3 

27 

173.9 

145.9 

87 

219.9. 

184.5 

4S 

36.8 

3o.9 

08 

82.7 

69.4 

68 

128.7 

108.0 

28 

174-7 

146.6 

88 

220.6 

i85.i 

49 

37.5 

3i.5 

09 

83.5 

70.1 

69 

129.5 

108.6 

29 

175.4 

l47-2 

89 

221.4 

i85.8 

5o 

38.3 

32.1 

10 

84.3 

70.7 

70 

l30.2 

109.3 

3o 

176.2 

147-8 

90 

222.2 

186.4 

5i 

39., 

32.8 

III 

85.0 

71.3 

171 

i3i  .0 

109.9 

23l 

177.0 

148.5 

291 

222.9 

187.1 

52 

39.8 

S6.A 

12 

85.8 

72.0 

72 

i3r.8 

no. 6 

32 

177.7 

149.1 

92 

223.7 

187.7 

53 

40.6 

34.1 

i3 

86.6 

72.6 

73 

i32.5 

II 1 .2 

33 

178.5 

149.8 

93 

224.5 

188.3 

54 

41.4 

34.7 

i4 

87.3 

73.3 

74 

i33.3 

III. 8 

34 

179.3 

i5o.4 

94 

225.2 

189.0 

55 

42. 1 

35.4 

i5 

88.1 

73.9 

75 

i34.i 

112. 5 

35 

180.0 

i5i.i 

95 

226.0 

189.6 

56 

42.9 

36.0 

16 

88. q 

74.6 

76 

i34.8 

ii3.i 

36 

180.8 

i5i.7 

96 

226.7 

190.3 

57    43.7  1 

36.6 

17 

89.6 

75.2 

77 

i35.6 

ii3.8 

37 

181.6 

i52.3 

97 

227.5 

190.9 

58 

44.4 

37.3 

18 

90.4 

75.8 

78 

i36.4 

114.4 

38 

182.3 

i53.o 

98 

228.3 

191.6 

59 

45.2 

37.9 

19 

91.2 

76.5 

79 

137. 1 

ii5.i 

39 

i83.i 

1 53.6 

99 

229.0 

192.2 

b.i 

46. 0 

38.6 

20 

91.9 

77-1 

80 

137.9 

ii5.7 

40 

183.9 

1 54.3 

3oo   229.8 

192.8 

ni.i. 

1),.,,. 

Lat. 

Dist. 

Dep. 

Lat. 

Dist.     Dep.  1 

Lat. 

Dist. 

Dep. 

Lat. 

Dist.    Dep.      Lat.   | 

[For  50  Degrees. 

TABLE  IL 

[Page  57 

Difference  of  Latitude  and  Departure  for  41 

Degrees 

Disl. 

Lat. 

Dep. 

Dist. 

Lat. 

Dep. 

Dist. 

Lat.       D.'p.   I 

Dist 
"i^i 

Lai. 

Dep. 

Disl. 

Lat.  I  Dep.  ] 

I 

00.8 

00.7 

61 

46. 0 

4o.o 

121 

91.3 

79-4 

1 36.6 

118.7 

241 

181.9 

i58.i 

2 

or. 5 

01 .3 

62 

46.8 

40.7 

22 

92. 1 

80.0 

82 

137.4 

119.4 

42 

182.6 

i58.8 

3 

02.3 

02.0 

63 

47.5 

4i.3 

23 

92.8 

80.7 

83 

i38.i 

1 20. 1 

43 

i83.4 

159.4 

4 

o3.o 

02.6 

64 

48.3 

42.0 

24 

93.6 

81.4 

84 

i38.9 

120.7 

44 

184.1 

160.1 

fi 

o3.8 

o3.3 

65 

49.1 

42.6 

25 

94.3 

82.0 

8b 

139.6 

1 2 1. 4 

45 

1S4.9 

160.7 

6 

04.5 

o3.9 

66 

49.8 

43.3 

26 

95.1 

82.7 

86 

140.4 

122.0 

46 

185.7 

161.4 

o5.3 

04.6 

67 

5o.6 

44.0 

27 

95.8 

83.3 

87 

i4i.i 

122.7 

47 

186.4 

162.0 

S 

06.0 

o5.2 

68 

5i.3 

44.6 

28 

96.6 

84.0 

88 

141.9 

123.3 

48 

187.2 

162.7 

9 

06.8 

o5.9 

69 

52.1 

45.3 

29 

97-4 

84.6 

89 

142.6 

124.0 

49 

187.9 

i63.4 

10 

1 1 

07.5 

osTS" 

06.6 
07.2 

70 

71 

52.8 

45.9 

3o 

98.1 

85.3 

90 

143.4 

124.7 

5o 

188.7 

164.0 
164.7 

53.6 

46.6 

i3i 

98.9 

85-9 

191 

144.1 

125.3 

25l 

189.4 

12 

09.1 

07.9 

72 

54.3 

47.2 

32 

99.6 

86.6 

92 

144.9 

126.0 

52 

190.2 

1 65.3 

1 3 

09.8 

08.5 

73 

55.1 

47-9 

33 

100.4 

87.3 

93 

145.7 

126.6 

53 

190.9 

166.0 

1 4 

10.6 

09.2 

74 

55.8 

48.5 

34 

loi  .1 

87.9 

94 

146.4 

127.3 

54 

191-7 

166.6 

i') 

II. 3 

09.8 

75 

56.6 

49.2 

35 

loi  .9 

88.6 

95 

147-2 

127.9 

55 

192.5 

167.3 

i6 

12. 1 

10.5 

76 

57.4 

49.9 

36 

102.6 

89.2 

96 

i47-9 

128.6 

56 

193.2 

168.0 

17 

12.8 

II. 2 

77 

58.1 

5o.5 

37 

io3.4 

89.9 

97 

148.7 

129.2 

t>7 

194.0 

168.6 

i8 

i3.6 

II. 8 

78 

58.9 

5l.2 

38 

io4.i 

90.5 

98 

1 49 -4 

129.9 

58 

194.7 

169.3 

'9 

i4.3 

12.5 

79 

59.6 

5i.8 

39 

104-9 

91 .2 

99 

l5o.2 

i3o.6 

59 

195.5 

169.9 

2(> 

i5.i 

i3.i 

80 

60.4 

52.5 

4o 

io5.7 

91.8 

200 

1 50.9 

l3l.2 

60 

196.2 

170.6 

21 

i5.8 

i3.8 

81 

61. 1 

53.1 

i4i 

106.4 

92.5 

201 

1 5 1.7 

i3i.9 

261 

197.0 

171.2 

22 

16.6 

14.4 

82 

61 .9 

53.8 

42 

107.2 

93.2 

02 

i52.5 

i32.5 

62 

197.7 

171.9 

23 

17-4 

i5.i 

83 

62.6 

54.5 

43 

107.9 

93.8 

o3 

i53.2 

i33.2 

63 

198.5 

172.5 

24 

18. 1 

l5.7 

84 

63.4 

55.1 

44 

10S.7 

94.5 

04 

154.0 

i33.8 

64 

199.2 

173.2 

25 

18.9 

16.4 

85 

64.2 

55.8 

45 

109.4 

95.1 

o5 

154.7 

i34.5 

65 

200.0 

173.9 

26 

19.6 

17. 1 

8& 

64.9 

56.4 

46 

no. 2 

95.8 

06 

155.5 

i35.i 

66 

200.8 

174.5 

27 

20.4 

17-7 

87 

65.7 

57.1 

47 

no. 9 

96-4 

07 

i56.2 

i35.8 

67 

201.5 

175.2 

28 

21 .1 

18.4 

88 

66.4 

57.7 

48 

III. 7 

97.1 

08 

157.0 

1 36.5 

68 

202.3 

175.8 

29 

21.9 

19.0 

89 

67.2 

58. 4 

49 

112. 5 

97.8 

09 

157.7 

137.1 

69 

2o3.o 

176.5 

3o 
3i 

22.6 

19.7 

90 

67.9 

59.0 

5o 

Il3.2 

98.4 

10 

i58.5 

137.8 

70 
271 

2o3.8 

177.1 

23.4 

20.3 

91 

68.7 

59.7 

i5i 

114.0 

99.1 

211 

159.2 

i3S.4 

204.5 

177.8 

32 

24.2 

21 .0 

92 

69.4 

60.4 

52 

114.7 

99-7 

12 

160.0 

139.1 

72 

2o5.3 

178.4 

33 

24.9 

21 .6 

93 

70.2 

61 .0 

53 

ii5.5 

100.4 

i3 

160.8 

139.7 

73 

206.0 

1 79. 1 

34 

25.7 

22.3 

q4 

70.9 

61.7 

54 

116.2 

101 .0 

i4 

161.5 

i4o.4 

74 

206.8 

179.8 

35 

26.4 

23.0 

q5 

71.7 

62.3 

55 

1 17.0 

101 .7 

i5 

162.3 

i4i-i 

7^ 

207.5 

180.4 

36 

27.2 

23.6 

96 

72.5 

63. 0 

56 

117. 7 

102.3 

16 

i63.o 

141.7 

7b 

20&.3 

181. 1 

37 

27. Q 

24.3 

97 

73.2 

63.6 

57 

118.5 

io3.o 

17 

i63.8 

142.4 

77 

209.1 

181.7 

3.H 

28.7 

24.9 

98 

74.0 

64.3 

58 

119. 2 

io3.7 

18 

164.5 

i43.o 

78 

209.8 

182.4 

39 

29.4 

25.6 

9Q 

74.7 

64.9 

59 

120.0 

104.3 

19 

i65.3 

143.7 

79 

210.6 

i83.o 

4o 

3o.2 

26.2 

100 

75.5 

65.6 

60 

120.8  '  io5.o 

20 

221 

166.0 

144.3 

80 

211.3 

183.7 

4i 

3o.9 

26.9 

101 

76.2 

66.3 

161 

121.5 

io5.6 

166.8 

145.0 

281 

212.1 

184.4 

42 

3. .7 

27.6 

02 

77.0 

66.9 

62 

122.3 

106.3 

22 

167.5 

145.6 

82 

212.8 

i85.o 

43 

32.5 

28.2 

o3 

77-7 

67.6 

63 

123.0 

106.9 

23 

168.3 

i46.3 

83 

21 3.6 

:85.7 

44 

33.2 

28.9 

o4 

78.5 

68.2 

64 

123.8 

107.6 

24 

169.1 

147.0 

84 

214.3 

186.3 

45 

34.0 

29.5 

o5 

79.2 

68.9 

65 

124.5 

108.2 

25 

169.8 

147.6 

85 

2l5.I 

187.0 

46 

34.7 

3o.2 

06 

80.0 

69.5 

66 

125.3 

108.9 

26 

170.6 

i48.3 

86 

2 1 5.8 

187.6 

47 

35.5 

3o.8 

07 

80.8 

70.2 

67 

126.0 

109. 0 

27 

171. 3 

148.9 

87 

216.6 

188.3 

48 

36.2 

3i.5 

08 

81.5 

70.9 

68 

126.8 

no. 2 

28 

172.1 

149.6 

88 

217.4 

188.9 

49 

37.0 

32.1 

09 

82.3 

71.5 

69 

127.5 

110.9 

29 

172.8 

l5o.2 

89 

218. 1 

1896 

5u 
5i 

37.7 
38.5 

32.8 
33.5 

10 

83.0 

72.2 

70 

128.3 

in  .5 

3o 

173.6 

i5o.9 

90 

218.9 

190.3 

II I 

83.8 

72.8 

171 

129.1 

112.2 

23l 

174.3 

i5i.5 

291 

219.6 

190.9 

52 

39.2 

34.1 

12 

84.5 

73.5 

72 

129.8 

112.8 

32 

175.1 

l52.2 

92 

220.4 

191.6 

53 

4o.o 

34.8 

i3 

85.3 

74.1 

73 

i3o.6 

ii3.5 

33 

175.8 

152.9 

93 

221.1 

192.2 

54 

40.8 

35.4 

i4 

86.0 

74.8 

74 

i3i.3 

n4-2 

34 

176.6 

i53.5 

94 

221.9 

192.9 

55 

4i.5 

36.1 

i5 

86.8 

75.4 

75 

l32.1 

114. 8 

35 

177.4 

i54.2 

95 

222.6 

193.5 

56 

42.3 

36.7 

i6;87.5.76.i 

76    i32.8 

n5.5 

36 

178.1 

i54.8 

96 

223.4 

194.2 

57 

43.0 

37.4 

17I88.3 

76.8 

77    i33.6 

116.1 

37 

178.9 

i55.5 

97 

224.1 

194.8 

58 

43.8 

38.1 

18 

89.1 

77-4 

78;i34.3 

116. 8 

38 

179.6 

i56.i 

98 

224.9 

195.5 

59 

44.5 

38.7 

19 

89.8 

78.1 

79 

i35.i 

II7-4 

39 

180.4 

i56.8 

99 

225.7 

'9^^ 

60 

45.3 

39.4 

20 

90.6 

78.7 

80 

i35.8 

118. 1 

4o 

181. 1 

157.5 

3oo 

226.4 

196.8 

Dcp. 

Lat. 

Dist 

Dcp. 

Lat. 

Dist 

Dop.  1    Lat. 

Dist 

Dep. 

Lat. 

Dist. 

Dep. 

Lat. 

1                                                                                                                              [For  49  Degrees. 

Page  58J 

TABLE  II. 

Difference  of  Latitude  and  Departure  for  42  Decrees.                       • 

1 

Dlst. 
I 

Lai. 

Dep. 

Disi. 

Lai.     Dep. 

Dist. 
121 

Lat. 

Dep. 

Dist. 

Lat. 

Dep. 

Dist.     Lat.  [ 

D>  p.  1 

00.7 

00.7 

61 

45.3    40.8 

89.9 

81 .0 

181 

i34.5 

121.1 

241 

179-1 

161  3 

2 

01.5 

01.3 

62 

46.1 

4i.5 

22 

90.7 

81.6 

82 

i35.3 

121.8 

42 

179-8 

161.0 

3 

02.2 

02.0 

63 

46.8 

42.2 

23 

91.4 

82.3 

83 

i36.o 

122.5 

43 

1806    162.6 

4 

o3.o 

02.7 

64 

47-6 

42.8 

24 

92.1 

83.0 

84 

i36.7 

123.1 

44 

181.3    i63.3 

5 

o3.7 

o3.3 

65 

48.3 

43.5 

25 

92.9 

83.6 

85 

i37.5 

123.8 

45 

182.1 

163.9 

6 

o4.5 

o4.o 

66 

49.0 

44.2 

26 

93.6 

84.3 

86 

i38.2 

124.5 

46 

182.8 

164.6 

7 

05.2 

04.7 

67 

49-8 

44.8 

27 

94.4 

85.0 

87 

139.0 

12D.I 

47 

i83.6 

i65.3 

8 

05.9 

o5.4 

68 

5o.5 

45.5 

28 

95.1 

85.6 

88 

139.7 

125.8 

48 

184.3 

165.9 

9 

06.7 

06.0 

69 

5i.3 

46.2 

29 

95.9 

86.3 

89 

i4o.5 

126.5 

49 

i85.o 

166.6 

lO 

II 

07.4 
08.2 

06.7 
07.4 

70 

52.0 

46.8 

3o 

96.6 

87.0 

90 

l4l.2 

127. 1 

5o 

i85.8 

167.3 

71 

52.8 

47.5 

i3i 

97-4 

87.7 

191 

i4i.9 

127.8 

25l 

186.5 

168.0 

12 

08.9 

08.0 

72 

53.5 

48.2 

32 

98.1 

88.3 

92 

142.7 

128.5 

52 

187.3 

168.6 

i3 

09.7 

08.7 

73 

54.2 

48.8 

33 

98.8 

89.0 

93 

143.4 

129.1 

53 

188.0    169.3 

i4 

10.4 

09.4 

74 

55.0 

49-5 

34 

99.6 

89.7 

94 

i44.2 

129.8 

54 

188.8    170.0 

lb 

II  .1 

10. 0 

75 

55.7 

5o.2 

35 

100.3 

90.3 

95 

144.9 

i3o.5 

55 

1S9.5 

170.6 

i6 

11.9 

10.7 

76 

56.5 

50.9 

36 

lOI.I 

91 .0 

96 

145.7 

i3i.i 

56 

190.2 

171.3 

17 

12.6 

11.4 

77 

57.2 

5i.5 

37 

101.8 

91.7 

97 

146.4 

i3i.8 

57 

191.0 

172.0 

i8 

i3.4 

12.0 

78 

58.0 

52.2 

38 

102.6 

92.3 

98 

i47-J 

i32.5 

58 

191.7 

172.6 

19 

14. 1 

12.7 

79 

58.7 

52.9 

39 

103.3 

93.0 

99 

147-9 

i33.2 

59 

192.5 

173.3 

20 

14.9 

i3.4 

80 

59.5 

53.5 

40 

104.0 

93.7 

200 

148.6 

1 33.8 

60 

193.2 

174.0 

21 

i5.6 

14. 1 

81 

60.2 

54.2 

i4i 

104.8 

94.3 

201 

149-4 

i34.5 

261 

194.0 

174.6 

22 

lb. 3 

14.7 

82 

60.9 

54.9 

42 

io5.5 

95.0 

02 

i5o.i 

i35.2 

62 

194.7 

175.3 

23 

17. 1 

i5.4 

83 

61.7 

55.5 

43 

106.3 

95.7 

o3 

1 50.9 

i35.8 

63 

195.4 

176.0 

24 

17.8 

16. 1 

84 

62.4 

56.2 

44 

107.0 

96.4 

04 

i5i.6 

i36.5 

64 

196.2 

176.7 

25 

18. b 

16.7 

85 

63.2 

56.9 

45 

107.8 

97.0 

o5 

i52.3 

137.2 

65 

196.9 

177-3 

26 

19.3 

17.4 

86 

63.9 

57.5 

46 

108.5 

97-7 

06 

i53.i 

i37.8 

66 

197-7 

178.0 

27 

20.1 

18.1 

87 

64.7 

58.2 

47 

109.2 

98.4 

07 

i53.8 

i38.5 

67 

198-4 

178.7 

28 

20.8 

18.7 

88 

65.4 

58.9 

48 

IIO.O 

99.0 

oS 

1 54 .6 

139.2 

68 

199.2 

179.3 

29 

21 .6 

19.4 

89 

66.1 

59.6 

49 

no. 7 

99-7 

09 

i55.3 

139.8 

69 

199.9 

180.0 

3o 

22.3 

20.1 

90 
91 

66.9 

67.6 

60.2 
60.9 

5o 

III. 5 

100.4 

ID 

i56.i 

i4o.5 

70 

200.6 

180.7 

3i 

23. 0 

20.7 

i5i 

112.2 

101 .0 

211 

i56.8 

l4l-2 

271 

201.4 

181.3 

32 

23.8 

21.4 

92 

68.4 

61.6 

52 

ii3.o 

101 .7 

12 

157.5 

i4i-9 

72 

202.1 

182.0 

33 

24.5 

22. 1 

93 

69.1 

62.2 

53 

1 13.7 

102.4 

■  l3 

i58.3 

142.5 

73 

202.9 

182.7 

34 

25.3 

22.8 

94 

69.9 

62.9 

54 

114.4 

io3.o 

i4 

159.0 

i43.2 

74 

2o3.6 

i83.3 

35 

26.0 

23.4 

95 

70.6 

63.6 

55 

ll5.2 

io3.7 

i5 

159.8 

143.9 

75 

204.4 

184.0 

36 

26.8 

24.1 

96 

71.3 

64.2 

56 

1 15.9 

104.4 

16 

160.5 

i44.5 

76 

205.1 

184.7 

37 

27.5 

24.8 

97 

72.1 

64.9 

57 

116.7 

io5.i 

17 

161.3 

i45.2 

77 

205.9 

i85.3 

38 

28.2 

25.4 

98 

72.8 

65.6 

58 

II7-4 

io5.7 

18 

162.0 

145.9 

■    78 

206.6 

186.0 

39 

29.0 

26.1 

99 

73.6 

66.2 

59 

118.2 

106.4 

19 

162.7 

146.5 

79 

207.3 

186.7 

40 

29.7 

26.8 

100 

74.3 

66.9 

60 

118.9 

107. 1 

20 

i63.5 

147-2 

80 

208.1 

187.4 

4i 

3o.5 

27.4 

101 

75.1 

67.6 

161 

119.6 

107.7 

221 

164.2 

147-9 

281 

208.8 

188.0 

42 

3l.2 

28.1 

02 

75.8 

68.3 

62 

120.4 

10S.4 

22 

i65.o 

148.5 

82 

209.6 

188.7 

43 

32. 0 

28.8 

o3 

76.5 

68.9 

63 

121.1 

109. 1 

23 

165.7 

149.2 

83 

210.3 

189.4 

U 

32.7 

29.4 

o4 

77.3 

69.6 

64 

121.9 

109.7 

24 

166.5 

149.9 

84 

211. 1 

190.0 

45 

33.4 

3o.i 

o5 

78.0 

70.3 

65 

122.6 

lie. 4 

25 

167.2 

i5o.6 

85 

211.8 

190.7 

46 

34.2 

3o.8 

06 

78.8 

70.9 

66 

123.4 

III  .1 

26 

168.0 

l5l.2 

86 

212.5 

191.4 

47 

34.9 

3i.4 

07 

79.5 

71.6 

67 

124. 1 

III. 7 

27 

168.7 

1 5 1.9 

87 

2i3.3 

192.0 

48 

35.7 

32.1 

oS 

80.3 

72.3 

68 

124.8 

112.4 

28 

169.4 

i52.6 

88 

214.0 

193.7 

49 

36.4 

32.8 

09 

81.0 

72.9 

69 

125.6 

1  i3.i 

29 

170.2 

i53.2 

89 

214.8 

193.4 

60 
5i 

37.2 

33.5 

10 

81.7 

73.6 

70 

126.3 

ii3.8 

3o 

23l 

1709 
17)  7 

153.9 
1 54.6 

90 

2i5.5 

194.0 

37-9l34.i 

III 

82.5 

74.3 

171 

127. 1 

114. 4 

291 

216.3 

194-7 

52 

38.6 

34.8 

12 

83.2 

74.9 

72 

127.8 

ii5.i 

32 

172.4 

i55.2 

92 

217.0 

195.4 

53 

39.4 

35.5 

i3 

84. 0 

75.6 

73 

128.6 

ii5.8 

33 

173.2 

155.9 

93 

217.7 

196.1 

54 

4o.i 

36.1 

i4 

84.7 

76.3 

74 

129.3 

116.4 

34 

173.9 

i56.6 

94 

218.5 

.96.7 

\      Kr, 

40.9 

36.8 

i5 

85.5 

77.0 

75 

i3o.i 

117. 1 

35 

174.6 

157.2 

95 

219.2 

.97-4 

56 

4i.6 

37.5 

16 

86.2 

77.6 

76 

i3o.8 

117.8 

36 

175-4 

157-9 

96 

220.0 

198.1 

5? 

42.4 

38.1 

17 

86.9 

78.3 

77 

i3i.5 

118.4 

37 

176.1 

1 58.6 

97 

220.7 

198.7 

58 

43.1 

38.8 

18 

87.7 

79.0 

78 

i32.3 

119. 1 

38 

176.9 

159.3 

98 

221.5 

199.4 

59 

43.8 

39.5 

19 

88.4 

79-6 

79 

i33.o 

119.8 

39 

177.6 

159.9 

99 

222.2 

200.1 

bo 

44.6 

4o.  I 

20 

89.2 

80.3 

80 

i33.8 

120.4 

4o 

178.41  160.6 

3uo 

222.9 

200.7 

Dist. 

Dcp. 

I.al. 

Disl.|  Dep. 

Lat. 

Dist.l    Dep.  !    Lat. 

Dist. 

Dep. 

1   Lat. 

Dist 

Dep. 

Lat. 

[ 

For  48  Degrees. 

TABLE     II.                                                                        [P-Se59 

Difference  of  Latitude  and  Departure  for  43  Degrees. 

Dist. 

Lat. 

Dep. 

Dist. 

Lai. 

Dep. 

Dist. 

Lat. 

Dep. 

Dist. 

Lat. 

Dep. 

Dist 

241 
42 
43 

44 
45 
46 
47 
48 

5o 

25l 
52 

53 

54 
55 
56 

57 
58 

60 

Lat. 

176.3 

177.0 

1777 
178.5 
179.2 
179.9 
180.6 
181.4 
182.1 
182.8 
1 83 .6 
184.3 
i85.o 
i85.8 
186.5 
187.2 
188.0 
188.7 
189.4 
190.2 

Dep. 

164.4 
i65.o 
165.7 
166.4 
167.1 
167.8 
168.5 
169.1 
169.8 
170.5 
171.2 

171.9 
172.5 
173.2 
173.9 
174.6 
175.3 
176.0 
176.6 
177-3 
178.0 
178.7 
1-9.4 
180.0 
180.7 
181.4 
182.1 
182.8 
i83.5 
1 84.1 
184.8 
i85.5 
186.2 
186.9 
187.5 
188.2 
188.9 
189.6 
190.3 
191.0 

I 

2 

3 
4 
5 
6 

7 
8 

9 

10 

00.7 
01 .5 
02.2 
02.9 
o3.7 
o4.4 
o5.i 
05.9 
06.6 
07.3 

00.7 
01 .4 
02.0 
02.7 
o3.4 
o4. 1 
04.8 
o5.5 
06.1 
06.8 

61 
62 
63 
64 
65 
66 

67 
68 
69 
70 

71 

72 

73 

74 
75 
76 

77 
78 

Z9 
80 

44.6 
45.3 
46.1 
46.8 
47-5 
48.3 
49.0 
49-7 
5o.5 

5l.2 

5i  .9 

52.7 
53.4 
54.1 
54.9 
55.6 
56.3 
67.0 
57.8 
58.5 

4i.6 
42.3 
43.0 
43.6 
44.3 
45.0 
45.7 
46.4 
47.1 
47.7 
48.4 
49.1 
49.8 
5o.5 
5i.i 
5i.8 
52.5 
53.2 
53.9 
54.6 
55.2 
55.9 
56.6 
57.3 
58.0 
58.7 
59.3 
60.0 
60.7 
61.4 

121 
22 

2  3 

24 

25 

26 

27 
28 

=9 
3o 

88.5 
89.2 
90.0 
90.7 
91.4 
92.2 
92.9 
93.6 
94.3 
95.1 

82.5 

83.2 
83.9 
84.6 
85.2 
85.9 
86.6 
87.3 
88.0 
88.7 

181 
82 
83 
84 
85 
86 

87 
88 
89 
90 

i32.4 
i33.i 
i33.8 
1 34.6 
i35.3 
i36.o 
i36.8 
137.5 
i38.2 
139.0 

123.4 
124.1 
124.8 
125.5 
126.2 
126.9 
127.5 
128.2 
128.9 
129.6 

12 

i3 
i4 
i5 
i6 
17 
i8 

19 

20 

08.0 
08.8 
09.5 
10.2 
11 .0 
11.7 
12.4 

l3.2 

.J. 9 

i4.6 

07.5 
08.2 
08.9 
09.5 
10.2 
10.9 
II. 6 

12.3 

i3.o 
i3.6 

i3i 

32 

33 
34 
35 
36 

37 
38 
39 
4o 

95.8 

96.5 

97.3 

98.0 

98.7 

99.5 

100.2 

100.9 

101 .7 

102.4 

89.3 
90.0 
90.7 
91.4 
92. 1 
92.8 
93.4 
94.1 
94.8 
95.5 

191 

9^ 
93 

94 

95 
96 

97 
98 

99 

200 

.39.7 
140.4 

l4l.2 

I4..9 

142.6 
143.3 
144.1 
144.8 

145.5 
i46.3 

i3o.3 
i3o.9 
i3i.6 
i32.3 
i33.o 
i33.7 
i34.4 
i35.o 
]35.7 
i36.4 

21 
22 

23 

24 

25 

26 

27 

23 

=9 

3o 

i5.4 
16.1 
16.8 
17.6 
18.3 
19.0 
19.7 
20.5 
21 .2 
21 .9 

14.3 
i5.o 
i5.7 
16.4 
17.0 
17.7 
18.4 
19. 1 
19.8 
20.5 

81 
82 
83 
84 
85 
86 

87 
88 
89 
90 

59.2 
60.0 
60.7 
61.4 
62.2 
62.9 
63.6 
64.4 
65.1 
65.8 

i4i 
42 
43 
44 
45 
46 
47 
48 

49 
5o 

io3.i 
103.9 
104.6 
io5.3 
106.0 
106.8 
1 07 . 5 
108.2 
109.0 
109.7 

96.2 

96.8 

97.5 

98.2 

98.9 

99.6 

100.3 

100.9 

loi  .6 

102.3 

201 
02 
o3 
o4 
o5 
06 
07 
08 
09 
10 

147.0 
147-7 
148.5 
149.2 
149.9 
i5o.7 
i5i.4 

l52.I 

152.9 
1 53.6 

137.1 
137.8 
i38.4 
139. 1 
139.8 
140.5 

l4l.2 

141.9 
142.5 
143.2 

261 
62 
63 
64 
65 
66 

67 
68 
69 

70 

190,9 
1 9 1 .() 
192.3 
.93.1 
193.8 
194.5 
195.3 
196.0 
1 96.7 
197.5 

3i 

32 

33 
34 
35 
36 
37 
38 
39 
4o 

22 .7 
23.4 
24.1 
24.9 
25.6 
26.3 
27.1 
27.8 
28.5 
29.3 

21. 1 

21.8 
22.5 

23.2 
23.9 
24.6 
25.2 
25.9 
26.6 
27.3 

9> 

93 
94 

96 

97 

98 

99 
100 

66.6 
67.3 
68.0 
68.7 
69.5 
70.2 
70.9 

71-7 
72.4 
73.1 

62.1 
62.7 
63.4 
64.1 
64.8 
65.5 
66.2 
66.8 
67.5 
68.2 

i5i 

52 

53 
54 
55 
56 
57 
58 

60 

110.4 
II 1 .2 
III  .9 
112. 6 
ii3.4 
ii4-i 
114. 8 
ii5.6 
116.3 
117.0 

io3 .0 
io3.7 
104.3 
io5.o 
105.7 
106.4 
107. 1 
107.8 
108.4 
1 09 . 1 

211 
12 
i3 
i4 
i5 
16 

17 
18 

19 

20 

i54.3 
i55.o 
i55.8 
i56.5 
157.2 
i58.o 
158.7 
159.4 
160.2 
160.9 

143.9 
144.6 
i45.3 
145.9 
i46.6 
147.3 
i48.o 
i48.7 
149.4 
i5o.o 

271 
72 
73 
74 
75 
76 
77 
78 

Z9 
80 

198.2 
198.9 
199.7 
200.4 
201.1 
201.9 
202. () 
2o3.3 
204.0 
204.8 

4i 
42 
43 
44 
45 
46 
47 
48 
49 
5o 

5i 

52 

53 

54 
55 
56 

57 
58 
59 
60 

3o  .0 
00.7 
3i.4 

32.2 

32.9 
33.6 
34.4 
35.1 

35.8 
36.6 

37.3 
38.0 
38.8 
39.5 
4o.2 
4 1 .0 
41.7 
42.4 
43.1 
43.9 

28.0 

28. 6 
29.3 
3o.o 
3o.7 
3i.4 

32.1 

32.7 
33.4 
34.1 
34.8 
35.5 
36.1 
36.8 
37.5 
38.2 
38-9 
39.6 
4o.2 
40.9 

lOI 

02 
o3 
04 
o5 
06 

07 
08 
09 

ID 

73.9 
74.6 
75.3 
76.1 
76.8 
77.5 
78.3 
79.0 
79-7 
80.4 

68.9 
69.6 
70.2 
70.9 
71.6 
72.3 
73.0 
73.7 
74.3 
75.0 

161 
62 
63 
64 
65 
66 

67 
68 
69 

70 

117.7 
118.5 
119. 2 
119. 9 
120.7 

121  .4 
122.1 
122.9 
123.6 
124.3 

109.8 
110.5 
III  .2 
111.8 
112. 5 

Il3.2 

113.9 
114.6 
ii5.3 
115.9 

221 
22 

23 

24 

25 

26 

27 
28 

If. 

161.6 
162.4 
i63.i 
1 63.8 
164.6 
i65.3 
166.0 
166.7 
167.5 
168.2 

1 50.7 
i5i.4 

l52.1 

i52.8 
i53.4 
i54.i 
i54.8 
i55.5 
1 56.2 
i56.9 

281 
82 

83 
84 
85 
86 
87 
88 
89 
90 

2()5.5 
206.2 
207.0 
207.7 
208.4 
209.2 
209.9 
210.6 
211.4 
212.1 

191.6 
192.3 
193.0 
193.7 
194.4 
195. 1 

196.4 
197.1 
197.8 

U  I 
12 

i3 
i4 
i5 
16 

17 
18 

19 

20 

81.2 
81.9 
82.6 
83.4 
84.1 
84.8 
85.6 
S6.3 
87.0 
87.8 

75.7 
76.4 
77-1 
77-7 
78.4 
79.1 
79.8 
80.5 
81.2 
81.8 

171 

72 
73 
74 
75 
76 
77 
78 

Z9 
80 

125. I 

125.8 
126.5 
127.3 
128.0 
128.7 
129.4 

l3o.2 

1 30.9 
i3i.6 

116.6 
117. 3 
118.0 
118.7 
119. 3 
120.0 
120.7 
121.4 
122. 1 
122.8 

23  1 
32 

33 

34 
35 
36 

37 
38 

39 
40 

168.9 
169.7 
170.4 
171. 1 
171.9 
172.6 
173.3 
1 74. 1 
174.8 
175.5 

157.5 
i58.2 
i58.9 
159.6 
160.3 
161.0 
161.6 
162.3 
1 63.0 
i63.7 

291 

92 
93 

95 
96 

9^ 
98 

,99 

3oo 

212.8 
2i3.6 
214.3 
2 1 5.0 
2.5.7 
216.5 
217.2 
217.9 
218.7 
219.4 

198.5 
199.1 
199.8 
200.5 
201.2 
201.9 
202.6 

203.2 
203.9 

2o4  6 
Lni. 

Dist. 

Dq,. 

T>at. 

Dist. 

Dep. 

I.at. 

Dist. 

Dep. 

Lat. 

Dist. 

Dep. 

Lat. 

Dist. 

Dep. 

[For  47  Degrees.     1 

Pago  60] 

TABLE   IL 

Difference  of  Latitude  and 

Departure  for  44  Degree^;. 

Dist. 

I.at. 

Dep. 

Dist. 

Lat. 

Dep. 

Dist. 

Lat. 

Do  p. 

Dist.     Lat. 

Dep. 

Dist. 

Lat. 

Dep. 

I 

00.7 

00.7 

61 

43.9 

42.4 

121 

87.0 

84.1 

181 

i3o.2 

125.7 

241 

173-4 

'"   4 

2 

01 .4 

CI  .4 

62 

44.6 

43.1 

22 

87.8 

84.7 

82 

i3o.9 

126.4 

42 

174.1 

j>^.i 

3 

02.2 

02.1 

63 

45.3 

43.8 

23 

88.5 

85.4 

83 

i3i.6 

127. 1 

43 

174-8 

168.8 

4 

02.9 

02.8 

64 

46. 0 

44.5 

24 

89.2 

86.1 

84 

i32.4 

127.8 

A^ 

175.5 

169.5 

5 

o3.b 

o3.5 

65 

46.8 

45.2 

25 

89.9 

86.8 

85 

i33.i 

128.5 

45 

176.2 

170.2 

6 

o4.3 

04.2 

66 

47.5 

45.8 

26 

90.6 

87.5 

86 

i33.8 

129.2 

46 

177.0 

170.9 

7 

o5.o 

04.9 

67 

48.2 

46.5 

27 

91.4 

88.2 

87 

i34.5 

129.9 

47 

177-7 

1 7 1. 6 

8 

ob.8 

o5.6 

68 

48.9 

47.2 

28 

92. 1 

88.9 

88 

i35.2 

i3o.6 

48 

178.4 

172.3 

9 

06.5 

06.3 

69 

49-(3 

47-9 

29 

92.8 

89.6 

89 

i36.o 

i3i.3 

49 

179.1 

173.0 

10 

07.2 

06.9 

70 

5o.4 

48.6 

3o 

93.5 

90.3 

90 

i36.7 

l32.0 

5o 

179-8 

173.7 

II 

07.9 

07.6 

71 

5i.i 

49-3 

i3i 

94.2 

91 .0 

191 

137.4 

i32.7 

25l 

180.6 

174.4 

12 

08.6 

08.3 

72 

bi,8 

5o.o 

32 

95.0 

91.7 

92 

i38.i 

i33.4 

52 

181.3 

175  1 

i3 

09.4 

09.0 

73 

52.5 

50.7 

33 

95.7 

92.4 

93 

i38.8 

i34.i 

53 

182.0 

175.7 

i4 

10. 1 

09.7 

74 

53.2 

5i.4 

34 

96.4 

93.1 

94 

139.6 

i34.8 

54 

182.7 

176.4 

lb 

10.8 

10.4 

75 

54.0 

52.1 

35 

97.1 

93.8 

95 

i4o.3 

i35.5 

5b 

i83.4 

177.1 

i6 

II. 5 

II  .1 

76 

54.7 

52.8 

.36 

97.8 

94.5 

96 

i4i.o 

i36.2 

56 

184.2 

177.8 

17 

12.2 

II. 8 

77 

55.4 

53.5 

37 

98.5 

95.2 

97 

141.7 

1.36.8 

57 

184.9 

178.5 

18 

12.9 

12.5 

78 

56.1 

54.2 

38 

99.3 

95.9 

98 

142.4 

137.5 

58 

i85.6 

179.2 

19 

i3.7 

l3.2 

79 

56.8 

54.9 

39 

100. 0 

96.6 

99 

143.1 

i38.2 

59 

186.3 

179.9 

20 

14.4 

.3.9 

80 

57.5 

55.6 

40 

100.7 

97.3 

200 

143.9 

i38.9 

60 

187.0 

180.6 

21 

i5.i 

i4.6 

81 

58.3 

56.3 

i4i 

loi  .4 

97-9 

201 

144.6 

139.6- 

261 

187.7 

181.3 

22 

i5.8 

i5.3 

82 

59.0 

57.0 

42 

102. 1 

98.6 

02 

145.3 

i4o.3.^ 

62 

188.5 

182.0 

23 

16.5 

16.0 

83 

59.7 

57.7 

43 

102.9 

99.3 

o3 

146.0 

i4i-o 

63 

189.2 

182.7 

24 

17.3 

16.7 

a4 

60.4 

58.4 

M 

io3.6 

1 00.0 

04 

146.7 

i4i.7 

64 

189.9 

i83.4 

2b 

18.0 

17.4 

85 

61. 1 

59.0 

45 

104.3 

100.7 

o5 

i47-5 

142.4 

65 

190.6 

184.1 

26 

18.7 

18. 1 

86 

61 .9 

59.7 

46 

io5.o 

101.4 

06 

i48.2 

143.1 

66 

191. 3 

184.8 

27 

19.4 

18.8 

87 

62.6 

60.4 

47 

io5.7 

102.1 

07 

148.9 

i43.8 

67 

192. 1 

i85.5 

28 

20.1 

19.5 

88 

63.3 

61. 1 

48 

106.5 

102.8 

08 

i49-6 

144.5 

68 

192.8 

1S6.2 

29 

20.9 

20. 1 

89 

64.0 

61.8 

49 

107.2 

io3.5 

09 

i5o.3 

i45-2 

69 

193.5 

186.9 

Jo 

21 .6 

20  8 

90 
91 

64.7 
65.5 

62.5 
63.2 

5o 

107.9 

104.2 

10 

i5r.i 

145.9 

70 

194.2 

1S7.6 

3i 

22.3 

21.5 

i5i 

108.6 

104.9 

21 1 

i5i.8 

146.6 

271 

194.9 

188.3 

32 

23.0 

22.2 

92 

66.2 

63.9 

52 

109.3 

io5.6 

12 

i52.5 

147.3 

72 

195.7 

188.9 

33 

23.7 

22.9 

93 

66.9 

64.6 

53 

no. I 

106.3 

i3 

i53.2 

i48.o 

73 

196.4 

189.6 

M 

24.5 

23.6 

94 

67.6 

65.3 

54 

no. 8 

107.0 

i4 

153.9 

148.7 

74 

197-1 

190.3 

3b 

25.2 

24.3 

95 

68.3 

66.0 

55 

III  .5 

107.7 

i5 

154.7 

149.4 

7^ 

197.8 

191.0 

36 

2b. 9 

25.0 

96 

69.1 

66.7 

56 

112. 2 

108.4 

16 

i55.4 

i5o.o 

76 

198.5 

191.7 

^7 

26.6 

25.7 

97 

69.8 

67.4 

57 

112.9 

109.1 

17    i56.i 

1 50.7 

77 

199.3 

192.4 

38 

27.3 

26.4 

98 

70.5 

68.1 

58 

ii3.7 

109.8 

18 

i56.8 

i5i.4 

78 

200.0 

193.1 

39 

28.1 

27.1 

99 

71.2 

68.8 

59 

114.4 

110.5 

19 

157.5 

l52.1 

79 

200.7 

193.8 

4o 

28.8 

27.8 

100 

71.9 

69.5 

60 

ii5.i 

III  .1 

20 

1 58.3 

i52.8 
i53.5 

80 

201.4 

194.5 

4i 

29.5 

28.5 

lOI 

72.7 

70.2 

161 

ii5.8 

HI  .8 

221 

159.0 

281 

202.1 

195.2 

42 

3o.2 

29,2 

02 

73.4 

70.9 

62 

116.5 

112. 5 

22 

159.7 

i54.2 

82 

202.9 

195.9 

43 

3o.9 

29.9 

o3 

74.1 

71.5 

63 

117. 3 

I  l3.2 

23 

160.4 

154.9 

83 

2o3.6 

196.6 

M 

31.7 

36.6 

04 

74.8 

72.2 

64 

118. 0 

113.9 

24 

161. 1 

i55.6 

84 

2o4-3 

197.3 

^^ 

32.4 

3i.3 

o5 

75.5 

72.9 

65 

118.7 

114.6 

25 

161.9 

i56.3 

85 

2o5.o 

198.0 

46 

33.1 

32. 0 

06 

76.3 

73.6 

66 

119.4 

ii5.3 

26 

162.6 

157.0 

86 

2o5.7 

198.7 

47 

33.8 

32.6 

07 

77.0 

74.3 

67 

120.1 

1 16.0 

27 

i63.3 

157.7 

87 

206.5 

199-4 

48 

34.5 

33.3 

08 

77-7 

75.0 

68 

120.8 

116.7 

28 

164.0 

i58.4 

88 

207.2 

200.1 

49 

35.2 

34.0 

09 

78.4 

75.7 

69 

121 .6 

117.4 

29 

164.7 

159.1 

89 

207.9 

200.8 

bo 
5i 

3b. 0 
36.7 

34.7 
35.4 

10 
1 1 1 

79.1 

76.4 

70 

122.3 

118. 1 

3o 

i65.4 

159.8 

90 

208.6 

201.5 

79.8 

77-1 

171 

123.0 

118.8 

23l 

166.2 

160.5 

291 

209.3 

202.1 

b2 

37.4 

36.1 

12 

80.6 

77.8 

72 

123.7 

1 19.5 

32 

166.9 

161.2 

92 

210.0 

202.8 

53 

38.1 

36.8 

i3 

81.3 

78.5 

73 

124.4 

120.2 

33 

167.6 

161.9 

93 

210.8 

2o3.5 

54 

38.8 

37.5 

i4 

82.0 

79.2 

74 

125.2 

120.9 

M 

168.3 

162.6 

94 

2»JI.5 

2o4.2 

CK 

39.6 

38.2 

i5 

02.7 

79-9 

75 

125.9 

121 .6 

35 

169.0 

i63.2 

95 

212.2 

204.9 

56 

4c. 3 

38.9 

16 

83.4 

80.6 

76 

126.6 

122.3 

36 

169.8 

163.9 

96 

212.9 

2o5.6 

57 

4i.o 

39.6 

17 

84.2 

81.3 

77 

127.3 

123. 0 

37 

170.5 

164.6 

97 

2l3.6 

206.3 

58 

41-7 

40.3 

18 

84.9 

82.0 

78 

128.0 

123.6 

38 

171.2 

165.3 

98 

214.4 

207.C 

59 

42.4 

4i  .0 

19 

85.6 

82.7 

79 

128.8 

124.3 

39 

171-9 

166.0 

99 

2l5.I 

207.7 

60 

43.2 

41.7 

20 

86.3 

83.4 

80 

129.5 

125.0 

40 

172.6 

166.7 

3  00 

2i5.8 

208.4 

Dist. 

Dep. 

I. at. 

Dist. 

Dep. 

Lat. 

Dist. 

Dep. 

Lat. 

Dist. 

Dep. 

Lat. 

Dist. 

Dep. 

Lat. 

[ 

''or  46  Degrees. 

TABLE  II. 

[Page  61 

Difference  of  Latitud 

e  and 

Departure  for  45  Degrees. 

Dist. 

Lat. 

Dep. 

Dist. 

Lat. 

Dep. 

Dist. 

Lat. 

Dep. 

Dist. 

Lat. 

Dep. 

Dist. 

Lat. 

Dep. 

I 

00.7 

00.7 

61 

43.1 

43.1 

121 

85.6 

85.6 

181 

128.0 

128.0 

241 

170.4 

170.4 

2 

01 .4 

01 .4 

62 

43.8 

43.8 

22 

86.3 

86.3 

82 

128.7 

128.7 

42 

171.1 

171. 1 

3 

02.1 

02.1 

63 

44.5 

44.5 

23 

87.0 

87.0 

83 

129.4 

129.4 

43 

171. 8 

171.8 

4 

02.8 

02.8 

64 

45.3 

45.3 

24 

87.7 

87.7 

84 

i3o.i 

i3o.i 

44 

172.5 

1-2.5 

5 

o3.5 

o3.5 

65 

46.0 

46.0 

25 

88.4 

88.4 

85 

i3o.8 

i3o.8 

45 

173.2 

173.2 

6 

o4.2 

04.2 

66 

46.7 

46.7 

26 

89.1 

89.1 

86 

i3i.5 

i3i.5 

46 

173.9 

173.9 

04.9 

04.9 

67 

47-4 

47-4 

27 

89.8 

89.8 

87 

l32.2 

l32.2 

47 

174.7 

174.7 

8 

0D.7 

o5.7 

68 

48.1 

48.1 

28 

90.5 

90.5 

88 

132.9 

132.9 

48 

175.4 

175-4 

9 

06.4 

06.4 

69 

48.8 

48.8 

29 

91 .2 

91.2 

89 

i33.6 

i33.6 

49 

170.1 

176.1 

10 

07.1 

07.1 

70 

49.5 

49-t5 

3o 

91.9 

91.9 

90 

1 34. 4 

134.4 

5o,  176.8 

176.8 

11 

07.8 

07.8 

71 

5o.2 

5o.2 

i3i 

92.6 

92.6 

191 

i35.i 

I35.I 

25l 

177-5 

177.5 

12 

08.5 

08.5 

72 

5o.9 

50.9 

32 

93.3 

93.3 

92 

i35.8 

i35.8 

52 

178.2  1  178.2 

i3 

09.2 

09.2 

73 

5i.6 

5i.6 

33 

94.0 

94.0 

93 

i36.5 

i36.5 

53 

178.9    178.9 

i4 

09.9 

09.9 

74 

52.3 

52.3 

M 

94.8 

94.8 

94 

137.2 

137.2 

54 

179.6 

179.6 

i5 

10.6 

10. 0 

75 

53.0 

53.0 

35 

95.5 

95.5 

95 

137.9 

137.9 

55 

180.3 

180.3 

i6 

11.3 

II. 3 

76 

53.7 

53.7 

36 

96.2 

96.2 

96 

i38.6 

i38.6 

56 

181.0 

181.0 

J7 

12.0 

12.0 

77 

54.4 

54.4 

37 

96.9 

96.9 

97 

139.3 

139.3 

57 

181.7 

181.7 

i8 

12.7 

12.7 

78 

55.2 

55.2 

38 

97.6 

97.6 

98 

i4o.o 

i4o.o 

58 

182.4 

182.4 

19 

i3.4 

i3.4 

79 

55.9 

55.9 

39 

98.3 

98.3 

99 

140.7 

140.7 

59 

i83.i 

i83.i 

20 

14.1 

i4.i 

80 

56.6 

56.6 

40 

99.0 

99.0 

200 

i4i.4 

141.4 

60 

i83.8 

i83.8 
184.6 

21 

i4.8 

i4.8 

81 

57.3 

57.3 

i4i 

99-7 

99-7 

201 

142.1 

142.1 

261 

184.6 

22 

i5.6 

i5.6 

82 

58.0 

58.0 

42 

100.4 

100.4 

02 

142.8 

142.8 

62 

i85.3 

i85.3 

23 

16.3 

16.3 

83 

58.7 

58.7 

43 

101 .1 

101. 1 

o3 

143.5 

143.5 

63 

186.0 

186.0 

24 

17.0 

17.0 

84 

59.4 

59.4 

44 

101.8 

101.8 

04 

144.2 

144.2 

64 

186.7 

186.7 

25 

17-7 

17.7 

85 

60.1 

60.1 

45 

102.5 

102.5 

o5 

145.0 

145.0 

65 

187.4 

187.4 

26 

18.4 

18.4 

86 

60.8 

60.8 

46 

io3.2 

io3.2 

06 

145.7 

145.7 

66 

188. 1 

188. 1 

27 

19. 1 

19. 1 

87 

61.5 

61.5 

47 

103.9 

io3,9 

07 

146.4 

146.4 

67 

188.8 

188.8 

28 

19.8 

19.8 

88 

62.2 

62.2 

48 

104.7 

104.7 

08 

i47-i 

147-1 

68 

189.5 

189.5 

29 

20.5 

20.5 

89 

62.9 

62.9 

49 

io5.4 

io5.4 

09 

147-8 

147.8 

69 

1 90 . 2 

190.2 

3o 

21.2 

21 .2 

90 
91 

63.6 
64.3 

bi.b 
64.3 

5o 

1 06 . 1 

106.1 

10 

148.5 

148.5 

70 
271 

190.9 
191 .6 

190.9 
1 9 1. 6 

3i 

21 .9 

21 .9 

i5i 

106.8 

106.8 

211 

149.2 

149.2 

32 

22.6 

22.6 

92 

65.1 

65.1 

52 

107.5 

107.5 

12 

149.9 

149-9 

72 

192.3 

192.3 

33 

23.3 

23.3 

93 

65.8 

65.8 

53 

108.2 

108.2 

i3 

i5o.6 

i5o.6 

73 

1 93 . 0 

193.0 

34 

24.0 

24.0 

94 

66.5 

66.5 

54 

108.9 

108.9 

i4 

i5i.3 

i5i.3 

74 

193.7 

193.7 

35 

24.7 

24.7 

95 

67.2 

67.2 

55 

109.6 

109.6 

i5 

l52.0 

l52.0 

7^ 

194.5 

194.5 

36 

25.5 

25.5 

96 

67.9 

67.9 

56 

110.3 

110.3 

16 

i52.7 

152.7 

1^ 

195.2 

195.2 

37 

26.2 

26.2 

97 

68.6 

68.6 

57 

III  .0 

III.O 

17 

i53.4 

i53.4 

77 

195.9 

19D.9 

38 

26.9 

26.9 

98 

69.3 

69.3 

58 

III. 7 

111.7 

18 

i54.i 

1 54.1 

78 

196.6 

196.6 

39 

27.6 

27.6 

99 

70.0 

70.0 

59 

112.4 

1 12.4 

19 

154.9 

154.9 

79 

197.3 

197.3 

4o 

28.3 

28.3 

100 

70.7 

70.7 

60 

ii3.i 

ii3.i 

20 
221 

i55.6 

i55.6 

80 

198.0 

198.0 

4i 

29.0 

29.0 

lOI 

71.4 

71-4 

161 

ii3.8 

ii3.8 

i56.3 

i56.3 

281 

198.7 

198.7 

42 

29.7 

29.7 

02 

72.1 

72.1 

62 

114.6 

114.6 

22 

157.0 

157.0 

82 

199-4 

199-4 

43 

3o.4 

So.  4 

o3 

72.8 

72.8 

63 

ii5.3 

ii5.3 

23 

157.7 

157.7 

83 

200. 1 

2C0.I 

44 

3i.i 

3i.i 

o4 

73.5 

73.5 

64 

116.0 

116.0 

24 

i58.4 

1 58.4 

84 

2(J0.8 

200.8 

45 

3i.8 

3i.8 

o5 

74.2 

74.2 

65 

116.7 

116.7 

25 

159.1 

1D9.1 

85 

201  .5 

201.5 

46 

32.5 

32.5 

06 

75.0 

75.0 

66 

117.4 

117.4 

26 

159.8 

159.8 

86 

202.2 

202.2 

47 

33.2 

33.2 

07 

75.7 

75.7 

67 

118.1 

118.1 

27 

160.5 

160.5 

87 

202.9 

202.9 

48 

33. 9 

33.9 

08 

76.4 

76.4 

68 

118. 8 

118.8 

28 

161 .2 

161.2 

88 

2o3.6 

203.6 

49 

34.6 

34.6 

09 

77.1 

77-1 

69 

119.5 

119.5 

29 

161 .9 

161.9 

89 

204.4 

204.4 

5o 

35.4 

35.4 

10 

77.8 

77.8 

70 

120.2 

120.2 

3o 

162.6 

162.6 
i63.3 

90 

205.1 

205.1 

5i 

36.1 

36.1 

III 

78.5 

78.5 

171 

120.9 

120.9 

23l 

i63.3 

291 

2o5.8 

2o5.S 

52 

36.8 

36.8 

12 

79.2 

7Q.2 

72 

121 .6 

121.6 

32 

164.0 

164.0 

92 

206.5 

206  5 

53 

37.5 

37.5 

i3 

79-9 

79-9 

73 

122.3 

122.3 

33 

164.8 

164.8 

93 

207.2 

207  2 

54 

38.2 

38.2 

i4 

80.6 

80.6 

74 

123.0 

I23.0 

34 

i65.5 

i65.5 

94 

207.9 

207.9 

55 

38. Q 

38.9 

i5 

81.3 

81.3 

75 

123.7 

123.7 

35 

166.2 

166.2 

9i 

208.6 

2&8.6 

56 

39.6 

39.6 

16 

82.0 

82.0 

76 

124.5 

124.5 

36 

166.9 

166.9 

96 

209.3 

209.3 

57 

4o.3 

40.3 

17 

82.7 

82.7 

77 

125.2 

125.2 

37 

167.6 

167.6 

97 

210.0 

210.0 

58 

4i  .0 

4i  .0 

18 

83.4 

83.4 

78 

125.9 

125.9 

38 

168.3 

168.3 

98,210.7 

210.7 

59 

41.7 

41.7 

19 

84.1 

84.1 

79 

126.6 

126.6 

39 

169.0 

169.0 

99 

211  .4 

211.4 

bo 

42.4 

42.4 

20 

84-9 

84-9 

80 

127.3 

127.3 

4o 

169.7 

169.7 

3oo 

212. 1 

2 1 2.1 

Dist. 

Dep. 

Lai. 

Dist. 

Dep. 

Lat. 

Dist. 

Dep. 

Lat. 

Dist. 

Dep. 

Lat. 

Dist. 

D..p. 

Lat. 

For  45  Degrees. 

Page  62] 

TABLE  III. 
Meridional  Parts. 

M. 

0° 

1° 

2' 

3° 

40 

5° 

6° 

7° 

8° 

9° 

10° 

11° 

12° 

13° 

M. 

0 

0 

60 

120 

180 

240 

3oo 

36  r 

421 

482 

542 

6o3 

664 

725 

787 

I 

I 

6i 

121 

i8i 

241 

3oi 

362 

422 

483 

543 

6o4 

665 

726 

788 

I 

2 

2 

62 

122 

182 

242 

302 

363 

423 

484 

544 

6o5 

666 

727 

789 

2 

3 

3 

63 

123 

i83 

243 

3o3 

364 

424 

485 

545 

606 

667 

728 

790 

3 

4 

5 

4 

64 

124 

184 

244 

3o4 
3o5 

365 

425 

486 

546 

607 

668 

729 

791 

4 
5 

5- 

65 

125 

i85 

245 

366 

426 

487 

547 

608 

669 

73o 

792 

fj 

6 

66 

126 

186 

246 

3o6 

367 

427 

488 

548 

609 

670 

73  ( 

793 

6 

7 

7 

67 

127 

187 

247 

3o7 

368 

428 

489 

549 

610 

671 

732 

794 

7 

S 

8 

68 

128 

188 

248 

3o8 

369 

429 

490 

55o 

611 

672 

734 

793 

8 

10 

9 

69 

129 

189 

249 
25o 

3o9 

370 

43o 

491 

55i 

612 

673 

735 

796 

_9 

ID 

10 

70 

i3o 

190 

3io 

371 

43 1 

4o2 

552 

6i3 

674 

736 

797 

II 

II 

71 

i3i 

191 

25l 

3ii 

372 

432 

493 

553 

6i4 

675 

737 

798 

:i 

'12 

12 

72 

l32 

192 

252 

3l2 

373 

4i^ 

494 

554 

6i5 

676 

738 

799 

12 

i3 

i3 

73 

i33 

193 

253 

3i3 

374 

4M 

495 

555 

6i6 

677 

739 

800 

i3 

i4 
i5 

i4 

74 

1 34 

194 

254 

3i4 

375 

435 

496 

556 

617 

678 

740 

801 

i4 
i5 

i5 

75 

i35 

195 

255 

3i5 

376 

436 

497 

557 

618 

679 

74 1 

802 

i6 

16 

76 

1 36 

196 

256 

3i6 

377 

437 

498 

558 

619 

680 

742 

8o3 

16 

17 

17 

77 

i37 

197 

257 

3i7 

378 

438 

499 

559 

620 

681 

743 

8o4 

17 

i8 

18 

78 

1 38 

198 

258 

3i8 

379 

439 

5oo 

56o 

621 

682 

744 

8o5 

18 

12 

20 

19 
20 

79 
80 

,39 

199 

259 

319 

38o 

44o 

5oi 

56 1 

622 

683 

745 

806 

11 
20 

i4o 

200 

260 

320 

38 1 

44 1 

502 

562 

623 

684 

746 

807 

21 

21 

81 

i4i 

201 

261 

321 

382 

442 

5o3 

564 

624 

685 

747 

808 

21 

22 

22 

82 

142 

202 

262 

322 

383 

443 

5o4 

565 

625 

687 

748 

809 

22 

23 

23 

83 

143 

203 

263 

323 

384 

444 

5o5 

566 

626 

688 

749 

810 

23 

24 
25 

24 

84 

1 44 

204 

264 

324 

385 

445 

5o6 
507 

567 

627 

6S9 

75o 

811 

24 

25 

25 

85 

i45 

205 

265 

325 

386 

446 

568 

628 

690 

75i 

812 

26 

26 

86 

1 46 

206 

266 

326 

387 

44i 

5o8 

569 

629 

691 

752 

8i3 

26 

27 

27 

87 

i47 

207 

267 

327 

388 

44^ 

509 

570 

63 1 

692 

753 

8i5 

27 

28 

28 

88 

1 48 

208 

268 

328 

389 

449 

5io 

571 

632 

693 

7^4 

816 

28 

29 

3o 

29 

89 

149 
i5o 

209 
210 

269 
270 

33o 

390 

45o 

5ii 

572 

633 

694 

755 

817 

29 

3o 

3o 

90 

33i 

39. 

45i 

5l2 

573 

634 

695 

756 

818 

3i 

3i 

9' 

i5i 

211 

271 

332 

392 

452 

5i3 

574 

635 

696 

757 

819 

3i 

32 

32 

92 

I  52 

212 

372 

333 

393 

453 

5i4 

575 

636 

697 

758 

820 

32 

33 

33 

93 

1 53 

2l3 

273 

334 

394 

454 

5i5 

576 

637 

698 

759 

821 

33 

34 
35 

34 

94 

1 54 

2l4 

274 
275 

335 

395 

455 

5i6 
5i7 

i>77 
578 

638 

699 

760 
761 

822 
823 

34 
35 

35 

95 

1 55 

2l5 

336 

396 

456 

639 

700 

36 

36 

96 

1 56 

216 

276 

337 

397 

457 

5i8 

^79 

64o 

701 

762 

824 

36 

37 

37 

97 

1 57 

217 

277 

338 

398 

458 

5.9 

58o 

64 1 

702 

763 

825 

37 

38 

38 

98 

1 58 

218 

278 

339 

399 

459 

520 

58 1 

642 

7o3 

764 

826 

38 

4o 

39 

99 

.59 

219 

279 
280 

340 
341 

4"o 

460 

J2I 

582 

643 

704 

765 

827 

39 

40 

4o 

100 

160 

220 

4ot 

46 1 

522 

583 

644 

7o5 

766 

828 

4r 

4i 

lOI 

161 

221 

281 

342 

402 

462 

523 

584 

645 

706 

767 

829 

4i 

42 

42 

102 

162 

222 

282 

343 

4o3 

463 

524 

585 

646 

707 

768 

83o 

42 

43 

43 

io3 

i63 

223 

283 

344 

4o4 

464 

525 

586 

647 

708 

769 

83  r 

4-i 

44 
45 

44 

io4 

1 64 

224 

284 

345 

4o5 

465 

526 

587 

648 

709 

770 

8J2 

44 
45 

45 

io5 

i65 

225 

285 

346 

4o6 

466 

527 

588 

649 

710 

77' 

833 

46 

46 

106 

166 

226 

286 

347 

407 

467 

528 

589 

65o 

711 

772 

834 

46 

47 

47 

107 

167 

227 

287 

348 

4o8 

468 

529 

590 

65  [ 

712 

773 

835 

47 

48 

48 

108 

168 

228 

288 

349 

409 

469 

53o 

591 

652 

7i3 

774 

83b 

48 

49 

5o 

49 

109 

169 

229 

289 

35o 

4io 

470 

53 1 

592 

653 

714 

77b 

837 

49 
5o 

5o 

110 

170 

23o 

290 

35i 

4ii 

471 

532 

593 

654 

7i5 

777 

838 

5i 

5i 

1 1 1 

171 

23l 

291 

352 

4l2 

472 

533 

594 

655 

716 

778 

839 

5i 

52 

52 

112 

172 

232 

292 

353 

4i3 

473 

534 

595 

656 

717 

779 

84o 

52 

53 

53 

ii3 

173 

233 

293 

354 

4i4 

474 

535 

596 

657 

7.8 

780 

84 1 

53 

54 

55 

54 

ii4 

174 

934 

294 

355 

4i5 

476 

536 

597 

658 

719 

781 

842 

54 
55 

55 

ii5 

175 

235 

295 

356 

4i6 

477 

537 

598 

659 

720 

782 

843 

56 

56 

116 

176 

236 

296 

357 

417 

478 

538 

599 

660 

721 

783 

844 

56 

5- 

57 

117 

177 

237 

297 

358 

4i8 

479 

539 

600 

661 

722 

784 

845 

^7 

5H 

58 

118 

178 

238 

298 

359 

419 

480 

540 

601 

662 

723 

785 

846 

d8 

59 

119 

179 

239 

299 

36o 

420 

48 1 

54 1 

602 

663 

724 

786 

847 

M. 

0° 

1° 

2° 

3° 

40 

5° 

G° 

7° 

8° 

9° 

10° 

11° 

12° 

13° 

TABLE  III. 

Meridional  Parts. 

fPage  63 

M. 

o 

14° 

15° 

1G° 

17° 

18° 

19° 

20° 

21° 

22° 

23° 

24° 

25° 

2G° 

27° 

1684 

M. 

0 

848 

910 

973 

io35 

109S 

1161 

1225 

1289 

1 354 

1419 

1484 

i55o 

1616 

I 

85o 

911 

974 

36 

99 

63 

26 1  90 

55 

20 

85 

5i 

18 

85 

T 

? 

85i 

9i3 

975 

37 

IIOO 

64 

27 

91 

56 

21 

86 

52 

19 

86 

2 

3 

852 

914 

976 

38 

01 

65 

28 

92 

57 

22 

87 

53 

20 

87 

3 

4 
5 

853 

9.5 

977 

39 

02 

66 

29 

93 

58 

23 

88 

54 

21 

88 

4 

5 

854 

916 

978 

io4i 

no3 

1167 

I23o 

1295 

i359 

1424 

1490 

1 556 

1622 

1689 

6 

855 

9'7 

979 

42 

o5 

68 

32 

96 

60 

25 

91 

57 

23 

90 

6 

7 

856 

918 

980 

4i 

06 

69 

33 

97 

61 

26 

92 

58 

24 

9' 

7 

8 

857 

919 

981 

44 

07 

70 

■M 

98 

62 

27 

93 

59 

25 

93 

8 

lO 

858 
~85^ 

920 

982 

45 

08 

71 

35 

99 

63 

28 

94 

60 

26 

94 

_? 

10 

921 

983 

1046 

1 109 

1172 

1236 

i3oo 

1 364 

i43o 

1495 

i56i 

1628 

1695 

1 1 

860 

922 

9«4 

47 

10 

73 

37 

01 

66 

3i 

96 

62 

29 

96 

II 

I? 

861 

923 

985 

48 

II 

74 

38 

02 

67 

32 

97 

63 

3o 

97 

12 

i3 

862 

924 

986 

49 

12 

7i 

39 

o3 

68 

33 

98 

64 

3i 

98 

i3 

i4 
i5 

863 

925 

987 

5o 

i3 

76 

4o 

o4 

69 

34 
1435 

99 

65 

32 

99 

1700 

i4 
i5 

864 

926 

988 

io5i 

iii4 

1177 

I24[ 

i3o5 

1370 

i5oo 

1 567 

i633 

iti 

8()5 

927 

989 

52 

i5 

78 

42 

06 

71 

36 

02 

68 

34 

01 

16 

17 

866 

928 

990 

53 

16 

79 

43 

07 

72 

37 

o3 

69 

35 

o3 

17 

i8 

867 

929 

991 

54 

17 

81 

44 

08 

73 

38 

o4 

70 

37 

04 

18 

19 

20 

868 

930 

993 

55 

18 
II 19 

82 

45 

10 

74 

39 

o5 

71 

38 

o5 
17(^6 

19 
20 

8tJ9 

93. 

994 

io56 

ii83 

1246 

i3ii 

1375 

1 440 

i5o6 

1572 

1639 

21 

870 

932 

995 

57 

20 

84 

48 

12 

76 

4i 

07 

73 

4o 

07 

21 

22 

871 

933 

996 

58 

21 

85 

49 

i3 

77 

43 

08 

74 

4i 

08 

22 

23 

S72 

9^4 

997 

59 

22 

86 

5o 

i4 

79 

44 

09 

7!) 

42 

09 

23 

24 
25 

873 

9J5 

998 

60 

23 

87 

5i 

i5 

80 

45 

10 

77 

43 

II 

24 

25 

874 

936 

999 

1061 

1125 

1188 

1252 

i3i6 

i38i 

1 446 

i5ii 

1578 

1644 

1712 

26 

«75 

9^7 

louo 

63 

26 

89 

53 

17 

82 

47 

i3 

79 

45 

i3 

26 

2? 

876 

93s 

01 

64 

27 

90 

54 

18 

83 

48 

i4 

80 

47 

i4 

27 

28 

«77 

939 

02 

65 

28 

9' 

55 

19 

84 

49 

i5 

81 

48 

i5 

28 

29 

3o 

878 
879 

941 

o3 

66 

29 

92 

56 

20 

85 

5o 

16 

82 

49 

16 

29 

3o 

942 

1004 

1067 

ii3o 

1 193 

1257 

l32I 

1 386 

i45i 

.517 

i583 

i65o 

1717 

3! 

880 

943 

o5 

68 

3i 

94 

58 

22 

87 

52 

18 

84 

5i 

18 

3i 

32 

882 

944 

06 

69 

32 

95 

59 

24 

88 

53 

19 

85 

52 

20 

32 

33 

883 

945 

07 

70 

33 

96 

60 

25 

89 

55 

20 

86 

53 

21 

33 

34 
35 

884 

946 

08 

71 

34 
ii35 

98 

6, 

26 

90 

56 

21 

88 

54 

22 
1723 

34 
35 

885 

9-^7 

1009 

1072 

1199 

1262 

1327 

1392 

1457 

l522 

1589 

1 656 

36 

886 

948 

10 

73 

36 

1200 

64 

28 

93 

58 

24 

90 

i)7 

04 

36 

37 

8S7 

949 

1 1 

74 

37 

01 

65 

29 

94 

59 

25 

91 

58 

25 

37 

38 

888 

95o 

12 

75 

38 

02 

66 

3o 

95 

60 

26 

92 

59 

26 

38 

39 

4o 

889 

93. 

i3 

76 

39 

o3 

67 

3i 

96 

61 

27 

93 

60 

27 

39 
4o 

890 

952 

ioi4 

1077 

I  i4o 

1204 

1268 

i332 

1397 

1462 

i528 

1594 

1661 

1729 

4i 

89. 

953 

i5 

7a 

4i 

o5 

69 

33 

98 

63 

29 

95 

62 

3o 

4^ 

4.-- 

892 

9^)4 

16 

79 

42 

oG 

70 

34 

99 

64 

3o 

96 

63 

3i 

42 

43 

893 

9^5 

18 

80 

44 

07 

71 

35 

i4oo 

65 

3i 

98 

64 

32 

43 

44 
45 

894 
895 

956 

19 

81 

45 

08 

72 

36 

01 

67 

32 

99 

66 

33 

44 
45 

g'-v 

1020 

1082 

ii46 

1209 

1273 

i338 

l4o2 

1 468 

i533 

1600 

1667 

1734 

46 

896 

938 

21 

84 

47 

10 

74 

39 

o3 

69 

35 

01 

68 

35 

46 

47 

897 

959 

22 

85 

48 

1 1 

75 

4o 

o5 

70 

36 

02 

69 

36 

47 

48 

898 

9fjt) 

23 

86 

49 

I? 

76 

4i 

06 

71 

37 

o3 

70 

38 

48 

49 
5o 

899 

96 1 

24 

87 

5o 

i3 

77 

42 

07 

72 

38 

04 

71 

39 

49 
5o 

900 

962 

1025 

1088 

ii5i 

I2l5 

1278 

i343 

i4o8 

1473 

1539 

iCo5 

1672 

1740 

5i 

9f)i 

963 

26 

89 

52 

16 

80 

44 

09 

74 

4o 

06 

73 

4i 

5i 

52 

902 

964 

27 

90 

53 

17 

81 

45 

10 

75 

4i 

08 

7!5 

42 

52 

53 

903 

9*)  5 

28 

9' 

54 

18 

82 

46 

II 

76 

42 

09 

76 

4^ 

53 

54 
55 

904 

966 

29 

92 

55 

19 

83 

4i 

12 

77 

43 

10 

77 

44 

54 
55 

9o5 

968 

io3o 

1093 

ii56 

1220 

1284 

1 348 

i4i3 

i479 

1 544 

1611 

1678 

1746 

56 

906 

969 

3i 

94 

57 

21 

85 

49 

i4 

80 

46 

12 

79 

47 

56 

:i7 

9"7 

970 

32 

95 

58 

22 

86 

5o 

i5 

81 

47 

i3 

80 

48 

57 

d8 

908 

97' 

33 

96 

59 

23 

87 

52 

16 

82 

48 

i4 

81 

49 

58 

59 
M. 

909 

972 

34 

97 

60 

24 

88 

53 

18 

83 

49 

i5 

82 

5o 
27° 

5? 
M 

14° 

15° 

1G° 

17° 

18° 

19° 

20° 

21° 

22° 

23° 

24° 

25° 

2C° 

Page  64] 

TABLE  III 

Meridional  Parts. 

M. 

o 

28° 

29° 

30° 

3J° 

32° 

33° 

34° 

35° 

36° 

37° 

38° 

39° 

40° 

41° 

-M. 
0 

1 75 1 

1819 

1S88 

1958 

2028 

2100 

2171 

2244 

23i8 

2393 

2468 

2545 

2623 

2702 

I 

52 

21 

90 

59 

3o 

01 

73 

46 

19 

94 

70 

46 

24 

o3 

I 

2 

53 

22 

91 

bo 

3i 

02 

74 

47 

20 

95 

71 

48 

25 

o4 

2 

J 

55 

23 

92 

62 

32 

o3 

75 

48 

22 

96 

72 

49 

27 

06 

3 

4 
5 

56 

24 

93 

63 

33 

04 

76 

49 

23 

98 

73 

5o 

28 

07 
2708 

4 

5 

1757 

1825 

1894 

1964 

2o34 

2I05 

2178 

225o 

2324 

2399 

2475 

255 1 

2620 

b 

58 

2b 

95 

65 

35 

07 

79 

52 

25 

2400 

76 

53 

3i 

10 

6 

7 

59 

27 

96 

bb 

37 

08 

80 

53 

27 

01 

77 

54 

32 

II 

7 

b 

60 

29 

98 

b7 

38 

OQ 

81 

54 

28 

o3 

78 

55 

33 

12 

8 

_9 

10 

6; 
1762 

3o 

99 

b9 

39 

10 

82 

55 

29 

o4 

80 

57 

34 

i4 
2715 

_9 

10 

i83i 

1900 

1970 

204o 

2III 

2184 

2257 

233o 

24o5 

2481 

2558 

2636 

II 

64 

32 

01 

71 

4i 

i3 

85 

58 

32 

06 

82 

59 

37 

16 

n 

12 

65 

33 

02 

72 

43 

i4 

86 

59 

33 

08 

84 

60 

38 

18 

I? 

iJ 

66 

34 

o3 

73 

44 

i5 

87 

60 

34 

09 

85 

62 

4o 

19 

1 3 

i4 
i5 

67 

35 

o5 

74 

45 

16 

88 

61 

35 

10 

86 

63 

4i 

20 

i4 
i5 

1768 

i837 

1900 

1976 

2o46 

2117 

2190 

2263 

2337 

241 1 

2487 

2564 

2642 

2722 

lb 

69 

38 

07 

77 

47 

19 

91 

64 

38 

i3 

89 

66 

44 

23 

16 

17 

70 

39 

08 

7» 

48 

20 

92 

65 

39 

i4 

90 

67 

45 

24 

17 

i8 

72 

40 

09 

79 

5o 

21 

93 

66 

40 

i5 

91 

68 

46 

26 

18 

£9 

20 

li 

4i 

10 

80 

5i 

22 

94 

68 

42 

16 

92 

69 

48 

27 

£9 

20 

1774 

1842 

1912 

1981 

2o52 

2123 

2196 

2269 

2343 

2418 

2494 

2571 

2649 

2728 

21 

7^ 

AS 

i3 

83 

53 

25 

97 

70 

44 

19 

95 

72 

5o 

29 

21 

22 

76 

45 

i4 

84 

54 

26 

98 

71 

45 

20 

96 

73 

5i 

3i 

22 

2J 

77 

46 

i5 

85 

56 

27 

99 

72 

46 

22 

98 

75 

53 

32 

23 

24 
25 

7S 

47 

lb 

86 

67 

28 

2200 

74 

48 

23 

99 

76 

54 

33 

24 

25 

1780 

1848 

1917 

1987 

2o58 

2129 

2202 

2275 

2349 

2424 

25oo 

2577 

2655 

2735 

2b 

81 

49 

18 

88 

59 

3i 

o3 

76 

5o 

25 

01 

78 

57 

36 

26 

27 

82 

5o 

20 

90 

60 

32 

o4 

77 

5i 

27 

o3 

80 

58 

37 

27 

28 

83 

52 

21 

91 

61 

33 

o5 

79 

53 

28 

04 

81 

59 

39 

28 

29 

3o 

84 

53 

22 

92 

63 

34 

07 

80 

54 
2355 

29 

•  o5 

82 

61 

4o 

29 

3o 

1785 

i854 

1923 

1993 

2064 

2i35 

2208 

2281 

2430 

25o6 

2584 

2662 

2742 

3i 

86 

55 

24 

94 

65 

37 

09 

82 

56 

32 

08 

85 

63 

43 

3i 

32 

H7 

56 

25 

95 

66 

38 

10 

83 

58 

33 

oq 

86 

65 

44 

32 

33 

89 

57 

27 

97 

67 

39 

II 

85 

59 

34 

10 

88 

66 

46 

33 

34 
35 

90 

■  58 

28 

98 

69 

4o 

i3 

86 

60 

35 

12 

89 

67 

47 

34 
35 

1791 

i860 

1929 

1999 

2070 

2l4l 

2214 

2287 

236i 

2437 

25i3 

2590 

2669 

2748 

3b 

92 

61 

3o 

2000 

71 

43 

i5 

88 

63 

38 

i4 

91 

70 

5o 

36 

37 

93 

62 

3i 

01 

72 

44 

16 

90 

64 

39 

i5 

93 

71 

5i 

37 

38 

94 

63 

32 

02 

73 

45 

17 

91 

65 

4o 

17 

94 

73 

52 

38 

39 
40 

95 

64 

34 

04 

75 

46 

19 

92 

66 

42 

18 

95 

74 

54 

39 
40 

1797 

i865 

1935 

2005 

2076 

2147 

2220 

2293 

2368 

2443 

25i9 

2597 

2675 

2755 

4i 

98 

66 

36 

06 

77 

49 

21 

95 

69 

44 

21 

98 

76 

56 

4i 

42 

99 

68 

37 

07 

7» 

5o 

22 

9(3 

70 

45 

22 

99 

78 

58 

42 

AS 

1800 

69 

38 

08 

79 

5i 

24 

97 

71 

47 

23 

2601 

79 

59 

43 

44 
45 

01 

70 

39 

10 

80 

52 

25 

98 

73 

48 

24 

02 

80 

60 

44 
45 

1802 

1871 

1941 

201 1 

2082 

2i53 

2226 

2299 

2374 

2449 

2526 

2603 

2682 

2762 

4fa 

o3 

72 

42 

12 

83 

55 

27 

23oi 

75 

5i 

27 

o4 

83 

63 

46 

47 

o5 

73 

AS 

i3 

84 

56 

28 

02 

76 

52 

28 

c6 

84 

64 

47 

48 

06 

7!) 

A4 

i4 

85 

57 

3o 

o3 

78 

53 

3o 

07 

86 

66 

48 

49 

5o 

07 

76 

45 

i5 

86 

58 

3i 

04 

79 

54 

2456 

3i 

08 

87 

67 

49 
5o 

1808 

1877 

1946 

2017 

2088 

2 1 59 

2232 

23o6  238o 

2532 

2610 

2688 

2768 

5i 

09 

7S 

48 

18 

89 

61 

33 

07   81 

57 

33 

II 

90 

70 

5i 

:i2 

10 

79 

49 

19 

90 

62 

35 

08   83 

58 

35 

12 

91 

71 

52 

d3 

11 

80 

5o 

20 

91 

63 

36 

09 

84 

59 

36 

i4 

92 

72 

53 

b4 

55 

i3 

"I'sTJ 

81 

5i 

21 

92 

64 
2i65 

37 

2238 

11 

23l2 

85 

61 

37 

i5 

94 

74 

54 

55 

i883 

1952 

2022 

2094 

2386 

2462 

2538 

2616 

2605 

2775 

56 

i5 

84 

53 

24 

95 

67 

39 

j3 

88 

63 

40 

17 

96 

76 

56 

57 

16 

85 

65 

25 

96 

68 

4i 

i4 

89 

64 

4i 

19 

98 

78 

57 

58 

17 

86 

56 

26 

97 

69 

42 

16 

90 

66 

42 

20 

99 

79 

58 

59 
M. 

18 

«7 

57 

27 

98 

70 

33° 

43 

17 
35° 

9' 
36° 

67 

44 

21 

2700 

80 

59 
M. 

28° 

29° 

30° 

31° 

32° 

34° 

37° 

38°  39° 

40° 

41° 

TABLE  III.                        [Page 

Meridional  Parts. 

65 

M. 

0 

I 

2 

3 
4 
5 
6 

7 
8 
9 

10 

11 

12 

i3 

i4 
i5 
i6 
17 
i8 

!9 

20 
21 
22 
23 
24 
25 
26 
27 
28 

29 

3o 
3i 

32 

33 
34 
35 
36 

37 
38 
39 

4o 
4i 
42 
43 
44 
45 
46 
47 
48 

49 
5o 
5i 

52 

53 
54 

55 
56 
57 
58 
59 

M. 

42° 

43° 

44° 

45° 

46° 

47° 

48° 

3292 

93 

95 
96 
98 

49° 

3382 
84 
85 
87 
88 

50° 

3474 
76 
78 

79 
81 

51° 

52° 

3665 
67 
68 
70 
72 

53° 

54° 

55° 

M. 

0 

I 
2 
3 
4 
5 
6 

7 
8 

^ 
10 
II 
12 
i3 
r4 
i5 
16 

17 
18 

19 
20 
21 
22 

23 

24 

25 

26 

27 
28 
29 

3i 

32 

33 
34 

35- 
36 

37 
38 
39 
4o 
4i 
42 
43 
AA 
45 
46 
47 
48 

49 
5o 
5i 

52 

53 
54 
55 
56 
57 
58 
59 

M. 

27S2 
83 
84 
86 
87 

2863 
64 
66 
67 
69 

2946 
47 

5o 
5i 

3o3o 
3i 
33 

36 

3ii6 
17 
18 
20 
21 

3i23 
24 
26 

27 
29 

32o3 
o4 
06 
07 
09 

3569 
70 
72 
74 
75 

3764 
65 
67 
69 

70 

3865 
66 
68 
70 
71 

3968 
70 
71 
73 
75 

2788 
90 

91 
92 

94 

2870 
71 
73 
74 
75 

2953 
54 
56 

57 
58 

3o37 
38 
40 
4i 

3210 
12 
i3 
i4 
16 

3^99 
33oi 
02 
o3 
o5 
33o6 
08 
09 
II 
12 

3390 
93 

96 

3397 

3400 
02 

o3 

3482 
84 
85 
87 
88 

3577 
78 
80 
82 
83 

3673 
75 
77 
78 
80 

3772 
74 
75 
77 
79 

3780 
82 
84 
85 
87 

3873 
75 
77 
78 
80 

3977 
78 
80 
82 
84 

2795 

9Z 
98 

o99 
2801 

2877 
78 
80 
81 
82 

2960 
61 
63 
64 
65 

3o44 
46 
47 

48 
5o 

3i3o 
3i 
33 

36 

3217 

19 
20 
22 

23 

3490 

9^ 
93 
95 
96 

3585 
86 
88 
90 
9' 

3593 

96 
98 
99 

368 1 
83 
85 
86 
88 

3882 
83 
85 
87 
89 

3985 

87 

89 

91 
92 

2802 
o3 
o5 
06 

07 

2884 
85 
86 
88 
89 

2967 
68 

70 
71 
72 

3o5i 
53 
54 
55 

57 

3i37 
39 
40 
42 

•43 

3 144 

t 

3225 

26 
28 

=9 
3i 

33i4 
16 
17 

19 

20 

34o5 
07 
08 
10 
II 

3498 

35oi 
o3 

04 

3690 

9^ 

9^ 
96 

3789 
90 
92 
94 
95 

3890 
92 

9i 
95 

97 

3994 

9^ 
98 

x99 

4001 

4oo3 
o5 
06 
08 
10 

2809 

ID 
II 

i3 
i4 

2891 

96 

2974 
75 
76 
78 
79 

3o58 
60 
61 
63 
64 

3232 

M 
35 

37 
38 

3322 
23 
25 

26 
28 

34i3 
i4 
16 
17 
19 

35o6 
07 
09 
10 
12 

35i4 
i5 
17 
18 
20 

36oi 
02 
04 
06 
07 

3609 
10 
12 
i4 
i5 

3698 

99 

3701 

o3 

04 

3797 

.n99 

38oo 
02 
o4 

3899 

3901 

02 

o4 

06 

2815 
17 
18 
20 
21 

2897 

99 

2900 

02 

o3 

2981 
82 
83 
85 
86 

3o65 
67 
68 
70 
71 

3i52 
53 
55 
56 
57 

3i59 
60 
62 
63 
65 

3240 
4i 
42 

45 

3329 
3i 

32 

34 
35 

3337 
38 
4o 
4i 
43 

3420 
22 

23 
25 

27 

3706 
08 
09 
II 
i3 

38o6 
07 
09 
II 
12 

3907 
09 
II 
i3 

i4 

4012 
i4 
i5 
17 
19 

2822 
24 

25 

26 
28 

2904 
06 
07 
08 
10 

2988 
89 
91 
95 
93 

3073 

74 
75 
77 
78 

3247 
48 
5o 
5i 
53 

3428 

3o 
3i 
33 

M 

3521 

23 
25 

26 
28 

0617 
18 
20 
22 

23 

3714 
16 

17 

19 
21 

38i4 
16 

17 

19 
21 

3822 

24 
26 

27 
29 

3916 
18 

19 
21 

23 

3925 
26 
28 
3o 

32 

4021 
22 
24 
26 
28 

4029 
3i 
33 
35 
37 

2829 
3o 

32 

33 
34 

2911 
i3 

i4 
i5 

17 

2995 

96 
98 

3ooo 

3o8o 
81 
83 
84 
85 

3i66 
68 
69 
71 
72 

3254 
56 
57 

60 

3344 
46 
47 

3436 
37 
39 
40 
42 

3443 
45 
47 
48 
5o 

3529 
3i 

32 

36 

3625 
26 
28 
3o 
3i 

3722 
24 
26 
27 
29 

3731 

32 

34 
36 

37 

2836 

37 
39 
40 
4i 

2Ql8 

'19 
21 
22 
24 

3002 

o3 
o5 
06 

07 

3087 
88 
90 

93 

3173 
75 
76 
78 
79 

3262 
63 
65 
66 
68 

3352 
53 
55 
56 
58 

3537 

39 
4o 
42 
43 

3633 
34 
36 
38 
39 

383 1 

32 

34 
36 
38 

3933 

35 
37 
38 
4o 

4o38 
4o 
42 
AA 
45 

2843 

45 
47 
48 

2849 
5i 

52 

54 
55 

2925 
26 
28 

3? 

3009 
10 
12 
i3 
i4 

3094 
95 

97 

98 

3ioo 

3i8i 
82 
84 
85 
87 

3269 
71 
72 

74 
75 

3359 
61 
62 
64 
65 

345 1 
53 
54 
56 

57 

3545 

47 

48 
5o 
5i 

364 1 

AA 
46 
47 

3739 
4i 
42 
AA 
46 

3839 
4i 
43 
AA 
46 

3942 
AA 
45 
47 
49 

4047 

^9 
5i 

52 

54 

2932 

33 
35 
36 
37 

3oi6 
17 

19 
20 
21 

3ioi 
o3 
o4 
o5 
07 

3i88 
90 

91 
92 

94 

3277 
78 
80 
81 
83 

3367 
68 

70 
71 
73 

3459 
60 
62 
64 
65 

3553 
55 
56 
58 
59 

3649 
5i 

52 

54 
55 

3747 

^9 
5o 

52 

54 

3848 

^9 
5i 

53 

54 

3951 

52 

54 
56 
58 

4o56 
58 
60 
61 
63 

a856 
58 

60 
62 

2939 
4o 
42 
43 

3o23 
24 
26 

27 
29 

3io8 
10 
II 
i3 
i4 

3195 
97 
98 

3200 
01 

3284 
86 
87 

89 
90 

3374 
76 

78 

3467 
68 
70 
71 
73 

356i 
62 
64 
66 
67 

3657 

60 
62 
64 

3755 
57 

^.9 
60 

62 

3856 
58 
60 
61 
63 

3959 
61 
63 
64 
66 

4o65 

67 
69 
70 

72 

42° 

43° 

44° 

45° 

46° 

47° 

48° 

49° 

50° 

51° 

52° 

53'^ 

54° 

55° 

P^g*  66]                 TABLE  III. 

Meridional  Parts. 

o 

56° 

57° 

58° 

59° 

60° 

61° 

4649 

62° 

4775 

63° 

4905 

64° 

5o39 

65° 

66° 
5324 

67° 

5474 

68° 
563 1 

69-^ 

5795 

—  1 
M.; 

0 

4074 

4i83 

4294 

4409 

4527 

5i79 

I 

7b 

84 

9b 

II 

29 

5i 

77 

07 

42 

81 

26 

77 

33 

97 

I 

2 

77 

86 

98 

i3 

3i 

53 

79 

09 

44 

84 

28 

79 

36 

58oo 

2 

3 

79 

88 

43oo 

i5 

33 

55 

81 

12 

46 

86 

3i 

82 

39 

o3 

3 

j4 

5 

81 

90 

02 

17 

35 

57 

84 

i4 

49 
5o5i 

88 

33 

84 

42 

06 
5809 

4 

5 

4o83 

4192 

43o4 

4419 

4537 

4660 

4786 

4916 

5191 

5336 

5487 

5644 

b 

85 

94 

06 

21 

39 

62 

88 

18 

53 

93 

38 

89 

47 

II 

6 

7 

86 

95 

08 

23 

4i 

64 

90 

20 

55 

95 

4i 

92 

5o 

14 

7 

8 

88 

97 

09 

25 

43 

66 

92 

23 

58 

98 

43 

95 

52 

17 

8 

lO 

90 

99 

II 

27 

45 

68 

94 

25 

60 

5200 

46 

97 

55 

20 

_9 
10 

4092 

4201 

43i3 

4429 

4547 

4670 

4796 

4927 

5062 

52o3 

5348 

55oo 

5658 

5823 

II 

94 

o3 

i5 

3i 

49 

72 

98 

29 

65 

o5 

5i 

02 

60 

25 

11 

12 

95 

o5 

17 

33 

5i 

74 

4801 

3i 

67 

07 

53 

o5 

63 

28 

12 

i3 

97 

07 

19 

34 

53 

76 

o3 

34 

69 

10 

56 

07 

66 

3i 

i3 

i4 
i5 

99 

08 

21 

36 

55 

78 

o5 

36 

71 

12 
5214 

58 
536i 

10 
55i3 

68 
5671 

34 
5837 

14 
i5 

4ioi 

4210 

4323 

4438 

4557 

4680 

4807 

4938 

5074 

lb 

o3 

12 

25 

40 

59 

82 

09 

4o 

76 

.17 

63 

i5 

74 

39 

16 

17 

04 

14 

27 

42 

62 

84 

II 

43 

78 

19 

66 

18 

76 

42 

17 

i8 

Ob 

16 

28 

44 

64 

87 

i4 

45 

81 

22 

68 

20 

79 

45 

18 

£9 

20 

08 

18 

3o 

46 

66 

89 

16 

47 

83 

24 

71 

23 

5526 

82 

5685 

48 
585i 

!9 
20 

4iio 

4220 

4332 

4448 

4568 

4691 

4Si8 

4949 

5o85 

5226 

5373 

21 

12 

21 

M 

5o 

70 

93- 

20 

5i 

88 

29 

76 

28 

87 

54 

21 

22 

i3 

23 

36 

52 

72 

95 

22 

54 

90 

3i 

78 

3i 

90 

56 

22 

2j 

i5 

25 

38 

54 

74 

97 

24 

56 

92 

34 

80 

33 

93 

59 

23 

24 
25 

17 

27 

4o 

56 

76 

99 

26 

58 

95 

36 

83 

36 

95 

62 
5865 

24 

25 

4u9 

4229 

4342 

4458 

4578 

4701 

4829 

4960 

5097 

5238 

5385 

5539 

56q8 

2b 

21 

3i 

44 

60 

80 

o3 

3i 

63 

99 

4i 

88 

41 

5701 

68 

26 

27 

22 

32 

46 

62 

82 

o5 

33 

65 

5l02 

43 

90 

44 

04 

71 

27 

28 

24 

M 

47 

64 

84 

07 

35 

67 

o4 

46 

93 

46 

06 

74 

28 

29 

3o 

26 
4128 

3b 

49 

66 

86 
4588 

10 

37 

69 

06 

48 

95 

49 

09 

76 

29 

3o 

4238 

435i 

4468 

4712 

4839 

4972 

5io8 

525o 

5398 

5552 

5712 

5879 

3i 

3o 

40 

53 

70 

90 

i4 

42 

74 

II 

53 

5401 

54 

i5 

82 

3i 

J2 

32 

42 

55 

72 

92 

16 

44 

76 

i3 

55 

o3 

57 

17 

85 

32 

33 

33 

44 

57 

74 

94 

18 

46 

78 

i5 

58 

06 

59 

20 

88 

33 

34 
35 

35 

4b 

59 

76 

96 

20 

48 

81 

18 

60 

08 
54ii 

62 

23 

91 

34 
35 

4i37 

4247 

436 1 

4478 

4598 

4722 

485o 

4983 

5l20 

5263 

5565 

5725 

5894 

3b 

39 

49 

63 

80 

4600 

24 

52 

85 

22 

65 

i3 

67 

28 

96 

36 

^7 

4i 

5i 

65 

82 

02 

26 

55 

87 

25 

67 

16 

70 

3i 

99 

37 

38 

42 

53 

67 

84 

o4 

28 

57 

90 

27 

70 

18 

73 

34 

5902 

38 

39 
4o 

44 

55 

69 

86 

06 
46o8 

3i 

59 

92 

29 

72 

5275 

21 
5423 

75 

36 

o5 

39 
4o 

4i46 

4257 

4370 

4488 

4733 

4861 

4994 

5i32 

5578 

5739 

5908 

4i 

48 

59 

72 

90 

10 

35 

63 

96 

■34 

77 

26 

80 

42 

II 

4i 

42 

5o 

bo 

74 

92 

12 

37 

65 

99 

36 

80 

28 

83 

45 

i4 

42 

43 

52 

62 

76 

94 

i4 

39 

68 

5ooi 

39 

82 

3i 

86 

47 

17 

43 

44 
45 

53 
4i55 

64 

78 

95 

16 
4618 

4i 

70 

o3 

4i 

84 

33 

88 

5o 

19 
5922 

44 
45 

4266 

438o 

4497 

4743 

4872 

5oo5 

5x43 

5287 

5436 

5591 

5753 

4b 

57 

68 

82 

99 

20 

45 

74 

08 

46 

89 

38 

94 

56 

25 

46 

47 

59 

70 

84 

45oi 

23 

47 

76 

10 

48 

92 

4i 

06 

58 

28 

47 

48 

61 

72 

86 

o3 

25 

5o 

79 

12 

5i 

94 

43 

99 

61 

3i 

48 

49 
5o 

62 

74 

88 

o5 

27 

52 

81 

i4 

53 

97 

46 

56o2 

64 

34 

49 

5o 

4i64 

4275 

4390 

4507 

4620 

4754 

4883 

5oi7 

5i55 

5299 

5448 

56o4 

5767 

5937 

5i 

66 

77 

92 

09 

3i 

56 

85 

19 

58 

53oi 

5i 

07 

70 

40 

5i 

52 

68 

79 

94 

II 

33 

58 

87 

21 

60 

o4 

54 

10 

72 

43 

52 

53 

70 

81 

96 

i3 

35 

60 

90 

23 

62 

06 

56 

12 

75 

46 

53 

54 
55 

72 

83 

98 

i5 

37 
4639 

62 

92 

26 

65 
5167 

09 

59 

i5 

78 

48 
5951 

54 
55 

417^ 

4285 

4399 

45i7 

4764 

4894 

5028 

53ii 

5461 

56i7 

5781 

5b 

7t) 

87 

4401 

19 

4i 

66 

96 

3o 

69 

i4 

64 

20 

83 

54 

56 

57 

77 

89 

o3 

21 

43 

69 

98 

33 

72 

16 

66 

23 

86 

57 

57 

58 

Z9 

91 

o5 

23 

45 

71 

4901 

35 

74 

19 

69 

25 

89 

60 

58 

59 

81 

92 

07 

25 

47 

73 

o3 

37 

76 

21 

71 

28 

92 

63 

M. 

56° 

57°  4  58°  1 

59° 

60° 

61° 

62° 

63° 

64° 

65° 

66° 

67°  68°  1 

69° 

TABLE  III.                  rragoc? 

Meridional  Parts. 

M. 

o 

70° 

71° 

6i46 

72° 

73° 

74° 

75° 

7G° 

77° 

78° 

79° 
8o46 

80° 

8375 

81° 

82° 

83° 

0 

5966 

6335 

6534 

6746 

6970 

7210 

7467 

7745 

8739 

9145 

9606 

I 

09 

49 

33 

38 

49 

74 

i4 

7'^ 

49 

5i 

81 

45 

53 

14 

I 

•} 

72 

52 

4i 

4i 

53 

78 

18 

76 

54 

5b 

87 

52 

bo 

22 

2 

^ 

75 

55 

45 

45 

57 

82 

22 

81 

59 

61 

93 

5b 

67^   3,| 

3 

4 

78 
5981 

58 

48 

48 

60 

86 

27 

85 

64 

67 

98 

65 

74 

39 
9647 

4 
5 

6161 

635 1 

6552 

6764 

6990 

723l 

7490 

7769 

8072 

84o4 

8771 

9182 

ti 

84 

64 

54 

55 

68 

94   35 1 

94 

74 

77 

10 

78 

89 

55 

b 

7 

86 

67 

58 

58 

71 

97 

39 

98 

78 

83 

lb 

84 

.9b 

64 

7 

8 

89 

70 

61 

62 

75 

7001 

43 

75o3 

83 

88 

22 

91 

9203 

72 

b 

_? 

lO 

92 
5995 

73' 

64 

65 

79 
6782 

o5 

47 

07 

88 

93 

27 

97 
8804 

11 

80 

_9 
u 

6177 

6367 

6569 

7009 

7252 

75i2 

7793 

8099 

8433 

9218 

9(389 

1 1 

98 

80 

71 

72 

8b 

i3 

56 

16 

98 

8104 

39 

10 

25 

97 

1 1 

1? 

6001 

83 

74 

76 

90 

17 

60 

21 

7803 

09 

45 

17 

33 

9706 

12 

rl 

o4 

86 

77 

79 

93 

21 

64 

25 

08 

i5 

5i 

23 

40 

14 

i3 

i4 
i5 

07 

89 

80   83 

97 

25 

68 

3o 

i3 

20 

57 

do 

48 

23 

i4 
i5 

60  I  G 

6192 

6384 

6586 

6801 

7029 

7273 

7535 

7S17 

8125 

8463 

8836 

9255 

973 1 

i6 

i3 

95 

87 

90 

04 

33 

77 

39 

22 

3i 

69 

43 

62 

40 

lb 

16 

98 

90 

93 

08 

37 

81 

44 

27 

3b 

74 

P 

70 

48 

17 

i8 

19 

6201 

94 

97 

12 

4i 

85 

48 

32 

4i 

80 

56 

77 

57 

18 

.'9 

30 

22 

6025 

o5 

97 

6600 

i5 

45 

89 

53 

37 

47 

8b 

63 

85 

65 

£9 

20 

6208 

6400 

66o3 

6819 

7048 

7294 

7557 

7842 

8i52 

8492 

8869 

9292 

9774 

21 

28 

II 

o3 

07 

23 

52 

98 

62 

47 

58 

98 

7b 

9300 

83 

21 

}7 

3i 

i4 

07 

10 

26 

56 

73o2 

66 

52 

63 

85o4 

83 

07 

91 

22 

23 

34 

17 

10 

i4 

3o 

60 

06 

7' 

57 

68 

10 

89 

i5 

9800 

23 

24 

i5 

37 
6o4o 

20 
6223 

i3 

17 

34 

6838 

64 

II 

76 

62 
7S67 

74 

lb 

96 

22 

09 

24 

25 

6417  6621 

7068 

73i5 

7580 

8179 

8522 

8903 

9330 

9817 

26 

43 

26 

20 

24 

4t 

72 

19 

85 

72 

85 

28 

09 

^7 

26 

2b 

27 

46 

3o 

23 

28 

45 

76 

23 

89 

77 

90 

34 

16 

45 

35 

27 

28 

49 

33 

27 

3i 

49 

80 

28 

94 

82 

96 

40 

23 

53 

44 

2b 

2y 
3o 

52 

6o55 

36 

3o 

35 

53 

84 

32 

99 

«7 
7892 

8201 

4b 

3o 

60 

52 

29 

3o 

6239 

6433 

6639 

6856 

7088 

7336 

7603 

8207 

8552 

8936 

9368 

9861 

3i 

58 

42 

37 

42 

60 

92 

4i 

08 

97 

12 

58 

43 

7b 

70 

3i 

32 

61 

45 

4o 

46 

64 

96 

45 

12 

7902 

18 

65 

5o 

83 

79 

32 

33 

64 

49 

43 

49 

68 

7100 

49 

17 

07 

23 

71 

57 

91 

88 

33 

34 
35 

67 

52 

47 

53 

71 

04 

53 

22 

12 

29 

77 

63 

99 

97 

34 
35 

6070 

6255 

645o 

6656 

6875 

7108 

7358 

7626 

7917 

8234 

8583 

8970 

9407 

9906 

3fi 

73 

58 

53 

60 

79 

12 

62 

3i 

22 

40 

89 

77 

14 

i5 

3b 

37 

76 

61 

57 

63 

83 

16 

66 

36 

■27 

45 

95 

84 

22 

24 

37 

38 

79 

64 

60 

67 

86 

20 

71 

40 

32 

bi 

8601 

91 

3o 

33 

3ii 

39 

4o 

82 
6o85 

68 

63 

70 

90 
6894 

24 

75 

45 

37 

56 

07 

98 

38 

42 

i9 
4o 

6271 

6467 

6674 

7128 

7379 

765o 

7942 

8^862 

8614 

9005 

9445 

9951 

4i 

88 

74 

70 

77 

98 

32 

84 

54 

48 

67 

20 

12 

53 

60 

4i 

42 

9' 

77 

73 

81 

6901 

36 

88 

59 

53 

73 

2b 

18 

61 

69 

42 

43 

94 

80 

77 

85 

o5 

4o 

92 

64 

58 

79 

32 

25 

69 

78 

43 

An 
4'i 

97 

83 

80 

88 

09 

45 

97 

68 

63 

84 

08 

32 

77 

87 

44 
45 

6 1 00 

6287 

6483 

6692 

6913 

7149 

7401 

7673 

7968 

8290 

8644 

9039 

9485 

9996 

46 

o3 

90 

87 

95 

17 

53 

06 

78 

73 

95 

5i 

4b 

.9^ 

iooo5 

4b 

47 

06 

93 

90 

99 

20 

57 

10 

83 

78 

83oi 

57 

53 

9501 

iooi5 

47 

48 

09 

96 

94 

6702 

24 

61 

i4 

87 

83 

07 

63 

bo 

09 

10024 

48 

49 

5o 

12 

61 1 5 

99 

97 

06 

28 

65 

19 

92 

89 

12 

69 

67 

17 

ioo33 
10043 

49 
5o 

63o3 

65oo 

6710 

6932 

7169 

7423 

7697 

7994 

83i8 

8676 

9074 

9525 

5i 

18 

06 

04 

i3 

36 

73 

27 

7702 

99 

24 

82 

81 

33 

ioo52 

5i 

52 

21 

09!  07'.   17 

4o 

77 

32 

06 

8004 

29 

88 

88 

4i 

1 006 1 

52 

53 

24 

12 

11'   20 

43 

81 

36 

II 

09 

35 

95 

9b 

49 

1 007 1 

53 

54 
55 

27 
6i3o 

i5 

i4 

24 

47 

85 

4i 

16 

i4 

4i 

8701 

9103 

57 

10080 

54 
55 

63i9 

65i7 

6728 

6951 

7189 

7445 

7721 

8020 

8347 

8707 

9110 

9565 

10089 

56 

33'   22 

21 

3i 

55 

94 

49 

25 

25 

52 

i4 

17 

73 

10099 

5b 

57 

36 

25'  24 

35 

59 

98 

54 

3o 

3o 

58 

20 

24 

81 

10108 

57 

58 

40 

28 

28 

38 

63 

7202 

58 

35 

35 

64 

26 

3i 

89 

10118 

58 

59 
M. 

43 
70° 

32 

3i 

42 

66 

06 

63 

4o 

4o 

69 

33 

38 

98 

10127 

55 

M. 

71° 

72° 

73° 

74° 

75° 

76° 

77° 

78° 

79° 

80° 

81° 

82° 

83° 

Page  68] 

TABLE   IV 

TheS 

un's  Declination  for  App 

irent  Noon  at  Greenwich 

for  the  year  1848,     1 

which  wi 

1  answer  nearly  for  the  years  1852,  1856,  1860. 

T 

JAN. 

FEB. 

WAR. 

APRIL 

MAY. 

JUNE. 

JULY. 

AUG. 

SEPT. 

OCT. 

NOV. 

DEC. 

1 
P 

South. 

South. 

South, 

Jforth. 

Jforth. 

JVorth. 

J\rorth. 

Jforth. 

Jforth. 

South 

Smith. 

South. 

0      1 

0      1 

0     1 

0      ( 

O      1 

0      1 

o     / 

0     / 

O       1 

O       1 

o     / 

0     / 

21.53 

23.  3 

17.15 

7.25 

4.42 

15.12 

22.  7 

23.  6 

17.57 

8.  9 

3.20 

14.35 

2 

22.59 

16.58 

7.  2 

5.  0 

15.30 

22.15 

23.  2 

17.41 

7.47 

3.44 

14.54 

22.  2 

2 

3 

22.53 

16.41 

6.39 

5.28 

15.48 

22.22 

22.57 

17.26 

7.25 

4.  7 

15.12 

22.11 

3 

4 

22.47 

16.23 

6.16 

5.51 

16.  5 

22.29 

22.52 

17.10 

7.  3 

4.30 

15.31 

22.19 

4 

6 
S 

22.41 

16.  5 

5.53 

6.14 

16.22 

22.36 

22.46 

16.54 

6.41 

4.53 

15.49 
16.  7 

22.27 
22.34 

5 

6 

22.34 

15.47 

5.29 

6.37 

16.39 

^2.42 

22.40 

16.37 

6.19 

5.16 

7 

22.27 

15.28 

5.  6 

6.59 

16.56 

22.48 

22.34 

16.20 

5.56 

5.39 

16.25 

22.41 

7 

8 

22.19 

15.10 

4.43 

7.22 

17.12 

22.53 

22.27 

16.  3 

5.34 

6.  2 

16.42 

22.47 

8 

<■) 

22.11 

14.51 

4.19 

7.44 

17.28 

22.59 

22.20 

15.46 

5.11 

6,25 

17.  0 

22.53 

9 

10 
U 

22.  3 

14.31 

3,56 

8.  6 
8.28 

17.44 

23.  3 

22.13 

15.28 

4.48 

6.48 

17.17 

22.58 

10 
TT 

21.54 

14.12 

3.32 

17.59 

23.  7 

22.  5 

15.11 

4.25 

7.11 

17.33 

23.  3 

12 

21.44 

13.52 

3.  9 

8.50 

18.14 

23.11 

21.56 

14.53 

4.  2 

7.33 

17.49 

23.  8 

12 

13 

21.35 

13.32 

2.45 

9.12 

18.29 

23.15 

21.48 

14.34 

3.39 

7.56 

18.  5 

23.12 

13 

U 

21.24 

13.12 

2  21 

9.33 

18.44 

23.18 

21.39 

14.16 

3.16 

8.18 

18.21 

23.15'  14  1 

16 

21.14 

12.52 

1.58 

9.55 

18.58 

23.20 

21.29 

13.57 

2.53 

8.40 

18.37 
18.52 

23.18 
23.2r 

15 
16 

21.  3 

12.31 

1.34 

10.16 

19.12 

23.23 

21.20 

13.38 

2,30 

9.  2 

17 

20.51 

12.10 

1.10 

10.37 

19.25 

23.24 

21.  9 

13.19 

2.  7 

9.24 

19.  6 

23,23 

17 

IS 

20.39 

11.49 

0.47 

10.58 

19.38 

23.26 

20.59 

13.  a 

1.44 

9.46 

19.21 

23.25 

18 

19 

20.27 

11.28 

0.23 

11.19 

19.51 

23  27 

20.48 

12.40 

1.20 

10.  8 

19.35 

23.26 

19 

20 
21 

20.14 
20.  1 

11.  7 

0.  lA^. 

11.39 

20.04 

23.27 

20.37 

12.20 

0.57 

10.30 

19.48 

23.27 
23.27 

20 
21 

10.45 

0.24 

12.00 

20.16 

23.27 

20.25 

12.  0 

0.34 

10.51 

20.  2 

22 

19.48 

10.23 

0.48 

12.20 

20.28 

23.27 

20.13 

11.40 

0.10 

11.12 

20.14 

23.27 

22 

23 

19.34 

10.  2 

1.12 

12.40 

20.40 

23.26 

20.  1 

11.20 

0.13  S, 

11.33 

20.27 

23.27 

23 

24 

19.20 

9.40 

1.35 

13.00 

20.51 

23.25 

19.49 

10.59 

0.37 

11.54 

20.39 

23.25 

24 

2-5 
26 

19.  6 

9.17 

1.59 

13.19 
13,39" 

21.  1 

23.24 

19.36 

10.39 

1.  0 

12.15 

20.51 

23.24 

25 
'26 

18.51 

8.55 

2,22 

21.12 

23.22 

19.23 

10.18 

1.23 

12.36 

21.  2 

23.22 

27 

18.36 

8.33 

2.46 

13.58 

21.22 

23.20 

19.  9 

9.57* 

1.47 

12.56 

21.13 

23.19 

27 

2.S 

18.20 

8.10 

3.  9 

14.17 

21.32 

23.17 

18.55 

9.36 

2.10 

13.16 

21.24 

23.16 

28 

20 

18.  5 

7.48 

3.33 

14.35 

21.41 

23.14 

18.41 

9.14 

2.34 

13.36 

21.34 

23.13 

29 

30 

17.48 

3.56 

14.54 

21.50 

23.10 

18.27 

8.53 

2.57 

13.56 

21.44 

23.  9 

30 
3l 

31 

17.32 

4.19 

21.59 

18.12 

8.31 

14.15 

23.  5 

Table  ] 

V.    A 

—  Tl 

le  Equ 

ation 

Df  Time  for  j 

4ppar€ 

;nt  No 

on  at  ( 

jrreenwich,  for 

1848, 

or  ne 

arly  fo 

r  1852 

,  1856 

I,  1860.     To 

be  ap 

plied  1 

o  the 

App.  Time. 

JAN. 

FEB. 

MAR. 

APRIL. 

aiAY. 

JUNE. 

JULY. 

AUG. 

SEPT. 

OCT. 

NOV. 

DEC. 

Jldd  to 

.Sdd  to 

.ddd  to 

Add  to 

Sub.fr. 

Sub.fr. 

Add  to 

Add  to 

Sub.fr. 

Sub.fr. 

Sub.fr. 

Sub.fr. 

>. 

J3pp. 

.   Jlpp. 

Jipp. 

Jlpp. 

Jlpp. 

Jlpp. 

App. 

App. 

App. 

App. 

App. 

App. 

>> 

P 
T 

Time. 

Time. 

Time. 

Time. 

Tune. 

Time. 

Time. 

Time. 

Time. 

Time. 

'Time. 

Time. 

P 

M.  S. 

M.  S. 

M.  S. 

M.  S. 

M.  S. 

M.  S. 

M.  S. 

M.  S. 

M.  S. 

M.  S. 

M.  S. 

M.  S. 

3.36 

13.50 

12.31 

3.51 

3.  5 

2.28 

3.31 

6.   0 

0.14 

10.25 

16.16 

10.36 

1 

2 

4.  4 

13.58 

12.19 

3.33 

3.12 

2.19 

3.42 

5.56 

0.33 

10.44 

16.17 

10.13 

2 

3 

4.33 

14.  5 

12.  6 

3.16 

3.19 

2.  9 

3.53 

5.52 

0.52 

11.  3 

16.17 

9.49 

3 

4 

5.  0 

14.11 

11.53 

2.58 

3.25 

1.59 

4.  4 

5.47 

1.12 

11.21 

16.16 

9.24 

4 

6 
6 

5.28 

14.17 

11.39 
11.25 

2.40 

3.30 

1.49 

4.15 

5.41 

1.31 

11.38 

16.14 

8.59 
8.34 

5 
6 

5.55 

14.22 

2.23 

3.35 

1.38 

4.25 

5.35 

1.51 

11.56 

16.11 

7 

6.21 

14.26 

11.11 

2.  6 

3.39 

1.27 

4.35 

5.28 

2.11 

12.13 

16.  8 

8.  8 

7 

8 

6.47 

14.29 

10.56 

1.49 

3.43 

1.16 

4.44 

5.20 

2.32 

12.30 

16.  4 

7.42 

8 

9 

7.13 

14.31 

10.40 

1.32 

3.46 

1.  4 

4.53 

5.12 

2.52 

12.46 

15.58 

7.15 

9 

10 
11 

7.38 

14.33 

10.25 

1.15 
0.59 

3.49 

0.53 

5.  2 

5.  3 

3.13 

13.  2 

15.52 

6.47 

10 
TI 

8.02 

14,33 

10.  9 

3.51 

0.41 

5.10 

4.54 

3.34 

13.17 

15.46 

6.20 

12 

8.26 

14.33 

9,52 

0.43 

3.52 

0.29 

5.17 

4.44 

3,55 

13.32 

15.38 

5.52 

12 

13 

8.49 

14.32 

9.36 

0.27 

3.53 

0.16 

5.25 

4.34 

4.16 

13.46 

15.30 

5.23 

13 

14 

9.11 

14.31 

9.19 

0.12 

3.54 

0.  4 

5.31 

4.23 

4.37 

14.  0 

15.20 

4.55 

14 

15 
16 

9.33 

14.28 

9.  2 

.S.O.  3 

3.54 

.4.0.  9 

5.38 

4.11 

4.58 

14.13 

15.10 

4.26 

15 

To 

9.54 

14.25 

8.44 

0.18 

3.53 

0.21 

5.43 

3.59 

5.19 

14.26 

14.59 

3.56 

17 

10.15 

14.21 

8.26 

0.32 

3.52 

0.34 

5.48 

3,46 

5.41 

14.38 

14.47 

3.27 

17 

18 

10.34 

14.16 

8.  9 

0.46 

3.50 

0.47 

5.53 

3.33 

6.02 

14.49 

14.34 

2,57 

18 

19 

10.53 

14.11 

7.51 

0.59 

3.47 

1.  0 

5.57 

3.20 

6.23 

15.  0 

14.21 

2.27 

19 

20 
21 

11.12 
11.29 

14.  5 

7.32 

1.13 

3.45 

1.13 

6.  1 

3.  6 

6.44 

15.10 

14.  6 

1,57 

20 
21 

13,58 

7.14 

1.25 

3.41 

1.26 

6.  4 

2.51 

7.  5 

15.20 

13.51 

1.27 

22 

11.46 

13.51 

6.56 

1.38 

3.37 

1.39 

6.  6 

2,36 

7.26 

15.29 

13.35 

0.57 

22 

23 

12.  2 

13.43 

6.37 

1.49 

3.32 

1.52 

6.  8 

2.21 

7.46 

15.37 

13.18 

0.27 

23 

24 

12.17 

13.35 

6,19 

2.  1 

3.27 

2.  4 

6.10 

2.  5 

8.  7 

15.44 

13.  0 

.4.0.  3 

24 

25 
26 

12.31 
12.45 

13.25 

6.  0 

2.11 

3.22 

2.17 

6.11 

1.49 

8.27 

15.51 

12.42 

0.33 

25 
26 

13.16 

5.42 

2  22 

3.15 

2.30 

6.11 

1.33 

8.48 

15,57 

12  22 

1.  3 

27 

12.58 

13.  5 

5.23 

2.31 

3.  9 

2  42 

6.11 

1.16 

9.  8 

16.  2 

12.  2 

1  33 

27 

28 

13.10 

12.55 

5.  5 

2.41 

3.  2 

2,55 

6.10 

0.58 

9.27 

16.  6 

11.42 

2.  2 

28 

29 

13.21 

12.43 

4.46 

2.49 

2.54 

3.  7 

6.  8 

0.41 

9.47 

16.10 

11.20 

2.32 

29 

30 

13.31 

4.28 

2.57 

2.46 

3.19 

6.  6 

0.23 

10.  6 

16.13 

10.58 

3.  1 

30 

31 

13.41 

4.10 

2.37 

6.  4 

0.  5 

16.15 

3.29  1 31 j 

TABLE  IV 

Page  69] 

The  Sun's  Declination  for 

Apparent  Noon 

It  Greenwich,  for  the  year  1849,  1 

which  \vi 

1  answer  nearly  for  the  years  1853,  1857.  1861. 

a' 
O 

T 

JAN. 

FEB. 

MAR. 

APRIL 

MAY. 

JUNE. 

JULY. 

AUG. 

SEPT. 

OCT. 

NOV. 

DEC. 

Q 

1 

Soiit/t. 

South. 

South. 

JVort/i. 

J^ort/t. 

M'orth. 

JVorth. 

J\l'orth. 

J^orth. 

South. 

South. 

Smith. 

0      f 

0     / 

0    > 

.      0      / 

o     ( 

O       1 

o     / 

o     / 

0     / 

o    / 

0      / 

0      / 

23.  0 

17.  2 

7.30 

4.37 

15.  8 

22.  5 

23.  7 

18.  0 

8.15 

3.15 

14.30 

21.51 

? 

22.54 

16.45 

7.  7 

5.  0 

15.26 

22.13 

23.  3 

17.45 

7.53 

3.38 

14.49 

22.  0 

2 

3 

22.49 

16.27 

6.44 

5.23 

15.43 

22.20 

22.58 

17.30 

7.31 

4.  1 

15.  8 

22.  9 

3 

4 

22.43 

16.10 

6.21 

5.46 

16.  1 

22.28 

22.53 

17.14 

7.  9 

4.24 

15.26 

22.17 

4 

5 

22.36 

15.51 

5.58 

6.  8 

16.18 

22.34 

22.48 

16.58 

6.46 

4.48 

15.45 

22.25 

5 

6 

22.29 

15.33 

5.35 

6.31 

16.35 

22.41 

22.42 

16.41 

6.24 

5.11 

16.  3 

22.32 

6 

7 

22.21 

15.14 

5.12 

6.54 

16.51 

22.47 

22.36 

16.24 

6.  2 

5.34 

16.21 

22.39 

7 

S 

22.13 

14.55 

4.48 

7.16 

17.  8 

22  52, 

22.29 

16.  8 

5.39 

5.57 

16.38 

22.45 

8 

9 

22.  5 

14,36 

4.25 

7.38 

17.24 

22.57 

22.22 

15.50 

5.16 

6.20 

16.56 

22.51 

9 

10 
11 

21.56 

14.17 

4.  1 

8.  1 

17.40 

23.  2 

22.14 

15.33 

4.54 

6.42 
7.  5 

17.13 

17729 

22.57 
23.  2 

10 
11 

21.47 

13.57 

3.38 

8.23 

17.55 

23.  6 

22.  7 

15.15 

4.31 

12 

21.37 

13.37 

3.14 

8.45 

18.10 

23.10 

21.58 

14.57 

4.  8 

7.28 

17.46 

23.  7 

12 

13 

21.27 

13.17 

2.51 

9.  6 

18.25 

23.14 

21.50 

14.39 

3.45 

7.50 

18.  2 

23.11 

13 

14 

21.16 

12.57 

2.27 

9.28 

18.40 

23.17 

21.41 

14.21 

3.22 

8.13 

18.17 

23.14 

14 

15 

21.  5 

12.36 

2.  4 

9.50 

18.54 

23.20 

21.32 

14.  2 

2.59 

8.35 
8.57 

18.33 
"  18^48 

23.18 
"2372r 

15 
16 

20.54 

12.15 

1.40 

10.11 

19.  8 

23.22 

21.22 

13.43 

2.36 

17 

20.42 

11.54 

1.16 

10.32 

19.22 

23.24 

21.12 

13.24 

2.12 

9.19 

19.  3 

23.23 

17 

18 

20.30 

11.33 

0.52 

10.53 

19.35 

23.25 

21.  2 

13.  5 

1.49 

9.41 

19.17 

23.25 

18 

19 

20.18 

11.12 

0.29 

11.14 

19.48 

23.26 

20.51 

12.45 

1.26 

10.  3 

19.31 

23.26 

19 

20 
21 

20.  5 
19.51 

10.50 

0.  5 

11.34 
11.55 

20.  1 
20.13 

23.27 

20.40 
20.28 

12.25 

1.  3 

10.24 

19.45 
"19758" 

23.27 
23.27 

20 
21 

10.29 

0.19A^. 

23.27 

12.  5 

0.39 

10.46 

•79 

19.38 

10.  7 

0.42 

12.15 

20.25 

23.27 

20.16 

11.45 

0.16 

11.  7 

20.11 

23.27 

22 

23 

19.24 

9.45 

1.  6 

12.35 

20.37 

23.27 

20.  4 

11.25 

0.  8S. 

11.28 

20.24 

23.27 

23 

24 

19.  9 

9.23 

1.30 

12.55 

20.48 

23.26 

19.52 

11.  4 

0.31 

11.49 

20.36 

23.26 

24 

2-5 
2G 

18.55 

9.  1 

1.53 

13.15 

20.59 

23.24 

19.39 

10.44 

0.54 

12.10 

20.48 

23.24 

25 
26 

18.40 

8.38 

2.17 

13.34 

21.  9 

23.22 

19.26 

10.23 

1.18 

12.31 

21.  0 

23.22 

27 

18.24 

8.16 

2.40 

13.53 

21.20 

23.20 

19.12 

10.  2 

1.41 

12.51 

21.11 

23.20 

27 

28 

18.  8 

7.53 

3.  4 

14.12 

21.29 

23.18 

18.59 

9.41 

2.  5 

13.11 

21.21 

23.17 

28 

29 

17.52 

3.27 

14.31 

21.39 

23.14 

18.45 

9.19 

2.28 

13.31 

21.32 

23.14 

29 

30 

17.36 

3.50 

14.49 

21.48 

23.11 

18.30 

8.58 

2.51 

13.51 

21.42 

23.10 

30 

31 

17.19 

4.14 

21.57 

18.15 

8.36 

14.11 

23.  6 

31 

Table  1 

V.   A 

—  T\ 

le  Equ 

ation 

DfTim 

e  for  J 

"ipparf 

mt  No 

on  at  Green\ 

vich,  for 

1849, 

or  ne 

arly  fo 

r  1853 

,  185- 

',  186] 

.     To 

be  ap 

plied  1 

o  the  App.  r 

rime. 

JAN.. 

FEB. 

MAR. 

APRIL. 

MAY. 

JUNE. 

JULY. 

AUG. 

SEPT. 

OCT. 

NOV. 

DEC. 

^dd  to 

Add  to 

Add  to 

Add  to 

Siib.fr. 

Sub  fr. 

Add  to 

Add  to 

Sub.fr. 

Sub.fr. 

Sub.  fr. 

Sub.fr. 

^ 

Jlpp. 

App. 

App. 

App. 

App. 

App. 

App. 

App. 

App. 

App. 

App. 

App. 

t-. 

T 

Time. 

Time. 

Time. 

Time. 

Time. 

Time. 

Time, 

Time. 

Tone. 

Time. 

Time. 

Time. 

P 

JM.  S. 

M.  S. 

M.  S. 

M.  S. 

M.  S. 

M.  S. 

M.  S. 

M.  S. 

M.   S. 

M.  S. 

M.  S. 

M.  S. 

3.58 

13.56 

12.35 

3.56 

3.  3 

2.31 

3.27 

6.  1 

0.10 

10.21 

16.16 

10.41 

2 

4.26 

14.  4 

12.22 

3.38 

3.11 

2.22 

3.39 

5.57 

0.29 

10.39 

16.17 

10.18 

?. 

3 

4.54 

14.10 

12.10 

3.20 

3.17 

2.13 

3.50 

5.52 

0.48 

10.58 

16.17 

9.54 

3 

4 

5.21 

14.16 

11.56 

3.  2 

3.24 

2.  3 

4.  1 

5.47 

1.  7 

11.16 

16.16 

9.30 

4 

■5 
"6 

5.48 

14.20 

11.43 

2.44 

3.29 

1.53 

4.11 

5.42 

1.27 

11.34 

16.14 

9.  5 

5 
6 

6.15 

14.24 

11.28 

2.26 

3.35 

1.42 

4.21 

5.35 

1.47 

11.52  1  16.12 

8.40 

7 

6.41 

14.27 

11.14 

2.  9 

3.39 

1.31 

4.31 

5.28 

2.  7 

12.  9 

16.  8 

8.14 

7 

8 

7.  6 

14.30 

10.59 

1.52 

3.43 

1.20 

4.40 

5.21 

2.28 

12.26 

16.  4 

7.48 

8 

9 

7.31 

14.31 

10.43 

1.35 

3.47 

1.  9 

4.49 

5.18 

2.48 

12.42 

15.59 

7.21 

9 

10 

7,55 

14.32 

10.28 

1.18 

3.50 

0.57 

4  58 

5.  4 

3.  9 

12.58 

15.53 

6.53 

10 

11 

8.19 

14.32 

10.12 

1.  2 

3.52 

0.45 

5.  6 

4.55 

3.29 

13.13 

15.47 

6.26 

11 

12 

8.42 

14.31 

9.55 

0.46 

3.54 

0.33 

5.14 

4.46 

3.50 

13.28 

15.39 

5.58 

12 

13 

9.  5 

14.30 

9.39 

0.30 

3.55 

0.21 

5.22 

4.35 

4.11 

13.42 

15.31 

5.29 

13 

14 

9.27 

14.28 

9.22 

0.15 

3.55 

0.  8 

5.28 

4.25 

4.32 

13.56 

15.21 

5.  0 

14 

15 
16 

9.48 

14.25 

9.  5 

S.O.  1 

3.55 

AO.  4 

5.35 

4.13 

4.53 

14.  9 

15.11 

4.31 

15 
16 

10.  9 

14.21 

8.47 

0.15 

3.54 

0.17 

5.41 

4.  2 

5.14 

14.22 

15.  0 

4.  2 

17 

10.29 

14.17 

8.30 

0.30 

3.53 

0.30 

5.46 

3.49 

5.35 

14.34 

14.48 

3.32 

17 

18 

10.48 

14.12 

8.12 

0.43 

3.51 

0.43 

5.51 

3.37 

5.56 

14.45 

14.35 

3.  3 

18 

19 

11.  6 

14.  6 

7.54 

0.57 

3.49 

0.56 

5.56 

3.23 

6.17 

14.56 

14.22 

2.33 

19 

20 
21 

11.24 

14.  0 

7.36 

1.10 

3.46 

1.  9 

6.  0 

3.10 

6.38 

15.  6 

14.  8 

■  2.  3 

20 
21 

11.41 

13.53 

7.18 

1.23 

3.42 

1.22 

6.  3 

2.55 

6.59 

15.16 

13.53 

1.33 

22 

11.58 

13.45 

7.  0 

1.35 

3.38 

1.35 

6.  6 

2.41 

7.20 

15.25 

13.37 

1.  3 

22 

23 

12.13 

13.37 

6.42 

1.47 

3.34 

1.48 

6.  8 

2.25 

7.40 

15.33 

13.20 

0.33 

23 

24 

12.28 

13.28 

6.23 

1.58 

3.29 

2.  1 

6.10 

2.10 

8.  1 

15.41 

13.  3 

0.  3 

24 

2.5 

12.42 

13.18 

6.  5 

2.  9 

3.23 

2.14 

6.11 

1.54 

8.21 

15.48 

12.45 

^.0.27 

25 

26 

26 

12.55 

13.  8 

5.46 

2.19 

3.17 

2.27 

6.11 

1.37 

8.42 

15.54 

12.26 

0.57 

27 

13.  7 

12.58 

5.28 

2.29 

3.10 

2.39 

6.11 

1.20 

9.  2 

15.59 

12.  6 

1.26 

27 

28 

13.19 

12.46 

5.10 

2.38 

3.  3 

2.52 

6.10 

1.  3 

9.22 

16.  4 

11,46 

1.56 

28 

29 

13.29 

4.51 

2.47 

2.56 

3.  4 

6.  9 

0.45 

9.42 

16.  8 

11.25 

2.25 

29 

30 

13.39 

4.33 

2.55 

2.48 

3.16 

6.  7 

0.27 

10.  1 

16.11 

11.  3 

2.54 

30 

81 

13.48 

4.14 

2.40 

6.  4 

0.  9 

16.14 

3.23 

31 

Page  70] 

TABLE  IV 

I 
1 

The  Sun's  Declination  for 

Apparent  Noon  at  Greenwich,  for  the  year  1850.  1 

which  wi 

I  answer  nearly  for  the  years  1854,  1858,  1862. 

T 

JAN. 

FEB. 

MAR. 

APRIL. 

MAY. 

JUNE. 

JULY. 

AUG. 

SEPT. 

OCT. 

NOV. 

DEC. 

South. 

South. 

South. 

JVortA. 

JVorth. 

JVortA. 

JVorth. 

JVorth. 

JVorth. 

.^outh. 

Sonth. 

South. 

0      / 

o     / 

O     1 

0     / 

o     ' 

0     / 

o     / 

o     / 

o    / 

o    / 

0      1 

0      1 

23.   1 

17.  6 

7.36 

4.31 

15.  3 

22.  3 

23.  8 

18.   4 

8.20 

3.  9 

14.25 

21.49 

1 

2 

22.56 

16.49 

7.13 

4.54 

15.21 

22.11 

23.  4 

17.49 

7.58 

3.32 

14.44 

21.58 

9 

3 

22.50 

16.32 

6.50 

5.17 

15.39 

22.19 

22.59 

17.33 

7.36 

3.56 

15.  3 

22.  7 

3 

4 

22.44 

16.14 

6.27 

5.40 

15.56 

22.26 

22.54 

17.18 

7.14 

4.19 

15.22 

22.15 

4 

5 

22.38 

15.56 

6.  4 

6.  3 

16.14 

22.33 

22.49 

17.  2 

6.52 

4.42 

15.40 

22.23 

5 

6 

6 

22.31 

15.37 

5.41 

6.26 

16.31 

22.39 

22.43 

16.45 

6.30 

5.  6 

15.59 

22.30 

7 

22.23 

15.19 

5.18 

6.48 

16.47 

22.45 

22.37 

16.29 

6.  7 

5.28 

16.16 

22.37 

7 

8 

22.15 

15.  0 

4.54 

7.11 

17.  4 

22.51 

22.31 

16.12 

5.45 

5.51 

16.84 

22.44 

8 

9 

22.  7 

14.41 

4.31 

7.33 

17.20 

22.56 

22.24 

15.55 

5.22 

6.14 

16.51 

22.50 

9 

10 
11 

21.58 

14,21 

4.  7 
3.44 

7.55 

17.36 

23.  1 

22.16 

15.37 

4.59 

6.37 

17.  8 

22.56 

10 

21.49 

14.  2 

8.17 

17.51 

23.  5 

22.  9 

15.19 

4.36 

7.  0 

17.25 

23.  1 

11 

12 

21.39 

13.42 

3.20 

8.39 

18.  7 

23.  9 

22.  0 

15.  2 

4.14 

7.22 

17.42 

23.  6 

12- 

13 

21.29 

13.22 

2.57 

9.  1 

18.22 

23.13 

21.52 

14.43 

3.51 

7.45 

17.58 

23.10 

13 

14 

21.19 

13.  2 

2.33 

9.23 

18.. 36 

23.16 

21.43 

14.25 

3.28 

8.  7 

18.14 

23.14 

14 

15 
16 

21.  8 

12.41 
12.20 

2.  9 
1.46 

9.44 

18.51 

23.19 

21.34 

14.  6 

3.  4 

8.30 

18.29 

23.17 

15 
16 

20.57 

10.  6 

19.  5 

23.22 

21.24 

13.48 

2.41 

8.52 

18.44 

23,20 

17 

20.45 

11.59 

1.22 

10.27 

19.19 

23.24 

21.14 

13.29 

2.18 

9.14 

18.59 

23.22 

17 

18 

20.33 

11.38 

0.58 

10.48 

19.32 

23.25 

21.  4 

13.  9 

1.55 

9.36 

19.14 

23.24 

18 

19 

20.21 

11.17 

0.34 

11.  9 

19.45 

23.26 

20.53 

12.50 

1.32 

9.58 

19.28 

23.26 

19 

20 
21 

20.  8 
19.55 

10.56 

0.11 

11.29 

19.58 

23.27 

20.42 

12  30 

1.  8 

10.19 
"10741 

19.42 
"19755" 

23.27 
23.27 

20 
2T 

10.34 

0.13. V. 

11.50 

20.10 

23.27 

20.31 

r..io 

0.45 

22 

19.41 

10.12 

0.37 

12.10 

20.22 

23.27 

20.19 

11.50 

0.22 

11.  2 

20.  8 

23.27 

22 

23 

19.27 

9.50 

1.  0 

12.30 

20.34 

23.27 

20.  7 

11.30 

0.  2S. 

11.23 

20.21 

23.27 

23 

24 

19.13 

9.28 

1.24 

12.50 

20.45 

23.26 

19.55 

11.10 

0.25 

11.44 

20.33 

23.26 

24 

25 
26 

18.58 

9.  6 

1.47 

13.10 
13.29 

20.56 

23.25 

19.42 

10.49 

0.49 

12.  5 

20.45 

23.25 

25 

26 

18.43 

8.44 

2.11 

21.  7 

23.23 

19.29 

10.28 

1.12 

12.26 

20.67 

23.23 

27 

18.28 

8.21 

2.34 

13.48 

21.17 

23.21 

19.16 

10.  7 

1.35 

12.46 

21.  8 

2.3.21 

27 

28 

18.12 

7.59 

2.58 

14.  7 

21.27 

23.18 

19.  2 

9.46 

1.69 

13.  6 

21.19 

2.3.18 

28 

29 

17.56 

3.21 

14.26 

21.37 

23.15 

18.48 

9.25 

2.22 

13.26 

21.29 

23.15 

29 

30 

17.40 

3.45 

14.45 

21.46 

23.12 

18.34 

9.  3 

2.46 

13.46 

21.39 

2,3.11 

30 

31 

17.23 

4.  8 

21.54 

18.19 

8.42 

14.  6 

23.  7 

31 

Table  I 

V.   A 

—  Tl 

le  Equ 

ation  of  Tin 

e  for  . 

4ppar6 

mt  No 

on  at  Green\ 

vich,  for 

1850 

or  ne 

arly  fo 

r  1854 

[,  1858,  186'^ 

J.     To 

be  ap 

plied  \ 

o  the  App.  ''. 

^ime. 

JAN. 

FEB. 

MAR. 

APRIL. 

MAY. 

JUNE. 

JULY. 

AUG. 

SEPT. 

OCT. 

NOV. 

DEC. 

Md  to 

Md  to 

Add  to 

Add  to 

Sub.fr. 

Sub.fr. 

Add  to 

Add  to 

Sub.fr. 

Sub.fr. 

Sub.fr. 

Sub.fr. 

>> 

App. 

.Spp. 

App. 

App. 

App. 

App. 

App. 

App. 

App. 

App. 

App. 

App. 

>i 

o 
T 

Tims. 

Time. 

Time. 

Tune. 

Time. 

Time. 

Time. 

Time. 

Time. 

Time. 

Time. 

Time. 

M.  S. 

U.  S. 

M.  S. 

M.  S. 

M.  S. 

M.  S. 

M.  S. 

M.  S. 

M.  S. 

M.  S. 

M.  S. 

M.  S. 

3.51 

13.54 

12.37 

3.59 

3.  3 

2.35 

3.23 

6.   1 

0.  6 

10.17 

16.16 

10,47 

?. 

4.19 

14.  1 

12.24 

3.41 

3.10 

2.26 

3.35 

5.57 

0.25 

10.36 

16.17 

10.24 

9 

3 

4.47 

14.  8 

12.12 

3.23 

3.17 

2.16 

3.46 

5.53 

0.44 

10.64 

16.17 

10.  0 

3 

4 

5.15 

14.14 

11.58 

3.  5 

3.24 

2.  6 

3.57 

5.48 

1.  3 

11.12 

16.16 

9.. 36 

4 

5 

5.42 

14.19 

11.45 

2.47 

3.29 

1.56 

4.  8 

5.43 

1.23 

11.30 

16.15 

9.11 

5 
6 

fi 

6.  8 

14.23 

11.31 

2.30 

3.35 

1.45 

4.18 

5.37 

1.42 

11.48 

16.12 

8.40 

7 

6.34 

14.26 

11. M 

2.12 

3.39 

1.35 

4.28 

6.30 

2.  2 

12.  6 

16.  9 

8,20 

7 

8 

7.  0 

14.29 

11.  2 

1.55 

3.43 

1.23 

4.38 

5.23 

2.22 

12.22 

16.  5 

7.54 

S 

9 

7.25 

14.31 

10.46 

1.38 

3.47 

1.12 

4.47 

5.15 

2.43 

12.38 

16.  0 

7.27 

9 

10 

TT 

7.50 

14.32 

10.31 

1.22 

3.49 

1.  0 

4  56 

5.  7 

3.  3 

12.54 

15.55 

7.  0 

10 
U 

8.14 

14.32 

10.15 

1.  5 

3.52 

0.48 

5.  5 

4.. 58 

3.24 

13.  9 

15.48 

6.32 

n 

8.37 

14.32 

9.59 

0.49 

3.53 

0.36 

5.13 

4.48 

3.45 

13.24 

15.41 

6.  4 

12 

13 

9.  0 

14.31 

9.43 

0.34 

3.54 

0.24 

5.20 

4.38 

4.  6 

13.39 

15.33 

5,36 

13 

u 

9.22 

14.29 

9.26 

0.18 

3.55 

0.11 

5.27 

4.28 

4.27 

13.53 

15.24 

5,  7 

14 

15 

16 

9.44 

14.26 

9.  9 

0.  3 

3.55 

.4.0.  2 

5.34 

4.17 

4.48 

14.  6 

15.14 

4.38 

15 
16 

10.  5 

14.23 

8.52 

8.0.12 

3.54 

0.14 

5.40 

4.  5 

6.  9 

14.19 

15.  3 

4.  9 

17 

10.25 

14.18 

8.34 

0.26 

3.53 

0.27 

5.46 

3.53 

6.30 

14.31 

14.52 

3.40 

17 

18 

10.45 

14.14 

8.17 

0.40 

3.52 

0.40 

5.51 

3.40 

6.51 

14.43 

14.39 

3.11 

IS 

iq 

11.  3 

14.  8 

7.59 

0.54 

3.49 

0.53 

5.55 

3.26 

6.12 

14.64 

14.26 

2.41 

19 

20 

11.21 

14.  2 

7.41 

1.  7 

3.47 

1.  6 

6.59 

3.13 

6.33 

16.  5 

14.12 

2.11 

20 
21 

11.33 

13.55 

7.23 

1.20 

3.43 

1.19 

6.  2 

2.58 

6.55 

15.15 

13..57 

1.41 

?,■?, 

11.55 

13.47 

7.  4 

1.32 

3.40 

1.32 

6.  5 

2.44 

7.15 

16.24 

13.42 

1.11 

22 

93 

12.10 

13.39 

6.46 

1.44 

3.35 

1.45 

6.  7 

2.28 

7.36 

16.32 

13.25 

0.41 

23 

''4 

12.25 

13.30 

6.27 

1.56 

3.31 

1.57 

6.  9 

2.13 

7.57 

15.40 

13.  8 

0.11 

24 

2.5 

12.39 

13.20 

6.  9 

2.  7 

3.25 

2.10 

6.10 

1.57 

8.18 

15.47 

12.50 

.1.0.18 

2.3 

26 

9f\ 

12.52 

13.10 

6.50 

2.17 

3.19 

2.23 

6.10 

1.40 

8.38 

15.54 

12,32 

0,48 

V 

13.  4 

13.  0 

5.32 

2.28 

3.13 

2.35 

6.10 

1.23 

8.58 

15.59 

12.12 

1.18 

27 

?,8 

13.16 

12.48 

5.13 

2.37 

3.  6 

2.47 

6.  9 

1.  6 

9.18 

16.  4 

11.52 

1.48 

28 

^9 

13.27 

4.54 

2.46 

2.59 

3.  0 

6.  8 

0.49 

9.38 

16.  8 

11.31 

2.17 

29 

30 
31 

13.36 

4.36 

2.55 

2.51 

3.11 

6.  6 

0.31 

9.57 

16.12 

11.  9 

2.46 

30 

13.45 

4.17 

2.43 

6.  4 

0.13 

16.14 

3.15 

31 

TABLE   IV 

[Page  7] 

The  Sun's  Declination  for 

Anna 

rent  Noon  at  Greenwich,  for  the  year  1851,  1 

wh 

ich  \vi 

1  answer  nearl}'  for  the  years  1855,  1859,  1863. 

1 
P 

JAN. 

FEB. 

aiAR. 

APRIL. 

l\L\v. 

JUNE. 

JULY. 

AUG. 

SEPT. 

OCT. 

NOV. 

DEC. 

P 

Soxah. 

South. 

South. 

JVurth. 

jVurth. 

Xorlh. 

JVorth. 

JVorth. 

A'orlh. 

South. 

South. 

South. 

0      ' 

0      1 

O     1 

O      1 

o     / 

0     / 

o     / 

0      ' 

o    / 

0     ' 

0      1 

O      1 

1 

23.  2 

17.11 

7  41 

4.25 

14.59 

22.  1 

2.3.  9 

18.  8 

8.25 

3.  3 

14.21 

21.47 

I 

f. 

22..57 

16..53 

7.19 

4.49 

15.17 

22.  9 

2.3.  5 

17.53 

8.  3 

3.27 

14.40 

21.56 

2 

3 

22..i;J 

16.36 

6.56 

.5.12 

15.. 35 

22.17 

23.  1 

17.37 

7.42 

3.-50 

14.59 

22.  5 

3 

4 

22.46 

16.18 

6..33 

5.35 

15.52 

22  24 

22..56 

17.21 

7.19 

4.13 

15.17 

22.13 

4 

5 
6 

22.39 

16.  0 

6.10 

5.57 

16.10 

22.31 

22.50 

17.  5 

6.57 
6.35 

4.36 
4..59 

15.36 
15.54 

22.21 

"22729" 

5 
6 

22.32 

15.42 

5.46 

6.20 

16.27 

22.38 

22.45 

16.49 

7 

92  2.5 

15.23 

5.23 

6.43 

16.44 

22.44 

22.39 

16.33 

6.13 

5.23 

16.12 

22.36 

7 

8 

22.17 

1.5.  5 

5.  0 

7.  5 

17.  0 

22.50 

22.32 

16.16 

5.-50 

5.46 

16.30 

22.42 

8 

0 

22.  9 

14.46 

4.36 

7.28 

17.16 

22.55 

22.25 

15.59 

5.27 

G.  8 

16.47 

22.49 

9 

10 
11 

22.  0 

14.26 

4.13 

7.50 

17.32 

23.  0 
23.  4 

22.18 

15.41 

5.  5 

6.31 

17.  4 

22.54 
23.  0 

10 
IT 

21..51 

14.  7 

3.49 

8.12 

17.48 

22.10 

15.24 

4.42 

6..54 

17.21 

12 

21.42 

13.47 

3.26 

8.34 

18.  3 

23.  9 

22.  2 

15.  6 

4.19 

7.17 

17.38 

23.  4 

12 

13 

2 1.. 32 

13.27 

3.  2 

8.-56 

18.18 

23.12 

21..54 

14.48 

3.-56 

7.39 

17..54 

23.  9 

13 

14 

21.22. 

13.  7 

2.39 

9.18 

18.33 

2-3.16 

21.45 

14.30 

3.33 

8.  2 

18.10 

23.13 

14 

1.5 
16 

21.11 

12.46 

2.15 

9.39 

18.47 

23.19 

21.36 

14.11 

3.10 

8.24 

18.25 

23.16 

15 
16 

21.  0 

12.25 

1.51 

10.  1 

19.  2 

23.21 

21.27 

13.52 

2.47 

8.46 

18.41 

2.3.19 

17 

20.43 

12.  5 

1.28 

10.22 

19.15 

23.23 

21.17 

13.33 

2.24 

9.  8 

18..56 

23  22 

17 

18 

20.. 36 

11.44 

1.  4 

10.43 

19.29 

23.25 

21.  7 

13.14 

2.  1 

9.30 

19.10 

23.24 

18 

19 

20.24 

11.22 

0.40 

11.  4 

19.42 

23.26 

20..56 

12..55 

1.37 

9.52 

19.24 

23.25 

19 

20 
21 

20.11 
19..58 

11.  1 

0.17 

11.24 

19.55 

23.27 

20.45 

12.35 

1.14 

10.14 

19.38 

23.27 

20 
21 

10.39 

0.  IN. 

11.45 

20.  7 

23.27 

20.34 

12.15 

0.51 

10.35 

19.52 

23.27 

22 

19.44 

10.18 

0.31 

12.  5 

20.19 

23.27 

20.22 

11.-55 

0.27 

10..57 

20.  5 

23.27 

22 

23 

19.. 31 

9..56 

0.,54 

12.25 

20.31 

23.27 

20.10 

11.35 

0.  4 

11.18 

20.18 

23.27 

23 

24 

19.16 

9.34 

1.18 

12.45 

20.43 

23.26 

19.58 

11.14 

0.20  S. 

11. .39 

20.30 

23.26 

24 

2,5 
20 

19.  2 

9.12 

1.42 
2.  5 

13.  5 

20.54 

23.25 

19.45 

10.54 

0.43 

12.  0 

20.42 

23.25 

25 
26 

18.47 

8.49 

13.25 

21.  4 

23.23 

19.32 

10.33 

1.  6 

12.21 

20.54 

23.23 

27 

18.32 

8.27 

2.29 

13.44 

21.15 

23.21 

19.19 

10.12 

1.30 

12.41 

21.  5 

23.21 

27 

28 

18.16 

8.  4 

2..52 

14.  3 

21.25 

23.19 

19.  5 

9.51 

1..53 

13.  1 

21.16 

23.19 

28 

29 

18.  0 

3.16 

14.22 

21.34 

2.3.16 

18.51 

9.30 

2.17 

13.22 

21.27 

23.16 

29 

30 
31 

17.44 

3.39 

14.40 

21.44 

23.13 

18.37 

9.  9 

2.40 

13.41 

21.37 

23.12 

30 

17.27 

4.  2 

21.52 

18.23 

8.47 

14.  1 

23.  8 

31 

Table  1 

V.   A 

—  Tl 

le  Equ 

at  ion 

of  Time  for  . 

A.ppar( 

5nt  No 

on  at 

Green) 

mch,  for 

1851 

or  ne 

arly  fo 

r  1855 

.,  185f 

),  1863.     To 

be  ap 

plied  1 

o  the 

App.l 

^ime. 

JAN. 

FEB. 

MAR. 

APRIL. 

MAY. 

JUNE. 

JULY. 

AUG. 

SEPT. 

OCT. 

NOV. 

DEC. 

Ji'ld  to 

.Bdd  to 

.add  to 

Add  to 

Sub.  fr. 

Sub.fr. 

Add  to 

Add  to 

Sub.fr. 

Siib.fr. 

Sub.fr. 

Sub.fr. 

Jlpp. 

Jipp. 

j]pp. 

App. 

App. 

App. 

App. 

App. 

App. 

App. 

App. 

App. 

>^ 

T 

Time. 

Time. 

Time. 

Tune. 

Time. 

Time. 

'lime. 

Time. 

Time. 

Time. 

Time. 

Time. 

P 

M.  S. 

M.  S. 

M.  S. 

HL  S. 

M.  S. 

M.  S. 

M.  S. 
3.23 

M.  S. 

M.  S. 

M.  S. 

M.  S. 
16.15 

M.  S. 
10.52 

3.44 

13.52 

12.40 

4.  4 

2.-59 

2.34 

6.  3 

0.  0 

10.12 

2 

4.12 

14.   0 

12.28 

3.46 

3.  7 

2.25 

3.34 

6.  0 

0.19 

10.31 

16.17 

10.30 

2 

3 

4.40 

14.  7 

12.15 

3.28 

3.14 

2.16 

3.46 

5.56 

0.38 

10.49 

16.17 

10.  6 

3 

4 

5.  8 

14.13 

12.  2 

3.11 

3.20 

2.  6 

3.57 

5.51 

0..57 

11.  8 

16.17 

9.42 

4 

5 
6 

5.3.5 

14.18 

11.49 

2.53 

3.26 

1.-56 
1.46 

4.  8 

5.46 

1.17 

11.26 

16.16 

9.18 

5 
6 

6.  2 

14.23 

11.35 

2.35 

3.31 

4.18 

5.40 

1.37 

11.44 

16.14 

8.53 

7 

6.28 

14.26 

11.21 

2.18 

3.36 

1..35 

4.28 

5.33 

1.-57 

12.  1 

16.11 

8.27 

7 

8 

6.-54 

14.29 

11.  6 

2.  1 

3.40 

1.24 

4.38 

5.26 

2.17 

12.18 

16.  7 

8.  1 

8 

9 

7.20 

14.31 

10.51 

1.44 

3.44 

1.13 

4.47 

5.18 

2.38 

12.35 

16.  3 

7.35 

9 

10 

7.44 

14.32 

10..36 

1.27 

3.47 

1.  1 

4.56 

5.10 

2.-58 

12.51 

15.-57 

7.  8 

10 

11 

8.  8 

14.33 

10.20 

1.10 

3.49 

0.49 

5.  4 

5.  1 

3.19 

13.  6 

15.51 

6.40 

11 

12 

8.32 

14.32 

10.  4 

0.54 

3.51 

0..37 

5.12 

4.51 

3.40 

13.21 

15.44 

6.12 

12 

13 

S..55 

14.31 

9.47 

0.38 

3.53 

0.25 

5.20 

4.41 

4.  1 

13.36 

15.36 

5.44 

13 

14 

0.17 

14.29 

9.30 

0.23 

3.-54 

0.13 

5.27 

4.30 

4  22 

13.50 

15.27 

5.16 

14 

15 
16 

9.. 38 

14.26 

9.13 

0.  7 

3.54 

0.  0 

5.33 

4.19 

4.43 

14.  4 

15.18 

4.47 

15 
16 

9.-59 

14.23 

8..56 

S.O.  8 

3..54 

A  0.12 

5.39 

4.  8 

5.  4 

14.17 

1-5.  7 

4.18 

17 

10.20 

14.19 

8.. 39 

0.22 

3..53 

0.25 

5.45 

3.-55 

5.26 

14.29 

14.-56 

3.48 

17 

18 

10.39 

14.14 

8.21 

0.36 

3.51 

0.38 

5.-50 

3.43 

5.47 

14.41 

14.44 

3.19 

18 

19 

10..58 

14.  9 

8.  3 

0.50 

3.49 

0.51 

5.-54 

3.30 

6.  8 

14.52 

14.30 

2.49 

19 

20 
21 

11.16 

14.  2 

7.45 

1.  4 

3.47 

1.  4 

5..59 

3.16 

6.29 

15.  3 

14.17 

2.19 

20 
2T 

11..33 

13..56 

7.27 

1.16 

3.44 

1.17 

6.  2 

3.  2 

6.-50 

15.13 

14.  2 

1.49 

22 

11.50 

13.48 

7.  8 

1.29 

3.40 

1.30 

6.  5 

2.47 

7.11 

15.22 

13.46 

1.19 

22 

23 

12.  5 

13.40 

6.-50 

1.41 

3.36 

1.43 

6.  7 

2.32 

7.31 

15.31 

1-3.30 

0.49 

23 

24 

12.20 

13.31 

6.32 

1.53 

3.31 

1..56 

6.  9 

2.17 

7.52 

15.39 

1.3.13 

0.19 

24 

25 
26 

12.35 

13.22 

6.13 

2.  4 

3.25 

2.  8 

6.11 

2.  1 

8.12 

15.46 

12..55 

^.0.11 

25 

26 

12.48 

1.3.12 

5.55 

2.14 

3.20 

2.21 

6.11 

1.45 

8.33 

15..52 

12.36 

0.41 

27 

13.  1 

13.  2 

5.36 

2.24 

3.13 

2.34 

6.12 

1.28 

8.53 

15..58 

12.17 

1.11 

27 

28 

1.3.13 

12.51 

5.18 

2.34 

3.  6 

2.46 

6.11 

1.11 

9.13 

16.  3 

11..57 

1.41 

28 

29 

13.24 

4.-59 

2.43 

2..59 

2..59 

6.10 

0..54 

9.33 

16.  7 

11.36 

2.10 

29 

30 
31 

13.34 
13.4'^ 

4.41 

2.51 

2.51 

3.11 

6.  8 

0.36 

9.-52 

16.11 

11.14 

2.40 

30 
31 

4.23 

2.43 

6.  6 

0.18    1 

16.13 

3.  9 

1   I'age  72J 

TABLE 

V. 

1  For  reducing  the  Sun's  Declination,  as  given  in  the  Nautical  Almanac  for  Noon 

at  Greenwich,  to  any  other  Time  under  any  other  Meridian. 

Add  aft.  N. 

Sub.  aft.  N. 

H.M 

H.M 

H.M 

H.M 

H.M 

H.M 

H.M 

H.M  1  Sub.  aft.  N. 

Add  aft.  N. 

Sub.  bef.  N. 

Add  bef.  N. 

0.2c 
5 

0.  4C 
10 

1.    0 
15 

1.  20 

20 

1.  40 

25 

2.    0 
30 

2.  20 
35 

2.  40 
40 

Add  bef.  N. 

Sub.  bef.  N. 

Add  in  W. 

Sub.  in  W. 

Sub.  in  W. 

Add  in  W. 

Sub.  in  E. 

Add  in  E. 

Deg 

m"!s" 

0.  c 

Deg. 
M.S 

0.   0 

Dcg 
M.S 

0.   0 

Deg. 

3I.S. 
0.   0 

Deg. 
M.S. 

0.   0 

Deg. 
M.S. 
0.   0 

Deg. 
M.S. 
0.  0 

Deg. 
M.S. 

0.   0 

Add  in  E. 

Sub.  in  E. 

Days. 

Days. 

Days. 

Days. 

i  Decemb.  21 

Decemb.  21 

21   June 

21   June 

20 

22 

0.  0 

0.    I 

0.    I 

0.    I 

0.   2 

0     2 

0.   2 

0.  3 

?2 

20 

19 

23 

0.  0 

0.    I 

0.   2 

0.    2 

0.  3 

0.  4 

0.  5 

0.  6 

23 

19 

18 

24 

0.   I 

0.   2 

0.  3 

0.  4 

0.  6 

0.  7 

0.  8 

0.    q 

24 

18 

17 

25 

0.   I 

0.  3 

0.  4 

0.  6 

0.  7 

0.  9 

O.II 

0.12 

25 

17 

i6 

26 

0.  2 

0.  4 

0.  5 

0.  7 

0.  9 

O.II 

o.i3 

o.i5 

26 

r6 

i5 

27 

0.  2 

0.  5 

0.  6 

0.  8 

O.II 

o.i3 

o.i5 

0.18 

37 

i5 

i4 

28 

0.  3 

0.  6 

0.  7 

O.IO 

0. 12 

o.i5 

0.18 

0.21 

28 

14 

i3 

^9 

0.  3 

0.  7 

0.  9 

0.12 

o.i5 

0.18 

0.21 

0.24 

29 

i3 

12 

3o 

0.  3 
0.  4 

0.  7 
0.  8 

O.IO 
O.II 

o.i3 
o.i5 

0.17 
0. 19 

0.20 
0.22 

0.23 
0.26 

0.27 
o.3o 

3o  June 

12 

II 

Decemb.  3i 

I   July 

11 

10 

January       i 

0.  4 

0.  8 

0.12 

0.16 

0.20 

0.24 

0.28 

0.32 

2 

10 

9 

2 

0.  4 

0.  8 

o.i3 

0.17 

0.21 

0.26 

o.3o 

0.35 

3 

9 

8 

3 

0.  6 

0.  9 

o.i4 

0.19 

0.24 

0.29 

0.33 

0.38 

4 

8 

7 

4 

0.  5 

O.IO 

O.lb 

0.21 

0.26 

o.3i 

0.36 

o.4i 

5 

7 

6 

5 

0.  5o.ii 

0.16 

0.22 

0.28 

0.33 

0.38 

0.44 

6 

6 

5 

6 

0.  60.12 

0.17 

0.24 

o.3o 

0.35 

0.41 

0.47 

7 

5 

4 

7 

0.  60.12 

0.18 

0.25 

o.3i 

0  37 

0.43 

0.49 

8 

4 

3 

8 

0.  60. i3 

0.19 

0.26 

0.33 

0.39 

0.45 

0.52 

9 

3 

2 

9 

0.  7 
0.  7 

o.i4 
o.i4 

0.20 
0.21 

0.27 
0.29 

0.34 
oT36 

0.41 
0.43 

0.48 
o.5o 

0.55 
0.57 

10 

2 

Decemb.     i 

10 

II 

I  June 

Novemb.  So 

II 

0.  7 

o.i5 

0.22 

o.3o 

0.37 

0.45 

0.52 

I.  0 

12 

3 1  May 

29 

12 

0.  8 

0.16 

0.23 

o.3i 

0.39 

0.47 

0.55 

I.  3 

i3 

3o 

28 

i3 

0.  8 

0.16 

0.24 

0.33 

o.4i 

0.49 

0.57 

1.  6 

i4 

29 

27 

i4 

0.  8 

0.17 

jo.  25 

0.34 

0.42 

o.5i 

0.59 

I.  8 

i5 

28 

26 

i5 

0.  9 

0.18 

0.26 

0.35 

0.44 

0.53 

1 .  2 

1 .11 

16 

27 

25 

16 

0.  9 

0.18 

0.27 

0.37 

0.46 

0.55 

I.  4 

i.i3 

17 

26 

24 

17 

0.  9 

0.19 

0.28 

0.38 

0.47 

0.57 

I.  6 

1. 16 

18 

25 

23 

18 

o.io 

0.20 

0.29 

0.39 

0.49 

0.58 

I.  9 

1. 19 

19 

24 

22 

19 

O.IO 
O.IO 

0.20 
0.21 

o.3o 
o.3i 

0.40 
0.41 

o.5o 
o.5i 

1 .  0 
1 .   2 

1 .10 
1 .12 

1.20 
1 .22 

20 

23 

21 

20 

21 

22 

20 

21 

O.II 

0.22 

0.32 

0.43 

0.53 

I.  4 

i.i4 

1.25 

22 

21 

19 

22 

O.II 

0.22 

0.33 

0.44 

0.55 

I.  6 

1. 17 

1.28 

23 

20 

18 

23 

O.II 

0.23 

0.34 

0.45 

0.56 

I-   7 

i.ig 

i.3o 

24 

19 

17 

24 

0.12 

0.23 

0.34 

0.46 

0.57 

I.    Q 

1 .21 

1.32 

25 

18 

16 

25 

0.12 

0.24 

0.35 

0.47 

0.59 

I  .11 

1.23 

1.35 

26 

17 

i5 

26 

0.12 

0.24 

0.36 

0.48 

I.  0 

1. 12 

1.24 

1.36 

27 

16 

i4 

27 

0.12 

0.25 

0.37 

0.49 

I.  2 

i.i4 

1 .26 

1 .39 

28 

l5 

i3 

.    28 

o.i3 

0.26 

0.38 

o.5i 

1.  4 

1. 16 

1.28 

r.4T 

29 

i4 

If 

January    3o 

o.i3 
o.i3 

0.26 
0.27 

0.39 
0.41 

0.53 
0.55 

I.  6 
I.  9 

1. 19 
r  .22 

1.33 

1.36 

1.45 
i.5o 

3 1  July 

12 

9 

February     i 

2  August 

10 

7 

3 

o.i4 

0.28 

0.42 

0.57 

1 .1 1 

1.25. 

1.39 

1.53 

4 

8 

5 

5 

o.i4 

0.29 

0.43 

0.58 

i.i3 

1.27 

1.42 

1.56 

6 

6 

3 

7 

o.ib 

o.3o 

0.45 

I.  0 

i.i5 

i.3o 

1.44 

i.5g 

8 

4 

Novemb.     i 

9 

o.i5 

o.3i 

0.46 

1 .   2 

1. 17 

1.32 

1.47 

2.  3 

10 

2  May 

October    3o 

II 

0.16 

0.32 

0.47 

I.  3 

1. 19 

1.35 

i.5o 

2.  6 

12 

3o  April 

28 

i3 

0.16 

0.32 

0.48 

I.  5 

1 .21 

1.37 

1.53 

2.     Q 

i4 

28 

26                   i5| 

0.16 

0.33 

0.49 

I.  6 

1 .22 

I.3q 

1.56 

2.12 

16 

26 

24 

17 

0.17 

0.34 

o.5o 

I-   7 

1.24 

1.41 

1.58 

2.l5 

18 

24 

21 

20 

0.17 
0.17 

c.  34 
0.35 

0.52 

0.53 

I.   9 
1 .11 

1.27 
1 .20 

1.44 
1.46 

2.   I 

2.  4 

2.19 
2.22 

21 

21 

18 

23 

24 

18 

i5 

February  26 

0.18 

0.36 

0.54 

i.i3 

i.3i 

r.4o 

2.  7 

2.25 

27 

i5 

12 

March         i 

0.18 

0.37 

0.55 

i.i4 

1.32 

1.5. 

2.    q 

2.28 

3o  August 

12 

9 

4 

0.19 

0.38 

0.56 

i.i5 

1.34 

1.53 

2.1; 

2.3o 

2  Sept. 

9 

6 

7 

0.19 

0.38 

0.57 

1. 16 

1.35 

1.54 

2.ii 

2.32 

5 

6 

October       3 

10 

0.19 

0.38 

0.57 

1. 17 

1.36 

1.55 

2.14 

2.34 

8 

3  April 

Septem.    3o 

i3 

0.19 

0.39 

0.58 

1. 17 

1.37 

1.56 

2.l5 

2.35 

II 

3 1   March 

27 

16 

0.19 

0.39 

T.5M 

1. 18 

1.38 

1.57 

2.16 

2.36 

i4 

28 

24 

i; 

0-20 

0.39 

j.5i 

1. 18 

1.38 

I., 57 

2.16 

2.36 

17 

25 

After 

Before 

0.20 

o.4o 

0.59 

I.IQ 

i.3q 

1.58 

2.17 

2.36 

Before 

After 

Equinox. 

Equinox. 

1 

1 

Equinox. 

Equinox. 

TABLE   V. 

[Pago  73 

For  reducing  the  Sun's  Declination,  as  given  in  the  Nautical  Ahnanac  for  Noon 

at  Greenwich,  to  any  other  Time  under  any  other  Merit] 

ian. 

Add  aft.  N. 

Sub.  aft 

N. 

H.M 

H.M 

H.31 

H.M 

H.M 

H3I 

H.M 

Sub.  aft.  N. 

Add  aft.  N. 

Sub.  bef.  N. 

Add  bef.  N. 

3.    C 
45 

3.  20 
50 

3.  40 
55 

4.    0 
CO 

4.  20 
65 

4.  40 
70 

5.    0 

75 

Add  bef:  N. 

Sub.  bef.  N. 

Add  in  W. 

Sub.  in  W. 

Sub,  in  W. 

Add  in  W. 

Sub.  in  E. 

Add  in 

E. 

Deg 
M.S 

0.  c 

Ucg. 
M.S 
0.  c 

De- 
M.S 

0.   0 

Dcg. 
M.S. 
0.  0 

Deg 

M.S. 
0.   0 

Deg 
M.S. 

0.  0 

Deg. 

M.S. 
0.  0 

Add  in  E. 

Sub.  in  E. 

Days. 

Days. 

Daj's. 

Days. 

December  21 

December  21 

21  June 

21   June 

20 

22 

0.  3 

0.  3 

0.  A 

0.  4 

0.  4 

0.  5 

0.  5 

22 

20 

19 

23 

0.  6|0.  7 

0.  8 

0.  9 

0.  9 

o.io 

O.II 

23 

19 

18 

24 

O.IO 

0,11 

0.12 

o.i3 

o.i4 

o.i5 

0.16 

24 

18 

17 

25 

o.iii 

O.ID 

0.16 

0.18 

0.19 

0.20 

0.22 

25 

'7 

16 

26 

0.16 

0.18 

0.20 

0.22 

0.24I0.26 

0.27 

26 

16 

i5 

27 

0.20 

0.22 

0.24 

0.26 

0.29 

o.3i 

0.33 

27 

i5 

i4 

28 

0.23 

0.25 

0.28 

o.3i 

0.34 

0.36 

0.38 

28 

i4 

i3 

2Q 

0.26 

0.29 

0.32 

0.35 

0.38 

o.4i 

0.44 

29 

i3 

12 

3o 

o.3o 
0.33 

0.33 
0.37 

0.36 
0.40 

0.40 
0.44 

0.43 
0.48 

0.46 
o.5i 

o.5o 
0.55 

3o  June 

12 

II 

Decembe 

r  3i 

I   July 

II 

10 

January- 

I 

0.36 

0.40 

0.44 

0.48 

0.53 

0.57 

I.   I 

2 

10 

9 

2 

0.39 

0.44 

0.48 

0.53 

0.67 

I.  2 

I.  6 

3 

9 

8 

3 

0.43 

0.48 

0.53 

0.57 

I.  2 

I-  7 

I. II 

4 

8 

7 

4 

0.46 

o.5i 

0.56 

I.   1 

I-   7 

1. 12 

1. 17 

5 

7 

6 

5 

0.49 

0.55 

I.  0 

I.  6 

I. II 

1. 17 

1 .22 

6 

6 

5 

6 

0.52 

0.58 

I.  4 

1. 10 

1.16 

1 .22 

1.27 

7 

5 

4 

7 

0.55 

I.   I 

I.    7 

1. 14 

1 .20 

1 .26 

1.32 

8 

4 

3 

8 

0.58 

I.  5 

I. II 

1. 18 

1 .24 

i.3i 

1.37 

9 

3 

2 

9 

I.   I 

I.  4 

I.  8 
1. 12 

i.i5 
I.iq 

1.22 
1 .26 

i.29]i.36 
i.33|i.4i 

1.43 
1.48 

10 

2 

December    i 

10 

II 

I  June 

November  3o 

II 

I-   7 

i.i5 

1.23 

1 .3o 

1.37 

1.45 

1.52 

12 

3 1  May 

29 

12 

1. 10 

1. 18 

1.26 

1.34 

1.4^ 

i.5o 

1.57 

i3 

3o 

28 

i3 

i.i3 

1 .22 

i.3o 

1.38 

1.46 

1.54 

2.    2 

i4 

29 

27 

i4 

1. 16 

1.25 

1.34 

1 .42 

i.5o 

1.58 

2.    7 

i5 

28 

26 

i5 

1. 19 

1.28 

1.37 

1.46 

1.55 

2.  3 

2.12 

16 

27 

25 

16 

1.22 

i.3i 

i.4o 

1.49 

1.59 

2.  8 

2.17 

17 

26 

24 

17 

1.25 

1.35 

1.44 

1.53 

2.  d 

2.12 

2.21 

18 

25 

23 

18 

1.28 

1.38 

1.47 

1.57 

2.  7 

2.16 

2.26 

19 

24 

22 

19 

i.3o 
1.33 

i.4i 
1.44 

i.5i 
1.54 

2.   I 
2.  4 

2. II 

2.21 

2.3l 

2.35 

20 

23 

21 

20 

2.l5  2.25 

21 

22 

20 

21 

1.36 

1.47 

1.57 

2.  8 

2.192.29 

2.40 

22 

21 

19 

22 

1.39 

i.5o 

2.  0 

2. II 

2.22  2.33 

2.44 

23 

20 

18 

23 

1.41 

1.53 

2.  4 

2.l5 

2,26 

2.37 

2.48 

24 

19 

17 

24 

1.43 

1.55 

2.   7 

2.18 

2.3o 

2.41 

2.52 

25 

18 

16 

25 

1.46 

1.58 

2.10 

2.21 

2.33 

2.45 

2.56 

26 

17 

i5 

26 

1.48 

2,   I 

2.l3 

2.25 

2.37 

2.49 

3.   I 

27 

16 

i4 

27 

i.5i 

2.  4 

2.16 

2.28 

2.40 

2.52 

3.  5 

28 

l5 

i3 

28 

1.54 

2.  7 

2.19 

2.3l 

2.44 

2.56 

3.  9 

29 

i4 

II 

January 

3o 

1.58 
2.  3 

2. II 

2.17 

2.24 
2.3o 

2.37 
2.43 

2.5l 

2.57 

3.  4 
3. II 

3.17 
3.24 

3 1  July 

12 

9 

February 

I 

2  August 

10 

7 

3 

2.  7 

2.21 

2.35 

2.49 

3.  3 

3.17 

3.32 

4 

8 

5 

5 

2. II 

2.25 

2.40 

2.54 

3.  9 

3.23 

3.38 

6 

6 

3 

7 

2.14 

2.29 

2.44 

2.59 

3.i4 

3.29 

3.44 

8 

4 

November    i 

9 

2.18 

2.33 

2.49 

3.  4 

3.19 

3.35 

3.5o 

10 

2  May 

October      3o 

II 

2.22 

2.38 

2.53 

3.  9 

3.25 

3.4i 

3.56 

12 

3o  April 

28 

i3 

2.25 

2.41 

2.58 

3.14 

3.3o 

3.46 

4.  3 

i4 

28      ^ 

26 

i5 

2.29 

2.45 

3.  2 

3.18 

3.35 

3.5i 

4.  8 

16 

26 

24 

17 

2.32 

2.49 

3.  5 

3.22 

3.39 

3.56 

4.i3 

18 

24 

21 

20 

2.36 
2.40 

2.53 
2.58 

3. II 
3.i5 

3.s8 
3.33 

3.45 
3.5i 

4.  3 
4.  8 

4.20 
4.26 

21 

21 

18 

23 

24 

18 

i5 

February 

26 

2.43 

3.   I 

3.20 

3.38 

3.56 

4.14 

4.32 

27 

i5 

12 

March 

I 

2.46 

3.   5 

3.23 

3.42 

4.   I 

4.19 

4.38 

3o  August 

12 

9 

4 

2.49 

3.  8 

3.26 

3.45 

4.  4 

4.23 

4.4i 

2  SepL 

9 

6 

7 

2.5l 

3.10 

3.29 

3.48 

4.  7 

4.26 

4.45 

5 

6 

October        3 

10 

2.53 

3.i3 

3.32 

3.5i 

4.10 

4.215 

4.49 

8 

3  April 

September  3o 

i3 

2.55 

3.14 

3.33 

3.53 

4.i3 

4.32 

4.5i 

II 

3 1  March 

27 

16 

2.56; 

3.i5 

3.34 

3.54 

4.14 

4.33 

4.52 

i4 

28 

^^ 

19 

2.56' 

3.i5 

3.35 

3.55 

4.i5 

4.33 

4.52 

17 

25 

After               [Before 

2.56'3.i5 

3.35 

3.55 

4.i5 

^.34 

4.53  Before 

After 

Equinox.         .Equinox. 

1 

1 

Equinox. 

Kquinox. 

10 

Page  74] 

TABLE  V. 

For  reducing  the  Sun's  Declination,  as  given  in  the  Nautical  Almanac  for  Noon 

at  Greenwich,  to  any  other  Time  under  any  other  Merid 

ian. 

Add   afl.  N. 

Sub.  aft. 

N. 

H.M 

H.M 

H.M 

HM 

H.M 

H.M 

H.M 

Sub.  afl.  N. 

Add   aft.  N. 

Sub.  bef.  N. 

Add  bef.  N. 

5.  20 
80 

5.  40 
86 

6.    0 
90 

6.  20 
95~ 

6.  40 
100 

7.    0 
10.5 

7.  20 
110 

Add  bef.  N. 

Sub.  bef  N, 

Add  in  W. 

Sub.  in  W. 

Sub.  in  W. 

Add  in  W 

Sub.  in  E. 

Add  in  E. 

Deg^. 

mTs^ 

0.   0 

Deg. 
M.S. 

0.   0 

Deg-. 
M.S. 

0.   0 

Deg. 
M.S. 
0.   0 

Deg. 

M.S. 
0.   0 

Deg. 
M.S. 

0.   0 

Deg. 
M.S. 

0.  0 

Add  in  E. 

Sub.  in  E 

Days. 

Days. 

Days. 

Days. 

December  21 

Decembei 

21 

21   June 

21   June 

20 

22 

0.  5 

0.  6 

0.  6 

0.   7 

0.  8 

0.  8 

0.  8 

22 

20 

19 

23 

O.II 

0.12 

o.i3 

o.i4 

o.i5 

o.i5 

0.16 

23 

19 

18 

24 

0.17 

0.19 

0.20 

0.2] 

0.22  0.23 

0.24 

24 

18 

17 

25 

0.23 

0.25 

0.26 

0.28 

0.2910.31 

0.32 

25 

17 

16 

26 

0.29 

o.3i 

0.33 

0.35 

0.37 

0.38 

0.40 

26 

.6 

i5 

27 

0.35 

0.38 

0.40 

0.42 

0.44 

0.46 

0.49 

27 

.5 

i4 

28 

0.41 

0.43 

0.46 

0.49 

o.5i 

0.54 

0.57 

28 

14 

i3 

29 

0.47 

o.5o 

0.53 

0.56 

0.59 

I.  2 

I.  5 

29 

i3 

12 

3o 

0.53 
0.59 

0.56 
1 .   2 

0.59 
I.  6 

I.  3 
1 .10 

I.  6 
i.i3 

I.  9 

1. 17 

1 .12 
1 .21 

3o  June 

12 

II 

Decembei 

3i 

I   July 

II 

10 

January 

I 

I.  5 

I.  9 

i.i3 

1. 17 

1 .21 

1.25 

1 .29 

2 

10 

9 

2 

1 .11 

i.ib 

1. 19 

1 .24 

1.28 

1.32 

1.37 

3 

9 

8 

3 

I. lb 

1. 21 

1 .26 

I  .'i 

1. 3b 

1.40 

1.45 

4 

8 

7 

4 

1 .22 

1.27 

1.32 

1.3" 

1.42 

1.47 

1.53 

5 

7 

6 

5 

1.27 

1.33 

1.38 

1.44 

1.49 

1.54 

2.  0 

6 

6 

5 

6 

1.33 

1 .39 

1.45 

i.5i 

1.57 

2.     2 

2.  8 

7 

5 

4 

7 

1.39 

1.45 

i.5i 

1.57 

2.  3 

2.    9 

2.16 

8 

4 

3 

8 

1.44 

1 .5o 

I  bi 

2.  4 

2.10 

2.16 

2.23 

9 

3 

2 

9 

i.5o 
1.55 

1.56 
2.  2 

2.  3 
2.  9 

2.10 
2.16 

2.17 
2.23 

2.23 

2.3o 

2.3o 
1738 

10 

2 

December    i 

10 

II 

I  June 

November  So 

II 

2.  0 

2.  7 

2.l5 

2.22 

2.3o 

2.37 

2.45 

12 

3 1  May 

29 

12 

2.  b 

2.l3 

2.21 

2.29 

2.37 

2.44 

2.52 

i3 

3o 

28 

i3 

2.10 

2.19 

2.27 

2.35 

2.43 

2.5. 

3.  0 

i4 

29 

27 

i4 

2.16 

2.25 

2.33 

2.42 

2.5o 

2.58 

3.  7 

i5 

28 

26 

i5 

2.21 

2.3o 

2.38 

2.47 

2.56 

3.  5 

3.i3 

16 

27 

25 

16 

2.26 

2.35 

2.44 

2.53 

3.   2 

3. II 

3.21 

17 

26 

24 

17 

2.3l 

2.4o 

2.5o 

2.59 

3.  9 

3.18 

3.28 

18 

25 

23 

18 

2.36 

2.46 

2.55 

3.  5 

3.i5 

3.24 

3.34 

19 

24 

22 

19 

2.41 

2.46 

2.5l 

2.56 

3.   I 
3.  6 

3.11 

3.17 

3.21 
3.27 

3.3i 
3.37 

3.4\ 

3.48 

20 

23 

21 

20 

21 

22 

20 

21 

2.5o 

3.  2 

3.12 

3.23 

3.33 

3.44 

3.55 

22 

21 

19 

22 

2.55 

3.  6 

3.17 

3.28 

3.39 

3.5o 

4.   I 

23 

20 

18 

23 

3.  0 

3. II 

3.22 

3.33 

3.45 

3.56 

4.    7 

24 

19 

17 

24 

3.  4 

3.16 

3.27 

3.3q 

3.5o 

4.   I 

4.i3 

25 

18 

16 

25 

3.  8 

3.20 

3.32 

'•i-44 

3.56 

4.  7 

4.19 

26 

17 

i5 

26 

3.i3 

3.25 

3.37 

3.4q 

4.   I 

4.i3 

4.26 

27 

16 

i4 

27 

3.17 

3.29 

3.42 

3.54 

4.  6 

4.1Q 

4.3. 

28 

i5 

i3 

28 

3.22 

3.34 

3.47 

4.  0 

4.12 

4.25 

4.38 

29 

i4 

II 

January 

3o 

3.3o 
3.38 

3.43 
3.5i 

3.56 
4.  5 

4.  9 
4.18 

4.22 
4.32 

4.36 
4.46 

4.49 
4.59 

3 1  July 

12 

9 

February 

I 

2  August 

10 

7 

3 

3.46 

4.  0 

4.14 

4.28 

4.42 

4.56 

5. 10 

4 

8 

5 

5 

3.52 

4.  6 

4.21 

4.36 

4.5o 

5.  5 

5.19 

6 

6 

3 

7 

3.59 

4.14 

4.29 

4.44 

4.59 

5.i4 

5.29 

8 

4 

November    i 

9 

4.  5 

4-21 

4.36 

4.52 

i>.  7 

5.23 

5.38 

10 

2  May 

October       3o 

II 

4.12 

4.28 

4.44 

5.  0 

5.16 

5.3i 

5.47 

12 

3o  April 

28 

i3 

4.194.35 

4.5i 

5.  7 

5.23 

5.40 

5.56 

i4 

28 

26 

i5 

4.244.41 

4.57 

5.14 

5.3o 

5.47 

6.  3 

16 

26 

24 

17 

4.304.47 

5.  3 

5.21 

5.38 

5.55 

6.12 

18 

24 

21 

20 

4.37 
4.44 

4.55 
5.  2 

5.12 
5.19 

5.29 
5737 

5.47 
5.55 

6.  4 
6.i3 

6.21 
6.3i 

21 

21 

18 

23 

24 

18 

i5 

February 

26 

4.5o 

5.  8 

5.26 

5.44 

6.  2 

6.20 

6.38 

27 

i5 

12 

March 

I 

4.56 

5.i5 

5.33 

5.52 

6.10 

6.29 

6.47 

3o  August 

12 

9 

4 

5.  0 

5.19 

5.38 

5.57 

6.16 

6.34 

6.53 

2  Sept. 

9 

6 

7 

5.  4 

5.23 

5.42 

6.    I 

6.20 

6.3q 

6.58 

5 

6 

October         3 

10 

5.  8 

5.27 

5.46 

6.   5 

6.25 

6.44 

7.  3 

8 

3  April 

September  3o 

i3 

5. II 

5.3o 

5.4g 

6.  8 

6.28 

6.47 

7.  6 

II                    1 

3 1  March 

27 

16 

5.12 

5.3i 

5.5. 

6. II 

6.3i 

6.5o 

7.  9 

i4 

28 

24 

19 

5.12 

5.32 

5.52 

6.12 

6.32 

6.5i 

7.H 

17 

25 

After 

Before 

^  5.i3 

5.33 

5.53 

6.i3 

6.33 

S.52 

7. II 

Before 

After 

Equinox.           Equinox. 

Equinox. 

Equinox. 

TABLE   V. 

f Pago  75 

For  reducing  the  S 

an's  Declination,  as  given  in  the  Nautical  Almanac  for  Noon 

at  Greenwich,  to  any  o;her  Time  under  any  other  Merit 

ian. 

Add  aft.  N. 

Sub.  aft 

N. 

H.M 

H.31 

H.M 

H.M 

H.Rl 

H.M 

H.M 

Sub.  aft.  N. 

Add   aft.   N. 

Sub.  bef.  N. 

Add  bef.  N. 

7.  40 
115 

8.    0 
120 

3.  20 
125 

8.  40 
130 

9.    C 
135 

9.  20 
140 

9.  40 
145 

Add  bef.  N. 

Sub.  bef.  N. 

Add  in  W. 

Sub.  in  \V. 

Sub.  in  W. 

Add  in  W. 

Sub.  in  E. 

Add  in 

E. 

Deg 
M.S 

0.   0 

Deg. 
M.S. 

0.   0 

Deg 
M.S 

0.   0 

Deg. 
M.S. 
0.   0 

Deg- 
M.S 

0.   0 

Deg 
M.S 
0.   0 

Deg. 
31. S. 

0.   0 

Add  in  E. 

Sub.  in  E. 

Days. 

Days. 

Days. 

Days. 

December  21 

JDeceinber  21 

21  June 

21   June 

20 

22 

0.   9 

0.   9 

0.   9 

O.IO 

o.io|o.io 

o.io 

22 

20 

19 

23 

0.17 

o.ib 

0.18 

0.19 

0.19 

0.20 

0.21 

23 

19 

10 

24 

0.25 

0.26 

0.27 

0.28 

0.29 

o.3o 

o.3i 

24 

18 

17 

25 

0.34 

0.35 

0.36 

0.38 

0.39 

o.4i 

0.43 

25 

17 

16 

26 

0.42 

0.44 

0.46 

0.48 

0.49 

o.5i 

0.53 

26 

16 

ID 

27 

o.5i 

0.53 

0.55 

0.57 

0.59 

I .   I 

I.  3 

27 

i5 

i4 

28 

0.59 

I.   2 

I.  5 

I-   7 

I.  9 

1. 12 

i.i4 

28 

14 

i3 

29 

1 .  8 

I. II 

1. 14 

1. 17 

1. 19 

1 .22 

1.25 

29 

i3 

12 

3o 

1. 16 
1.24 

1. 19 
1.28 

1.23 

1.02 

1 .26 
1.35 

1 .29 
1 .39 

1.32 

1.43 

1.35 
1.46 

3o  June 

12 

II 

December  3i 

I   July 

1 1 

10 

January 

I 

1.33 

1.37 

1.41 

1.45 

1.49 

1. 53 

1.57 

2 

10 

9 

2 

1.42 

1. 4b 

i.5i 

1.55 

1.59I2.  3 

2.  7 

3 

9 

8 

3 

1.49 

1.54 

1.59 

2.  4 

2.  9 

2.l3 

2.18 

4 

8 

7 

4 

1.68 

2.  3 

2.  8 

2.l3 

2.19 

2.23 

2.28 

5 

7 

6 

5 

2.  5 

2. II 

2.16 

2.22 

2.28 

2.33 

2.39 

6 

6 

5 

6 

2.  i4 

2.20 

2.26 

2.32 

2.38 

2.43 

2.49 

7 

5 

4 

7 

2.22 

2. 28 

2.34 

2.4l 

2.47 

2.53 

2.59 

8 

4 

3 

8 

2.29 

2.36 

2.43 

2.49 

2.56 

3.  3 

3.  9 

9 

3 

2 

9 

2.37 
2.45 

2.44 

2.52 

2.5l 

2.59 

2.58 
3.  6 

3.  5 
3.14 

3.12 
3.21 

3.19 
3.28 

10 

2 

December     i 

10 

II 

I  June 

November  So 

TI 

2.52 

3.  0 

3.   7 

3.i5 

3.23 

3.3o 

3.38 

12 

3 1  May 

29 

12 

3.  0 

3.  8 

3.16 

3.24 

3.32 

3.39 

3.47 

i3 

3o        ^ 

28 

I  3 

3.  8 

3.16 

3.24 

3.32 

3.40 

3.49 

3.57 

i4 

29 

27 

i4 

3.i5 

3.24 

3.32 

3.4i 

3.49 

3.58 

4.  6 

i5 

23 

26 

t5 

3.22 

3.3i 

3.40 

3.49 

3.58 

4.  7 

4.16 

16 

27 

25 

16 

3.3o 

3.39 

3.48 

3.57 

4.  7 

4.16 

4.25 

17 

26 

24 

17 

3.37 

3.46 

3.56 

4.  6 

4.16I4.24 

4.'M 

18 

25 

23 

18 

3.44 

3.54 

4.  4 

4.14 

4.24 

4.33 

4.43 

19 

24 

22 

19 

3.5i 

3.58 

4.   I 
4.  8 

4. II 
4.19 

4.21 
4.29 

4.3i 
4.39 

4.41 
4.5o 

4.5i 
5.  0 

20 

23 

21 

20 

21 

22 

20 

21 

4.  5 

4.16 

4.27 

4.37 

4.484.59 

5.  9 

22 

21 

10 

22 

4.12 

4.23 

4.M 

4.45 

4.56 

5.  7 

5.18 

23 

20 

18 

23 

4.19 

4.3o 

4.41 

4.53 

5.  4 

5.i5 

5.26 

24 

19 

17 

24 

4.25 

4.36 

4.48 

5.  0 

5.12 

5.23 

5.34 

25 

18 

16 

25 

4.3i 

4.43 

4.55 

5.  7 

5.19 

5.3o 

5.42 

26 

17 

i5 

2fi 

4.38 

4.5o 

5.  2 

5.14 

5.26 

5.38 

5.5o 

27 

16 

i4 

27 

4.43 

4.56 

5.  8 

5.21 

5.33 

5.46 

5.58 

28 

l5 

i3 

28 

4.5o 

5.  3 

5.16 

5.28 

5.40 

5.54 

6.  6 

29 

i4 

1 1 

January 

3o 

5.  2 
5.i3 

5.i5 
5.27 

5.28 
5.4o 

5.41 
5.54 

5.54 
6.  8 

6.  8 
6.22 

6.21 
6.35 

3 1  July 

12 

9 

February 

I 

2  August 

10 

7 

3 

5.24 

5.38 

5.52 

6.   6 

6.20 

6.35 

6.49 

4 

8 

5 

5 

5.34 

5.49 

6.  4 

6.18 

6.33 

6.47 

7.  2 

6 

6 

3 

7 

5.44 

5.59 

6.14 

6.29 

6.44 

6.59 

7.14 

8 

4 

November    i 

9 

5.53 

6.  q 

6.24 

6.40 

6.55 

7. II 

7.26 

10 

2  May 

October      3o 

II 

6.  3 

6.18 

5.34 

6.5o 

7.  6 

7.21 

7.37 

12 

3o  April 

28 

i3 

6.12 

5.28 

(3.44 

7.  0 

7.16 

7.32 

7.48 

i4 

28 

26 

i5 

6.20 

5.36 

6.53 

7.10 

7.26 

7.42 

7.58 

16 

26 

24 

17 

6.29 

5.45 

7.  2 

7.19 

7.36 

7.52 

3.  9 

18 

24 

21 

20 

6.39 

6.4s 

3.56 
7.  6 

7.i3 
7-24 

7.3i 
7.42 

7.48 
8.  0 

8.  5 
8.17 

3.22 
3.34 

21 

21 

18 

23 

24 

18 

i5 

February 

26 

6.57 

7.i5 

7.34 

7.52 

8.10 

3.28 

3.46 

27 

i5 

12 

March 

I 

7-  6 

7.24 

7.42 

3.    I 

8.20 

3.38 

3.57 

3o  August 

12 

9 

4 

7-12 

7.3. 

7.5o 

3.   9 

8.28 

3.46 

7.  6 

2  Sept. 

9 

6 

7 

7-17 

7.36 

7.55 

^.14 

8.33 

3.53 

5.12 

5 

6 

October         3 

10 

7-237.42 

i.    I 

3.20 

8.39 

3.59 

,.i8 

8 

3  April 

September  3o 

i3 

7-267.45 

i.  4 

i.24 

8.43 

9.  3 

^.22 

II 

3 1  Maroh 

27 

16 

7.29,7.48. 

i.   7 

3.27 

8.47 

7.  6 

9.25 

i4 

28 

24 

19 

7.3o  7.5oi 

3.10 

3.29 

8.49 

9.  8 

7-27 

17 

25 

After 

Before 

7.3il7.5oi 

3.10 

3.3o 

8.5o 

?.    Q 

7.28 

Before 

After 

Equinox. 

Equinox. 

1         1 

V-, 

Equinox. 

Equinox. 

Page  76]                                                                   TABLE 

V. 

For  reducing  the  Sun's  Declination,  as  given  in  the  Nautical  Almanac 

foi  Noon 

at  Greenwich,  to  any  other  Time  under  any  other  Meridian. 

Add  aft.  N. 

Sub.  aft.  N. 

H.IM 

H.  M. 

H.  M. 

li.  M. 

H.  M. 

H.  M. 

H.  M. 

Sub.  aft.  N. 

Add  aft.  N. 

Sub.  bef.  N. 

Add  bef.  N. 

10  0 

10.  20 

10.  40 

11.    0 

11.  20 
170 

11.  40 

12.    0 

Add  bef  N. 

Sub.  bef  N. 

Add  in  W. 

Sub.  in  W. 

130 

155 

160 

165 

175 

ISO 

Sub.  in  W. 

Add  inTvT 

Sub.  in  E. 

Add  in  E. 

Dejx 

Deg. 

Deg. 

Deg-. 

Beg. 

Deg. 

Deg. 

Add  in  E. 
Days. 

Sub.  in  E. 
Days. 

Days. 

Daj's. 

M.S. 

M.   S. 

M.  S. 

M.  S. 

M.  S. 

M.  S. 

M.  S. 

Decemb.21 

Decemb.   21 

0.  0 

0.   0 

0.   0 

0.   0 

0.   0 

0.  0 

0.   0 

21  June 

21   June 

20 

22 

O.II 

O.ll 

0.12 

0.12 

0.12 

o.i3 

o.i3 

22 

20 

19 

23 

0.22 

0.23 

0.24 

0.24 

0.25 

0.26 

0.26 

23 

19 

18 

24 

0.33 

0.34 

0.35 

0.36 

0.37 

0.38 

0.39 

24 

18 

17 

25 

0.44 

0.46 

0.47 

0.48 

o.5o 

o.5i 

0.53 

25 

'7 

16 

26 

0.55 

0.57 

0.58 

I.  0 

I.   2 

I.  4 

I.  6 

26 

16 

i5 

27 

I.  6 

I.  8 

1 .11 

i.i3 

i.i5 

1. 17 

1. 19 

27 

i5 

i4 

28 

1. 17 

1 .20 

1.23 

1.25 

1.27 

i.3o 

1.32 

28 

i4 

i3 

29 

1.28 

i.3i 

1.34 

1.37 

i.4o 

1.43 

1.46 

29 

i5 

12 

3o 

1.39 

i.5o 

1 .42 

1.45 

1.49 

I  .52 

1.55 

1.59 

3o  June 

12 

II 

Decemb.  3i 

1.54 

1.57 

2.   I 

2.     5 

2.  8 

2.12 

I  July 

11 

10 

January      i 

2.   I 

2.  5 

2.    9 

2,l3 

2.17 

2.21 

2.25 

2 

10 

n 

2 

2.12 

2.16 

2.20 

2.25 

2.3o 

2.34 

2.38 

3 

9 

g 

3 

2.23 

2.27 

2.32 

2.37 

2.42 

2.47 

2.5l 

4 

8 

7 

4 

2.34 

2.39 

2.44 

2.49 

2.54 

2.59 

3.  4 

5 

7 

6'                   5 

2.44 

2.5o 

2.55 

3.  0 

3.  6 

3.12 

3.17 

6 

6 

5i                   6 

2.55 

3.   I 

3.  6 

3.12 

3.18 

3.24 

3.3o 

7 

5 

4 

7 

3.  5 

3. II 

3.17 

3.23 

3.29 

3.36 

3.42 

8 

4 

3 

8 

3.i5 

3.21 

3.28 

3.34 

3.4i 

3.48 

3.54 

9 

3 

2 

9 

3.25 
3.35 

3.32 

3.38 

3.45 
3.56 

3.52 

3.59 

4.  6 

10 

2 

Decemb.  i 

10 

3.42 

3.49 

4.  4 

4.11 

4.18 

II 

I  June 

Novemb.So 

11 

3.45 

3.52 

3.59 

4.  7 

4.i5 

4.22 

4.3o 

12 

3 1  Mav 

29 

12 

3.55 

4.  3 

4.10 

4.18 

4.26 

4.34 

4.42 

i3 

3o 

28 

i3 

4.  5 

4.i3 

4.21 

4.29 

4.38 

4.46 

4.54 

i4 

29 

27 

i4 

4.i5 

4.23 

4.3i 

4.40 

4.49 

4.57 

5.  5 

i5 

28 

26 

i5 

4.24 

4.33 

4.4i 

4.5o 

4.59 

5.  8 

5.17 

16 

27 

25 

16 

4.34 

4.43 

4.52 

5.   I 

5.10 

5.19 

5.28 

17 

26 

24 

17 

4.43 

4.53 

5.  2 

5. II 

5.21 

5.3o 

5.4o 

18 

25 

23 

18 

4.52 

5.  2 

5.12 

5.22 

5.32 

5.41 

5.5i 

19 

24 

22 

19 

5.  I 

5.12 

5.22 

5.32 

5.42 
5.53 

5.52 

6.  2 

20 

23 

21 

20 

5.10 

5.21 

5.3i 

5.42 

6.  3 

6.i3 

21 

22 

20 

21 

5.20 

5.3i 

5.41 

5.52 

6.  3 

6.14 

6.24 

22 

21 

19 

22 

5.29 

5.4o 

5.5i 

6.   2 

6.i3 

6.24 

6.34 

23 

20 

18 

23 

5.37 

5.49 

6.  0 

6. II 

6.23 

6.34 

6.44 

24 

19 

17 

24 

5.45 

5.5? 

6.  9 

6.20 

6.32 

6.43 

6.54 

25 

18 

/            \t 

25 

5.54 

6.  6 

6.17 

6.29 

6.4i 

6.53 

7.  4 

26 

17 

26 

6.  2 

6.i4 

6.26 

6.38 

6.5i 

7.  3 

7-i4 

27 

16 

i4 

27 

6.10 

6.22 

6.34 

6.47 

7.  0 

7.12 

7.24 

28 

i5 

i3 

28 

6.19 

6.3i 

6.43 

6.56 

7-  9 

7.22 

7-34 

29 

i4 

II 

January    3o 

6.34 

6.47 

7.  0 

7.i3 

7.26 

7-4o 

7.53 

3i  July 

12 

9 

February     i 

6.49 

7.  3 

7.16 

7.3o 

7-43 

7.57 

8. II 

2  August 

10 

7 

3 

7.  3 

7-17 

7.3. 

7.45 

7.59 

8.i3 

8.28 

4 

8 

5 

5 

7.16 

7.3i 

7.45 

8.  0 

8.14 

8.28 

8.43 

6 

6 

3 

7 

7.29 

7-44 

7.59 

8.i4 

8.28 

8.43 

8.58 

8 

4 

Novemb.  i 

9 

7-4i 

".56 

8.12 

8.27 

8.42 

8.58 

9.13 

10 

2  May 

October  3o 

11 

7.53 

8.  8 

8.24 

8.40 

8.56 

9. 12 

9.28 

12 

3o  April 

28 

i3 

8.  4 

8.20 

8.36 

8.53 

9.  9 

9.23 

9.42 

i4 

28 

26 

i5 

8.i5 

8.32 

8.48 

9.  5 

9.21 

9-38 

9-54 

16 

26 

24 

17 

8.26 

8.43 

9.  0 

9.17 

9-34 

9.5o 

10.  7 

18 

24 

21 

■    20 

8.40 

8.57 

9.14 

9.32 

9.49 

10.  6 

10.24 

21 

21 

18 

23 

8.52 

9.10 

9.28 

9.46 

10.  3 

10.21 

10.39 

24 

18 

i5 

February  26 

9-  4 

9.22 

9.40 

9.58 

10.16 

10.34 

10.53 

27 

i5 

12 

March         i 

9.15 

9.33 

9.5i 

10.10 

10.29 

10.47 

II.  6 

3o  August 

12 

9 

4 

9.24 

9.43 

10.   I 

10.20 

10.39 

10.58 

1 1 .16 

2  Sept. 

9 

6 

7 

9.30 

9.50 

ro.   9 

10.28 

10.47 

II.  6 

11.24 

5 

6 

October     3 

10 

9.37 

9.56 

10.16 

10.35 

10.54 

11. i3 

II  .32 

8 

3  April 

Septem.  3o 

i3 

9.41 

10.  0 

10.21 

10.40 

10.59 

11.18 

11.38 

II 

3 1  March 

27 

16 

9-45 

10.  4 

10.24 

10.44 

II .  3 

11.22 

II  .42 

i4 

28 

24 

19 

9-47 

10.  6 

[O.26 

10.46 

II.  5 

11.24 

11.44 

17 

25 

After 

Before 

9-48 

10.  7 

10.27 

10.47 

II.  6 

11.25 

11.45 

Before 

After 

Equinox. 

Equinox. 

Equinox. 

Equinox. 

TABLE  VI.                                             [P=^ge77 

Sun's  Right  Ascension. 

A 

JAN. 

FEB. 

.MAU. 

APR. 

.AIAY. 

JUNE. 

JULY 

AUG. 

SEPT. 

OCT. 

NOV. 

DEC. 

Q 

1 

A.  in. 

h.  m. 

h.  in. 

h.  m. 

h.  TO. 

k.  m. 

h.  m. 

h.  TO. 

h.  TO. 

h.  TO. 

II.   TO. 

h.  m. 

i8.46 

20. 58 

22.48 

0.42 

2.33 

4.36 

6.40 

8.45 

10.41 

12.29 

14.25 

16.29 

1 

2 

i8.5o 

21 .   2 

22.52 

0.46 

2.37 

4.40 

6.44 

8.49 

10.45 

12.33 

14.29 

16.33 

2 

3 

1 8. 55 

21.6 

22.56 

0.49 

2.41 

4.44 

6.48 

8.53 

10.48 

12.36 

14.33 

16. 38 

3 

4 

10.59 

21 .  10 

23.00 

0.53 

2.45 

4.48 

6.53 

8.57 

10.52 

12.40 

14.37 

16.42 

4 

5 

6 

19.  4 

21  .14 

23.  3 

0.57 

2.48 

4.52 

6.57 

9.  0 

10.56 

12.44 

14. 4i 

16.47 

5 
6 

19.   8 

21.19 

23.  7 

1 .  0 

2.52 

4.56 

7-   I 

9.  4 

10.59 

12.47 

14.45 

16. 5i 

7 

19. 12 

21  .23 

23.  11 

I.  4 

2.56 

5.  I 

7.   3 

9.  8 

II.  3 

12. 5i 

14.49 

16.55 

7 

8 

19.17 

21.27 

23.14 

I-   7 

3.  0 

5.  5 

7-  9 

9.12 

11.  6 

12.55 

14.53 

17.  0 

8 

iJ 

19.21 

21  .3o 

23.18 

I .  II 

3.  4 

5.  9 

7.i3 

9.16 

II  .10 

12.58 

14.57 

17-  4 

9 

10 
11 

19.25 

21  .34 

23.22 

i.i5 

3.  8 

5.i3 

7.17 

9.20 

11.14 

i3.  2 

i5.   1 

17.  8 

10 
11 

19.30 

21.38 

23.25 

1.18 

3.12 

5.17 

7.21 

9.23 

11.17 

i3.  6 

i5.  5 

17.13 

12 

19.34 

21.42 

23.29 

1 .22 

3.16 

5.21 

7.25 

9.27 

11.21 

i3.  9 

i5.  9 

17.17 

12 

13 

19.38 

21.46 

23.33 

1 .26 

3.20 

6.25 

7.29 

9.3i 

II  .24 

i3.i3 

i5.i3 

17.22 

13 

14 

19.43 

21 .5o 

23.36 

1 .3o 

3.24 

5.29 

7.33 

9.35 

11.28 

i3. 17 

15.17 

17.26 

14 

15 
16 

19.47 

21.54 

23.40 

1.33 

3.27 

5.34 

7.37 

9.38 

11.32 

1 3 . 2 1 

l5.22 

17. 3i 

17.35 

15 
16 

19.51 

21.58 

23.44 

1.37 

3.3i 

5.38 

7.41 

9.42 

11.35 

i3.24 

15.26 

17 

19.56 

22.  2 

23.47 

i.4i 

3.35 

5.42 

7-46 

9-46 

11 .39 

13.28 

i5.3o 

17.39 

17 

18 

20.  0 

22.  6 

23. 5i 

1.44 

3.39 

5.46 

7.5o 

9.50 

11.42 

i3.32 

i5. 34 

17-44 

18 

19 

20.  4 

22.10 

23.55 

1.48 

3.43 

5.5o 

7.54 

9.53 

11.46 

i3.35 

i5.38 

17-48 

19 

20 
21 

20.  8 

22. i3 

23.58 

1.52 

3.47 

5.54 

7.58 

9.57 

II. 5o 

13.39 

i5.42 

17-53 

20 

20. i3 

22.17 

0.   2 

1.55 

3.5i 

5.59 

8.   2 

10.  I 

11.53 

13.43 

15.47 

17.57 

21 

22 

20.17 

22.21 

0.  6 

1.59 

3.55 

6.  3 

8.  6 

10.  4 

11.57 

13.47 

i5.5i 

18.    2 

22 

23 

20.21 

22.25 

0.  9 

2.  3 

3.59 

6.  7 

8.10 

10.  8 

12.  0 

i3.5i 

i5.55 

18.  6 

23 

24 

20.25 

22.29 

o.i3 

2.  7 

4.  3 

6.11 

8.14 

10.12 

12.  4 

i3.54 

1 5. 59 

18.10 

24 

25 
26 

20.29 

22.32 

0.17 

2. 10 

4.  7 

6.i5 

8.18 

10.16 

12.  7 

i3.58 

16.  3 

18. i5 

25 
~26" 

20.34 

22.36 

0.20 

2.14 

4. II 

6. 19 

8.21 

10.19 

12. II 

i4.  2 

16.  8 

18.19 

27 

20.38 

22.40 

0,24 

2.18 

4.i5 

6.24 

8.25 

10.23 

12. i5 

i4.  6 

16.12 

18.24 

27 

28 

20.42 

22.44 

■0.27 

2.22 

4.20 

6.28 

8.29 

10.27 

12.18 

14.10 

16.16 

18.28 

28 

29 

20.46 

22.46 

0.3! 

2.26 

4.24 

6.32 

8.33 

io.3o 

12.22 

14. i4 

16.21 

18.33 

29 

30 
31 

20.  5o 

0.35 

2.29 

4.28 

6.36 

8.37 

10.34 

12.26 

14. 18 

16.25 

18.37 

30 
31 

20.54 

0.38 

4.32 

8.41 

10.37 

l4.2I 

18.4 

H 

This  Table  gives  nearly  the  Sun's  Right  Ascension  Co 

r  the  ^-ears  18.33,  1834,  1835,  and  1836,  and  is 

sufficiently  exact  for  finding  when  any  Star  comes  to  th 

e  meridian.      But  in  all  calculations  for  deter- 

niinlna;-  the  longitude  by  celestial  observations,  the  Sun's 

Right  Ascension  must  be  taken  from  the  Nauti- 

cal  Almanac,  where  it  is  calculated  to  a  greater  degree  o; 

accuracy. 

Table  VI. 

A. 

Correction  for  the  daily  variation  of  the  Equati 

on  of  Time  found  in  Table  IV.  A. 

Find  the  daily  variation  of  Equation  of  Time  at  the 

top,  the  hour  at  Greenwich  at  the  side. 

3 

n 

// 

//       1 

/     // 

II 

;/ 

//    ; 

II   1 

1   II 

// 

// 

// 

'/ 

// 

II 

II   1 

II 

II 

II 

II 

// 

// 

II 

// 

// 

II 

ti 

0 

1 

0 

3     A 

I   5 

G 

7 

8   £ 

101 

112 

13 

14 

15 

16 

17 

18 

192 

121 

22 

23 

24 

25 

26 

27 

28 

29 

30 

Q 

1 

o 

0 

0     c 

)   0 

0 

0 

0    c 

0 

0    1 

I 

I 

I 

1 

1 

I 

I 

I    I 

I 

I 

1 

I 

1 

1 

I 

I 

I 

15 

2 

0 

0 

0    c 

)   0 

1 

I 

I     1 

1 

I    1 

1 

I 

I 

1 

I 

2 

2 

2    2 

2 

2 

2 

2 

2 

2 

2 

2 

3 

30 

3 

0 

0 

0     1 

1 

I 

1 

1    I 

I 

1    2 

2 

2 

2 

9 

2 

2 

2 

3    3 

3 

3 

3 

3 

3 

3 

4 

4 

4 

45 

4 

o 

0 

I     1 

1 

1 

I 

I    2 

2 

2    2 

2 

2 

3 

3 

3 

3 

3 

3   4 

4 

4 

4 

4 

4 

5 

5 

5 

5 

60 

5 

0 

0 

1     1 

1 

I 

I 

2    2 

2 

9    3 

3 

3 

3 

3 

4 

4 

4   < 

i   4 

5 

5 

5 

5 

5 

6 

6 

6 

6 

75 

6 

0 

I 

1 

2 

2 

2    2 

3 

3    3 

; 

4 

4 

4 

4 

5 

5 

5   5 

6 

6 

6 

6 

7 

7 

7 

7 

8 

90 

7 

o 

I 

1 

2 

2 

2    3 

3 

3   4 

4 

4 

4 

5 

5 

5 

6 

3   6 

6 

7 

7 

8 

8 

8 

8 

9 

105 

8 

o 

I 

2 

2 

2 

3    3 

3 

4   4 

4 

5 

5 

5 

6 

6 

6 

7    7 

7 

8 

8 

8 

9 

9 

9 

10 

10 

120 

CI 

0 

2 

2 

3 

3   3 

4 

4    5 

5 

5 

6 

6 

6 

7 

7 

3    8 

8 

g 

Q 

Q 

10 

10 

1 1 

1 1 

II 

135 

10 

o 

I     : 

2 

3 

3 

3   A 

4 

5    5 

5 

6 

6 

7 

7 

8 

8 

3    Q 

9 

10 

10 

10 

10 

1 1 

1 1 

12 

12 

i3 

150 

11 

o 

2 

3 

3 

4   A 

5 

5   6 

6 

6 

7 

7 

8 

8 

Q 

?  10 

11 

II 

1 1 

12 

12 

i3 

i3 

i4 

1G5 

12 

2     : 

3 

3 

4 

4   5 

5 

6   6 

7 

7 

8 

8 

9 

9 

10  I 

J  1 1 

II 

12 

12 

i3 

i3 

i4 

i4 

i5 

i5 

180 

13 

2     : 

3 

3 

4 

4   E 

5 

6    7 

7 

8 

8 

9 

9 

10 

10  I 

I  1 1 

12 

12 

i3 

i4 

i4 

i5 

i5 

16 

16 

195 

14 

2      2 

3 

4 

4 

5   5 

6 

6    -7 

8 

8 

Q 

9 

10 

II 

II  I 

2  12 

i3 

r3 

i4 

i5 

i5 

16 

16 

17 

18 

210 

15 

2 

3 

4 

4 

5  e 

6 

7    8 

8 

Q 

9 

10 

II 

II 

12  I 

3i3 

i4 

i4 

i5 

16 

16 

17 

18 

18 

19 

225 

u; 

2     ^ 

3 

4 

5 

5  e 

7 

7   8 

9 

9 

10 

II 

1 1 

12 

i3  I 

3i4 

i5 

i5 

16 

17 

17 

18 

19 

■9 

20 

240 

17 

2     I 

4 

4 

5 

6   C 

7 

8   p 

9 

10 

11 

1 1 

12 

i3 

i3  I 

^i5 

16 

16 

17 

18 

18 

19 

20 

21 

21 

255 

18 

2 

1     i 

4 

5 

5 

6   7 

8 

8    9 

10 

11 

II 

12 

i3 

i4 

i4i 

5  16 

17 

17 

18 

19 

20 

20 

21 

22 

23 

270 

19 

2 

2     .: 

4 

5 

6 

6   7 

8 

9  10 

10 

II 

12 

i3 

i3 

i4 

i5  I 

3  17 

17 

18 

>9 

20 

21 

21 

22 

23 

24 

285 

20 

2 

3     I 

4 

5 

6 

7    8 

8 

9  10 

II 

12 

i3 

i3 

i4 

i5 

161 

718 

18 

•9 

20 

21 

22 

23 

23 

24 

25 

300 

21 

2 

3     ^ 

4 

5 

6 

7   8 

9  I 

0  11 

11 

12 

i3 

i4 

i5 

16 

171 

3ia 

19 

20 

21 

^■^ 

23 

24 

25 

25 

26 

315 

22 

2 

3    I 

5 

6 

6 

7    8 

91 

0  II 

12 

i3 

14 

i5 

16 

17 

17  1 

3lQ 

20 

21 

22 

23 

24 

25 

26 

27 

28 

330 

23 

2 

3    A 

5 

6 

7 

8    9 

10  I 

I  12 

12 

i3 

i4 

i5 

16 

17 

18  I 

5!2o 

21 

22 

23 

24 

25 

26 

27 

28 

29 

345 

24 

2 

3    I 

5 

6 

7 

8    9 

io|i 

I  12 

i3 

i4 

i5 

16 

17 

18 

19  2 

^21 

22 

23 

24 

25 

26 

27 

28 

29 

3o 

360 

1 

Page 

78J                                         TABLE   VII. 

Amplitudes. 

O 

< 
1— 1 
O 

Q 

Lat. 

0 
«     «  CO  NJin  !£) 

r~-co   On  O   "  CS 

no  NsTiD  O   t--00 

OnO   11   CM  no  N^ 

ID  o   r~oo  On  0 

CM     CM     CN     m     CM  CO 

^   a  CO 

o 

00 
CO 

Q 

loo    O  O    «  N5-^o 
CN    CN  ro  ro  CO  ro 

Onoo    t~-  -'  O    -I 

r^Nq-   ►,  00  o  in 

«    CM    CM  no  NJ- 

N^NJu-,  X)  OO     " 

ID          «    CM  no  ID 

NTOOnooOinnoi-    O    n 
-no  -^        o  |nt        CN 

ro  oo  ro  no  ro  CO 

CM     CN     CT     C-)     «     Ol 

no  no  no  no  no  Nq- 

m    ni    «    c^    m   n 

^^•^N^^N^ 

Ng-ID  ID  ID  ID  ID 

to  to  to  to   l>  t^ 

\^'?.i 

o 

^ 

O     "     CN  xq-^O  CO 

►-.  -^00  no    ooo 
M    «    w    oi    o  no 

CO  ID    CM    On  t^iD 
no  Nq-LD  ID          >-■ 

ID  NTin  ID    Cn  On 
M  no  NTID           n 

1  ni  to    1-  to    CM    On 
CO  N<T          -  no  NT 

r^to  lo 

OiNq- 

fo  (D  no  no  CO  ro 
M   oi   n)   cN    n   CN 

no  no  no  no  no  CO 
ni    n)    CT    n    o)    « 

no  no  no  no  NsT  Ng. 

NT  NT  NT  NT- ID  in 
ni    CM   rt    CM   CM    m 

in  in  to  to  O  to 

t~~    tN.    tN 

o 

^ 

1  O     >->    CN  OO  LO  CO 

O  ^vT  r^  -  O    - 
M   «   «   ni    o  no 

1    r^CO     On^D  NT    CN 

CO    Nq-N^ID                  l-l 

0     O     On  O     -  CO 

cN  CO  CO  in         " 

in  CO   CM  to  "CO 
CM  no  in         CM  no 

liD  no   ni 
ID   1  no 

C-iCSCSCSWOilCSDnMtNCI 
<Nni«(MCN0l|cN01tNCN«01 

ni   CM   CM   CM  en  no 

no  CO  no  no  Nq- NT 

CM     CM     CM     CM     CM     CM 

NT  NT  NT  ID  in  in 

CM     CM     CM     ni     ni     CM 

ID  <o  to 

o 

Q 

O  i-t  CT  ro  lO  t^ 

O  no  O   cm   O 
«   1-1   «   CN   ci  no 

ID      O       r^nO      •xOOIOlDVg^Nq.i^y-) 

noNq-Nq-uo              |-iCMnoN<Tin 

CO   ono   t^-   r-coooo 
-  no  NTin   «   CM    NT        -, 

CM   CM    CM   CM    m  no 

CM     CM     CM     CM     CM     CN 

nonononoNTNT   Nq-i^in 
CTmmcMcMCMicNnini 

Ol     CS     «     «     CN     CS 

m    m    ni    n)   n)    ni 

CM     CM     m     CM     CM     CM 

0 

o 

s 

O   "  CN  no  vo   t-- 

On  CM  yD    Onco  CO 

|no  CO  N<3-  ►-    r--in 
CO  no  ^in  in 

CM    "    OnOnOnOnIO    niNT-r-«tO|i    l-^NT  1 
-icMnicoNTin|«cMnoNT        -.|cont       | 

--^    -.    ^    --,    ^    „ 

O    O    O    O    O    O 
m    ni   CN    cN    m    CN 

^  ^    ~    ^    ~     * 

ni   ni   o   mnonolcocoNT 
IcMcMcMcMnimlmmm 

Q  1  «  S  S  ?J  «  S 

n)   CM    CM    ni   CM   « 

o 

1—1 

O   "   o)  no  uo  r^ 

On  ni  ID  CO   m  ^o 
f,   „   „   ci    a 

1    1  O     CM  CO  XT    - 

no  no  >i  Nq-iD 

CO  to  ID  no  no  no 
M   «  no  NTLD 

no  NT  to  CO   1  in     ONin  ►. 
►1   CM  CO  in          -  no  in 

O  On  O  Om>  O 

On  On  On  On  On  O 

OnCnOnO-OnD 

o'  o'  d  d  <D  o' 

"►i»i-iini     niCMn) 

o 

00 

I— 1 

O    -<   cs  no  NT(0 

00    -1  ^sT   l^  "  ID 
M     1-1     ►-     M     CS 

On  NT    OniD    "  CO 

CM  no  no  Nq- in  in 

ID   CM   C  CO  T^to 

M     Oi     «  CO  NT 

to    r^co    On  «  Nq. 

ID             n     CM  NJID 

00   ni    t-- 

CM  CO 

CO  CO  CO  00  CO  CO 

00  CO  CO  OO  00  00 

CO  CO  00  00  CO  CO 

On  On  On  On  On  On 

On  O   O   O   O   O 

1     «     - 

o 

1—1 

S 

r^ 

O   "   "  no  Njo 

00    OnoO    Oxrlco    CN    r^cNCON^Ii-iCOiDno    "    O 
MnicsooinonoN^-^un                 ^mnoNj- 

IOnOnOnOcmnt      r^QNT 
NT-in             CMCONTlDnCM 

r-  f^  r-  r^  r^  i> 

t^r^t^r^r^r^lr^t^r^r-r~-r^  cooooocococo 

ODCO    OnOnOnOn     OnO    0 

0 

CO 

I— I 

O  f-i  -1  n  ^m 

r^  O   ni  ID  CO   m 

O  c  in  o  ID  NM 

CM  CO  CO  NsT  NT  ID 

f^no    O  CO.  ID  NX 
ID           w    w    CM  no 

nicMnMCTNTiincO'- 
N<TiD         "   CM  no    xrm   >i 

O  O  O  O  O  O 

O  O  O  O  O  O 

O  O  O  ^  O  'O 

to  r^  i>  t^  1^  i> 

t^  r~-00  OO  CO  CO     CO  CO    On 

o 

I— 1 

o   M   M  (N  Nyj-uo 

r^  On  «  Ng.    (^   „ 

NT  CO  CO  r~-  CM  r^  1  no  onvo  no  o  r- 
cMCMnonoNTNTiniD         mcmcm 

•O"^noconono  IXTto    On 
no  NTin         11   CM    no  NTin 

lT)  ID  tn  uo  uo  m 

urj  ir>  iT^  i^  ^D  ^^ 

IDLDlDiniDlD     IDUOO^OOO 

tototo  r~-t^r-ir^i^i> 

0 

1—1 

S 

n 

O  1-1  1-1  cs  CO  in 

(O  00   "1  no  to   On 

no  (O   O  ID   On-^ 
CM   o  CO  no  no  NT 

CNID    M    r^Nq.  „ 
NTlD                 n    ni 

ONr^lDNTCOCO    |ntnt^ 

ninoNTiD         wniCONT 

xr  Njj- Nq- v^  N^  vq- 

■^•^^N^a-^N^ 

NT  ^  NT  Nq- -^  Nq. 

N^NTiD  ID  ID  ID 

IDIDIDID(0^    ItOtOtO 

o 

CO    S 

00"«ro'<3-!r)030cNinoO 

mntcom^q^    Onions    Onid 
omcMnonoNg-^iDiD              n 

CM    C    r^to  NTCO   |nO  no  NT 
mnonoNTiD           -ic^co 

mocnnorono    noronononono 

nononoconono    nononoNjjNq-vq. 

NT  Nq- NT  NT  NT  ID    in  in  in 

o 

r-i 

O   O   "   Ci  no  NT 

ID  r^  On«  N^yD 

On  CN  O    Onco  CO 
«    CM    CM    o  no  CO 

CM    i^  CM    t^no   ON|tO   ni   o    t^iD  no  1  M    n   w 
NTNTiDiD                 "    niConoNjiD           «   o 

«   d   cs   n)   cs   cs 

oi   M   «   n)   ni   CM 

CM     n)     CM     CM     ni     CM 

Oi    ni   CM   CMCono    conocononono    NTNq-Nq- 

o 

I— I 
1—1 

S 
Q 

O   O   -   c\  fv)  Nq- 

>D  r^oo  O  CO  in 

00   O  NT  r-  «  Ng- 
M   m   CM   m  CO  no 

ONno  00  no  00  CO 
no  NT  NT  ID  ID 

ONin  «   cnO  NT 
«    CM    CM  CO  Nq- 

ni    O    On 
ID 

CM     CM     m     CM     CM     « 

«  no  no 

o 

o 

O   O   I-"   I-'   «  no 

in  O  CO    On  -.  "^ 

(O    On  -1  NT  CO    - 

1   -1   ni   CM   CM  no 

uo   ONno  CO   CM    I-- 

CO  no  ^T' NT  ID  ID 

no  00  NT  «    r^N:^ 
.-    CM    CM  no 

1-    On  r- 
NqNTiD 

O   O   O   O   O   O 

o*  o*  o  o  d  o' 

O   O   O   O   O   0 

o  d  o  o  d  d 

-   —  1-   ».  1_   11 

►-■..- 

o 

O   O   «   "   CN  no 

•^LD    C^-CO    O    O) 

-T  [^  Cn  m  in  CO 

M     «     «     ni     CM     CM 

-  in   Onco    t^  cm 

CO  no  CO  NT  «^  ID 

to   -   r^  ni  00  NT 

ID                     -1     n     M 

I  00  ID 

CO  CO  Nq- 

On  0>  CN  Ov  O  O 

On  On  On  On  On  On 

On  On  On  On  On  On 

On  On  On  On  On  On 

OnC  o  c  o  o 

O   O   O 

0 

00 

S 

O   O   «   w   CN  no 

N^riD  <o    l>  On  ~ 

no  in    r^  On  CM  ID 

00    -  NTCO    «  to 
m  no  CO  no  NT  XT 

O  NT    On  NT   Onid 
ID  ID  ID                   — 

-   r~no 
CM    o  no 

Q 

CO  00  00  00  00  00 

00  00  00  00  00  CO 

CO  CO  CO  00  OO  CO 

00  CO  00  CO  CO  CO 

CO  CO  OO    On  On  On 

On  On  On 

o 

t^ 

S 

O  O  "  "  cs   o 

no  Nq-iD    r-oo    On 

■-1  CO  ID    r^  On  CM 
1     •-•«•-     1-1     CM 

NT  t—  O  no  to    O 
CM   CM  no  no  CO  NT 

NTtO    CM  to    "  in 
NT  NT  ID  in 

OtO    1 

n     -     CM 

Q 

r--  r^  t^  r^  t^  1^ 

t^  t^  t^  r--  t^  r- 

r^  r-~  r~-  t--  t^  t-- 

r^  f^  r^  r^  1^  r^ 

r^  r^  r--  r^co  oo 

00  00  00 

o 

CO 

S 

0    O    O    '-'    >-   « 

no  NsjiD  o   r-co 

O    "  no  ID    C^  On 

-   C1-)   O   CO      1    Nq- 
CM     CM     CM     CM  CO  no 

r--  -  xrco  CM  to 
CO  NT Nq- NTin  in 

0  ID    O 

Q 

^O  vo  ^  O  O  O 

o  o  o  o  o  o 

o  to  o  o  o  o 

to  ^o  to  to  o  to 

to  to  to  ^o  o  o 

r-  r^  r^ 

o 

lO 

S 

0   O   O   "   >-   CN 

ct  no  N^iD  o   r- 

00    On  —    CM  NT  ID 

t^  On  -  NTtO  00 

«"     CM     CM     CM     CM 

-  NT  r-~  c  no   r^ 

no  no  no  NT  Nq- NT 

O  NTOO 
ID  ID  ID 

Q 

ID  LD  to  CO  m  in 

in  in  ID  in  in  ID 

ID  ID  ID  ID  ID  ID 

ID  ID  ID  ID  ID  ID 

ID  ID  in  ID  ID  ID 

ID  ID  in 

o 

'^ 

^ 

O   O   O   "   "   " 

CM   cs  fo  NsTNq-iD 

O   oco  O   11   m 

NfiD    O  On  1  no 

„   «   M   «   m   CM 

ID  r^ONCMNTt^   ocoto 

CM    m   n:  no  no  CO    ntntn<t 

a 

^  ^  N^   Ni^  ^l-  ^q- 

~^^N^N^N3-^ 

NT  NT  NT  NT  NT  NT 

NT  NT  NT  NT  NT  NT 

Nq-NrNq-Nq-N-NT    ntntnj- 

o 

CO 

* 

O    O    O    O    "    " 

«   oi    nt  no  no  XT 

ID  O  to    C^OO    On 

O    «  no  NTtO    [^ 

'O  CO  no 

Q 

ro  n->  no  no  no  no 

no  no  no  no  no  no 

no  CO  no  no  CO  CO 

"O  CO  no  CO  no  CO 

CO  CO  no  no  no  no 

no  CO  CO 

o 

(M 

S 

O   0   O   O   O   - 

«     «     «    d     CM  DO 

V)  NT -T  ID  ID  <0 

t--co   On  OnO   « 

CM   NT  ID  <0     I"-  On 

CN   ni   ni 

Q 

m   (N   «   cs   «   « 

CT    CM    CM    «    n)    m 

CM    CM    CM    ng    CM    CM 

ni   n<   ni    CM    CM   o 

CM     m     CM     CM     CM     n 

CM     CM     CM 

o 

1—1 

S 

O   O   O   O   O   O 

O   I   1   1   1   -1 

m    CM   CM   CM  no  no 

-O  NTNq-iD  ID  O  1 

o   r-  r--co  On  On 

C    "1   ni 

L 

it. 

O 
~  CM  no  ^^n  o 

r-OO   On  O   "   m 

•ONTiDtO   t^cOONQ    1-c   ntnoN^ 

D  to    t~-CO    On  O      11    CM  no 

1 

.    CM   dcn.no  |no  CO 

! 

TABLE   VII 

Amplitudes. 


[Page  7;i 


Lat. 


1^00   o  o   •- 


S5 


iro^oo^ro  \cr>  en  m  cr>  CO  en  ^  en  en  en  en  ^<t  "^ 


^ 


1-H       "^ 


uo  m  o   O  1--  [^  t^co  OO 


k4 

o 

e=; 

(^f 

Q 

* 

o 

* 

>-H 

d 

~<T  ^  in  m  in  to  O  O  r-^ 


°  =:^ 


"^  ^  "T  in  in  in 


O  O  r^  t^co  O 


°  ;ii 


°  s 


I^D   t~^  r-co  CN  o 


r~co  o  "  f^  >n 


°  ;=; 


o   O   o   >-   «   « 


-I  O^  r^ro  CO  ro   O 
00  in  o  in  "^^ 


-q-in  in  !D  r^  r-- 


°  ^ 


OO    O  CTv  O  O    O 


°    S 


^CC  CO  CO    O 


ooo  o  o  « 


o  '-^ 


Q 


ooo  r^  r^  t-^ 


I  ^O    CN 

I  in   -  ro 

I  00   Ov  o 


■  I  Ooo  cs   cN   o  in 

■  jfo  ro  x:  in  «  ro 


OOO""" 


°  ^ 


^^•^xrininminino 


fo  ^  "nT  ^  ^  '<g- 


O  c>j   t^"^oo  m 


o  o  r^  t-~-  r--oo 


in  in  in  in  o  O 


CO  00   O  On  O    C 


^  o  om  o  r- 
in  -<  ro        o  >n 

o  r^  t^co  CO  OO 


°  ^ 


o    -= 

Q 


cN  m  n  ro 


rooo-q-Njxrin  lu-iinooo 


I  ro  o  —  I  r~in  o  < 


r^oo  OS  o  o  o 


irororo    ro^^q-N^inin 


io  in  CM 
in  «  Ng- 
in  o  O 


r^  r~co 


CO   O  O    O   "   « 


o  o  o  o  o  o 


cs   cs   cs  ro  ro  ro 


N^  -^t  ^  m  in  o 


SOOO   OOOO    OnOOO   O   O 


mom 
t^  r^  r^ 


r--  r^  t^  r^  r-oc   co  co  oo  co  oo  oo 


O    O   "    I-    ■-    " 


OnOOnQnOO     OO'-'^'-'CN 


O    t^  r~-CO    On  o 

^     i-i     X     r-     -.     tS 

Vr  ro  xr  CO  o  o 
m       ro        Njj  „ 

N^rm  m  o  o  t^ 

r--  CN     OnCO     Onn^ 

cs  in  I-  Nj  I-.  m 


O  O  O  O  o  O 


oot--t-^r-r^   r-~t^  r^co  cooo  coco  onOnQno 


O   O    "   >-   - 


in  in  in  in  in  uo   m  m  m  m  in  m  '  o  O  O  O  O  O   O  r^r^r^r-oo'cocooo  CnqsOn 


CO  en  en  rrt  m  ^^  '^^^'^^^^^^  "^  ^^  "^-r  ^^  "^  u^  mmminmo 
'  O  cN  N^  r^  On  cN    NT*  r^oooo   ONro  r^^in  On^    o^m  c  r^ro  o 
ronrorofy^vq-Ng'-'q-u-jininin  i-.-h«im     oro"^N^m 


o  o  o  o  t^  r^ 


rn  m  rn  en  rn  en   en  en  en  en  en  ^^  \~<t'^'^'^-t^^ 
■^00  cs  r^  f\ 


Lat. 


Page  80] 


TABLE  VIII. 


Right  Ascensions  and  Declinations  of  some  of  the  principal  Fixed  Stars,  adapted 
to  the  beginning  of  the  Year  1830,  with  their  Annual  Variations. 


Names  and  Situations  of  the  STARS. 


Cassiopeiae. , , 

Pegasi Mgenib 

Phcenicis 

Andromedae 

CassiopeicB Schedir 

Ceti    Dencb  Kaitos 

Cassiopeiae 

Polar  Star,  tail  of  the  Little  Bear . . 

Ceti 

AndromediB Mirach 

Cassiopeiae 

Ceti 

Eridani Jlchernar 

Cassiopeiae 

Ceti Baten  Kaitos 

Trianguli. ._ 

Arietis 

Andromedos .Mamak 

*Arietis 

Ceti 

Ceti 

Arietis 

Eridani    

Ceti    Menkar 

Persei Jllgol 

Eridani 

Persei  . . . : Mgenib 

Persei 

Eridani 

Tauri Pleiadum 

Persei 

Tauri 

Tauri *Aldebaran 

Eridani 

Eridani 

Aurigae.    Capellse Alajoth 

Orionis Rigcl 

Tauri 

Orionis .  Bellatrix 

Orionis 

Leporis 

Orionis 

Orionis 

Orionis 

Columbae 

Orionis 

Columbae   

AurigsB 

Orionis Betergueze 

AurigBB 


2.3 

2 
2.3 

3 
3 

2.3 

3 

2.3 

3.4 

2 

3 
3 
I 

3.4 

3 

3.4 


3 
3 
3 

2 
2.3 

3.4 

2 

3.4 

3.4 

3 

3.4 
3.4 


3.4 
3.4 


3 

3 

3.4 


Right 
Ascension. 


H.  M.  S. 
O.  O.  9 
o.  4 -So 
o.iy.Si 
o . 3o . 1 5 
o.3o.55 
o.35.o3 
o . 46 . 3 1 
0.59.J1 
I .00.02 
I .00.14 
1. 14. 46 
I  .i5.32 
I .3i .22 

1 .43.15 
I .43.05 
1.43.25 

1. 45. 16 


1.53.30 
1.57.36 
2.10.46 
2.34.30 
2.40.00 
2.48.07 
2.53.24 
2.57.08 
3.o4.5i 
3.12.14 
3.3o.5i 
3.35.07 
3.37.24 
3.43.28 
4.10.08 
4. 26. 1 1 
4.28.57 


4.59.30 
5.04.09 
5.06.22 
5.15.33 
,16.01 

.23.20 

.25. i4 
.27.07 
.27.35 

.32.11 

.33.3o 

.39,42 

.44.58 

5.45.32 

5.45.58 

5.47-04 


Annual 
Vaiiat. 
R.  A. 
Add 
after 
1830. 


3.12 

3.08 
2.97 
3.17 
3.33 
3.00 
3.53 
i5.52 
3.00 
3.3i 
3.83 
3.00 
2.24 
4.19 
2.95 
3.39 
3.28 


3.63 
3.34 
3.02 
3. II 
3.5o 
2.92 
3.12 
3.86 
2.56 
4.22 
4.22 
2.87 
3.54 
3.74 
3.39 
3.42 
2.33 


2.95 
4.40 
2.88 
3.78 
3.21 
3.06 
2.64 
2.93 
3.04 
3.02 
2.17 
2.84 
2. II 
4.92 
3.24 
4.40 


Declination. 


58. i3  N. 
14. i4  N. 

43.14  S. 

29.56  N. 
55.36  N. 

18.55  S. 

59.48  N. 
88.24  N. 
IX. o5  b. 
34.43  N. 
59.21  N. 
9.04  S. 
58.06  S. 
62.50  N. 
II. II  S. 
28.45  N. 
19.58  N. 


4i.3i  N. 

22.39  N. 
3.45  S. 

2.3i  N. 

26.33  N. 

9.35  S. 

3.25  N. 
40.18  N. 

29.40  S. 
49.15  N. 
47.14  N. 

10.21  S. 

23.34  N. 

31.22  N. 

i5.i3  N. 
16.10  N. 
30.55  S. 


Annual 
Variation. 


-(-  20.0 
-f-  20.0 
—  20.0 
9.9 


5.19  S. 
45  .'49  ]y- 

8.24  s. 
28.27  N. 

6. II  N. 

0.26  S. 
17.57  S. 

6.02  S. 

I. 19  S. 

2.02  s. 
34.10  s. 

9.44  s. 

35. 5i  S. 

54.16  N. 

7.22  N. 

44.55  N. 


—  7- 


+ 
+ 
+ 


9.9 
9.8 
9.6 
9-4 
9-4 
9-4 
9.0 
9.0 
8.5 
8.1 


7.6 
7.5 
6.9 
5.7 
5.4 
4.9 
4.6 
4.3 
4.7 
3.4 


1-7 


5.2 

4.8 
4.6 
3.9 
3.8 

3.2 

3.0 

2.8 

2.4 

2.3 

1.8 
1.3 
1.3 
1.2 
1 .1 


TABLE  VIIL 


[Page  81 


Right  Ascensions  and  Declinations  of  some  of  the  principal  Fixed  Stars,  adapted 
to  the  beginning  of  the  Year  1830,  with  their  Annual  Variations. 


Names  and  Situations  of  the  STARS. 


Geminorum • 

Canis  Majoris 

Canis  Majoris 

Argus Canopus 

Geminorum 

Geminorum 

Canis  Majoris Sirius 

Canis  Majoris 

Canis  Majoris 

Canis  Majoris 

Geminorum 

Canis  Majoris 

Canis  Minoris 

Geminorum Castor 

Canis  Minoris Procynn 

Geminorum   *Poli.ux 

Argus 

Argus 

Argus 

Argus 

Ursse  Majoris 

Argus 

H  ydrfE Jilplinrd 

Ursce  Majoris 

Leonis 

Leonis 

Leonis 

Leonis *Pi,egui-us 

Urste  Majoris 

Leonis 

Ursre  INLijoris 

UrsfE  Majoris 

UrsfB  Majoris    Diihhe 

UrsfE  Majoris 

lyoonis 

Loonis 

Hydra;  and  Cratcris 

Draconis 

Leonis Dcnchola 

Virginis 

Urste   Majoris 

UrsEB   RLijoris 

Corvi 

Virginis 

Crucis 

Corvi 

Crucis 

Corvi Algorah 

Draconis 

Crucis 


11 


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Right 
Ascension. 


II.  M.  S. 
6.I2.4I 
6.13.4? 
6.i5.i3 

6.20. I  I 

6.27.53 
6.33.28 

6.37.39 

6.51.57 
6.54.57 

7.01 .29 
7.09.58 
7.17.22 
7. 17.56 
7.23.44 
7.30.24 

7.34.54 
7.57.37 


8.00.18 
8.04.18 

8.40.01 

8.47.32 
9. II. 19 
9. 19.14 

9.21 .27 
9 .  36 . 1 1 
9.43.05 
9.58.03 
9.59.19 
10.06.49 
10.10.35 
10. 12. 10 
io.5i.32 
10.53.10 
II .oo.o5 


II .o5.o3 
II .05.19 
II .io.5i 
II  .21  .i3 
II .40.23 
II .41 .5i 
1 1 . 44 • 5 1 
12. 06. 58 
12.07.05 
12. 1 1 .13 
12. 17. II 
12.21 .o5 
12.21 .48 
12.25.28 
1 2. 26. 1 1 
12.37.52 


Annual 
Variat. 
R.  A. 

Add 
after 
1830. 


3.62 

22.36 

2.3o 

3o.oo 

2.64 

17.53 

1.33 

52.36 

3.46 

16.32 

3.69 

25.17 

2.64 

16.29 

2.35- 

28.45 

2.39 

27.42 

2.44 

26.08 

3.59 

22.17 

2.37 

28.59 

3.26 

8.38 

3.86 

32. i5 

3.i4 

5.39 

3.68 

28.26 

2. II 

39.32 

2.56 
1 .85 
1.66 
4.i3 
0.73 
2.95 
4.06 
3.43 
3.45 
3.28 
3.22 
3.68 
3.3o 
3.62 
3.68 
3.81 
3.42 


3.19 
3.16 
3.00 
3.70 
3.06 
3.12 
3.19 
3.00 
3.08 
3.07 
3.26 
3.10 
3.26 
3.i3 
2.60 
3.43 


Declination. 


23.49  S- 

46. 50  S. 

54.05  S. 

48.42  N. 
69.01  S. 
7.56  S. 
52.27  N. 
24.33  N. 
26.48  N. 
17.35  N. 
12.48  N. 
43.46  N. 
20.42  N. 
42.21  N. 
57.17  N. 
62.40  N. 
45.25  N. 


21 .27  N. 
16.21  N. 
i3.52  S. 
70.16  N. 
i5.3i  N. 

2.43  N. 
54.38  N. 
57.59  N. 
16. 36  S. 

0.17  N. 
62.09  S. 
i5.34  S. 
56.09  S. 
22.27  s. 

70.44  N. 

58.45  S. 


Annual 
Variation. 


1.3 
1.8 
2.4 
2.9 

A. A 
4.5 
4.8 
5.3 
6.0 
6.6 
6.7 
7.2 
8.7 


+    9-8 


0.0 
0.3 
2.9 
3.4 

4.9 
5.3 
6,0 
6.2 
6.6 
7.3 
7.3 
7.6 
7.8 

7-9 
9.2 
9.2 
9.4 


— '19.5 

—  19.5 

+  19-6 

—  19.8 

—  20.0 

—  20.0 

—  20.0 

—  20.0 
-j-  20.0 

—  20.0 
-\-  20.0 
-f-  20.0 

4-  20.0 

+  19-9 
—  19.9 

+  19-8 


Page  82] 


TABLE  VIII. 


Right  Ascensions  and  Declinations  of  some  of  the  principal  Fixed  Stars,  adapted 
to  the  beginning  of  the  Year  1830,  with  their  Annual  Variations. 


Names  and  Situations  of  the  STARS. 


Ursae  Majoris AUoth 

Canum  Venatis Cor  Caroli 

Centauri 

Virginis *Spica 

UrscB  Majoris 

UrsoB  Majoris Benetnasch 

Bootis 

Centauri 

Centauri 

Draconis 

Bootis Arcturus 

Centauri 

Bootis Mirac 

Libroe   Zubertcschamah 

Librre Zubenclgubi 

Bootis 

Libree Zubenelgemuhi 

Draconis 

Serpentis 

Corona  Borealis Gemma 

Serp  litis 

Ser  ,entis 

Serpentis 

Scorpii 

Scorpii 

Scorpii 

Ophiuchi 

Ophiuchi 

Herculis 

Scorpii *Antares 

Draconis 

Herculis 

Ophiuchi 

Herculis 

Herculis 

Scorpii , . . 

Herculis 

Ophiuchi 

Herculis 

Draconis 

Scorpii Lesath 

Draconis 

Ophiuchi 

Ophiuchi   '. 

Draconis Etanin 

Sagittarii 

Lyrse Wega 

LytiB 

Sagittarii 

LyrsB 


Il 

o 

o 

3 

a 

to 

CS 

U 

S 

s 

2.3 

2.3 

{ 

3 

a 

I 

L 

2.3 

V 

2 

n 

3 

? 

2 

9 

2 

a 

3 

a 

I 

ofi 

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3 

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2.3 

3:4 

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3 

(i 

2.3 

I 

3 

d 

3 

a 

2 

a 

2 

t 

3 

Y 

3 

S 

3 

^' 

2 

^' 

5.6 

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3 

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3 

Y 

3.4 

a 

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3 

(i 

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L 

3.4 

? 

3 

V 

3 

t 

3 

t 

3 

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2.3 

a 

2 

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3 

x 

3 

? 

2 

a 

2 

? 

3 

Y 

2 

e 

2.3 

a 

I 

? 

3 

a 

3 

Y 

3 

Right 
Ascension. 


H 


M.  S. 
2.46.32 
2.48.04 
3. II .o4 
3.16.15 
3 . 1 7 . o4 
3.4o.5o 
3.46.36 
3.51.55 
3.56.42 

3.59.47 
4.07.55 
4.28.14 
4.37.34 
4.41.29 
4.54.09 
4.55.33 
5.07.52 


5.p: .09 
5.26.41 
5.27.30 
5.35.54 
5.42.21 
5.48.36 
5.5o.i8 
5.55.34 
5.55.34 
6.05.27 
6.09.20 
6. i4.25 
6 . 1 9 . 00 
6.21 .42 
6.22.55 
6.27.48 
6.34.53 


6.37.04 
6.39. 10 

6.53.47 
7.00.38 
7.06.54 
7.08.19 
7.22;o5 
7.26.36 
7.27.03 
7.35.05 
7.62.40 
8.12.53 
8.3I.II 
8.43.48 
8.44.43 
8.52.35 


Annual 
Variat. 
R.  A. 
Add 
after 
1830. 


2.66 
2.84 
3.36 
3.i5 
2.42 
2.35 
2.86 
4. 1 3 

3.49 
1.63 
2.73 

4.47 
2.62 
3.3i 
3.49 
2.26 


I  .32 

2.86 
2.53 
2.94 
2.97 

74 
53 
47 
47 
i4 
16 

2.64 

3.66 

0.79 
2.58 
3.29 

2.25 


2.o5 
3.87 
2.29 

2.73 
o.  i5 
4.06 
1.35 
2.77 
2.96 
1 .39 
3.98 
2.01 
2.21 
3.72 
2.24 


Declination. 


56.53 
39.14 
35.49 
10.16 

55.49 
5o.io 
19.15 
59.33 
35.32 
65.11 
20.04 
60.08 
27.48 
1 5. 20 
24.36 
4i.o4 
8.45 


59.34 

11 .07 

27.18 

6.58 

5.00 

16. i3 

22.08 

19.20 

19.20 

3.i5 

4.16 

19.33 

26.03 

61.54 

21 .52 

10.  i3 

31.55 


39.15  N. 
33.59  S. 
3i.ii  N. 
i5.3o  S. 
14.35  N. 
65.55  N. 
36.59  S. 

52.26  N. 
12. 4i  N. 

4.39  N. 
5i.3i  N. 

34.27  S. 
38.38  N. 
33.10  N. 
26.30  S. 

32.28  N. 


+ 


TABLE  VIII. 


[Page  83 


Ritrht  Ascensions  and  Declinations  of  some  of  the  principal  Fixed  Stars,  adapted 
to  the  beginning  of  the  Year  1830,  with  their  Annual  Variations. 


Names  and  Situations  of  the  STARS. 


Aquiloe   

Aqui!iB 

Draconis 

Cygni 

Aquilce    

Aquiloe *Athair 

Capricorni 

Pavonis 

Cygni 

Delphini 

Delphini 

Cygni Dencb 

Cygni 

Cephei 

Cygni. 

Cephei Alderamin 

Aquarii 

Cephei  

Pegasi   

Capricorni 

Aquarii 

Gruis 

Pegasi 

Pegasi 

Aquarii 

Piscis  Australis *Fomalhaut 

Pegasi 

Pegasi ,  .*Markab 

Cephei 

Andromed;^' 


3 
3.4 
3.4 

I 

3 
3.4 

3 

3 

3 


^ 

3 

s 

2.3 

s 

3.4 

a 

3 

a 

2 

L 

3 

'; 

3 

S 

3 

Right 
Ascension. 


H.  M.  S. 
18.57.14 
18.57.36 
19. I2.30 
19.23.52 
19.38. 1 1 
19.42 .29 
20.08.37 

20. 12 .09 
20. 16.08 

20. 3i .45 
2o.35.3i 
20.35.38 
20.39.20 
20.41 .49 
21 .05.43 
21 .i4.3i 

21 .22.36 


21 .26.26 
21 .35.5o 
21 .37.39 
21 .57.03 
21 .57.29 
22.32.59 
22.35.o3 
22.45.37 
22.48. i4 
22.55.33 

2  2.56. 18 
23.32 .26 
23.59.37 


Annual 
Variat. 
R.  A. 
Add 
after 
1830. 


3.18 
2.75 
0.02 
2.42 
2.85 
2.92 
3.33 
4.81 

2.l5 

2.78 
2.80 
2.04 
2.39 
1 .22 
2.55 
1 .42 
3.16 


0.81 
a. 94 
3.3o 
3.08 
3.82 
2.98 
2.80 
3.20 
3.3i 


2.39 

3.07 


Declination. 


5.08  S. 
13.37  N. 
67.22  N. 
27.36  N. 
10.12  N. 

8.26  N. 
i3.o4  S. 
57.16  S. 
39.43  N. 

15.19  N. 
14.28  N. 
44.41  N. 

33.20  N. 
61. II  N. 
29.32  N. 
61.52  N. 

6.19  S. 


69.49  N. 

9.06  N. 
16.54  S. 

I .09  S. 
47-47  S. 

9.57  N. 
29.20  N. 

16.43  S. 

3o.3i  S. 
27.10  N. 
14.18  N. 
76.41  N. 
28.09  N. 


Annual 
Variaiion. 


5.0 
5.0 
6.2 
7.2 
8.3 
8.7 

—  10.7 

—  10.9 
-f-  II  .2 
+  12.3 
+  12.6 
+  12.6 

4-  12.8 

-h  i3.8 

+  14.5 

4-  i5.o 

—  i5.5 


+  i5.7 
4-  16.2 

—  16.3 

—  17.2 

—  17.2 
+  18.6 
+  18.7 

—  19.0 

—  19. 1 
+  19-3 
+  19-3 
+  19-9 
-f-  20.0 


Note. — If  the  places  of  these  stars  are  wanted  for  any  time  before  the  beginning  of  the 
year  1S30,  multiply  the  annual  variation,  in  right  ascension,  by  the  number  of  years  before 
1830,  and  subtract  the  product  from  the  right  ascension  standing  in  the  table ;  but  the  prod- 
uct of  tlie  annual  variation  in  declination  by  the  number  of  years  before  1S30  must  be  added 
to,  or  .subtracted  from  the  declination,  according  as  the  siim  —  or  -f-  is  marked  in  the  Table  ; 
but  for  any  years  after  1830,  the  annual  variation  in  right  ascension,  multiplied  by  the  num- 
ber of  years  after  1830,  must  be  added  to  the  right  ascension  in  the  Table,  and  the  annual 
variation  in  declination,  multiplied  by  the  number  of  years  after  1830,  must  be  either  added 
to,  or  subtracted  from  the  declination,  according  to  the  signs  in  the  Table.  The  Annual 
Variation  is  set  down  for  seconds  and  decimals  of  a  second.  An  asterisk  is  prefixed  to  the 
stars  wliose  distances  from  the  moon  are  given  in  the  Nautical  Almanac.  When  very  great 
accuracy  is  required,  the  corrections  found  in  Tables  XLII.  and  XLIII.,  for  aberration  and 
nutation,  are  to  be  applied  to  the  numbers  deduced  from  Table  VIII.;  but  these  corrections 
are  generally  not  of  much  importance  in  nautical  calculations.  The  corrected  values  are, 
however,  given  in  the  Nautical  Almanac  for  100  of  the  bright  stars  of  this  catalogue  for  every 
ten  days  in  the  year,  and  these  values  are  always  to  be  preferred. 


Page  84] 

TABLE   IX. 

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oooooo 

oooooo 

oooooo  lo  00 

o    t^ 

s 

o    -    -- 
OOO 

«   CN  0-)  ro  Ncj  vq. 

OOOOOO 

U-)  in  0  0  r  ~  CO 

oooooo 

CO    On  ON  0    0     - 

c   0  0  -  -^  ^ 

-   CN  en  en  Nq-^]- 

in  0  0 

H 

OOO 

oooooo 

oooooo 

oooooo 

OOOOOO 

000 

0   O 

^ 

O    -    " 
OOO 

«    «  ro  ro  ro  xr 

OOOOOO 

N^uTinooo     r^r^cocoONCNOC"-cNCN 

oooooo     COCCOC     --.-.«-- 

en  en  NT 

h- 

tsi^o^D 

oooooo 

oooooo  oooooo  oooooo 

000 

o   lO 

S 

O   "    - 

OOO 

-    c-i    CN    tN  ro  m 

OOOOOO 

Nq-v-i-Nq-n-iu-ju-,    .^0    r^r--r--CO    co    OnCnCnO    C 
OOOOOO      cooooo      0000-- 

«  «  « 

mh 

OOO 

d  d  0  0  d  d 

000000  oooooo  oooooo 

d  d  d 

O     "-^ 

g 

S5S 

—     I-H     M     (M     CS  en 

OOOOOO 

tn  en  m  N^  Nq.  vj- 

oooooo 

iniotnoooiO  r^t^p^ooco 

oooocooooooo 

On  On  On 

0    0    0 

a 

OOO 

OOOOOO 

oooooo 

oooooo  10  00000 

0  do 

o   CO 

a 

OO-        ----<M« 

OOO      OOOOOO 

oi  cNcncncncn   m^^~<rN.-rNrru-) 
OOOOOO    oooooo 

to  m  in  0  0  0 
OOOOOO 

0  t--  1^ 

000 

2 

OOO  oooooo 

oodooo  oooooo 

oooooo 

000 

O     C^{ 

s 

OCO      -._„--- 
960      OOOOOO 

-cNCNcsriCN     rNcsrocncncnimcnNq-N^vTNj 

oooooo    000000000000 

Nq-Nj-in 

000 

2 

■■6  ^'6     r^-^^'^^  d 

oooooo  10  00000  loo  0000 

do  0 

O    I— i 

3 

000    oooo-« 
000    oooooo 

oooooo 

oooooo 

oooooo 

CN    cs     r: 

COO 

3 

000  oooooo 

oooooo 

oooooo 

oooooo 

dod 

o   O 

s 

000 

OOO 

OOOCOOIOCOOCC 

000000000000 

cooooo 
oooooo 

OCO  OCO    coo 

oooooo     OCO 

2 

d  d  d 

0  0  0  0  0  0  lo  0  d  0  d  0 

oooooo 

oooooo  000 

Lat. 

0 

Nq-in  0  r^co  On 

0  -  cs  en  ^in 

0  r^co  On  0  - 

1 

csenNq-ino  r^co  OnC 

i 

TABLE   IX. 


[Page  85 


II 
2  -3 


o 

n 

o 

a 

o 

rt 

-a 

_, 

a 

O 

(I)- 

rt 

-a 

C 

O 

•j^ 

o 

Q 

-O 

0) 

a 

rt 

0) 

— 

tS 

Ci 

3 

-* 

tn 

^ 

2 

a; 

M 


75 


j; 

o 

jH 

G 

O 

H 

♦J 

O"! 

tj: 

J3 

■4^ 

^ 

-d 

O 

C 

C^i::3,i^i>t>    r^r^r-r-r^t^j 


t^  i^  i>  i>  r^  r^ 


-»  2 


^  P 


m  vn  CO 

LD  UT  LT) 


r-^r~-t^t^t~-r-^lr^i^r~r^c^r^ 


So-o(olr^r-~r^r^i~^f~    r^t^r^-r-r^f- 


C^     ^ 


i  ^lO  oi  _  ^  r^  o  c 


t-  t^  t^OO  CO  OO 


r-  t^  r~-  t^oo  00 


r-  r--  r--  r^  t-~co 


00  CO  CO  CO  CO  00 


00  CO  00  OO  03  OO 


OO  CO  OO  00  CO  CO 


t-^  t^  r^  r^  r'  r~-    r^oo  co  oo  co  oo 


1  in  osro  r^  I  -  o 


oovo   oo  r^r-r-r^lr-r-^r^t^r^r^    r^r~i-^r^t~-t^lr^r^oooococo   oooooo 


CO  oo  co_ 
ro  ro  -^ 


°    ^ 


CO    C    ol        .    _     _     _ 
~^Lnm   Ln  in  lo  coo 


oi 


_o_o_o 
io_o  o 


ooot-^t^t~-   r^t-^i^r^r^r- 


r^  r-  t--  r-  r-  r^ 


r^  r^  (^  t^co  00   00  oo  co 


o o oo  O  1-^ 


r-  r^  r^  r^  r^  r^ 


r-  r-  r-  1^  r^  1^ 


r-  r^  r--  r-  r^oo   co  co  oo 


0_0_OlO  O  OOP  o 


Ovr^  -^o  o  - 


o  r^  r^  r^  r--  r^ 


N:r  r^  o  cN  in  o 

^     «     N-     CN     CN     O) 


t^  t^  i-^  r^  t^  r- 


°  ^ 


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^C    O  00    o 


oo  oooo  oo  t--r-~r--r^ 


t^  r-  t^  r-  r-  r- 


r-r~-t~-r-~t~-r~-lr~-r^r^ 


°   Hi 


o  o  o  o  o  o  'o  o  o  o  r-  r^ 


r^  t^  t^  r^  r^  r- 


r^t^r^i~~t^r~-    r-~i~^i^ 


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I  o  o  o  o  o  ^ 


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r~-  r^  r--  r-~  r^  r- 


-<Tr^Of^r~-o    -^ooCT 
—   —   tNCNnro    roro~^ 


t--  r--  r^  r-  r~-  r-- 


o  o  oooo  I  oooooo 


o  o  o  t^  r-  r^ 


On  «  N3-  t^  O  ro 


r^  r^  r^  r^  t^  r- 


o  oojoo  o  o  o  o 


^  \D  'O  '^  T>  ■•Si 


o  o  o  o  o  r-^ 


r-  r-~  r~-  r^  r^  r^ 


r^  r^  r~- 

O  ro  O 

CS     CS     CN 


o    "^^ 


O  O  O  OOP 


OOOOOO 


OOOOOO 


o  [^  t~~  r^  r-  t-~- 


oooooo'oooooo 


m  in  in  in  o  O 


ooooooloooo  r~-c-~ 


Oi    OJ    cs 

'o  o  o 


OJ     O)     O) 

o  o  o 


r^co  o^ 

CN     04     O) 

o  o  o 


oooooo 


oooooo 


oooooo 


in  o  O 
o  r^  r^ 


I  w;  I  ^  3 


I  ro  ro  ^^  in  O 


—  lo  O  O  I o  o  o  o  o  o 


o  o  oooo  'oooooo 


oooooo 


oooooo  I  o  o  o  o  o  o 


oooooo 


!  O  OjOOO' 


I  _•  l-^in  O 


I  n  'o  o  o 


o  r~  CO  oo  o  o 


o  o  o  ooo 


in  o  r^oc  O^  o 


oooooo 


oooo  o_oJo  oo 


vjNTininoOlt^r^oo   On^O 


S  I  o  O  O   O  O  O  O  O  O  I  o  o  o  o  o  o 


oooooo 


O    I^CO    On  O    - 


.  oo  ro    ro  oo  CO 


OOOOOO  iOOO 


ojoooo   on~^^^ininO|r^r--ooco  OvOI—  >-■  oioo'-^m 


o  o  o  oooooo 'ooo  ooo  oooooo 


ooo  '^  '-O  o  o  o  o 


OOOOOO 


O  o  -■  - 


oooooo 


ifOooN^^iniino  r^r>co  C^ 


o  o  o  o  o  o  lo  o  o  o  o  o  ooo 


I  ^  lin  m  un 
So  o  o 


CO    On  On  On  O    C' 


12  =  - 


«    0{  lOO  ro  ■ 


OOOO  O  O  lO  o  o  o  o  o 


~.\  o  o  o 


oooooo 


oooo  oo 


I  =  i  o  o  o 


OOOOOCIOCOOCC 
OOOOOOOOOOOO 

oooooo  'o  o  oooo 


oooooo  iQooooo  ooo 
"-sr^^^^a-mininiinmooooio  t^r^ 

OOOOOCOOOOOOOOO 

oooooo  ^oooooo  Iqoo 

lOCOOOO      OOOOOOIOOO 
OOOOOO      OOOOCOOOO 

loooooo    OOOOOO'OOP 


Lat.   00  1^  00 


Pago  I 


TABLE  X. 

For  finding  the  Distance  of  Terrestrial  Objects  at  Sea,  in  Statute  Miles. 


Heighl 

Distance. 

Height 

Distance. 

Height 

Distance. 

Heiglit 

Distance. 

Heiglit 

Distance. 

Height 

Distance. 

Height 

Distance. 

infeeu 

MU.  Dec. 

in  feet 

Mil.  Dec. 

in  feel. 

MU.  Dec. 

in  feeu 

MU.  Dec. 

in  feet. 

MU.  Dec. 

in  feet. 

MU.  Dec. 

in  feet. 

MM.  Dec. 

I 

1.32 

26 

6.75 

55 

9.81 

210 

19.17 

460 

28.37 

920 

4o.i3 

3l00 

73.7 

2 

1.87 

27 

6.87 

60 

10.25 

220 

19.62 

470 

28.68 

940 

40.56 

3200 

74.8 

3 

2.29 

28 

7.00 

65 

10.67 

23o 

20.06 

480 

28.98 

960 

40.99 

33oo 

76.0 

4 

2.65 

29 

7.12 

70 

11.07 

240 

20. 5o 

490 

29.29 

980 

41.42 

3400 

77-1 

b 

2.96 

3o 

7.25 

7b 

11.46 

25o 

20.92 

5oo 

29.58 

1000 

41.80 

35oo 

78.3 

6 

3.24 

3i 

7-37 

80 

11.83 

260 

21.33 

520 

3o.i7 

IIOO 

43.90 

36oo 

79-4 

7 

3.5o 

32 

7.48 

85 

12.20 

270 

21.74 

54o 

30.74 

1200 

45.80 

3700 

80.5 

8 

3.74 

33 

7.60 

90 

12.55 

280 

22.14 

56o 

3i.3i 

i3oo 

47.70 

38oo 

81.6 

9 

3.97 

34 

7.71 

95 

12.89 

290 

22.53 

58o 

31.86 

1 400 

49.50 

3900 

82.6 

lO 

4.x8 

35 

7.83 

100 

i3.23 

3oo 

22.91 

600 

32. 4i 

i5oo 

5l.20 

4000 

83.7 

II 

4.39 

36 

7-94 

io5 

i3.56 

3io 

23.29 

620 

32.94 

1600 

52.90 

4ioo 

84.7 

12 

4.58 

37 

8.o5 

no 

i3.88 

320 

23.67 

64o 

33.47 

1700 

54.50 

4200 

85.7 

i3 

4.77 

38 

8.16 

ii5 

14.19 

33o 

24.03 

660 

33.99 

1800 

56.10 

43oo 

86.8 

i4 

4.95 

39 

8.26 

120 

14.49 

340 

24.39 

680 

34.50 

1900 

57.70 

44oo 

87.8 

i5 

5.12 

40 

8.37 

125 

14.79 

35o 

24.75 

700 

35.00 

2000 

59.20 

45oo 

88.7 

i6 

5.29 

41 

8.47 

i3o 

i5.o8 

36o 

25.10 

720 

35. 5o 

2100 

60.60 

4600 

89.7 

17 

5.45 

42 

8.57 

i35 

15.37 

370 

25.45 

740 

35.99 

2200 

62.10 

4700 

90.7 

i8 

5.61 

43 

8.68 

i4o 

i5.65 

J80 

25.79 

760 

36.47 

23oo 

63.40 

4800 

91.7 

19 

5.77 

44 

8.78 

i45 

15.93 

390 

26.13 

780 

36.95 

2400 

64.80 

4900 

92.6 

20 

5.92 

45 

8.87 

i5o 

16.20 

4oo 

26.46 

800 

37.42 

25oo 

66.10 

5ooo 

93.5 

21 

6.06 

46 

8.97 

160 

16.73 

4io 

26.79 

820 

37.88 

2600 

67.50 

IraUe 

96.1 

22 

6.21 

47 

9.07 

170 

17.25 

420 

27.11 

84o 

38.34 

2700 

68.70 

23 

6.34 

48 

9.17 

180 

17.75 

43o 

27.43 

860 

38. 80 

2800 

70.00 

24 

6.48 

49 

9.26 

190 

18.24 

440 

27.75 

880 

39.25 

2900 

71 .20 

25 

6.61 

5o 

9.35 

200 

18.71 

45o  1  28.06 

900 

39.69 

3ooo 

72.50 

TABLE  X.      A. 

Parallax  in  Altitude  of  a  Planet. 


Horizontal  Parallax  of  a  Planet. 


II 

II 

1 

II 

// 

// 

II 

II 

II 

// 

;/ 

II 

// 

II 

II 

// 

// 

II 

/; 

// 

II 

// 

// 

II 

// 

n 

// 

// 

II 

// 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

1] 

12 

13 

14 

15 

IC 

17 

18 

19 

20 

21 

22 

23 

24 

25 

26 

27 

28 

30 

35 

2 

3 

4 

5 

6 

7 

8 

9 

^ 

II 

12 

i3 

i4 

i5 

16 

17 

18 

iq 

20 

21 

22 

23 

M 

25 

26 

27 

28 

3o 

35 

2 

3 

4 

5 

6 

7 

8 

9 

10 

II 

12 

i3 

1 4 

i5 

16 

17 

18 

IQ 

20 

21 

22 

23 

24 

25 

26 

27 

28 

3o 

35 

2 

3 

4 

5 

6 

7 

8 

8 

9 

10 

II 

12 

i3 

i4 

i5 

16 

17 

18 

19 

20 

21 

22 

23 

23 

24 

a5 

26 

28 

33 

2 

3 

3 

4 

5 

6 

7 

8 

9 

10 

10 

II 

12 

i3 

i4 

i5 

16 

16 

17 

18 

19 

20 

21 

22 

23 

2.3 

24 

26 

3o 

2 

2 

3 

4 

5 

6 

7 

7 

8 

9 

10 

II 

II 

12 

i3 

i4 

i5 

16 

16 

17 

18 

19 

20 

20 

21 

22 

23 

25 

29 

2 

2 

3 

4 

5 

5 

6 

7 

8 

8 

9 

10 

II 

u 

12 

i3 

i4 

i5 

i5 

16 

17 

18 

18 

19 

20 

21 

21 

23 

27 

2 

3 

4 

4 

5 

6 

7 

7 

8 

10 

10 

II 

12 

12 

i3 

14 

i5 

i5 

16 

17 

18 

18 

19 

20 

20 

22 

26 

2 

3 

3 

4 

5 

6 

6 

7 

8 

8 

9 

lO 

10 

II 

12 

i3 

i3 

14 

i5 

i5 

16 

17 

17 

18 

19 

19 

21 

24 

2 

3 

3 

4 

5 

5 

6 

7 

7 

8 

0 

9 

10 

10 

II 

12 

12 

i3 

i4 

i4 

i5 

16 

16 

17 

18 

18 

20 

23 

2 

2 

3 

4 

4 

5 

6 

6 

7 

7 

8 

9 

9 

10 

10 

II 

12 

12 

i3 

i4 

i4 

i5 

i5 

16 

17 

17 

18 

22 

2 

2 

3 

3 

4 

5 

5 

6 

6 

7 

7 

8 

9 

9 

10 

10 

II 

II 

12 

i3 

i3 

i4 

i4 

i5 

i5 

16 

17 

20 

2 

2 

3 

3 

4 

4 

5 

5 

6 

6 

7 

7 

8 

8 

9 

10 

10 

II 

11 

12 

12 

i3 

i3 

i4 

i4 

i5 

16 

19 

0 

2 

2 

3 

3 

4 

4 

5 

5 

6 

6 

7 

7 

8 

8 

9 

9 

10 

10 

II 

II 

12 

12 

i3 

i3 

i4 

i5 

J7 

0 

2 

2 

3 

3 

4 

4 

4 

5 

5 

6 

6 

7 

7 

7 

8 

8 

Q 

9 

10 

10 

II 

11 

II 

12 

12 

i3 

i5 

0 

2 

2 

2 

3 

3 

4 

4 

4 

5 

5 

5 

6 

6 

7 

7 

7 

8 

8 

9 

9 

9 

10 

10 

II 

II 

12 

i4 

0 

2 

2 

2 

3 

3 

3 

4 

4 

4 

5 

5 

5 

6 

6 

6 

7 

7 

8 

8 

8 

9 

Q 

9 

10 

10 

12 

0 

2 

2 

2 

2 

3 

3 

3 

4 

4 

4 

5 

5 

5 

6 

6 

6 

6 

7 

7 

7 

^ 

8 

8 

g 

9 

II 

0 

2 

2 

2 

2 

3 

3 

3 

4 

4 

4 

4 

5 

5 

5 

6 

6 

6 

6 

7 

7 

7 

7 

8 

10 

0 

0 

2 

2 

2 

2 

3 

3 

3 

3 

4 

4 

4 

4 

5 

5 

5 

5 

6 

6 

6 

6 

7 

7 

7 

8 

0 

0 

I 

2 

2 

2 

2 

2 

3 

3 

3 

3 

4 

4 

4 

4 

4 

5 

5 

5 

5 

5 

6 

6 

6 

7 

0 

0 

I 

I 

2 

2 

2 

2 

2 

2 

3 

3 

3 

3 

3 

3 

4 

4 

4 

4 

4 

5 

5 

5 

5 

6 

0 

0 

0 

I 

I 

1 

I 

2 

2 

2 

2 

2 

2 

2 

3 

3 

3 

3 

3 

3 

3 

3 

4 

4 

4 

4 

5 

0 

0 

0 

0 

I 

I 

I 

I 

I 

I 

I 

I 

2 

2 

2 

2 

2 

2 

2 

2 

2 

3 

3 

3 

3 

3 

3 

4 

0 

0 

0 

0 

0 

0 

0 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

1 

I 

I 

2 

2 

2 

2 

2 

2 

2 

2 

2 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

I 

I 

I 

I 

I 

I 

1 

I 

I 

I 

I 

I 

I 

I 

I 

I 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

.90 


90 


TABLE  XI.                fi'-s«67 

Seek  the  nearest  number  to  the  reduced  time  in  the  top  column,  and  the  difference  of 

parallax,  proportional  logarithm 

,  or  semi-diameter  for  12  hours  in  the  side  column  ;  under  1 

the  former,  and  opposite 

the  latter,  is  the  correction  to  be  applied  to  the  number,  marked 

first 

in  the  Nautical  Almanac, 

additive  if  increasing,  subtraclive  if  decreasing. 

5  S 

Reduced  Time. 

h 

h 

h 

h 

h 

h 

h 

h 

h 

h 

h 

h 

h 

h 

h 

h 

h 

h 

h 

h 

h 

h 

h 

h 

£i 

1 

ii 

2 

_^i 

3 

_31 

4 

ii 

5 

§_ 

G 

i^ 

7 

3_ 

8 

3l 

9 

_^1 

10 

m 

11 

i^l 

12 

S25 

h 

1^ 

h 

h 

h 

"h 

h 

~h 

h 

h 

h 

h 

h 

h 

h 

h 

h 

h 

h 

h 

h 

h" 

h 

h 

>• 

12i 

13 

13i 

14 

14i 

15 

15i 

IG 

lOi 

17 

l!i 

18 

18;^ 

19 

19^ 

20 

20i 

21 

21d 

22 

22h 

23 

2^ 

24 

I 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

I 

I 

I 

I 

I 

I 

I 

] 

I 

1 

1 

1 

2 

0 

0 

0 

0 

0 

0 

I 

I 

I 

1 

I 

I 

I 

I 

I 

I 

I 

I 

2 

2 

2 

2 

2 

2 

3 

0 

0 

0 

0 

I 

I 

I 

I 

1 

I 

I 

2 

2 

2 

2 

2 

2 

2 

2 

3 

3 

3 

3 

4 

0 

0 

0 

I 

I 

I 

1 

I 

2 

2 

2 

2 

2 

2 

3 

3 

3 

3 

3 

3 

4 

4 

4 

5 

0 

0 

I 

I 

I 

2 

2 

2 

2 

2 

3 

3 

3 

3 

4 

4 

4 

4 

4 

5 

5 

5 

6 

0 

0 

I 

1 

2 

2 

2 

2 

~T 

~3 

3 

3 

4" 

"4 

4 

4 

5 

5 

5 

5 

"6 

6 

7 

0 

I 

2 

2 

2 

3 

3 

3 

3 

4 

4 

4 

5 

5 

5 

6 

6 

6 

6 

7 

7 

8 

0 

I 

2 

2 

2 

3 

3 

3 

4 

4 

4 

5 

5 

5 

6 

6 

6 

7 

7 

7 

8 

8 

9 

0 

I 

2 

2 

3 

3 

3 

4 

4 

4 

5 

5 

6 

6 

6 

7 

7 

7 

8 

8 

9 

9 

10 

11 

0 

2 

2 

2 

3 

3 

4 
4 

4 
5 

5 
5 

5 
5 

5 
6 

6 

6 

7 

_7 
7 

7 
8 

7 
8 

8 
9 

8 
9 

_9. 
10 

_9 
10 

10 
II 

10 
11 

0 

2 

2 

^ 

12 

0 

2 

2 

3 

3 

4 

4 

5 

5 

6 

6 

7 

7 

8 

8 

9 

9 

10 

10 

11 

II 

12 

i3 

2 

2 

3 

3 

4 

4 

5 

5 

6 

6 

7 

8 

8 

9 

9 

10 

10 

11 

II 

12 

12 

i3 

1 4 

2 

2 

3 

3 

4 

5 

5 

6 

6 

7 

8 

8 

9 

9 

10 

10 

II 

12 

12 

i3 

i3 

i4 

i5 

2 

2 

3 

4 

4 

5 

6 

6 

7 

7 

8 

_9 

_9_ 

10 

II 

II 

12 

12 

i3 

i4 

i4 

i5 

i6 

2 

T 

3 

4 

5 

5 

6 

7 

7 

8 

9 

9 

10 

1 1 

II 

12 

~iT 

73 

T4" 

i5 

i5 

16 

17 

2 

3 

4 

4 

5 

6 

6 

7 

8 

8 

9 

10 

II 

1 1 

12 

i3 

i3 

i4 

i5 

16 

16 

17 

18 

2 

3 

4 

4 

5 

6 

7 

7 

8 

9 

10 

ID 

II 

12 

i3 

i3 

i4 

i5 

16 

16 

17 

18 

'9 

2 

2 

3 

4 

5 

6 

6 

7 

8 

9 

9 

10 

11 

12 

'^ 

i3 

i4 

i5 

16 

17 

17 

18 

19 

20 
21 

2 
2 

2 

3 
T 

4 
4 

5 
"5 

6 

_7 

7 

8 

9 

10 

1 1 

12 

12 

i3 

i4 

i5 

16 
17 

12 
17 

17 
"18 

18 
'9 

19- 
20 

20 
21 

6 

7 

8 

9 

10 

10 

II 

12 

Ti 

T4 

i5 

76 

22 

2 

3 

4 

5 

5 

6 

7 

8 

9 

10 

II 

12 

l3 

i4 

i5 

16 

16 

17 

18 

19 

20 

21 

22 

23 

2 

3 

4 

5 

6 

7 

8 

9 

10 

II 

II 

12 

l3 

i4 

i5 

16 

17 

18 

•9 

20 

21 

22 

23 

24 

2 

3 

4 

5 

6 

7 

8 

9 

10 

II 

12 

i3 

i4 

i5 

16 

17 

18 

19 

20 

21 

22 

23 

24 

25 

2 

3 

4 

5 

6 

7 

8 

9 

10 

II 

12 

i4 

i5 

16 

n 

18 

19 

20 

21 

22 

23 

24 

25 

26 

T 

3 

4 

5 

~6 

8 

9 

10 

1 1 

12 

73 

i4 

i5 

~T6 

17 

18" 

'9 

21 

22 

u 

24 

25 

26 

27 

2 

3 

4 

6 

7 

8 

9 

10 

1 1 

12 

i3 

i5 

16 

17 

18 

19 

20 

21 

22 

24 

25 

26 

27 

28 

2 

3 

5 

6 

7 

8 

9 

10 

12 

i3 

i4 

i5 

16 

17 

'9 

20 

21 

22 

23 

24 

26 

27 

28 

29 

2 

4 

5 

6 

7 

8 

10 

II 

12 

i3 

i4 

16 

17 

18 

19 

21 

22 

23 

24 

25 

27 

28 

29 

3o 

2 

4 

5 

6 

7 

_9_ 

10 

11 

12 

i4 

i5 

16 

17 

J-9. 

20 

21 

22 

24 

25 

26 

27 

29 

3o 

3i 

T 

4 

T 

"6" 

8 

9 

10 

12 

73 

14 

i5 

17 

iS 

'9 

21 

22 

^ 

25 

26 

27 

28 

3o 

3i 

32 

3 

4 

5 

7 

8 

9 

r  I 

12 

i3 

i5 

16 

17 

'9 

20 

21 

23 

24 

25 

27 

28 

29 

3i 

32 

33 

3 

4 

5 

7 

8 

10 

1 1 

12 

i4 

i5 

16 

18 

'9 

21 

22 

23 

25 

26 

27 

29 

3o 

32 

33 

34 

3 

4 

6 

7 

8 

10 

u 

i3 

14 

16 

17 

18 

20 

21 

23 

24 

25 

27 

28 

3o 

3i 

33 

34 

35 

3 

4 

6 

7 

_9 

10 

12 

i3 

i5 

16 

17 

_'9. 

20 

22 

23 

25 

26 

28 

29 

3i 

32 

34 

35 

36~ 

3 

4 

6 

7 

9 

ID 

12 

i3 

15 

16 

18 

19 

21 

22 

^ 

25 

27 

28 

3o 

3i 

33 

34 

36 

37 

2 

3 

5 

6 

8 

9 

II 

12 

i4 

i5 

•7 

18 

20 

22 

23 

25 

26 

28 

29 

3i 

32 

34 

35 

37 

38 

2 

3 

5 

6 

8 

9 

II 

i3 

i4 

16 

'7 

'9 

21 

22 

24 

25 

27 

28 

3o 

32 

33 

35 

36 

38 

39 

2 

3 

5 

6 

8 

10 

II 

i3 

i5 

16 

18 

'9 

21 

23 

24 

26 

28 

29 

3i 

32 

34 

36 

37 

39 

4o 

2 

3 

5 

7 

8 

10 

12 

i3 

i5 

17 

18 

20 

22 

23 

25 

27 

28 

3o 

32 

33 

35 

37 

38 

40 

4i 

2 

T 

5 

7 

9 

10 

12 

~4 

lb 

17 

'9 

20 

22 

I4 

26 

27 

^ 

3i 

32 

34 

l6 

38 

1^ 

4i 

42 

2 

3 

5 

7 

9 

10 

12 

i4 

16 

17 

'9 

21 

23 

24 

26 

28 

3o 

3i 

33 

35 

37 

38 

40 

42 

43 

2 

4 

5 

7 

9 

1 1 

l3 

i4 

16 

18 

20 

21 

23 

25 

27 

29 

3o 

32 

34 

36 

38 

39 

4\ 

43 

M 

2 

4 

5 

7 

9 

II 

i3 

i5 

i5 

18 

20 

22 

od 

26 

27 

29 

3i 

33 

35 

37 

38 

4o 

42 

44 

45 

2  4 

6 

7 

9 

11 

i3 

i5 

17 

'9 

21 

22 

24 

26 

28 

3o 

32 

34 

36 

37 

39 

4i 

43 

45 

TABLE  XVII 

[Page  83 

When  a  Star,  or  either  (»f  the  Planets  Jupiter  or  Saturn,  is  observed. 

Parallax  0" 

►A  p. 
Alt. 
DM 

"■).   0 

Cor. 

Log'. 

*Ap. 
Alt. 

Cor. 

Log-. 

A.r.-|Cor. 

Log. 

'Ai>. 

Alt. 

Cor. 

Log. 

-Vp. 
Alt. 

Cor. 

Log. 

1.7517 

D  ftl 

M    S 

U  M 

M   S 

D  M 

M   S 

D  M 

ftl   S 

5o.  8 

0.9581 

;o.  0 

54.45 

1.2277 

i3.i5 

56.  2 

1.3433 

19.  0 

57.16 

1.4925 

36.3o 

58.43 

lO 

5o.24 

0.9700 

3 

54.47 

1.2297 

20 

56.  3 

1.3459 
1.3485 

10 

57.17 

1 .4960 

37.  0 

58.45 

1.7568 

20 

50.39 

0.9S17 

6 

54.48 

1.23.7 

25 

56.  5 

20 

57.19 

1.4996 

3o 

58-46 

1.7618 

3o 

5o.54 

0.9930 

9 

54. 5o 

1.2337 

3o 

56.  6 

1.3511 

3o 

57.20 

i.5o3i 

38.  0 

58-47 

r.7668 

4o 

5 1.  8  1. 004 1 

12 

54.5 1 

1.2357 

35 

56.  8 

1.3537 

4o 

57.22 

i.5o66 

3o 

58-49 

1.7717 

5o 
6.  0 

5l.2I 

5 1. 34 

i.oi5o 

i5 

54.53 

1.2377 

4o 
i3.45 

56.  9 

1.3562 

5o 

57.23 

i.Sioi 

39.  0 

58. 5o 

1.7766 

1.02  55 

10.18 

54.54 

1.2397 

56. 10 

1.3587 

20.  0 

57.25 

i.5i36 

39.30 

58. 5i 

I -78 10 

10 

51.46 

i.o36o 

21 

54.56 

1.2417 

5o 

56.12 

i.36i2 

10 

57.26 

1.5170 

4o.  0 

58.52 

1.7854 

20 

51.57 

1.0462 

24 

54.57 

1.2437 

55 

56. 1 3 

1.3637 

20 

57.27 

1.5204 

3o 

58.53 

1 .7900 

3o 

52.   9 

i.o562 

27 

54.58 

1.2457 

14.  0 

56.15 

1.3662 

3o 

57.29 

1.5238 

4i.  0 

58.55 

1 .7946 

4o 

52.19 

1 .0660 

3o 

55.  0 

1.2476 

5 

56.16 

1.3687 

4o 

57.30 

1.5271 

3o 

58.56 

1-7987 

5o 

D2.29 

1.0755 

33 

33.    I 

1.2496 

10 

56.17 

1. 3711 

1.3735 

5o 
21.  0 

57.31 
57.33 

i.53o4 

42.  0 

58.57 

1.8028 

7-  0 

52. 3o 

1.0849 

10.36 

55.  3 

i.25i5 

i4.t5 

56.19 

1.5338 

42-3o 

58-58 

1.8070 

10 

52.49 

1 .094 1 

39 

55.  4 

1.2534 

20 

56. 20 

1.3759 

10 

57.34 

1.5370 

Ai.  0 

58-59 

1.8112 

20 

52.58 

I.I032 

42 

55.  5 

1.2553 

25 

56.21 

1.3783 

20 

57.35 

1.5401 

3o 

59.  0 

i.8i52 

3o 

53.  6 

1.1120 

45 

55.  7 

1.2572 

3o 

56.22 

1.3807 

3o 

57.36 

1.5432 

4A.  o|59.  1 

1.8192 

35 

53.11 

1.1164 

48 

55-.  8 

1.2591 

35 

56  24 

1.383 1 

4o 

57.37 

1.5463 

3o 

59.  2 

1.8230 

4o 
7.45 

53. i5 
53.19 

I.I  207 

5i 

55.9 

1.2610 

4o 

56.25 

1.3855 

5o 

57.39 

1.5494 

45.  0 

59.  3 

1.8268 

I.I25o 

10.54 

55.11 

1.2629 

14.45 

56.26 

1.3878 

22.  0 

57.40 

1.553  5 

46. 

59.  5 

1.8338 

48 

53.2  1 

1. 1275 

57 

55.12 

1.2648 

5o 

56.28 

1.3901 

10 

57.41 

1.5556 

47- 

59.  7 

1.841 1 

31 

53.24 

i.i3oi 

11.  0 

55.13 

1.2667 

55 

56.29 

1.3924 

20 

57-42 

1.5586 

48. 

59.  9 

r-8478 

54 

53.26 

1. 1 326 

3 

55.14 

1.26S6 

1 5.  0 

56.3o 

1.3947 

3o 

57.43 

i.56i6 

49- 

39.11 

1-8547 

57 

53.28 

i.i35i 

6 

55.16 

1.2705 

5 

56.3i 

1.3970 

4o 

57.44 

1.5646 

5o. 

59.12 

1.8611 

8.  0 
873 

53.3o 
53.33 

I.I3-6 
i.i4oi 

_9 

I  i.i;- 

55.17 

1.2724 

10 
i5.i5 

56.32 

1.3993 

5o 

57-45 

1.5676 

5i. 

59.14 

1-8676 

5'5.i8 

1.2742 

56.33 

1.4016 

23.    0 

57-46 

1.5706 

52. 

59.16 

1.8734 

6 

53.35 

1.1425 

ID 

55.19 

1 .2760 

20 

56.34 

1.4039 

10 

57.47 

1.5736 

53. 

59.17 

1-8794 

9 

53.37 

i.i45o 

18 

55.21 

1.2778 

25 

56.36 

1.4061 

20 

57.49 

1.5765 

54. 

59.19 

1.8846 

12 

53.39 

1. 1474 

21 

55.22 

r.2796 

3o 

56.37 

i.4o83 

3o 

57.50 

1.5794 

55. 

59.20 

1 .8900 

i5 

f3.42 

1. 1499 

24 

55.23 

1.28:4 

35 

56.38 

i.4io5 

4o 

57.51 

1.5822 

5b. 

59.22 

1.8956 

i8 

8.21 

53.44 
53.46 

i.i523 
I.I547 

27 

55.24 

1.2832 

r.^57; 

4o 

56.39 

1.4127 

5o 

57-52 

i.585o 

57. 

59.23 

1 .9003 

ii.3o 

55.25 

15.45 

56.40 

1.4149 

24.  0 

57.53 

1.5879 

58. 

59.24 

1.9050 

24 

53.48 

1. 1 57 1 

33 

55.27 

1.2S68 

5o 

56.41 

1.4171 

10 

57-54 

1.5907 

59- 

59.26 

1.9102 

27 

53.5o 

1. 1595 

■66 

55.28 

1.2886 

55 

56.42 

1.4193 

20 

57-55 

1.5935 

60. 

59-27 

1.9142 

3o 

53.52 

1.1619 

39 

55.29 

1.2904 

16.  0 

56.43 

1.4214 

3o 

57.56 

1.5963 

61. 

59.28 

1.9183 

33 

53.54 

1. 1 642 

4? 

55. 3o 

1.2922 

5 

56.44 

1.4236 

40 

57.56 

1 .5990 

62. 

59.30 

1.9226 

36 
8.39 

53.56 
53.58 

1. 1 666 

45 

55. 3i 

1.2940 

10 

56.45 

1.4258 

5o 

57.57 

1.6017 

63. 

59.3  r 

1.9270 

1. 1 689 

11.48 

55.32 

1.2957 

i6.i5 

56.46 

1.4279 

25.   0 

57-58 

1  -6044 

64. 

59.32 

1.9302 

42 

54.  0 

1.1713 

5i 

55.34 

1.2974 

20 

56.47 

1.4300 

20 

58.  0 

1-6097 

65. 

59-33 

1-9335 

45 

54-  2  1.1735 

54 

55.35 

1.2991 

25 

56.48 

1.4321 

4o 

58.  2 

1.6149 

66.- 

59.35 

1 .9369 

48 

54.  4 

1.1758 

57 

55.36 

i.3oo8 

3o 

56.49 

1.4342 

26.00 

58.  4 

1.6201 

67. 

59.36 

1 .9404 

5i 

54.  6 

1.1781 

1 1.  0 

55.37 

i.3o25 

35 

56. 5o 

1.4363 

20 

58.  5 

1.6251 

68. 

59.37 

..9438 

54 

8.57 

54.08 
54^ 

1.1804 

3 
12.  6 

55.38 

i.3o42 

4^^ 

56.5 1 

1.4384 

40 

58.  7 

i.63oi 

69. 

59.38 

1 .9471 

1. 1826 

55.39]i.3o59 

16.45 

56.52 

i.44o5 

27.  0 

58.  9 

I.6350 

70. 

59.39 

1.9501 

9.  0 

54.12 

1. 1. 849 

9 

55.40 

1 .3076 

5o 

56.53 

1.4425 

20 

58.10 

1 .6400 

71- 

59.40 

1.9528 

3 

54.13 

1.1871 

12 

55.41 

1.3093 

55 

56.54 

1.4445 

4o 

58.12 

1.6449 

72. 

59.42 

1.9553 

6 

54.15 

1. 1 893 

ID 

55.42 

i.3iio 

17.  0 

56.55 

1.4465 

28.  0 

58- 1 3 

1 .6498 

73. 

59.43 

..9578 

9 

54.17 

1. 1916 

18 

55.43 

1.3127 

5 

56.56 

1.4486 

•  20 

58-i5 

1.6545 

74. 

59.44 

1.9603 

12 

54.19 

1. 1938 

21 

55.44 

i.3i44 

10 
17.. 5 

56.57 

1.4506 

Ao 

58.16 

1. 659 1 

75. 

59.45 

1.9625 

V.i5 

54.21 

1.1960 

12.24 

55.45 

i.3i6i 

56.58 

1.4526 

29.  0 

58.18 

1.6635 

76. 

59.46 

1.9643 

18 

54.22 

1.1982 

27 

55.46 

1.3178 

20 

56.59 

1.4546 

3o 

58. 20 

1 .6702 

77- 

59.47 

1 .9660 

2J 

54.24 

i.atx)3 

3o 

55.47 

1.3194 

2  5 

57.0 

1.4566 

3o.  0 

58.22 

1 .6769 

78. 

59.48 

1 .9676 

24 

54.26 

1. 2025 

33 

55.48 

1.32II 

3o 

57.   I 

1 .4586 

3u 

58.24 

1 .6833 

79. 

59.49 

1 .9692 

27 

54.28 

I.2o4~ 

36 

55.49 

1.3227 

35 

57.  2 

1 .4606 

3i.  0 

58.25 

1 .6896 

80. 

59.50 

1 .9706 

3o 
9.33 

54.29 

1.2068 

39 

55. 5o 

1.3243 

40 

57.  31.4626 

3o 

58.27 

1.6957 

81. 

59.51 

1.9714 

54.3 1 

1.2089 

12.42 

55. 5i 

1.3259 

17.45 

57.  4 

1 .4646 

32.   0 

58.29 

1.7018 

82. 

59.52 

.9722 

36 

54.33 

1. 2 1 10 

45 

55.52 

1.3273 

5o 

57.  4 

1.4665 

3o 

58.3 1 

1-7079 

83. 

59.53 

1.9729 

39 

'MM 

r.2i32 

48 

55.53 

1.3291 

55 

57.  5 

1 .4684 

33.  0 

58.33 

1.7140 

84. 

59.54 

1-9734 

42 

54-36 

I.2I53 

5i 

55.54 

i.33o7 

18.  0 

57.  G 

1.4703 

3o 

58.34 

1.7202 

85. 

59.55 

1-9737 

45 

54.37 

1.2173 

54 

55.55 

1.3323 

10 

57.  81.4741 

34.  0 

58.36 

1-7263 

86. 

59.56 

1.9739 

48 

54.3^ 

1. 2194 

57 

55.56 

1.3339 

20 

57.  9 

1.4778 

3o 

58.37 

I -73 1 2 

87. 

59.57 

1.9741 

9.51I54.41 

I.22[5 

i3.  0 

55.57 

1.3355 

i8.3o 

57.11 

1.481 5 

35.  cj< 

58.39  1.7362 

88. 

59.58 

1.9742 

54  54.42 

1.2236 

5 

55.59 

1.3381 

4o 

57.13 

1.4852 

3o 

58. 4o  1.7414 

89. 

59.59 

1.9742 

57  54.44 

1.2257 

10 

56.  0 

1.3407 

5o  57.14 

1.4888 

36.  058.42I1. 74661 

90. 

60.  0 

1.9742 

12 


Page  90]                                                     TABLE   XVII.                                                                      ! 

When  the  Planet  Venus  or  Mars  is  used,  and  the  i'arallax  is  nearly  equal  to  5". 

Parallax  5". 

-Ap. 
All. 

Cor. 

Log 

*Ap. 
Alt. 

Cor. 

Log. 

*Ap. 
Alt. 

D  M 

Cor. 
M    S 

Log. 

*Ap. 
All. 

Cor. 

Log. 

*Ap. 
Alt. 

D   31 

36.3o 

37.  0 
3o 

38.  0 
3o 

39.  0 

"~3^ 

40.  0 
3o 

4i.  0 

3o 

42.  0 

Cor. 

31   S 

58.47 
58.49 
58.5o 
58.5i 
58.52 
58.54 
58.55 
58.56 
58.57 
58.58 
59.00 
59.01 

Log. 

1.775c 
1 .7803 
1.7855 
1.7907 
1 .7953 
1 .8008 

DiM 

M    S 

U  31 

M    S 

D  M 

31   S 

5.  0 

10 
20 

3o 
4o 
5o 

5o.i3 
50.29 
5o.44 
50.59 
5i.i3 
51.26 

0.961/^ 
0.9733 
0.9851 
0.9966 
1 .0078 
1. 01 88 

10.  0 
3 
6 

9 
12 

i5 

54.5o 
54.52 
54.53 
54.55 
54.56 
54.58 

3342 
2363 
2383 
2{o4 

2,)25 

2445 

i3.i5 
20 

25 

3o 
35 
4o 

13.45 
5o 
55 

i4.  0 

5 

ic 

56 .07 
56.08 
56.10 
56.11 
56.12 
56.14 

I.352I 

1.3547 
1.3574 
i.36oo 
1.3626 
1.3652 

19.  0 
10 

20 
3o 
4o 
5o 

57.20 
57.22 
57.23 
57.25 
57.26 
57.28 

1.5049 
i.5o86 
I.5I23 
i.5i59 
1.5195 

1. 5201 

6.  0 

10 
20 

3o 
4o 
5o 

51.39 
5i.5i 
52.03 
52.14 
52.24 
52.34 

1.0295 
I  .o4oo 
i.o5o3 
1 .0604 
1 .0703 
1.0800 

10.18 
21 
24 
27 
3o 
33 

54.59 
55.00 
55.02 
55.03 
55.05 
55.06 

2465 
2485 
2  5o5 

2525 

2545 
2565 

56.15 
56.17 
56.18 
56.19 
56.21 

56.22 

1.3678 
1 .3704 
1.3729 
1.3754 
1.3779 
i.38o4 

20.  0 

10 
20 
3o 
4o 
5o 

57.29 
57.31 
57.32 
57.33 
57.35 
57.36 

1.5266 

I.5301 
1.5336 
1.5371 
i.54o5 
1.5439 

1 .8o56 
1. 8 104 
i.8i5i 
1.8198 
1.8244 
1.8289 

7-  o 

lO 
20 

3o 
35 
4o 

52.44 
52.54 
53.03 
53.11 
53.16 
53.20 

1.0895 
1 .0988 
1. 1079 
I.I  169 

I.I2l3 

1. 1257 

I0.36 
39 

42 
45 
48 
5i 

55.07 
55.09 
55.10 
55.12 
55.13 
55.14 

2585 
2605 
2624 
2643 
2662 
2681 

14. i5 
20 

25 

3o 
35 
4o 

56.23 
56.25 
56.26 
56.27 
56.29 
56.3o 

1.3829 
1.3854 
1.3878 
1.3902 
1.3927 
1.3951 

21.  0 
10 
20 
3o 
4o 
5o 

57.37 
57.39 
57.40 
57.41 
57.42 
57.43 

1.5473 
1.5507 
1.5540 
1.5573 
i.56o5 
1.5637 

42.3o 

43.  0 
3o 

44.  0 
3o 

45.  0 

59.02 
59.03 
59.04 
59-05 
59.06 
59.07 

1.8333 
1.8376 
1.8419 
1.84G1 
i.85o2 
1.8543 

7-45 
48 
5i 
54 
57 

8.  0 

53.24 
53.26 
53.28 
53. 3o 
53.32 
53.35 

i.i3oo 
1.1326 
i.i352 
1. 1378 
i.i4o4 
1. 1429 

10.54 
57 

II.  0 
3 
6 
9 

55.15 
55.17 
55.18 
55.19 
55.21 

55.22 

2701 
2720 
2739 
2758 

2777 
2796 

14.45 
5o 
55 

i5.  0 

5 

10 

56.3 1 
56.32 

56.34 
56.35 
56.36 
56.37 

1.3975 
1.3999 
1.4023 
1 .4o46 
1 .4070 
1 .4093 

22.  0 

10 
20 
3o 
4o 
5o 

57.44 
57.46 
57.47 
57.48 

57.49 
57.50 

1.5669 
1. 5701 
1.5733 
1 .5764 
1.5-95 
1.5826 

1.5857 
1.5887 
1.5917 
1 .5947 
1.5977 
1 .6007 

46. 

47. 
48. 
49. 
5o. 
5i. 

59.09 
59.10 
59.12 
59.14 
59.15 
59.17 

1.8622 
1 .8699 
1.8773 
1 .8844 
1. 8914 
1. 898 1 

8.  3 

6 

9 

12 
i5 
18 

53.38 
53.40 
53.42 
53.44 
53.47 
53.49 

1. 1454 
1. 1479 
i.i5o4 
1. 1529 
I.I554 
1. 1578 

11.12 

i5 
18 
21 
24 

27 

55.23 
55.?4 
55.26 
55.27 
55.28 
55.29 

2815 
2833 

2852 

2871 
2889 
2907 

i5.i5 
20 

25 

3o 
35 
40 

1 5.45 
5o 
55 

16.  0 

5 
10 

56.38 
56.39 
56.40 
56.41 
56.43 
56.44 

1.4116 
1.4139 
1.4162 
i.4i85 
1.4208 
1.4231 

23.    0 

10 

20 

3o 
40 
5o 

57.51 
57.52 
57.53 
57.54 
57.55 
57.56 

32. 

53. 

54. 

55. 
56. 

57. 

59.19 
59.20 
59.22 
59.23 
59.24 
59.26 

1 .9045 
1. 9107 
I. 9167 
1.9226 
1.9282 
1.9336 

8.21 

24 
27 
3o 
33 
36 
8.39 
42 
45 
48 
5i 
54 

53. 5i 
53.53 
53.55 
53.57 
53.59 
54.01 
54.03 
54.05 
54.07 
54.09 
54.11 
54.13 

i.i6o3 
1. 1627 
i.i65i 
1. 1675 
1. 1 699 
1.1722 

ii.3o 
33 
36 
39 
42 
45 

55. 3o 
55.32 
55.33 
55.34 
55.35 
55.36 

2925 
2944 
2962 
298c 

=998 
3oi6 

3^3 
3o5o 
3o68 
3o86 
3io4 

3l22 

56.45 
56.46 
56.47 
56.48 
56.49 
56. 5o 

1.4253 
1.4276 
1.4298 
1.4320 
1.4342 
1.4364 

24.  0 
10 
20 
3o 
40 
5o 

57.57 
57.58 
57.59 
58.00 
58.01 
58.02 

i.6o36 
i.6o65 
1 .6094 
1.6122 
i.6i5i 
1.6179 

58. 
59. 
60. 
61. 
62. 
63. 

59.27 
59.28 
59.30 
59.31 
59.32 
59.33 

1.9388 
1.9438 
1 .9486 
1.9532 
1.9577 
1. 9619 

1. 1746 
1. 1770 
..1793 
,.1817 
1.1840 
1. 1 863 

11.48 
5i 
54 
57 

12.  0 
3 

12.  6 

9 
12 

i5 
18 
21 

55.37 
55.38 
55.39 
55.41 
55.42 
55.43 

i6.i5 
20 

25 

3o 

4o 

16.45 
5o 
55 

17.  0 

5 

10 

56.5 1 
56.52 
56.53 
56.54 
56.55 
56.56 

56.57 
56.58 
56.59 
57.00 
57.01 
57.02 

1 .4386 
1 .4408 
1.4430 
I.445I 
1.4472 
1  ..^493 

1.4:^4 
1.4535 
1.4556 
1.4577 
1.4597 
1.4618 

25.  0 

20 
4o 

26.  0 
20 
4o 

27.  0 
20 
40 

28.  0 
20 
4o 

58.03 
58.05 
58.06 
58.08 
58. 10 
58.11 

1 .6207 
1.6262 
1.6317 
1.6371 
1.6424 
1.6476 

64. 
65. 
66. 
67. 
68. 
69. 

59.34 

5q.36 

59.37 
59.38 
59.39 
59.40 

1 .9660 
1 .9700 
1.9738 
1 .9773 
1 .9807 
1.9839 

8.57 

6 

9 
12 

9.15 
18 
21 
34 

27 
3o 

54.15 

54.17 
54.18 
54.20 
54.22 
54.24 
54.25 
54.27 
54.29 
54.3 1 
54.32 
54.34 

1 .  1 886 
1 .  1 908 
1 .  1 93 1 
1.1953 
1.1975 
1. 1998 

1 .2021 
1 .2043 
i.2o65 
1.2087 
1.2109 

1.2l3() 

55.44 
55.45 
55.46 
55.47 
55.48 
55.49 

3i39 
3i56 
3173 
3190 
3207 
3224 

58.13 
58.15 
58.16 
58.18 
58.19 
58.21 

1.6527 
1.6578 
1.6629 
1 .6678 
1.6727 
1.6775 

70. 

71. 
72. 
73. 
74. 
75. 

59.41 
59.42 
59.43 
59.44 
59.45 
59.46 

1 .9870 
1.9899 
1.9927 
1 .9953 

1.9977 
2.0000 

12.24 

27 
3o 
33 
36 
39 
12.42 
45 
48 
5i 
54 
57 

55. 5o 
55.5i 
55.52 
55.53 
55.54 
55.55 

55.56 
55.57 
55.58 
55.59 
56.00 
56.01 

3242 
3259 
3276 
3293 
33io 
3326 

17.15 
20 

25 

3o 
35 
40 

57.03 
57.04 
57.05 
57.06 
57.07 
57.07 

1.4639 
1 .4660 
1.4680 
1 .4700 
1.4720 
1 .4740 

29.  0 
3o 

3o.oo 
3o 

3i.  0 
3o 

58.22 

58.24 
58.26 
58.28 
58.3o 
58.32 

1.6823 
1.6893 
1 .6962 
1.7029 
1.7095 
1. 7160 

76. 

77. 
78. 

80. 
81. 

59.47 
59.48 
59.49 
59.50 
59.51 
59.52 

2.0022 
2.0042 
2.0060 
2.0077 
2.0092 
2.0106 

9.33 
36 
39 

42 
45 
48 

54.36 
54.37 
54.39 
54.41 
54.42 
54.44 

1.2l5l 

1.2173 
1. 2195 
1.2216 

1.2237 

1.2258 

3343 
3359 
3376 
3392 
3408 
3424 

17-45 
5o 
55 

18.  0 
10 
20 

i8.3o 
40 
5o 

57.08 
57.09 
57.10 
57.11 
57.13 
57.14 
57.16 

57.17 
57.19 

1 .4760 
1.4780 
1 .4800 
1.4820 
1.4859 
1 .4898 

32.  0 

3o 

33.  0 
3o 

34.  0 
3o 

58.33 
58.35 
58.37 
58.38 
58.4o 
58.4i 

1.7224 
1.7287 
1.7349 
1 .7409 
1.7468 
1.7526 

82. 
83. 
84. 
85. 
86. 
87. 
88. 
89. 
90. 

59.53 
59.54 
59.55 
59.55 
59.56 
59.57 
59.58 
5?.59i 
So. 00 

2.0118 
2.0129 
2.0139 

2.0147 
2.oi53 
2.0158 
2.0162 
2.C164 
2.0164 

9.51 
54 
57 

54.45 
54.47 
54.49 

1.2279 

i.23oo 

1.2321 

i3.  0 
5 

TO 

56.02 
56.o3 
56.o5 

3440 
3468 
3495 

1.4936 
1 .4974 

1.5oi2 

35.  0 
3o 

36.  0 

58.43 
58.44 
58.46 

1.7584 
1 .7640 
1.7695 

TABLE  XVI] 

, 

[Page  91 

When  the  Planet  Venus  or  Mars  is  used,  and  the  Parallax  is  nearly  equal  to  KV. 

Parallax  10''' 

*AP. 
All. 

DM 

Cor. 

Log. 

*Ap. 
All. 

Cor. 

Log. 

*Ap. 
Alt. 

Cor. 

Log. 

*Ap. 
All. 
D  I\l 

Cor. 

Log. 

-Ap. 
All. 
D  IM 

Cor. 

Log. 

M  i< 

D  JVI 

M   S 

D  IM 

JM  S 

M  S 

M    S 

5.  0 

5o.i8 

0.9650 

10.  0 

54.55 

1.2412 

i3.i5 

56.11 

i.36i3 

19.  0 

57.25 

1.5 1 80 

36.3o 

58.5 1  1.7996 

10 

50.33 

0.9771 

3 

54.57 

1.2433 

20 

56.13 

1 .3640 

10 

57.27 

1.5218 

37.  0 

58.53  1.8052 

20 

50.48 

0.9890 

6 

54.58 

1.2454 

25 

56.14 

1.3667 

20 

57.28 

1.5256 

3o 

58.54 

1.8108 

3o 

5i.o3 

1 .0006 

9 

55.00I1.2475 

3o 

56.16 

1.3693 

3o 

57.30 

1.5293 

38.  0 

58.55 

i.8i63 

4o 

5l.I7 

i.oi  19 

12 

55.01 

1.2496 

35 

56.17 

1.3720 

40 

57.31 

i.533o 

3t. 

58.56 

1.821-7 

5(. 
6.  0 

5i.3i 
51.43 

I.023o 

1.0338 

i5 

55.02 

i.25i6 

40 
i3.45 

56.19 

1.3746 

5o 

57.33 

1.5367 

39.  0 

58.58 
58.59 

1.8269 
1.8321 

10.18 

55.04 

1.2536 

56.20 

1.3773 

20.  0 

57.34 

1.5404 

39.30 

10 

5i.55 

1.0444 

21 

55.05 

1.2557 

5o 

56.22 

1.3799 

10 

57.35 

1.5440 

4o.  0 

59.00 

1.8372 

20 

52.07 

1 .0549 

24 

55.07 

1.2577 

55 

56.23 

1.3825 

20 

57.37 

1.5476 

3o 

59.0. 

1.8422 

3o 

52.18 

i.o65i 

27 

55.08 

1.2597 

14.  0 

56.24 

i.385i 

3o 

57.38 

1.5512 

4i.  0 

59.02 

1.8472 

4o 

52.29 

1. 075 1 

3o 

55.10 

1. 2617 

5 

56.26 

1.3877 

40 

57.39 

1.5547 

3o 

59.03 

1. 852 1 

5o 

52.39 

1.0849 

33 

55.11 

1.2638 

10 
14. i5 

56.27 

1.3902 

5o 

57.41 

1.5582 

42.  0 

59.04 
59.05 

1.8569 
1.8616 

7-  o 

52.49 

1 .0945 

10.36 

55.12 

1. 2658 

56.28 

1.3927 

21.  0 

57.42 

1.5617 

42. 3o 

10 

52.58 

1 .  1 039 

39 

55.14 

1.2678 

20 

56.3o 

1.3952 

10 

57.43 

1.5652 

43.  0 

59.06 

1.8662 

90 

53.07 

i.ii3i 

42 

55.15 

1 .2698 

25 

56.3 1 

1.3977 

20 

57.44 

1.568^ 

3o 

59.07 

1.8708 

3o 

53.15 

1. 1222 

45 

55.16 

1.2718 

3o 

56.32 

1 .4002 

3o 

57.45 

1.5720 

44.   0 

59.08 

1.8753 

35 

53.19 

1. 1 267 

48 

55.18 

1.2737 

35 

56.33 

1.4027 

4o 

57.47 

1.5754 

3o 

59.09 

1.8797 

4o 
7-45 

53.24 
53.28 

i.i3ii 
I.I355 

61 

55.19 

1.2756 

40 
14.45 

56.35 
56.36 

i.4o52 

.  5o 

57.48 

1.5787 

45.  0 

59.10 

1 .8840 

10.54 

55.20 

1.2776 

1 .4077 

22.  0 

57.49 

1.5820 

46. 

59.12 

1.8925 

4« 

53.3; 

i.i38i 

57 

55.22 

1.2795 

5o 

56.37 

1.4  lOI 

10 

57.5o 

1.5853 

47- 

59.14 

1 .9007 

5i 

53.33 

1. 1 407 

11.  0 

55.23 

1.2815 

5:) 

56.38 

1.4125 

20 

57.51 

1.5886 

48. 

59.15 

1 .9087 

54153.30 

1. 1433 

3 

55.24 

1.2835 

i5.  0 

56.39 

1.4149 

3o 

57.52 

1.5919 

49. 

59.17 

1. 9 1 64 

5753.38 

1. 1459 

6 

55.26 

1.2854 

5 

56.4 1 

1.4173 

40 

57.54 

1.5951 

5o. 

59.19 

1.9238 

8.  0 

53.40 

I.I485 

9 
II. 12 

55.27 
55.28 

1.2873 
1.2892 

10 
i5.i5 

56.42 

'■4197 
1.4221 

5o 
23.  0 

57.55 
57.56 

1.5983 

5i. 

59.20 

1. 9310 

8.  3 

53.43 

i.i5ii 

56.43 

1.601 5 

52. 

59.22 

1.9380 

6 

53.45 

I.I536 

i5 

55.29 

1.291 1 

20 

56  44 

1.4245 

10 

57.57 

1 .6046 

53. 

59.23 

1.9448 

9 

53.47 

i.i56i 

18 

55.3o 

1.2930 

25 

56.45 

1.4269 

20 

57.58 

1 .6077 

54. 

59.25 

1.9513 

12 

53.49 

1. 1 586 

21 

55.32 

1.2949 

3o 

56.46 

1.4292 

3o 

57.59 

1.6108 

55. 

59.26 

1.9576 

i5 

53.5i 

1.1611 

24 

55.33 

1.2968 

35 

56.48 

i.43i5 

40 

58. 00 

1. 6 1 39 

56. 

59.27 

1.9637 

i8 

53.53 

I.I636 

27 

55.34 

1.2987 
1 .3oo5 

4o 
i5.45 

56.49 

1.4338 

5o 

58.01 

1.6170 

57. 

59.28 
59.30 

_i^60 
1.9751 

8.21 

53.56 

1. 1 661 

1 1 .3o 

55.35 

56. 5o 

1.436] 

24.  0 

58.02 

1 .6200 

58. 

24 

53.58 

1. 1686 

33 

55.36 

1 .3o24 

5o 

56.5i 

1.4384 

10 

58.03 

1.6230 

59. 

59.3 1 

1 .9806 

27 

54.00 

1.1710 

36 

55.38 

i.3o42 

55 

56.52 

1 .4407 

20 

58. o4 

1 .6260 

60. 

59.32 

1.9858 

3o 

54.f)2 

1. 1734 

39 

55.39 

1 .3o6o 

16.  0 

56.53 

1.4430 

3o 

58.05 

1.6290 

61. 

59.33 

1 .9909 

33  54.04 

1. 1758 

42 

55.40 

1.3078 

5 

56.54 

1.4453 

4o 

58. 06 

I.6320 

62. 

59.34 

1.9958 

36 
8.39 

54.06 

1. 1782 

45 

55.41 

1.3096 

10 

56.55 

1.4476 

5o 

58.07 

1.6349 

63. 

59.36 

2.0005 

54.08 

1. 1806 

11.48 

55.42 

i.3ii4 

i6.i5 

56.56 

1.4498 

25.  0 

58. 07 

1 .6378 

64. 

59.37 

2.0049 

4i 

54.10 

i.iS3o 

5i 

55.43 

i.3i33 

20 

56.57 

1.4520 

20 

58.09 

1.6435 

65. 

59.38 

2.0092 

45 

54. 1 2 

I.I854 

54 

55.44 

i.3i5i 

25 

56.58 

1.4542 

4o 

58.11 

1 .6492 

66. 

59.39 

2.oi33 

48 

54.14 

1.1878 

5? 

55.45 

1.3169 

3o 

56.59 

1.4564 

26.  0 

58.1 3 

1.6548 

67. 

59.40 

2.0172 

5i 

54.16 

1.1901 

12.  0 

55.47 

i.3]87 

35 

57.00 

1 .4586 

20 

58.14 

1 .6604 

68. 

59.41 

2.0209 

54 

54.18 

1. 1 925 

3 

55.48 

i.32o5 

4o 

37.01 

1 .4608 

4o 

58.16 

1 .6659 

69. 

59.42 

2.0245 

8.57 

54.20 

1. 1 948 

12.  6 

55.49 

1.3223 

16.45 

57.02 

1 .463() 

27.  0 

58.17 

1 .67 1 2 

70. 

59.43 

2.0279 

9.  (. 

54.22 

1-1971 

9 

35. 5o 

1.3241 

5o 

57.03 

1.4652 

20 

58.19 

1 .6765 

71- 

59.44 

2.o3l  I 

3 

54.23 

1. 1994 

12 

55.5 1 

1.3258 

55 

57.04 

1.4673 

40 

58.21 

1.6818 

72. 

59.45 

2.o34i 

(i 

54.25 

1.2017 

i5 

55.52 

1.3275 

17.  0 

57.05 

1.4694 

28.  0 

58.22 

1 .6870 

73. 

59.46 

2.0370 

9 

54.27 

1 .2o4o 

18 

55.53 

1 .3293 

5 

57.06 

i.47ir> 

20 

58.24 

1.6921 

74. 

59.46 

2.0397 

12 

^5 

54.29 

54.3(. 

1.2063 

21 

5:).54 

i.33io 

10 

57.07 

•  .4737 

4o 

58.25 

1. 697 1 

75. 
76. 

59.47 

2.0422 

1  .2085 

12.24 

55.55 

1.3327 

17.15 

57.08 

1.4758 

29.  0 

58.26 

1. 702 1 

59.48 

2.0446 

18 

54.32 

1.2  I  08 

27 

55.56 

1.3345 

20 

57.09 

1 .4779 

3o 

58.28 

1 .7094 

77- 

59.49 

2.0468 

21 

54.34 

1.2I0I 

3o 

55.57 

1.3362 

25 

57.10 

1 .4800 

3o.  0 

58.3o 

1.7166 

78. 

59.50  2.0488 1 

24 

54.36 

1. 2  1 53 

33 

55.58 

1.3379 

3o 

57.10 

1.4821 

3o 

58.32 

1.7237 

79. 

59.51 

2.0  507 

27 

54.37 

1. 2175 

36 

55.59 

1.3396 

35 

57... 

1.4842 

3i.  0 

58.34 

1 .7307 

80. 

59.52 

2.0524 

3o 

54.39 

1. 2197 

39156.00 

i.34i3 

4o 

57.12 

1 .4863 
1.4883 

3o 

58.36 

1.7375 

81. 

59.53 

2.0539 

Q.33 

54-41 

1. 2219 

12.42  56.01 

1 .3430 

17.45 

57.13 

32.  0 

58.38 

1.7442 

82. 

59.53 

2.0553 

36 

54.42 

1.2241 

45(56.02 

1.3447 

5o 

57.14 

1 .4904 

3o 

58.39 

1.7508 

83. 

59.54 

2.0565 

39 

54.44 

1.2263 

48  56.03  1 .3464 

55 

57.15 

1.4924 

33.  0 

58.4 1 

..7573 

84. 

59.55 

2.0575 

42 

54.46 

1.2285 

5 1  56.04 

1. 34s  I 

18.  0 

57.16 

1.4944 

3o 

58.43 

1.7637 

85. 

59.56 

2.o58/i 

45 

54.47 

i.23o6 

54  56.o5 

1.3498 

10 

57.17 

1.4984 

34.  0 

58.44 

1 .7699 

86. 

59.57 

2.0592 

48 

54.49 

1.2328 

57  56.06 

i.35i4 

20 

57.19 

i.5o24 

3o 

58.46 

i.776(; 

87. 
88. 

59.58 
59.58 

2.0597 

9.5. 

54.5o 

1.2349 

1 3.  0  56.07 

i.353o 

i8.3o 

57.21 

i.5o63 

35.  0 

58.47 

1.7821 

2.0601 

54 

54.52 

1.2370 

5|  56.08 

1.3558 

40 

57.22 

I.5l02 

3o 

58.49 

1. 788 1 

89. 

59.59 

2.o6o3 

57 

54.54 

I.239I 

10  56. 10 

L_ 

1.3586 

5o 

57.24 

i.5i4i 

36.  0 

58. 5o 

1.7939 

90. 

60.00 

2.o6o3 

Page  92] 

TABLE  XVII 

When  the  Planet  Venus  or  Mars  is  used,  and  the  Parallax  is  nearly  e 

qual  to  15". 

Parallax  15' 

»Ap. 
Alt. 
DIM 
5.  o 

Cor. 
M  S 

5o.2  2 

Log. 

*Ap. 
Alt. 

Cor. 

Log:. 

Alt. 

Cor. 

hog. 

*Ap. 
Alt. 

Cor. 

Log. 

-A  p. 
Alt. 

Cor. 

Log. 

1.8257 

D  M 

M  S 

D  M 

M  S 

D  31 

M   S 

D  M 

M  te 

0.9688 

10.  o!55,oo 

1.2483 

i3.i5 

56.16 

1.3706 

19.  0 

57.30 

i.53i4 

36.3o 

58.55 

10 

5o.38 

0.9810 

3 

55.C2 

i.25o4 

20 

56.18 

1.3734 

10 

57.31 

1.5353 

37.  0 

58.57 

:.83i7 

20 

50.53 

0.9929 

6 

55.03 

1.2525 

25 

56.iq 

1.3762 

20 

57.33 

1.5392 

3o 

58.58 

1.8376 

3o 

5i.o8 

1 .0046 

9 

5b.o5 

1.2546 

3o 

56.21 

1.3789 

3o 

57.34 

1.543: 

38.  0 

58.59 

1.S434 

4o 

5l.22 

1. 0160 

12 

55.06 

1.2567 

35 

56.22 

i.38i6 

40 

57.36 

1.5470 

3o 

59. 0( 

1.849: 

5o 

51.35 

1.0272 

i5 

55.07 

1.2588 

4o 
13.45 

56.24 

1.3843 

5o 

57.37 

i.55o8 

39.  0 

59.02 

1 .8548 

6.  o 

5 1. 48 

I.0382 

10.18 

55.09 

1.2609 

56.25 

1.3870 

20.  0 

57.39 

1.5546 

39.30 

59.03 

1.8603 

10 

5  2. CO 

1 .0490 

21 

55.10 

i.263o 

5o 

56.26 

1.3897 

10 

57.40 

1.5583 

4o.  0 

59.04 

I.S658 

20 

52.12 

1.0595 

24 

55.12 

i.265i 

55 

56.28 

1.3923 

20 

57.41 

1.5620 

3o 

59.05 

1.8712 

3o 

52.23 

1.069S 

27 

55.13 

1.2672 

14.  0 

56.29 

1.3949 

3o 

57.43 

1.5657 

4i.  0 

59.06 

1.8765 

4o 

52.34 

1 .0799 

3o 

55. i5 

1.2693 

5 

56.3o 

1.3975 

40 

57.44 

1.5694 

3o 

59.07 

:.88i7 

5o 

52.44 

1.0898 

36 

55.16 

1. 2713 

10 

56.32 

1.4001 

5o 

57.45 

1.5730 

42.  0 

59.08 

1.8868 

7-  o 

52.54 

1.0995 

I0.36 

55.17 

1.2733 

i4.i5 

56.33 

1.4027 

21.  0 

57.47 

1.5766 

42.3o 

59.09 

1.89:8 

ID 

53.03 

1. 1090 

39 

55.19 

1.2754 

20 

56.34 

i.4o53 

10 

57.48 

i.58o2 

43.  0 

59.10 

1 .8968 

20 

53.12 

1. 1184 

42 

55.20 

1.2774 

25 

56.36 

1 .4079 

20 

57.49 

1.5837 

3o 

59.1 : 

1.90:7 

3o 

53.21 

1. 1 276 

45 

55.21 

1.2794 

3o 

56.37 

i.4io5 

3o 

57.50 

1.5872 

44.  0 

59.12 

1 .9065 

35 

53.25 

I.j322 

48 

55.23 

1. 2814 

35 

56.38 

i.4i3o 

40 

57.51 

1.5907 

3o 

59.13 

1.9:12 

4o 

7-45 

53.29 

53.34 

I.I366 

5i 

55.24 

1.2834 

4o 

56.4o 

i.4i55 

5() 

57.53 

1.5942 

45.  0 

59.14 

[.9159 

i.i4io 

10.54 

55.25 

1.2854 

14.45 

56.41 

1.4180 

22.  0 

57.54 

1.5977 

46. 

59.15 

1.9251 

4ti 

53.36 

1. 1437 

57 

55.27 

1.2874 

5o 

56.42 

1.4205 

10 

57.55 

1 .601 1 

47. 

59.17 

:  .9340 

5i 

53.38 

I.I464 

II.  0 

55.28 

1.2893 

55 

56.43 

1.4230 

20 

57.56 

1 .6045 

48. 

59.19 

1.9426 

54 

53.41 

1. 1490 

3 

55.29 

1.2913 

i5.  0 

56.44 

1.4255 

3o 

57.57 

1 .6079 

49. 

59.20 

:  .9509 

57 

53.43 

i.i5i6 

6 

55.3o 

1.2933 

5 

56.46 

1.4279 

40 

57.58 

1.6112 

5o. 

59.22 

1.9590 

8.  o 
8.  3 

53.45 

1.1542 

9 

55.32 

1.2952 

10 
i5.i5 

56.47 

i.43o4 

5o 

57.59 

1.6145 

5i. 

59.23 

1 .9668 

53.48 

I.I567 

1 1 . 1 2 

55.33 

1.2971 

56.48 

1.4329 

23.  0 

58.00 

1.6178 

52. 

59.25 

1.9744 

6 

53. 5o 

1. 1593 

i5 

55.34 

1.2990 

20 

56.49 

1.4353 

10 

58.0I 

1.62:1 

53. 

59.26 

1.9817 

Q 

53.52 

1. 1619 

18 

55.35 

i.3oio 

25 

56. 5o 

1.4377 

20 

58.02 

1.6244 

54. 

59.27 

1.9888 

12 

53.54 

1. 1644 

21 

55.37 

1.3029 

3o 

56.5i 

1. 440 1 

3a 

58.03 

1.6276 

55. 

59.29 

1.9957 

i5 

53.57 

1 .  1 669 

24 

55.38 

i.3o48 

35 

56.52 

1.4425 

4o 

58.04 

i.63o8 

56. 

59.30 

2.0023 

i8 

53.59 

1. 1695 

27 

55.39 

1.3067 

4o 

56.53 

1.4449 

5o 

58.05 

I.6340 

57- 

59.31 

2.0087 

8.21 

54.01 

1. 1720 

ii.3o 

55.40 

i.3o86 

i5.45 

56.54 

1.4473 

24.  0 

58. 06 

1. 637 1 

58. 

59.32 

2.0149 

24 

54.03 

1. 1745 

33 

55.41 

i.3io5 

5o 

56.56 

1 .4497 

10 

58.07 

1.6402 

59. 

59.34  2.0209 1 

27 

54.05 

1.1770 

36 

55.42 

i.3i24 

55 

56.57 

1.4520 

20 

58.08 

1.6433 

60. 

59.35 

2.0267 

3o 

54.07 

1-1795 

39 

55.44 

i.3i43 

16.  0 

56.58 

1.4543 

3o 

58.09 

1.6464 

bi. 

59.3b 

2.o322 

33 

54.09 

1.1819 

42 

55.45 

1.3:62 

5 

56.59 

1.4566 

4o 

58.10 

1.6495 

62. 

59.37 

2.0375 

36 

54.11 

1.1843 

45 

55.46 

i.3i8o 

10 

57.00 

1.4589 

5o 

58.11 

1.6526 

63. 

59.38 

2.0427 

8.3g 

54.13 

1.1868 

11.48 

55.47 

1. 3198 

i6.i5 

57.01 

1.4612 

25.  0 

58.12 

1.6556 

64. 

59.39 

2.0476 

42 

54.15 

1.1892 

5i 

55.48 

1 .3217 

20 

57.02 

1.4635 

20 

58.14 

1.6616 

65. 

59.40 

2.o523 

45 

54.17 

1. 1916 

54 

55.49 

1.3235 

25 

57.03 

1.4658' 

4o 

58.i6 

1.6675 

66. 

'J9-4i 

2.0568 

48 

54.19 

1. 1940 

57 

55.5o 

1.3253 

3o 

57.04 

1 .468 1 

26.  0 

58.17 

1.6734 

67. 

59.42 

2.0612 

5i 

54.21 

1. 1964 

12.  0 

55.5i 

1 .3271 

35 

57.o5 

1.4704 

20 

58.19 

1.6792 

68. 

59.-43 

2.0654 

54 

54.23 

1. 1988 

3 

55.53 

1.3289 

4o 

57.06 

1.4726 
1.4748 

40 

58. 20 

:  .6849 

69. 

59.44 

2.0693 

8.57 

54.25 

1.201 2 

12,  6 

55.54 

1.3307 

16.45 

57.07 

27.  0 

58.22 

1 .6905 

70. 

59.44 

2.0730 

9-  o 

54.26 

I.2035 

9 

55.55 

1.3325 

5o 

57.08 

1.4770 

20 

58.24 

1 .6960 

71. 

59.4:) 

2.0766 

3 

54.28 

1 .2058 

12 

55.56 

1.3343 

55 

57.09 

1 .4792 

40 

58.25 

1.7015 

72. 

59. 461 2. 0800 

6 

54.3o 

1.2081 

i5 

55.57 

I.336I 

17.  0 

57.10 

1 .4814 

28.  0 

58.26 

1.7069 

73. 

59.47  2.0832 

9 

54.32 

I. 2104 

18 

55.58 

1.3379 

5 

57.11 

1.4836 

20 

58.28 

1. 7123 

74. 

59.48 

2.0862 

12 

54.34 

1.2127 

21 

55.59 

1.3397 

10 

57.12 

1.4858 

4o 

58.29 

1.7176 

75. 

59.49 

2.0890 

9.i5 

54.35 

1.2l5o 

12.24 

56. 00 

i.34i5 

17.15 

57.12 

1.4880 

29.  0 

58.3i 

1.7228 

76. 

59.49 

2.09:6 

i8 

54.37 

1.2173 

27 

56. 01 

1.3433 

20 

57.13 

1.4902 

3o 

58.33 

i.73o5 

77- 

59.50 

2.0940 

21 

54.3q 

1.2196 

3o 

56. 02 

1.3450 

25 

57.14 

1.4924 

3o.  0 

58.35 

1.7381 

78. 

59.51 

2-0962 

24 

54.41 

1. 2219 

33 

56.03 

1.3468 

3o 

57.15 

1.4945 

3o 

58.37 

1.7456 

79- 

59.52 

2-0983 

27 

54.42 

1.2242 

36 

56.04 

1.3485 

35 

57.16 

1.4966 

3i.  0 

58.38 

1.7529 

80. 

59.53 

2-ioo3 

3o 

54.44 

1.2264 

39 

56.05 

i.35o3 

4o 

57.17 

i.49«7 

3o 

58 .40 

1.7601 
1 .7671 

81. 

59.53 

2.1021 

9.33 

54.46 

1.2286 

12.42 

56.06 

1 .3520 

17.45 

57.18 

i.5oo8 

32.  0 

58.42 

82. 

59.54 

2.io36 

36 

54.47 

i.23o8 

45 

56.07 

1.3537 

5o 

57.19 

1.5029 

3o 

58.44 

1 .7740 

83. 

59.55 

2.1049 

39 

54.49 

I.2330 

48 

56.08 

1.3554 

55 

57.19 

i.5o5o 

33.  0 

58.45 

1.7809 

84. 

59.56 

2.1060 

42 

54. 5i 

1.2352 

5i 

56.09 

1.3571 

18.  0 

57.20 

1.5071 

3o 

58.47 

1.7876 

85. 

59-56 

2.1070 

45 

54.52 

1.2374 

54 

56.10 

I.358S 

10 

57.22 

I.5lI2 

34.  0 

58.48 

1.7942 

86. 

59-57 

2.1078 

48 

54.54 

1.2396 

57 

56.11 

i.36o5 

20 

57.24 

1.5 1 53 

3c 

38.5o 

1 .8007 

87. 

59-58 

2.11)84 

9.5i 

54.55 

1.2418 

i3.  0 

56.12 

1.3622 

18.30j57.25 

1.5194 

35.  0 

58.5i 

1 .8071 

88. 

59-5Q 

2.1089 

54 

54.57 

1.2440 

5 

56.13 

I.3650 

4057.27 

1.5235 

3o 

58.53 

i.8i34 

89. 

59.59 

2.1092 

57 

54.58 

1. 2461 

io56.i5| 

1 .3678 

5o  57.29 

1.5275 

36.  0 

58.54 

1 .8106 

90. 

So.oo 

2.1093 

TABLE  XVII.                     ■                 [Page  93 

When  the  Planet  Venus  or  Mars  is  used,  and  thi 

i  Parallax  is  nearly  equal  to  20". 

Pakallax  20'' 

• 

Alt. 
DM 

Cor. 
M    t: 

Log-. 

*Ap. 
Alt. 

Cor. 

Log. 

*Ap. 

Alt. 

Cor. 

Log. 

*Ap. 
Alt. 

D  J\l 

Cor. 

1 
Log. 

*Ap. 
Alt. 
Dl\I 

36.3c 

Cor. 

Log. 
1.8535 

D   M  M    S 

D  I\l 

M   S 

31    S 

58.5c; 

5.  0 

50.27 

0.9725 

10.  0  55. o5 

1.2554 

i3.i5 

56.2  1 

i.38o2 

19.  0 

57.35 

1.5453 

10 

50.43  0.9848 

3 

55.06 

1.2576 

20 

56.22 

i.383o 

10 

57.37 

1.5494 
i.55j4 

37.  0 

59.01 

1.8599 

20 

50.58 

0.9969 

6 

55.08 

1.2597 

25 

56.23 

1.3858 

20 

57.38 

3o 

59.02 

1 .8662 

3o 

5i.j3 

1 .0087 

9 

55.09 

1.2619 

3o 

56.25 

1.3886 

3o 

57.39 

1.5574 

38.  0 

59.03 

1.8724 

4c) 

51.27 

1.0202 

12 

55.11 

1.2640 

35 

56.27 

1.3914 

4o 

37.41 

i.56i4 

3o 

59.04 

1.8785 

5o 

5 1.40 

i.o3i5 

i5 
10.18 

55.12 

1.2662 

4o 
13.45 

56.28 
56.3o 

1 .394 1 
1.3969 

5o 

57.42 

1.5653 

39.  0 
39  3o 

59.05 

1 .8846 

6.  0 

51.53 

1.0426 

55.14 

1.2683 

20.  0 

57.43 

1 .5692 

59.1 !( 

1 .8906 

10 

52. o5 

1.0535 

21 

55.15 

1.2704 

5o 

56.3 1 

1 .3996 

10 

57.45 

1.5731 

4o.  0 

59.07 

1 .8964 

20 

52.17 

1.0641 

24 

55.17 

1.2725 

55 

56.32 

1.4023 

20 

57.46 

1.5770 

3o 

Sg.og 

1.9021 

3o 

62.28 

1.0745 

27 

55.18 

1.2746 

i4-  0 

56.33 

1 .4o5o 

3o 

57.47 

1 .58o8 

4i.  0 

So.ic 

1.9078 

4o 

52.39 

1.0847 

3o 

55.20 

1.2767 

5 

56.34 

1 .4077 

4o 

57.49 

1 .5846 

00 

59.11 

1.9134 

5o 

7-  o 

02.49 
52.59 

1 .0947 
1 . 1 046 

33 

55.21 

1.2788 

10 

i4.i5 

56.36 
56.38 

1.4104 
1.4  i3i 

5o 

57.50 

1.5884 

42.  0 

59.12 

1.9189 
1.9243 

I0.36 

55.22 

1.2809 

21.  0 

57.51 

1.5921 

42.3o 

59.13 

10 

53.08 

1.1142 

39 

55.24 

i.283o 

20 

56.39 

i.4i57 

10 

57.53 

1.5958 

43.  0 

59.1411.9297! 

20 

D3.I7 

1.1237 

42 

55.25 

i.285i 

25 

56.4o 

i.4i83 

20 

57.54 

1.5995 

3o 

59.15 

1.9350 

3o 

33.26 

i.i33o 

45 

55.26 

1.2871 

3o 

56.42 

1.4209 

3o 

57.55 

1 .6o3 1 

44-  0 

59.15 

1.9402 

35 

53.3o 

1. 1376 

48 

55.28 

1. 2891 

35 

56-43 

1.4235 

4o 

57.56 

1 .6067 

3o 

59.16 

1 .9454 

4o 

53.35 

1.1422 

5i 

55.29 

1.291 1 

40 
14.45 

56-44 
56.46 

1.4261 
1.4287 

5o 

57.57 

i.6io3 

45.  0 

59.17 

1.950.1 

7-45 

53.39 

1.1467 

10.54 

55.3o 

1.2932 

22.  0 

57.58 

1.6139 

46. 

59.19 

1 .9604 

4« 

53.41 

1.1494 

57 

55.32 

1.2952 

5o 

56.47 

i.43i3 

10 

57.59 

1.6175 

47- 

59.21 

1 .9700 

5i 

•)3.43 

I.l520 

11.  0 

55.33 

1.2972 

55 

56.48 

1.4339 

20 

58.01 

1.6210 

48. 

59.22 

1 .9793 

54 

33.40 

1.1547 

3 

55.34 

1.2992 

i5.  0 

56.49 

1 .4364 

3o 

58.02 

1 .6245 

49. 

59.24 

1 .9884 

57 

53.48 

1.1574 

6 

55.35 

1.3oi2 

5 

56.5o 

1 .4389 

4o 

58.03 

1.6280 

5o. 

59.25 

1-9972 

S.  0 

53. 5o 

1. 1 600 

9 

55.37 

i.3o32 

10 
i5.i5 

56.52 
56.53 

I -44 1 4 
1.4439 

bo 

23.   0 

58.04 
58.o5 

i.63i4 

5i. 

59.27 

2.0057 

8.  3 

53.53 

1. 1 626 

11.12 

55.38 

i.3o52 

1.6348 

52. 

59-28 

2.0140 

6 

53.55 

I.I652 

i5 

55,3q 

1.3071 

20 

56.54 

1.4464 

.    IC. 

58. 06 

1.6382 

53. 

59.29 

2.0221 

9 

53.57 

1.1678 

18 

55.40 

1.3091 

25 

56.55 

1.4489 

20 

58.07 

1.6416 

54. 

59-30 

2.0299 

12 

53.59 

1. 1704 

21 

55.42 

i.3iii 

3o 

56.56 

i.45i4 

3o 

58.08 

1 .645o 

55. 

59-32 

2.0374 

ID 

5401 

1.1729 

24 

55.43 

i.3i3o 

35 

56.57 

1.4539 

40 

58.09 

1 .6483 

56. 

59.33 

2-0447 

l8 

54.04 

1. 1755 

27 
ii.3o 

55.44 

1 .3:49 
i.3i68 

40 
1 5.45 

56.58 
56.59 

1.4563 

5o 

58.10 

1.65 1 6 

57. 
58. 

59-34 
59.35 

2.c)5i8 
2.0587 

8.21 

54.06 

1. 1 780 

55.45 

1 .4587 

24.  0 

58.11 

1 .6549 

24 

54.08 

i.i8o5 

33 

55.46 

1 .3187 

5o 

57.01 

1.461 1 

10 

58.12 

1.6582 

59. 

59.36 

2.o653 

27 

54.10 

i.i83o 

36 

55.47 

1.3206 

55 

57.02 

1.4635 

20 

58.13 

1.661 5 

60. 

59.37 

2.0717 

3o 

54.12 

1.1855 

39 

55.49 

1.3225 

16.  0 

57.03 

1.4659 

3o 

58.14 

1 .6647 

61. 

59.38 

a.c>779 

33 

54.14 

1.1880 

42 

55. 5o 

1.3245 

6 

57.04 

1.4683 

4o 

58.15 

1 .6679 

62. 

59.39 

2.0838 

36 
8.3q 

54.16 
54.18 

1. 1 905 

45 

55.5i 

1.3264 

7T3r8'3 

10 

57.05 

1 .4707 

5o 

58.16 

1.6711 

63. 

59.402.08951 

1.1930 

11.48 

55.52 

16. i5 

57.06 

1 .4730 

25.    0 

58.16 

1 .6742 

64. 

59-41 

2.0950 

42 

54.20 

I.iq55 

61 

55.53 

I.3302 

20 

57.07 

1.4754 

20 

58.18 

i.68o5 

65. 

59-42 

2 . 1 oo3 

45 

54-2  2 

1.1979 

54 

55.54 

1.3321 

25 

57.08 

'.4777 

40 

58.20 

1.6867 

66. 

59-43 

2.io54 

48 

54-24 

1.2004 

57 

55.55 

1.3340 

3o 

57.09 

1 .4800 

26.    0 

58.22 

1.6928 

07. 

59-44 

2.1102 

5i 

54.26 

1.2028 

12.  0 

55.56 

1.3358 

35 

57.10 

1.4823 

20 

58.23 

1 .6988 

68. 

59-45 

2.ri48 

54 
8.57 

54.2.8 
54.3(. 

1.2052 

3 

55.57 

1.3377 

40 

57.11 

1 .4846 

40 

58.25 

1 .7047 

69. 

59.45 

2.1192 

1.2076 

12.  6 

55.59 

1 .3396 

16.45 

57.12 

1.4869 

27.  0 

58.26 

i.7if)6 

70. 

59.46 

2.1234 

9.  o 

54-3i 

1.2100 

9 

56.o(j 

1 .3414 

5o 

57.13 

1.4S92 

20 

58.28 

1.7164 

71. 

59.47 

2.1274 

3 

54-33 

1.2124 

12 

56.01 

1.3432 

55 

5714 

1.4915 

40 

58. 3o 

1.7222 

72. 

59-48 

2.l3l2 

('. 

54.35 

1.2147 

i5 

56. 02 

1 .3450 

17.  0 

57.14 

1.4938 

28.  0 

58.3i 

1.7279 
1.7335 

70. 

59.48 

2.134s 

9 

54-37 

1.2170 

18 

56. o3 

1.3468 

5 

57.15 

1.4961 

20 

58.32 

74. 

^9-49 

2.1382 

12 

54-39 

1.2194 

21 

36-04 

1 .3486 

10 

57.16 

1.49S3 

4o 

58.34 

1.7391 

75. 

59-5C, 

2.i4i4 

9.i5 

54.40 

1.2217 

12.24 

56. o5 

1 .35o4 

17. i5 

57.17 

i.5oo5 

29.    0 

58.35 

1.7446 

76.     159.51 

2.1444 

i8 

'^4.4^ 

1.2240 

27 

56.o6 

1.3522 

20 

57.18 

1.5028 

3o 

58.37 

1.7527 

77.     i59-5i 

2.1471 

21 

54.44 

1.2263 

3o 

56.07 

1.3540 

25 

57.19 

i.5c5o 

3o.  0 

58.39 

1 .7607 

78.     59.52 

2.1496 

24 

54.46 

1.2286 

33 

56.08 

1.3558 

3o 

57.20 

1.5072 

3o 

58.4i 

1.7685 

79.     59.53 

2.l520 

27 

54-47 

1.2309 

36 

56.09 

1.3576 

35 

57.21 

1.5094 

3i.  0 

58.43 

1.7762 

80.     59.54 

2.1542 

3o 
9.33 

54-49 
54.5i 

1.2332 

1.2355 

39 

56.10 

1.3594 
1 .3612 

40 

57.22 

i.5ii6 

3o 

32.   0 

58.44  1.7838 
58.46  1.7913 

81.     |59.54  2.i56i 

12.42 

56.11 

17-45 

57.23 

i.5i37 

82.      59.55'2.i578 

36 

54-52 

1.2377 

45 

56.12 

1.3629 

5o 

57.23 

i.5i59 

3o 

58.48^ 

1.7987 

83.      59.55I2.1594I 

39 

54.54 

1.2400 

48 

56.13 

1.3647 

55 

57.24 

i.5i8i 

33.  0158.49 

1.8059 

84. 

59-56j2.i6o7 

42 

54.56 

1.2423 

5i 

56.14 

1.3664 

18.  0 

57.25 

1.5202 

3o58.5i 

i.8i3o 

85. 

59.57  2  1618 

45 

54.57 

1.2445 

54 

56.15 

1 .368 1 

10 

57.27 

1.5245 

34.  o!58.52 

1.8200 

86. 

59.57 

2,1627 

48 

54.59 

1.2467 

57 

56.16 

1.3698 

20 
i8.3o 

57.28 

1.5287 

3o!58.54 

1.8269 

87. 

59-58 

2.1634 

9.5i 

55.00! 

1.2489 

i3.  0 

56.1- 

1.3715 

57.30 

1.5329 

35.  0^58.55 

1.8337 

88. 

59-59 

2.1639 

54  55.02 

I.25l  I 

5 

56.18 

1 .3744 

40 

57.39 

1. 5371 

3o;58.57i.84o4 
36.  0,58.58,1.8470 

89. 

59.59 

2.1642 

57  55.03 

1.2533 

TO  56. 20] 

1.3773 

5o 

57.33 

1.5412 

90. 

So.oo 

2.1643 

Page  94] 

TABLE  XVII 

, 

When  the  Planet  Venus  or  Mars  is  used,  and  the  Parallax  is  nearly  equal  to  25^'. 

Parallax  25'' 

• 

•Ap 

Alt. 

Cor. 

Log. 

*Ap. 
Alt. 

Cor. 

Log. 

*Ap. 
Alt. 

Cor. 

Log-. 

*Ap. 
Alt. 

Cor. 

Log. 

*Ap. 
Alt. 

Cor. 

Log. 

DM 

M   S 

D  M 

M    S 

D  M 

M   S 

D  M 

M   S 

D  M 

M   S 

5.  0 

5o.32  0.9763 

10.  0 

55.10 

1.2628 

i3.i5 

56.26 

1.3899 

19.  0 

67.39 

1.5697 

36.3o 

69.03 

1.8832 

lO 

5o.48  0.9887 

3 

55.11 

i.265o 

20 

56.28 

1.3928 

10 

67.41 

1.5639 

37.  0 

b9.ob 

1.8901 

20 

5i.o3 

1 .0009 

6 

55.13 

1. 2671 

25 

56.29 

1.3967 

20 

57.42 

1. 668 1 

3o 

59.06 

1.8968 

Jo 

5i.i8 

1. 01 28 

9 

55.14 

1.2693 

3o 

56.3 1 

1.3986 

3o 

67.44 

1.6722 

38.  0 

69.07 

1.9035 

4o 

bi.32 

1.0245 

12 

55.16 

1. 2715 

35 

56.32 

1.4014 

4o 

67.45 

1.5763 

3o 

69.08 

1. 9100 

5o 

51.45 

1.0359 

i5 

55.17 

1.2737 

40 

56.33 

1 .4042 

5o 

57.47 

i.58o4 

39.  0 

69.09 

1.9166 

6.  o 

51.58 

1. 047 1 

10.18 

55.19  1-2759 

i3.45 

56.35 

1 .4070 

20.  0 

67.48 

1.5844 

39.30 

69.10 

1.9229 

10 

52.11 

i.o58i 

21 

55.20  1. 2781 

5o 

56.36 

I.  ■1098 

10 

67.49 

1.5884 

4o.  0 

69.11 

1.9292 

20 

52.22 

1.0688 

24 

55.22 

1.2802 

55 

56.38 

1. 4126 

20 

57.61 

1.5924 

3o 

59.12 

1.9354 

3o 

52.34 

1 .0793 

27 

55.23 

1.2823 

i4-  0 

56.39 

i.4i54 

3o 

67.62 

1.6964 

4i.  0 

69.13 

1.94x6 

4o 

52.44 

1.0896 

3o 

55.24 

1.2844 

5 

56.4o 

1.4182 

40 

67.53 

1 .6oo3 

3o 

b9.i4 

1 .9477 

bo 

52.54 

1 .0998 

33 

55.26 

1.2865 

10 

56.42 

1.4209 

5o 
21.  0 

67.66 
67.66 

1 .6042 
1.6081 

42.  0 

69.16 

1.9537 

7-  0 

53.04 

1. 1098 

10.36 

55.27 

1.2886 

i4-i5 

56.43 

1.4236 

42.3o 

69'.  1 6 

1.9696 

lO 

53.13 

1.1195 

39 

55.29 

1.2907 

20 

56.44 

1.4263 

10 

67.57 

1.6119 

43.  0 

69.17 

1.9654 

20 

53.22 

1.1291 

42 

55.3o 

1.2928 

2b 

b6.4b 

1.4290 

20 

67.68 

1.6167 

3o 

59.18 

1.971 1 

3o 

53.3i 

I.I385 

45 

55.3i 

1.2949 

3o 

56.47 

1.4317 

3o 

58 .00 

1.6195 

44.  0 

69.19 

1.9768 

3h 

53.35 

r.i432 

48 

55.33 

1.2970 

3b 

56.48 

1.4344 

4o 

58.01 

1.6233 

3o 

69.20 

1.9824 

4o 

53.40 

i.i47« 

5i 

55.34 

1.2991 

4o 

b6.49 

1 .4370 

60 
22.  0 

58.02 
58.03 

1.6271 

45.  0 

69.21 

1.9879 

7-45 

53.44 

I.l522 

10.54 

55.35 

I.3oi2 

14-45 

56.5o 

1.4397 

i.63o8 

46. 

69.22 

1 .9987 

4» 

53.46 

i.i55o 

57 

55.36 

i.3o32 

5o 

56.62 

1.4423 

10 

68.04 

t.6346 

47. 

69.24 

2.0092 

5i 

53.48 

1. 1 577 

II.  0 

55.38 

i.3o52 

55 

56.53 

1-4449 

20 

68.06 

1.6382 

48. 

69.26 

2.0194 

54 

53.5i 

1.1604 

3 

55.39 

1.3073 

i5.  0 

56.54 

1-4475 

3o 

58.06 

1.6418 

49- 

69.27 

2.0294 

67 

53.53 

i.i63i 

6 

55.4o 

1.3093 

5 

56.55 

1.4501 

40 

58.07 

1.6454 

60. 

69.28 

2.0391 

8.  o 

53.55 

I.I658 

9 
II. 12 

55.41 

i.3ii3 

10 

56.56 

1.4627 

60 

23.    0 

68.08 

1 .6490 

61. 

59.3c 

2.0485 

8.  3 

53.58 

1. 1684 

55.43 

i.3i33 

i5.i5 

56.57 

1.4552 

68.09 

1.6626 

62. 

69.31 

2.0677 

6 

54.00 

1.1711 

i5 

55.44 

i.3i53 

20 

66.69 

1.4578 

10 

58.10 

1.6661 

63. 

69.32 

2.0666 

9 

54.02 

1-1737 

18 

55.45 

1.3173 

25 

67.00 

1 .4604 

20 

58.11 

1.6696 

54- 

69.33 

2.0762 

12 

b4.o4 

1.1763 

21 

55.46 

1.3193 

3o 

67.01 

1.4629 

3o 

58.12 

1. 663 1 

55. 

69.34 

2.0836 

i5 

54.06 

1. 1790 

24 

55.48 

I.32I3 

35 

57  02 

1.4654 

4o 

58.13 

1.6666 

66. 

69.36 

2.0918 

i8 

54.09 

1. 1816 

27 

55.49 

1.3233 

40 

67.03 

1 .4679 

5o 

68.14 

1. 670 1 

67. 

59.37 

2.0997 

8.21 

54.11 

1. 1842 

ii.3o 

55. 5o 

1.3253 

1 5.45 

67.04 

1 .4704 

24.  0 

68. i5 

1.6735 

68. 

59.3s 

2.1073 

24 

54.13 

1. 1868 

33 

55. 5i 

1.3272 

5o 

67.06 

1.4729 

10 

68.16 

1 .6769 

59. 

69.39 

2.1 147 

27 

54.15 

1. 1893 

36 

55.52 

1.3292 

bb 

67.06 

1-4754 

20 

58.17 

1 .6803 

60. 

69.40 

2.1219 

3o 

54.17 

1.1918 

39 

55.53 

i.33ii 

16.  0 

67.07 

i-477« 

3o 

68.18 

1.6837 

61. 

69.41 

2.1288 

33 

54.19 

1. 1943 

42 

55.55 

I.333I 

5 

67.08 

i.48o3 

4o 

68.19 

1.6870 

62. 

59.41 

2.1356 

36 
8.39 

54.21 

1. 1968 

45 

55.56 

I.3350 

10 
i6.i5 

67.09 

1.4827 

60 

25.    0 

68.20 
68.21 

1 .6903 
7I3936 

63. 

69.42 

2.1419 

54.23 

1. 1993 

11.48 

55.57 

1.3370 

67.10 

1.4862 

64. 

59.43 

2.1481 

42 

b4.2b 

1.2018 

bi 

55.58 

1.3389 

20 

67.1. 

1.4876 

20 

58.23 

1.7002 

65. 

59-44 

2.1641 

4b 

54.27 

r.2o43 

54 

55.59 

1.3408 

25 

67.12 

1 .4900 

40 

58.25 

1.7067 

66. 

69.46 

2.1699 

48 

54.29 

1.2068 

57 

56. 00 

1.3427 

3o 

67.13 

1.4924 

26.    0 

68.26 

i.7i3i 

67. 

69.46 

2.1654 

bi 

54.3 1 

1.2092 

12.  0 

56.01 

1.3446 

35 

57-14 

1.4948 

20 

68.28 

1.7194 

68. 

69.46 

2.1707 

b4 
8.57 

54.33 

r.2117 

3 

56.02 

1.3465 

4o 

57-15 

1.4972 

4o 

68.29 

1.7266 

69. 

59.47 

2.1767 

54.35 

I.2l4l 

12.  6 

56.03 

1.3484 

16.45 

57.16 

1.4996 

27.  0 

68.3 1 

1.7318 

70. 

69.48 

2.i8o5 

9.  0 

54.36 

I.2I65 

9 

56.04 

I.3502 

5o 

67.17 

1.6019 

20 

58.32 

1.7379 

71. 

69.49 

2.i85i 

3 

54.38 

r.2189 

12 

56.05 

I.352I 

55 

67.18 

i.5o42 

40 

58.34 

1.7440 

72. 

69.49 

2.1894 

6 

b4.4o 

r.22i3 

i5 

56.06 

1.3539 

17.  0 

67.19 

i.6o65 

28.  0 

58.36 

1.7600 

73. 

,59.60 

2.1935 

9 

b4.42 

1.2237 

18 

56.08 

1.3558 

5 

67.20 

i.5o88 

20 

58.37 

1.7669 

74. 

69.61 

2.1973 

I? 

9.. 5 

b4.43 

1. 2261 

21 

56.09 

1-3577 

10 

67.21 

1.6111 

4o 

58.38 

1.7617 

75. 
76. 

59.6. 

2.2009 

54.45 

1.2285 

12.24 

56. 10 

r.3596 

17-15 

67.22 

i.5i34 

29.  0 

68.39 

1.7675 

59.62 

2.2043 

18 

54-47 

1 .2309 

27 

56.11 

r.36i4 

20 

67.23 

1.6167 

3o 

58.4i 

1 .7760 

77. 

69.53 

2,2076 

21 

54.49 

1.2333 

3o 

56.12 

1.3632 

25 

67.24 

i.5i8o 

3o.  0 

68.43 

1.7845 

78. 

69.53 

2.2Io5 

24 

54.51 

1.2356 

33 

56.13 

1.3650 

3o 

67.26 

I.5203 

3o 

68.45 

1.7928 

79- 

69.54 

2.2l32 

27  54.52 

1.2379 

36 

56.14 

1 .3668 

35 

67.261.5226 

3i.  0 

58.47 

1. 8010 

80. 

69.54 

2.2l56 

3o 

54.54 

1.2402 

39 

56.15 

1 .3686 

40 

67.27 

1.6249 

3o 

58.49 

1 .8090 

81. 

69.55 

2.2178 

9.33 

54.56 

r.2425 

12.43 

56.16 

1.3704 

17-45 

57-27 

1.6271 

32.    0 

58. 5o 

1.8169 

82. 

59.66 

2.2198 

36 

54.57 

1.2448 

45 

56.17 

1.3722 

bo 

67-28 

1.5293 

3o 

68.62 

1.82,47 

83. 

69.56 

2.2216 

39 

54.59 

1-2471 

48 

56.18 

1.3740 

55 

67-29 

i.53i6 

33.  0 

58.54 

1.8324 

84. 

59.57 

2.2232 

42  55.00 

1-2494 

5i 

56.19  f.3758 

18.  0 

67.30 

1.5337 

3o 

58.55 

1 .8400 

85. 

69.67 

2.2245 

45|55.02 

i.25i6 

54 

56. 20  1.3776 

10 

57.32 

1.5381 

34.  0 

58.66 

1.8474 

86. 

59.68 

2.2266 

48 

55.04 

1.2538 

57 

56.2  1 

1.3794 

20 

67.33 

1.5426 

3o  58.68 

1.8547 

87. 

59.68 

2.2264 

9.5. 

55.05 

i.256i 

i3.  0 

56.21 

i.38ri 

i8.3o 

67.35 

1.5468 

35.  068.59 

1.8620 

88. 

69.69 

2.2269 

54  55.07 

1.2584 

5 

56.23 

1. 384 1 

40 

57.36 

1.6611 

3o  69.01 

1.8692 

89. 

69.692.2273 

57  55.08  1 .2606 

10  56.25  1.3870 

5o57.38ji.5554 

36.  0  69.02 

1 .8763 

90. 

60.00j2.2274 

TABLE   XVII 

[Page  95 

When  the  Planet  Venus  is  used,  and  the  P 

arallax  is  nearly  equal  to  30". 

Parallax  30' 

All. 
D.M 

("or. 

Log. 

'A  p. 

Alt. 

U  M 

Cor. 

Log. 

'iiS: 

Cor. 

Log. 

*Ap. 
All. 

Cor. 

Log. 

*Ap. 
Alt. 

Cor. 

Log. 

M    S 

1)  M 

M   S 

D  M 

M    S 

U  M 

M   S 

5.  o 

5<).37 

0.9801 

10.  o55.i5 

1.2702 

i3.i5 

56.3 1 

1 .3999 

19.  0 

57.44 

1.5745 

36.3o 

59.07  I  91 5i 

lo  5o.53 

0.9926 

3 

55.1b 

1.2724 

20 

56.32 

1 .4029 

10 

5746 

1.5789 

37.  0 

59.08  1.9225 

20 

5 1. (.8 

1.0049 

6 

55.18 

1.2746 

25 

56.34 

1 .4o58 

20 

57-47 

1.5832 

3o 

59.10  1.9297 

3(.- 

51.23 

1.0170 

9 

55.19 

1.2769 

3o 

56.35 

1.4087 

3c 

57.49 

1.5875 

38.  0 

59.11  1.9369 

4<) 

51.37 

1.0288 

12 

:)D.2i 

1.2791 

35 

36.37 

1.4116 

40 

57.50 

1. 5918 

3o 

39.12  1.9440 

5.- 

5i.5<. 

1  .o4o3 

i5 

DD.22 

1.2813 
i.28"35 

40 

56.38 

1.4145 

5o 

57.52 

1.5960 

39.  0 

59.13^1.9511 

0.  0 

52.03 

i.o5i6 

10.18 

55.24 

13.45 

56.4o 

1.4174 

20.  0 

57.53 

1 .6002 

39.30 

59.14 

1 .9580 

10 

52.15 

1.0627 

21 

55.2^) 

1.2857 

5c 

56.4 1 

1.4203 

10 

57.54 

1 .6044 

4o.  0 

59.15 

1 .9648 

n> 

52.27 

1.0735 

24 

55.26 

1.2879 

55 

56-42 

1. 423 1 

20 

57.56 

i.6o85 

3o 

59.16 

1.9716 

3n 

52.38 

1 .0842 

27 

55.28 

1.2900 

14.  0 

56.44 

1.4259 

3o 

57.57 

1.6126 

4i.  0 

59.17 

1.9783 

4o 

52.49 

1 .0947 

3o 

33.29 

1.2922 

5 

5645 

1.4287 

4o 

57.58 

1.6167 

3o 

59.18 

1 .9849 

5(. 

52.59 

i.i(»l9 

6i 

53. 3i 

1.2944 

10 

36.46 

i.43i5 

5o 

57.59 

1.6207 

42.  0 

39.19 

1.9914 

7.  0 

53.09 

1.1149 

10.36 

55.32 

1.2966 

14. i5 

56.48 

1.4343 

21.  0 

58.0I 

1.6247 

42. 3o 

59.20 

1 .9978 

10 

53.18 

1. 1248 

39 

55.33 

1.2987 

20 

56.49 

1. 437 1 

10 

58.02 

1.6287 

43.  0 

59.21 

2.0042 

PCI 

53.27 

I.I345 

42 

55.35 

i.3oo8 

25 

56. 5o 

1 .4399 

20 

58.03 

1.6327 

3o 

39.22 

2.oio5 

3i. 

53.36 

i.i44i 

45 

55.36 

1.3029 

3o 

56.52 

1-4427 

3o 

58.04 

1.6366 

44.  0 

59.23 

2.0167 

35 

53.40 

1.1488 

48 

55.37 

i.3o5o 

35 

56.53 

1-4454 

40 

58.05 

1 .64o5 

3o 

59.24 

2.0228 

40 

53.45 

i.i534 

61 

55.39 

1. 3071 

4o 

56.54 

1. 448 1 

5o 

58.07 

16444 

45.  0 

59.24 

2.0289 

7-45 

53.49 

I.I579 

10.54 

55.40 

1.3092 

14.45 

56.55 

1.4508 

22.  0 

58. 08 

1 .6483 

46. 

59.26 

2.o4o8 

4« 

53.5i 

1. 1 608 

57 

55.41 

i.3ii3 

5o 

56.57 

1.4535 

10 

58.09 

1.6522 

47- 

39.27 

2.0524 

5i 

53.53 

I.I636 

II.  0 

55.43 

i.3i34 

55 

56.58 

1.4562 

20 

58. 10 

1 .656o 

48. 

39.29 

2.0637 

54 

53.56 

I.I663 

3 

55.44 

i.3i55 

i5.  0 

56.59 

1.4589 

3o 

58.11 

1.6598 

49. 

59.30 

2.0747 

57 

53.58 

1 . 1 690 

6 

55.45 

1.3176 

5 

57.00 

1.4616 

4o 

58.12 

1.6636 

5o. 

59.32 

2.0855 

8.  0 

54.00 

1-1717 

9 

55.46 

1.3197 

10 

37.01 

1.4643 

5o 

58.i3 

1.6673 

5i. 

59.33 

2.0960 

8.  3 

54.02 

1. 1744 

11.12 

55.48 

1. 3217 

i5.i5 

57.02 

1.4669 

23.    0 

58.14 

1.6710 

52. 

59.34 

2.1062 

6 

54.05 

1.1771 

i5 

55.49 

1.3237 

20 

57.03 

1.4695 

10 

58. i5 

1.6747 

53. 

59.35 

2.1162 

9 

54.07 

1. 1798 

18 

55. 5o 

1.3258 

25 

57.05 

1. 472 1 

20 

58.i6 

1.6784 

54. 

59.36 

2.1259 

12 

54.09 

1. 1824 

21 

55.5i 

1.3278 

3o 

57.06 

1-4747 

3o 

58.17 

1.6820 

55. 

59.37 

2.i353 

i5 

54.11 

i.i85i 

24 

55.53 

1.3298 

35 

57.07 

1 .4773 

4o 

58.18 

1.6857 

56. 

59.38 

2.1445 

18 

54.13 

1.1877 

27 

55.54 

i.33i8 

40 

57.08 

1.4799 

5o 
24.  0 

58.19 

1 .6893 

57. 

59.39 

2.1534 

8.21 

54.16 

1. 1 903 

II. 3o 

55.55 

1.3338 

i5.45 

57.09 

1.4824 

58. 20 

1.692Q 

58. 

59.40 

2.1621 

24 

54.18 

1. 1 929 

33 

55.56 

1. 3358 

5o 

57.10 

1.4850 

10 

58.21 

1.6965 

59. 

59.41 

2.I-705 

27 

54.20 

1. 1955 

36 

55.57 

1.3378 

55 

57.11 

1.4876 

20 

58.22 

1 .7000 

60. 

59.42 

2.1787 

3(. 

54.22 

1.1981 

39 

55.58 

1.3398 

16.  0 

57.12 

1.4901 

3o 

58.23 

1.7035 

61. 

59.43 

2.1866 

3il 

54.24 

I;  2007 

42 

55.59 

1.3418 

5 

57.13 

1.4926 

4o 

58.24 

1 .7070 

62. 

59.44 

2.1942 

36 
8.39 

54.26 
54.28 

I.2o32 

45 

56.01 

1.3438 

10 

57.14 

1. 495 1 

5o 

58.25 

1.7105 

63. 

59.45 

2.2016 

i.2o57 

11.48 

56.02 

1.3458 

i6.i5 

57.15 

1.4976 

25.   0 

58.26 

1 .7140 

64. 

59.45 

2.2087 

42 

54.30 

1.2082 

5i 

56.03 

1.3478 

20 

57.16 

i.5ooi 

20 

58.27 

1.7208 

65. 

59.46 

2.2  I  56 

45 

54.32 

1.2107 

54 

56.o4 

1.3498 

25 

57.17 

1.5026 

4o 

58.29 

1.7276 

66. 

59.47 

2.222i 

48 

54.34 

[.2l32 

57 

56.05 

1.3517 

3o 

57.18 

i.5o5o 

26.  0 

58.3 1 

1.7343 

67. 

59.48 

2.228(3 

31 

54.36 

I.2l57 

12.  0 

56. 06 

1.3536 

35 

57.19 

1.5075 

20 

58.32 

1.7410 

68, 

59.48 

2.2347 

54 

54.38 

1. 2182 

3 

56.07 

1.3555 

4o 

57.20 

1. 5 1 00 

4o 

58.34 

1.7476 

69. 

59.49 

2.24o5 

8.57 

54.39 

1.2207 

12.  .6 

56.08 

1.3574 

16.45 

57.21 

1.5124 

27.  0 

58.35 

1 .754 1 

70. 

59.50 

2.2461 

9.  0 

54.41 

1.2232 

9 

56.09 

1.3593 

5o 

57.22 

i.5i48 

20 

58.37 

1 .7605 

71. 

59.50 

2.25l4 

3 

54.43 

1.2257 

12 

56.10 

I.36I2 

55 

57.23 

1. 5172 

40 

58.38 

1 .7669 

72. 

59.51 

2.2  565 

•    () 

54.45 

1. 2281 

i5 

56.11 

1. 363 1 

17.  0 

57.24 

1.5196 

28.  0 

58 .40 

1.7732 

73. 

59.51 

2.2613 

9 

54.47 

i.23o5 

18 

56.12 

1.3650 

5 

57.25 

1.5220 

20 

58.41 

1.7794 

74. 

59.52 

2.2658 

12 

54.48 

r.2329 

21 

56.13 

1.3669 

10 

57.26 

1.5244 

40 

58.43 

1.7856 

75. 

59.52 

2.2701 

54.5o 

1.2353 

12.24 

56.14 

1.3688 

17.15 

57.27 

1.5268 

29.  0 

58.44 

1.7917 

76. 

59.53 

2.2741 

18 

54.52 

1.2377 

27 

56. 1 5 

1.3707 

20 

57.28 

1.5292 

3o 

58.46 

1 .8007 

77. 

59.54 

2.2778 

21 

54.54 

1. 240 1 

3o 

56.16 

1.3725 

25 

57-29 

i.53i5 

3o.  0 

58.48 

1.8097 

78. 

59.54 

2.2812 

24 

54.55 

1.2425 

33 

56.17 

1:3744 

3o 

57-29 

1.5338 

3o 

58. 5o 

i.8i85 

79. 

59.55 

2.2844 

27 

54.57 

1.2449 

36 

56.18 

1.3763 

35 

57.30 

1.5362 

3i.  0 

58. 5i 

1.8272 

80. 

59.55 

2.2873 

3o 

54.59 

1.2472 

39 

56.19 

1.3782 

.  4o 

57.31 

1.5385 

3o 

58.53 

1.8357 

81. 

59.56 

2.2899 

9.33 

55.00 

1.2495 

12.42 

56. 20 

i.38oo 

17-45 

57.32 

1.5408 

32.   0 

58.54 

1.8441 

82. 

59.562.29231 

36 

55.02 

i.25i8 

45 

56.21 

i.38i8 

5o 

57.33 

1. 543 1 

3o 

58.56 

1.8524 

83. 

59.57 

2.2944 

39 

55.04 

1.2541 

48 

56.22 

1.3837 

55 

57.34 

1.5454 

33.  0 

58.58 

1.8606 

84. 

59.57 

2.2962 

42 

55.05 

1.2  564 

5i 

56.23 

1.3855 

18.  0 

57.35 

1.5477 

3o  58.59 

1.8687 

85. 

39.58 

2.2977 

45 

55.07 

1.2587 

54 

56.24 

1.3873 

10 

57.36 

1.5523 

34.  0  59.01 

1.S767 

86. 

59.58 

2.2990 

48 

55.09 

1. 2610 

57 

56.25 

1.3891 

20 

57.38 

1.5568 

3o 

59.02 

1.8846 
1.8924 

87. 

59.58 

2 .3ooo 

9.5, 

55.10 

1.2633 

i3.  0 

56.26 

1.3909 

18. 3o 

57.39 

i.56i3 

35.  0 

59.03 

88. 

59.59 

2.3007 

54 

55.12 

1.2656 

5 

56.28 

1.3939 

4o 

57.41 

1.5657 

3o 

59.05 

1.9000 

89. 

59.59  2.301  I  1 

57155. i3|i. 2679 

10 

56.29 

1.3969 

5o 

57.43,1.5701 

36.  0 

59.06 

1.9076 

90. 

5o.oo  2.3o3i 

Page  96] 

TABLE  XVII 

When  the  Planet  Venus  is  used,  and  the  Parallax  is  nearly  equal 

to  35''. 

Parallax  35'^ 

• 

*Ap. 
Alt. 
DM 

Cor. 
31    S 

Log. 

*Ap. 
Alt. 

Cor. 

Log-. 

*Ap. 

Alt. 

Cor. 

Log. 

Alt. 

Cor. 

Log. 

A.?.-    Cor. 
D   M  M   S 

Log. 

D   JM 

M    S 

D  mIM    S 

D  M 

M   S 

5.  o 

5o.42 

0.9840 

ID.    0 

55.20 

1.2777 

i3.i5l56.36 

1.4101 

19.  0 

57.49 

1.5899 

36.3o 

59.114.9495 1 

10 

50.58 

0.9966 

3 

55.21 

1.2800 

20I56.37 

i.4i3i 

10 

57.50 

1 .5944 

:?7.  0 

59.12 

1.9575 

20 

5i.i3 

1 .0090 

6 

55.23 

1.2823 

25 

56.39 

1.4161 

20 

57.52 

1.5989 

3o 

59.14 

1 .9654 

3o 

51.28 

1.0212 

9 

55.24 

1.2846 

3o 

56 .40 

1.4191 

3o 

57.53 

i.6o33 

38.  0 

59.15 

1.9732 

4o 

5i.42 

i.o33i 

12 

55.26 

1.2860 

35 

56.42 

1.4221 

4o 

57.55 

1 .6077 

3o 

59.16 

1 .9809 

5o 

5i.55 

1.0447 

i5 

55.27 

1.2891 

4o 

56.43 

1.4251 

5o 

57.56 

1.6121 

39.  0 

59.17 

1 .9886 

6.  0 

52.o8 

i.o56i 

10.18 

55.29 

1.2913 

i345 

56.44 

1.4281 

20.  0 

57.57 

i.6i65 

39.30 

59.18 

1. 996 1 

10 

52.2a 

1.0673 

21 

55.3o 

1.2936 

5o 

56.46 

1.4310 

10 

57.59 

1.6208 

4o.  0 

59.19 

2.oo36 

■20 

52.32 

1.0783 

24 

55.3i 

1.2958 

55 

56.47 

1.4339 

20 

58.00 

1.6251 

3o 

59.20 

2.0110 

3o 

52.43 

1.0891 

27 

55.33 

1.2980 

i4.  0 

56.49 

1 .43G0 

3o 

58.02 

1.6294 

4i.  0 

59.21 

2.0183 

4o 

52.54 

1.0997 

3o 

55.34 

1.3002 

5 

56.5o 

1.4397 

4o 

5$.o3 

1.6336 

3o 

59.22 

2.0255 

5o 

53.04 

I.I  100 

33 

55.36 

i.3o24 

10 

56.5i 

1.4426 

5o 

58. 04 

1.6378 

42.  0 

59.23 

2.0327 

7-  0 

53.14 

1.1202 

10.36 

55.37 

i.3o46 

i4.i5 

56.53 

1.4455 

21.  0 

58.05 

1.6420 

42. 3o 

59.24 

2.0398 

10 

53.23 

l.l302 

39 

55.38 

i.3o68 

20 

56.54 

1.4483 

10 

58.00 

1.6462 

43.  0 

59.25 

2.0468 

20 

53.32 

i.i4oo 

42 

55.40 

1.3090 

25 

56.55 

1.4511 

20 

58.08 

i.65o3 

3o 

59.26 

2.0537 

3o 

53.4i 

1. 1497 

45 

55.41 

i.3iii 

3o 

56.56 

1.4540 

3o 

58.09 

1.6544 

44.  0 

59.26 

2.0606 

35 

53.45 

1.1545 

48 

55.42 

i.3i32 

35 

56.58 

1 .4568 

4o 

58.IO 

1.6585 

3o 

59.27 

2.0674 

4o 

53.49 

1.1592 

5i 

55.44 

i.3i54 

4o 

56.59 

1.4596 

5o 

58.11 

1.6626 

45.  0 

59.28 

2.0741 

7-45 

53.53 

I.I638 

10.54 

55.45 

1.3176 

14.45 

57.00 

1.4624 

22.  0 

58.12 

1.6666 

46. 

59.29 

2.0873 

4» 

53.56 

1.1667 

57 

55.46 

1.3197 

5o 

57.01 

1.4652 

.10 

58.13 

1 .6706 

47. 

59.31 

2.ioo3 

5i 

53.58 

1.1695 

11.  0 

55.48 

1.3218 

55 

57.02 

1 .4679 

20 

58.14 

1.6746 

48. 

59.32 

2.1 129 

54 

54-01 

1.1753 

3 

55.49 

1.3239 

i5.  0 

57.04 

1 .4706 

3o 

58.i5 

1.6785 

49. 

59.33 

2.1253 

57 

54.03 

1. 1751 

6 

55. 5o 

1.3260 

5 

57.05 

1.4734 

4o 

58.17 

1.6824 

5o. 

59.35 

2.1375 

8.  o 

54.05 

1.1778 

9 

55.5i 

1.3281 

10 

57.06 

1.4761 

.50 

58.18 

1.6863 

5i. 

59.36 

2.1493 

8.  3 

54.07 

i.i8o5 

11.12 

55.53 

i.33o2 

i5.i5 

57.07 

1 .4788 

23.  0 

58.19 

1 .6902 

52. 

59.37 

2.1609 

6 

54.10 

1.1832 

i5 

55.54 

1.3323 

20 

57.08 

i.48i5 

10 

58.20 

1. 694 1 

53. 

59.38, 

2.1723 

Q 

54.12 

1.1859 

18 

55.55 

1.3344 

25 

57.09 

1.4842 

20 

58.21 

1 .6980 

54. 

59.39 

2.1833 

12 

54.14 

1.1886 

21 

55.56 

1.3365 

3o 

57.10 

1.4869 

3o 

58.22 

1.7018 

55. 

59.40 

2.1940 

i5 

54.16 

1. 1913 

24 

55.58 

1.3386 

35 

57. 1 2I  1. 4896 

40 

58.23 

1.7056 

56. 

59.41 

2.2045 

j8 

54.18 

1. 1939 

27 

55.59 

1.3406 

4o 

57.13 

1.4922 

5o 

58.24 

1.7094 

57. 

59.42 

2.2148 

8.21 

54.21 

1 . 1 966 

1 1 .3o 

56.00 

1.3426 

i5.45 

57.14 

1.4948 

24.  0 

58.25 

1.7132 

58. 

59.43 

2.2248 

24 

54.23 

1.1993 

■63 

56.01 

1.3447 

5o 

57.15 

1.4975 

10 

58.26 

1.7169 

59. 

59.44 

2.2346 

27 

54.25 

1. 2019 

36 

56.02 

1.3467 

55 

57.16 

i.Sooi 

20 

58.27 

1.7206 

60. 

59.45 

2.2440 

3o 

54.27 

1.2045 

39 

56.03 

1.3487 

16.  0 

57.17 

1.5027 

3o 

58.28 

[.7243 

61. 

59.45 

2.2.532 

33 

54.29 

1. 2071 

42 

56.04 

1.3507 

5 

57.18 

i.5o53 

4o 

58.29 

1.7280 

62. 

59.46 

2.2621 

36 
8.3q 

54.3i 
54.33 

1.2097 

45 

56.06 

1.3527 

10 

57.19 

1.5079 

5o 

58.29 

1.7317 

63. 

59.47 

2.2708 

1.2123 

11.48 

56.07 

1.3547 

16. i5 

57.20 

i.5io4 

25.  0 

58.3o 

1.7353 

64. 

59.48 

2.2792 

42 

54.35 

1.2148 

5i 

56.08 

1.3567 

20 

57.21 

i.5i3o 

20 

58.32 

1.7425 

65. 

59.48 

2.2873 

45 

54.37 

1.2173 

54 

56.09 

1.3587 

25 

57.22 

i.5i56 

40 

58.34 

1.7496 

66. 

59.49 

2.2951 

48 

54.39 

1.2199 

57 

56. 10 

1 .3607 

3o  57.23 

i.5i8i 

26.  0 

58.35 

1.7567 

67. 

59.49 

2.3026 

5i 

54.41 

1.22Vf 

12.  0 

56.11 

1.3627 

35  57.24 

1.5206 

20 

58.37 

1.7637 

68. 

59.50 

2.3099 

54 

54.43 

1.2249 

3 

56.12 

1.3647 
1.3667 

4o  57.25 

I.523I 

4o 

58.38 

1.7706 

69. 

59.51 

2.3 1 68 

8.57 

54.44 

1.2275 

12.  6 

56. 1 3 

16.45 

57.26 

1.5256 

27.  0 

58.40 

1.7775 

70. 

59.51 

2.3235 

9.  0 

54.46 

1 .23oo 

9 

56.14 

1.3686 

5o 

57.27 

1.5281 

20 

58.4 1 

1.7843 

71- 

59.52 

2.3299 

3 

54.48 

1.2325 

12 

56.15 

1.3706 

55 

5728 

1 .53o6 

40 

58.43 

1.7910 

72. 

59.52 

2.3359 

6 

54.50 

i.235o 

i5 

56.16 

1.3725 

17.  0 

57.29 

1.5331 

28.  0 

58.44 

1.7977 

70. 

59.53 

2.3417 

9 

54.52 

1.2374 

18 

56.17 

1.3745 

5 

57.30 

1.5356 

20 

58.46 

1.8043 

74. 

59.53 

2.3471 

12 

54.53 

1.2399 

21 
12.24 

56.18 
56.19 

1.3764 
1.3783 

10 
17.15 

57.31 

1.5381 

4o 

58.47 

1.8108 

73. 

59.54 

2.3523 

9.i5 

54.55 

1.2423 

57.32 

i.54o5 

29.  0 

58.48 

1.8173 

76. 

59.55 

2.3571 

i8 

54.57 

1.2448 

27 

56.20 

i.38o2 

20 

57.33 

1.5430 

3o 

58.5o 

1.8269 

77- 

50.55 

2.36i6 

21 

54.59 

1.2472 

3o 

56.21 

I.382I 

2  5 

57.34 

1.5454 

3o.  0 

58.52 

1.8364 

78. 

59.55 

2.3658 

24 

55.00 

1.2496 

33 

56.22 

1 .384o 

3o 

57.34 

1.5478 

3o 

58.54 

1.8457 

79- 

59.56  2.3697 

27 

55.02 

1.2520 

36 

56.23 

1.3859 

35 

57.35 

i.55o2 

3i.  0 

58.55 

1.8550 

80. 

59.56,2.3732 

3o 

55.04 

1.2544 

39 

56.24 

1.3878 

40 

57.36 

1.5526 

3o 

58.57 

1.8641 

81. 

59.57I2.3764 

9.33 

55.05 

1.2568 

12.42 

56.25 

1.3897 

17.45 

57.37 

1.5550 

32.  058.59 

1.8732 

82. 

59.57,2.3793 

36 

55.07 

1.2592 

45 

56.26 

1.3916 

5o 

57.38  1.5574 

3o|59.oi 

1.8821 

83. 

59.572.3819 

39 

55.09 

1.2615 

48 

56.27 

1.3935 

55 

57.39 

1.5598 

33.  059.02 

1 .8909 

84. 

59.582.3841 

42 

55.10 

1.2639 

5i 

56.28 

1.3954 

18.  0 

57.39 

1.562  1 

30|59.o3 

1 .8996 

85. 

59.58  2.3859 

45 

55.12 

1.2662 

54 

56.29 

1.3973 

10 

57.41 

1.5668 

34.  0,59.05 

1..9081 

86. 

59.582.3874 

48 

55.14 

1.2685 

57 

56.3o 

1.3991 

20 

57.43 

1.5715 

3059.06 

1.9165 

87. 

59.59'2.3886 

9.5i 

55.15 

1.2708 

i3.  0 

56.3 1 

1 .4009 

18. 3o 

57.441.5761 

35.  0J59.08 

1.9249 

88. 

59.592.3895 

54 

55.17 

1. 2731 

5  56.33  i.4o4o 

4o 

57.461.5807 

3o  59.09  1.9332 

89. 

59.592.3900 

57 

55.18 

1.2754 

iO|56.34'i.4o7i 

5o 

57.48  1.5853 

36.  059.10I1.9414 

90. 

60.002.3903 

TABLE  XVIII.                        LPage  97 

When  the  Sun  is  used. 

©Ap 
Alt. 

Cor. 

Log. 

©Ap 
All. 

Cor. 

Log. 

0Ap. 
Alt. 

Cor. 

Log. 

0Ap 
Alt. 
D  31 

19.  c 

Cor. 
M  & 

Log. 

0Ap 
Alt. 

Cor. 

Log. 

D  M 

5.  o 

JI  & 

D  M 

M  S 

D  M 

M  fe 

D  IVI 

M  S 

5o.i6 

0.9645 

10.  0 

54.54 

1.2397 

i3.i5 

56.10 

1.3592 

57-24 

i.5i49 

36.3o 

58.5o 

1.7934 

10 

5o.32 

0.9766 

3 

54.55 

1.2418 

20 

56.12 

1.3619 

IC 

57-25 

1.5187 

37.  0 

58.5i 

1 .7990 

20 

5o.48 

0.9885 

b 

54-57 

1.2439 

25 

56.13 

1 .3646 

20 

57-27 

1.5225 

3o 

58.53 

1.8045 

3o 

5i.  3 

1 .0000 

9 

54.58 

1.2460 

3o 

56.15 

1.3672 

3o 

57-28 

1.5262 

38.  0 

58.54 

x.8xoo 

4o 

5i.i6 

I.QIl3 

12 

55.  0 

1. 248 1 

35 

56.16 

1 .3699 

4o 

57.30 

1.5299 

3o 

58.55 

1.8x54 

5o 
6.  o 

5i.3o 

I.0223 

i5 

55.  I 

i.25oi 

4o 

56.17 

1.3725 

5o 

57.31 

1.5336 

39.  0 

58.57 

1.8206 
1.8257 

51.42 

i.o33o 

io.i8 

55.  3 

1.2522 

i3.45 

56.19 

1.3751 

20.  0 

57.33 

1.537.2 

39.30 

58.58 

10 

51.55 

1 .0437 

21 

55.  4 

1.2543 

5o 

56. 20 

1.3777 

10 

57.34 

1.5408 

40.  0 

58.59 

x.83o7 

20 

52.  6 

r.o54i 

24 

55.  6 

1.2563 

55 

56.22 

i.38o3 

20 

57.35 

1.5444 

3o 

59.  0 

1.8357 

3o 

52.17 

1.0643 

27 

55.  7 

1.2583 

i4-  0 

56.23 

1.3828 

3o 

57.37 

1.5480 

4i-  0 

59.  I 

X.8406 

4o 

52.28 

1 .0742 

3o 

55.  8 

1.2603 

5 

56.24 

1.3853 

40 

57.38 

i.55i5 

3o 

59.  2 

X.8454 

5o 

52.38 

I  .oS4o 

33 

55.10 

1.2623 

10 

56.26 

1.3878 

5o 

57.39 

I.5550 

42.  0 

59.  3 

i.85oo 

7-  0 

52.48 

1.0935 

I0.36 

55.11 

1.2643 

i4-i5 

56.27 

1 .3904 

21.  0 

57-41 

1.5585 

42. 3o 

59.  4 

1.8546 

ID 

52.57 

r.1029 

39 

55.13 

1.2663 

20 

56.28 

1.3929 

10 

57.42 

1.5619 

43.  0 

59.5 

1.8593 

20 

53.  6 

1.1122 

42 

55.14 

1.2683 

25 

56.3o 

1.3954 

20 

57.43 

1.5653 

3o 

59.  6 

1.8638 

3o 

53.15 

1.1212 

45 

55.15 

1.2702 

3o 

56.3i 

1.3979 

3o 

57-44 

1.5686 

44.  0 

59.  7 

1.8683 

35 

53.19 

1. 1257 

48 

55.17 

1.2722 

35 

56.32 

1 .4004 

4o 

57-46 

1.5719 

3o 

59.  8 

1.8726 

4o 

53.23 

i.i3oi 

5i 

55.18 

1.2742 

40 
14.45 

56.33 

1.4029 

5o 

57.47 

1.5752 

45.  0 

59.  9 

1.8768 

7-45 

53.27 

I.I345 

10.54 

55.19 

1. 2761 

56.35 

i.4o53 

22.  0 

57.48 

1.5784 

46. 

59.x  X 

1.8848 

48 

53.3o 

i.i37i 

57 

55.20 

1.2780 

5o 

56.36 

1-4077 

10 

57-49 

1.5817 

47. 

59.x3 

1.8928 

6i 

53.32 

1. 1397 

II.  0 

55.22 

1.2799 

55 

56.37 

1.4101 

20 

57.50 

1.5849 

48. 

59.X5 

1 .9004 

54 

53.35 

1. 1423 

3 

55.23 

1. 2818 

i5.  c 

D6.38 

1.4125 

3o 

57.51 

i.588i 

49. 

59.16 

1 .908 1 

57 

53.37 

I.I448 

6 

55.24 

1.2837 

5 

56.39 

1.4149 

4o 

57.52 

1. 5913 

5o. 

59.18 

X.9X54 

«.  o 

53.39 

1. 1474 

9 

55.26 

1.2856 

10 

56.4i 

1.4173 

5o 

57.53 

1.5945 

5i. 

59.19 

1.9225 

8.  3 

53.41 

1-1499 

II. 12 

55.27 

1.2875 

i5.i5 

56.42 

1-4197 

23.  0 

57-54 

1.5976 

52. 

59.21 

1.9294 

6 

53.44 

i.i524 

i5 

55.28 

1.2894 

20 

56.43 

1.4221 

10 

57.56 

1 .6008 

53. 

59.22 

X.9362 

9 

53.46 

i.i55o 

18 

55.29 

1. 2913 

25 

56.44 

1.4244 

20 

57-57 

1.6039 

54. 

59.24 

1.9424 

12 

53.48 

1. 1575 

21 

55.3o 

1.2932 

3o 

56.45 

1.4267 

3o 

57-58 

1.6070 

55. 

5Q.25 

1.9484 

i5 

53. 5g 

1. 1599 

24 

55.32 

1. 2951 

35 

56  46 

1.4290 

4o 

57-59 

1.6101 

56. 

59.26 

1 .9544 

i8 

53.52 

1.1624^ 

27 

55.33 

1.2970 

4o 

56.47 

i.43i3 

5o 

58.  0 

1. 61 3 1 

57. 

59.28 

1 .9602 

8.21 

53.55 

1. 1649 

ii.3o 

55.34 

1.2988 

15.45 

56.49 

1.4336 

24.  0 

58.  I 

1.6161 

58. 

59.29 

X.9658 

=4 

53.57 

1. 1673 

33 

55.35 

1 .3007 

5o 

56.5o 

1.4359 

10 

58.  2 

1.6191 

59. 

59.30 

X.9715 

27 

53. 5q 

1. 1698 

36 

55.36 

i.3o25 

55 

56.5i 

1.4382 

20 

58.  3 

1. 622 1 

60. 

59.31 

1.976X 

3o 

54.  I 

1. 1722 

39 

55.38 

i.3o43 

16.  0 

56.52 

1 .4404 

3o 

58.  3 

1.6250 

61. 

59.33 

X.9807 

33 

54.  3 

1. 1746 

42 

55.39 

i.3o6i 

5 

56.53 

1.4427 

40 

58.  4 

1.6279 

62. 

59.34 

1.9854 

36 

54.  5 

1-1770 

45 

55.40 

1 .3079 

10 

56.54 

1.4449 

5o 
25.  0 

58.  5 

i.63o8 

63. 

59.35 

1.9901 

8.39 

54.  7 

1. 1794 

11.48 

55.41 

1.3097 

i6.i5 

56.55 

1. 447 1 

58.  6 

1.6336 

64. 

59.36 

1 .9946 

42 

54.  9 

1.1818 

61 

55.42 

i.3ii5 

20 

56.56 

1.4493 

20 

58.  8 

1.6393 

65. 

59.37 

1 .9986 

45 

54.11 

i.i84i 

54 

55.43 

i.3i33 

25 

56.57 

i.45i5 

40 

58.10 

1.6449 

66. 

59.38 

2.0025 

48 

54.i3 

I.I865 

57 

55.44 

i.3i5i 

3o 

56.58 

1.4537 

26.  0 

58.11 

i.65o5 

67. 

59.39 

2 .0064 

5i 

54.15 

1.18S8 

12.  0 

55.45 

1 .3169 

35 

56.59 

1.4559 

20 

58.13 

1.6559 

68. 

59.40 

2.0x00 

54 

54.16 

1.1912 

3 

D5.46 

1 .3187 

4o 

57.  0 

1. 458 1 

4o  58.15 

1. 661 2 

69. 

59.4X 

2.0x36 

8.57 

54.18 

1. 1935 

12.  6 

55.48 

i.32o5 

16.45 

57-  I 

J  .4602 

27.  0 

58.16 

1.6665 

70. 

59.42 

2.0173 

9.  0 

54.20 

1. 1958 

9 

D5.49 

1.3223 

5o 

57.  2 

1.4624 

20 

58.18 

1 .6718 

71. 

59.43 

2.0208 

3 

54.22 

1.1981 

12 

55.5o 

1.3240 

55 

57.  3 

1.4646 

40 

58.19 

1 .677 1 

72. 

59.44 

2.0238 

6 

54.24 

1.2004 

i5 

55.5t 

1.3257 

17.  0 

57-  4 

1.4667 

28.  0 

58.21 

1.6824 

73. 

59.45 

2.0268 

9 

54.26 

1.2026 

18 

55.52 

1.3275 

5 

57-  5 

1 .4688 

20 

58.22 

1.6874 

74. 

59.46 

2.0296 

12 

54.27 

1.2049 

21 

35.53 

1.3292 

10 

37-  6 

1 .4709 

4o 

58.24 

1.6923 

75. 

59.47 

).0322 

..o343 

9.. 5 

54.29 

1. 2071 

12.24 

55.54 

1.3309 

17-15 

57.  6 

1 .4730 

29.  0 

58.25 

1.6972 

76. 

59.48  : 

18 

54.31 

1.2094 

27 

55.55 

1.3326 

20 

^7.  7 

1-4751 

3o 

58.27 

1 .7046 

77- 

59.49 : 

..o363 

21 

54.33 

r.2ii6 

3o 

55.56 

1.3343 

25 

57.  8 

1.4772 

3o.  0 

58.29 

1.71x7 

78. 

59.50 : 

.0382 

24 

54.34 

1. 2139 

33 

55.57 

i.336o 

3o 

57.  9 

1-4793 

3o 

58.3i 

1.7187 

79- 

59.51  : 

.o4oo 

27 

54.36 

I.2l6l 

36 

55.58 

1.3377 

35 

37. 10 

1.4814 

3i.  0 

58.33 

1.7255 

80. 

59.52  : 

.0417 

3o 

54.36 

I.2J83 

39 

55.59 

1 .3394 

4o 

37.11 

1.4835 

3o 

58.35 

1 .732 1 

81. 
82. 

59.52  : 

59.53  : 

.0432 
.0446 

9.33 

54.39 

1.2205 

12.42 

56.  0 

1 .34 1 1 

17-45 

37.12 

1.4855 

32.  0 

58.36 

1.7387 

36 

54.41 

1.2227 

45 

56.  I 

1.3427 

5o 

37.13 

1.4876 

3o 

58.38 

1.7454 

83. 

59.54  : 

.o45o 

39 

54.43 

1.2248 

48 

56.  2 

1-3444 

55 

37-14 

1.4896 

33.  0 

58. 4o 

1.7520 

84. 

59.55  : 

.0453 

42 

54.44 

1.2270 

5i 

56.  3 

1.3461 

18.  0 

37-i4 

1.4916 

3o 

58.4i 

1.7582 

85. 

59.56  : 

.0456 

45 

54.46 

1. 2291 

54 

36.  4 

1.3478 

10 

37.16 

1.4956 

34.  0 

58.43 

1 .7643 

86. 

59.57  : 

.0458 

48 

54.48 

i.23i3 

57 

56.  5 

1.3494 

20 

37.18 

1 .4995 

3o 

58.44 

1.7702 

87. 

59.57  s 

.0460 

9.5i 

54.49 

1.2334 

i3.  0 

56.  6 

1.35x0 

i8.3o 

37.19 

i.5o34 

35.  0 

58.46 

1.7762 

88. 

59.58  : 

.o46x 

54 

54.5i 

1.2355 

5 

56.  7 

1.3538 

40 

37.21 

1.5073 

3o 

58.47 

1.7821 

89. 

59.59  = 

.0462 

57 

54.52 

1.2376 

10  56.  9I 

1.3565 

5o 

37.22 

i.Siii 

36.  0 

58.49 

1.7878 

90. 

60.  0  2 

.0462 

13 


rsge98]             TABLE  XIX.  Correction. 

ii    r- 

Table  A. 

Tai;leB. 

D  's  Horizontal  Parallax. 

Proportional  part  for  Seconds 
of  Parallax. 

For  Min. 
of  Alt. 

<^ 

Add. 

Add. 

D.A 

5 

I.  54' 

0  i4-35  I 

55' 
"3T35 

50' 
12. 3f 

57' 
11.36 

58' 

59' 

60' 

8^36 

61'  S.0"l"2" 
7.37^595857 

3" 
56 

4" 
55 

5" 
54 

6' 

5: 

1711 
T2 

8"|9"| 

M. 
0 

S. 
12 

10. 36 

9.36 

5i 

5o 

I 

0  l4.20  I 

3.20 

12.2c 

II  .20 

10.21 

9.21 

8.21 

7.21  10494847 

46 

45 

44 

4; 

42 

4i 

4o 

2 

1 1 
9 

2 

o  i4.  5  I 

3.  5 

12.  t 

II .  6 

10.  6 

9.  6 

8.  7 

7-  7  20  39  38  37 

36 

35 

34 

33 

32 

3i 

3o 

4 

8 

3 

0  i3.5i  I 

2.52 

II  .52 

10.52 

9.52 

8.53 

7.53 

6.53  3 

0  29  2 

827 

26 

25 

24 

2C 

22 

21 

20 

5 

6 

4 

o  1 3. 38  1 

2.39 

II  .3^ 

10.39 

9.39 

8.4o 

7.40 

6.40  4 

0  19  I 

817 

16 

i5 

i4 

IC 

12 

11 

10 

7 

4 
3 

5 
6^ 

0  i3.26  I 

2.26 

11.27 

10.27 

9.27 

8.27 

7.28 

6.28  5 

0  9 

8  7 

6 

5 

4 
54 

53 

2 

5I 

I 
57 

0 
5^ 

8 
9 

0 

2 

0 

9 

0  r3.i7  I 

2.18 

ii.jfc 

10.18 

9.19 

8.19 

7.19 

6.20 

o59  58|57l 

56 

55 

I 

0  i3.  6  I 

2.  7 

11.  - 

10.  7 

9.  8 

8.  8 

7.  « 

6.  8  io49|48  47 

46 

45 

44 

43 

42 

4i 

4o 

2 

7 

2 

0  12.55  I 

1.56 

10. 5e 

9.57 

8.57 

7.57 

6.58 

5.58  20  39138  37  36 

35 

34 

33 

32 

3i 

3o 

3 

3 

0  12.45  I 

1.46 

10. 4t 

9.46 

8.47 

7-47 

6.48 

5.48  3 

0  29  2 

827 

26 

25 

24 

23 

22 

21 

20 

5 

4 

4 

0  12.36  I 

1.36 

10.37 

9-37 

8.37 

7.38 

6.38 

5.39  4o  19  I 

817 

16 

i5 

14 

i3 

12 

II 

10 

7 

3 
2 

5 

7 

0  12.27  I 
0  12.18  I 

1.27 

10.2& 

9.28 
9.19 

8.28 

7.29 

6.29 

5.3o  5o  9 

8  7 

6 

5 

4 

3 

2 

I 

0 

8 
9 

1 
0 

1.19 

10.19 

8.20 

7.20 

6.21 

5.21 

1 

TABLE  XIX.  Logarithms. 

1^  '■< 

Table  C. 

Cor.  for  Sec 

Apparent  Altitude  of  ])  's  Centre. 

of  Par. 

«Ph 

Add. 

0  / 

0  1 

0  1 

0  / 

0  /  0  / 

0  ; 

0  / 

0  / 

0  / 

0  / 

0  / 

0  / 

• 

"54" 

S. 
0 

5  0 

3o84 

510 

3o58 

5  20 

3o33 

5  30 

3oo9 

5  40 

2987 

5  50 

2966 

6  0 

2946 

6  10 

2926 

6  20 

2908 

6  30 

6  40 

6  50 

7  0 

Sec. 

Cor. 

2891 

2874 

2859 

2844 

0 

i4 

10 

3o68 

3o4i 

3oi6 

2993 

2971 

2950 

2930 

291 1 

2892 

2875 

2859 

2843 

2828 

I 

i3 

20 

3o5i 

3o25 

3ooo 

2977 

2955 

2934 

2914 

2895 

2877 

2860 

2843 

2827 

2813 

2 

II 

3o 

3o35 

3009 

2984 

2961 

2939 

2918 

2898 

2879 

2861 

2844 

2828 

2812 

2797 

3 

9 

40 

3019 

299J 

2968 

2945 

2923 

2902 

2883 

2864 

2846 

2828 

2812 

2797 

2782 

4 

8 

5o 

3oo3 

2977 

2952 

2929 

2907 

2887 

2867 

2848 

283o 

2813 

2797 

2781 

2767 

5 

6 

55 

0 

2987 

2961 

2936 

2913 

2891 

2871 

285i 

2833 

2815 

2798 

2781 

2766 

2751 

6 

5 

10 

2971 

2945 

2921 

2898 

2876 

2855 

2836 

2817 

2799 

2782 

276b 

2751 

2736 

7  ■ 

3   1 

20 

2955 

2929 

2905 

2882 

2860 

2840 

2820 

2802 

2784 

2767 

2751 

2736 

2721 

8 

I 

3o 

2939 

2918 

2889 

2866 

2845 

2824 

2805 

2786 

2769 

2752 

2736 

2720 

2706 

9 

0 

/\n 

292J 
2907 
2891 

2897 
2882 

2866 

0H73 

o85i 

2829 
2814 
2798 

2809 

2790 
2774 
2759 

2771 

275'^ 

2737 

2721 

2705 

2691 

2676 

2661 

56 

5o 
0 

2858 
2842 

2835 
2820 

2793 
2778 

2756 
2741 

2738 
2723 

2721 
2706 

2706 
2690 

2690 
2675 

Sec. 
0 

Cor. 

i4 

10 

2876 

285i 

2827 

2804 

2783 

2763 

2744 

2725 

2708 

2691 

2676 

2660 

2646 

I 

i3 

20 

2860 

2835 

2811 

2789 

2768 

2748 

2729 

2710 

2693 

2676 

2661 

2646 

263 1 

2 

II 

3o 

2844 

2820 

2796 

2774 

2752 

2732 

2714 

2695 

2678 

2661 

2646 

263 1 

2617 

3 

10 

40 

2829 

2804 

2780 

2758 

2737 

2717 

2698 

2680 

2663 

2647 

263 1 

2616 

2602 

4 

8 

5o 

2813 

2789 

2765 

2743 

2722 

2702 

2683 

2665 

2648 

2b32 

2616 

2601 

2587 

5 

7 

57 

0 

2798 

2773 

2750 

2728 

2707 

2687 

2669 

265o 

2633 

2617 

2601 

2587 

2573 

6 

5 

10 

2783 

2758 

2735 

2713 

2692 

2672 

2654 

2636 

2618 

2602 

2587 

2572 

2558 

7 
8 

4 

20 

2767 

2743 

2720 

2698 

2677 

2657 

2639 

2621 

2604 

2588 

2572 

2557 

2543 

3o 

2752 

2728 

2705 

2683 

2662 

2642 

2624 

2606 

2589 

2573 

2558 

2543 

2529 

9 

I 

"58" 

4o 
5o 

0 

2737 
2722 

2707 

2713 
2698 

2683 

2690 
2675 
2660 

2668 
2653 
2638 

2647 

2632 

2618 

2628 
2613 
2  5q8 

2609 
2595 
2  58o 

2591 

2577 

2562 

2574 
256o 

2545 

2558 
2544 

2543 
2529 

2528 
25i4 

25i5 

2500 

Sec. 

Cor. 
i4 

2529 

25i4 

25oo 

2486 

0 

10 

2692 

2668 

2645 

2623 

2603 

2  584 

2  565 

2548 

253i 

25i5 

2  5oo 

2485 

2472 

I 

i3 

20 

2677 

2653 

263o 

2609 

2588 

2569 

255i 

2533 

25i6 

250I 

2485 

2471 

2457 

2 

II 

3o 

2662 

2638 

2Ci5 

2594 

2574 

2554 

2536 

2519 

2502 

2486 

2471 

2457 

2443 

3 

10 

40 

2647 

2623 

2601 

2579 

2559 

2540 

2522 

2  5o4 

2488 

2472 

2457 

2443 

2429 

4 

8 

5o 

2632 

2608 

2  586 

2  565 

2  544 

2525 

25o7 

2490 

2473 

2458 

2443 

2428 

24i5 

5 

7 

5q 

0 

2617 

2594 

2571 

255o 

253o 

25ll 

2493 

2476 

2459 

2444 

2429 

2414 

2401 

6 

5 
4 

10 

2603 

2579 

2557 

2536 

25i6 

2497 

2479 

2461 

2445 

2429 

24i5 

2400 

2387 

7 
8 

20 

2588 

2565 

2542 

2521 

250I 

2482 

2465 

2447 

243i 

24i5 

2400 

2386 

2373 

3o 

2573 

2550 

2528 

2507 

2487 

2468 

245o 

2433 

2417 

2401 

2386 

2372 

235q 

9 

IkT 

4o 
5o 

0 

2559 
2544 
2  53o 

2535 

2521 
2507 

25i3 
2499 
2485 

2492 
2478 
2464 

2473 

2458 
2444 

2454 
2440 

2426 

2436 
2422 

2408 

2419 
24o5 

2391 

24o3 
2389 
2375 

2387 
2373 

2373 
2359 

2358 
2345 

2345 
233 1 

Sec. 

Cor. 

2359 

2345 

233i 

23l7 

0 

i4 

10 

25i5 

2492 

2470 

2450 

243o 

241 1 

2394 

2377 

236i 

2345 

233i 

23i7 

23o4 

I 

i3 

20 

25oi 

2478 

2456 

2435 

2416 

2397 

238o 

2363 

2347 

2332 

23i7 

23o3 

2290 

2 

II 

3o 

2487 

2464 

2442 

2421 

2402 

2383 

2366 

2349 

2333 

23i8 

23o3 

2290 

2276 

3 

10 

40 

2472 

245o 

2428 

2407 

2388 

2869 

2352 

2335 

2819 

23o4 

2290 

2276 

2  263 

4 

8 

5o 

2458 

2435 

24i4 

2393 

2374 

2356 

2339 

2321 

23o6 

2290 

2276 

2262 

2249 

5 

7 

6i 

0 

2444 

2421 

2400 

2379 

236o 

2342 

2325 

23o8 

2292 

2277 

2262 

2249 

2236 

5 

b 

10 

2430 

2407 

2386 

2  365 

2  346 

2328 

23l  I 

2294 

2278 

2263 

2249 

2235 

2222 

7 
8 

4 
3 

20 

2416 

2393 

2872 

235i 

2332 

23(4 

2297 

2280 

2265 

225o 

22J0 

2222 

2209 

30 

2402 

2879 

2358  2338 

23i8 

23oo 

2283 

2167 

225l 

2  236 

2222 

2208 

2195 

9      \ 

TABLE  XIX.                fPaseoo 

Correction. 

I  ^  S 

Table  A. 

Table  B. 

D  's  Horizontal  Parallax. 

Proportional  part  for  Seconds 
of  Parallax. 

For  Min. 

of  Alt. 

^^ 

Add. 

Add. 

D. 

31. 

54' 

55' 

56' 

57' 

5S' 
8.23 

59' 

7.23 

60' 
6.24 

61' 

5.24 

S. 
0 

0"l"2"3"4"j 
59  58  5^56  551 

5" 
54 

6"  7" 

8"  9"  M. 
5i  5o  0 

S. 
c 

7 

o 

12.2 

11 .22 

10.22 

9.22 

53 

52 

10 

12. I. 

5ii.i3 

10. i4 

9.14 

8.i5 

7.i5 

6.16 

5.16 

10 

4948474 

645 

^A 

43 

42 

4i4 

ol  2 

5 

?.o 

12.  J 

)  11.  5 

10.  6 

9.  6 

8.  7 

7-  7 

6.  8 

5.  8 

20 

39  38  37  36|35| 

34 

33 

32 

3i  3o  i 

4 
3 

3o 

II. 5t 

5  10.58 

9.59 

8.59 

8.  0 

7.  0 

6.  1 

5.  I 

3a 

29 

2f 

3  27  2 

625 

24 

23 

22 

21  20  s 

4o 

II.  5c 

) 10. 5i 

9.5i 

8.52 

7.53 

6.53 

5.54 

4.54 

4o 

19 

I 

3  17  I 

6i5 

i4 

i3 

12 

II  10  I 

1 

5c 

9 

i 

3  7 

6  5 

4 

3 

2 

1 

0  9 

0 

TABLE  XIX.  Logarithms. 

o  '<' 

Table  C. 

>2  i2 

Apparent  Altitude  of  j)  's  centre. 

Correction  for  Seconds 
of  Parallax. 

«A, 

Add. 

0    / 

0  / 

0    / 

0  / 

0  / 

0  /  lo  / 

0  / 

0  / 

0  / 

0  / 

0  / 

M. 

54 

S. 

0 

7  3 

2841 

7  6 
2836 

7  9 
283i 

7  12 

2827 

7  15 

2823 

7  187  21 

7  24 

281 1 

7  27 

7  30 

7  33 

7  30 

Sec. 

Cor. 

2819 

2815 

2807 

2803 

2799 

2795 

0 

i3 

10 

2825 

2821 

2816 

2812 

2808 

2804 

2800 

2796 

2791 

2787 

2783 

2780 

I 

12 

20 

28  UJ 

2805 

2800 

2796 

2792 

2788 

2784 

2780 

2776 

2772 

2768 

2765 

2 

10 

3o 

2794 

2790 

2785 

2781 

2777 

2773 

2769 

2765 

2761 

2757 

2753 

2749 

3 

9 

4o 

2779 

2774 

2770 

2766 

2762 

2758 

2754 

2750 

2746 

2742 

2738 

2734 

4 

7 

5o 

2764 

2759 

2755 

2751 

2747 

2743 

2739 

2735 

2731 

2727 

2723 

2719 

5 

6 

55 

0 

2748 

2744 

2739 

2735 

2731 

2727 

2723 

2719 

2716 

2712 

2708 

2704 

6 

4 
3 

lO 

2733 

2729 

2724 

2720 

2716 

2712 

2708 

2704. 

2700 

2696 

2692 

2689 

7 
8 

20 

2718 

2714 

2709 

2705 

2701 

2697 

2693 

2689 

2685 

2681 

2677 

2674 

I 

3o 

2703 

2699 

2694 

2690 

2686 

2682 

2678 

2674 

2671 

2667 

2663 

2659 

9 

"56" 

4o 
5o 

0 

2688 
2673 
2658 

2684 
2669 

2654 

2679 
2664 
2649 

2675 
2660 

2645 

2671 
2656 

2641 

2667 
2653 
2638 

2663 
2649 
2634 

2659 
2645 
263o 

2656 
2641 
2626 

2652 
2637 
2622 

2648 
2633 
2618 

2644 
263o 

26:5 

Sec. 

Cor. 

0 

i3 

lO 

2643 

2639 

2635 

263 1 

2627 

2623 

2619 

261 5 

2611 

2607 

2603 

2600 

I 

12 

20 

2628 

2624 

2620 

2616 

2612 

2608 

2604 

2600 

2597 

2593 

2589 

2  586 

2 

10 

3o 

2614 

2610 

2606 

2602 

2598 

2594 

2590 

2586 

2582 

2578 

2574 

257. 

3 

9 

4o 

2599 

2595 

2591 

2587 

2583 

2579 

2575 

2571 

2567 

2  563 

2559 

2556 

4 

7 

5o 

2  584 

258o 

2576 

2572 

2  568 

2  564 

256o 

2556 

2553 

2549 

2545 

2542 

5 

6 

57 

0 

2570 

2  566 

2562 

2558 

2554 

255o|2546 

2542 

2538 

2534 

253o 

2527 

6 

4 

10 

2555 

255i 

2547 

2543 

2539 

2535 

253i 

2527 

2524 

2520 

25i6 

25i3 

7 

3 

20 

a  540 

2536 

2532 

2528 

2524 

2521 

25x7 

25i3 

25l0 

25o6 

2502 

2499 

8 

I 

3o 

2526 

2522 

25i8 

25i4 

25lO 

25o6 

25o3 

2499 

2495 

2491 

2487 

2484 

9 

0 

~5'8" 

4o 
5o 

o 

25l2 

2497 

2483 

25o8 

2493 

2479 

25o4 
2489 

2475 

25oo 
2485 

2471 

2496 

2481 

2467 

2492 

2478 

2463 

2488 
2474 
2460 

2484 
2470 
2456 

2481 
2467 
2452 

2477 
2463 

2448 

2473 
2459 

2444 

2470 
2456 

2441 

Sec. 

Cor. 

0 

i3 

10 

2469 

2465 

2461 

2457 

2453 

2449 

2445 

2441 

2438 

2434 

243o 

2427 

I 

12 

20 

2454 

245o 

2446 

2442 

2438 

2435 

243 1 

2427 

2424 

2420 

2416 

24i3 

2 

10 

3o 

2440 

2436 

2432 

2428 

2424 

2421 

2417 

24i3 

24lO 

2406 

2402 

2399 

3 

9 

7 

4o 

2426 

2422 

2418 

2414 

2410 

2407 

24o3 

2399 

2396 

2392 

2388 

2385 

4 

5o 

2412 

2408 

24o4 

240D 

2396 

2393 

2389 

2385 

2382 

2378 

2374 

2371 

5 

6 

59 

o 

2398 

2394 

2390 

2386 

2382 

2379 

2375 

2371 

2  368 

2364 

236o 

2357 

6 

5 

10 

2384 

23So 

2376 

2372 

2368 

2365 

236i 

2357 

2354 

235o 

2346 

2343 

7 

3 

20 

2370 

2366 

2362 

2358 

2354 

235i 

2347 

2343 

2340 

2336 

2332 

2329 

8 

2 

3o 

2356 

2352 

2348 

2345 

234i 

2337 

2334 

233o 

2327 

2323 

23i9 

23i6 

9 

I 

6^ 

4o 
5o 

o 

2342 
2328 

23i4 

2338 
2324 

23ll 

2334 

2320 

2307 

233i 

23i7 

23o3 

2327 
23i3 
2299 

2323 

2309 
2296 

2320 

2  3o6 
2292 

23i6 

23o2 
2289 

23i3 
2299 

2286 

2309 
2295 
2282 

23o5 
2291 

2278 

2302 
2288 

2275 

See. 

Cor. 

0 

l3 

10 

23oi 

2297 

2293 

2290 

2286 

22S2 

2279 

2275 

2272 

2268 

2264 

2261 

I 

20 

2287 

2283 

2279 

2276 

2272 

226S 

2265 

2261 

2258 

2254 

225o 

2247 

2 

3o 

2273 

2270 

2266 

2262 

2258 

2255 

225l 

2248 

2245 

2241 

2237 

2234 

3 

Ao 

2260 

2256 

2252 

2249 

2245 

2241 

2238 

2234 

223l 

2227 

2223 

2220 

4 

5o 

2246 

2243 

2239 

2235 

223l 

2228 

2224 

222T 

2218 

22l4 

2210 

2207 

5 

6 

6i 

0 

2233 

2229 

2225 

2222 

2218 

22l4 

22II 

2207 

2  2o4 

2200 

2196 

2193 

6 

5 

10 

2219 

2216 

2212 

2209 

22o5 

2201 

2198 

2194 

2I9I 

2187 

2i83 

2180 

7 

4 

20 

2206 

2202 

2198 

2195 

2I9I 

2187 

2l84 

2181 

2178 

2174 

2170 

2167 

8 

2 

3o 

2192 

2189 

2i85 

2182 

2I78I2I74 

2i7i|2i67 

2164 

2  J  60 

2l57 

2i54 

9 

I 

P-^geioo]                TABLE  XIX. 

Correction. 

-•  -• 

Table  A. 

Table B. 

j>  's  Horizontal  Parallax. 

Proportional  part  for  Seconds 
of  Parallax. 

For  Min. 
of  Alt. 

Add. 

Add. 

D. 

7 

M. 

3o 

!34' 

55' 

56' 

57' 

58' 
8.  07 

59' 
.  06 

60' 
.  I 

61' 

5.  I 

S.  0"1"2"3" 
1^5^58  5^56 

4" 
55 

5"  6" 
5453 

7" 

52 

8" 
57 

3" 
5o 

M. 

0 

S. 
6 

II. 5 

3  10.58 

9.598 

.59 

4o 

II. 5 

■)  10. 5i 

9.51  8 

.52 

7.536 

.535.54 

4.54 

104948  4746 

45 

44 

43 

42 

4i 

io 

2 

5 

5o 

II. 4 

i  10.44 

9.458 

.45 

7.466.465 

.47 

4.47 

20393837, 

36 

35 

34 

33 

32 

3j 

io 

i 

4 

8 

0 

II. 3 

7  10.38 

9.388 

.39 

7.406.40  5 

.41 

4.4i 

3()  29  2 

8  27 

lb 

25 

24 

23 

22 

21 

20 

5 

6 

7 

2 

lO 

II. 3 

I  10.32 

9.328 

.33 

7.336.345 

.35 

4.35 

4o  19  I 

8  17 

6 

i5 

i4 

i3 

12 

II 

!0 

2 
1 

20 

I  I  .2 

5 10.26 

9.268 

•  27 

7.286 

.28|5 

.29 

4.3o 

5o  9 

8  7 

6 

5 

4 

3 

2 

I 

0 

9 

0 

0 

TABLE  XIX.  Logarithms. 

o  2 

Table  C. 

^1 

Apparent  Altitude  of  5  's  centre. 

Correction  for  Seconds 
of  Parallax. 

«p: 

Add. 

0  / 

0  / 

0   / 

0  / 

0  / 

0  / 

0  / 

0   / 

0  / 

0  / 

0  1 

0  / 

M. 

S. 
o 

7  39 

2791 

7  42 

2787 

7  45 

2783 

7  48 
2780 

7  51 

2777 

7  54 

2774 

7  57 

2770 

8  0 

2766 

8  3 

8  6 

8  9 

8  12 

Sec. 

Cor. 

2762 

2759 

2756 

2753 

0 

i3 

lO 

2776 

2772 

2768 

2765 

2762 

275c; 

2755 

2751 

2747 

2744 

2741 

2738 

I 

I? 

20 

2761 

2757 

2753 

2750 

2747 

2743 

2739 

2736 

2782 

2729 

2726 

2723 

2 

10 

3o 

i745 

2741 

2737 

2734 

2731 

2727 

2724 

2721 

2717 

2714 

2711 

2708 

3 

9 

4o 

2780 

2726 

2722 

2719 

2716 

2712 

2709 

2706 

2702 

2699 

2696 

2693 

4 

7 

5o 

2715 

271 1 

2707 

2704 

2701 

2697 

2694 

2691 

2687 

2b84 

268 1 

2678 

5 

6 

55 

o 

2700 

2696 

2692 

2689 

2686 

2682 

2679 

2676 

2672 

2669 

2666 

2663 

6 

4 
3 

10 

2685 

2681 

2677 

2674 

2671 

2667 

2664 

2661 

2657 

2654 

2b5i 

2048 

7 
8 

20 

2670 

2666 

2662 

2659 

2656 

2652 

2649 

2646 

2642 

2639 

2636 

2633 

I 

JO 

2655 

265 1 

2647 

2644 

2641 

2637 

2634 

263i 

2627 

2624 

2621 

2bl8 

9 

~5"6 

4o 

5o 

o 

2640 
2626 
261 1 

2637 
2622 

2607 

2633 
2618 
2603 

263o 
2615 
2600 

2627 
2612 

2597 

2623 
2608 
2593 

2619 
2605 
2590 

2616 
2602 

2587 

2612 
2598 

2609 
2595 

2606 
2592 

2603 
2589 

Sec. 

Cor. 

2583 

258o 

2577 

2574 

0 

i3 

10 

2596 

2592 

2589 

2586 

2583 

2579 

2575 

2572 

2569 

2566 

2563 

256o 

I 

12 

20 

2582 

2578 

2574 

2571 

2568 

2564 

256i 

2558 

2554 

255i 

2548 

2545 

2 

10 

3o 

2  567 

2563 

2559 

2556 

2553 

2549 

2546 

2543 

2540 

2537 

2534 

253i 

3 

9 

4o 

2552 

2548 

2545 

2542 

2539 

2535 

2532 

2529 

2525 

2522 

25i9 

25lb 

4 

7 

5o 

2538 

2534 

253o 

2527 

2524 

2520 

25i7 

25i4 

25ll 

25o8 

25o5 

2502 

5 

6 

57 

o 

2523 

2519 

25i6 

25i3 

25l0 

25o6 

25o3 

25oo 

2496 

2493 

2490 

2487 

6 

4 

10 

2509 

25o5 

25C2 

2499 

2496 

2492 

2489 

2486 

2482 

2479 

2476 

2473 

7 
8 

3 

20 

24q5 

2491 

2487 

2484 

2481 

2477 

2474 

2471 

2468 

2465 

2462 

2459 

I 

3o 

2480 

2476 

2473 

2470 

2467 

2453 

2460 

2437 

2454 

245i 

2448 

2445 

9 

0 

T8 

4o 
5o 

0 

2466 
2452 

2438 

2462 
2448 
2434 

2459 
2445 
243 1 

2456 
2442 

2428 

2453 
2439 

2425 

2449 
2435 

2421 

2446 
2432 

2418 

2443 
2429 

24i5 

2440 
2425 
2411 

2437 
2422 

2408 

2434 
2419 
24o5 

2431 
2416 
2402 

Sec. 

Cor. 

0 

i3 

10 

2424 

2420 

2416 

24i3 

2410 

2407 

24o4 

2401 

2397 

2894  2391 

2388 

I 

12 

20 

2410 

2406 

2402 

2399 

2896 

2393 

2390 

2387 

2383 

^So 

2377 

2374 

2 

10 

3o 

2396 

2392 

2388 

2385 

2382 

2379 

2376 

2373 

2369 

2366 

2363 

236o 

3 

9 

4o 

2382 

P.378 

2375 

2372 

2369 

2365 

2362 

2359 

2356 

2353 

235o 

2  347 

4 

7 

5o 

2368 

2364 

236 1 

2358 

2355 

235i 

2348 

2345 

2342 

2339 

2336 

2333 

5 

6 

5q 

o 

2354 

235o 

2347 

2344 

234i 

2337 

2334 

233i 

2328 

2325 

2322 

23i9 

6 

5 

10 

234o 

2336 

2333 

233o 

2827 

2323 

23  20 

23i7 

23i4 

23l  I 

23o8 

23o5 

7 

3 

20 

2326 

2322 

23i9 

23i6 

23 1 3 

23lO 

2807 

23o4 

23oi 

2298 

2295 

2292 

8 

2 

3o 

23l2 

23o8 

23o5 

2302 

2299 

2296 

2293 

2290 

2287 

2284 

2281 

2278 

9 

I 

"6^ 

4o 
5o. 

0 

2299 
2285 

2271 

2295 
2281 

2268 

2292 
2278 

2265 

2289 
2275 
2262 

2286 
2272 

2259 

2282 
2269 

2255 

2279 

2266 

2252 

2276 

2  263 

2249 

2273 
2260 

2270 
2257 

2267 
2254 

2264 

225l 

Sec. 

Cor. 

2  246 

2243 

2240 

2237 

0 

l3 

10 

2258 

2254 

225l 

2248 

2245 

2242 

2239 

2236 

2233 

223o 

2227 

2224 

I 

12 

20 

2244 

2240 

2237 

2234 

223l 

2228 

2225 

2222 

2219 

2216 

22l3 

2210 

2 

'O 

3o 

223l 

2227 

2224 

2221 

2218 

22l5 

2212 

2209 

2206 

22o3 

2  200 

2197 

3 

9 

4o 

2217 

22l4 

22II 

2208 

22o5 

2  201 

2198 

2195 

2192 

2189 

2186 

2i83 

4 

8 

5o 

2204 

2  200 

2197 

2194 

2191 

2l8b 

2i8d 

2182 

2179 

2176I2173 

2170 

5 

0 

6i 

0 

2190 

2187 

2184 

2181 

2178 

2175 

2172 

2169 

2166 

2i63  2160 

2l57 

6 

5 

10 

2177 

2174 

217I 

2168 

2i65 

2I6I 

2i58 

2i55 

2l52 

2149I2147 

2i44 

7 
8 

4 

20 

2164 

2160 

2i57 

2i54 

2l5l 

2i4b 

2145 

2142 

2189 

2i36|2i33 

2i3o 

2 

3o 

2l5l 

2l47 

2i44 

2l4l 

2i38 

2i35 

2l32 

2129 

2126 

2123I2120 

2II7 

9 

I 

1 

TABLE  XIX.                [i-^se  101 

Correction. 

X'  a 

Table  A. 

Table B. 

5  's  Horizontal  Parallax. 

Proportional  part  for  Seconds 
of  Parallax. 

For  Min. 
of  Alt. 

<r 

Add. 

Add. 

D. 

M. 

10 

54' 

55' 

56' 

57' 

58' 
7.35 

59' 

6.36 

GO' 

5.37 

01' 

4.37 

S. 
0 

0" 

58 

1' 

5- 

2"  3"  4"  5" 
56  55  54  53 

6" 
5I 

7" 

8"  9"  M. 
5o49  1 

S. 

5 
4 
4 

II. 3; 

10.34 

9-34 

8.35 

20 

II. 2- 

10.28 

9.28 

8.29 

7.3o 

6.3o 

5.3i 

4.32 

10 

48 

4- 

46  45  44 

0 

42 

4i 

4o  39  2 

3o 

II. 2: 

10.22 

9.23 

8.24 

7.24 

6.25 

5.25 

4.26 

20 

38 

3- 

36  35  34 

33 

32 

3i 

3o  2 

?   4 

3 

4o 

11. It 

) 10.17 

9.18 

8.18 

7.19 

6.20 

5.20 

4.21 

3o 

28 

2- 

26  2 

524 

23 

22 

21 

20  I 

V  }i 

DO 

II  .11 

10.12 

9.13 

8.i3 

7.14 

d.i5 

5.i5 

4.16 

4o 

18 

I- 

16  I 

5i4 

i4 

i3 

12 

II  io|  7 

1 

5o 

9 

f 

7 

6  5 

4 

3 

2 

1 

o\    § 

i 

TABLE  XIX.  Logarithms. 

c  ■"' 

Table  C. 

Apparent  Altitude  of  j)  's  centre. 

Correction  for  Seconds 
of  Parallax. 

«£ 

Add. 

o   / 

0  / 

010/ 

0  / 

0  / 

0  / 

0  / 

0  / 

0  / 

0  / 

0  / 

M. 

'57 

s. 

O 

8  15 

2749 

8  18 
2746 

8  21 

2743 

8  24 
2740 

8  27 
2737 

8  30 

2734 

8  33 

2731 

8  36 

2728 

8  39 
2725 

8  42 
2722 

8  45 
2719 

8  48 
2716 

Sec. 

Cor. 

0 

i3 

10 

2734 

2731 

2728 

2725 

2722 

2719 

2716 

2713 

2710 

2707 

2704 

2701 

I 

12 

20 

2719 

2716 

2713 

2710 

2707 

2704 

2701 

2698 

2695 

2692 

2689 

2686 

2 

10 

3o 

2704 

2701 

2698 

2695 

2692 

2689 

2686 

2683 

2680 

2677 

2674 

2671 

3 

Q 

4o 

26S9 

2686 

2683 

2680 

2677 

2674 

2671 

2668 

2665 

2662 

2659 

2656 

4 

7 

5o 

2675 

2671 

2668  2665 

2662 

2659 

2656 

2653 

265o 

2647 

2644 

2641 

5 

6 

55 

0 

2659 

2656 

2653  265o 

2647 

2644 

2641 

2638 

2635 

2632 

2629 

2627 

6 

4 

10 

2644 

2641 

2638  2635 

2632 

2629 

2626 

2623 

2620 

2617 

2614 

2612 

7 

3 

20 

2629 

2626 

2623 

2620 

2617 

2614 

261 1 

2609 

2606 

2603 

2600 

2597 

8 

I 

3o 

2615 

2612 

2609 

2606 

2603 

2600 

2597 

2594 

2591 

2588 

2585 

2583 

9 

0 

~5"6" 

4o 
5o 

0 

2600 
2586 

2571 

2597 

2582 

2568 

2594 
2579 
2565 

2591 

2576 

2562 

2588 
2573 
2559 

2585 
2570 

2582 

2567 

2553 

2579 
2564 
255o 

2576 

2562 

2547 

2573 
2559 

2  544 

2570 
2556 

254i 

2568 
2553 

2539 

Sec. 

Cor. 

0 

i3 

lO 

2556 

2553 

255o 

2547 

2  544 

254i 

2538 

2535 

2533 

253o 

2527 

2524 

I 

12 

20 

2542 

2539 

2536 

2  533 

253o 

2527 

2524 

2521 

25i8 

25i5 

25l2 

25lO 

2 

10 

3o 

2527 

2524 

2521 

25  18 

25i5 

25l2 

2509 

25o6 

2  5o4 

25oi 

2498 

2496 

3 

9 

4o 

25i3 

25lO 

2507 

25o4 

250I 

2498 

2495 

2492 

2490 

2487 

2484 

2481 

4 

7 

5o 

2499 

2496 

2493 

2490 

2487 

2484 

2481 

2478 

2475 

2472 

2469 

2467 

5 

6 

5? 

o 

2484 

2481 

2478 

2475 

2472 

2469 

2466 

2463 

2461 

2458 

2455 

2453 

6 

4 

10 

2470 

2467 

2464 

2461 

2458 

2455 

245s 

2449 

2Z|47 

2444 

2441 

2439 

7 

3 

20 

2456 

2453 

245o 

2447 

2444 

2441 

2438 

2435 

2433 

243o 

2427 

2425 

8 

I 

3o 

2442 

2439 

2436 

2433 

243o 

2427 

2424 

2421 

2419 

2416 

24i3 

2411 

9 

0 

58 

4o 

5o 

o 

2427 
24 1 3 
2399 

2424 

2410 
2396 

2421 
2407 

2393 

24i8 
2404 
2390 

2416 
2402 

2388 

24i3 
2399 

2385 

2410 
2396 

2382 

2407 
2393 

2379 

24o5 
2391 

2377 

2402 
2388 

2374 

2399 
2385 
2371 

2397 

2383 
2369 

Sec. 

Cor. 

0 

i3 

10 

2385 

2382 

2379 

2376 

2374 

2371 

2368 

2365 

2363 

236o 

2357 

2355 

I 

12 

20 

2371 

2368 

2o65 

2362 

236o 

2357 

2354 

235i 

2349 

2346 

2343 

2341 

2           If)     1 

3o 

2357 

2355 

2352 

2349 

2346 

2343 

2340 

2337 

2335 

2332 

2329 

2327 

3 

9 

7 
6 

4o 

2344 

234. 

2338 

2335 

2332 

2329 

2326 

2323 

2321 

23i8 

23i5 

23i3 

4 

5o 

233o 

2327 

2324 

2321 

23i8 

23i5 

23l2 

2309 

23o7 

23o4 

2302 

23oo 

5 

5s) 

0 

23i6 

23i3 

23lO 

23o7 

23o5 

2302 

2299 

2296 

2294 

2291 

2289 

2286 

6 

5 

10 

2302 

2299 

2296 

2293 

2291 

2288 

2285 

2282 

2280 

■-277 

2275 

2272 

7        3 

2() 

2289 

2286 

2283 

2280 

2277 

2274 

2271 

2268 

2266 

2263 

2261 

2259 

8        2 

3o 

2275 

2272 

2269 

2266 

2264 

2261 

2258 

2255 

2253 

225o 

2248 

2245 

9 

I 

"6^ 

4o 
5o 

0 

2261 

224s 

2  234 

2259 

2245 

2232 

2256 

2242 

2229 

2253 

2239 

22  j6 

225o 

2247 

2244 

223l 

2217 

2241 
2228 

2214 

2239 
2226 
2212 

2236 
2223 

2209 

2234 
2221 

2207 

2232 
2218 

2205 

2237  2234 
2223  2220 

Sec. 

Cor. 

0 

i3 

lO 

2221 

2218 

22l5 

2213 

22092207 

2204 

2201 

2199 

2196 

2194 

219I 

I 

12 

20 

2207 

22o5 

2202 

219^7 

2  1 96  2 1 93 

2190 

2188 

2186 

2i83 

2181 

2178 

2 

■3o 

2194 

2191 

2188 

2i85 

2i83  2180 

2177 

2175 

2172 

2169 

2167 

2i65 

3 

9 
8 

Uo 

2180 

2178 

2175 

2172 

2170  2167 

2164 

2162 

2159 

2i56 

2i54 

2 1  52 

4 

5o 

2167 

2i65 

2162 

2159 

2i57|2i54 

2l5l 

2149 

2i46 

2143 

2l4l 

2i38 

5 

6 

6: 

o 

2i54 

2l5l 

2 1 48 

2i45 

2l43j2l4o 

2i37 

2i35 

2i33 

2i3o 

2128 

2125 

6 

5 

10 

2l4l 

2i38 

2i35 

2l32 

2i3o  2127 

2124!2122 

2119 

2II6 

2Il4 

2112 

7 

4 

20 

2127 

2125 

2122 

2119 

2117  2Il4 

211  I  2109 

2106 

2I03 

2I0I 

2099 

8 

2 

3o 

21  I  i 

2lI2l2I09|2I06 

2Io4  2101 

2098  2096 

2093 

2090 

2088 

I2086 

9 

I 

Page  102] 


TABLE   XIX 

Correction 


D  's  Horizontal  Parallax. 


D. 


3o 


54' 


•  / 
.  3 
10.59 
10.54 


55' 


10. i3 
10.  8 
10.  4 
9.59 
9.55 


56' 


9-14 
9.  9 

9.  4 
9.  o 
8.56 


57' 


8.i4 
8.10 
8.  5 
8.  I 
7.57 


58' 


7.i5 
7.10 
7.  6 
7.  2 
6.58 


59' 


6.16 
6. II 
6.  7 
6.  2 
5.58 


60' 
5.16 

5.12 

5.  7 
5.  3 

61' 

4.17 
4.i3 
4.  8 
4.  4 

.594. 


Table  A. 

Proportional  part  for  Seconds 

of  Parallax. 

Add. 


0"1" 

5857 

48  47 
3837 
28  27 
[918 
9I  8 


5"|6" 

7" 

53 

52 

5i 

43 

42 

4i 

33 

32 

3i 

23 

22 

22 

i4 

i3 

12 

4 

3 

2 

8" 
5^ 
40 
3o,  . 

21   2U 

I  ilio 


Table  B. 

For  Min. 

of  Alt. 

Add. 


M. 


TABLE  XIX.     Lo2arithms. 


E  = 


M. 

"57 


55 


56 


57 


58 


59 


60 


61 


3o 


Apparent  Altitude  of  5  's  centre. 


8  51 


2718 
2698 
2683 
2658 
2653 
2638 

2624 
2609 
2594 
2  58o 
2  565 
255o 
'2336 

2521 

2507 
2493 
2478 

2464 


245o 
2436 
2422 
2408 
2894 
2880 


2366 

2352 

2338 
2324 

23lO 

2297 


2  283 

2269 

2256 

2242 
2229 

22l5 


2  202 
2188 
2175 
2162 

2i49 
2i35 


2122 
2109 
2096 
2o83 


8  54 


2710 
2695 
2680 
2665 
265o 
2635 
2621 
2606 
2591 
2577 
2562 
£547 
2533" 
2519 
2  5o4 
2490 
2476 
2/161 


2447 
2433 
2419 
24o5 
2891 
2877 


2363 
2349 
2335 
2822 
23o8 
2294 


2281 
2267 

2253 

2240 
2226 

22l3 


2200 
2186 
2173 
2160 

2i46 
2t33 


2120 
2107 
2094 
208 1 


8  57 


2708 
2693 
2678 
2663 
2648 
2633 
2619 
2604 
2589 
2575 
256o 
2545 


9  0 


2705 
2690 
2675 
2660 
2645 
2680 


253i 
25i7 

25o2 

2488 
2474 
2459 


2445 
243 1 
2417 
24o3 
2389 
2375 

2061 
2347 
2333 
2820 
2806 
2292 


2279 
2265 

225l 
2  238 

2224 

22II 


2198 
2184 
217I 

2i58 
2i44 

2l3l 


2II8 

2io5 
2092 
2079 


2616 
2601 
2586 
2572 
2557 
2542 

2528 
25i4 
2499 
2485 
2471 
2456 


2442 
2428 
2414 
2400 
2386 
2872 


2358 
2344 
2880 
2817 
23o3 


2276 
2262 
2248 

2235 
2221 
2208 


2195 
2181 
2168 

2i55 

2l4l 

2128 

2Il5 
2102 
2089 
2076 


9  3 


2702 
2687 
2672 
2657 
2642 
2627 


2618 
2598 
2583 
2569 
2554 
2539 


2525 
25ll 

2496 
2482 

2468 
2453 


2439 
2425 
2412 
2398 
2384 
2870 

■2356 
2342 
2828 
23i4 
2800 
22S7 


2278 
2260 
2246 
2282 
2218 

2205 


2192 

2178 

2 1 65 

2l52 

2i33 
2125 


2099 

2086 
2073 


9  6 


2699 
2685 
2670 
2655 
2640 
2625 


261 1 

2596 
258i 
2566 
255i 
2537 


2523 

2509 
2494 
2480 

2466 
245i 


2437 
2423 
2409 
2895 
2881 
2867 
"2353 
2339 
2835 
2812 
2298 
2284 


2270 
2257 
2243 
2280 
2216 

2203 


2190 
2176 

2i63 

2l5o 

2 1 36 

2123 


21 10 
2097 
2084 
2071 


9  9 


2697 
26S2 
2667 

2652 

2637 
2622 


2608 

2598 

2578 

2564 
2549 
2535 


2520 

2  5o6 
2492 
2477 
2463 
2449 


2435 
2421 
2407 
2898 

2879 
2365 

235i 
2887 
2823 
2809 
2296 
2282 


2205 
2255 

2241 
2228 

22l4 
2201 


21!: 


2174 
2161 

2i48 
2i34 


2108 
2095 
2082 
2069 


912 


2679 
2664 
2649 
2634 
2619 


2605 
2590 
2575 
256i 
2546 

2532 


25i7 
25o3 
2489 
2474 
2460 
2446 


2482 
2418 
2404 
2890 
2876 
2862 


2348 
2334 
2820 
2806 
2298 
2279 

2265 
2252 
2288 
2225 
2211 
2 1 98 


2i85 
2171 
2i58 
2i45 

2l3l 


2I05 

2092 

2079 
2066 


915 


2691 
2676 
2661 
2646 
2682 
2617 


2602 
2588 
2578 
2558 
2  544 
2529 


25i5 
25oi 
2486 
2472 
2458 
2444 


2430 
24 1 5 
2401 
2887 
2873 
236o 


2346 
2882 
2818 
2804 
2291 
2277 


2268 

225o 

2286 
2228 
2209 


9  18 

2688 
2678 
2658 
2643 
2629 
2614 


2599 
2585 
2570 
2555 
254i 
2526 


25l2 
2498 

2483 
2469 
2455 
2441 


2427 
24i3 
2899 
2385 
2871 
2357 


2843 
2829 
23i5 
23oi 
2288 
2274 


2260 
2247 

2233 
2220 
2207 
2193 

2i8q 
2166 
2 1 53 

2l4o 

2127 

2Il4 


2IOI 

2088 
2074 
2061 


9  21 


2686 
2671 
2656 
2641 
2627 
2612 


2D97 
2583 
2  568 
2553 
2539 
2524 


25l0 

2496 

24s  I 
2467 
2453 
2439 


2425 
2410 
2896 

2882 

2368 
2355 


2341 
2827 
2818 
2299 
2286 
2272 

2258 

2245 

223l 
2218 
22o5 
219I 


2178 
2i64 
2l5l 

2i38 

2125 
2II2 


2099 
2086 


iOJC) 


9  24 


2669 
2654 
2689 
2625 
2610 


2595 
258  r 
2566 
255i 
2537 
2522 


25o8 
2494 
2479 
2465 
245i 
2437 


2428 
2408 
2894 
2880 
2366 
2353 


2889 

2325 
23ll 

2297 

2284 

2270 


2256 

2243 
2229 
2216 

2203 


2176 

2162 
2i49 
2  1 36 
2128 

21  10 


2097 
2084 

2070 
2057 


92; 


2682 
2667 
2652 
2687 
2622 

2608 


2598 

2578 

2564 
2549 
2535 
2520 


■■25o6 
2492 

24-7 
2463 
2449 
2434 


2420 
24o6 
2892 
2878 
2364 
285i 


2887 
2828 
2809 
2295 
2282 
2268 


2254 
2241 
2227 

22l4 
2201 
2187 

2174 
2160 
2l47 

2i34 
2121 
2108 


2095 
2082 
2068 
2o55 


TABLE  XIX.                            [Page  103 

Correction. 

-''  q 

Table  A. 

Table B. 

])'s  Horizontal  Parallax. 

Proportional  part  for  Seconds 
of  Par.allax. 

For  Min. 
of  Alt. 

<'^ 

Add. 

Add. 

,._. 

D. 

9 

M. 

3o 

54' 

55'  50' 

57' 

7.58 

53 

59' 

60' 

61'  1 

S. 
0 

0" 
58 

I" 

37 

2"  3"  4" 
56  55  54 

5"l6" 

7" 
57 

8" 
5"^ 

9" 

49" 

M. 

0 

S. 

10.55 

9.568.57 

6.5 

95.59 

5.  04.  I 

53  52 

3 

4o 

10. 5i 

9.528 

.53 

7.54 

6.5 

55.56 

4.563.57 

10 

48 

47 

(6  4 

544 

4342 

4i 

40 

39 

2 

2 

5o 

10.48 

9.498 

.49 

7.5o 

6.5 

I  5.52 

4.533.54 

20 

38 

37 

36  35|34 

i. 

32 

3i 

3o 

29 

4 

2 

to 

0 

lo. 44 

9.458 

.46 

7.47 

6.4 

8  5.49 

4.5o3.5o 

3o 

28 

27 

26  2 

524 

24  23 

22 

21 

20 

5 

2 

lO 

10.4' 

9.428 

.4-3 

7-44 

6.4 

55.45 

4.46  3.47 

40 

'9 

18 

17  ' 

6  lb 

i4  i3 

12 

II 

10 

7 

1 

20 

10.38 

9.398 

.40 

7.41 

6.4 

r  5.42 

4.433.44 

5o 

9 

8 

7 

J  5 

4  3 

2 

I 

0 

9 

1 

TABLE  XIX.  Logarithms. 

s  i 

Table  C. 

=1 

Apparent  Altitude  of  D  's  centre. 

Cor.  Sec. 
of  Par. 

«3^ 

Add. 

0          f     0          1 

0   / 

0  1 

0   /  0   /  0   /  0   / 

0   / 

0   / 

0  ^0  /  0  /  1 

iM. 

54" 

s. 

o 

9  30  9  34 

9  38 

9  42 

9  46  9  50  9  54  9  58 

10  2 

10  6 

10101014 

1018 

Sec. 

Cor. 

2679 

2676 

2673 

2669 

2666 

2663 

2660 

2657 

2654 

265 1 

2648 

2645 

2642 

0 

i3 

lO 

2664 

2661 

2658 

2654 

265i 

2648 

2645 

2642 

2640 

2637 

2634 

263 1 

2628 

I 

12 

20 

2649 

2646 

2643 

2639 

2636 

2633 

263o 

2627 

2625 

2622 

2619 

2616 

2613 

2 

10 

3o 

2634 

263 1 

2628 

2624 

2621 

2618 

2615 

2612 

2610 

2607 

2604 

2601 

2598 

3 

9 

4o 

2619 

2616 

2613 

2610 

2607 

2604 

2601 

2598 

2595 

2592 

2589 

2587 

2584 

4 

7 

55 

5o 
o 

2605 

2602 

2599 

2595 

2592 
2577 

2589 

2586 

2583 

258i 
2  566 

2578 

2575 

2572 

2569 

5 
6 

6 
4 

2590 

2587 

2584 

258o 

2574 

2571 

2  568 

2563 

256o 

2557 

2554 

10 

2575 

2572 

2569 

2566 

2  563 

256o 

2557 

2554 

2552 

2549 

2  546 

2543 

2540 

7 

3 

20 

256i 

2558 

2555 

255i 

2548 

2545 

2542 

2539 

2537 

2534 

253 1 

2528 

2525 

8 

2 

3o 

2  546 

2543 

2540 

2537 

2534 

253i 

2528 

2525 

2523 

2520 

25 1 7 

25i4 

25ll 

y 

0 

4o 

2532 

2529 

2526 

2523 

2520 

25i7 

25i4 

25ll 

2  5o8 

2  5o5 

2502 

25oo 

2497 

56 

5o 

0 

25i7 

25i4 

25ll 

25o8 

25o5 

2502 

2499 

2496 

2494 

2491 

2488 

2485 

2482 

Sec. 

Cor. 

25o3 

2500 

2497 

2494 

2491 

2488 

2485 

2482 

2480 

2477 

2474 

2471 

2468 

0 

i3 

10 

2489 

2486 

2483 

2480 

2477 

2474 

2471 

2468 

2465 

2462 

2459 

2457 

2454 

I 

12 

20 

2474 

2471 

2468 

2465 

2462 

2459 

2456 

2453 

245i 

2448 

2445 

2443 

2440 

2 

10 

3o 

2460 

2457 

2454 

245 1 

2448 

2445 

2442 

243o 

2437 

2434 

243 1 

2428 

2425 

3 

9 

4o 

2446 

2443 

2440 

2437 

2434 

243i 

2428 

2425 

2423 

2420 

2417 

2414 

2-4 1 1 

4 

7 

57 

5o 

0 

2432 

2429 

2426 

2423 

2420 

2417 

24i4 

241 1 

2409 

2406 

24o3 

2400 

2397 
2383 

5 
6 

6 

2418 

24i5 

2412 

2409 

2406 

24o3 

2400 

2397 

23q5 

2302 

2389 

2386 

lO 

2404 

2401 

2398 

2395 

2392 

2389 

2386 

2383 

238 1 

2378 

2375 

2372 

2369 

7 

3 

20 

2390 

2387 

2384 

238 1 

2378 

2375 

2372 

2369 

2367 

2364 

236i 

2358 

2355 

8 

2 

3o 

23-6 

2373 

2370 

2367 

2364 

236 1 

2358 

2355 

2353 

235o 

2347 

2345 

2342 

y 

0 

4o 

2362 

2359 

2356 

2353 

235o 

2347 

2344 

234! 

2339 

2336 

2333 

233i 

2328 

58" 

5o 
o 

2348 

2345 

2342 

2339 

2336 

2333 

233o 

2327 

2325 

2322 

23i9 

23i7 

23i4 

Sec. 
0 

Cor. 
i3 

2334 

233i 

2328 

2325 

2322 

2319 

23 1 6 

23i3 

23ll 

23o8 

2  3o6 

23o3 

23oo 

10 

2320 

23i7 

23i4 

23ll 

23o8 

23o6 

23o3 

23oo 

2298 

2295 

2292 

2290 

2287 

I 

12 

20 

23o6 

23o3 

23oo 

2297 

2294 

2292 

2289 

2286 

2284 

2281 

2278 

2276 

2273 

2 

10 

3o 

2293 1 2290 

2287 

2284 

22Si 

2278 

2275 

2272 

2270 

2267 

2264 

2262 

2259 

3 

y 

4o 

2279 

2276 

2273 

2270 

2267 

2  265 

2262 

2259 

2257 

2254 

225l 

2249 

2246 

4 

8 

5q 

5o 

0 

2265 

2262 

2259 

2257 

2254 

225l 

2248 

2245 

2243 

224o 

2237 

2235 

2232 

5 
6 

6 
5 

2252 

2249 

2246 

2243 

2240 

2238 

2235 

2232 

2230 

2227 

2224 

2222 

2219 

lO 

2238 

2235 

2233 

223o 

2227 

2224 

2221 

2218 

2216 

22l3 

2210 

2208 

2  2o5 

7 
8 

4 

20 

2225 

2222 

2219 

2216 

22l3 

22II 

2208 

2205 

2203 

2200 

2197 

2195 

2192 

3o 

22II 

2208 

2206 

2203 

2  200 

2197 

2194 

219I 

2189 

2186 

2184 

2182 

2179 

y 

I 

4o 

2198 

2195 

2192 

2189 

2186 

2184 

2181 

2178 

2176 

2173 

2170 

2168 

2i65 

6.7 

5o 

0 

2i85 

2182 

2179 

2176 

2173 
2160 

217I 

2168 

2i65 

2i63 

2160 

2  I  57 

2i55 

2l52 

Sec. 
0 

Cor. 
"71 

2I7I 

2168 

2166 

2i63 

2i58 

2i55 

2l52 

2i5o 

2i47 

2i44 

2142 

2  I  39 

10 

2i58 

2i55 

2 1  52 

2149 

2i46  2i44 

2l4t 

2i38 

2i36 

2i33 

2l3l 

2129 

2126 

I 

12 

20 

2i45 

2l42 

2139 

2i36 

2i33  2i3i 

2128 

2125 1 2123 

2120 

2II7 

2n5 

2II2 

2 

10 

3o 

2l32 

2129 

2126 

2  123 

2120 

2118 

2Il5 

2II2 1 2II0 

2107 

2I04 

2102 

2099 

3 

9 

4o 

2II8 

2Il5 

2Il3 

2II0 

2107 

2I05 

2102 

2099 

2097 

2094 

2091 

2089 

2086 

4 

8 

5o 

2io5 

2102 

2100 

2097 

2094 

2092 

2089 

2086 

2084 

2081 

2078 

2076 

2073 

5 
6 

6 

5 

6i   o 

2092 

2089 

2087 

2084 

2081 

2079 

2076 

2073 

2071 

2068 

2o65 

2o63 

2060 

10 

2079 

2076 

2074 

2071 

2068 

2o56 

2o63 

2060 

2o58 

2o55 

2o52 

2o5o 

2o47 

7 
8 

4 

20 

2066 

2o63 

2061 

2o58 

2o55 

2o53  2o5o 

2047 

2045 

2o42 

2039 

2037 

2o34 

2 

3o 

2o53 

2o5o 

2048 

2u45 

2042 

2o4o  2o37 

2o34 

2032 

2029 

2027 

2025 

2022 

y 

I 

■ . 1_ 

P-seio4]                 TABLE  XIX. 

Correction. 

ii  a 

Table  A. 

Table  B. 

D  's  Horizontal  Parallax. 

Proportional  part  for  Seconds 
of  Paralla.x. 

For  Min. 
of  Alt 

<^ 

Add. 

Add. 

D. 

lO 

M. 

20 

54' 

55' 

56' 

57' 

58' 
.41 

5'' 

GO 

4.4 

61' 

3  3.44 

S. 
0 

0" 

58 

1" 

57 

2" 

56 

3"  4" 
5554 

5 

5" 

'  G"J7' 

8" 
5^ 

9'' 
4q 

M.  1   S. 

I0.38 

9.39 

8.40 

7.41  t 

S  52 

5x 

0   ,   3 

3o 

10. 3b 

9.36 

8.37 

7.38  e 

.39 

5.4c 

4.4 

1  3.42 

10 

48 

47 

46 

4 

544 

4342 

4i 

4o 

3q 

2 

2 

4o 

10.32 

9.33 

8.34 

7.35  t 

.36 

5.3- 

4.3 

y3.3g 

20 

38 

37 

36 

35134 

33  32 

3i 

3o 

ig 

3 

^ 

5o 

10.29 

9.30 

8.3i 

7.32  t 

.33 

5.34 

4.3 

5  3.36 

3o 

29 

28 

27 

2 

3  25 

2 

4  23 

22 

21 

2U 

5 

2 

II 

o 

10.27 

9.28 

8.29 

7.3o  t 

.3i 

5.32 

4.3 

3  3 .  34 

4o 

19 

x8 

17 

I 

5x5 

X 

ii3 

X2 

1 1 

10 

6 

7 

1 

TO 

10.24 

9.25 

8.26 

7.27  t 

.29 

5.3o 

4.3 

I  3.32 

5o 

9 

8 

7 

5  5 

i    3 

2 

I 

0 

8 
9 

1 
I 

TABLE  XIX.  Logarithms. 

o  d 

Table  C. 

ffil 

Apparent  Altitude  of  5  's  centre. 

Corr.  for  Sec 
of  Parallax. 

«3; 

Add. 

0  / 

0  / 

0  / 

0  / 

0  / 

0  /  0  / 

0  / 

0  / 

0  / 

0  / 

0   / 

M. 

54 

S. 
o 

10  22 

2639 

10  2G 

10  30 

10  34 

263 1 

10  38 

10  4210  46 

10  50 

10  54 

10  58 

11  2 

11  6 

Sec. 

Cor. 
i3 

2637 

2634 

2628 

2626 

2623 

2621 

2618 

2616 

2614 

26x1 

0 

10 

2625 

2622 

2619 

2616 

261 3 

261 1 

2608 

2606 

2603 

2601 

2599 

2597 

I 

X2 

20 

2610 

2608 

2605 

2602 

2599 

2597 

2594 

2591 

2588 

2586 

2584 

2582 

2 

XO 

3o 

2595 

2593 

2590 

2587 

2584 

2582 

2579 

2577 

2574 

2572 

2570 

2567 

3 

9 

4o 

2S81 

2578 

2575 

2572 

2570 

2568 

2565 

2562 

256o 

2557 

2555 

2552 

4 

7 

"5T 

5o 

0 

2  566 

2  564 
2549 

256i 

2558 

2555 

2553 

255o 
2535 

2548 
2533 

2545 
253o 

2543 

254i 

2538 

5 
6 

6 

4 

255: 

2546 

.2543 

2540 

2538 

2528 

2526 

2523 

10 

2537 

2535 

2532 

2529 

2526 

2524 

2521 

2519 

25x6 

25x4 

25X2 

2509 

7 

3 

20 

2522 

2520 

2517 

25i4 

25l2 

25l0 

25o7 

2  5o4 

2502 

2499 

2497 

2495 

8 

2 

3o 

2  5o8 

25o6 

25o3 

25oo 

2497 

2495 

2492 

2490 

2487 

2485 

2483 

2480 

9 

0 

56" 

4o 
5o 

0 

24y4 
2479 

2.'192 

2477 

2489 
2474 

2486 
2471 

2483 
2469 

2481 
2467 

2478 
2464 

2476 
2462 

2473 
2459 

2471 
2457 

2469 
2455 

2466 
2452 

Sec. 
0 

Cor 

2465 

2463 

2460 

2457 

2455 

2453 

245o 

2447 

2445 

2442 

2440 

2438 

i3 

10 

245 1 

2449 

2446 

2443 

2440 

2438 

2435 

2433 

243o 

2428 

2426 

2424 

I 

12 

20 

2437 

2435 

2432 

2429 

2426 

2424 

2421 

24l§ 

24x6 

24i4 

2412 

2410 

2 

XO 

3o 

2423 

2420 

2418 

24i5 

2412 

2410 

2407 

240D 

2402 

2400 

2398 

2396 

3 

9 

4o 

2409 

2406 

2404 

24oi 

2398 

2396 

2393 

2391 

2388 

2386 

2384 

2382 

4 

7 

"57 

5o 

0 

2395 

2392 

2390 

2387 

2384 

2382 

2379 

2377 

2374 
2  36o 

2372 
2358 

2370 
2356 

2368 
2354 

5 
6 

6 
5 

2  38 1 

2378 

2376 

2373 

2370 

2368 

2365 

2363 

10 

2367 

2364 

2362 

2359 

2356 

2354 

235i 

2349 

2346 

2344 

2342 

2340 

7 

3 

20 

2353 

2  35o 

2348 

2345 

2342 

2340 

2337 

2335 

2333 

233i 

2329 

2326 

8 

2 

3o 

2339 

2337 

2334 

233i 

2329 

2327 

2324 

2322 

23x9 

23x7 

23i5 

23X2 

9 

0 

"58" 

4o 

5o 

o 

2325 
23  I  I 

2323 

2309 

2320 
23o6 

23i7 
23o3 

23i5 

23oi 

23i3 
2299 

23lO 

2296 

23o8 
2294 

23o5 
229X 

23o3 
2289 

23oi 

2287 

2299 
2285 

Sec. 

Cor. 

2298 

2295 

2293 

2290 

2288 

2286 

2283 

2281 

2278 

2276 

2274 

2271 

0 

i3 

10 

2284 

2282 

2279 

2276 

2274 

2272 

2269 

2267 

2264 

2262 

2260 

2258 

I 

12 

20 

2270 

2268 

2265 

2262 

2260 

2258 

2255 

2253 

225x 

2249 

2247 

2244 

2 

10 

3o 

2257 

2254 

2252 

2249 

2247 

2245 

2242 

2240 

2237 

2235 

2233 

223x 

3 

9 

4o 

2243 

2  24  I 

2238 

2235 

2233 

223l 

2228 

2226 

2224 

2222 

2220 

22X7 

4   '   8  1 

^' 

5o 

0 

2  9  30 

2227 

2225 

2222 

2220 

2218 

22l5 

22l3 

2210 

2208 

2206 

2204 

5 
6 

6 
5 

2216 

22l4 

221  I 

2208 

2206 

2204 

2201 

2199 

2197 

2195 

2X-93 

2x91 

10 

22o3 

2200 

2198 

2195 

2193 

2I9I 

2188 

2186 

2x83 

2x8x 

2x79 

2177 

7 

4 

20 

2190 

2187 

2i85 

2182 

2180 

2178 

2175 

2173 

2170 

2168 

2x66 

2x64 

8 

2 

3o 

2176 

2174 

2171 

2168 

2166 

2164 

2161 

2x59 

2x57 

2x55 

2x53 

2x5x 

9 

I 

t3o 

4o 

5o 

0 

2 1 63 
2i5o 

2160 
2l47 

2i58 
2145 

2i55 

2142 

2x53 

2l4o 

2l5l 

2i38 

2i48 
2i35 

2x46 
2x33 

2x44 

2l3o 

2142 
2128 

2X40 
2x26 

2x38 
2x24 

Sec. 

Cor. 

2 1 37 

2i34 

2 1  32 

2129 

2127 

2125 

2122 

2X20 

21  X7 

2Xl5 

2Xl3 

2X1  X 

0 

l3 

10 

2123 

2121 

2II8 

2Il5 

21l3 

2III 

2109 

2107 

2X04 

2102 

2100 

2098 

I 

12 

20 

2110 

2107 

2I05 

2102 

2100 

2098 

2096 

2094 

2091 

2089 

2087 

208b 

2 

10 

3o 

2097 

2094 

2092 

2089 

2087 

2o85 

2o83 

2081 

2078 

2076 

2074 

2072 

3 

9 

4o 

2084 

2081 

2079 

2076 

2074 

2072 

2070 

2068 

2o65 

2o63 

2061 

2o59 

4 

8 

"gT 

bo 
o 

2071 

2068 

2066 

2o63 

2061 

2o59 

2o57 

2o55 

2o52 

2o5o 

2o48 

2o46 

5 
6 

6 
5 

2o58 

2o55 

2o53 

2o5o 

204S 

2o46 

2o44 

2042 

2039 

2o37 

2o35 

2o33 

10 

2o45 

2042 

2o4o 

2o37 

2o35 

2o33 

2o3l 

2029 

2026 

2024 

2022 

2020 

7 

4 

20 

2032 

2029 

2027 

2024 

2022 

2020 

2018 

20x6 

20X4 

2012 

2010 

2008 

8 

2 

3o 

2020 

2017 

20 1 5 

2012 

2009 

2007 

2oo5 

20o3 

2001  X999 

1997 

1995 

9 

I 

TABLE  XIX.                        [Page  105 

Correction. 

-'i  i 

Table  A. 

Table B. 

<!2 

D  's  Horizontal  Parallax. 

Proportional  part  for  Seconds 
of  Parallax. 

ForMin. 
of  Alt. 

<'^ 

Add. 

Add. 

D. 
II 

M. 

10 

54' 

55' 

56' 

57' 

58' 
6.3o 

59' 
5.3i 

GO' 

4.32 

Gl' 
3.33 

S. 
0 

0"] 

58f 

7 

2"  3"  4" 

56  55  54 

5"  G" 

53  5i 

7' 
57 

3 
5o 

49 

M. 

S. 
2 

10.25 

9.26  t: 

.27 

7.28 

0 

20 

10.23 

9.24  fi 

.25 

7.26 

6.28 

5.25 

4.3o 

3.3i 

xo 

48- 

'7 

46  4 

544 

4 

342 

4i 

4o 

39 

2 

2 

3o 

10.21 

9.22  6 

.23 

7.24 

6.26 

5.27'4.28 

3.29 

20 

38  37 

36  35|34 

33|33 

32 

3i 

3o 

3 
4 
5 

4o 

10.19 

9.20  L 

.21 

7.22 

b.24 

5.254.26 

3.27 

3o 

29: 

8 

27  2 

625 

2 

4  23 

22 

2X 

20 

5o 

10.17 

9.18  f: 

.19 

7.21 

6.22 

5.234.24 

3.26 

40 

19 

8 

17  I 

6i5 

I 

4i3 

12 

X  I 

10 

6 

12 

0 

10. i5 

9.i6g 

.i8 

7.19 

6.20 

5.224.23 

3.24 

5o 

9 

8 

7 

6  5 

4  3 

2 

I 

0 

§ 

0 

u 

TABLE  XIX.  Logarithms. 

S  ^' 

Table  C. 

51 

Apparent  Altitude  of  d  's  centre. 

Cor.  for  Sec. 
of  Parallax. 

Ap^ 

Add. 

0  / 

0  / 

0  1 

0  1 

0  / 

0  / 

0  / 

0  /  0  / 

0/0   / 

M. 

54 

S. 

0 

1110 

2609 

1115 

1120 

1125 

2600 

1130 

1135 

1140 

11451150 

11  55 

12  0 

Sec. 

Cor. 

2606 

2603 

2597 

2594 

2592 

2589 

2586 

2584 

258i 

0 

i3 

10 

2594 

2591 

2588 

2585 

2582 

2579 

2577 

2574 

257X 

2569 

2566 

I 

12 

20 

2579 

2576 

2573 

2571 

2568 

2565 

2563 

256o 

2557 

2555 

2552 

2 

10 

3o 

2565 

2562 

2559 

2556 

2553 

255o 

2548 

2545 

2542 

254o 

2537 

3 

9 

4o 

255o 

2547 

2544 

2542 

2539 

2536 

2534 

253x 

2528 

2526 

2523 

4 

7 

33" 

5o 
o 

2536 

2533 

253o 

2527 

2524 

252X 

25x9 

25 1 6 

25i3 

25l  I 

25o8 

5 
6 

6 
4 
3 

2521 

25i8 

25i5 

25i3 

25l0 

25o7 

25o5 

2502 

2499 

2497 

2494 

10 

2507 

25o4 

25oi 

2498 

2495 

2492 

2490 

2487 

2485 

2482 

2480 

7 
8 

20 

2493 

2490 

2487 

2484 

2481 

247« 

2476 

2473 

2470 

2468 

2465 

2 

3o 

247« 

2475 

2472 

2470 

2467 

2464 

2462 

2459 

2456 

2454 

245 1 

9 

0 

le" 

4o 
5o 

0 

2464 
2450 

2461 
2447 

2458 
2444 

2456 
2442 

2453 
2439 

2450 
2436 

2448 
2434 

2445 

243  X 

2442 
2428 

2440 
2426 

2437 
2423 

Sec. 

Cor. 

2436 

2433 

243o 

2427 

2424 

2421 

2419 

2416 

2414 

24XX 

2409 

0 

1 3 

10 

2422 

2419 

2416 

24i3 

2410 

2407 

24o5 

2402 

2400 

2397 

2395 

I 

12 

20 

2408 

24o5 

2402 

2399 

2396 

2393 

239X 

2388 

2386 

2383 

238  X 

2 

xo 

3o 

2394 

2391 

2J88 

2385 

2382 

2379 

2377 

2374 

2372 

2369 

2367 

3 

9 

4o 

238o 

2377 

2374 

2371 

2  368 

2365 

2363 

236o 

2358 

2355 

2353 

4 

7 

TT 

5o 
o 

2366 

2363 

236o 

2357 
2  344 

-J 

354 

2352 

2349 

2347 

2344 

2342 

2339 

5 
6 

6 
5 

2352 

2349 

2346 

34  X 

2338 

2336 

2333 

233o 

2328 

2325 

10 

2338 

2335 

2332 

233o 

2327 

2324 

2322 

23x9 

23i7 

23i4 

23l2 

7 
8 

3 

20 

2324 

2321 

23i8 

23i6 

23x3 

23X0 

2  3o8 

23o5 

23o3 

23oo 

2298 

2 

3o 

23lO 

23o7 

23o4 

2302 

2299 

2297 

2294 

2292 

2289 

2287 

2284 

9 

0 

"58" 

4o 

5o 

o 

2297 

2283 

2294 
2280 

2291 

2277 

2289 
2275 

2?86 
2272 

2283 

2269 

228X 

2267 

2278 
2264 

2276 
2262 

2273 
2259 

2271 
2257 

Sec. 

Cor. 

2269 

2266 

2263 

2261 

2258 

2256 

2253 

225l 

2248 

2246 

2243 

0 

x3 

10 

2256 

2253 

225o 

2248 

2245 

2242 

2240 

2237 

2235 

2232 

2  23o 

I 

12 

20 

2242 

2239 

2236 

2234 

223x 

2229 

2226 

2224 

222X 

2219 

2216 

2 

10 

3o 

2229 

2226 

2223 

2221 

2218 

22l5 

22l3 

2210 

2208 

2205 

2203 

3 

9 

4o 

22l5 

2212 

2209 

2207 

2204 

2202 

2199 

2x97 

2195 

2192 

2x90 

4 

8 

3^ 

5o 
o 

2202 

2199 

2196 

2194 

219I 

2189 

2186 

2X84 

2181 

2179 

2x76 

5 
6 

6 
5 
4 

2189 

2186 

2l83 

2j8l 

2178 

2x75 

2173 

2170 

2168 

2i65 

2l63 

lO 

21-5 

2172 

2169 

2167 

2164 

2162 

2x59 

2x57 

2i55 

2 1  52 

2l5o 

7 

20 

2162 

2  I  59 

2 1 56 

2x54 

2l5l 

2x49 

2i46 

2i44 

2x4l 

2x39 

2i36 

3o 

2i49 

2i46 

2143 

2x4l 

2i38 

2x35 

2i33 

2l30 

2128 

2125 

2X23 

9 

"6o 

4o 
5o 

0 

2x36 

2122 

2i33 
2119 

2i3o 

21  17 

2x28 

2Il4 

2X25 
2XX2 

2122 
2109 

2120 
2x07 

2II7 

2I04 

21X5 
2X02 

2XX2 

2099 

2II0 
2097 

Sec. 

Cor. 

2109 

2106 

2I04 

2IOI 

2099 

2096 

2094 

2091 

20S9 

2086 

2084 

0 

i3 

lO 

2096 

2093 

2091 

2088 

2086 

2083 

2081 

2078 

2076 

2073 

2071 

I 

12 

20 

2o83 

2080 

2078 

2075 

2073 

2070 

2068 

2o65 

2o63 

2060 

2o58 

2 

10 

3o 

2070 

2067 

2o65 

2062 

2060 

2o57 

2o55 

2052 

2050 

2047 

2045 

3 

9 

4o 

2o57 

2o54 

2052 

2049 

2o47 

2o44 

2o42 

2039 

2o37 

2o34 

2o32 

4 

8 

fiT 

5o 

0 

2o44 

204l 

2o39 

2o36 

2o34 

203l 

2029 

2026 

2024 

2021 

2019 

5 
6 

6 
5 

203l 

2028 

2026 

2023 

2021 

2018 

2016 

20X3 

2011 

2008 

2006 

lO 

2018 

20 1 5 

20l3 

2010 

2008 

2006 

2003 

200X 

'999 

X996 

1994 

7 
8 

4 

20 

2006 

2oo3 

2000 

1998 

1995 

1993 

1990 

1988 

1986 

1983 

X981 

2 

3o 

1993 

1990 

1987 

1985 

1982 

1980 

1977 

1975 

.973 

1970 

X968 

9     '  j 

14 


PageiGG]                TABLE  XIX. 

Correction. 

w-   - 

Table  A. 

Table  B. 

<  t 

])  's  Horizontal  Parallax. 

Proportional  part  for  Seconds 
of  Parallax. 

For  Mm. 
of  Alt. 

<."" 

Add. 

Add. 

D. 

12 

M. 

o 

54' 

55' 

56' 

57' 

58 
6.2 

5^ 
I  5.23 

GO' 

4.24: 

Gl' 

.25 

S. 
0 

0"1 
58  5 

'  2" 

7  56 

3' 

55 

4" 
54 

5' 

5a 

G" 
5^ 

7" 
5'i 

8" 
5^ 

9 

49 

M. 

0 

S. 

10.  i6 

Q.178 

.19 

7.20 

10 

io.i5 

9.168 

•  17 

7.19 

6.2 

0  5.21 

4.231 

.24 

10 

48  47  46 

45 

44 

43 

42 

4i 

40 

39 

2 

20 

10. i3 

9:148 

.16 

7-17 

6.1 

9  5.20 

4.21  1 

.23 

20 

38  37  37 

3t 

35 

3^ 

33 

32 

3i 

3o 

3 

3o 

10.12 

9.i38 

.i4 

7.16 

6.1 

7  5.19 

4.20  C 

.22 

3o 

292 

827 

■it 

25 

24 

23 

22 

21 

20 

5 

4o 

10.10 

9.128 

.i3 

7.i5 

6.1 

65.18 

4.193 

.21 

40 

191 

817 

iC 

i5 

1^ 

i3 

12 

11 

10 

0 

5o 

10.  9 

9. II  8 

.12 

7.14 

6.1 

55.17 

4.18: 

.20 

5o 

9 

8  7 

i 

5 

I 

3 

2 

I 

0 

1 

0 

TABLE  XIX.  Logarithms. 

^  x" 

Table  C.   | 

M  2 

Apparent  Altitude  of  j)  's  centre. 

Cor.  for  Sec. 
of  Parallax. 

«2^ 

Add. 

0    / 

0  / 

0  / 

0  / 

0  / 

0  /  1  0  / 

0  / 

0  / 

0  / 

0  / 

M. 

54 

S. 

0 

12  5 

1210 

1215 

12  20 

1225 

12  30 

2566 

12  35 

12  40 

12  45 

12  50 

12  55 

Sec. 

Cor. 

2578 

2576 

2573 

2571 

2568 

2564 

256i 

2559 

2557 

2554 

0 

i3 

10 

2  564 

256i 

2559 

2bb6 

25b4 

255i 

2549 

2546 

2544 

2542 

2540 

I 

12 

20 

2549 

2547 

2544 

2542 

2539 

2537 

2535 

2532 

253o 

2528 

2525 

2 

10 

3o 

2  534 

2532 

2529 

2527 

2524 

2522 

2520 

25  I  7 

25i5 

25i3 

25ll 

3 

9 

4o 

2520 

25i8 

25i5 

25i3 

25lO 

2  5o8 

25o6 

25o3 

250I 

2499 

2496 

4 

7 

5o 

25o6 

25o3 

250I 

2499 

2496 

2494 

2492 

2489 

2487 

2485 

2482 

5 
6 

6 
4 
3 

55 

0 

2491 

2489 

2486 

2484 

2481 

2479 

2477 

2474 

2472 

2470 

2468 

10 

2477 

2475 

2472 

2470 

2467 

2465 

2463 

2460 

2458 

2456 

2454 

7 

20 

2463 

2461 

2458 

2456 

2453 

2451 

2449 

2446 

2444 

2442 

2439 

8 

2 

3o 

2449 

2447 

2444 

2442 

2439 

2437 

2435 

2432 

2430 

2428 

2425 

9 

0 

16" 

4o 

5o 

o 

2435 
2421 

2433 
2419 

2430 
2416 

2428 
24i4 

2425 
24II 

2423 
2409 

2421 
2407 

2418 

2416 
2402 

24i4 
2400 

24II 

2397 

2 
2 

4o4 

Sec. 

Cor. 

2407 

24o5 

2402 

2400 

2397 

2395 

2393 

390 

2388 

2386 

2383 

0 

i3 

10 

2393 

2391 

2388 

2386 

2383 

238i 

2379 

2376 

2374 

2372 

2369 

1 

12 

20 

2379 

iZ-ji 

2374 

2372 

2369 

2367 

2365 

2362 

236o 

2358 

2355 

2 

10 

3o 

2365 

2363 

2000 

2358 

2355 

2353 

235i 

2348 

2346 

2344 

234 1 

3 

9 

4o 

235i 

2349 

2346 

2344 

234i 

2339 

2337 

2334 

2332 

233o 

2328 

4 

7 

5? 

5o 
o 

2337 

2335 

2332 

233o 

2327 

2325 

2323 

2320 

23i8 

23i6 

23i4 

5  • 
6 

6 
5 

2323 

2321 

23i8 

23i6 

23i3 

23ll 

2309 

23o7 

23o5 

23o3 

23oo 

10 

23lO 

23o7 

23o4 

2302 

23oo 

2298 

2296 

2293 

2291 

2289 

2287 

7 
8 

3 

20 

2296 

2294 

2291 

2289 

2286 

2284 

2282 

2279 

2277 

2275 

2273 

2 

3o 

2282 

2280 

2277 

3273 

2272 

2270 

2268 

2266 

2264 

2262 

2259 

9 

0 

58 

4o 

5o 

o 

2269 

2255 

2266 
2252 

2263 
225o 

2261 
2248 

2259 
2245 

2257 

2243 

2255 

2241 

2252 
2239 

2250 
2237 

2248 

2235 

2246 

2232 

Sec. 

Cor. 

2241 

2238 

2236 

2234 

2232 

223o 

2228 

2225 

2223 

2221 

2219 

0 

i3 

10 

2228 

2225 

2223 

2221 

2218 

2216 

22l4 

2212 

2210 

2208 

2205 

I   1   12   1 

20 

22l4 

22II 

2209 

2207 

22o5 

22o3 

2201 

2198 

2196 

2194 

2192 

2 

10 

■ 

3o 

2201 

2198 

2196 

2194 

2191 

2189 

2187 

2i85 

2183 

2181 

2179 

3 

9 

4o 

2188 

2i85 

2i83 

2181 

2178 

2176 

2174 

2172 

2170 

2168 

2i65 

4 

8 

5-7 

5o 

0 

2174 

2171 

2169 

2167 

2i65 

2l5l 

2i63 

2161 

2i58 

2i56 

2i54 

2l52 

5 
6 

6 
5 

2161 

2i58 

2i56 

2  I  54 

2149 

2l47 

2145 

2143 

2l4l 

2139 

10 

2i48 

2145 

2 1 43 

2l4l 

2i38 

2i36 

2i34 

2l32 

2i3o 

2128 

2126 

7 
8 

4 

2() 

2i34 

2l32 

2l3o 

2128 

2125 

2123 

2I2I 

2119 

2II7 

21l5 

2Il3 

2 

3o 

2 1 21 

2II8 

2116 

2Il4 

2112 

2II0 

2108 

2106 

2104 

2102 

2100 

9 

I 

677 

4o 
5o 

0 

2108 
2095 

2I05 

2092 

2io3 
2090 

2I0I 

2088 

2099 
2086 
2073 

2097 
2084 

2095 
2082 

2092 
2079 

2090 
2077 

2088 
2075 

2086 

2073 

Sec. 

Cor. 

2082 

2079 

2077 

2075 

2071 

2069 

2066 

2064 

2062 

2060 

0 

i3 

lO 

2069 

2066 

2064 

2062   2060 

2o58 

2o56 

2o54 

2o52 

2o5o 

2o48 

I 

12 

20 

2o56 

2o53 

205l 

2049 

2o47 

2045 

2043 

204l 

2039 

2o37 

2o35 

2 

ID 

3o 

2o43 

2o4o 

2o38 

2o36 

2o34 

2032 

2o3o 

2028 

2026 

2024 

2022 

3 

9 

4o 

2o3o 

2027 

2025 

2023 

2021 

2019 

2017 

20l5 

20l3 

201 1 

2009 

4 

8 

67 

5o 

0 

2017 

20l5 

20l3 

201  I 

2008 

2006 

2004 

2002 

2000 

1998 

1996 

5 
6 

6 
5 
4 

2004 

2002 

2000 

1998 

1995 

1993 

I99I 

1989 

1987 

1985 

1983 

JO 

1992 

198Q 

1987 

1985 

1983 

1981 

1979 

1977 

1975 

1973 

1970 

7 
8 

20 

1979 

1976 

1974 

1972 

1970   1968 

1966 

1964 

1962 

i960 

1958 

2 

3o 

1966 

1964 

1962 

i960 

1957   1955 

1953 

I  95 1 

1949 1 1947 

1945 

9 

I 

TABLE  XIX. 

[Page  107 

Correction. 

-■  a 

Table  A. 

Table B. 

D  's  Horizontal  Parallax. 

Proportional  part  for  Seconds 
of  Parallax. 

For  Min. 
of  Alt. 

<^ 

Add. 

Add. 

D. 

i3" 

M. 

o 

54' 

55' 

5G' 

57' 
l7iZ 

58' 

59' 

GO' 

Gl' 

S. 
0 

0" 

57 

1" 

^6 

2"  :3"-4" 
55545I 

5"  6" 

7" 
5F) 

8" 
49 

9' 

48 

M. 

0 

S. 

0 

10.10 

9.l2t 

5.;3 

6.16 

5.1b 

4.19 

3.21 

10 

lO.  q 

9. II  t 

i.I2 

7.14 

6.i5 

a. 17 

4.19 

3.20 

10 

47 

^6 

454443 

4 

M4i 

4o 

3q 

39 

2 

0 

20 

10.  8 

9.10  ' 

5.12 

7.i3 

6.i5 

5.16 

4.18 

3.20 

20 

38  37 

36  35  M 

33|32 

3i 

3o 

29 

3 

0 
0 

3o 

lo.  8 

9.  9t 

i.ii 

7.i3 

6.i4 

5.16 

4.17 

3.19 

3o 

28 

27 

26  2 

524 

2 

322 

21 

20 

19 

5 

0 

4o 

10.  7 

9.  9t 

i.IO 

7.12 

6.14 

D.l5 

4.17 

i.19 

40 

18 

7 

161 

5i4 

i3  12 

1 1 

10 

9 

6 

0 

5o 

io»  6 

9.  8t 

i.IO 

7.12 

6.i3 

5.i5 

4.17 

3.18 

5o 

8 

7 

6 

5  4 

4  3 

2 

' 

0 

I 

0 
0 

TABLE  XIX.  Logarithms. 

c  ^' 

Table  C. 

■  Apparent  Altitude  of  ]>  's  centre. 

Cor.  Sec. 
of  Par. 

A;S 

Add. 

0    1 

0   / 

0  / 

0  / 

0  1 

0  , 

0  1 

0  / 

0  / 

0  / 

0   / 

0  / 

M. 

54 

S. 

0 

13  0 

13  5 

1310 

1315 

13  20 

13  25 

13  30 

13  35 

1 

3  40 

13  45 

13  50 

253i 

13  55 

2529 

Sec. 
0 

Cor. 
i3 

2552 

255o 

2548 

2545 

2543 

2541 

2539 

2537 

535 

2533 

10 

2538 

2536 

2534 

253i 

2529 

2527 

2525 

2523 

2521 

25i9 

25i7 

25i5 

I 

12 

20 

2523 

2521 

2519 

25i7 

25i5 

25i3 

25ll 

2509 

2507 

25o5 

25o3 

25oi 

2 

10 

3o 

2509 

25o7 

25o5 

2502 

25oo 

2498 

2496 

2494 

2492 

2490 

2488 

2486 

3 

9 

4o 

2494 

2492 

2490 

2488 

2486 

2484 

2482 

2480 

2478 

2476 

2474 

2472 

4 

7 

35" 

5o 

0 

24S0 

2478 

2476 

2474 

\ 

472 

2470 

2468 

,2466 

2464 

2462 

2460 

2458 

5 
6 

6 
5 

2466 

2464 

2462 

2460 

458 

2456 

2454 

2452 

2450 

2448 

2446 

2444 

10 

2452 

2450 

2448 

2446 

2444 

2442 

2440 

2438 

2436 

2434 

2432 

2430 

7 

3 

20 

2437 

2435 

2433 

243 1 

2429 

2427 

2425 

2423 

2421 

2419 

2417 

24i5 

8 

2 

3o 

2423 

2421 

2419 

2417 

24i5 

24i3 

241 1 

2409 

2407 

24o5 

24o3 

2401 

y 

0 

4o 

2409 

2407 

24o5 

24o3 

2401 

2399 

2397 

2395 

2393 

2391 

2389 

2387 

"56" 

5o 
o 

2395 

2393 

2391 

23S9 

2387 

2385 

2383 

238i 

2379 
2365 

2377 

2375 

2373 

Sec. 
0 

Cor. 
i3 

238i 

2379 

2377 

2375 

2373 

2371 

2369 

2367 

2364 

2362 

236o 

lO 

2367 

2365 

2363 

236 1 

2359 

2357 

2355 

2353 

235i 

235o 

2348 

2  346 

I 

12 

20 

2353 

235i 

2349 

2347 

2345 

2343 

234: 

2339 

2337 

2336 

2334 

2332 

2 

ID 

3o 

2339 

2337 

2335 

2333 

233i 

2329 

2327 

2325 

2323 

2322 

2320 

23i8 

3 

9 

4o 

2326 

2334 

2322 

2320 

23i8 

23 1 6 

23i4 

23l2 

23X0 

23o8 

23o6 

23o4 

4 

8 

^ 

5o 
o 

23l2 

23lO 

23o8 

23o6 

23o4 

2302 

23oo 

2298 

2296 

2295 

2293 

2291 

5 
6 

6 
5 

2298 

2296 

2294 

2292 

2290 

2288 

2286 

2284 

2282 

2281 

2279 

2277 

10 

2285 

2283 

22SI 

2279 

2277 

2275 

2273 

2271 

2269 

2267 

2265 

2263 

7 

3 

20 

2271 

2269 

2207 

2265 

2263 

2261 

2259 

2257 

2255 

2254 

2252 

225o 

8 

2 

3o 

2257 

2255 

2253 

225l 

2249 

2247 

2245 

2243 

2241 

2240 

2238 

2236 

9 

I 

4o 

2244 

2242 

2240 

2238 

2236 

2234 

2232 

223o 

2228 

2227 

2225 

2223 

^ 

5o 

0 

223o 

2228 

2226 

2224 

2222 

2220 

221S 

2216 

22l4 

22l3 

22II 

2209 

Sec. 
0 

Cor. 
i3 

2217 

22l5 

22l3 

221  t 

2209 

2207 

2205 

2203 

2201 

2200 

2198 

2196 

lO 

2  203 

2201 

2199 

2198 

2196 

2194 

2192 

2190 

2188 

2187 

2l85 

2l83 

I 

12 

20 

2190 

2188 

2186 

2184 

2182 

2180 

2178 

2176 

2174 

2173 

217I 

2169 

2 

10 

3o 

2177 

2175 

2173 

217I 

2169 

2167 

2i65 

2i63 

2161 

2160 

2i58 

2i56 

3 

Q 

4o 

2i63 

2l6l 

21^9 

2i58 

2i56 

2i54 

2l52 

2i5o 

2i48 

2l47 

2145 

2i43 

4 

s 

^ 

5o 
o 

2l5o 

2i48 

2i46 

2i45 

2 
2 

143 

2l4l 

2  1 39 

2  I  37 

2i35 

2i34 

2l32 

2l3o 

5 
6 

6 
5 

2137 

2i35 

2i33 

2l3l 

129 

2127 

2125 

2123 

2122 

2120 

2118 

2117 

10 

2124 

2122 

2120 

2II8 

2116 

2Il4 

2II2 

2H0 

2108 

2107 

2io5 

2103 

7 

4 

20 

2111 

2109 

2107 

2I05 

2I03 

2I0I 

2099 

2097 

2095 

2094 

2092 

2090 

8 

3 

3o 

2098 

2096 

2094 

2092 

2090 

2088 

2086 

2084 

2082 

2081 

2079 

2077 

y 

I 

4o 

2o85 

2o83 

2081 

2079 

2077 

2075 

2073 

2071 

2069 

2068 

2066 

2064 

"fc 

5o 
o 

2072 

2070 

2068 

2066 

2064 

2062 

2060 

2o58 

2o56 

2o55 

2o53 

205l 

Sec. 
c 

Cor. 
i3 

2059 

2o57 

2o55 

2o53 

205l 

2049 

2o47 

2045 

2o43 

2042 

2040 

2o38 

10 

2o46 

2o44 

2042 

2o4o 

2o38 

2o36 

2o34 

2o32 

203l 

2029 

2027 

2026 

I 

12 

20 

2o33 

203l 

2029 

2027 

2025 

2023 

202  1 

2019 

2018 

2016 

20l4 

20l3 

2 

10 

3o 

2020 

2018 

2016 

20l4 

2012 

2010 

2008 

2006 

2005 

2003 

2001 

2000 

3 

9 

4o 

2007 

2005 

2003 

2002 

2000 

1998 

1996 

1994 

1992 

1990 

1989 

1987 

4 

8 

Tf 

5o 

0 

1994 

1992 

1990 

1989 

1987 

1985 

1983 

I981 

1979 

1977 

1976 

1974 

5 
6 

7 
5 

1 98 1 

1979 

1977 

1976 

1974 

1972 

1970 

1968 

1967 

1965 

1963 

1962 

10 

1969 

1967 

1965 

1963 

1961 

1959 

1957 

1955 

1954 

1952 

1950 

1949 

7 
8 

4 
3 

20 

1956 

104 

1952 

1961 

1949 

1947 

1945 

1943 

1942 

iq4o 

1938 

1937 

•^0 

1943 

1941 

[939 

1938 

1936 

I9M 

1932 

1930 

1929 

1927 

1925 

1924 

y 

2 

Page  108]                        TABLE  XIX. 

1  > 

Correction. 

■^  i 

Table  A. 

TableB. 

])  's  Horizontal  Parallax. 

Proportional  part  for  Seconds 
of  Parallax. 

For  Mill. 
of  Alt. 

<'^ 

Add. 

Add. 

D. 

i4 

M. 

o 

54' 

55' 

56' 

57' 

58' 
6.i3 

59' 
5.i5 

GO' 

61' 

S. 
0 

0"  1" 
5^56 

2" 
55 

3"  4" 
5453 

5"  G" 
5I57 

7" 
55 

8" 

49 

9" 
4F 

M. 

0 

S. 
0 

10.  6 

9.  88 

•  9 

7. II 

4.17 

3.18 

10 

10.  5 

9.  7a 

•  9 

7. II 

6.i3 

5.i5 

4.16 

3.18 

10 

47  46 

45 

4443 

4 

242 

4i 

40 

39 

2 

0 

20 

lo.  5 

9.  7a 

•  9 

7. II 

6.i3 

5.14 

4.16 

3.18 

20 

38 

57 

36 

3534 

33|32 

3i 

3o 

29 

4 

0 
0 

3o 

10.  5 

9.  7a 

•  9 

7. II 

6.i3 

5.14 

4.16 

3. 18 

3o 

28 

27 

26 

25  24 

2 

3  22 

21 

20 

19 

5 

6 

0 

4o 

10.  5 

9.  7a 

•  9 

7. II 

6.i35.i5 

4.16 

3.18 

40 

18 

7 

16 

i5  i4 

I 

3  12 

II 

II 

10 

0 

5o 

10.  5 

9.  78.  9 

7. II 

6.J35.I5 

4.17 

3.19 

5o 

9 

8 

7 

6  5 

41  3 

9 

I 

0 

8 
9 

0 
0 

TABLE  XIX.  Logarithms. 

o^ 

Tablk  C. 

Apparent  Altitude  of  D  's  centre. 

Cor.  Sec. 
of  Par. 

«;ih 

Add. 

0   / 

0 

0  / 

0  / 

0  / 

0  / 

0  / 

0  / 

0  / 

0  / 

0  , 

0   / 

M. 

S. 
o 

14  0 

14  5 

1410 

14  15 

14  20 

14  25 

14  30 

14  35 

14  40 

14  45 

14  50 

14  55 

2  5o8 

Sec. 
0 

Cor. 
18 

2527 

2525 

2523 

2521 

2520 

25i8 

25i6 

25i4 

25i3 

25ll 

2509 

10 

2bi3 

25ll 

2509 

2507 

25o6 

25o4 

2502 

25oo 

2499 

2497 

24q5 

2494 

I 

12 

20 

2499 

2497 

249b 

2493 

2492 

2490 

2488 

2486 

2485 

2483 

2481 

2480 

2 

10 

3o 

2484 

2482 

2480 

2478 

2477 

2475 

2473 

2471 

2470 

2468 

2466 

2465 

3 

9 

4o 

2470 

2468 

2466 

2464 

2463 

2461 

2459 

2457 

2456 

2454 

2452 

245i 

4 

7 

~5T 

5o 
o 

24bb 

2454 

2452 

2450 

2449 

2447 

2445 

2443 

2442 

2440 

2488 

2487 

5 
6 

6 
5 

2442 

2440 

2438 

2436 

2435 

2433 

243  I 

2429 

2428 

2.426 

2424 

2428 

10 

2428 

2426 

2424 

2422 

2421 

2419 

2417 

24 1 5 

24x4 

2412 

2410 

2409 

7 

3 

20 

24i3 

2412 

2410 

2408 

2406 

24o5 

24o3 

2401 

2400 

2898 

2396 

2895 

8 

2 

3o 

2399 

2398 

2896 

2394 

2892 

2891 

2389 

2887 

28S6 

2384 

2882 

2881 

9 

0 

4o 

23«b 

2384 

2382 

238o 

2378 

2377 

2375 

2873 

2872 

2870 

2868 

2867 

1j6 

bo 

0 

2871 

2870 

2368 

2366 

-^ 

364 

2363 

236i 

2359 

2858 

2856 

2354 

2358 

Sec. 
0 

Cor. 
t3 

2358 

2356 

2354 

2352 

35i 

2349 

2347 

2845 

2844 

2842 

2840 

2889 

10 

2344 

2342 

2340 

2  338 

2337 

2335 

2333 

233i 

2880 

2828 

2826 

2825 

I 

12 

20 

233o 

2328 

2326 

2324 

2323 

2321 

23  I Q 

2817 

2816 

23i5 

2818 

2812 

2 

10 

3o 

23i6 

23i4 

23i3 

23ll 

2809 

23o8 

2806 

23o4 

2808 

2801 

2299 

2298 

3 

4o 

2302 

23oo 

2299 

2297 

2295 

2294 

2292 

2290 

2289 

2287 

2285 

2284 

4 

8 

^ 

bo 
o 

2289 

2287 

228b 

2283 

2282 

2280 

2278 

2276 

2275 

2274 

2272 

2271 

5 
6 

6 
5 

2275 

2273 

2272 

2270 

2268 

2267 

2265 

2268 

2262 

2260 

2258 

2257 

10 

2261 

2259 

22b8 

2256 

2254 

2253 

225l 

2249 

2248 

2246 

2244 

2248 

7 

3 

20 

2248 

2246 

224b 

2243 

2241 

2240 

2238 

2286 

2235 

2288 

2281 

2280 

8 

2 

Jo 

2234 

2282 

2  23  I 

2229 

2227 

2226 

2224 

2222 

2221 

2219 

2217 

2216 

9 

I 

4o 

2221 

2219 

2218 

2216 

22l4 

22l3 

22II 

2209 

2208 

2206 

2204 

2  203 

^ 

bo 

0 

2207 

22o5 

2204 

2202 

2  200 

2199 

2197 

2195 

2194 

2198 

2I9I 

2190 

Sec. 
0 

Cor. 
i3 

2194 

2192 

2191 

2189 

2187 

2186 

2184 

2182 

2181 

2179 

2177 

2176 

10 

2181 

2179 

2178 

2176 

2174 

2173 

217I 

2169 

2168 

2166 

2164 

2168 

I 

12 

20 

2167 

2ibb 

2164 

2162 

2160 

2159 

2  I  57 

2i55 

2i54 

2i53 

2l5l 

2l50 

2 

10 

3o 

2i54 

2162 

2l5l 

2149 

ai47 

2 1 46 

2i44 

2l42 

2l4l 

2l4o 

2i38 

2187 

3 

9 

4o 

2l4l 

2139 

21^8 

2i36 

2 1 34 

2i33 

2l3l 

2129 

2128 

2126 

2124 

2123 

4 

8 

^ 

bo 

0 

2128 

2126 

2125 

2123 

2121 

2120 

2II8 

21  16 

2Il5 

2Il3 

2III 

2II0 

5 
6 

6 
5 

2Il5 

2Il3 

21  12 

2II0 

2108 

2107 

2I05 

2io3 

2102 

2100 

2098 

2097 

lO 

2I0I 

2099 

2098 

2096 

2094 

2093 

2091 

2089 

2088 

2087 

2o85 

2084 

7 

4 

20 

2088 

2086 

208b 

2o83 

2081 

2080 

2078 

2076 

2075 

2074 

2072 

2071 

8 

3 

3o 

2075 

2073 

2072 

2070 

2068 

2067 

2o65 

2o63 

2062 

2061 

2059 

2o58 

9 

I 

4o 

2062 

2060 

2059 

2o57 

2o55 

2o54 

2052 

2o5o 

2049 

2o48 

2046 

2o45 

1 

&7 

bo 
o 

2049 

2047 

2o46 

2o45 

2043 

2042 

2o4o 

2088 

2087 

2o35 

2o33 

2082 

Sec.  Cor. 

2o36 

2o34 

2o33 

2o32 

2o3o 

2029 

2027 

2025 

2024 

2022 

2020 

2019 

0 

i3 

lO 

2024 

2022 

2021 

2019 

2017 

2016 

20l4 

2012 

2CII 

2010 

2008 

2007 

I 

12 

20 

201 1 

2009 

2008 

2006 

2004 

2003 

2001 

1999 

1998 

1997 

1995 

1994 

2 

10 

Jo 

1998 

199b 

199b 

1993 

I99I 

1990 

1988 

1986 

1985 

1984 

1982 

1981 

3 

9 

40 

1985 

1983 

1982 

1980 

1978 

1977 

1975 

1978 

1972 

1971 

1969 

1968 

4 

8 

~67 

bo 

0 

1972 
i960 

1970 

1969 

1968 

1966 

i9bb 

1963 

I  96  I 

i960 

1959 

1957 

1956 

b 
6 

7 
5 

1958 

1957 

,955 

1953 

1952 

1950 

1948 

1947 

1946 

1944 

1943 

lO 

1947 

194b 

1 9 14 

1942 

1940 

19^9 

1987 

193b 

1934 

1933 

1981 

1980 

7 

4 

20 

1935 

1933 

1982 

1930 

1928 

1927 

1925 

1928 

1922 

1921 

I9I9 

1918 

8 

0 

Jo 

1922 

1920 

1919 

I9I7 

r9i5 

I9I4  1 I9I2 

1910 

1909 

1908 

190b 

1903 

9 

TABLE  XIX.  Correction. 

[Page 

109 

<i 

Table  A. 

TableB. 

Proportional  part  for  Seconds 

For  Min. 

i» 

])  s  Horizontal  Parallax. 

of  Parallax. 

of  Alt. 

Add. 

Add. 

D. 
"i5 

M. 

0 

54' 

55' 

50' 

57' 

58' 
6.1 

59' 
35. i5 

GO' 

4.17; 

Gl' 

.19 

S. 
0 

0"1"2"3" 
5^50  55  54 

4" 
53 

5"G" 

7" 

5^ 

8" 
49 

9" 

48" 

M. 

0 

s. 

0 

10.  5 

9-  7 

8.  9 

7.11 

52 

5i 

lO 

10.  5 

9-  7 

8.  9 

7.11 

6.1 

35. i5 

4.17: 

i.19 

ID 

47  46  45i44 

Ai 

43 

42 

4 1 

40 

39 

2 

0 

20 

10.  5 

9-  7 

8.  9 

7.11 

6.1 

J5.i6 

4.18: 

.20 

20 

38  3 

73635 

34 

33 

32 

3i 

3o 

29 

4 

0 

Jo 

10.  5 

9-  7 

8.10 

7.12 

6.1 

45.16 

4.18: 

.20 

3o 

28  27I26 

25 

24 

23 

22 

21 

20 

19 

5 

0 

4o 

10.  6 

9.  8 

8.10 

7.12 

6.1 

45.17 

4.19: 

.21 

40 

181 

7,6 

16 

i5 

i4 

i3 

12 

11 

10 

7 

0 

l6 

5o 
o 

10.  6 

9..  8 

8.10 

7.i3 

6.1 

55.17 

4.19: 

.22 

5o 
0 

9  8  7 

57  56  55 

6 

54 

5 
53 

4 

52 

3 
57 

2 

5^ 

49 

0 

48 

8 
9 
0 

0 
0 
0 

10.  6 

9.  98.11 

7.i3 

6.1 

35.184.20: 

.23 

lO 

10.  7 

9.  98.12 

7.14 

6.1 

3  5.194.21  I 

.23 

10 

47  46  45  45 

U 

43 

42 

4i 

4o 

39 

0 

20 

10.  7 

9.108.12 

7.15 

6.1 

7  5.20  4.22 : 

i.24 

20 

38  37  36  35 

M 

33 

32 

3i 

3o 

29 

4 

0 

Jo 

10.  8 

9.11  8.i3 

7.i5 

6.1 

3  5.20  4-23  C 

i.25 

3o 

2S2 

7  26  25 

24 

2J 

22 

21 

21 

20 

5 

0 

4o 

10.  9 

9.1x8.14 

7.16 

6.1 

9  5.21  4-24 : 

i.26 

4o 

191 

8i7ie 

i5 

x4 

i3 

12 

11 

10 

7 

0 

i)o 

10.10 

9.12  8.i5 

7.17 

6.2 

J 5.22  4-55 -. 

5.27 

5o 

9' 

8  7  e 

5 

4 

3 

2 

1 

0 

8 
9 

1 
1 

TABLE  XIX.  Logarithms. 

«i 

Table  C. 

s= 

Cor.  Sec. 

Apparent  Altitude  of  D  's  centre. 

of  Par. 

«A- 

Add. 

0    10      1 

0  /  0  / 

0  / 

0   /  0   / 

0 

0/0    / 1 

0  / 

0  / 

M. 

'54 

S. 
o 

15  01510 

15  20  15  30 

15  40 

15  50 

16  0 

10  10 

1G20 

16  30 

16  40 

16  50 

2474 

See. 
0 

Cor. 
i3 

25o6 

25o3 

25oo 

2497 

2494 

2491 

2488 

2485 

2482 

2479 

2476 

10 

2492 

2488 

2485 

2482 

2479 

2476 

2473 

2471 

2468 

2465 

2462 

2460 

I 

12 

20 

2478 

2474 

2471 

2468 

2465 

2462 

2459 

2457 

2454 

245i 

2448 

2446 

2 

xo 

3o 

24-63 

2460 

2457 

2454 

2451 

2448 

2445 

2442 

2439 

2436 

2434 

243 1 

3 

9 

4o 

2449 

2446 

2443 

2440 

2437 

2434 

243 1 

2428 

2425 

2422 

2420 

2417 

4 

7 

Ts 

5o 

0 

2435 

2432 

2429 

2426 

2423 

2420 

2417 

2414 

2411 

2408 

2406 

24o3 

5 
6 

6 
5 

2421 

2418 

24i5 

2412 

2409 

2406 

24o3 

2400 

2397 

2394 

2392 

2389 

10 

2407 

2404 

2401 

2398 

23q5 

2392 

2389 

2386 

2383 

238o 

2378 

2375 

7 
8 

3 

20 

2393 

2390 

2387 

2384 

238i 

2378 

2375 

2372 

2369 

2366 

2364 

236x 

2 

3o 

2379 

2376 

2373 

2370 

2367 

2364 

236i 

2358 

2355 

2352 

235o 

2347 

y 

0 

4o 

2365 

2362 

2359 

2356 

2353 

235o 

2347 

2344 

2342 

2339 

2336 

2334 

^6 

5o 
o 

235i 

2348 

2345 

2342 

2339 

2336 

2333 

233o 

2328 

2325 

2322 

2320 

Sec. 

Cor. 

2337 

2334 

233i 

2328 

2325 

2322 

23 1 9 

23i6 

23i4 

23ll 

23o8 

23o6 

0 

l3 

10 

2323 

23  20 

23i7 

23i4 

23ll 

23o8 

23o5 

2302 

2  3oo 

2297 

2295 

2292 

I 

12 

20 

23lO 

23o7 

23o4 

23oi 

2298 

2295 

2292 

2289 

2287 

2284 

2281 

2279 

2 

10 

3o 

2296 

2293 

2290 

2287 

22S4 

2281 

2278 

2275 

2273 

2270 

2267 

2265 

3 

9 

4o 

2282 

2279 

2276 

2273 

2270 

2267 

2264 

2261 

2259 

2256 

2254 

225l 

4 

8 

"tl 

5o 

0 

2269 

2265 

2262 

2259 

2246 

2257 

2254 

225l 

2248 

2246 

2243 

2240 

2238 

5 
6 

6 
5 

2255 

2252 

2249 

2243 

2240 

2237 

2234 

2232 

2229 

2227 

2224 

lO 

2241 

2238 

2235 

2232 

223o 

2227 

2224 

2221 

2219 

2216 

22X3 

221  I 

7 
8 

3 

20 

2228 

2225 

2222 

2219 

2216 

22l3 

2210 

2207 

2205 

2203 

2200 

2198 

2 

3o 

22l4 

2211 

2208 

22o5 

2  203 

2200 

2197 

2194 

2192 

2189 

2187 

2184 

y 

I 

4o 

2201 

2198 

219-3 

2192 

2190 

2187 

2184 

2181 

2179 

2176 

2173 

2171 

Ts" 

5o 
o 

2188 

2l85 

2182 

2179 

2176 

2173 

2170 

2167 

2i65 

2162 

2160 

2  I  57 

Sec. 
0 

Cor. 
x3 

2174 

217I 

2168 

2 1 65 

2i63 

2160 

2  I  57 

2i54 

21 52 

2149 

2147 

2144 

TO 

2161 

2i58 

2i55 

2l52 

2l5o 

2l47 

2144 

2l4l 

2139 

2x36 

2  1  34 

2i3x 

X 

12 

20 

2i48 

2145 

2  1  \l 

2139 

2i37 

2i34 

2l3l 

2128 

2126 

2123 

2121 

2118 

2 

xo 

3o 

2 1 35 

2l32 

2i;>9 

2126 

2123 

2120 

2II7 

21l4 

2112 

2110 

2108 

2io5 

3 

9 

4o 

2  I  21 

2119 

2116 

2Il3 

2110 

2107 

2I04 

2101 

2099 

2097 

2094 

2092 

4 

8 

'5^ 

6o 

O 

2108 

2106 

2I03 

2  100 

2097 

2o84 

2094 

2091 

2088 

20S6 

2084 

2081 

2079 

5 
6 

6 
5 
4 

2095 

2092 

2089 

2086 

2081 

2078 

2075 

2073 

2071 

2068 

2066 

10 

2082 

2079 

2076 

2073 

2071 

2068 

2o65 

2062 

2060 

2o58 

2o55 

2o53 

8 

20 

2069 

2066 

2o63 

2060 

2o58 

2o55 

2052 

2049 

2047 

2045 

2042 

2o4o 

3 

3o 

2o56 

2o53 

2o5o 

2o47 

2045 

2042 

2o39 

2o36 

2o34 

2o32 

2o3o 

2027 

y 

X 

4o 

2()43 

2o4l 

2o38 

2o35 

2o32 

2029 

2026 

2023 

2021 

2019 

20x7 

20l4 

&)" 

5o 
o 

2()3o 

2028 

2025 

2022 

2020 

2017 

20T4 

20II 

2009 

2006 

2004 

2001 

Sec,  Cor.| 

2017 

201  5 

2012 

2009 

2007 

2004 

2001 

1998 

1996 

1993 

1991 

1988 

0 

i3 

10 

20o5 

2002 

1999 

1996 

1994 

I99I 

1988 

I9S5 

I9S3 

1981 

1979 

1976 

I 

12 

2() 

1992 

1989 

1986 

1983 

1981 

1978 

1975 

1972 

1970 

1968 

1966 

1903 

2 

10 

3c. 

1979 

1977 

1974 

1971 

1909 

1966 

1963 

i960 

1958 

1955 

1953 

1950 

3 

9 

4o 

1966 

1964 

1961 

1958 

1955 

1953 

1950 

1947 

1945 

1942 

1940 

19^7 

4 

8 

"67 

5o 
o 

1954 

I  95  I 

1948 

1945 

1943 

1940 

1937 

1934 

1932 

1930 

1928 

1925 

5 
6 

■7 
5 
4 

1 94 1 

1939 

1936 

1933 

1931 

1928 

1925 

1922 

1920 

I9I7 

1915 

I9I2 

lO 

1928 

1926 

1923 

1920 

1918 

1916 

1912 

1909 

1907 

1905 

1902 

1900 

7 
8 

20 

I9I6 

I914 

1911 

1908 

1906 

1903 

1900  1897 

1895 

1892 

1890 

1887 

3o 

1903 

1901 

1898 

1895 

1893  1  1890 

1887  i885 

i883 

1880 

1878 ;  1875 

y 

2 

P'tgeiio]            TABLE  XIX.  Correction.                  | 

^  c 

Table  A. 

TableB. 

<  S 

Proportional  part  for  Seconds 

For  Min. 

c.-" 

J)  's  Horizontal  Parallax. 

of  Parallax. 

of 

A.lt. 

<^ 

Add. 

Add.  1 

D. 

17 

M. 

0 

54'  !  55'  1 

5G' 

57' 
7.19 

58' 

59' 

60' 

61' 

S. 
0 

0"1" 
56  55 

2"  3"  4" 
54535^ 

5' 
5^ 

G" 
5^ 

7" 
49 

S" 
48 

9" 

47 

M. 

0' 

S. 
0 

10.11 

9.14s 

•17 

6.22 

5.24 

4. -27 

3.3o 

10 

10.12 

9.ibfc 

.18 

7.20 

6.23 

5.26 

4.28 

3.3i 

10 

46 

i6 

454. 

143 

42 

4i 

4o 

39 

38 

0 

20 

10. i3 

9.i6i 

.19 

7.21 

6.24 

5.27 

4.3o 

3.32 

20 

37 

36 

35  34|33 

3:j 

3i 

3o 

29 

28 

4 

0 

:3o 

10. i4 

9.17^ 

.20 

7.23 

6.25 

5.28 

4.3i 

3.34 

3o 

27 

26 

25  2 

3  24 

2,- 

22 

21 

20 

19 

5 

6 

7 

40 

10. i5 

9.1st 

.21 

7.24 

6.27 

5.29 

4.32 

3:35 

4o 

18 

17 

161 

5i4 

IC 

12 

11 

10 

9 

78" 

bo 
0 

10.16 

9.i9t 

.22 

7.25 

6.28 
6.29 

5.3i 
5.32 

4.34 
4.35 

3.37 
3.38 

5o 
0 

8 
56 

7 
55 

6 

5"4  5 

5  5 
3  5^ 

/ 
5" 

3 

"5o 

2 

4q 

I 
48 

0 

47 

8 
9 

0 

10.18 

9.21  t 

.23 

7.26 

0 

10 

10.19 

9.22  f: 

i.25 

7.28 

6.3: 

5.34 

4.37 

3.40 

10 

47 

46 

4544143 

42I41 

4o 

39 

38 

•2. 

0 

20 

10.20 

9.23t 

.2b 

7.29 

6.32 

5.35 

4.38 

3.4i 

20 

37 

36 

3534I33 

3; 

3i 

3o 

29 

28 

4 

3o 

10.22 

9.25  i 

i.28 

7.3. 

6.34 

5.37 

4.40 

3.43 

3o 

28 

27 

26  2 

524 

28  22 

21 

20 

19 

5 
6 
7 

4o 

10.23 

9.26  J 

i.29 

7.32 

6.36 

5.39 

4.42 

3.45 

40 

18 

17 

161 

5i4 

i3  12 

II 

10 

9 

bo 

10.24 

9.28t 

i.3i 

7-34 

6  37 

5.40 

4.44 

3.47 

5o 

9 

8 

7 

6  5 

4  3 

2 

I 

0 

8 

a 

TABLE  XIX.-  Logarithms.                 | 

°  i 

TaelkCI 

^4 

Cor 

Rpc 

Apparent  Altitude  of  ])  's  centre. 

of 

Par, 

«a- 

Add.  1 

0   / 

0  f 

0  / 

0  / 

0  / 

0  / 

0  / 

0  / 

0  / 

0  / 

0  1 

0  / 

M. 

'54 

S. 
0 

17  0 

1710 

17  20 

17  30 

1 

7  40 

17  50 

18  0 

1810 

18  20 

18  30 

2450 

18  40 

2448 

18  50 
2446 

Sec. 
0 

Cor. 
l3 

2471 

2469 

2466 

2464 

1462 

2459 

2457 

2455 

2452 

10 

2457 

2454 

2452 

2449 

2447 

2444 

2442 

2440 

2438 

2436 

2434 

2431 

I 

12 

20 

2443 

2440 

2438 

2435 

2433 

2430 

2428 

2426 

i4i4 

2422 

2420 

2417 

2 

10 

3o 

2429 

2426 

2424 

2421 

2419 

2416 

2414 

2412 

2410 

2408 

2406 

24o3 

3 

9 

4o 

24i5 

2412 

2410 

2407 

24o5 

2402 

2400 

2898 

2896 

2894 

2892 

2889 

4 

7 

"55" 

5o 
0 

2401 

2398 

2396 

2393 

2391 

2388 

2386 

2384 

2882 
2368 

2880 

2878 

2875 

5 
6 

6 
5 

2387 

2384 

2382 

2379 

2377 

2374 

2372 

2870 

2366 

2364 

2861 

10 

2373 

2370 

2368 

2365 

2363 

236o 

2358 

2356 

2354 

2352 

235o 

2  348 

7 

3 

20 

2359 

2356 

2354 

235i 

2349 

2346 

2344 

2342 

2340 

2338 

2336 

2334 

8 

2 

3o 

2345 

2342 

2340 

2337 

2335 

2333 

233i 

2829 

2826 

2024 

2822 

2820 

9 

0 

4o 

233r 

2329 

2326 

2324 

2322 

2319 

23  1 7 

23i5 

23l2 

2810 

2808 

23o6 

56 

5o 
0 

23i7 

23i5 

23l2 

23ro 

23o8 

23o5 

23o3 

2801 

2299 

2297 

2295 

2298 

Sec. 
0 

Cor. 
l3 

23o3 

23oi 

2298 

2296 

2294 

2291 

2289 

2287 

2285 

2283 

2281 

2279 

10 

2290 

2287 

2285 

2282 

2280 

2278 

2276 

2274 

2271 

2269 

2267 

2265 

I 

12 

20 

2276 

2274 

2271 

2269 

2267 

2264 

2262 

2260 

2258 

2256 

2254 

2252 

2 

10 

3o 

2262 

2260 

2257 

2255 

2253 

225o 

2248 

2246 

2244 

2242 

2240 

2238 

3 

^ 

4o 

2249 

2247 

2244 

2242 

2240 

2237 

2235 

2233 

2281 

2229 

2227 

2225 

4 

8 

"57" 

5o 
0 

2235 

2233 

223o 

2228 

2226 

2  2  23 

22  21 

2219 

2217 

22l5 

22l3 

22II 

b 
6 

6 
5 

2222 

2220 

2217 

22l5 

22l3 

2210 

2208 

2206 

2204 

2202 

2200 

2198 

10 

2208 

2206 

22o3 

2201 

2199 

2197 

2195 

2193 

2190 

2188 

2186 

2i85 

7 

3 

20 

2195 

2193 

2190 

2188 

2186 

2l83 

2181 

2179 

2177 

2175 

2x78 

2171 

8 

2 

3o 

2182 

2180 

2177 

2175 

2173 

2170 

2168 

2166 

2164 

2162 

2160 

2i58 

9 

I 

40 

2168 

2166 

2i63 

2161 

2  I  59 

2i57 

2i55 

2i53 

2l5l 

2149 

2l47 

2i45 

Is" 

5o 
0 

2i55 

2i53 

2l5o 

2i48 

2 1 46 

2i43 

2l4l 

2189 

2187 

2i35 

2i33 

2l3l 

Sec. 
0 

Cor. 
i3 

2142 

2l4o 

2i37 

2i35 

2i33 

2i3o 

2128 

2126 

2  124 

2122 

2120 

2118 

10 

2129 

2127 

2124 

2122 

2120 

2II7 

2Il5 

2Il3 

2111 

2109 

2107 

2I05 

I 

12 

20 

2116 

2114 

211 1 

2109 

2107 

2I04 

2102 

2  100 

2098 

2096 

2094 

2092 

2 

10 

3o 

2102 

2100 

2097 

2095 

2093 

2091 

2089 

2087 

2o85 

2o83 

2081 

2079 

3 

9 

4o 

2089 

2087 

2084 

2082 

2080 

2078 

2076 

2074 

2072 

2070 

2068 

2066 

4 

8 

"5^ 

5o 
0 

2076 

2074 

2071 

2069 

2067 

2o65 

2o63 

2061 

2059 

2057 

20f 

5 

2o53 
2o4o 

5 
6 

6 
5 

2o63 

2061 

2o58 

2o56 

2o54 

2o52 

2o5o 

2048 

2046 

2o44 

■xoi 

10 

2o5o 

2o48 

2046 

2044 

2042 

2039 

2o37 

2o35 

2o33 

2o3r 

2029 

2027 

7 

4 
3 

20 

2o37 

2o35 

2o33 

so3i 

2029 

2026 

2024 

2022 

2020 

2018 

2016 

20l5 

8 

3o 

2025 

2023 

2020 

2018 

2016 

20l3 

20II 

2009 

2007 

2005 

2003 

2002 

y 

4o 

2012 

2010 

2007 

2oo5 

2003 

2001 

1999 

1997 

1995 

1993 

I99I 

1989 

6^ 

5o 
0 

1999 

1997 

1994 

1992 

1990 

19S8 

1986 

1984 

1982 

1980 

1978 

1976 

Sec. 
0 

Cor. 
l3 

1986 

1984 

1981 

1979 

1977 

1975 

i973 

1 9-' I 

1969 

1967 

1965 

1964 

10 

1973 

1971 

1969 

1967 

1965 

1962 

i960 

1958 

1956 

1954 

1932 

1951 

I 

12 

20 

1 96 1 

1959 

1956 

19^4 

1952 

1950 

1948 

19-16 

1944 

1942 

1940 

1988 

2 

ID 

3g 

1948  1946 

1943 

1941 

1939 

1937 

1935 

1933 

1981 

1929 

1927 

1926 

3 

9 

4o 

1935 :  1933 

1 93 1 

1929 

1927 

1925 

1923 

1921 

1919 

1917 

1915 

1918 

4 

8 

eT 

bo 
0 

.923 

1921 

1918 

1916 
1904 

1914 

1902 

I9I2 

1899 

1910 

1908 

1906 

1904 

1902 

I90I 

b 
6 

7 
5 
4 

1910 

1908 

1906 

1897 

1895 

1894 

1892 

1890 

1888 

10 

1898 

1896 

1893 

1891 

1889 

1887 

i885 

i883 

1881 

.879 

1877 

1876 

7 
8 

20 

i885 

1 883 

1881 

1879 

1877 

1875 

1873 

1871 

1869 

1867 

i865 

i8b3 

.3 

3o 

1873 

1871  ,  >868 

1866 

1864 

1862 

i860 

i858 

i856 

i854 

i8b2 

i85i 

9 

2 

TABLE  XIX.  Correction.             [Page  in 

^ 

r. 

Tabli:  a. 

TableB. 

•<  4) 

Proportional  part  for  Seconds 

For  Min. 

a.  M 

D  '3  Horizontal  Parallax. 

of  Parallax. 

of  Alt. 

<'^ 

Add. 

Add. 

D. 
19 

M. 

0 

54' 

55' 

56' 

57' 

58 
6.3 

59'  60' 
95.424.46 

61' 

3.49 

S. 
0 

0" 
56 

1" 
5'5 

2"  3 
545 

'4"}5 
3  5"^  15 

'G" 
1  5o 

7// 
49 

d"9" 
48'48" 

M.  1  S. 

10.26 

9.29  S 

.32 

7.36 

0 

0 

10 

10.28 

9.3it 

.34 

7-37 

6.4 

i5.44'4.47 

J.5i 

10 

47 

46 

45  44!43|4 

241 

4o 

39138 

2 

0 

20 

12.29 

9.33i 

.36 

7.39 

6.4 

35. 46'4. 49 

J. 53 

20 

37 

36 

35  3 

4  33  3 

2J1 

3i 

3029 

4 

'^ 

3o 

10. 3i 

9.34i 

.38 

7.41 

6.4 

55. 484.51 

J.bb 

3o 

28 

27 

26  2 

5  24  2 

J  22 

21 

20  19 

5 

6 
7 

4o 

10.33 

9.366 

.3q 

7-43 

6.4 

65.5o|4.53 

3.57 

40 

18 

17 

i6i5li4|i 

4  1-3 

12 

11 

10 

20 

5o 
0 

10.34 

9.38  6 

.41 

7.45 

6.4 

85.52 

4.56 

J. 59 

5o 
0 

9 

55 

8 
54 

7 
53  5 

6  5  4  3 
2  5i  5o49 

2 

48 

I 
4^ 

0 

47 

9 
0 

2 
0 
0 
0 

10.37 

9-41  6 

■  44 

7.48 

6.5 

I  5.55 

4.59 

4.  2 

10 

10.39 

9.43  6 

.46 

7.5o 

6.5 

45.57 

5.  1 

4.  5 

10 

4b 

4b 

444342  4 

1  40 

39 

38 

37 

2 

20 

10. 4i 

9-456 

.46 

7.52 

6.5 

65.59 

5.  3 

4.  7 

20 

36 

35 

34  33  33  3 

2J1 

3o 

29 

28 

4 

1 

3o 

10.43 

9.476 

.50 

7.54 

6.5 

86.  2 

5.  5 

4.  9 

3o 

27 

26 

25  2 

4  2-3  22I21 

20 

•9 

18 

5 

1 

4o 

10.45 

9.496 

.52 

7.56 

7- 

06.  4 

5.  8 

4.12 

40 

17 

17 

161 

^1' 

3  12 

1 1 

10 

9 

7 

5o 

10.47 

9.516 

.55 

7-59 

7- 

26.  6 

3. 10 

4.14 

5o 

8 

7 

6 

3  2 

2 

1 

0 

8 
9 

2 
2 

TABLE  XIX.  Logarithms. 

c  ^ 

Table  C. 

W^rt 

Cor.  Sec. 

Apparent  Altitude  of  D 's  Centre. 

of  Par. 

«a, 

Add. 

■  0  / 

0  / 

0  / 

0  / 

0  / 

0  / 

0  / 

0  / 

0  / 

0  1 

0  / 

0  1 

M. 

31 

S. 
0 

19  0 

19  10 

19  20 

19  30 

19  40 

19  50 

20  0 

20  10 

20  20 

20  30 

20  40 

20  50 

Sec. 

Cor. 

2445 

2443 

2441 

2439 

2437 

2435 

2433 

243i 

2429 

2427 

2426 

2424 

0 

12 

10 

243o 

2428 

2427 

2425 

2423 

2421 

2419 

2417 

24i5 

24i3 

2412 

2410 

I 

II 

20 

2416 

2414 

2412 

2410 

2408 

2407 

24o5 

24(33 

2401 

2399 

2398 

2396 

2 

0 

3o 

2402 

2400 

2398 

2396 

2J94 

239J 

2391 

2389 

2387 

2385 

2384 

2382 

3 

8 

40 

2388 

2386 

2384 

2382 

2J80 

2J79 

2377 

2J7b 

2373 

2371 

2370 

2368 

4 

6 

^5" 

5o 
0 

2374 

2372 

2370 

2368 

2366 
2353 

2  36  5 

2JbJ 

236i 

2359 

2357 
2344 

2356 
2342 

2354 
234 1 

5 
6 

5 

4 

236o 

2358 

2357 

2355 

235i 

2349 

2335 

2347 

2346 

10 

2347 

2345 

2343 

234 1 

2339 

2337 

2333 

2332 

233o 

2328 

2327 

7 

2 

20 

2333 

233 1 

2329 

2327 

2325 

2323 

2321 

23i9 

23i8 

23i6 

23i4 

23i3 

8 

I 

3o 

23i9 

23i7 

23i5 

23i3 

23ll 

23lO 

23o8 

23o6 

23o4 

2302 

23oi 

2299 

9 

0 

4o 

23o5 

2jo3 

23oi 

2299 

2297 

2296 

2294 

2292 

2291 

2289 

2287 

2286 

"56" 

bo 
0 

229a 

2290 

2288 

2286 

22S4 

2282 

2280 

2278 

2277 

2275 

2273 

2272 

Sec 

Cor. 

2278 

2276 

2274 

2272 

2270 

2269 

2267 

2265 

2264 

2262 

2260 

2259 

0 

12 

10 

2264 

2262 

2261 

2259 

2257 

2255 

2253 

225l 

22  5o 

2248 

2246 

2245 

I 

11 

20 

225l 

2249 

2247 

2245 

2243 

2242 

2240 

2238 

2  236 

2234 

2233 

223l 

2 

9 

3o 

2237 

2235 

2233 

223l 

2229 

2228 

2226 

2224 

2223 

2221 

2219 

2218 

3 

8 

4o 

2224 

2222 

2220 

2218 

2216 

22l5 

22l3 

22II 

2210 

2208 

2206 

2205 

4 

7 

^ 

DO 
0 

2210 

2208 

2207 

22o5 

2  203 

2201 

2199 

2197 

2196 

2194 

2192 

2191 

5 
6 

5 
4 
3 

2197 

2195 

2193 

219I 

2189 

2188 

2186 

2184 

2i83 

2181 

2179 

2178 

10 

2l84 

2182 

2180 

2178 

2176 

2175 

2173 

2171 

2170 

2168 

2166 

2i65 

7 

20 

2170 

2168 

2167 

2i65 

2i63 

2161 

2159 

2i57 

2i56 

2i54 

2l52 

2l5l 

8 

I 

3o 

2  I  57 

2i55 

2i53 

2l5l 

2149 

2i48 

2i46 

2i44 

2143 

2l4l 

2139 

2i38 

9 

0 

4o 

2i44 

2l42 

2l40 

2i38 

2i36 

2i35 

2i33 

2l3l 

2i3o 

2128 

2126 

2125 

"58" 

5o 
0 

2i3o 

2128 

2127 

2125 

2123 

2122 

2120 

2118 

2117 

2Il5 

2Il3 

2112 

Sec. 

Cor. 

2II7 

2Il5 

2Il4 

2II2 

2110 

2109 

2107 

2io5 

2Io4 

2102 

2  100 

2099 

0 

12 

10 

2I04 

2102 

2IOI 

2099 

2097 

2095 

2093 

2091 

2090 

2088 

2087 

2086 

I 

II 

20 

2091 

2089 

2088 

20S6 

2084 

2082 

2080 

2078 

2077 

2075 

2074 

2073 

2 

9 

Jo 

2078 

2076 

2075 

2073 

2071 

2069 

2067 

2o65 

2064 

2062 

2061 

2060 

3 

8 

40 

2o65 

2o63 

2062 

2060 

2o58 

2o56 

2o54 

2052 

205l 

2049 

2o48 

2047 

4 

7 

5? 

5o 
0 

2052 

2050 

2049 

2047 

2045 

2043 

204l 

2039 

2o38 

2o36 

2o35 

2o34 

5 
6 

6 
4 
3 

2039 

2037 

2o36 

2o34 

2032 

2o3i 

2029 

2027 

2026 

20^4 

2022 

2021 

10 

2026 

2024 

2023 

2021 

2019 

2018 

2016 

20l4 

20l3 

2011 

2009 

2008 

8 

20 

20 1 4 

2012 

2010 

2008 

2006 

2003 

2003 

2001 

2000 

1998 

1996 

1995 

3o 

2001 

1999 

1997 

1995 

1993 

1992 

1990 

1988 

1987 

1985 

1984 

1983 

y 

4o 

1988 

198b 

198b 

1983 

1981 

1979 

1977 

1975 

1974 

1972 

1971 

1970 



"6^ 

bo 
0 

1975 

1973 

1972 

1970 

1968 

1967 

1965 

1963 

1962 

i960 

1958 

1957 

Sec. 

Cor 

1963 

I  96  I 

1959 

1957 

1955 

1954 

1952 

1950 

1949 

1947 

1946 

1945 

0 

12 

10 

I9D0 

1948 

1940 

1944 

1942 

1941 

J9J9 

1937 

1936 

1934 

1933 

1932 

I 

II 

20 

1937 

1935 

I9J4 

1932 

1930 

1929 

1927 

1925 

1924 

1922 

1920 

1919 

2 

0 

Jo 

192b 

1923 

I  92  I 

I9I9 

1917 

1916 

1914 

I9I2 

I9II 

1909 

1908 

1907 

J 

8 

40 

I9I2  I9IO 

1909 

1907 

I90D 

1904 

1902 

1900 

1899 

1897 

1895 

1894 

4 

7 

67 

5o 
0 

1900  1898 
1887  i885 

1896 
1884 

1894 
1882 

1892 

I89I 

1889 

1887 

1886 

1884 

i883 

1882 

5 
6 

6 
5 
3 

1880 

1879 

1877 

1875 

1874 

1872 

1871 

1870 

10 

1875  1873 

187I 

1869 

1867 

1866 

1864 

1862 

1 86 1 

i859 

i858 

i857 

/ 
« 

20 

1862  i860 

.859 

i857 

i8b5 

i854 

i852 

i85o 

1849 

1 847 

i846 

1845 

2 

3o 

i85o 

1 848 

1847 

1845 

1843 

1842 

i84o 

1 838 

i837 

i835  i834l 

i833   9  1  ij 

f  Page  112] 


TABLE   XIX. 

Correction. 


D.    M. 


23 


24 


D  's  Horizontal  Parallax. 


54' 


10.49 
10. 5i 
10.53 
10.55 
10.58 
II . 


9.53 
9.55 
9.57 


10. 


10.  7 
10.  9 
10.12 
10. 1 
10.17 
10. 19 


10.23 
10.26 
10.29 
10. 3i 
10.34 
10.37 


10.40 
10.43 
10.46 
10.49 
10.52 
10.55 


5G' 


8.57 
8.59 

9' 
9-  4 
9.  6 
9.  9 


9' 

9.14 

9.16 

9.19 

9.21 

9.24 


9.28 
9.31 
9.33 
9-36 
9.39 
9-42 
9-45 


8.  3 
8.  6 
8.  8 
8. 10 
8.i3 


8.i5 


8.23 
8.26 
8.29 


8.33 
8.35 
8.38 
8.41 
8.44 
8.47 


8.5o 
8.53 
8.56 
9.  o 
9.  3 
9.  6 


58'    59'    GO'    61' 


i5 


7-17 


7.20 
7.22 
7.25 
7.28 
7.3i 
7-34 
7.37 
7.40 
7.43 
7-46 
7-49 
7.52 

7T55 
7.59 
8.  2 
8.  5 
8.  8 
8.12 


.16 


.14 


Table  A. 

Proportional  part  for  Seconds 

of  Parallax. 

Add. 


Table  B 
For  Min 

of  alt. 

Add. 


1//  -2"  3"  4" 


6" 


5o 


8"  19" 


M. 


S. 


Explanation  of  Table  XIX. 
This  table  consists  of  two  parts,  for  finding  a  correction  of  the  moon's  distance  and  a  loga- 
rithm corresponding :  they  are  both  in  the  same  page  from  the  beginning  of  the  t^able  to 
the  altitude  of  21  degrees,  after  which  the  correction  is  on  the  left  hand  page,  and  the 
logarithm  on  the  right,  both  being  found  at  the  same  opening  of  the  book,  in  the  following 
manner. 

To  find  the  Correction  of  Table  XIX. 

1.  Enter  the  table  marked  Correction,  and  find  in  the  side  column  the  moon's  apparent 
altitude,  or  the  altitude  next  less,  if  there  be  any  units  of  miles  in  the  altitude  ;  opposite  to 
this,  and  under  the  minutes  of  the  moon's  horizontal  parallax,  will  be  the  approximate  cor- 
rection. 

2.  Enter  table  A,  abreast  of  the  approximate  correction,  and  find  the  seconds  of  the 
moon's  horizontal  parallax,  viz.  the  tens  of  seconds  at  the  side,  and  the  units  at  the  top, 
under  the  latter,  and  opp(5site  the  former  will  be  the  correction  of  table  A. 

3.  Enter  table  B,  abreast  of  the  approximate  correction,  and  find  the  units  of  miles  in  the 
moon's  apparent  altitude  (neglected  above),  opposite  to  which  will  be  a  number  of  seconds, 
which,  being  added  to  the  corrections  found  from  table  XIX.  and  from  table  A,  will  give 
the  sought  correction. 

To  find  the  Logarithm  of  Table  XIX. 

Enter  the  table  marked  Logarithms,  in  the  column  titled  at  the  top  with  the  degrees  and 
minutes  nearest  to  the  moon's  apparent  altitude,  and  find  the  logarithm  corresponding  to 
the  moon's  horizontal  parallax  in  the  side  column,  or  the  next  less  parallax,  if  there  be 
units  of  seconds  in  it.  Abreast  of  this  in  the  table  C,  opposite  the  units  of  seconds  of  par- 
allax neglected,  will  be  a  correction,  to  be  added  to  the  former  logarithm,  to  obtain  the 
logarithm  sought. 

It  was  observed  in  a  former  part  cf  his  work,  that  in  fij.mg  these  tables  so  as  to  render  the 
corrections  of  the  tables  A,  B,  C,  additive,  it  had  been  found  necessary  to  make  the  great- 
est corrections  correspond  to  0"  of  parallax  and  0'  of  altitude,  so  that  ichen  you  find  the  ex- 
act parallax  and  altitude  in  the  side  and  top  columns  of  table  XIX.  it  icill  still  be  necessary  to 
refer  to  the  tables  .4,  B,  or  C,  to  take  out  the  corrections  corresponding  to  0"  of  parallax  or  0' 
of  altitude.  This  is  evident  from  the  inspection  of  the  tables,  but  it  was  proper  to  make 
this  remark  as  a  caution  to  prevent  mistakes.  To  illustrate  these  rules,  the  following  ex- 
amples are  given,  in  which  all  the  corrections  are  put  down  and  added  together;  but  after  a 
little  practice  it  will  be  very  easy  to  take  the  numbers  from  the  table  by  inspection  and  add 
them  together  without  the  trouble  of  writing  them  down  separately. 


TABLE  XIX. 

[Page  1:3 

Logarithms. 

s  i 

Table  C. 

k| 

Cor.  Sec 

-"^  rt 

Apparent  Altitude  of  5  's  centre. 

of  Par. 

'^■Ch 

Add. 

0   / 

0  / 

0  /  0  1 

0   / 

0  / 

0   / 

0  / 

0  / 

0  /[  0  /  1  0  1 

M. 

T4 

S. 
0 

21  0 

2120 

214022  0 

22  '-20 

22  40 

23  C 

23  20 

23  40 

24  C 

24  2C 

24  40 

2891 

Sec. 
0 

Cor. 
12 

2422 

2419 

2416 

24i3 

2410 

2407 

2404 

2401 

2399 

2896 

2894 

10 

2408 

24o5 

2402 

2399 

2896 

2893 

2890 

2887 

2385 

2382 

2880 

2877 

I 

II 

20 

2394 

2391 

2388 

2385 

2382 

2879 

2876 

2878 

23-1 

2368 

2  366 

2363 

2 

9 

3o 

238.) 

2377 

2374 

2371 

2368 

2365 

2862 

2860 

2357 '2355 

2352 

235o 

3 

H 

4o 

2  366 

2363 

236o 

2357 

2354 

2352 

2349 

2346 

2344 

2341 

2889 

2325 

2336 

4 

6 

T-r 

5o 
0 

2352 

233o 

2349 
2335 

2346 

2343 

2340 

2338 

2335 

2882 

233o 

2327 

2822 

5 
6 

5 
4 

2332 

2329 

2326 

2824 

2321 

2818 

2816 

2818 

2811 

2808 

10 

2325 

2322 

23i9 

23i6 

23i3 

2810 

2807 

2804 

2802 

2299 

2297 

2294 

7 

2 

20 

23ll 

2j0« 

23o5 

2302 

2299 

2296 

2293 

2291 

2288 

2286 

2284 

2281 

8 

I 

3o 

2297 

2294 

2291 

2288 

2285 

2283 

2280 

2277 

2275 

2272 

2270 

2267 

9 

0 

4o 

2284 

2281 

2278 

2275 

2272 

2269 

2266 

2268 

2261 

2  258 

2  256 

2253 

l6 

5o 
0 

2270 

2267 

2264 

2261 

2258 

2256 

2253 

2250 

2248 

2245 

2243 

2240 

Sec. 
0 

Cor. 
12 

2257 

2253 

2250 

2247 

2244 

2242 

2289 

2286 

2234 

2281 

2229 

2226 

10 

2243 

2240 

2237 

2234 

223l 

2229 

2226 

2228 

2221 

2218 

22IO 

22l3 

I 

II 

20 

2229 

2226 

2223 

2220 

2217 

22l5 

2212 

2210 

2207 

22o5 

2203 

2200 

2 

9 

3c, 

2216 

22l3 

2210 

2207 

2204 

2202 

2199 

2196 

2194 

2191 

2189 

2186 

3 

8 

4o 

2  2o3 

2200 

2197 

2194 

2191 

2188 

2i85 

2l83 

2180 

2178 

2176 

2178 

4 

7 

^ 

5o 
0 

2189 

2186 

2l83 

2180 

2177 

2.75 

2172 

2170 

2167 
2 1 54 

2i65 

2l5l 

2i63 

2149 

2160 

2i46 

5 
6 

5 
4 

2176 

2173 

2170 

2167 

2164 

2162 

2i59 

2i56 

10 

2i63 

2160 

2i57 

2i54 

2l5l 

2149 

2i46 

2143 

2l4l 

2i38 

2i36 

2i33 

7 

3 

20 

2149 

2i46 

2i44 

2l4l 

2i38 

2i35 

2182 

2l3o 

2128 

2125 

2123 

2120 

8 

I 

3o 

2i36 

2i33 

2l30 

2127 

2  124 

2122 

2119 

2117 

2Il5 

2112 

2no 

2107 

9 

0 

4" 

212.3 

2120 

2II7 

2Il4 

2III 

2109 

2106 

2104 

2IOI 

2099 

2097 

2094 

"58" 

bo 
0 

21  10 

2107 

2I04 

2I0I 

2098 

2o85 

2096 

2098 

2091 
2078 

2088 

2086 
2078 

2084 
2071 

2081 
2068 

Sec. 
0 

Cor. 
12 

2097 

2094 

2091 

2088 

2o83 

2080 

3075 

10 

2084 

2081 

2078 

2075 

2072 

2070 

2067 

2o65 

2062 

2060 

2o58 

2o55 

I 

II 

20 

2071 

2068 

2065 

2062 

2o59 

2o57 

2o54 

2052 

2049 

2o47 

2045 

2042 

2 

9 

3c 

2o58 

2o55 

2o52 

2049 

2046 

2044 

204 1 

2089 

2o36 

2o34 

2082 

2029 

3 

8 

4o 

2o45 

2042 

2o39 

2o36 

2o33 

2o3l 

2028 

2026 

2028 

2021 

2019 

2016 

4 

7 

59" 

5o 
0 

2o32 

2029 

2026 

2023 

2020 

2018 

20l5 

20l3 

2010 

2008 

2006 

2oo3 

5 
6 

6 

4 

2019 

2016 

20l3 

2010 

2008 

2006 

2oo3 

2000 

1998 

1995 

1993 

'99' 

10 

2006 

2oo3 

2001 

1998 

1995 

1993 

1990 

1987 

1985 

T982 

1980 

1978 

7 
8 

3 

?o 

1993 

1990 

1088 

1985 

1982 

1980 

1977 

1975 

1972 

1970 

1968 

1965 

2 

3o 

1981 

1978 

1975 

1972 

1969 

1967 

1964 

1962 

1959 

1957 

1955 

1953 

y 

0 

4o 

1968 

1955 

1962 

1959 

19^7 

19^4 

1952 

1949 

1947 

1944 

1942 

1940 

"6^ 

60 
0 

1955 

1952 

1950 

1947 

1944 

1942 

1939 

1926 

1937 
1924 

1934 

1982 

1980 

1927 

Sec. 
0 

Cor. 
12 

1943 

1940 

1937 

1934 

1 93 1 

1929 

1921 

1919 

1917 

1915 

10 

1930 

1927 

1925 

1922 

1919 

1917 

I9I4 

I9I2 

1909 

1907 

1905 

1902 

I 

II 

20 

I9I7 

1914 

I9I2 

1909 

1906 

1904 

I90I 

1899 

1896 

1894 

1892 

1890 

2 

10 

3o 

1905 

1902 

1900 

1897 

1894 

1892 

1889 

1887 

1884 

1B82 

1880 

1877 

3 

8 

4o 

1892 

1889 

1887 

1 884 

1881 

1879 

1876 

1874 

1871 

1869 

1867 

ib65 

4 

7 

61 

5.) 
0 

1880 

1877 

1 865 

1875 
1862 

1872 
1859 

1869 
"i85f 

1867 

1864 

1862 

1859 

1857 

i855 
1843 

i853 
i84o 

5 
6 

ft 

5 

1 868 

1 854 

i852 

i85o 

1847 

1845 

10 

1 855 

i852 

i85o 

1847 

1844 

1842 

1889 

1887 

i834 

1882 

i83o 

1828 

7 

3 

20 

i843 

i84o 

i838 

1 835 

i832 

i83o 

1827 

1825 

1822 

1820 

1818 

1816 

8 

2 

3o 

i83r  1828  1 

1825  1822  1 

1820 

1817 

i8i5 

i8i3 

1810  1808  1 

1806 

i8o3 

9 

I 

15 


Page  114] 

TABLE   XIX. 

Correction. 

it 

Table  A. 

Tab 

leB. 

])  's  Horizontal  Parallax. 

Proportional  part  for  Seconds 
of  Parallax. 

For  Min. 
of  alt. 

<;'=* 

Add. 

Add. 

D 

^5 

ai. 

0 

54 

55' 

56' 

57' 

58' 

59' 

CO' 

61' 

S. 
0 

0" 

53 

1" 
5i 

2" 
57 

3" 
53 

4" 
49 

5" 
48 

6" 

48 

7" 
47 

8" 
46 

9" 

45 

JM. 

"~o~ 

S. 
0 

11.53 

10.59 

to.  5 

9.10 

8.16 

7.22 

6.27 

5.33 

i(j 

11.57 

II.  2 

10.  8 

9.14 

8.i97.25j6.3i 

5.36 

10 

^A 

Ai 

42 

4i 

4o 

39 

39 

38 

37 

36 

2 

1 

20 

12.  0 

II.  b 

lO.II 

9.17 

8.237.286.34 

5.40 

20 

Jb 

M 

33 

32 

3i 

3o 

3o 

20 

28 

27 

4 

I 

Jo 

12.  J 

II.  9 

io.i5 

9.20 

8.26 

7.32 

6.3fc 

5.44 

3o 

26 

25 

24 

23 

22 

21 

20 

20 

19 

lb 

5 

6 

7 

2 
2 

40 

12.   6 

II  .12 

10.18 

9.24 

8.3o 

7.36 

6.41 

5.47 

4o 

17 

16 

i5 

i4 

i3 

12 

11 

II 

10 

9 

26 

bo 
0 

12.   9 

1 1  .i5 

10.21 

9.27 

8.33 

7.39 
7.43 

6.45 

5.5i 
5.55 

5o 
0 

8 

53 

7 

52 

6 
57 

5 

5o 

4 

49 

3 

4^ 

2 

48 

I 

I 

46 

0 

45 

8 
9 
0 
1 

3 
3 
0 
0 
1 

12. ]3 

II. 19 

10.25 

9.3. 

8.37 

6.4c; 

10 

12.16 

II  .22 

10.28 

9-M 

8.40 

7.47 

b.bJ 

b.59 

K) 

44 

4J 

42 

4i 

4o 

4o 

39 

38 

37 

36 

20 

12.19 

II  .25 

10.32 

9.38 

8.44 

7.5o 

6.5e 

6.  3 

20 

35 

J4 

33 

32 

32 

3i 

3o 

2Q 

28 

27 

3 

1 

Jo 

12.23 

II  .29 

10.35 

Q.41 

8.48 

7.54 

7.  c 

6.  7 

3o 

26 

25 

24 

23 

23 

22 

21 

20 

19 

18 

5 

2 

4o 

12.26 

I  I  .32 

10.39 

Q.45 

8.5i 

7.58 

7.  4|6.ii 

4o 

'7 

16 

i5 

i4 

14 

i3 

12 

II 

10 

9 

7 

3 

27 

bo 
0 

12.29 

11.36 

10.42 

9.49 

8.55 

8.   2 

7.  8 

6.i5 
6.20 

5o 

0 

8 

52 

7 
5i 

6 

5^ 

6 

5 

48 

4 

48 

3 

2 

46 

45 

0 

8 
9 

0 

3 

3 
0 

12.34 

II  .40 

10.47 

9.53 

9.  0 

8.  6 

7.i3 

ID 

12.37 

11.44 

10. 5i 

9.57 

9-  4 

8.10 

7-17 

6.24 

lo 

4J 

42 

4i 

40 

40 

39 

38 

37 

36 

35 

1 

20 

12. 4l 

11.48 

10.54 

10.   I 

9.  8 

8.i4 

7.21 

6.28 

20 

M 

33 

:ii 

32 

3i 

3o 

29 

28 

27 

26 

3 

4 
5 

1 

2 

Jo 

12.44 

II. 5i 

10.58 

10.  5 

9. 11 

8.18 

7.25 

6.32 

Jo 

2  b 

24 

24 

23 

22 

21 

20 

19 

18 

17 

2 
2 
3 

4o 

12.4s 

11.55 

II .  2 

10.  9 

9.i5 

8.22 

7.29 

6.36 

4o 

17 

16 

i5 

i4 

i3 

12 

II 

10 

9 

9 

6 

^8 

bo 

0 

t2.52 

II  .59 

II.  5 

10.12 

9.19 

8.26 

7.33 

6.40 
6.44 

5o 

0 

8 
5^ 

7 
5i 

6 

5o 

5 

49 

4 

48 

3 

48 

2 

47 

I 
46 

I 

45 

0 

8 
9 

0 

3 
3 

12.55 

12.  2 

II.  9 

10.16 

9.23 

8.3o 

7.37 

10 

12.59 

12.  6 

II. i3 

10.20 

9.27 

8.34 

7.42 

6.49 

10 

A'i 

42 

4i 

4i 

40 

39 

38 

37 

36 

35 

h 

1 

20 

iJ.  3 

12.10 

II. 17 

10.24 

9.3i 

8.39 

7.46 

6.53 

20 

M 

34 

JJ 

32 

3i 

3o 

29 

28 

27 

26 

4 

2 

Jo 

i3.  6 

12. i4 

II. 21 

10.28 

9.36 

8.43 

7.5o 

6.57 

3o 

26 

25 

24 

23 

22 

21 

20 

19 

t9 

18 

5 

2 
2 
3 

40 

i3.io  12.18 

II  .25 

10.32 

9.40 

8.47 

iM 

7.   2 

4o 

17 

16 

i5 

i4 

i3 

12 

12 

II 

10 

9 

7 

29 

bo 

0 

!3.i4  12.22 

11.29 

10. 36 

9-44 

8.5i 

7.59 

7.  6 

5o 

8 

7 

6 

5 

5 

4 

47 

3 

46 

2 

45 

I 

0 

43 

9 
0 

4 

i3. 19 

12.26 

11.34 

10.42 

9.49 

8.57 

8.  4 

7.12 

0 

5i 

5o 

49 

48 

48 

0 

(O 

1J.2J 

12. 3o 

11.38 

10.46 

9.53 

9.   I 

8.  8 

7.16 

10 

42 

41 

4i 

4o 

39 

38 

37 

36 

35 

34 

2 

1 

20 

i3.27 

12.34 

11.42 

io.5o 

9.57 

9.  5 

8.i3 

7.21 

20 

34 

33 

32 

3i 

3o 

29 

28 

27 

27 

26 

4 

1 
2 

Jo 

iJ.Ji 

12.38 

11.46 

10.54 

10.  2 

9.10 

8.17 

7.25 

Jo 

25 

24 

23 

22 

21 

21 

20 

19 

18 

17 

5 

2 

4o 

i3.35 

12.43 

[i.5o 

10.58 

10.  6 

9.148.22 

7.3o 

4o 

16 

i5 

14 

i4 

i3 

12 

II 

10 

9 

8 

7 

3 

bo 

13.39 

12.4- 

11.55 

II.  3 

10.10 

9.188.26 

7.34 

5o 

7 

7 

6 

5 

4 

3 

2 

I 

0 

0 

8 
9 

3 
4 

EXAMPLE    I. 

Given  the  moon's  apparent  a 

titude  44°  27',  and  her  horizontal  parallax  5G'  55",     Rec 

pircd 

the 

correction  and  log-arithm  1 

For  the  Correctior 

[. 

For  the  Logarithm. 

In  Tab.  xix.  to  alt.  44°  20'  and  pa 

r.  6G'  is     19'  54" 

In  Tab.  xix.  to  nearest  alt.44.i°  and  par.  5C 

'  .^n" 

20RR 

. ,  Tab.  A.  55"  parallax 

3 

..  Tab.  C.  5"  parallax 

.. . . 

5 

..Tab.  B.  7' altitude 

5 

Sought  correction 

Sought  logarithm 

2093 

.  20'    2" 

EXAMPLE    II. 

Given  the  moon's  apparent  altit 

ude  50°  IG',  and  horizontal  parallax  59'  0".     Required  the 

corre( 

tion 

and  log-arithm  ? 

For  the  Correction 

. 

For  the  Logarithm. 

In  Tab.  xix.  to  aii.  50°  10'  and  pa 

r.  69'is    22'    3" 

In  Tab.  xix.  to  alt.  50°  and  par.  59'  0". . 

... 

913 

..  Tab.  A.O"paral 
. .  Tab   B    6'  altitu 

ax 

38 
4 

. .  Tab.  G.  0"  parallax  . 

12 

Je 

... 

1925 

0)0)/ 

45"| 
AMP 

LE    III. 

EX 

Given  the  moon's  apparent  altiti 

ide  28°  27',  and  horizontal  parallax  54'  10".     Required  the 

correc 

tion 

and  log-arithm  ? 

For  the  Correctior 

. 

For  the  Loerarithm. 

In  Tab.  xix.  to  alt.  28'^  20'  and  pa 

r.  54'is    13'    3" 

Tab.  xix.  to  nearest  alt.  28°  30'  and  par.  54' 

10"  i 

J354 

..  Tab.  A.  10"  pnrallav 

d.-^ 

Table  C.  0"  parallax 

12 

..  Tab.  B.  7'  altituc 

e 

3 

;3G6 

Sought  logarithm 

Sr>noh»  rorrPfllr 

n' 

49"  1 

/ 

TABLE  XIX.                 [rise  us 

Logarithms. 

o   ts 

Table  C. 

y  (0 

Apparent  Altitude  of  D  's  Centre. 

Cor.  Sec. 
of  Par. 

CiCn 

Add. 

0  / 

0  / 

0  ?  I  0  / 

0   / 

0  / 

0  / 

0  / 

0  / 

0   / 

0  / 

0  / 

M. 

i4 

S. 
0 

25  0 

•25  2'J 

25  4C 

2G  0 

2387 

2d  20 

•2G40 

27  0 

27  30 

28  0 
2871 

28  30 

2368 

29  0 

2865 

29  30 
2862 

Sec 
0 

Cor. 
12 

2J89 

2387 

2384 

2880 

2878 

2876 

2874 

10 

2375 

2J7J 

2871 

2369 

2867 

2364 

2862 

2860 

2357 

2854 

235i 

2348 

I 

1 1 

20 

2861 

2359 

2jb7 

2355 

2jbJ 

235i 

2849 

2846 

2848 

2841 

2388 

2335 

2 

3o 

2J47 

2345 

2J4J 

234i 

2889 

2887 

2335 

2332 

2829 

2827 

2824 

2821 

8 

40 

2334 

23J2 

2329 

2827 

2825 

2828 

2821 

2818 

2815 

23i3 

2810 

2807 

4 

6 

55 

5o 
0 

2820 

23i8 

23lb 

23i3 

28  i  I 

2809 

23o7 

2804 

2801 

2299 

2296 

2298 

5 
6 

5 
4 

23o6 

23o4 

23oi 

2299 

2297 

2295 

2298 

2291 

2288 

2286 

2283 

2280 

10 

2292 

2290 

2288 

22S6 

2284 

2282 

2280 

2277 

2274 

2272 

2269 

2266 

7 

2 

20 

2279 

2277 

2274 

2272 

2270 

2268 

2266 

2264 

2261 

2258 

2255 

2  258 

8 

I 

Jo 

2265 

2  263 

2261 

2269 

2267 

2255 

2258 

225o 

2247 

2245 

2242 

2289 

9 

0 

40 

225[ 

2249 

2247 

2245 

2248 

2241 

2289 

2286 

2233 

223l 

2228 

2226 

ItT 

bo 

0 

2  2  38 

2206 

22J4 

2282 

2280 

2228 

2226 

2228 

2220 

2218 

22l5 

22l3 

Sec. 

Cor. 

2224 

2222 

2220 

2218 

2216 

2214 

2212 

2210 

2207 

2205 

2202 

2199 

0 

12 

10 

22II 

2209 

2207 

2205 

22o3 

2201 

2199 

2196 

2198 

2I9I 

2188 

2i85 

I 

II 

20 

2198 

2190 

2193 

2I9I 

2189 

2187 

2i85 

2188 

2180 

2178 

2175 

2172 

2 

9 

8 

Jo 

2l84 

2182 

2180 

2178 

2176 

2174 

2172 

2170 

2167 

2i65 

2162 

2169 

8 

40 

2I7I 

2169 

2167 

2i65 

2168 

2161 

2159 

2i56 

2i58 

2l5l 

2148 

2i46 

4 

7 

^ 

bo 
0 

2ib8 

2  I  bo 

2ib3 

2l5l 

2149 

2147 

2i46 

2148 

2l4o 

2i38 

2i85 

2182 

5 
6 

5 
4 

2i44 

2142 

2l40 

2i38 

2i36 

2184 

2182 

2l3o 

2127 

2125 

2122 

2119 

10 

2l3l 

2129 

2127 

2125 

2128 

2I2I 

2119 

2II7 

2ll4 

2II2 

2109 

2106 

7 

3 

20 

2118 

2I16 

2Il4 

2112 

2110 

2108 

2106 

2I04 

2I0I 

2099 

2096 

2093 

8 

I 

Jo 

2io5 

2103 

2I0I 

2099 

2097 

2095 

2093 

2091 

2088 

2086 

2088 

2080 

9 

0 

40 

2092 

2090 

2088 

2086 

2084 

2082 

2080 

2078 

2075 

2078 

2070 

2067 

^ 

bo 
0 

2079 

2077 

2075 

2078 

2071 

2069 

2067 

2o65 

2062 

2059 

2057 

2o54 

Sec. 

Cor 

2066 

2064 

2062 

2060 

2o58 

2o56 

2o54 

2052 

2049 

2046 

2044 

204 1 

0 

12 

10 

20bJ 

2o5l 

2q49 

2047 

2045 

2048 

204l 

2089 

2o36 

2o38 

2081 

2028 

I 

II 

20 

2o4o 

2o38 

2o36 

2084 

2082 

2080 

2028 

2026 

2028 

2020 

2018 

20l5 

2 

g 

Jo 

2027 

2025 

2023 

2021 

2019 

2017 

20l5 

20l3 

2010 

2007 

2005 

2003 

3 

s 

40 

20l4 

2012 

2010 

2008 

2006 

2005 

2008 

2000 

1997 

1994 

1992 

1990 

4 

7 

"57 

bo 
0 

2001 

1999 

1997 

199b 

1 99  J 

1992 

1990 

1987 

1984 

1982 
1969 

1980 
1967 

'977 
1964 

5 
6 

6 
4 
3 

1989 

1987 

1985 

1988 

1981 

1979 

1977 

1975 

1972 

10 

1976 

1974 

1972 

1970 

1968 

1966 

1964 

1962 

i9b9 

1956 

1954 

1952 

7 
8 

20 

1 9b  J 

1 96 1 

i9b9 

1967 

1955 

19^4 

1952 

1949 

1946 

1944 

1942 

1989 

2 

Jo 

I95I 

1949 

19^7 

1945 

1943 

I94I 

1939 

1987 

1934 

1981 

1929 

1927 

9 

40 

i9i8 

1936 

1934 

1982 

1980 

1928 

1926 

1924 

1921 

1918 

I9I6 

1914 

"6^ 

bo 
0 

1925 

1923 

I92I 

I9I9 

1917 

I9I6 

1914 

I9I2 

1909 

1906 

1904 

1902 

Sec. 

Cor. 

1913 

I9I1 

1909 

1907 

1905 

1908 

1 90 1 

1899 

1896 

1898 

I89I 

1889 

0 

12 

10 

1900 

1898 

189b 

1894 

1892 

1891 

1889 

I8S7 

1 884 

1881 

1879 

1877 

I 

II 

20 

1888 

1886 

1884 

1882 

1880 

1878 

1876 

1874 

1871 

1869 

1867 

1864 

2 

10 

3o 

187b 

1873 

I87I 

1869 

1867 

1866 

1864 

1862 

i859 

i856 

1 854 

1 852 

3 

8 

4o 

1 863 

I86I 

i8b9 

1857 

i855 

i8b4 

i852 

1849 

1846 

i844 

1842 

l88q 

4 

7 

61 

5o 
0 

i85i 

1849 

1847  1845  1 

1843 

i84r 

1889 

1887 

1884 

1882 

1880 

1827 

5 
6 

6 
5 

i838 

i836 

!834 

1882 

i83o 

1829 

1827 

1825 

1822 

1819 

1817 

i8i5 

10 

1826 

1824 

1822 

1820 

1818 

1817 

i8i5 

I8I8 

1810 

1807 

i8o5 

1808 

7 

3 

20 

i8i4 

1812 

1810 

1808 

1806 

i8o5 

i8o3 

1800 

1797 

179b 

1798 

1791 

8 

2 

3o 

1801 

1799 

1798 

1796   1794 [ 

1792 

1790 

1788 

1785  1788  1 

1781  1778 1  9  1 

1 

P«?«ii6]                                          TABLE   XIX. 

Correction. 

■^  = 

Table  A. 

Tab.B. 

J>  's  Horizoi^tal  Parallax. 

Proportional  part  for  Seconds 
of  Parallax. 

ForM. 
of  alt. 

<'^ 

Add. 

Add. 

D. 

3^ 

M. 

0 

54' 

55' 

56' 

57' 

58' 

59' 

GO' 

Gl' 

S. 

0" 
5i 

1" 

Vo 

2" 

49 

3" 
48 

4" 

48 

5" 
47 

6" 

46 

7" 
45 

8" '9" 

M. 

? 

s. 

0 
0 

i3.43 

12. 5i 

tl.5q 

II.  7 

10. i5 

9.23 

8.3. 

7.39 

0 

AA 

43 

lO 

i3.47 

12.55 

12.   C 

II. II 

10.19 

9.27 

8.35 

7-44 

10 

42 

42 

4i 

4o 

39 

38 

37 

3b 

ib 

35 

2 

1 

20 

i3.5i 

12.59 

12.  7 

II. 16 

10.24 

9.32 

8.40 

7.48 

20 

34 

33 

32 

3. 

3o 

29 

29 

2b 

27 

26 

4 

2 

3o 

i3.55 

i3.  3 

12.12 

II  .20 

10.29 

9.37 

8.45 

7.53 

3o 

2b 

24 

23 

23 

22 

21 

20 

19 

18 

17 

b 
6 

3 

4o 

i3.5q 

i3.  8 

12.16 

11.24 

10.33 

9.41 

8.49 

7.58 

4o 

17 

16 

lb 

14 

.3 

12 

II 

.0 

10 

V 

7 

3 

37 

bo 
o 

i4.  3 

l3.I2 

12.20 

1 1 .29 

10.37 

9-46 

8.54 

8.  3 

bo 

8 

7 

6 

b 

4 

4 
46 

3 
45 

2 

I 
Ai 

u 
Ao- 

_9_ 
0 

4 
0 
0 

i4.  9 

i3.i7 

12.26 

11.34 

10.43 

9.5i 

9.  0 

8.   9 

0 

5o 

49 

48 

47 

47 

10 

i4.i3 

l3.22 

12. 3o 

II  .39 

10.47 

9.56 

9.   5 

8.i3 

.0 

4i 

4i 

40 

39 

38 

37 

36 

ib 

35 

M 

1 

20 

I4.I7 

i3.26 

12.35 

11.43 

10.52 

10.   I 

9.10 

8.18 

20 

Si 

32 

3. 

Jo 

3o 

29 

28 

27 

26 

25 

4 

3o 

l4.2I 

i3.3o 

12,39 

11.48 

10.57 

10.  6 

9.14 

8.23 

^0 

24 

24 

23 

22 

21 

20 

19 

.8 

18 

17 

5 

5 

4o 

14.26 

i3.35 

12.44 

11.53 

II.     2 

10.10 

9.19 

8.28 

4o 

16 

lb 

14 

i3 

12 

12 

11 

.0 

9 

8 

7 

4 
4 
0 

3l 

bo 

0 

i4.3o 

i3.39 

12.48 

11.57 

II.  6 

10. i5 

9.24 

8.33 

bo 

7 

6 

6 

5 

4 

3 
46 

2 

45 

I 
AA 

0 
43 

0 
4^ 

H 
9 

0 

14.35 

i3. 44 

12.53 

12.   2 

II. II 

10.20 

9.29 

8.38 

0 

5o 

49 

48 

47 

47 

lO 

14.39 

i3.48 

12.57 

12.  7 

II. 16 

10.25 

9-34 

8.43 

lo 

42 

4i 

4o 

39 

38 

37 

36 

36 

35 

M 

2 

1 

20 

i4.43 

i3.53 

i3.  2 

12. II 

II. 21 

10. 3o 

9.39 

8.48 

20 

33 

32 

3i 

3i 

3o 

29 

28 

27 

26 

26 

4 

2 

3o 

i4.48 

13.57 

i3.  7 

12.16 

11.25 

10.35 

9-44 

8.54 

3o 

25 

24 

23 

22 

21 

20 

20 

19 

18 

17 

5 
6 

2 

40 

i4.52 

i4.  2 

i3.ii 

12.21 

II  .3o 

10. 4o 

9.49 

8.59 

4o 

16 

i5 

i5 

14 

i3 

12 

II 

10 

9 

9 

3 

33 

bo 

0 

.4.57 

14.  7 

i3.i6 

12.26 

11.35 

10.45 

9-54 

9-  4 

DO 

8 

7 

6 

5 

4 

4 
45 

3 

AA 

2 

43 

1 
4^ 

0 
~A\ 

9 
0 

4 

0 
0 

i5.   2 

l4.I2 

l3.22 

12. 3i 

II  .41 

10.5. 

ID.     I 

9.10 

0 

49 

48 

47 

46 

46 

10 

i5.  7 

14.17 

13.27 

12.36 

11.46 

10. 56 

ID.    6 

9.15 

.0 

41 

4o 

39 

38 

37 

36 

36 

35 

34 

33 

2 
3 
4 

1 

20 

lb. 12 

14.22 

i3.3i 

12.41 

11.5. 

II.   I 

10. II 

9.21 

20 

32 

3i 

3i 

3o 

29 

28 

27 

26 

26 

25 

2 

3o 

lb. 16 

14.26 

i3.36 

12.46 

11.56 

.1.  6 

10.16 

9.26 

3o 

24 

23 

22 

21 

21 

20 

19 

18 

17 

.6 

5 

2 

3 

40 

lb. 21 

14.3, 

i3.4i 

.2.5. 

12.   I 

II  .11 

10.21 

9.3. 

4o 

.6 

i5 

1 4 

.3 

.2 

II 

I . 

10 

9 

8 

3 

bo 

lb. 26 

14.36 

i3.46 

.2.56 

12.  6 

II. 17 

10.27 

9.37 

5o 

7 

6 

6 

5 

4 

3 
45 

2 

Ta 

I 
43 

I 

0 

9 
0 

4 
0 
1 

1 

34 

0 

i5.3i 

14. 4i 

i3.5i 

.3.   I 

12. II 

II  .22 

10.32 

9.42 

0 

49 

48 

47 

47 

46 

4242 

10 

lb. 3b 

14.46 

i3.56 

.3.  6 

12.17 

11.27 

.0.37 

9-48 

.0 

4i 

4o 

3q 

38 

37 

37 

36 

35 

3433 

2 

20 

ib.4o 

i4.5o 

i4.  I 

.3.11 

12.22 

II  .32 

.0.43 

9.53 

20 

32 

02 

3. 

■Jo 

29 

28 

28 

27 

26'25 

3 

4 

2 

3o 

lb. 45 

14.55 

i4.  6 

i3.i6 

12.27 

11.38 

.0.48 

9.59 

3o 

24 

23 

23 

22 

21 

20 

19 

18 

18  .7 

5 
6 

7 

3 

4o 

lb. bo 

i5.  0 

i4.n 

.3.22 

.2.32 

11.43 

.0.54 

10.  4 

4o 

.6 

i5 

i4 

.4 

.3 

12 

1 1 

10 

9   9 

4 

35 

bo 
o 

lb. 55 

i5.  5 

14.16 

.3.27 

12.38 

11.48 

10.59 

.0.10 

5o 

8 

7 

6 

5 

4 

4 

3 

43 

2 

4^ 

i|  0 
4T'4T 

8 
9 
0 
1 
2 

4 
5 
0 

16.  0 

i5.ii 

14.22 

.3.33 

12.44 

11.55 

11.  5 

10.16 

0 

48 

47 

46 

46 

45 

10 

16.  5 

i5.i6 

14.27 

i3.38 

12.49 

.2.    0 

II  .11 

.0.22 

10 

4o 

39 

38 

37 

37 

36 

35 

34 

33  33 

1 

20 

16.10 

l5.2I 

14.32 

i3.43 

12.54 

12.    6 

II. 17 

10.28 

20 

32 

3. 

3o 

29 

28 

28 

27 

26 

2524 

4 

2 

3o 

16. i5 

i5.26 

r4.38 

.3.49 

i3.  0 

.2.  II 

.  1 .22 

10.33 

3o  24 

23 

22 

21 

20 

19 

19 

18 

17!. 6 

5 
6 

7 

4o 

16.20 

i5.32 

14.43 

.3.54 

i3.  5 

12.17 

11.28 

10.39 

4oi5 

i5 

i4 

.3 

12 

II 

II 

10 

98 

4 

DO 

16.25  15.37I 

14.48 

.3.59 

l3.IT 

12.22 

11.33 

10.45 

5o    7 

6 

6 

5 

4 

3 

2 

2 

i|  0 

U 
9 

4 
5 

EXAMPLE    IV. 

Given    Ihe  moon's  apparent  altitude  76°  36',  and  her  horizontal  parallax  56'  18".      Required  the 

correction  and  logarithm  ? 

Fo)-  the  Correction. 

For  the  Logarithm. 

In  Tab.  xix.  to  alt.  76°  30'  and  par.  6G'  is      46'  37" 

In  Tah.xix.  to  nearest  alt.  77°  and  par.  56' 10"    2110 

..Tab.  A.  18"  parallax 10 

..Tab.  C.  8    parallax 2 

..Tab.  B.  6' altitude 6 

Soug-ht  logarithm 2112 

Souffht  correction 46'  53" 

EXAMPLE    V. 

Given  the  moon's  apparent  altitude  16°  25',  and  her  horizontal  parallax  58'  45' .     Required  the  correc- 

tion and  logarithm  1 

For  the  Correction.                            i 

For  the  LogaHthm. 

In  Tab.  xix.  to  alt.  16°  20'  and  par.  58'  is       6'  17" 

Tab.  xix.  to  nearest  alt.l6°  20'  and  par .58'  40" is  2099 

..  Tab.  A.  45"  parallax 14 

..Tab.B.  5'  altitude 0 

Tab.  C.  5"  parallax    6 

Sought  logarithm 

9Ifl'i 

..    fi' 

31" 

TABLE  XIX. 

[P 

.i,c  117 

Logarithms. 

i^ 

Table C. 

Apparent  Altitude  of  5  's  centre. 

Cor.  Sec. 
of  Par. 

«a. 

Add. 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

M. 

~54 

s. 
0 

30 

2  3Go 

30i 

2358 

31 

2356 

3U 

2354 

32 

S2h 

33 

33i 

34 

34i 

35 

35^ 

2338 

Sec. 

Cor. 

2J52 

2349 

2347 

2345 

2344 

2342 

234o 

0 

12 

JO 

2  346 

2344 

2342 

2340 

2JJ8 

2336 

2334 

2332 

233o 

2328 

2326 

2324 

1 

11 

■f.O 

23  J  J 

233o 

2328 

2326 

2J24 

2322 

2320 

23i8 

23i6 

23i4 

23 1 3 

23ll 

2 

9 
8 

3o 

23  10 

23i6 

23i4 

23l2 

2J10 

23o8 

23o6 

23o4 

23o2 

23oo 

2299 

2297 

3 

40 

23o5 

2  3o2 

23oO 

2298 

2296 

2294 

2292 

2290 

2289 

2287 

2285 

2283 

4 

7 

"55" 

bo 
0 

2291 

2289 

2287 

2285 

2283 

2281 

2279 

2277 

2275 

2274 

2272 

2270 

5 
6 

5 
4 

2278 

2275 

2273 

2271 

2269 

2267 

2265 

2  263 

2262 

2260 

2258 

2256 

10 

2264 

2262 

2260 

2258 

2  2  56 

2254 

2252 

225o 

2  248 

2246 

2245 

2243 

7 

2 

20 

225l 

2248 

2246 

2244 

2242 

2240 

2238 

2236 

2234 

2232 

223l 

2229 

8 

I 

Jo 

2237 

22J5 

223J 

223l 

2229 

2227 

2225 

2223 

2221 

2219 

2218 

2216 

9 

0 

"56" 

4o 

5o 

0 

2224 
22r  ! 

2221 
32gS 

2219 

2206 

2217 
2204 

22l5 
2202 

22l3 
2200 

221  I 
2198 

2209 
2196 

2208 
2194 

2206 
2192 

2204 
2I9I 

2202 
2189 

Sec. 

Cor. 

2197 

2194 

2192 

2190 

2188 

2186 

2184 

2182 

2181 

2179 

2177 

2175 

0 

12 

10 

2iS3 

2181 

2179 

2177 

2175 

2173 

2171 

2169 

2168 

2166 

2164 

2162 

I 

II 

20 

2170 

2lb8 

2166 

2164 

2162 

2160 

2168 

2x56 

2i54 

2  I  52 

2l5l 

2149 

2 

9 

8 

Jo 

2i57 

2i55 

2lbJ 

2l5o 

2i48 

2i46 

2145 

2143 

2l4l 

2139 

2i38 

2i36 

3 

4o 

2i44 

2l4l 

2139 

2i37 

2lJ5 

2IJJ 

2l3l 

2129 

2128 

2126 

2124 

2122 

4 

7 

^ 

bo 
0 

2i3o 

2128 

2126 

2124 

2122 

2120 

2118 

2116 

2Il5 

21l3 

2111 

2109 

5 
6 

5 
4 

2117 

2Il5 

211J 

2111 

2109 

2107 

2io5 

2I03 

2102 

2100 

2098 

2096 

10 

2104 

2102 

2100 

2098 

2096 

2094 

2092 

2090 

2089 

2087 

2085 

2083 

7 

3 

20 

2091 

2089 

2087 

2o85 

208J 

2081 

2079 

2077 

2076 

2074 

2072 

2070 

8 

I 

Jo 

2078 

2076 

2074 

2072 

2070 

2068 

2066 

2064 

2o63 

2061 

2059 

2o57 

9 

0 

Ts" 

40 

5o 
0 

2o65  2o63 
2o52  2o5o 

2061 
2048 

2o59 
2o46 

2o57 
2044 

2o55 
2042 

2o53 

2o4o 

2o5l 

2o38 

2025 

2o5o 
2o37 

2048 
2o35 

2o46 
2o33 

2o44 

2o3l 

Sec. 

Cor. 

2u39  2o37 

2o35 

2o33 

203l 

2029 

2027 

2024 

2022 

2020 

2018 

0 

13 

10 

202fj 

2024 

2022 

2020 

2018 

2016 

20l4 

20l3 

20  I  I 

2009 

2008 

2006 

1 

II 

20 

20l3 

2011 

2009 

2007 

2005 

2oo3 

2002 

2000 

1998 

1996 

1995 

1993 

2 

9 

Jo 

2001 

1998 

199b 

1994 

1 99  J 

1991 

1989 

1987 

1985 

1983 

1982 

1980 

3 

8 

4o 

I9SS 

1986 

19S4 

1982 

1980 

1978 

197b 

1974 

1973 

1971 

1969 

1967 

4 

7 

^ 

bo 
0 

•97^ 

•97^ 

1 97 1 

1 909 

1967 

1965 

i9b3 

1 96 1 

1900 

1958 

1957 

1955 

5 
6 

6 
4 

1962 

i960 

1958 

1956 

1954 

1952 

1 95 1 

1949 

1947 

1945 

1944 

1942 

10 

1950 

194s 

1946 

1944 

1942 

1940 

19J8 

1936 

1935 

1933 

1 93 1 

1930 

7 

3 

20 

1937 

ig.ib 

1933 

1931 

1929 

1927 

1925 

1923 

1922 

1920 

1919 

19.7 

8 

2 

Jo 

1925 

1923 

1921 

1919 

1917 

1915 

1913 

I9II 

I9I0 

1908 

1906 

1904 

y 

I 

"60" 

4o 

5o 

0 

1912 
1900 

1887 

1910 

1898 

1908 
1896 

1906 
1894 
1881 

1904 
1892 

1902 
1890 

1900 
1888 

1875 

1898 
1886 
1873 

1897 

i885 

1895 

i883 

1894 
1881 

1892 
1880 

Cor. 

Sec. 

i885 

i883 

1879 

1877 

1872 

1870 

1869 

1867 

0 

12 

10 

187b 

187J 

1871 

i860 

1867 

i865 

1 863 

I86I 

i860 

i858 

1 857 

i855 

I 

11 

20 

1862 

i860 

i858 

1 856 

1 854 

i852 

i85r 

1849 

1847 

1845 

1 844 

1842 

2 

10 

Jo 

i85o 

1 848 

1 846 

i844 

1842 

1840 

1 838 

1 8  36 

i835 

1 833 

i832 

i83o 

3 

8 

4o 

.837 

i835 

iS33 

i83i 

i83o 

1828 

1826 

1824 

1823 

1821 

1820 

i8r8 

4 

7 

1)T 

5o 
0 

1825 

1823 
i8u 

1821 
1809 

1819 
'I'Sof 

1817 

i8i5 

i8i4 

1812 

1 800 

1811 

1809 

1807 

1806 

5 
6 

6 
5 

i8o5 

i8o3 

1802 

1798 

1 796 

1795 

1793 

10 

1801 

1799 

1797 

1795 

1793 

1791 

[789 

1787 

1786 

1784 

1783 

1781 

7 

3 

20 

17H9 

•  7»7 

1783 

1 78  J 

1781 

1779 

1777 

177b 

1774 

1772 

1771 

1769 

8 

2 

1,,  ^^1 

■  77^) 

'774 

1772 

1770 

1769 

.767  1765  1 

1763 
1 . 

1762  1760  j 

1759 

1757 

I 

PaseliS]                                                  TABLE    XIX. 

Correction. 

-J 

_• 

Table  A. 

Tab.B. 

J)  'a  Horizontal  Parallax. 

Proportional  part  for  Seconds 
of  Parallax. 

ForM. 
of  alt. 

Add. 

Add. 

D. 

36 

M. 

0 

54' 

55' 

56' 

57' 
l4~6 

58' 

59' 

60' 

61' 

S. 
0 

0" 

4"7 

1" 

4"6 

2" 

45 

3" 

45 

4" 
44 

5"  6" 
434^ 

7  ■' 

4i 

d" 
4"i 

9" 
40 

M. 

"o" 
1 

S. 

0 

1 

iG.3i 

(5.43 

14.54 

I3.I7 

12.29 

11.40 

10. 5i 

lO 

i6.3(i 

i5.48 

i5.  0 

14. 11 

i3.23 

12.34 

11.46 

10.57 

10 

39 

38 

37 

37 

36 

35 

34 

33 

J3 

32 

1 
2 
2 

20 

16.42 

i5.53 

i5.  5 

14.17 

i3.28 

12.40 

II. 5i 

II.  3 

20 

Jl 

3o 

29 

28 

28 

27 

26 

25 

24 

24 

4 

3o 

16.47 

i5.58 

iS.io 

14.22 

i3.34 

12.45 

11.57 

II.  9 

3o 

23 

22 

21 

20 

20 

19 

18 

17 

16 

16 

5 
6 

3 
3 

40 

16.52 

16.  4 

i5.i6 

14.27 

i3.3q 

12. 5i 

12.  3 

II. i5 

4o 

lb 

i4 

i3 

12 

12 

II 

10 

Q 

c 

8 

7 

4 

3? 

bo 

0 

16.57 

16.  9 

l5.2I 

14.33 

i3.45 

12.67 

12.  9 

II  .21 

5o 

7 

6 

5 

4 

4 

0 
43 

2 

4^ 

I 
47 

0 
47 

>o 
4(1 

9 
0 

5 

17.   2 

16.14 

15.26 

14.38 

i3.5o 

i3.  3 

12. i5 

11.27 

0 

47 

46 

45 

45 

44 

10 

17.   7 

16.20 

i5.32 

14-44 

i3.56 

i3.  8 

12.20 

11.33 

10 

J9 

38 

37 

37 

J6 

35 

34 

34 

JJ 

32 

2 

1 
2 
2 

20 

i-.i3|i6.25 

15.37 

14. 5o 

i4.  2 

i3.i4 

12.26 

11.39 

20 

3i 

3o 

3o 

■iq 

28 

27 

26 

26 

2b 

24 

4 

3o 

17.18 

16. 3o 

15.43 

i4.55 

i4.  8 

l3.20 

12.32 

11.45 

3o 

23 

22 

22 

21 

20 

iq 

18 

18 

17 

16 

5 
6 
7 

4o 

.7.23 

16. 36 

1 5. 48 

i5.   I 

i4.i3 

i3.26 

12.38 

II. 5i 

4o 

i5 

i4 

i4 

i3 

12 

II 

10 

10 

9 

8 

4 

38 

bo 

0 

17.29 

16.41 

i5.54 

i5.  6 

.14.19 
14.26 

i3.32 

12.44 

11.57 
12.  4 

5o 
0 

7 
46 

7 
45 

6 

44 

5 

44 

4 

3 

42 

3 

47 

2 

47 

1 
40 

0 
3^ 

9 
0 

5 
0 

.7.35 

16.48 

16.  0 

i5.i3 

i3.38 

12. 5i 

lO 

17.40 

16.53 

16.  6 

15.19 

i4.32 

i3.44 

12.57 

12.10 

10 

38 

37 

37 

36 

35 

34 

33 

33 

J2 

3i 

2 

1 

20 

17-46 

16.59 

16.12 

i5.25 

14.37 

i3.5o 

i3.  3 

12.16 

20 

30 

3o 

29 

28 

27 

26 

26 

2  5 

24 

23 

■J 

4 

2 

2 

do 

17. bi 

17.  4 

.6.17 

i5.3o 

14.43 

i3.56 

l3.    q 

12.22 

3o 

22 

22 

21 

20 

19 

IQ 

iS 

17 

16 

i5 

5 
6 

7 

a 

3 

4 

4o 

17. b7 

17.10 

16.23 

i5.36 

14.49 

i4.  2 

i3.i5 

12.20 

4o 

i5 

i4 

i3 

12 

12 

1 1 

10 

9 

8 

8 

3^ 

bo 
o 

18.   2 

18.  8 

17.15 
17.21 

16.29 
16.34 

15.42 
1 5. 48 

i4.bb 

i4.  8 

l3.22 
13.28 

12.35 

5o 
0 

7 
46 

6 

45 

5 
44 

4 

4 
43 

3 

42 

2 

4"i 

I 
47 

I 
4^ 

0 
3q 

H 
9 

0 

5 
5 
0 

i5.   I 

14.  i4 

12.41 

10 

18. i3 

17.27 

16.40 

i5.54 

i5.  7 

l4.20 

13.34 

12.47 

10 

38 

37 

37 

36 

35 

34 

34 

33 

J2 

3 1 

2 

1 

20 

18.19 

17.32 

16. 46 

i5.5q 

ib.i3 

14.27 

i3.4o 

12.54 

20 

3i 

3o 

2Q 

28 

27 

27 

26 

2b 

24 

34 

4. 

2 

Jo 

18.24 

17.3s 

16.52 

16.  5 

i5.i9 

14.33 

1 3. 46 

i3.  0 

3o 

23 

22 

21 

20 

20 

IQ 

18 

17 

17 

16 

5 

3 
4 
4 

40 

18. 3o 

17-44 

16.57 

16.11 

i5.25 

i4.3q 

i3.53 

i3.  6 

4o 

i5 

i4 

i4 

i3 

12 

1 1 

10 

10 

9 

8 

7 

4^ 

bo 

0 

18. 36 

17.49 

17-  3 

16.17 

i5.3i 

14.45 

13.59 

i3.i3 

5o 

7 

7 

6 

5 

4 

3 
47 

3 

4^ 

2 

4^ 

1 

0 

B 
9 

0 

5 
5 
0 

18.42 

17.56 

17.10 

16.24 

i5.38 

14.52 

14.  6 

l3.20 

0 

45 

u 

43 

^i 

42 

39  38 

10 

18.48 

18.   2 

17.16 

16. 3o 

i5. 44 

14.59 

14. i3 

13.27 

10 

37 

37 

36 

35 

M 

34 

33 

32 

3i;3i 

20 

18.54 

18.  8 

17.22 

16. 36 

i5.5i 

i5.   5 

14.19 

13.33 

20 

3o 

2q 

28 

■il 

■21 

26 

25 

24 

2423 

y 

2 
2 
3 

Jo 

18. b9 

18. i4 

17.28 

16.42 

i5.57 

i5.ii 

14.25 

i3.4o 

3o 

22 

21 

21 

20 

iQ 

18 

18 

17 

i6|,5 

5 

40 

.9.  b 

18.19 

17.34 

16.48 

16.  3 

I5.I7 

14.32 

i3.46 

4o 

i5 

i4 

i3 

12 

II 

11 

10 

9 

8    8 

7 

4 

4 

4F 

bo 
o 

19.11 

18.25 

17.40 

16.55 

16.  9 

i5.24 

i4.38 

i3.53 

5o 
0 

7 

6 
43 

5 

42 

5 
4^ 

4 
4i 

3 

40 

2 

3^ 

2 

3^ 

I    0 

38  3^ 

8 
9 
0 

5 
6 
0 

19.18 

18.32 

17-47 

17.  2 

16.16 

i5.3i 

14.46 

14.  0 

10 

19.23 

18. 38 

17.  b3 

.7.  8 

16.23 

■  5.37 

.4.52 

14.    7 

ID 

36 

36 

35 

M 

33 

33 

32 

3i 

3o  3o 

2 

1 

20 

19.20 

i«.44 

17.59 

17-14 

16.29 

i5. 44 

14.59 

i4.i4 

20 

29 

28 

27 

27 

26 

25 

24 

24 

23  22 

4 

2 

Jo 

19.35 

18. bo 

18.  b 

17.20 

16.35 

i5.5o 

i5.  5 

14.20 

3o 

21 

21 

20 

IQ 

18 

18 

17 

16 

i5i5 

5 

3 

4o 

19.41 

18. 56 

18.11 

17.26 

16.42 

i5.57 

l5.12 

14.27 

4o 

i4 

i3 

12 

12 

1 1 

ID 

Q 

Q 

8    7 

7 

4 

DO 

19.47 

19.   2 

18.17 

17.33 

16.48 

16.  3 

.5.19 

14.34 

5o 

6 

6 

5 

4 

3 

3 

2 

I 

0   0 

8 
9 

5 
6 

EXAMPLE    VI. 

Given    the    moon's    apparent  altitude   11°  20',  and   horizontal    parallax   GO'  43''.       Required    the 

correction  and  logarithm  1 

Tojind  the  Correction. 

Tojind  the  Logarithm. 

In  Tab.  xix.  to  alt.  11°  20'  and  par.  GO  is    4'  30" 

Tab.  xix.  to  nearest  alt.  1]°20'  and  par.G0'40"   2052 

..Tab.  A.  43"  parallax 16 

Tab.   C.  3"parallax 9 

..Tab.  B.O' altitude 2 

Sought  logarithm 2061 

48" 

EX 

AMPLE    VII. 

Given  the  moon's  apparent  altitude  8°  40',   and   horizontal  parallax  5G'  20' .     Required  the  correc- 

tion and  logarithm  ? 

To  find  the  Correction.                         i                           Tojind  the  Logariih7n. 

InTab.  xix.  to  alt.  8°  40'  and  par.  5G'  is    9'  18" 

Tab.  xix.  to  nearest  alt.  8°  39'  and  par.  5C'  20 '     2518 

..Tab.  A.20"  parallax 38 

Tab.  C.  0"  parallax 13 

..  Tab.  B.  0'  altitude 5 

Sought  logarithm 2531 

Smin-ht  />nrrf.nt 

on 10' 

1" 

TABLE  XIX. 

rpa.,-  l:;.l 

Logarithms. 

5  i 

Tablk  C.  1 

Apparent  Altitude  of  5  's  centre. 

Cor 
of 

Sec. 
Far. 

«2- 

Add.   1 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

IM. 

'5T 

s. 

u 

33 

^37 

3Gi 

2335 

37 

37i 

38 

38i 

39 

39i 

40 

2326 

m 

2324 

41 

2323 

41^ 

232  2 

Sec. 

Cor. 

2334 

2332 

233i 

2329 

2328 

2327 

0 

12 

'  10 

232.3 

2321 

2320 

23i8 

23i7 

23i5 

23 1 4 

23i3 

23 12 

23 10 

23o9 

23o8 

1 

II 

20 

2309 

23o7 

23o6 

23o4 

23o3 

23o2 

23oi 

23oo 

2298 

2297 

2296 

2294 

2 

l3„ 

229(3 

2294 

2293 

2291 

2290 

2288 

2287 

2286 

2285 

2283 

2282 

2281 

3 

4o 

2282 

2280 

2279 

2277 

2276 

2274 

2273 

2272 

2271 

2270 

2268 

2267 

4 

7 

"5T 

5o 
0 

2  2b8 

2266 

2  265 

2264 

2263 

2261 

2260 

2258 

2257 

2256 

2255 

2254 

5 
6 

5 

4 

2255 

2253 

2252 

225o 

2249 

2247 

2240 

2245 

2244 

2242 

2241 

2240 

10 

2241 

2239 

2238 

2236 

2235 

2234 

2233 

2232 

2230 

2229 

222S 

2227 

7 

2 

ao 

2228 

2226 

2225 

2223 

2  22  2 

2220 

2219 

2218 

2217 

22l5 

2214 

22l3 

8 

I 

3o 

2214 

2212 

22II 

2210 

2209 

2207 

2206 

2204 

22o3 

2202 

2201 

2200 

9 

0 

3(3" 

4o 
0 

220I 
2188 

2199 
2186 

2198 
2l85 

2196 
2l83 

2195 
2182 

2193 
2180 

2192 

2179 

219I 

2178 

2190 

2177 

2189 

2175 

2188 
2174 

2186 
2173 

Sec. 

Cur 

2174 

2173 

217I 

2169 

2168 

2167 

2166 

2164 

2i63 

2162 

2161 

2160 

0 

12 

10 

2161 

2159 

2i58 

2i56 

2lb5 

2i53 

2l52 

2l5l 

2l5o 

2149 

2i48 

2l47 

I 

II 

20 

2l48 

2i46 

2i45 

2i43 

2l42 

2l4o 

2139 

2i38 

2i37 

2i35 

2i34 

2i33 

2 

9 
8 

3o 

2i3b 

2i33 

2l32 

2l3o 

2129 

2127 

2126 

2125 

2124 

2122 

2121 

2120 

3 

4o 

2I2T 

2119 

2118 

2117 

2I16 

21  l4 

2Il3 

2II2 

2111 

2109 

2108 

2107 

4 

7 

5? 

bo 
0 

2108 

2106 

2I05 

2I04 

2103 

2:01 

2100 

2099 

2097 
2084 

2096 

2o83 

2095 
2082 

2094 
2081 

5 
6 

5 
4 

2095 

2093 

2092 

2090 

2089 

2088 

2087 

2086 

lO 

2082 

2080 

2079 

2077 

2076 

2075 

2074 

2073 

2071 

2070 

2069 

2068 

7 

3 

20 

2069 

2067 

2066 

2064 

2o63 

2062 

2061 

2060 

2o58 

2o57 

2o56 

2o55 

8 

2 

3o 

20b6 

2o54 

20b3 

2o5l 

2o5o 

2049 

2048 

2o47 

2o46 

2044 

2043 

2042 

9 

0 

Ts" 

4" 
5o 

0 

2043 

2o3o 

204  r 
2028 

2040 

2039 

2o38 

2025 

2o36 

2023 

203b 

2022 
2009 

2o34 
2021 

200S 

2o33 
2020 

2007 

203l 

2o3o 

2029 
2016 
2oo3 

2027  1  2026 

2018  2017 
2006  2oo5 

Sec. 

Cor. 

2017 

2016 

20l5 

20 1 3 

2012 

2010 

0 

12 

10 

2oo5 

2oo3 

2002 

2000 

1999 

1997 

1996 

1995 

1994 

1993 

1992 

1991 

I 

II 

20 

1992 

1990 

1989 

1987 

1986 

1985 

1984 

1982 

1981 

1980 

1979 

1978 

2 

0 

Zk) 

■y/y 

1977 

1976 

1975 

1974 

1973 

1Q7I 

1970 

1969 

1967 

1966 

1965 

3 

8 

i 

4(. 

1966 

1965 

1964 

1962 

I  96  I 

i960 

1958 

1957 

1966 

1955 

1954 

1953 

4 

7 

1 
59 

bo 
0 

104 
I94I 

1952 
1939 

1 95 1 

^1938" 

1949 

1948 
1936 

1947 
1934 

1946 
1933 

1944 
1932 

1943 

1942 

1941 

1940 

5 
6 

6 
4 

1937 

1 93 1 

1929 

1928 

1927 

H) 

1929 

1927 

1926 

1924 

1923 

I92I 

1920 

1919 

1918 

I9I7 

1916 

i9.b 

7 

3 

20 

I9I6 

1914 

I9I3 

I9II 

I9IO 

1909 

1908 

1907 

1906 

1904 

1903 

1902 

8 

2 

3o 

1903 

1902 

1 90 1 

1899 

1898 

1896 

189b 

1894 

1893 

1892 

i8qi 

1890 

9 

I 

60 

4o 
5o 

0 

1891 
1879 

r866 

1889 
1877 

1888 
1876 

1 88b 
1874 
1862 

1 885 
1873 

1884 

I87I 

i8S3 
1870 

1882 
1S69 

i88i 
1868 

1879 
1867 

1878 
1866 

1877 
1 865 

Sec. 

Cor. 

1864 

i863 

1861 

1859 

i858 

1857 

1 856 

i855 

1 854 

i853 

0 

12 

10 

1 854 

i852 

i85i 

1849 

1 848 

1847 

i8-f6 

1845 

1844 

1843 

i84i 

i84o 

I 

II 

ao 

i84i 

i84o 

1839 

i837 

i836 

i834 

1833 

i832 

i83i 

i83o 

1S29 

1828 

2 

10 

3o 

1829 

1827 

1826 

182b 

1824 

1822 

1 82 1 

1820 

1819 

18)8 

1817 

1816 

3 

8 

4o 

1817 

i8i5 

1814 

1812 

1811 

1810 

1809 

1S08 

1807 

i8o5 

i8o4 

i8o3 

4 

7 

So 

i8o5 

i8o3 

1802 

1800 

1 79V 

1798 

1797 

1796 

1795 

1793 

1792 

179' 

5 

6 

61 

0 

1792 

1791 

1790 

1788 

1787 

1785 

1784 

1783 

1782 

,78. 

1780 

1779 

6 

5 

10 

1780 

1778 

1777 

1776 

1775 

1773 

1772 

1771 

1770 

1769 

1768 

1767 

7 

0 

20 

1768 

1766 

1765 

1764 

1763 

1 76 1 

1700 

1759 

1758 

1757 

1756 

1755 

8 

2 

00 

1756 

1754 

1753 

1752 

1751 

1749 

1748  1747  j 

1746  1745  1 

n44 

r743 

'^  1 

I 

Page  120]                                                          TABLE    XIX. 

Correction. 

ii  s 

Table  A. 

Tab.B- 

<  g 

D  's  Horizontal  Parallax. 

Proportional  part  for  Seconds 
of  Paralla.x. 

ForM. 
of  Alt. 

Add. 

Add. 

D 

4^ 

M. 

0 

54' 

55' 

5G' 

57' 

53' 

59' 

60' 

01' 

S. 
0 

0"  1" 

44  4'3 

2" 
43 

3" 
4^ 

4" 

47 

5" 
4) 

6" 
4^ 

7"  8" 
30  38 

9" 
37 

M. 
0 

s. 

IQ.53 

rg.  8 

18.24 

17.39 

16.54 

16.10 

i5.25 

i4.4i 

lO 

19.59 

19.14 

18. 3o 

17.45 

17.    I 

16.16 

i5.32 

14.47 

10 

37  36 

35 

34 

M 

33 

32 

3i 

3i 

3o 

2 

1 

20 

20.  5 

19.20 

i8'.36 

17.52 

17.  7 

16.23 

i5.39 

14.54 

20 

29  28 

28 

27 

2b 

2b 

25 

24 

23J23 

3 

3 

3o 

20.11 

19.26 

18.42 

17.58 

17.14  16.29 

i5.45 

i5.   1 

3o 

22  21 

20 

20 

19 

18 

17 

17 

ibi5 

5 

3 

4 
4 

.jo 

20.17 

19.33 

18.48 

18.  4 

17.20  16. 36 

i5.52 

i5.  8 

40 

i4 

i4 

i3 

12 

11 

1 1 

10 

9 

9    8 

I 

_9 
0 

5o 

2D    2  3 

19.39 

18.55 

18.11 

17.27  16.43 

15.59 

[5.i5 

5o 

7 

6 

6 

5 

4 

3 
39 

3 
39 

2 

38 

1    0 

3^36 

5 

6 
0 

43   0 

20. 3o 

19.46  19.   2 

18.18 

17.34  16. 5o 

16.  6 

i5.23 

0 

43 

42 

42 

4i 

40 

10 

20  36 

19.52119.  8 

18.25 

.7-41 

16.57 

16. i3 

i5.3o 

10 

36 

35 

M 

34 

33 

3^ 

3i 

3i 

3o  29 

0 

1 

20 

20.42 

19.58119.15 

18. 3i 

17-47 

17-  4 

16.20 

,5.37 

20 

28 

28 

27 

26 

26 

25 

94 

23 

23  29 

4 

3 

3o 

20.48 

20.  5 

19.21 

18. 38 

17.54  I7-II 

16.2- 

i5.44 

3o 

21 

21 

20 

19 

18 

18 

17 

lb 

i5 

i5 

6 

7 

3 

4o 

20.54 

20.11 

19.28 

18.44 

18.    I 

17.17 

16.34 

i5.5i 

4o 

14 

i3 

10 

12 

II 

10 

10 

9 

8 

7 

5 

44 

5o 

0 

21.    0 

20.17 

19.34 

18. 5i 

.8.  7 

17.24 

16. 4i 

i5.58 
16.  6 

5o 
0 

7 
42 

6 

4i 

5 
4~i 

5 

4o 

4 
39 

3 
38 

2 

38 

2 

3^ 

I 
36 

0 
35 

B 
9 

0 

5 
6 
0 

21.    8 

20.25 

19.41 

18. 58 

18. i5 

17.32 

16.49 

10 

21. i4 

20. 3l 

19.48 

19.   5 

18.22 

17.39 

16. 56 

16. i3 

10 

35 

34 

33 

33 

32 

3i 

3i 

30J29 

28 

2 

1 

20 

21  .20 

20.37 

19.54 

19. 11 

18.28 

17-46 

17.  3 

l6.20 

20 

28 

27 

2b 

26 

2  5 

24 

93 

2322 

21 

4 

^ 

3o 

21.26 

20.44 

20.   I 

19.18 

18.35 

17.52 

17.10 

16.27 

3o 

21 

20 

19 

18 

18 

17 

lb 

ibi5 

i4 

5 

4 

4o 

21.33 

20. 5o 

20.  7 

19.25 

18.42 

17.59 

17.17 

16.34 

4o 

i3 

i3 

12 

11 

1 1 

10 

9 

8 

8 

7 

7 

5 

45 

5o 

0 

21.39 

20.56 

20.14 

19.32 

18.49I18.  6 

17.24 

16. 4i 

5o 

6 

6 

5 

4 

3 

3 

37 

2 

37 

1 

1 

0 

9 
0 

6 
0 

21.46 

21.  4 

20.21 

19.39 

18.57 

18.14 

17-32 

16.49 

0 

4i 

4o 

4o 

39 

38 

36;35|35 

10 

21.53 

21 .  10 

20.28 

19.46 

19.  3 

18.21 

17.39 

16.57 

10 

M 

33 

33 

32 

3i 

3o^3o 

99  28 

28 

2 

1 

20 

21.59 

21.17 

20.35 

19.52 

19.10 

18.28 

17-46 

17-  4 

20 

27 

26 

2b 

25 

24 

23 

23 

99  21 

21 

4 

3 

3o 

22.    5 

21.23 

20.41 

19.59 

19.17 

18.35 

.7.53 

17.11 

3o 

20 

19 

19 

18 

17 

16 

lb 

i5i4 

'4 

5 

3 

4o 

22.12 

21.30 

20.48 

20.  6 

19.24 

18.42 

18.   0 

17. .8 

4o 

i3 

12 

12 

11 

10 

9 

9 

8 

7 

7 

7 

5 

46 

5o 
o 

22.18 

21.36 

20.55 

20. 1 3 

,9.31 
19.38 

18.49 

18.   7 
18. i5 

17.26 
.7.33 

Do 
0 

6 
4i 

5 

4^ 

5 
40 

4 
3o 

3 

38 

2 

38 

2 

3^ 

1 

0 

0 
35 

8 

9 

0 

6 
0 

22.25 

21.43 

21,   I 

20.20 

18. 56 

36!35 

10 

22. 3l 

2I.5o 

21.  8 

20.27 

19-45 

19.  3 

18.29 

17.40 

10 

34 

33 

33 

32 

3i 

3i 

3o 

9929 

28 

3 
4 

1 

20 

22.38 

21.56 

21.  i5 

20.33 

19.52 

19. II 

18.29 

17-48 

20 

27 

27 

26 

95 

24 

24 

23 

22  92 

21 

3 

3o 

22.44 

22,    3 

21 .22 

20.40 

19.59 

19.18 

18. 36 

17.55 

3o 

20 

20 

19 

18 

18 

17 

lb 

i5i5 

i4 

6 

4 
4 

4o 

22. 5l 

22.10 

21.28 

20.47 

20.  6 

19.25 

18.44 

18.  2 

4o 

iJ 

i3 

12 

II 

II 

10 

9 

9   8 

7 

7 

5 

47 

5o 
o 

22.57 

22.16 

21.35 

20.54 

20. 1 3 

19.32 

18. 5i 

18.10 

5o 

7 
4o 

6 
39 

5 
39 

4 
38 

4 
37 

3 

37 

2 

36 

2    I 
35'35 

0 

34 

9 
0 

_6_ 
0 

23.  5 

22.24 

21.43 

21 .   2 

20.21 

19.40 

18.59 

18.18 

0 

10 

23.11 

22. 3l 

21.50 

21.9 

20.28 

19-47 

19.  7 

18.26 

10 

33 

33 

32 

3i 

3i 

3o|29 

2828 

27 

2 

I 

?f) 

23.18 

22.37 

21 .57I21 .16 

20.35 

19-55 

19.14 

18.33 

20 

26 

26 

25 

24 

24 

23 

99 

22 

21 

20 

4 

3 

So 

23.25 

22.44 

22.  4 

2  1.23 

20.43 

20.  2 

19.21 

18.41 

3o 

20 

19 

18 

18 

17 

16 

lb 

i5 

M 

14 

5 

4 
4 

4o 

23. 3i 

22. 5l 

22.11 

2  1  .3o 

20. 5o 

20.  9 

19.29 

18.48 

4o 

i3 

12 

12 

11 

10 

10 

9 

8 

8 

7 

7 

6 
6 

0 

1 

48 

5o 
o 

23.38 

22.58 

22.18 

21  .37 

20.57 

20.17 

19.36 

i3.56 

5o 
0 

6 

3^ 

5 
38 

5 
38 

4 
37 

3 
36 

3 
36 

2 

35 

I 
3"4 

I 

34 

0 

33 

9 

0 
1 

23.46 

23.  6 

22.25 

21.45 

21.  5 

20.25 

19-45 

19.  5 

10 

23.52 

23.12 

22.32 

21.52 

21.12 

20.32 

19.52 

19.12 

10 

J2 

32 

3i 

3o 

3o 

29 

28 

9« 

27 

2b 

2 

3 
4 

1 
2 
3 

20 

23.59 

23.19 

22.39I22.  0 

21.20 

20.4o|20.    0 

19.20 

20 

2b 

25 

24 

24 

23 

22 

2  9 

2  1 

20 

20 

3o 

24.  6 

23.26 

22.46 

22.     7 

21.27 

20.47 

20.  7 

19.28 

3o 

19 

18 

18 

17 

16 

ibiib 

i4 

14 

i3 

b 
6 

4 
4 

io 

24.13 

23.33 

22.53 

22.14 

21.34 

20.55 

20.  j5 

19.35 

40 

12 

12 

11 

10 

10 

9;  8 

8 

7 

b 

7 

5 

4g 

5o 

0 

24.20 

23.40 

23.     I 

22.2! 
22.29 

21.42 

21.     2 

20.23 

19.43 

5o 
0 

6 

38 

5 
3^ 

4 

37 

4 
36 

3 
35 

2 

35 

2 

34 

1 
33 

0 

0 

8 
9 
0 

7 
0 
1 

24.27 

23.48 

23.  9 

21  .5o 

21 .10  20. 3l 

19.52 

33|32 

10 

24.34 

23.55 

23. 1622. 36 

2  I  .  57 

21.18  20.39 

20.  0 

10 

3i 

3i 

3o 

3099 

28 

28 

27 

2696 

2 

1 

2 
3 

Ut 

24.41 

24.  2 

23.93  22.44 

22.     5 

21  .26  20.46 

20.  7 

20 

25 

24 

24 

23  22 

29 

21 

20 

20 

19 

4 

3o 

24.48 

24.  9 

23. 3o  22. 5 1 

22.12 

21 .33  20.54 

20 . 1 5 

3o 

18 

18 

17 

17  i6 

l5 

i5 

i4 

i3 

i3 

b 

4 

io 

24.55 

24.16 

23.3722.58 

22.19 

21.41  21.     2 

20.23 

4o 

12 

1 1 

1 1 

10   9 

9 

8 

7 

7 

6 

7 

5 

5o 

25.     2 

24.23 

23.4423.  6 

22.27 

21.48  21  .     9 

20. 3 1 

5o 

5 

5 

4 

4   3 

2 

' 

I 

0 

0 

9 

7 

TABLE  XIX. 

[ra. 

e  101 

Logarithms. 

0  i 

TablkcI 

K=5 

Cor. 

'•'"  a 

Apparent  Altitude  of  D  's  centre. 

of 

Pnr 

Ai, 

Add.  1 

0 

0 

0  ,  0 

0 

0 

0 

0 

0 

0 

0 

0 

M. 

s. 

o 

42 

m 

43 

43i 

.  44 

m 

45 

45i 

46 

47 

48 

49 

23T)"&' 

Sec. 

0 

Cor. 
12 

2321 

2320 

23i9 

23i8 

23i7 

23i6 

23i5 

23l4 

23i3 

23ll 

2309 

lO 

23o7 

23o6 

23o5 

23o4 

23o3 

23o2 

23oi 

2  3  00 

2299 

2297 

2296 

2294 

I 

II 

20 

2293 

2292 

2291 

2290 

2289 

2288 

2287 

2286 

2285 

2284 

2282 

2280 

2 

9 

3o 

2280 

2279 

2278 

2277 

2276 

2275 

2274 

2273 

2272 

2270 

2268 

2267 

3 

8 

4() 

2266 

2265 

2264 

2263 

2262 

2261 

2260 

2259 

2258 

2256 

2255 

2253 

4 

7 

55" 

5o 

0 

2253 

225l 

2250 

2249 

2248 

2247 

2246 

2246 

2245 

2243 

224l' 

2240 

5 
6 

5 
4 

2239 

2238 

2237 

2236 

2235 

2234 

2233  1  2232 

223l 

2229 

2228 

2226 

10 

2226 

2224 

2223 

2222 

2221 

2220 

2219 

2219 

2218 

2216 

22l4 

22l3 

7 

3 

2o 

2212 

22II 

2210 

2209 

2208 

2207 

2206 

2205 

2204 

2  203 

2201 

2199 

8 

I 

3o 

2199 

2198 

2197 

2196 

2195 

2194 

2193 

2192 

2  I  91 

2189 

2188 

2186 

y 

0 

4o 

2i85 

2l84 

2l83 

2182 

2181 

2180 

2179 

2 '79 

2178 

2  1  76 

2174 

2173 

"56 

5o 
o 

2172 

2I7I 

2170 

2169 

2168 

2i55 

2.67 
2  1  54 

2166 

2i53 

2i65 

2164 

2i63 

21C1 

2159 

■Sec. 
0 

Cor. 

12 

2159 

2i58 

2i57 

2i56 

2l52 

2l5l 

2149 

2i48 

2146 

10 

2146 

2144 

2143 ' 2142 

2l4l 

2i4o 

2139 

2i38 

2i38 

2i36 

2i34 

2 1 33 

I 

II 

20 

2l32 

2l3l 

2i3o 

2129 

2128 

2127 

2126 

2125 

2125 

2123 

2121 

2120 

2 

9 

3o 

2119 

2II8 

2117 

2116 

2Il5 

2Il4 

2Il3 

2II2 

21  I  I 

2II0 

2io8 

2107 

3 

8 

4o 

2106 

2I05 

2104 

2I03 

2102 

2IOI 

2100 

2099 

2098 

2097 

2095 

2093 

4 

7 

5? 

5o 
o 

2093 

2092 

2091 

2090 

2089 

2088 

2087 

2086 

2o85 
2.072 

2084 

2082 

2080 

5 
6 

5 
4 

2080 

2079 

2078 

2077 

2076 

2075 

2074 

2073 

2071 

2069 

2067 

lO 

2067 

2066 

2o65 

2064 

2o63 

2062 

2061 

2060 

2059 

2o58 

2o56 

2o54 

7 

3 

20 

2o54 

2o53 

2052 

205l 

2o5o 

2049 

2o48 

2o47 

2046 

2o45 

2043 

2042 

8 

2 

So 

204l 

2o4o 

2039 

2o38 

2o37 

2o36 

2o35 

2o34 

2o33 

2032  2o3o 

2029 

9 

0 

4o 

2028 

2027 

2026 

2025 

2024 

2023 

2022 

2021 

2021 

2019 

:.TI7 

2016 

l8 

5o 
o 

20  r  5 

20l4 

20l3 

2012 

2011 

2010 

2009 

2008 
1996 

2008 

2006 

2005 

2003 

Sec. 
0 

Cor. 
12 

2002 

200  I 

2000 

1999 

1998 

1998 

1997 

1995 

1993 

1992 

1990 

ID 

1990 

1989 

1988 

1987 

1986 

1985 

1984 

1983 

1982 

1981 

1979 

1977 

I 

II 

20 

1977 

197b 

1975 

'974 

1973 

1972 

1971 

1970 

1970 

1968 

1966 

1965 

2 

Q 

3o 

1964 

I9b3 

1962 

1 96 1 

i960 

1959 

19^)8 

1957 

'9^7 

1955 

19^4 

1952 

3 

8 

4o 

1952 

1 95 1 

1950 

1949 

1948 

1947 

1946 

1945 

1944 

1943 

1 94 1 

1939 

4 

7 

59 

DO 

o 

1939 

1938 

1937 

1936 

1935 

19^4 

1933 

1932 

1932 

1930 
I917 

1928 

1927 

5 
6 

6 
4 
3 

1926 

1925 

1924 

1923 

1922 

1922 

1 92 1 

1920 

1919 

I9I6 

1914 

10 

1914 

1913 

I9I2 

1911 

1910 

1909 

1908 

1907 

1906 

1905 

1 903 

1902 

7 
8 

20 

1 90 1 

1900 

1899 

1898 

1897 

1897 

1896 

1895 

1894 

1892 

I89I 

1889 

2 

3o 

1889 

1888 

1887 

1886 

i885 

1 884 

1 883 

1882 

18S2 

1880 

1878 

.877 

y 

I 

4o 

1876 

1875 

1874 

1873 

1872 

1872 

1871 

1870 

1869 

1867 

1866 

1864 

r>:> 

5o 
o 

1 864 
1 852 

1 863 

1862 

1861 

i860 

1859 

i858 
1846 

i857 
1845 

i857 

1 855 

i854 

i852 
i84o 

Sec. 
0 

Cor. 
12 

i85i  i85o 

1849 

i848 

1847 

1844 

1843 

i84i 

lO 

1839 

1338  1837 

i836 

1 835 

i835 

1 834 

i833 

i832 

i83o 

1829 

1827 

I 

II 

20 

1S27 

1826  1825 

1824 

1823 

1822 

1821 

1820 

1820 

1818 

,8.7 

i8i5 

2 

10 

3o 

i8i5 

1814 

i8i3 

1812 

1811 

1810 

1809 

1808 

1808 

1806 

i8o4 

i8o3 

3 

8 

4o 

1802 

i8or 

iSoo 

1800 

1799 

1798 

1797 

1796 

'79'i 

1794 

1792 

1791 

4 

7 

"67' 

5o 
o 

1790 

1789 

1788 

1787 

1786 

,78b 

1 785 

1784 

1783 

1782 

1780 

1779 

5 
6 

6 
5 

1778 

1777 

1776 

1775 

1774 

1774 

1773 

1772 

177' 

1769 

1768 

1766 

10 

1766 

1765 

1764 

1763 

1762 

1 76 1 

1760 

1759 

1759 

.757 

1756 

1754 

I 

0 

20 

1754 

1753 

1752 

I75i 

1750 

1749 

1748 

1747 

1747 

1745 

1744 

1742 

2 

3o 

1742 

1 74 1 

1740 

1739 

1738 

1737 

1736 

1735 

1735  1733 

1732 

1730 

y 

I 

16 


.f*« 


Page  122]                                                            TABLE     XIX 

> 

Correction. 

^  a 

Table  A. 

Tab.B. 

J)  's  Horizontal  Parallax. 

Proportional  part  for  Seconds 
of  Parallax. 

For  M. 
of  Alt. 

^"^ 
^ 

Add. 

Add. 

D. 

5o 

M. 

0 

54' 

55' 

56' 

57' 

53' 

59' 

GO' 

61' 

S. 
0 

0" 

1'^ 

3'7 

2" 
3^ 

3" 
36 

4" 
35 

5" 
35 

G" 
34 

7"  8"  9" 
34  33  3^ 

M. 

^0 

s. 

IT" 

25.  9 

24.30 

23. 5i 

23.13 

22.34 

21.56 

21.17 

20.39 

10 

25.  i6 

24.37 

23.59 

23.20 

22.42 

22.  3 

21 .25 

20.47 

10 

32 

3i 

3o 

3o 

29 

28 

28 

27  27  26 

2 

1 

20 

25.23 

24.44 

24.  6 

23.28 

22.49 

22.11 

21.33 

20.54 

20 

25 

25 

24 

23 

23 

22 

21 

2i|2:-  19 

4 

5 

3 

3o 

25. 3o 

24. 5i 

24.13 

23.35 

22.57 

22.19 

21.41 

21 .   2 

3o 

IV 

18 

18 

17 

16 

16 

i5 

i4ii4  i3 

4 

4o 

25.37 

24.  5q 

24 . 2  J 

23.42 

2.3.  4 

22.26 

21.48 

21 .10 

4o 

12 

12 

11 

II 

10 

9 

9 

«'  7 

/ 

6 

7 

5 

57 

5o 

0 

25.44 

25.  6 

24.28 
24.36 

23. 5o 

23.12 
23.21 

22.34 
22.43 

21.56 

21.18 

DO 
0 

6 
37 

5 

36 

5 
36 

4 
35 

4 
35 

M 

2 

33 

2 

33 

I 

0 

8 
9 

b 

7 
0 

1 
2 

25.52 

25.14 

23.58 

22.  5 

21 .27 

32[3i 

10 

25.59 

25.21 

24.43 

24.  6 

23.28 

22. 5i 

22. i3 

21 .35 

10 

3i 

3o 

3o 

29 

28 

28 

27 

26 

26,25 

2 

20 

26.  6 

25.28 

24. 5i 

24.13 

23.36 

22.58 

22.21 

21.43 

20 

25 

24 

23 

23 

22 

21 

21 

20 

20 

ig 

4 

3 

3o 

26.13 

25.36 

24.58 

24.21 

23.43 

23.  6 

22.29 

21 .5i 

3o 

18 

18 

17 

16 

16 

i5 

i5 

i4 

i3 

i3 

5 
6 

7 

4 
5 
5 

4o 

26.20 

25.43 

25.  6 

24.28 

23. 5i 

23.14 

22.37 

21 .59 

4o 

12 

II 

11 

10 

10 

9 

8 

8 

7 

6 

5^ 

5o 
o 

26.27 
^."35 

25. 5o 
25.58 

25.13 

25.21 

24.36 

23.59 

24.  7 

23.22 
23. 3i 

22.45 
22.54 

22.  8 
22. 17 

5o 
0 

6 
36 

5 

35 

5 

35 

4 
34 

3 

34 

3 
33 

2 

¥2 

I 

3l 

1 
37 

0 
37 

~0~ 

6 
7 
0 

24.44 

10 

26.42 

26.  6 

25.29 

24.52 

24. i5 

23.38 

23.    2 

22.25 

10 

3o 

29 

29 

28 

27 

27 

26 

26 

25|24 

2 

2 

2() 

26.50 

26.13 

25.36 

25.    0 

24.23 

23.46 

23.10 

22.33 

20 

24 

23 

23 

22 

21 

21 

20J20 
14  i3 

I9I8 

4 

3 

3o 

26.57 

26.20 

25.44 

25.   7 

24.31 

23.54 

23.18 

22.41 

3o 

18 

17 

16 

16 

i5 

i5 

l3  12 

5 

4 

4o 

27.  4 

26.28 

25. 5i 

25.15 

24.39 

24.   2 

23.26 

22.49 

40 

12 

II 

10 

10 

9 

9 

8 

7 

7 

6 

7 

5 

53 

DO 
O 

27.11 

26.35 

25.59 

25.23 

24.46 
24.55 

24.10 
24.19 

23.34 

22.58 

DO 
0 

6 
35 

5 

34 

4 
34 

4 
33 

33 

2 
3I 

2 
37 

I 

37 

I 

3^ 

0 

3o 

8 
9 
0 

7 

0 
1 

27.20 

26.43 

26.  7 

25. 3i 

23.43 

23.    7 

10 

27.27 

26. 5 1 

26.15 

25.39 

25.  3 

24.27 

23. 5i 

23.15 

10 

29 

28 

28 

27 

27 

26 

25 

25 

24 

24 

2 

2 
2 
3 

20 

27.34 

26.58 

26.22 

25.47 

25.11 

24.35 

23.  5q 

23.23 

20 

23 

22 

22 

21 

21 

20 

20 

19 

18 

18 

4 

3o 

27.41 

27.  6 

26.30 

25.54 

25.19 

24.43 

24.  7 

23.32 

3o 

17 

17 

16 

i5 

i5 

i4 

i4 

i3 

12 

12 

5 
6 
7 

4 
5 

4o 

27.49 

27.13 

26.38 

26.  2 

25.27 

24.51 

24.15 

23. 4o 

4o 

II 

II 

10 

9 

9 

8 

8 

7 

6 

6 

5 

54 

5o 

0 

27.56 

27.21 

26.45 

26.10 

25.34 
25.43 

24.59 
25.  8 

24.24 
24.33 

23.48 
23.58 

5o 
0 

5 

34 

5 
33 

4 
33 

3 

32 

3l 

2 
37 

2 
37 

1 

35 

0 

29 

0 
29 

U 
9 
0 

7 
0 

28.  4 

27.29 

26.54 

26.19 

10 

28.12 

27.87 

27.  2 

26.26 

25. 5i 

25.16 

24.41 

24.  6 

10 

28 

28 

27 

26 

26 

25 

25 

24 

24 

23 

2 
3 
4 

2 
2 
3 

20 

28.19 

27.44 

27.  9 

26.34 

25.59 

25.24 

24.49 

24.14 

20 

22 

22 

21 

21 

20 

19I19 

18 

18 

17 

3o 

28.27 

27.52 

27.17 

26.42 

26.  7 

25.32 

24.58 

24.23 

3o 

17 

16 

i5 

i5 

i4 

i4 

i3 

12 

12 

II 

6 

4 

4o 

28.34 

27.59 

27.25 

26.50 

26.15 

25.41 

25.  6 

24.31 

4o 

II 

10 

10 

9 

8 

8 

7 

7 

*; 

5 

6 

55 

5o 

0 

28.42 

28.  7 

27.32 

26.58 

26.23 
26.32 

25.49 

25.14 

24.40 
24.49 

5o 
0 

5 
33 

4 

32 

4 

32 

3 
3i 

3 
3i 

2 

3^ 

I 
3^ 

I 

29 

0 

0 

9 
0 

6 
7 
0 

28.50 

28.16 

27.41 

27.  7 

25.58 

25.24 

lO 

28.58 

28.23 

27.49 

27.15 

26.4026.  6 

25.32 

24.58 

10 

27 

27 

26 

26 

25 

24 

24 

23 

23 

22 

2 

2 

20 

29.  5 

28.31 

27.57 

27.23 

26.49  26. 1 4 

25.40 

25.  6 

20 

22 

21 

21 

20 

19 

1918I18 

17 

17 

4 

3 

3-0 

29.  i3 

28.39 

28.  5 

27.31 

26.67  26.23 

25.49 

25.15 

3o 

16 

i5 

i5 

i4 

i4 

i3i3 

12 

11 

II 

5 
6 

7 

4 

4o 

29.20 

28.46 

28.12 

27.39 

27.  526.31 

25.57 

25.23 

4o 

10 

to 

9 

9 

8 

7 

7 

6 

b 

5 

6 

56 

5o 

0 

29.28 
29.35 

28.54 

28.20 

27.47 

27.13  26.39 

26.   5 

25.32 

DO 

5 

4 

3 

3 

2 

2 

33 

I 

3^ 

I 

29 

0 
29 

0 

9 
0 

7 
0 

29.  2 

28.28 

27.55 

27.21  26.47 

26.14 

25.40 

0 

33 

32 

32 

3i 

3i 

10 

29.43 

29.  9 

28.36 

28.  3 

27.29  26.56 

26.22  25.49 

10 

27 

27 

26 

26 

25 

25 

24 

24 

23 

22 

2 
3 
4 

2 
2 

3 

20 

29.50 

29.17 

28.44 

28.11 

27.3727.  4 

26. 3i  25.58 

20 

22 

21 

21 

20 

20 

19 

19 

18 

18 

17 

3o 

29.58 

29.25 

28.52 

28.19 

27.4627.12 

26.3926.  6 

3o 

16 

16 

i5 

i5 

i4 

i4 

i3 

i3 

12 

II 

b 
6 

4 

4o 

3o.  6 

29.33 

29.  0 

28.27 

27.54 

27.21 

26.4826.15 

4o 

II 

10 

£0 

9 

9 

8 

8 

7 

b 

b 

7 

6 

57 

5o 
o 

3o.i3 

29.40 

29.  8 

28.35 
28.44 

28.  2 

27.29 

26.56  26.23 
27.  626.33 

5o 
0 

5 
3^ 

5 
3i 

4 
3i 

4 
3o 

3^ 

3 
29 

2 

29 

I 

1 

21 

0 

27 

9 
0 

7 
0 

3o.22 

29.49 

29.17 

28.11 

27.39 

10 

3o.3o 

29.57 

29.25 

28.52 

28.19 

27.47 

27.14 

26.42 

10 

27 

26 

26 

25 

24 

24 

23 

23 

22 

22 

2 

2 

20 

30.37 

3o.   5 

29.33 

29.  0 

28.28 

27.55 

27.23 

26.51 

20 

21 

21 

20 

20 

19 

19 

18 

17 

17 

16 

4 

3 

3o 

3o.45 

3o.i3 

29.41 

29.  8 

28.36 

28.  4 

27.32 

26.59 

3o 

16 

i5 

i5 

i4 

14 

i3 

i3 

12 

12 

11 

b 
6 

4 

4o 

30.53 

3o.2I 

29.49 

29.16 

28.44 

28.12 

27.40 

27.  8 

4o 

10 

10 

9 

9 

8 

8 

7 

7 

6 

6 

7 

6 

58 

5o 

0 

3i.   I 
3i.  9 

30.29 

29.57 

29.25 

28.53 

28.21 

27.49 

27.17 

5o 

5 

5 

4 

3 

3 

2 

2 

Is 

I 

27 

I 
27 

0 

8 
9 
0 

0 

30.37 

3o.  6 

29.34 

29.  2 

28.30 

27.58 

27.27 

0 

3i 

3o 

3o 

29 

29 

10 

3i  .17 

30.45 

3o.i4 

29.42 

29.10 

28.39 

28.  7 

27.35 

10 

26 

25 

25 

24 

24 

23 

23 

22 

22 

21 

2 

3 

4 

2 
2 
3 

20 

3i.25 

30.53 

30.22 

29.50 

29.19 

28.47 

28.16 

27.44 

20 

21 

20 

19 

19 

18 

18 

17 

17 

lb 

lb 

3o 

3i.33 

3i.   I 

3o.3o 

29.59 

29.27 

28.56 

28.24 

27.53 

3o 

i5 

i5 

i4 

i4 

i3 

10 

12 

12 

11 

II 

b 

4 

4o 

3 1.40 

3i.  930.38 

3o.  7 

29.36 

29.  4 

28.33 

28.  2 

4o 

10 

10 

9 

8 

8 

7 

7 

6 

6 

5 

7 

6 

59 

5o 
o 

3 1. 48 
31.57 

3i  .1730.46 
31.26  30.55 

3o.i5 
3o.24 

29.44  29.13 

28.42 

28.11 

5o 
0 

5 
3o 

_4 
29 

29 

3 

28 

3 
28 

2 

27 

2 

27 

I 

1 

0 

9 
0 

7 
0 

29.53 

29.23 

28.52 

28.21 

10 

32.  5 

3i.34  3i.  3 

30.33 

3o.  2 

29.31 

29.  0 

28.30 

10 

25 

24 

24 

23 

23 

22 

22 

21 

21 

20 

2 

2 

20 

32. i3 

3i. 4231.12 

3o.4i 

3o.  10 

29.40 

29.  9 

28. 3q 

20 

20 

19 

19 

18 

18 

17 

17 

i6 

lb 

i5 

4 

3 

3o 

32.21 

3i.5o|3i  .20 

30.49 

30.19 

29.48  29.18 

28.48 

3o 

i5 

i4 

14 

i3 

i3 

12 

12 

II 

II 

10 

5 

4o 

32.29 

31.58 

31.28 

30.58 

3o .  27 

29.57  29.27 

28.56 

4o 

10 

9 

9 

8 

8 

7 

7 

6 

b 

5 

7 

6 

5o 

32.37 

32.  6 

3i.36 

3r.  6 

30.36 

3o.  629.36 

29.   5 

5o 

5 

4 

4   3 

3 

2 

I 

1 

0 

0 

8 
9 

8 

•■" ' 

TABLE  XIX.                        [Page  123 

Logarithms. 

Apparent  Altitude  of  ])  's  centre. 

Table  C. 
Cor.  for  Seconds 
of  Parallax- 
Add. 

M. 

"5T 

s. 

o 

lO 
20 

3o 
4o 
5o 

o 

lO 
20 

3o 
4o 
5o 

0 
10 
20 

3o 

4o 
5o 

0 
lO 
20 

3o 
4o 
5o 

0 
lO 
20 

3o 
4o 
5o 

o 

10 
20 

3o 
4o 
5o 

0 
lO 

20 

3o 
4o 
5o 

o 

lO 
20 

3o 

0 

50 

0 
51 

0 
52 

0 
53 

0 
54 

0 
55 

0 
56 

0 
57 

0 
58 

0 
59 

Sec. 

Cor. 

23u6 
2293 
2279 
2265 

2252 
2238 

23o5 
2291 
2277 
2264 

225o 
2237 

23o3 
2290 
2276 
22G2 
2249 

2235 

2302 
2288 
2275 
2261 
2248 

2234 

23oi 

2287 
2274 

2260 
2246 

2233 

23uO 

2286 

2272 
2259 
2245 

2232 

2298 

2285 

2271 

2258 

2244 

2230 

2297 
2284 
2270 
2257 

2243 
2230 

2296 

2283 

2269 

2256 

2242 
2229 

22o5 
2202 
2268 
2254 
2241 
2227 

0 
I 
2 
3 

i 

6 

7 
8 

9 

12 
II 

9 

8 

7 
5 
4 
3 
I 
0 

2225 
22II 
2198 
2184 
217I 

2i58 

2223 
2210 
2196 
2l83 
2x70 

2i56 

2222 
2208 
2195 
2182 
2168 

2i55 

2221 
2207 
2194 
2180 
2167 
2i54 

2219 

2206 
2193 
2179 

2I()6 

2i53 

2218 
2205 
219I 

2178 

2i65 

2l52 

2217 
2204 
2190 

2177 

2164 

2l5o 

2216 
2203 
2189 
2176 

2i63 
2149 

22l5 

2202 

2188 
2175 

2162 
2i48 

2214 
2201 
2187 
2174 
2l6l 
2l47 

Sec. 

Cor. 

2i45 

2l3l 

2II8 

2io5 
2092 

2079 

2i43 

2l3o 

2II7 
2104 
2091 

2078 

2142 
2129 
2II6 

2102 

2089 
2076 

2l4l 

2127 

2Il4 
2IOI 

2088 
2075 

2i39 
2126 

2Il3 

2100 
2087 
2074 

2i38 

2125 
2112 
2099 
2086 
2073 

2i37 
2124 

2III 

2098 
2o85 
2072 

2i36 

2123 
2II0 
2097 

2084 
2071 

2i35 
2122 
2109 
2096 
2o83 
2070 

2i34 
.2121 
2108 
2095 
2082 
2069 

0 
I 
2 
3 
4 
5 
6 

7 
8 

9 

12 
II 

9 

8 

7 
5 
4 
3 
2 
0 

20G6 
2o53 
2o4o 
2027 

20l4 
2002 

2o65 

2o52 

2o39 
2026 

2ol3 
2000 

2o63 
2o5o 
2o37 

2025 
2012 

1999 

2062 
2049 

2o36 

2023 

2010 

1998 

2061 
2048 
2o35 
2022 
2009 
1997 

2060 
2047 

2o34 

2021 
2008 
1996 

2059 
2o46 
2o33 

2020 

2007 

1994 

2o58 
2045 

2o32 
2019 
2006 
1993 

2057 
2044 

203l 

2018 

2005 

1992 

2o56 
2043 

2o3o 

2017 

2004 

1991 

Sec. 

Cor. 

1989 
1976 
1963 
1 95  I 
1938 
1926 

1987 
1975 

1962 

1949 
1937 
1924 

1986 
1973 

I96I 

1948 
1936 

1923 

1985 

1972 

i960 

1947 
1934 

1922 

1984 
1971 
1958 
1946 
1933 
1921 

1983 
1970 
1957 
1945 
1932 
1920 

1982 

1969 

195(3 
1944 

1931 

I9I8 

1981 
1968 
1955 
1943 
1930 
1917 

1980 

1967 
1954 
1942 

1929 
1917 

1979 

1966 

1953 
1941 
1928 

1916 

0 
1 
2 
3 
4 
5 
6 

7 
8 

9 

12 
II 

7 
6 
4 
3 
2 

I913 

I  90 1 
1888 
1876 

i863 
i85i 

7838" 
1826 
i8i4 
1802 
1789 
1777 

I9I2 

1899 
18S7 
1874 
1862 
1849 

I9IO 

1898 

i885 

1873 
1861 
1848 

1909 
1897 

1 884 
1872 
1859 
1847 

1908 
1896 
J  883 
1871 
i858 
1846 

1907 
1895 
1882 
1870 
1857 
1845 

1833 
1820 
1808 
1796 
1784 
1771 

1906 

1893 
1881 
1869 

i855 
i844 
i832 
1819 
1807 
1795 
1783 
1770 

1905 
1892 

i8«o 
1868 
i855 
1843 

1904 
1892 

1879 
1867 

i854 
1842 

1903 

I89I 

1878 
1866 

i853 
i84i 

Sec. 

Cor. 

1837 
1825 

i8i3 

1800 
1788 
1776 

i836 
1824 
1811 
1799 

1787 
1775 

i835 
1822 
1810 
1798 
1786 
1774 

1834 
1821 
1809 

1797 
1785 
1773 

i83i 
1818 
1806 

1794 
1782 
1770 

i83o 
1817 
i8o5 
1793 
1781 
1769 

1829 
1816 
1804 
1792 
1780 
1768 

0 
I 
2 

3 
4 
5 
6 

7 
8 

9 

12 
II 

10 
8 

7 
6 
5 
3 
2 
I 

1765 
1753 
1741 
1729 

1764 

1752 
1740 
1728 

1763 
i75i 
1739 
1727 

1761 

1749 
1737 
1725 

1760 
1748 
1736 
1724 

1759 
1747 
1735 
1723 

1758 
1746 
1734 
1722 

1757 
1745 
1733 
1 72 1 

1757 
1744 
1732 
1720 

1756 
1743 
1731 
1719 

Pageiaj]                                           TABLE   XIX. 

Correction. 

iS  a 

Table  A. 

Tab.B. 

<  S 

])  's  Horizontal  Parallax. 

P''oportional  part  for  Seconds 
of  Parallax. 

For  xM. 
of  Alt. 

i!^ 

Add. 

Add. 

D 

60 

M. 

0 

54' 

55' 

56' 
3i.45 

57' 
3i.i5 

53' 
3o.45 

59'  I 

60'  1   61'  1 

S.  0" 
0  29 

[II  2" 
2928 

3"  4"! 

27 

G" 
l6 

7" 
i6 

8" 

25 

9" 

3'5 

1 

s. 

0 

1 

32.45 

32. i5 

3o.  1 5  29.45  29.15 

^ 

27 

10 

32.53 

32. 24  3 1. 54 

3i.24 

3o.54 

3o. 24  29.54  29.24 

1024 

2423 

23 

22 

22 

21 

21 

20 

30 

2 

3 

20 1 

33.   I 

32.3232.    2 

31.32 

3i.  3 

3o.33  3o.  3 

29.34 

2019 

1918 

18 

17 

17 

16 

16 

i5 

i5    I 

3 

3o 

33.  9 

32.40  32.10 

3i.4i 

3i .  II 

30.42 

30.12 

29.43 

3o 

i4 

i4 

i3 

i3 

12 

12 

II 

II 

10 

It)     5 

4 
5 

4o 

33.1? 

32.48  32.19 

31.49 

3l.20 

3o.5o 

3o.2I 

39.52 

4o 

9 

9 

8 

8 

7 

7 

6 

6 

5 

5     ? 

6 

57 

5o 
0 

33.25 

33.34 

32.5632.27 

3i.58 

32.    7 

31.28 

30.59 

3o.3o 

3o.    I 

5o 
0 

4 
28 

4 
28 

3 

27 

3 

27 

2 

2 

I 

I 

0 

0  e 
24  i 

7 
8 

0 
1 

33.  5 

32.36 

3 1. 38 

3i.  9 

3o.4o 

3o.ii 

10 

33. 4i 

33.14 

32.45 

32.16 

3i.47 

3i  .18  3o.49 

3o .  20 

10 

23 

23 

22 

22 

21 

21 

20 

20 

19 

'9    S 

2 
3 
3 

20 

33. 5i 

33.22 

32.53 

32.24 

3i.55 

3i  .27 

30.58 

3o.39 

20 

18 

18 

17 

17 

'7 

16 

16 

i5 

i5 

14    4 

'3o 

33. 5o 

33. 3o 

33.   1 

32.33 

32.  4 

31.35 

3i.  7 

3o.38 

3o 

14 

i3 

i3 

12 

12 

11 

II 

10 

10 

9 

5 

4 
5 

;4o 

34.  7 

33.38 

33.10 

32. 4i 

32. i3 

31.44 

3i.i6 

30.47 

4o 

9 

8 

8 

7 

7 

6 

6 

5 

5 

5 

7 

6 

5o 

34. i5 

33.46 

33.18 

32. 5o 

32.21 

31.53 

3 1. 25 

3o.56 

5o 

4 

_4 

3 

3 

i 

2 

I 

I 

0 

0 

8 
9 

7 
8 

6l 

0 

34.24 

33.56 

33.27 

32.59 

32. 3 1 

32.  3 

31.35 

3i.  7 

0 

27 

27 

26 

26 

^ 

^ 

24 

T/i 

^ 

^ 

0 
1 

0 
1 

10 

34.32 

34.  4 

33.36 

33.  8 

32.40 

32.12 

31.44 

3i.i6 

10 

22  22 

21 

21 

21 

20 

20 

19 

'9 

18 

3 

4 

3 
3 

20 

34.40 

34.12 

33.44 

33.-I7 

32.49 

32.21 

31.53 

3i.25 

20 

18 

17 

17 

16 

16 

i5 

i5 

i5 

i4 

1 4 

3o 

34.48 

34.21 

33.53 

33.25 

32.57 

32. 3o 

32.    2 

3 1 .  34 

3o 

i3 

i3 

12 

12 

II 

II 

10 

10 

9 

9 

5 
6 

4 
5 

4o 

34.56 

34.29 

34.   I 

33.34 

33.  6 

32.39 

32.11 

31.44 

4o 

8 

8 

8 

7 

7 

6 

6 

5 

5 

4 

7 

6 

63 

5o 
0 

35.  5 
35.14 

34.37 
34.40 

34.10 
34.19 

33.42 
33.52 

33.15 
33.25 

32.48 
33.58 

32.20 

31.53 

5o 
0 

4 
26 

3 
26 

3 

25 

2 

2"5 

2 

2 

I 

I 
l3 

0 
22 

0    i_ 

32       0 

7 
8 

0 
1 

32. 3o 

32.  3 

10 

35.22 

34.55 

34.28 

34.    I 

33.34 

33.  7 

32.39 

32.12 

10 

22 

21 

21 

20 

20 

19  19 

18 

18 

17      2 

2 
3 

20 

35. 3o 

35.  3 

34.36 

54.  9 

33.42 

33.16 

32.49 

32.22 

20 

17 

17 

16 

i6 

i5 

i5i4 

i4 

i3 

i3 

4 

4 

3o 

35.38 

35.12 

34.45 

34.18 

33. 5i 

33.25 

32.58 

32. 3l 

3o 

i3 

12 

12 

II 

II 

10  10 

0 

0 

9 

5 

Q 

4 

40 

35.47 

35.20 

34.53 

34.27 

34.  0 

33.34 

33.  7 

32.40 

4o 

8 

8 

7 

7 

6 

6   5 

5 

5 

4 

7 

6 

5o 

35.55 

35.28 

35.   2 

34.36 

34.  9 

33.43 

33.16 

32. 5o 

5o 

4 

3 

3 

2 

2 

I 

I 

0 

0 

0 

8 

7 

8 

64 

0 

36.  4 

35.38 

35.12 

34.45 

34.19 

33.53 

33 .  26 

33.  0 

0 

25 

^ 

^ 

^ 

^ 

^ 

22 

22 

32 

21 

'  0 
1 

0 
1 

10 

36.12 

35.46 

35.20 

34.54 

34. 28 

34.   2 

33.36 

33.  9 

10 

21 

20 

20 

19 

^9 

19 

18 

18 

17 

17 

^ 

3 

4 

20 

36.21 

35.55 

35.29 

35.  3 

34.37 

34.11 

33.45 

33.19 

20 

16 

16 

16 

i5 

i5 

14 

i4 

i3 

i3 

13 

4 

3o 

36.20 

36.  3 

35.37 

35.12 

34.46 

34.20 

33.54 

33.28 

3o 

12 

12 

II 

II 

TO 

10 

9 

9 

9 

8 

5 
6 

4 
5 

4o 

36.37 

36.12 

35.46 

35.20 

34.55 

34.29 

34.  3 

33.38 

40 

8 

7 

7 

6 

6 

6 

5 

5 

4 

4 

7 
8 
9 

0 
1 

6 

65 

5o 
0 

36.46 
36.55 

36.20 
36. 3o 

35.55 
36.  4 

35.29 

35.39 

35.  4 
35.14 

34.38 

34.13 
34.23 

33.47 
3"3.57 

5o 
0 

3 
24 

3 

24 

3 

2 

l3 

2 

22 

I 

22 

I 
22 

0 
21 

0 
21 

0 
20 

7 
8 

0 
1 

34.48 

10 

37.  3 

36.38 

36.13 

35.48 

35.23 

34.57 

34.32 

34.   7 

10 

20 

19 
i5 

19 

19 

18 

18 

17 

17 

17 

16 

2 
3 

4 

3 

4 

20 

37.12 

35.47 

36.22 

35.57 

35.32 

35.  6 

34.41 

34.16 

20 

16 

i5 

14 

i4 

i4 

i3 

i3 

12 

13 

3o 

37.20 

36.55 

36. 3o 

36.   5 

35.41 

35.16 

34. 5i 

34.26 

3o 

12 

II 

II 

10 

10 

9 

9 

9 

8 

8 

5 

4 
5 

40 

37.28 

37.  4 

36.39 

36.14 

35.5ol35.25 

35.  0 

34.35 

4o 

7 

7 

7 

6 

6 

5 

5 

4 

4 

4 

7 

6 

66 

5o 
0 

37.37 
37.45 

37.12 
37.2, 

36.48 
36.5(i 

36.23 
36.32 

35.5935.34 
36.  8135.43 

35.  9 

34.45 

5o 
0 

3 
T4 

3 

2 
i3 

2 
i3 

2 
22 

I 
22 

I 
22 

0 
21 

0 
21 

0 
30 

8 
9 

0 
1 

7 
8 

0 
1 

35.19 

34.54 

10 

37.54 

37.29 

37.   5 

36.41 

36.1735.52 

35. 28 

35.  4 

10 

20 

20 

19 

19 

18 

18 

i8 

17 

n 

16 

2 

2 
3 

4 

20 

38.   2 

37.3s 

37.14 

36. 5o 

36.2636.   2 

35.38 

35.13 

20 

16 

16 

i5 

i5 

14 

i4 

i4 

i3 

i3 

13 

4 

3o 

33.11 

37.47 

37.23 

36.59 

36.35'36.ii 

35.47 

35.23 

3o 

12 

12 

II 

II 

10 

10 

10 

9 

9 

8 

5 
6 

4 
5 

4o 

38.10 

37.55 

37.31 

37.  8 

36.44 

36. 20 

35.56 

35.33 

4o 

8 

8 

7 

7 

6 

6 

6 

5 

5 

4 

7 

6 

5o 

38.27 

38.  4 

37.40 

37.17 

36.53 

36.29 

36.  6 

35.43 

5o 

4 

4 

3 

3 

3 

2 

2 

I 

I 

0 

8 
9 

7 
8 

67 

0 

38.37 

38.13 

37 .  5o 

37.27 

37.  3  36.40 

36. 16 

35.53 

0 

23 

23 

22 

22 

21 

21 

21 

20 

30 

30 

0 
1 

0 
1 

10 

38.45 

38.22 

37.59 

37.36 

37.  i2|36.49 

36.26 

36.  2 

10 

19 

19 

18 

18 

18 

17 

17 

16 

16 

16 

2 
3 

4 

2 

20 

38.54 

38. 3i 

38.   8 

37.45 

37.31  36.58 

36.35 

36.12 

20 

i5 

i5 

i5 

1 4 

i4 

i3 

i3 

i3 

12 

12 

4 

3o 

3.9.   2 

38.39 

38. T7 

37.54 

37.3i!37.  8 

36.45 

36.22 

3o 

II 

II 

II 

10 

10 

10 

9 

9 

8 

8 

5 
6 

5 

4o 

39.  u 

38.48 

38.2  5 

38.  3 

37.40 

37.17 

36.54 

36. 3i 

4o 

'8 

7 

7 

6 

6 

6 

5 

5 

5 

) 

7 

6 

68 

5o 
0 

39.19 

38.57 

38.34 

38.12 

38.22 

37.49 

37.26 

37.  4 

36. 4r 

5o 

_4 

3 

3 

3 

2 

2 
20 

2 
20 

I 

19 

I 
19 

19 

8 
9 

0 
1 

7 
8 

0 
1 

39.29 

39.  7 

38.44 

37.59 

37.37 

37.14 

36.52 

0 

22 

22 

21 

21 

21 

10 

39.38 

39.15 

38.53 

38. 3 1 

38.   8 

37.46 

37.24 

37.    I 

10 

18 

18 

18 

17 

17 

16 

16 

16 

i5 

i5 

2 

20 

39.46 

39.24 

39.   2 

38. 4o 

38. 18 

3^.55 

37.33 

37.11 

20 

i5 

i4 

■  4 

i4 

i3 

i3 

12 

12 

12 

11 

4 

3 

4 

3o 

39.55 

39.33 

39.11 

38.49 

38.37 

38.   5 

37.43 

37.21 

3o 

1 1 

II 

10 

10 

9 

9 

9 

8 

8 

8 

5 
G 

5 
5 

4o 

4o.  3 

39.41 

39 .  20 

38.58 

38.36 

38.14 

37.52 

37.30 

4o 

7 

7 

7 

6 

6 

5 

5 

5 

4 

4 

7 

6 

69 

5o 
0 

40.12 

39.50 

39.29 

39.   7 

38.45 
38.55 

38.3.4 

38.  2 

37.40 

5o 

4 

3 

3 

2 

2 

2 
19 

I 

I 
'9 

• 
18 

0 
78 

8 
9 

0 
I 

7 
8 
0 

1 
2 

4o.3I 

40.  0 

39.3s 

39.17 

38.34 

38.12 

37.5. 

0 

21 

2 1 

20 

20 

30 

10 

4o.3o 

4o.  9 

39.47 

39.26 

39.   5 

38.43 

38.2  2 

38.    1 

10 

17 

17 

17 

16 

16 

16 

ID 

i5 

i5 

i4 

20 

40.39 

4o.i8 

39.56 

39.35 

39.1^ 

38.53 

38.32 

38.10 

20 

i4 

i4 

i3 

i3 

i3 

12 

12 

12 

II 

1 1 

3 

4 

3 

4 

3o 

40.47 

40.26 

4o.  5 

39.44 

39.33 

39.   2 

38. 4 1 

38. 20 

3o 

to 

10 

10 

g 

9 

9 

8 

8 

8 

7 

5 

5 

4o 

40.56 

40.35 

4o.i4 

39.53 

39.33 

39. 12 

38. 5 1 

38. 3o 

4o 

7 

7 

6 

6 

6 

5 

5 

4 

4 

4 

7 

6 

5o 

4i.  5 

4o.4'i 

4n.23 

4o.  3 

39.43 

39. 21 

39.    , 

38. 40 

5o 

3 

3 

3 

2 

2 

2    I 

I 

I 

0 

8 
9 

7 
8 

TABLE  XIX. 

[Page  125 

Logarithms. 

5  2 

Tablk  C.     1 

Cor.  for 

Seconds 

Apparent  Altitude  of  ]>  's  centre. 

of  Pa 

rallax. 

'='Sh 

Add.       1 

0 

0 

0 

0 

0 

0 

0 

0 

c 

c 

^5T 

s. 

0 

GO 

Gl 

62 

G3 

64 

G5 

66 

67 

68 

69 

Sec. 

Cor. 

2295 

2294 

2293 

2292 

2291 

2291 

2290 

2289 

2289 

2288 

0 

12 

10 

22«I 

2280 

2279 

2278 

2278 

2277 

2276 

2276 

2273 

2274 

I 

II 

20 

2267 

2266 

2266 

2265 

2264 

2263 

2263 

2262 

2261 

2261 

2 

9 

«0 

2254 

2253 

22D2 

225l 

225o 

225o 

2249 

224s 

2248 

2247 

3 

8 

4o 

2240 

2239 

2238 

2238 

2237 

2236 

2236 

2235 

2234 

2234 

4 

7 

55" 

5o 
o 

2227 

222t) 

2225 

2224 

2223 

2223 

2222 

2221 

2221 

2S20 

5 
6 

5 
4 

22l3 

2212 

2212 

2211 

2210 

2209 

2209 

2208 

22CJ7 

2207 

10 

2200 

2199 

2198 

2197 

2197 

2196 

2195 

2195 

2194 

2194 
2180 

7 

3 

20 

2186 

2186 

2185 

2184 

2i83 

2i83 

2182 

2181 

2181 

8 

I 

Jo 

2173 

2172 

217I 

217I 

2170 

2169 

2169 

2168 

2167 

2167 

9 

0 

"56 

4o 

5o 

o 

2160 
2l47 

2159 
2  1 46 

2i58 
2145 

2i57 
2144 

2  I  57 
2143 

2i56 
2i43 

2l55 
2142 

2i55 

2l4l 

2  I  54 
2l4l 

2i54 

2l4o 

Sec. 

Cor. 

2i33 

2l33 

2l32 

2l3l 

2l3o 

2i3o 

2129 

2128 

2128 

2127 

0 

12 

10 

2120 

2119 

2119 

2118 

2117 

2116 

2I16 

2Il5 

21l4 

21l4 

I 

II 

20 

2107 

2106 

2io5 

2105 

2I04 

2io3 

2X03 

2102 

2101 

2101 

2 

9 

8 

3o 

2094 

2093 

2092 

2092 

20'91 

2090 

2090 

2089 

2088 

2088 

3 

4o 

20«1 

2080 

2079 

2078 

2078 

2077 

2077 

2076 

2075 

2075 

4 

7 

^ 

5o 
o 

2068 

2067 

2066 

2o65 

2o65 

2064 

2064 

2o63 

2062 

2062 

5 
6 

5 

4 

2o55 

2o54 

2o53 

2o53 

2052 

205l 

205l 

2o5o 

2049 

2049 

lO 

2042 

2o4l 

2040 

2o4o 

2039 

2o38 

2o38 

2o37 

2o36 

2o36 

7 

3 

30 

2029 

2028 

2028 

2027 

2026 

2025 

2025 

2024 

2024 

2023 

8 

2 

3o 

2016 

2016 

20 1  5 

20l4 

20l3 

20l3 

2012 

20  I  I 

2011 

2010 

9 

0 

T8 

4o 
5o 

0 

2004 
1991 

2oo3 

1990 

2002 
1989 

2001 
1988 

2000 

1988 

2000 

1987 

1999 
1986 

1999 

1986 

1998 
1985 

1998 
1985 

Sec. 

Cor. 

1978 

1977 

1976 

1976 

1975 

1974 

1974 

1973 

1972 

1972 

0 

12 

lO 

1965 

1965 

1964 

1963 

1962 

1962 

1961 

i960 

i960 

1959 

I 

II 

2o 

1953 

1952 

1 95 1 

1950 

1950 

1949 

1948 

1948 

1947 

1947 

2 

9 

3o 

1940 

1939 

1938 

1938 

1937 

1936 

1936 

1935 

1934 

1934 

3 

8 

4o 

1927 

1927 

1926 

1925 

1924 

1924 

1923 

1923 

1922 

1921 

4 

7 

^ 

bo 
o 

1915 

1914 

1913 

1912 

1912 

I91I 

1911 

1910 

1909 

1909 

5 
6 

6 

4 

1902 

1902 

I  90 1 

1900 

1899 

1899 

1898 

1898 

1897 

1896 

10 

1890 

1889 

1888 

1887 

1S87 

1886 

1886 

1885 

1884 

i884 

7 

3 

20 

1877 

1877 

1876 

1875 

1874 

1874 

1873 

1873 

1872 

1872 

8 

2 

3o 

i865 

1864 

i863 

i863 

1862 

i8bi 

1861 

i860 

i860 

1859 

9 

I 

()0 

4o 
5o 

0 

i853 

i84o 

i852 
i84o 

i85i 
1839 

i85o 
1 838 

i85o 

1837 

1849 
1837 

1848 

i836 

i848 
i836 

1847 
i835 

1847 
i834 

Sec. 

Cor. 

1S28 

1827 

1826 

1826 

1825 

1824 

1824 

1823 

1823 

1822 

0 

12 

lO 

1816 

i8i5 

i8i4 

i8i3 

i8i3 

1812 

1812 

1811 

1810 

1810 

1 

II 

20 

i8o3 

i8o3 

1802 

1801 

1801 

1800 

1799 

1799 

1798 

1798 

2 

10 

3o 

1791 

1791 

1790 

1789 

1788 

1788 

1787 

1787 

1786 

1786 

3 

8 

4o 

1779 

1778 

1778 

1777 

1776 

1776 

1775 

1774 

1774 

1773 

4 

7 

""bT 

5o 
o 

1767 
1755 

1766 
1754 

1765 
""i"753" 

1765 

1764 

1763 

1763 

"1762 

1762  1761 

5 
6 

6 
5 
4 

1753 

1752 

1751 

1751 

1750 

1750  1749 

10 

1743 

1742 

1741 

1740 

1740 

1739 

1739 

1733 

1737 

.737 

7 
8 

20 

i73i 

1730 

1729 

1728 

1728 

1727 

1727 

1726 

1725 

1725 

JO 

1719 

1718 

1717 

1716 

1716 

17x5 

I7i5 

1714 

17.3 

1713 

9 

I 

Pageias]                                         TABLE   XIX. 

Correction. 

ij  c 

Table  A. 

Tab.B. 

<  8 

D  's  Horizontal  Parallax. 

Correction  for  Seconds 
of  Parallax. 

For  M. 
of  Alt. 

-=;'■* 

Add. 

Add. 

D. 

M. 

54 

55 

56' 

57 

58' 
39.52 

59' 

60' 

61' 

S. 
0 

0" 
20 

1" 

20 

2" 
19 

3" 
19 

1" 
19 

5" 
78 

G" 
78 

7" 
78 

8" 

17 

9" 
17 

M. 
0 

s. 

0 

70 

0 

4i.i4 

40.54 

40.33 

4o.i3 

39.32 

39.11 

38. 5i 

10 

41.28 

4i.  3 

40.42 

4o.22 

40.  2 

39.41 

3o.2I 

39.   I 

10 

17 

16 

lO 

16 

i5 

i5 

i5 

i4 

1 4 

i4 

1 

2 

2 

20 

4i.32 

41.12 

4o.bi 

40. 3i 

4o.li 

39.51 

39.31 

39.10 

20 

i3 

i3 

i3 

12 

12 

12 

II 

11 

II 

10 

3 

4 

3o 

4i.4o 

41.20 

4i.  0 

4o.4o 

40.20 

40.  0 

39.40 

39.20 

3o 

10 

10 

9 

9 

9 

8 

8 

8 

7 

7 

.=. 

•  5 
6 
6 

40 

41.49 

41.29 

4i.  9 

4o.5o 

4o.3o 

40.10 

39.50 

J9.30 

4o 

7 

6 

6 

6 

5 

5 

5 

4 

4 

4 

6 

71 

bo 
0 

41.58 

41.38 

4i.i8 

40.59 

40.39 

40.19 

4o.  0 

39.40 

5o 

3 

3 

3 

2 

2 

2 

17 

1 
17 

I 

17 

1 
76 

0 
76 

0 

7 
8 
0 

42.  8 

41.48 

41.28 

4i-  9 

40.49 

4o.3o 

4o.io 

39.51 

0 

19 

19 

18 

18 

18 

10 

42.16 

41.57 

41.J8 

41-18 

40.59 

40.39 

40.20 

4o.  I 

10 

16 

lb 

i5 

lb 

i5 

i4 

i4 

i4 

]3 

i3 

2 

20 

42.25 

42.   6 

41.47 

41.27 

4i.  8 

40.49 

4o.3o 

40.11 

20 

i3 

12 

12 

12 

II 

11 

II 

10 

10 

10 

3 
4 

4 

Jo 

42.34 

42.1b 

4i.bfa 

41.37 

41.18 

40.59 

40.39 

40.20 

3o 

9 

9 

9 

8 

8 

8 

8 

7 

7 

7 

5 

5 

40 

42.43 

42.24 

42.  5 

41.46 

41.27 

4i.  8 

40.49 

40. 3o 

4o 

6 

6 

6 

b 

5 

5 

4 

4 

4 

3 

7 

7 

72 

bo 
0 

42. 5i 

42.33 

42.14 

41-55 

4i.36 

41.18 

40.59 

40. 4o 

5o 

3 

3 
78 

2 

17 

2 

17 

2 

17 

I 
76 

I 

76 

I 

76 

I 
76 

0 
75 

8 
9 
0 

7 
8 
0 

43.   I 

42.43 

42.24 

42.  5 

41-47 

41.28 

4i.io 

4o.5i 

0 

18 

10 

43.10 

42. 5i 

42. JJ 

42.15 

41.56 

4i.38 

4i  .20 

4i.  I 

10 

i5 

lb 

i4 

i4 

i4 

i3 

i3 

i3 

i3 

12 

k 

2 

20 

43.19 

43.  0 

42.42 

42.24 

42.  6 

41.48 

41.29 

4i.ii 

20 

12 

12 

II 

11 

11 

10 

10 

10 

10 

Q 

4 

4 

Jo 

43.27 

4':^.  9 

42. 5i 

42.33 

42.15 

41.57 

4i  .39 

4l.21 

3o 

9 

9 

8 

8 

8 

7 

7 

7 

7 

6 

5 
6 
7 

a 

40 

43.36 

4J-18 

43.  0 

42.43 

42.25 

42.  7 

41.49 

41-J1 

40 

6 

6 

5 

b 

5 

4 

4 

4 

3 

3 

7 

73 

bo 
0 

43.45 
43.55 

4J-27 

43.37 

43.10 
43.20 

42.52 
43.  2 

42.34 

42.17 

41.59 

4i-4i 

5o 

3 

3 

2 

2 

2 

I 
76 

1 
75 

I 
75 

0 
75 

0 
74 

8 
9 
0 

8 
8 

0 

42.45 

42.27 

42.10 

41.52 

0 

17 

17 

16 

16 

16 

10 

44.  4 

43-46 

43.29 

43.12 

42.54 

42.37 

42.19 

42.  2 

10 

i4 

i4 

14 

i3 

i3 

i3 

12 

12 

12 

12 

2 

2 

20 

44. iJ 

43.  bb 

43.38 

43.21 

43.  4 

42.47 

42.29 

42.12 

20 

11 

11 

11 

10 

10 

10 

10 

9 

9 

9 

4 

4 

Jo 

44-21 

44.  4 

43.47 

43. 3o 

43.13 

42.56 

42.39 

42.22 

3o 

8 

8 

8 

8 

7 

7 

7 

6 

6 

6 

5 

5 

40 

44. 3o 

44.13 

43.57 

43.40 

43.23 

43.  6 

42.49 

42.32 

4o 

6 

5 

5 

5 

4 

4 

4 

4 

3 

3 

7 

7 

74 

bo 
0 

44.39 

44.22 

44.  6 

43.49 

43.32 
43.43 

43.16 
43.26 

42.59 
43.10 

42.42 
42.53 

5o 
0 

3 
16 

2 
76 

2 

73 

2 
75 

2 
75 

I 
75 

I 
74 

I 
74 

0 
74 

0 
74 

8 

9 

0 

8 
0 

44.49 

44-32 

44-16 

43.59 

10 

44.58 

44.42 

44-25 

44.  9 

43.52 

43.36 

43.20 

43.  3 

10 

i3 

i3 

i3 

12 

12 

12 

12 

II 

u 

II 

2 

20 

45.  7 

44. 5i 

44.  S4 

44-18 

44.  2 

43.46 

43. 3o 

43.13 

20 

11 

10 

10 

10 

10 

9 

9 

9 

8 

8 

4 

4 

Jo 

45.16 

45.  0 

44.44 

44.28 

44-12 

43.56 

43.40 

43.23 

3o 

8 

8 

7 

7 

7 

7 

6 

6 

6 

5 

5 
6 

7 

5 

4o 

45.25 

45.  9 

44.53 

44.37 

44-21 

44.  5 

43.49 

43.34 

4o 

5 

5 

5 

4 

4 

4 

4 

3 

3 

3 

7 

75 

bo 
0 

45.34 
45.43 

45.18 

45.  2 

44-46 
44-57 

44. Ji 
44-41 

44.15 
44-26 

43.59 
44.10 

43.44 
43.55 

5o 
0 

3 

Is 

2 
75 

2 

74 

2 

74 

I 

74 

I 
74 

1 
73 

1 
73 

0 
73 

0 
73 

8 
9 
0 

8 
9 
0 

45.28 

45.12 

10 

45.52 

45.37 

45.22 

45.  6 

44-5i 

44-36 

44-20 

44.  5 

10 

12 

12 

12 

12 

II 

II 

11 

II 

10 

10 

2 

2 

20 

46.   I 

45.46 

45.31 

45.16 

45.   1 

44-45 

44 -3o 

44.15 

20 

10 

10 

9 

9 

9 

0 

8 

8 

8 

8 

4 

4 

Jo 

46.10 

4b.  55 

45.40 

45.25 

45.10 

44-55 

44-40 

44-25 

3o 

7 

7 

7 

7 

6 

6 

6 

6 

5 

5 

5 

5 
6 

7 

40 

46.19 

46.  4 

45.50 

45.35 

45.20 

45.  5 

44. 5o 

44-35 

4o 

5 

5 

4 

4 

4 

4 

3 

3 

3 

3 

7 

76 

bo 

0 

46.28 
46.38 

46.14 
46.24 

45.59 
46.  9 

45.44 
45.55 

45.29 
45.40 

45. i5 
45.26 

45.  0 

44-45 

5o 
0 

2 
I4 

2 

74 

2 

74 

2 
73 

I 

73 

I 

73 

I 

71 

I 
12 

0 
12 

0 

13 

8 
9 
0 

8 
9 
0 

45.11 

44-57 

10 

46.47 

46.33 

46.18 

46.  4 

45. 5o 

45.35 

45.21 

45.  7 

10 

12 

11 

11 

11 

11 

10 

IC 

10 

10 

10 

2 

2 

20 

46.56 

46.42 

46.28 

46.14 

45.59 

45.45 

45. 3i 

45.17 

20 

9 

9 

9 

9 

8 

8 

8 

8 

7 

7 

4 

4 

Jo 

47.  5 

4b. 5i 

46.  J7 

46.23 

46.  9 

45.55 

45.41 

45.27 

3o 

7 

7 

7 

6 

6 

6 

e 

5 

5 

5 

5 
6 

7 

5 
6 

7 

4o 

47.14 

47.  0 

46.46 

46.33 

46.19 

46.  5 

45. 5i 

45.37 

4o 

5 

4 

4 

4 

4 

3 

3 

3 

3 

3 

77 

bo 
0 

47-23 

47-  9 

46.56 
47-  6 

46.42 
46.53 

46.28 
46.39 

46. i5 
46.26 

46.   I 
46.12 

45.47 
45.59 

5o 
0 

2 
73 

2 
73 

2 
73 

2 
12 

I 

12 

I 
12 

I 
12 

1 
11 

0 
u 

0 
1 1 

8 
9 
0 

8 
9 
0 

47-33 

47.20 

lO 

47-42 

47-29 

47-15 

47-   2 

46-49 

46.35 

46.22 

46.  9 

10 

II 

II 

10 

10 

10 

10 

IC 

9 

9 

Q 

2 

2 

20 

47-5i 

47-38 

47-25 

47-12 

46.59 

46.45 

46.32 

46.19 

20 

9 

8 

8 

8 

8 

8 

7 

7 

7 

7 

3 
4 

3 

4 

Jo 

48.  0 

47-47 

47 -J4 

47-21 

47-   8 

46.55 

46.42 

46.29 

3o 

6 

6 

6 

6 

6 

5 

5 

5 

5 

5 

5 

5 

4o 

4B-  9 

47-56 

47-44 

47-31 

47.18 

47.  5 

46.52 

46.39 

4o 

4 

4 

4 

4 

3 

3 

3 

3 

3 

2 

7 

7 

78 

bo 
0 

48.18 
48.28 

48.  6 
48.16 

47-53I47-40 
48.  3I47-51 

47-28 

47-15 

47-   2 

46. 5o 
47.   1 

5o 
0 

2 
12 

2 
12 

2 
12 

I 

1 1 

1 
II 

I 
11 

I 
II 

I 
11 

0 
10 

0 
10 

8 
9 
0 

8 
9 
0 

47-38 

47-26 

47-i3 

10 

48  37 

48.25 

48. i3 

4«.  0 

47-48 

47-36 

47-24 

47-11 

10 

10 

10 

10 

9 

9 

9 

c 

9 

8 

8 

2 

20 

48.46 

48.34 

48.22 

48.10 

47-58 

47-46 

47-34 

47-21 

20 

b 

8 

8 

7 

7 

7 

7 

7 

6 

6 

4 

4 

Jo 

48.55 

48.44 

48.32 

48.20 

48.  8 

47-56 

47-44147-32 

3o 

6 

6 

6 

5 

5 

5 

5 

5 

4 

4 

5 

4o 

49.  5 

48.53 

48.41 

48.29 

48.17 

48.  6 

47-54 

47-42 

4o 

4 

4 

4 

3 

3 

3 

3 

3 

2 

2 

7 

7 

79 

5o 
0 

49.14 

49.  2 

48. 5o 

48.39 

48.27 
48.38 

48.16 
48.26 

48-  4 
48. i5 

47-52 
48.  4 

5o 
0 

2 
1 1 

2 
11 

2 
11 

1 
10 

I 
10 

1 
10 

1 
10 

I 

10 

0 
10 

0 
9 

8 
9 
U 

8 
9 
0 

49.24 

49.12 

49.   1 

48.49 

ID 

49.33 

49.22 

49-10 

48.59 

48.48 

48.36 

48.25 

48.14 

10 

9 

9 

9 

9 

8 

8 

8 

8 

8 

8 

2 

2 

20 

49-42 

49-3i 

49.20 

49-  9 

48.57 

48.46 

48.35 

48.24 

20 

7 

7 

7 

7 

7 

6 

6 

6 

6 

6 

3 
4 

3 

4 

Jo 

49-5i 

49.40 

49-29 

49-18 

49-   7 

48.56 

48.45 

48.34 

3c 

5 

5 

5 

5 

5 

5 

4 

4 

4 

4 

5 
6 

7 

5 

4o 

5o.  0 

49.49 

49-39 

49.28  49.17 

49-  6 

48.55 

48.45 

4o 

4 

3 

3 

3 

3 

3 

3 

2 

2 

2 

7 

li)o 

5o.  9 

49-59 

49.48 

49.37  49.27 

49-16 

49-  6 

48.55 

5o 

2 

2 

' 

I 

I 

I 

I 

1 

0 

0 

8 
9 

8 
9 

t 

TABLE  XIX. 
Logarithms. 

[Page  127 

s  g 

Tablk  C.     1 

33^ 

Cor.  for 

Seconds   1 

Jf  a 

Apparent  Altitude  of  5  's  Centre. 

of  Parallax. 

aCU 

Add. 

o 

0 

0 

0 

0 

0 

0 

0 

0 

0 

M. 

IT 

s. 

c 

70 

71 

72 

73 

2286 

74 

2285 

75 

2  285 

76 

2285 

77 

78 

79 

Sec. 

Cor. 

2287 

2287 

2286 

2284 

2284 

22S4 

0 

12 

lO 

2274 

2273 

2273 

2272 

2272 

2272 

2271 

2271 

2271 

2271 

I 

II 

20 

2260 

2260 

2259 

2259 

2258 

2258 

2258 

2257 

2257 

2257 

2 

9 

3o 

2247 

2246 

2246 

2245 

2245 

2245 

2244 

2243 

2243 

2243 

3 

8 

4o 

2233 

2233 

2232 

2232 

223l 

223l 

223l 

2230 

223o 

223o 

4 

7 

Is" 

5o 
o 

2220 

2219 

2219 

2218 

2218 

2218 

2217 

2217 

2216 

2216 

5 
6 

5 

4 

2206 

2206 

2205 

2205 

2204 

2204 

2204 

2  2o3 

22u3 

2  2o3 

lO 

2193 

2192 

2192 

2191 

2191 

219I 

2190 

2190 

2190 

2190 

7 
8 

3 

20 

2179 

2179 

2179 

2178 

2178 

2178 

2177 

2176 

2176 

2176 

I 

3o 

2166 

2166 

2lb5 

2i65 

2lb4 

2i64 

2164 

2i63 

2i63  '  2i63 

9 

0 

"56" 

4o 
5o 

0 

2i53 

2l40 

2l52 
2139 

2l52 

2i39 

2l52 

2i38 

2l5l 

2i38 

2l5l 

2i38 

2l50 

2i37 

2i5o 

2  1 37 

2)  5c  2:5o 

2i37 

2i37 

Sec. 

Cor. 

2  I  26 

2126 

2126 

2125 

2125 

2125 

2124 

2123 

2123 

2123 

0 

12 

lO 

2Il3 

2Il3 

2II2 

2II2 

2U2 

2111 

2III 

2110 

2110 

2II0 

I 

II 

20 

2100 

2100 

2099 

2099 

2098 

2098 

2098 

2097 

2097 

2097 

2 

Jo 

20S7 

2087 

2086 

20S6 

2o85 

2o85 

2o85 

2084 

2084 

2084 

3 

6 

4o 

2074 

2074 

2073 

2073 

2072 

2072 

2072 

2071 

2071 

2071 

4 

7 

37 

5o 
o 

206 1 

2061 

2060 

2060 

2059 

2059 

2o59 

2o58 

2o58 

2o58 

5 
6 

5 
4 

2048 

2048 

2047 

2047 

ao46 

2o46 

2o46 

2o45 

2045 

2045 

10 

2o35 

2o35 

2o34 

2o34 

2o34 

2o33 

2o33 

2o32 

2o32 

2032 

7 

3 

20 

2022 

2022 

2022 

2021 

2021 

2021 

2020 

2019 

2019 

2019 

8 

2 

3o 

2010 

2009 

2009 

2008 

2008 

2008 

2007 

2007 

2007 

2007 

9 

0 

4o 

1997 

1996 

1996 

1995 

1995 

1995 

1994 
1982 

1994 

1994 

1994 
I981 

"sF 

5o 

0 

1984 

1984 

1983 

1983 

19S2 

1982 

I98I 
1968 

I981 

Sec. 

Cor. 

1971 

I97I 

1970 

1970 

1970 

1970 

1969 

1968 

1968 

0 

12 

ID 

'9^9 

i9b8 

1958 

1957 

1957 

1957 

1956 

1956 

1956 

1956 

I 

II 

20 

1946 

1946 

1945 

1945 

1944 

1944 

1944 

1943 

1943 

1943 

2 

9 

Jo 

19J3 

1933 

1933 

1932 

1932 

1932 

I93I 

1931 

1930 

1930 

3 

8 

40 

1921 

1920 

1920 

1920 

1919 

1919  1919 

1918 

1918 

1918 

4 

7 

5^ 

bo 
o 

1908 
1896 

1908 
78"^ 

1907 
1895 

1907 
1895 

1907 
1894 

1907 
1894 

1906 

1893 

1905 

1905 
1893 

1905 
1893 

5 
6 

6 
4 

1893 

ID 

i883 

1883 

1882 

1882 

1882 

1882 

I88I 

1880 

1880 

1880 

7 

3 

20 

1871 

1870 

1870 

1870 

1869 

1869 

1869 

1868 

1868 

1868 

8 

2 

3o 

i858 

i858 

i8b8 

i857 

1857 

j857 

1 856 

i856 

1 856 

i856 

9 

I 

"fc 

4o 
5o 

0 

1 846 
i834 

1 846 
1 833 

1845 
i833 

1845 
i833 

1844 
i832 

1844 
i832 

1844 
i832 

1843 
i83i 

1843 
i83i 

1843 
i83i 

Sec. 

Cor. 

1822 

1821 

1821 

1820 

1820 

1820 

1819 

1819 

1819 

1819 

0 

12 

lO 

1809 

1809 

1808 

1S08 

1808 

1808 

1807 

1806 

1806 

1806 

I 

II 

20 

1797 

1797 

1796 

1796 

1795 

1795 

1795 

1794 

1794 

1794 

2 

10 

3o 

1783 

17S4 

1784 

1784 

1783 

1783 

1783 

1782 

1782 

1782 

3 

8 

4o 

1773  1772 

1772 

1771 

1771 

1771 

1770 

1770 

1770 

1770 

4 

7 

"gF 

5o 

0 

176. 

1760 

1760 

1759 

n^9 

1759 

1758 

1758 

1758 

1758 

5 
6 

6 

5 

1719 

1748 

1748 

1747 

1747 

1747 

!746 

1746 

1746 

1746 

lO 

1736 

1736 

1736 

1735 

1735 

1735 

1734 

1734 

1734 

1734 

8 

3 

20 

I7'4 

1724 

1724 

1723 

1723 

1723 

1722 

1722 

1722 

1722 

2 

3o 

I7I2 

1712 

1712 

1711 

1711  1711 

1710 

1710 

1710 

1710 

9 

1 

r-ige  123]                                          TABLE   XIX. 

Correction. 

. 

^  c 

Table  A. 

Tab.  B. 

<  S 

])  's  Horizontal  Parallax. 

Proportional  part  for  Seconds 
of  Parallax. 

ForM 
of  Alt. 

<" 

Add. 

Add. 

D 

8^ 

M. 

0 

54' 

55' 

56' 

57' 

58' 

59' 

60' 

CI' 

S. 
0 

0" 
10 

1" 
10 

2" 
10 

3" 
10 

4" 

5" 

6" 
~9 

711 
9 

9 

9^ 
9 

M. 

s. 

0 

1 

So.iySo.   Q 

49-58 

49-48 

49.38 

49.27 

49.17 

49.  6 

10 

5o. 2850.18 

5o.  8 

49-58 

49.47 

49.37 

49.27 

49-17 

10 

8 

8 

8 

8 

8 

8 

7 

7 

7 

7 

2 

1 

20 

50.38 

50.27 

5o.i7 

5o.  7 

49.57 

49-47 

49.37 

49.27 

20 

7 

7 

b 

6 

6 

6 

6 

6 

5 

5 

a 

4 

3a 

50.47 

50.37 

10.27 

50.17 

5o.  7 

49-57 

49.47 

49.37 

3o 

5 

5 

5 

5 

4 

4 

4 

4 

4 

4 

5 

6 

40 

50.56 

5o.46 

5o.36  50.27 

50.17 

5o.   7 

49.57 

49.48 

4o 

3 

3 

3 

3 

3 

3 

2 

2 

2 

2 

I 
9 
0 

7 

87 

10 
0 

5i.   5 

50.55 
5i.  6 

5o.46 
50.56 

5o.36 
5o.47 

50.27 

5o.i7 

5o.  8149. 58 

5o 

2 

2 

I 

I 

I 

I 
"8 

I 
~8 

I 
~8 

0 
~8 

0 

9 
0 

5i.i5 

50.37 

50.28 

50.19 

5o.  9 

0 

9 

9 

9 

9 

8 

10 

51.24 

5i.i5 

5i.  6 

5o.57 

50.47 

50.38 

00.29 

5o.20 

10 

8 

7 

7 

7 

7 

7 

7 

6 

6 

6 

2 

).<) 

51.33 

51.24 

5t.i5 

5i.  6 

5o.57 

50.48 

50.39 

5o.3o 

20 

6 

6 

6 

6 

5 

5 

5 

5 

5 

5 

4 

4 

3<i 

5i  .42 

5i  34 

5r.25 

5t.i6 

5i.  7 

50.58 

50.49 

5o.4o 

3o 

5 

4 

4 

4 

4 

4 

4 

3 

3 

3 

5 

5 
6 

4o 

5i.52 

51.43 

5i.34 

51.26 

G1.17 

5i.  8 

50.59 

5o.5i 

4o 

3 

3 

3 

3 

2 

2 

2 

2 

2 

2 

7 

7 

5o 

52.      I 

51.52 

5i.44|5i.35 

DI  .27 

5i.i8 

5i  .10 

5i.   I 

5o 

2 

I 

i)  I 

I 

I 

I 

0 

0 

0 

9 

8 
9 

82 

0 

52.  I  I 

52.  3 

5i.54 

51.46 

5i.38 

5i  .29 

5i  .21 

5i.i3 

0 

8 

8 

8 

81  7 

7 

7 

7 

7 

7 

0 

0 
1 

10 

52.20 

52.12 

52.  4 

51.56 

51.47 

5i.39 

5i.3r 

5i.23 

10 

7 

7 

6 

6 

6 

6 

6 

6 

6 

5 

2 

2 
3 

4 

20 

52.29 

52.21 

52.  i3 

52.  5 

5i.57 

5r.49 

5i.4i 

5i.33 

20 

5 

5 

5 

5 

5 

5 

5 

4 

4 

4 

4 

3o 

52.39 

52. 3l 

52.23 

52.15 

52.    7 

51.59 

51.52 

51.44 

3o 

4 

4 

4 

4 

4 

3 

3 

3 

3 

3 

b 
6 

5 
6 

4o 

52.48 

52.40 

52.32 

52.25 

52.17 

52.    9 

52.     2 

5i.54 

4o 

3 

3 

2 

2 

2 

2 

2 

2 

2 

2 

7 

7 

83 

5o 
0 

52.57 
53.  7 

52.49 

53.  0 

52.42 
52.52 

5  2.?  4 
52.45 

52 .  27 

52.19 

52.12 

52.  4 

5o 

I 

I 

I 

I 

I 

I 
~6 

I 

~6 

0 

"6 

0 

0 
"6 

8 
9 
0 

9 

0 
1 

52.38 

52. 3o 

52.23 

52.16 

0 

7 

7 

7 

7 

7 

10 

53.16 

53.  Q 

53.  2 

52.55 

52.48 

52.41 

52.33 

52.26 

10 

6 

6 

6 

6 

5 

5 

5 

5 

5 

5 

2 

2 
3 
4 

30 

53.25 

53.18 

53.11 

53.  5 

52.58 

52. 5i 

52.44 

52.37 

20 

5 

5 

4 

4 

4 

4 

4 

4 

4 

4 

4 

3o 

53.35 

53.28 

53.21 

53.14 

53.  7 

53.   I 

52.54 

52.47 

3o 

4 

3 

3 

3 

3 

3 

3 

3 

3 

3 

5 

5 

40 

53.44 

53.37 

53. 3i 

53.24 

53.17 

53.11 

53.  4 

52.57 

4o 

2 

2 

2 

2 

2 

2 

2 

2 

I 

I 

7 

7 

5o 

53.53 

53.47 

53.40 

53.34 

53.27 

53.2  1 

53.14 

53.  8 

5o 

I 

I 

I 

I 

I 

I 

I 

0 

0 

0 

9 

8 
9 

84 

0 

54.  3 

53.57 

53. 5i 

53.44 

53.38 

53.32 

53.26 

53.19 

0 

6 

6 

6 

6 

6 

6 

5 

5 

5 

5 

0 

0 

10 

54.12 

54.  6 

54.  0 

53.54 

53.48 

53.42 

53.36 

53.30 

10 

5 

5 

5 

5 

5 

5 

4 

4 

4 

4 

2 

2 

30 

54.22 

54.16 

54.10 

54.  4 

53.58 

53.52 

53.46 

53.40 

20 

4 

4 

4 

4 

4 

4 

3 

3 

3 

3 

4 

4 

3o 

54. 3i 

54.25 

54.19 

54-14 

54.  8 

54.  2 

53.56 

53.51 

3o 

3 

3 

3 

3 

3 

3 

2 

2 

2 

2 

b 
6 

5 

4o 

54.40 

54.35 

54.29 

54.23 

54.18 

54.12 

54.    7 

54.   I 

40 

2 

2 

2 

2 

2 

2 

2 

I 

I 

I 

7 

7 

5o 

54.49 

54.44 

54.39 

54.33 

54.38 

54.22 

54-17 

54.11 

5o 

I 

I 

I 

I 

I 

I 

I 

0 

0 

0 

9 

9 

85 

0 

55.  0 

54.54 

54.49 

54.44 

54-39 

54.33 

54.28 

54.23 

0 

5 

5 

5 

5 

5 

5 

5 

4 

4 

4 

0 
1 

0 
1 

10 

55.  Q 

55.  4 

54.59 

54.54 

54.49 

54.43 

54.38 

54.33 

10 

4 

4 

4 

4 

4 

4 

4 

4 

4 

3 

2 
3 
4 

2 

20 

55.18 

55.13 

55.  8 

55.  3 

54.58 

54.54 

54.49 

54.44 

20 

3 

3 

3 

3 

3 

3 

3 

3 

3 

3 

4 

3o 

55.27 

55.23 

55.18 

55.13 

55.  8 

55.  4 

54.59 

54.54 

3o 

3 

3 

2 

2 

2 

2 

2 

2 

2 

2 

b 

5 
6 

4o 

55.36 

55.32 

55.27 

55.23 

55.18 

55.14 

55.  9 

55.  5 

4o 

2 

2 

3 

2 

I 

I 

I 

I 

I 

I 

7 

7 

86 

5o 
0 

55.46 

55.41 

55.37 

55.33 
55.43 

55.28 
55.39 

55.34 
55.35 

55.20 
55. 3i 

55.15 
55.27 

5o 
0 

I 

I 
7 

I 

1 

I 
~1 

I 
~4 

I 

I 
"4 

0 
~4 

0 

"4 

0 
"3 

8 
9 

0 
1 

8 
9 

0 
1 

55.56 

55.52 

55.48 

10 

56.   5 

56.   I 

55.57 

55.53 

55.49 

55.45 

55.41 

55.37 

10 

3 

3 

3 

3 

3 

3 

3 

3 

3 

3 

2 

2 
3 

4 

20 

56.14 

56.11 

56.  7 

56.  3 

55.59 

55.55 

55.52 

55.48 

20 

3 

3 

3 

3 

3 

2 

2 

2 

2 

2 

4 

3o 

56.24 

56 .  20 

56.16 

56.13 

56.  9 

56.  5 

56.   2 

55.58 

3o 

2 

2 

2 

2 

2 

2 

2 

2 

2 

2 

6 

5 
6 

4o 

56.33 

56.29 

56.26 

56.22 

56.19 

56.15 

56.12 

56.  8 

4o 

2 

I 

I 

I 

I 

I 

I 

I 

I 

I 

7 

7 

8^ 

5o 
0 

56.42 
56.52 

56.39 

56.36 

56.32 
56.43 

56.29 

56.26 

56.2  2 

56.34 

56.19 
56. 3o 

5o 
0 

I 
1, 

I 
"3 

I 
~3 

I 
~3 

I 
~3 

I 

I 
1 

0 

0 

0 
~3 

9 
0 

9 
0 

56.49 

56.46 

56.40 

56.37 

10 

57.   2 

56.59 

56.56 

56.53 

56. 5o 

56.47 

56.44 

56. 4i 

10 

3 

3 

2 

2 

2 

2 

2 

2 

2 

2 

2 
3 

4 

2 

20 

57.11 

57.  8 

57.  5 

57.  3 

57.  0 

56.57 

56.54 

56. 5i 

20 

2 

2 

2 

2 

2 

2 

2 

2 

2 

2 

4 

3o 

57.20 

57.18 

57.15 

57.12 

57.10 

57.  7 

57.  4 

57.  2 

3o 

2 

2 

2 

2 

I 

I 

I 

I 

I 

I 

6 

5 
6 

4o 

57.29 

57.27 

57.25 

57.22 

57.20 

57.17 

57.15 

57.12 

4o 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

7 

7 

5o 

57.39 

57.36 

57.34 

57.32 

57 -3o 

57.27 

57.25 

57.23 

5o 

I 

I 

I 

I 

I 

I 

0 

0 

0 

0 

9 

9 

88 

0 

57.49 

57-47 

57.45 

57.43 

57.4. 

57.38 

57.36 

57.34 

0 

2 

2 

2 

2 

2 

2 

2 

2 

2 

3 

0 
1 

0 
1 

10 

57.58 

57.56 

57.54 

57.52 

57.50 

57.49 

57.47 

57.45 

10 

2 

2 

2 

2 

2 

2 

2 

2 

2 

I 

2 

2 

30 

58.  7 

58.  6 

58.  4 

58.   2 

58.  0 

57.59 

57.57 

57.55 

20 

I 

I 

1 

I 

I 

I 

I 

I 

I 

I 

4 

4 

3o 

58.17 

58. i5 

58.14 

58.12 

58.10 

58.  Q 

58.  7 

58.  6 

3o 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

5 
6 

5 
6 

4o 

58.26 

58.25 

58.23 

58.22 

58.20 

58.19 

58. 18 

58.16 

4o 

I 

I 

I 

I 

I 

I 

I 

I 

I 

1 

7 

7 

89 

5o 
0 

58.35 
58.45 

58.34 

58.33 

58.32 
58.42 

58. 3o 
58. 4i 

58.29 

58.28 

58. 3q 

58.27 
58.38 

5o 
0 

I 
I 

I 
I 

I 
I 

I 

I 

I 
I 

0 
I 

0 

I 

0 
I 

0 
I 

0 

I 

9 
0 

9 
0 

58.44i58.43 

58. 4o 

10 

58.55 

58.54  58.53 

58.52 

58. 5i 

58. 5o 

58.49 

58.49 

10 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

2 

2 

20 

59.  4 

59.  3i59.  3 

59.   0. 

59.   I 

59.  0 

58.59 

58.59 

20 

I 

I 

I 

J 

I 

I 

I 

I 

I 

I 

4 

4 

3o 

59.13 

59.13  59.12 

59.12 

59.11 

59.11 

59.10 

59.10 

3o 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

6 

5 
6 

40 

59.22 

59.22  59.22 

59.21 

59.21 

59.21 

59.20 

59.20 

4o 

I 

I 

I 

I 

I 

I 

I 

I 

I 

1 

7 

7 

Ibo 

59.32 

59.32,59.31 

59.31 

59.3. 

59.31 

59.3, 

59.31 

5o 

I 

I 

I 

I    .1 

0 

0 

0 

0 

0 

9 

9 

TABLE  XIX. 

iPago  129 

Logarithms. 

o 

4 

Table  C. 

Apparent  Altitude  of  5  s  centre. 

Cor.  for  Seconds 
of  Parallax. 

«Ai 

Add. 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

M. 

s. 

0 

60 

81 

82 

83 

2283 

84 

22b3 

85 

2253 

86 

2283 

87 

88 

89 

Sec. 

Cor. 

2283 

2283 

2  283 

22«3 

2  283 

2283 

0 

12 

10 

2270 

2270 

2270 

2270 

2269 

2269 

2269 

2269 

2269 

2269 

I 

11 

20 

2256 

22!^ 

2256 

2256 

2256 

2256 

2256 

2256 

2256 

2256 

2 

9 

3o 

2243 

2243 

2242 

2242 

2242 

2242 

2242 

2242 

2242 

2242 

3 

8 

4o 

2229 

2229 

2229 

2229 

2229 

2229 

2229 

2229 

2229 

2229 

4 

7 

IT 

5o 
0 

2216 

2216 

2216 

22l5 

22l5 

22lb 

22  I  5 

22l5 

22l5 

22  I  5 

5 
6 

5 
4 

2202 

2202 

2202 

2  202 

2202 

2202 

2202 

2202 

2202 

2  202 

10 

2189 

2189 

2189 

2180 

21S9 

2189 

2189 

2189 

2188 

2  1 88 

7 

3 

20 

2176 

2175 

2175 

2175 

2175 

2175 

2.75 

2175 

2175 

2175 

8 

I 

3o 

2162 

2162 

2162 

2162 

2x62 

2162 

2162 

2162 

2162 

2162 

9 

0 

"56" 

4o 
5o 

0 

2149 

2i36 

2149 
2i36 

2149 

2i36 

2149 

2i36 

2149 

2i36 

2149 

2i35 

2149 

2Io5 

2149 

2i35 

2149 

2i35 

2149 

2i35 

Sec. 

Cor. 

2123 

2123 

2122 

2122 

2122 

2122 

2122 

2122 

2122 

2122 

0 

12 

10 

2109 

2109 

2109 

2109 

2109 

2109 

2109 

2109 

2109 

210Q 

1 

11 

20 

2096 

2096 

2096 

2096 

2096 

2096 

2096 

2096 

2096 

2096 

2 

9 

3o 

2o83 

2o83 

2o83 

2o83 

2o83 

2083 

2o83 

2o83 

2o83 

2o83 

3 

8 

4o 

2070 

2070 

2070 

2070 

2070 

2070 

2070 

2070 

2070 

2070 

4 

7 

57 

5o 
0 

2o57 

2o57 

2057 

2057 

2o57 

2o57 

2o57 

2o57 

2o57 

2057 

5 
6 

0 

4 

2o44 

2044 

2o44 

2o44 

2044 

2044 

2044 

2044 

2044 

2o44 

10 

2032 

2o3l 

2o3f 

203l 

2o3l 

2o3l 

2o3l 

2o3l 

203l 

2o3l 

7 

3 

20 

2019 

2019 

2019 

2018 

2018 

2018 

2018 

2018 

2018 

2018 

8 

2 

3o 

2C06 

2006 

2006 

2006 

2006 

2006 

2006 

2ou5 

2oo5 

200J 

9 

0 

'58 

40 
5o 

0 

1993 

1980 

1993 

1980 

1993 
1980 

1993 

1980 

1993 

1980 

1993 

1980 

1993 

1980 

1993 
1980 

1993 

1980 

1993 
1980 

Sec. 

Cor. 

,968 

.968 

1967 

1967 

1967 

1967 

1967 

1967 

1967 

1967 

0 

12 

to 

1955 

19^5 

1955 

[955 

1955 

1955 

1955 

1955 

1955 

iq55 

I 

11 

20 

1942 

1942 

1942 

1942 

1942 

1942 

1942 

1942 

1942 

1942 

2 

9 

Jo 

1930 

1930 

1930 

1930 

1929 

1929 

1929 

1929 

1929 

1929 

3 

8 

40 

1917 

1917 

1917 

I9I7 

1917 

1917 

I9I7 

1917 

1917 

1917 

4 

7 

^ 

bo 
0 

1905 

1905 

1904 

1904 

1904 

1904 

1904 

1904 

1904 

1904 

5 
6 

6 

4 

1892 

1892 

1892 

1892 

1892 

1892 

1892 

1892 

1892 

1892 

10 

1880 

J  880 

1880 

1879 

1879 

1879 

1879 

1879 

1879 

1879 

7 

3 

20 

1867 

1867 

1867 

1867 

1867 

1867 

1867 

1867 

1867 

1867 

2 

3o 

i855 

i855 

i855 

i855 

i855 

i855 

i855 

iS55 

i855 

1 855 

9 

1 

"fk7 

4o 
5o 

0 

1843 
i83o 
1818 

1842 
i83o 

1842 
i83o 

1842 
i83o 
1818 

1842 
i83o 
1818 

1842 
i83o 
1818 

1842 
i83o 

"isTs 

1842 
i83o 

1842 
i83o 

1842 
i83o 

Sec. 

Cor. 

1818 

1818 

1818 

1818 

1818 

0 

12 

10 

18c  6 

1806 

1806 

1806 

i3o5 

i8o5 

i8o5 

i8o5 

i8o5 

i8o5 

I 

II 

20 

1793 

1793 

1793 

1793 

1793 

1793 

1793 

1793 

1793 

1793 

2 

10 

jO 

1781 

17S1 

1781 

1781 

1781 

1781 

1781 

1781 

1 78 1 

1781 

3 

8 

4o 

1769 

1769 

1769 

1769 

1769 

1769 

1769 

1769 

1769 

1769 

4 

7 

(JT 

5o 
0 

175- 
1745 

17^7 
1745 

1757 
1745 

1757 

1757 

1757 

11^7 

1757 

1757 

17^7 

5 
6 

6 
5 

1745 

1745 

1745 

1745 

1745 

1745 

1745 

10 

1733 

1733 

1733 

1733 

.733 

1733 

1733 

T733 

1733 

1733 

7 
8 

4 

20 

1721 

1721 

1721 

1721 

1721  1721 

1721 

1721 

1721 

1721 

3o 

1709 

1709 

1709 

1709 

1709  1709  ]  1709  1  1709 

1709 

1709 

9 

' 

17 


rage  1301  TABLE  XX. 

Third  Correction  of  the  first  method  of  working  a  Lunar  Observation,  additive. 


0*     I) 
Alt.    Alt. 

lO^"!  10° 
20 
30 
40 
30 
GO 
70 


20° 


30° 


40° 


50' 


50° 


70= 


80= 


10° 

20 

30 

40 

50 

60 

70 

80 


20 
30 
40 
50 
60 
70 


10° 

20 

30 

40 

50 

60 

70 

80 


10° 

20 

30 

40 

50 

60 

70 

80 


10° 

20 

30 

40 

50 

GO 

70 

80 


10° 

20 

30 

40 

50 

60 

70 

80 


10° 

20 

30 

40 

50 

60 

70 


True  Distance  of  the  Moon  from  the  Sun  or  a  Star. 


20° 


96" 

66 

18 


25° 


79" 
71 

39 


30° 

"67^ 

18 


35° 

~60" 
58 
46 
28 


19 


40= 


53" 

52 

44 

32 

18 


45= 


50° 


G0° 


23 


20 


70°    60° 


90° 

23" 

22 

22 

21 

20 

19 

18 

18 

"20" 
20 
20 
19 
19 
18 
18 


18 


100= 


19'' 

18 

18 

18 

18 

18 

18 


IS 


110° 120°     ^t 


15" 

15 

15 

16 

17 

18 


18 


0* 
Alt. 


10°   10= 

20 

30 

40 

50 

60 

70 

80 


10° 

20 

30 

40 

50 

GO 

70 

80 


10° 

20 

30 

40 

50 

60 

70 

SO 


10° 

20 

30 

40 

50 

60 

70 

80 


30° 


40° 


10° 

20 

30 

40 

60 

60 

70 

80 


50° 


10° 

20 

30 

40 

50 

60 

70 

80 


10° 

20 

30 

40 

50 

60 

70 

80 


60° 


an  I' 


In  using  this  Table,  it  will,  in  general,  be  sufficiently  accurate,  to  find  the  nearest  altitudes  and  distance, 
and  take  out  the  corresponding  correction,  without  the  trouble  of  making  a  proportion  for  the  neglected  de- 
grees and  minutes  ;  as  in  the  following  examples  : — 

Example  I.  Given  the  apparent  distance  47°  34',  the  star's  apparent  altitude  5Ci°  31',  and  the  moon's  ap- 
parent altitude  70°  47',  to  find  the  third  correction. 

Here  the  Table  may  be  entered  with  the  app.  dist.  50°,  ^^s  alt.  50°,  5  's  alt.  70° ;  the  corresponding  correc- 
tion is  20",  additive. 

Example  II.  Given  the  apparent  distance  81°  20',  sun's  apparent  altitude  12° 30',  moon's  apparent  altitude 
20°  38',  to  find  the  third  correction. 

Here  the  Table  may  be  entered  with  the  app.  dist.  80°,  Q's  alt.  10°,  D  's  alt.  20°  ;  the  corresponding  correc- 
tion is  27". 


1  ^ 
1 

TABLE  XXI. 

|Page  131 

For 

turnin 

g  Degrees  and  Minutes  into  Time,  an 

(i  the  contrary. 

D. 

H.  M. 

D. 

H.  iM. 

D. 

1  11.  M. 

D. 

H.  31. 

D. 

H.  31 

D. 

H.  M. 

M. 

RI.  S. 

M. 

M.  S. 

4.  4 

M. 

M.S. 

31. 

31.  S. 

31. 

31.  S. 

31, 

31.  S. 
20.  4 

I 

0.  4 

61 

121 

8.  4 

181 

12.  4 

241 

16.  4 

3oi 

2 

o.  8 

62 

4.  8 

122 

8.  8 

182 

12.  8 

242 

16.  8 

3o2 

20.  8 

3 

0.12 

63 

4.12 

123 

8.12 

i83 

12.12 

243 

16. 12 

3o3 

20.12 

/, 

o.i6 

64 

4.16 

124 

8.16 

184 

12.16 

244 

16.16 

3o4 

20.16 

5 

0.20 

65 

4.20 

120 

8.20 

i85 

12.20 

245 

16.20 

3o5 

20.20 

6 

0.24 

66 

4.24 

126 

8.24 

186 

12.24 

246 

16.24 

3o6 

20.24 

7 

0.28 

67 

4.28 

127 

8.28 

187 

12.28 

247 

16.28 

307 

20.28 

8 

0.32 

68 

4.32 

128 

8.32 

188 

12.32 

248 

16.32 

3oS 

20.32 

9 

C.36 

69 

4.36 

129 

8.36 

189 

12.36 

249 

16. 36 

309 

20.36 

10 

0.40 

70 

4.4o 

i3o 

8.4o 

190 

12.40 

25o 

16.40 

3io 

20.40 

I  ( 

0.44 

71 

4.44 

i3i 

8.44 

191 

12.44 

25l 

16.44 

3ii 

20.44 

12 

0.48 

72 

4.48 

l32 

8.48 

192 

12.48 

252 

16.48 

3l2 

20.48 

i3 

0.52 

73 

4.52 

i33 

8.52 

193 

12.52 

253 

16.52 

3i3 

20.52 

i4 

0.56 

74 

4.56 

i34 

8.56 

194 

12.56 

254 

16. 56 

3i4 

20. 56 

13 

1 .  0 

75 

5.  0 

i35 

9.  0 

195 

i3.  0 

255 

17-  0 

3i5 

21.  0 

i6 

I.  4 

76 

5.  4 

i36 

9-  4 

196 

i3.  4 

256 

17.  4 

3i6 

21.  4 

17 

I.  8 

77 

5.  8 

1 37 

9.  8 

197 

i3.  8 

257 

17.  8 

3i7 

21.  8 

iS 

1.12 

78 

5.12 

i38 

9.12 

198 

l3.I2 

258 

17.12 

3i8 

21 .12 

'9 

1. 16 

79 

5.16 

139 

9.16 

199 

13. 16 

259 

17.16 

319 

21.16 

20 

\  .20 

80 

5.20 

i4o 

9.20 

200 

l3.20 

260 

17.20 

320 

21.20 

21 

1.24 

81 

5.24 

i4i 

9.24 

201 

i3.24 

261 

17.24 

321 

21 .24 

32 

i.aS 

82 

5.28 

142 

9.28 

202 

13.28 

262 

17.28 

322 

21.28 

23 

1.32 

83 

5.32 

i43 

9.32 

203 

i3.32 

263 

17.32 

323 

21.32 

24 

1.36 

84 

5.36 

1 44 

9.35 

204 

i3.36 

264 

17.36 

324 

21.36 

25 

1 .40 

85 

5.4o 

i45 

9.40 

205 

i3.4o 

265 

17.40 

325 

21 .40 

26 

1.44 

86 

5.44 

i46 

9-44 

206 

13.44 

266 

17.44 

326 

21.44 

27 

1.48 

87 

5.48 

1 47 

9-48 

207 

i3.48 

267 

17.48 

327 

21.48 

28 

1.52 

88 

5.52 

i48 

9.52 

208 

i3.52 

268 

17.52 

328 

21 .52 

29 

1.56 

89 

5.56 

149 

9.56 

209 

i3.56 

269 

17.56 

829 

21.56 

3o 

2.  0 

90 

6.  0 

i5o 

10.  0 

210 

14.  0 

270 

18.  0 

33o 
33i 

22.  0 
22.  4 

3i 

2.  4 

91 

6.  4 

i5i 

10.  4 

211 

i4.  4 

271 

18.  4 

32 

2.  8 

92 

6.  8 

l52 

10.  8 

212 

i4.  S 

272 

18.  8 

332 

22.  8 

33 

2.12 

93 

6.12 

i53 

to. 12 

2l3 

l4.I2 

273 

18. r2 

333 

22. 12 

34 

2. 16 

94 

6.16 

1 54 

TO. 16 

2l4 

14.16 

274 

18.16 

334 

22.16 

35 

2.20 

95 

6.20 

i55 

10.20 

2l5 

14.20 

275 

18.20 

335 

22.20 

i'". 

2.?4 

96 

6  24 

i56 

10.24 

216 

14.24 

276 

18.24 

336 

22.24 

V 

2.28 

97 

0.28 

i57 

10.28 

217 

14.28 

277 

18.28 

337 

22.28 

3S 

2.32 

v8 

6.32 

1 58 

10.32 

218 

i4.32 

278 

18.32 

338 

22.32 

39 

2.36 

99 

6.36 

1 59 

10.36 

219 

i4.36 

279 

18. 36 

339 

22.36 

4o 

2.4') 

100 

6.40 

160 

10.40 

220 

14.40 

280 

18.40 

340 

22.40 

■A\ 

2.44 

lOI 

6.44 

161 

10.44 

221 

14.44 

281 

18.44 

341 

22.44 

^i 

2.48 

102 

6.48 

162 

10.48 

222 

14.48 

282 

18.48 

342 

22.48 

43 

2.52 

io3 

6.52 

i63 

10.52 

223 

14.52 

283 

18.52 

343 

22.52 

44 

2.56 

io4 

6.56 

i64 

10.56 

224 

14.56 

284 

18. 56 

344 

22.56 

45 

3.  0 

io5 

7.  0 

i65 

II.  0 

225 

i5.  0 

285 

19.  0 

345 

23.  0 

46 

3.  4 

106 

7-  4 

166 

11.  4 

226 

i5.  4 

286 

19.  4 

346 

23.  4 

4^ 

3.  8 

:o7 

7.  8 

.67 

II.  8 

227 

i5.  8 

287 

19.  8 

347 

23.  8 

48 

3.12 

:o8 

7.12 

168 

11.12 

228 

l5.I2 

288 

19.12 

348 

23.1  J 

49 

3.16 

109 

7.16 

169 

11.16 

229 

i5.i6 

289 

19.16 

349 

23.16 

bo 

3.20 

no 

7.20 

170 

II  .20 

23o 

l5.20 

290 

19.20 

■'-JO 

23.20 

5i 

3.24 

III 

7.24 

171 

u  .24 

23l 

i5.24 

291 

19.24 

35i 

23.24 

52 

3.28 

112 

7.28 

172 

11.28 

232 

15.28 

292 

19.28 

352 

23.28 

53 

3.32 

ii3 

7.32 

173 

I  I  .32 

233 

i5.32 

293 

19.32 

353 

23.32 

5  4 

3.36 

ii4 

7.36 

174 

11.36 

234 

i5.36 

294 

19.36 

354 

23.36 

55 

3.40 

n5 

7.40 

175 

II  .40 

235 

i5.4o 

295 

19.40 

355 

23. 4o 

56 

3.44 

116 

7.44 

176 

11.44 

236 

i5. 44 

296 

19.44 

356 

23.44 

57 

3.48 

117 

7.48 

177 

11.48 

237 

i5.48 

297 

19.48 

357 

23.48 

58 

3.52 

118 

7.52 

178 

II  .52 

238 

i5.52 

298 

19.52 

358 

23.52 

5^ 

3.56 

119 

7.56 

179 

11.56 

239 

i5.56 

299 

19.56 

359 

23.56 

6o 

4.  0 

120 

8.  0 

180 

12.  0 

240 

16.  0 

3oo 

20.  0 

36o 

24.  0 

Pago  132] 

TABLE  XXIL 

Proportional  Logarithms 

A  VI 

h      m 

h      m 

h      m 

h        VI 

h      VI 

h      m 

h      m 

h      m 

S. 
o 

0=    0' 

{p     1' 

0°  2' 

0°  3' 

0°  4' 

0°  5' 

0°     6' 

0°     7' 

0°  8' 

S. 
0 

2.2553 

1 .9542 

1.7782 

1.6532 

1.5563 

1.4771 

1.4102 

1.3522 

I 

4.0334 

2481 

9506 

7757 

65i4 

5549 

4759 

4091 

35i3 

I 

3 

3.7324 

2410 

9-171 

7734 

6496 

5534 

4747 

4o8i 

35o4 

2 

3 

55G3 

234i 

9435 

7710 

6478 

5520 

4735 

4071 

3495 

3 

4 
5 

43i4 

2272 

9400 

7686 

646o 

55o6 

4723 

4o6i 

3486 

4 

5 

3.3345 

2.2  2o5 

1.9365 

i.76(i3 

I .6443 

1.5491 

I.47II 

i.4o5o 

1.3477 

6 

2553 

2139 

9331 

7639 

642  5 

5477 

4699 

4o4o 

3460 

6 

7 

i883 

2073 

9296 

7616 

6407 

5463 

4688 

4o3o 

3459 

7 

8 

i3o3 

2009 

9262 

7593 

6390 

5449 

4676 

4o2o 

345o 

8 

9 

lO 

0792 

1946 

9228 

7570 

6372 

5435 

4664 

4oio 

3441 

9 

10 

3.0334 

2.I8S3 

I. 9195 

1.7547 

I .6355 

1. 5421 

1.4652 

1 . 4ooo 

1.3432 

II 

2 .9920 

1822 

9162 

7524 

6338 

5407 

464o 

3989 

3423 

II 

12 

9542 

1 76 1 

9128 

75oi 

6320 

5393 

4629 

3979 

34i5 

12 

i3 

9195 

1701 

9096 

7479 

63o3 

5379 

4617 

3969 

3406 

i3 

i4 
i5 

8873 

1642 

9f.63 

7456 

6286 

5365 

4606 

3959 

3397 

i4 
i5 

2.8573 

2. 1 584 

1. 903 1 

1.7434 

1 .6269 

I.535I 

I .4594 

1 .3940 

1.3388 

i6 

8293 

1 526 

8999 

7412 

6252 

5337 

4582 

3939 

3379 

16 

I? 

8o3o 

1469 

8967 

7390 

6235 

5324 

4571 

3929 

3371 

17 

i8 

7782 

i4i3 

8935 

7368 

6218 

53 10 

4559 

3919 

3362 

18 

J9_ 

20 

7547 

i358 

8904 

7346 

6201 

5296 

4548 

3910 

3353 

19 

20 

2.7324 

2.i3o3 

1.8873 

1.7324 

i.6i85 

1. 5283 

1.4536 

I .3900 

1.3345 

21 

7112 

1249 

8842 

7302 

6168 

5269 

4525 

3890 

3336 

21 

22 

6910 

1 196 

8811 

7281 

6i5i 

5256 

45i4 

388o 

3327 

22 

23 

6717 

1143 

8781 

7259 

6i35 

5242 

45o2 

3870 

3319 

23 

24 
25 

6532 

1091 

8751 

7238 

6118 

5229 

4491 

386o 

33io 

24 

25 

2.6355 

2.104u 

I. 8721 

1. 7217 

I .6102 

I.52I5 

1.4480 

I.3S5I 

I .33oi 

26 

6i85 

O9S9 

8691 

7196 

6oS5 

5202 

4468 

384 1 

3  2  93 

26 

^7 

6021 

0939 

8661 

7175 

6069 

5189 

4457 

383i 

3284 

27 

28 

5863 

0889 

8632 

7i54 

6o53 

5.75 

4446 

3821 

3376 

28 

29 

3o 

5710 

0840 

8602 

7133 

6037 

5i62 

4435 

38i2 

3267 

29 

3o 

2.5563 

2.0792 

1.8573 

I .7112 

1 .6021 

i.5i49 

1.4424 

1 .38o2 

1 .3259 

3i 

5421 

0744 

8544 

7091 

600  5 

5 1 36 

44i2 

3792 

32  5o 

3i 

32 

5283 

069(5 

85i6 

7071 

5989 

5i33 

44oi 

3783 

3242 

32 

3S 

5i49 

0649 

84S7 

7o5o 

5973 

5iio 

4390 

3773 

3233 

33 

34 
35 

5019 

o6()3 

8459 
I.843I 

7o3o 

5957 

5097 

4379 

3764 

3225 

34 
35 

2.4894 

2.0557 

I .7010 

1 .5941 

i.5o84 

1.4368- 

1.3754 

1 .32x6 

3'i 

4771 

05l2 

84o3 

6990 

5925 

^07 1 

4357 

3745 

3208 

36 

37 

^652 

0467 

8375 

6970 

5ooo 

5o58 

4346 

£735 

3199 

37 

38 

4536 

042  a 

8348 

6950 

5894 

5045 

4335 

3726 

3191 

38 

39 

40 

4424 

0378 

8320 

6930 

5878 

5o32 

4325 

3716 

3i83 

39 
40 

2.43i4 

2.0334 

1.8293 

I .6910 

1.5863 

I .5oio 

i.43i4 

1.3707 

I .3174 

4i 

4206 

0391 

8266 

6890 

5847 

50O7 

43o3 

3697 

3 1 66 

4i 

42 

4 1 02 

0248 

8239 

6871 

5832 

4994 

4292 

3688 

3i58 

42 

43 

4ooo 

0206 

8212 

685i 

58i6 

4981 

4281 

36-8 

3i49 

A'i 

44 
45 

3900 

0164 

8i8e 

6832 

58oi 

4969 

4270 

3669 

3j4i 

45" 

2.38o2 

2.0122 

1. 8159 

I. 6812 

1.5786 

1 .4956 

1 .4260 

I . 366o 

1.3:33 

46 

3707 

0081 

8i33 

6793 

5771 

4943 

4249 

365o 

3i24 

46 

47 

36i3 

oo4o 

8107 

6774 

5755 

4931 

4238 

364 1 

3ii6 

47 

48 

3522 

oono 

8081 

6755 

5740 

4918 

4228 

3632 

3io8 

48 

49 
5o 

3432 

1 .  99(10 

8o55 

6736 

5725 

4906 

4217 

3623 

3ioo 

49 
5o 

2.3345 

I .9920 

1 .8o3o 

1.6717 

1 ,5710 

1.4894 

1 .4206 

i.36i3 

I  3091 

5i 

3259 

9881 

8004 

6698 

5695 

4881 

4196 

36o4 

3o83 

5i 

b2 

3i74 

9842 

7979 

6679 

568o 

48(i9 

4i85 

3595 

3075 

52 

53 

3091 

9803 

7954 

666: 

5('-66 

4856 

417^; 

3586 

3u67 

53 

54 
'55 

3oio 

9765 

7929 

6642 

565 1 

4844 

4i64 
i.4i54 

3576 

3o59 
i.3o5i 

54 
55" 

2  2931 

1.9727 

I . 7904 

I .6624 

1.5636 

1. 4832 

I .3567 

56 

2852 

9690 
9652 

7879 

66o5 

5621 

4820 

4i43 

3558 

3o43 

56 

67 

2775 

7855 

6587 

56o7 

4808 

4i33 

3549 

3o34 

!>7 

58 

270C 

9615 

783o 

6568 

5592 

4795 

4l22 

3540 

3026 

58 

S. 

.2626 

9579 

7806 

655o 

5578 

4783 

4lI2 

353i 

3oi8 

59 
S. 

0°  (y 

0°  V 

0°  2' 

0°  ?,' 

0°  4' 

0"    5' 

0°  (V 

0°  7' 

0°  B' 

TABLE  XXIL 

[Pag 

e  133 

Proportional  Logarithms. 

s. 

o 

A  m 

h     m 

h     m 

h     m 

h     VI 

/(  VI 

h     m 

A  m 

A  711 

0°  9' 

0°   10' 

0°  11 

0°  12' 

0°  13' 

0°  14' 

0°  15' 

0°  16' 

0°  17' 

S. 
0 

1 .3(110 

1.2553 

I .2r39 

I .1761 

I  .i4i3 

I. 1091 

I .0792 

I .o5i2 

I .0248 

I 

3on2 

2545 

2l32 

1755 

1 408 

1086 

0787 

o5o7 

0244 

I 

2 

3094 

2538 

2126 

1749 

l402 

108 1 

0782 

o5o2 

0240 

2 

J 

29156 

2  D  3 1 

2119 

1743 

1397 

1076 

0777 

0498 

0235 

3 

4 
5' 

2978 

2524 

2Il3 

1737 

I39I 

1 07 1 

0773 

0493 

023l 

4 
5 

1 . 2970 

i.25i7 

I . 2  1 06 

I.I73I 

1.1386 

1 .1066 

I .0768 

1 .0489 

1 .0227 

6 

2962 

25lO 

2099 

1725 

i38o 

1061 

0763 

o484 

0223 

6 

7 

2954 

25o2 

2093 

1719 

1 374 

io55 

0758 

0480 

0219 

7 

8 

2946 

2495 

20S6 

1713 

i369 

io5o 

0753 

0475 

02l4 

8 

_9_ 

lO 

2939 

2488 

2080 

1707 

1 363 

1045 

0749 
1.0744 

0471 

0210 

9 

10 

1 .?93i 

I .2481 

I .2073 

I .1701 

I.I358 

I . io4o 

I .0467 

I .0206 

II 

29^3 

2474 

2067 

1695 

i352 

io35 

0739 

G.562 

0202 

II 

12 

P915 

2467 

2061 

1689 

1 347 

io3o 

0734 

o458 

0197 

12 

i3 

2907 

2460 

2o54 

1 683 

1 342 

1025 

0730 

0453 

0193 

i3 

i4 
i5 

2899 

2453 

2o48 

1677 

1 336 

1020 

0725 

0449 

0189 

14 
i5 

1 .2S91 

1.2445 

1 .  204 1 

1.1671 

i.t33i 

I .ioi5 

1 .0720 

1 .0444 

I.OI85 

i6 

2883 

2438 

20  3  5 

1 665 

1 325 

1009 

0715 

o44o 

0181 

16 

17 

2876 

243i 

2028 

1660 

1 3  20 

1004 

071 1 

0435 

0176 

17 

i8 

28(18 

2424 

2022 

i654 

i3i4 

0999 

0706 

043 1 

0172 

18 

'9 

20 

2860 

24-17 

2016 

1 648 

1 309 

0994 

0701 

0426 

0168 

''9 
20 

I .2852 

I .2410 

1 .2009 

1 .  1642 

I .i3o3 

I .0989 

I . 0696 

I .0422 

1 .0164 

21 

2845 

24o3 

2003 

i636 

1298 

0984 

0692 

o4i8 

0160 

21 

22 

2837 

2396 

1996 

i63o 

1292 

0979 

0687 

o4i3 

oi56 

22 

2j 

2829 

2389 

1990 

1624 

1287 

0974 

0682 

0409 

oi5i 

^3 

24 
25 

2821 

2382 

1984 

1619 

1282 

0969 

0678 

o4o4 

0147 

24 

25 

1. 2814 

I .2375 

1-1977 

i.i6i3 

1 .1276 

1 .0964 

1 .0673 

I .o4oo 

I .0143 

2b 

2806 

2368 

I97I 

1607 

1271 

0959 

0668 

0395 

0139 

26 

27 

2798 

2362 

1965 

iGoi 

1266 

0954 

o663 

0391 

oi35 

27 

28 

2791 

2355 

1955 

1595 

1200 

0949 

0659 

o387 

oi3i 

28 

29 

3o 

2783 

2348 

1952 

1 589 

1255 

0944 

o654 

o382 

0126 

29 

3o 

1.2775 

1 .2341 

I .1946 

1. 1 584 

1 .1249 

1 .0939 

I .0649 

1.0378 

I .0122 

Ji 

3768 

2334 

1939 

1578 

1244 

0934 

0645 

o374 

0118 

3i 

J  2 

2760 

2327 

1933 

1572 

1239 

0929 

o64o 

0369 

oii4 

32 

ciJ 

2753 

2320 

1927 

1 566 

1233 

0924 

o635 

o365 

0110 

33 

M 
35 

2745 

23i3 

1 92 1 

i56i 

1228 

0919 

o63i 

o36o 

0106 

34 
35 

1.2738 

I .23o7 

I.T9I4 

I.I555 

I .1223 

I. 0914 

1 .0626 

I .o356 

1 .0102 

36 

2730 

23oo 

1908 

1 549 

I217 

0909 

0621 

o352 

0098 

36 

^7 

2722 

2293 

1902 

1 543 

I2I2 

0904 

06 1 7 

o347 

0093 

37 

38 

27t5 

2286 

1896 

i538 

1207 

0899 

06 1 2 

o343 

0089 

38 

39 

40 

2707 

2279 

1889 

i532 

I20I 

0894 

0608 

0339 

008  5 

39 

40 

I .2700 

I .2272 

I.I883 

1.1526 

1  .  I  I  96 

1.08S9 

I .o6o3 

i.o334 

1 .0081 

4i 

2692 

2266 

1877 

I  520 

M91 

0884 

0598 

o33o 

0077 

4i 

^2 

2685 

2259 

1871 

i5i5 

1 1 86 

0880 

0594 

0326 

0073 

42 

43 

2678 

2252 

i865 

1 509 

1 180 

0875 

0589 

032I 

0069 

43 

44 
45 

2670 

2245 

1859 

i5o3 

1175 

0870 

o585 

o3i7 

oo65 

44 
45 

1.2663 

1 .2239 

I.I852 

1. 1498 

1 . 1 1 70 

I.0865 

i.o58o 

I .o3i3 

1 .0061 

46 

2655 

2232 

1 846 

1492 

1164 

0860 

0575 

o3o8 

00  5  7 

46 

47 

2648 

2225 

i84o 

i486 

1 1 59 

o855 

0571 

o3o4 

oo53 

47 

48 

2640 

2218 

i834 

i48i 

ii54 

o85o 

o566 

o3oo 

0049 

48 

49 
5o 

2633 

2212 

1828 

1475 

1 149 

0845 

o562 

0295 

0044 

49 
5o 

I .2626 

I .2205 

1. 1822 

1. 1469 

1. 1 143 

i.o84o 

1.0557 

1 .0291 

1 . 0040 

5i 

2618 

2198 

1816 

1464 

ii38 

c)835 

o552 

0287 

oo36 

5i 

52 

2611 

2192 

1809 

1458 

ii33 

o83i 

o548 

0282 

oo32 

52 

53 

2604 

2i85 

7  8o3 

i452 

1128 

0826 

o543 

0278 

0028 

53 

54 
55 

2596 

2178 

1797 

1447 

I  I  23 

0821 

0539 

0274 

0024 

54 

55' 

1.2589 

r .2172 

1.1791 

1.1441 

1.1H7 

I. 0816 

1.0534 

1.0270 

1 .0020 

56 

2582 

2i65 

1785 

i436 

1112 

081 1 

o53o 

0265 

0016 

56 

5-; 

2  574 

2159 

1779 

i43o 

1107 

0806 

o525 

0261 

0012 

57 

58 

2567 

2l52 

1773 

1424 

1102 

0801 

0D2I 

0257 

0008 

58 

59 
S. 

2  56o 

2r45 

1767 

1419 

1097 

0797 

o5i6 

0252 

ooo4 

59 

S. 

0°  9' 

0°  10' 

0°  11' 

0°  12' 

0°  13' 

0°  14' 

0^  15' 

0°  16' 

0°  17' 

P''g«i34]               TABLE  XXII. 

Proportional  Logarithms. 

S. 

o 
I 

2 

3 
4 
5 
6 

7 
8 

9 

lO 

II 

12 

i3 
i4 
i5 
i6 

17 
i8 

19 
20 
21 
22 

23 

24 

25 

26 
27 
28 
29 

3o 
3i 

32 

33 
34 
35 
35 

37 
38 
39 

40 
4i 
42 
43 

45 
46 
47 
48 

49 
5o 
5i 

52 

53 

54 
55 
56 
57 
58 
59 

S. 

h     m 

0°  18' 

h   m 
0°19' 

h   m 
0°20' 

h    m 
0°21' 

h  rn 
0°22' 

h  m 
0°23' 

h   m 
0°24' 

h   m 
0°25' 

h    VI 

0°26' 

h    m 
0°27' 

h    m 

0°28' 

h    m 
0°29' 

S. 

1 0000 
9996 

9988 
9984 

9765 
9761 
9758 
9754 
9750 

9542 
9539 
9535 
9532 
9528 

9331 
9327 
9324 
9320 
9317 

9128 
9125 
9122 
9119 
9115 

8935 
8932 

^9=? 
8926 
8923 

8751 
8748 
8745 
8742 
8739 

8573 
8570 
8568 
8565 
8562 

84o3 
84oo 
8397 
8395 
8392 

8239 
8236 
8234 
823i 
8228 

8081 
8079 
8076 
8073 
8071 

7929 
7926 
7924 
7921 
7919 

0 
I 
2 
3 
4 
'  5 
6 

8 
9 
10 
II 
12 
i3 
i4 
i5 
16 

17 
18 

19 
20 
21 
22 

23 

24 

25 

26 

27 
28 
29 

3o 
3i 

32 

33 
34 
35 
36 
37 
38 
39 

40 
4i 
42 
43 
A^ 
45 
A(> 
47 
48 

49 
5o 
5i 

52 

53 

54 

99S0 
9976 
9972 
996S 
9964 

9746 
9742 
9739 
9735 
973 1 

9524 
9521 
9517 
95i4 
9510 

93i3 
9310 
9306 
93o3 
9300 

9112 
9109 
9106 
9102 
9099 

8920 
8917 
8913 
8910 
8907 

8904 
8901 
8898 
8895 
8892 

8736 
8733 
8730 
8727 
8724 

8559 
8556 
8553 
855o 
8547 

8389 
8386 
8384 
838 1 
8378 

8226 
8223 
8220 
8218 
8215 

8068 
8066 
8o63 
8061 
8o58 

7916 
7914 
791 1 
7909 
7906 

9960 
9956 
9952 
9948 
9944 

9727 
9723 
9720 
9716 
9712 

9506 
95o3 

9499 
9496 
9492 

9296 
9293 
9289 
9286 
9283 

9096 
9092 
9089 
9086 
9083 

8721 
8718 
8715 
8712 
8709 

8544 
8542 
8539 
8536 
8533 

8375 
8372 
8370 
8367 
8364 

8212 
8210 
S207 
8204 
8202 

8o55 
8o53 
8o5o 
8o48 
8045 

7904 
7901 

7899 
7896 

7894 

9940 
9936 
9932 
9928 
9924 
9920 
9916 
9912 
9908 
9905 

9708 
97o5 
9701 
9697 
9693 

9488 
9485 
9481 
9478 
9474 

9279 
9276 
9272 
9269 
9266 

9079 
9076 
9073 
9070 
9066 

8888 
8885 
8882 
8879 
8876 

8706 
8703 
8700 
8697 
8694 

853o 
8527 
8524 

8522 

85i9 

836i 
8359 
8356 
8353 
835o 

8199 
8196 
8194 
8191 
8188 

8043 
8o4o 
8037 
8o35 
8o32 

7891 
7889 
7887 
7884 
7882 

9690 
9686 
9682 
9678 
9675 

9471 
9467 
9464 
9460 
9456 

9262 
9259 
9255 
9252 
9249 

9063 
9060 
9057 
9053 
90  5  0 

8873 
8870 
8867 
8864 
8861 

8691 
8688 
8685 
8682 
8679 

85i6 
85i3 
85io 
85o7 
85o4 

8348 
8345 
8342 
8339 
8337 

8186 
8i83 
8181 
8 1 78 
8175 

8o3o 
8027 
8025 
8022 
8020 

7879 
7877 
7874 
7872 
7869 

9901 
9897 
9893 
9889 
9885 

9671 
9667 
9664 
9660 
9656 

9453 
9449 
9446 
9442 
9439 

9245 
9242 
9238 
9235 
9232 

9047 
9044 
9041 
9037 
9034 

8857 
8854 
885 1 
8848 
8845 

8676 
8673 
8670 
8667 
8664 

85o2 

8499 
8496 
8493 
8490 

8334 
833 1 
8328 
8326 
8323 

8173 
8170 
8167 
8i65 
8162 

8017 
&oi4 
8012 
8009 
8007 

8004 
8002 

7999 
7997 
7994 

7867 
7864 
7862 
7859 
7857 
7855 
7852 
785o 
7847 
7845 

9881 
9877 
9873 
9869 
9865 

9652 
9649 
9645 
9641 
9638 

9435 
9432 
9428 
9425 
9421 

9418 
9414 
9411 
9407 
9404 

9228 
9225 
9222 
9218 
9215 

903 1 
9028 
9024 
9021 
9018 

8842 
8839 
8836 
8833 
883o 

8661 
8658 
8655 
8652 
8649 

8487 
8484 
8482 

8479 
8476 

8320 
83i8 
83i5 
83i2 
83o9 

8159 
8i57 
8i54 
8i52 
8149 

9861 
9358 
9854 
9850 
9846 

9634 
9630 
9626 
9623 
96'9 

9212 
9208 
9205 
9201 
9198 

9015 
9012 
9008 
9005 
9002 

8827 
8824 
8821 
8817 
8814 

8646 
8643 
864o 
8637 
8635 

8632 
8629 
8626 
8623 
8620 

8473 
8470 
8467 
8465 
8462 

83o7 
83o4 
83oi 
8298 
8296 

8i46 
8i44 
8i4i 
8i38 
8i36 

7992 

7989 
7987 

7984 
7981 

7842 
7840 
7837 
7835 
7832 

9842 
9838 
9S34 
983o 
9827 

9615 
9612 
9608 
9604 
9601 

9400 
9397 
993 
9390 
9386 

9195 
91 91 
9188 
9185 
9181 

9178 
9175 
9171 
9168 
9165 

8999 
8996 
S992 
8989 
8986 
89S3 
8980 

8977 
8973 
8970 

881 1 
8808 
88o5 
8802 
_8799_ 

8796 
8793 
8790 
8787 
8784 

8459 
8456 
8453 
845 1 
8448 

8293 
8290 
8288 
8285 
8282 

8i33 
8i3i 
8128 
8125 
8123 

7979 
7976 

7974 
7971 
7969 

7830 
7828 
7825 
7S23 
7820 

9823 
9819 
9815 
9811 
9807 

9597 
9593 
9590 
9586 
9582 

9383 

9^79 
9376 
9372 
9369 

8617 
86i4 
8611 
8608 
86o5 

8445 
8442 
8439 
8437 
8434 

8279 
8277 
8274 
8271 
8269 

8120 
8117 
8ii5 
8112 
8110 

7966 
7964 
7961 
7959 
7956 

7818 
7815 
7813 
781  r 
7808 

9803 
9800 
9796 
9792 
9788 

9579 
9575 
9571 
9568 
9564 

9365 
9362 
9358 
9355 
9351 

9162 
91 58 
9155 
9152 
9148 

8967 
8964 
8961 
8958 
8954 

8781 
8778 
8775 
8772 
8769 

8602 
8599 
8597 
8594 
8591 

843 1 
8428 
8425 
8423 
8420 

8266 
8263 
8261 
8258 
8255 

8107 
8104 
8102 
8099 
8097 

7954 
7951 

7949 
7946 
7944 

7806 
7803 
7801 

7798 
7796 

9784 
9780 

9777 
9773 
9769 

9561 
9557 
9553 
9550 
9546 

0°19' 

9348 
9344 
9341 
9337 
9334 

9145 
9142 
9 1 38 
9  35 
9132 

8951 
8948 
8945 
8942 
8939 

8766 
8763 
8760 
8757 
8754 

8588 
8585 
8582 
8579 
3576 

8417 
84i4 
84ii 
8409 
8406 

8253 
8250 
8247 
8244 
8242 

8094 
8091 
8089 
8086 
8084 

7941 

7939 
7936 

7934 
7931 

7794 

7791 
7789 
7786 
7784 

55 
56 

57 
58 
59 

0°  18' 

0°20^ 

0°21' 

0°  22'  0°  23' 

0°24' 

0°25' 

0°26' 

0°27' 

0°28'0°29'| 

S. 

TABLP  XXII.               li'='S'='35 
Proportional  Logarithms. 

S. 

o 

I 

2 

3 
4 
5 
6 

7 
8 

9 

10 

II 

12 

i3 
i4 
i5 
i6 
17 
i8 

19 
20 
21 
22 

23 

24 

25 

26 

27 
28 
29 

3o 
3i 

32 

33 
34 
35 
36 
37 
38 
39 

4o 
4i 
42 
43 
A^\ 
45 
46 
47 
48 
49 
5o 
5i 

52 

53 
54 
55 
56 

57 
58 

S, 

h    m 
0°30' 

k    m 
0°  31' 

h    m 
0°  32' 

h    m 
0°  33' 

h    m 
0°34' 

h    m 
0°  35' 

h    m 
0°3G' 

h    m 

0°37' 

h    in 

0°3S' 

h    m 
0°  39' 

h    VI 
0°  40' 

h    m 
0°41' 

S. 

0 

I 
2 
3 
4 
5 
6 

7 
8 

9 

10 
1 1 

12 

i3 
i4 
i5 
16 

17 
18 

19 
20 

21 
22 

23 
24 
25 

26 
27 
28 
29 

3o 
3i 

32 

33 
34 
35 
36 
37 
38 

40 
4i 
42 
43 
44 
45 
46 
47 
48 

49 
5o 
5i 

52 

53 
54 
55 
56 

57 
58 
59  .1 

S. 

7782 

7779 
7777 
7774 
7772 

7639 
7637 
7634 
7632 
763o 

75oi 

7499 
7497 
7494 
7492 

7368 
7365 
7363 
7361 
7359 

7238 
7236 
7234 
7232 
7229 

7112 
7110 
7108 
7106 
7104 

6990 
6988 
6986 
6984 
6982 

6871 
6869 
6867 
6865 
6863 

6755 
6753 
6751 
6749 
6747 

6642 
6640 
6638 
6637 
6635 
6633 
663 1 
6629 
6627 
6625 

6532 
653o 
6529 
6527 
6525 
6523 
6521 
65i9 
65i8 
65i6 

6425 
6423 
6421 
6420 
64i8 
64 16 
64i4 
64i3 
64n 
6409 

7769 
7767 
7765 
7762 
7760 

7627 
7625 
7623 
7620 
7618 

7490 
7488 
7485 
7483 
7481 

7357 
7354 
7352 
7350 
7348 

7227 
7225 
7223 
7221 
7219 

7102 
7100 
7098 
7096 
7093 

6960 
6978 
6976 
6974 
6972 

6861 
6859 
6857 
6855 
6853 

6745 
6743 
6742 
6740 
6788 

7757 
7755 
7753 
775o 
7748 

7616 
7613 
76 1 1 
7609 
7607 

7479 
7476 
7474 
7472 
7470 

7346 
7344 
7341 
7339 
7337 

7217 
7215 
7212 
7210 
7208 

7091 
7089 
7087 
7085 
7083 

6970 
6968 
6966 
6964 
6962 

685 1 
6849 
6847 
6845 
6843 

6736 
6734 
6732 
6730 
6728 

6624 
6622 
6620 
6618 
6616 

65i4 
65i2 
65 10 
65o9 
65o7 

6407 
6406 
64o4 
6402 
6400 

7745 
7743 
774 1 
7738 
7736 

7604 
7602 
7600 

7597 
7595 

7467 
7465 
7463 
7461 
7458 

7335 
7333 
7330 
7328 
7326 

7206 
7204 
7202 
7200 
7198 

7081 
7079 

7077 
7075 
7073 

6960 
6958 
6956 
6954 
6952 

684 1 
684o 
6838 
6836 
6834 

6726 
6725 
6723 
6721 
6719 

66i4 
6612 
661 1 
6609 
6607 

65o5 
65o3 
65oi 
65oo 
6498 

6398 
6397 
6395 
6393 
6391 

7734 
773 1 

7729 
7726 
7724 

7593 
7590 
7588 
7586 
7583 

7456 
7454 
7452 
745o 

7447 

7324 
7322 
7320 
7317 
73i5 

7196 
7193 
7191 
7189 
7187 

7071 
7069 
7067 
7065 
7063 

6950 
6948 
6946 
6944 
6942 

6832 
683o 
6828 
6826 
6824 

6717 
6715 
6713 
67 II 
6709 

66o5 
66o3 
6601 
6600 
6598 

6496 
6494 
6492 

6491 
6489 

6390 
6388 
6386 
6384 
6383 

7722 
7719 
7717 
7714 
7712 

7581 
7579 
7577 
7574 
7572 

7445 
7443 

744 1 
7438 
7436 

73i3 
73ii 
7309 
7307 
73o4 

7i85 
7i83 
7181 
7179 
7177 

7061 
7059 
7057 
7055 
7052 

6940 
6938 
6936 
6934 
6932 

6822 
6820 
6818 
6816 
68 1 4 

6708 
6706 
6704 
6702 
6700 

6596 
6594 
6592 
6590 
6589 

6487 
6485 
6484 
6482 
648o 

633 1 
6379 
6377 
6376 
6374 

7710 
7707 
7705 
77o3 
7700 

7570 
7567 
7565 
7563 
7560 

7434 
7432 
7429 
7427 
7425 

73o2 
73oo 
7298 
7296 
7294 

7175 
7172 
7170 
7168 
7166 

7o5o 
7048 
7046 
7044 
7042 

6930 
6928 
6926 
6924 
6922 

6812 
6810 
6809 
6807 
68o5 

6698 
6696 
6694 
6692 
6691 

6587 
6585 
6583 
658 1 
6579 

6478 
6476 
6475 
6473 
6471 

6372 
6371 
6369 
6367 
6365 

6364 
6362 
636o 
6358 
6357 

7698 
7696 
7693 
7691 
7688 

7558 
7556 
7554 
755 1 
7549 

7423 
7421 
7418 
74i6 
74i4 

7291 
7289 
7287 
7285 
7283 

7164 
7162 
7160 
7i58 
7i56 

7040 
7o38 
7o36 
7034 
7o32 

6920 
6918 
6916 
6914 
6912 

68o3 
6801 
6799 
6797 
6795 

6689 
6687 
6685 
6683 
6681 

6678 
6576 
6574 
6572 
6570 

6469 
6467 
6466 
6464 
646a 

7686 
7684 
7681 
7679 
7677 

7547 
7544 
7542 
7540 
7538 

74 1 2 
7409 
7407 
74o5 
74o3 

7281 

7279 
7276 
7274 
7272 

7i54 
7i52 
7149 
7147 
7145 

7o3o 
7028 
7026 
7024 
7022 

6910 
6908 
6906 
6904 
6902 

6793 
6791 
6789 
6787 
6785 

6679 
6677 
6676 
6674 
6672 

(>5()S 
6567 
6565 
65fi3 
656 1 

()559 
6558 
6556 
6554 
6552 

6460 
6459 
6457 
6455 
6453 

6355 
6353 
635i 
635o 
6348 

7674 
7672 
7670 
7667 
7665 

7535 
7533 
753i 
7528 
7526 

74oi 
7398 
7396 
7394 
7392 

7270 
7268 
7266 
7264 
7261 

7143 
7i4i 
7139 
7137 
7i35 

7020 
7018 
7016 
7014 
7012 

6900 
6898 
6896 
6894 
6892 

6784 
6782 
6780 
6778 
6776 

6670 
6668 
(■6f;6 

6663 

645 1 
645o 
6448 
6446 
6444 

6346 
6344 
6343 
634 1 
6339 

6338 
6336 
6334 
6332 
633i 

7663 
7660 
7658 
7655 
7653 

7524 
7522 
7519 

7517 
75i5 

7390 
7387 
7385 
7383 
738i 

7259 

7257 
7255 
7253 
725i 

7133 
7i3i 
7129 

7127 
7124 

7010 
7008 
7006 
7004 
7002 

6890 
6888 
6886 
6884 
6882 

6774 
6772 
6770 
6768 
6766 

6661 
6659 
6657 
6655 
6653 

655o 
6548 
6547 
6545 
6543 

6443 
644 1 
6439 
6437 
6435 

765i 
7648 
7646 
7644 
7641 

75i3 
75io 
7508 
7506 
75o3 

7379 
7376 

7374 
7372 
7370 

7249 
7246 
7244 
7242 
7240 

7122 
7120 
7118 
7116 
7ii4 

7000 
6998 
6996 
6994 
6992 

6881 
6879 
6877 
6875 
6873 

6764 
6763 
6761 
6759 
6757 

665 1 
665o 
f648 
6646 
6644 

654 1 
6539 
6538 
6536 
6534 

6434 
6432 
643o 
6428 
6427 

6329 

6327 
632  5 
6324 

6322 

0°80' 

0°31' 

0°  3-2' 

0°  33'|0°  34 

0°35' 

0°  3G'iO°  37' 

0°38' 

0°  39'i0°  40' 

0°  41' 

i'^?-  '3^]               TABLE  XXII. 

Proportional  Logarithms. 

S. 

0 

h   m  !  Ii   111 

h   m 

A  m 

h  VI   1  h  m 

h    m 

h   m 

h   m 

h   m 

h     VI 

h    m 

0°  42'  0^  43' 

0044/ 

0°45' 

0°  4G'  0°  47' 

0°48' 

0°49' 

0°50 

0°51' 

0=52' 

0°53' 

S. 
0 

6320 

6218 

6118 

602  X 

5925 

5832 

5740 

565 1 

5563 

5477 

5393 

53x0 

I 

63i9 

6216 

6xx7 

6019 

5924 

583o 

5739 

5649 

5562 

5476 

5391 

5309 

I 

2 

63i7 

6215 

6ix5 

6017 

5922 

5829 

^737 

5648 

556o 

5474 

5390 

53o7 

2 

3 

63 1 5 

6213 

6ii3 

6016 

5920 

5827 

5736 

5646 

5559 

5473 

5389 

53o6 

3 

4 
5 

63i3 

6211 

6x12 

6ox4 

5919 

5826 

5734 

5645 

5557 

5471 

5387 
5386 

53o5 
53o3 

5 

63 1 2 

6210 

6110 

60 1 3 

5917 

5824 

5733 

5643 

5556 

5470 

6 

63io 

6208 

6108 

60  X  I 

5916 

5823 

573  X 

5642 

5554 

5469 

5384 

53o2 

6 

7 

63oS 

6206 

6107 

6009 

5914 

5821 

5780 

564o 

5553 

5467 

5383 

53oo 

7 

8 

63o6 

6205 

6io5 

6008 

59x3 

58x9 

5728 

5639 

555i 

5466 

5382 

5299 

8  > 

9 

10 

63o5 

6203 

6io3 

6006 

59XX 

58x8 

5727 

5637 

555o 

5464 
5463 

538o 

5298 
"5l^6 

9  1 

in   1 

63o3 

6201 

6102 

6oo5 

5909 

58i6 

5725 

5636 

5549 

5379 

II 

63oi 

6200 

6100 

6oo3 

5908 

58x5 

5724 

5635 

5547 

546x 

5377 

5295 

1 1  ; 

12 

63oo 

6198 

6099 

6001 

5906 

58x3 

5722 

5633 

5546 

5460 

5376 

5294 

X2 

i3 

6298 

6196 

6097 

6000 

5905 

58x2 

5721 

5632 

5544 

5459 

5375 

5292 

i3 

i4 
i5 

6296 

6195 

6095 

5998 
5997 

5908 

58io 

57x9 

563o 

5543 

5457 

5373 

529X 
5290 

1 4 
x5 

6?94 

6193 

6094 

5902 

5809 

57x8 

5629 

554i 

5456 

5372 

i6 

6293 

6191 

6092 

5995 

5900 

58o7 

5716 

5627 

5540 

5454 

5370 

5288 

x6 

17 

6291 

6190 

6090 

5993 

58q8 

58o6 

57x5 

5626 

5538 

5453 

5369 

5287 

17 

i8 

6289 

6188 

6089 

5992 

5897 

58o4 

57x3 

5624 

5537 

5452 

5368 

5285 

18 

19 
20 

6288 

6186 

6087 
6o85 

5990 

5895 

58c3 

57x2 

5623 

5536 

5450 

5366 

5284 
5283 

20 

62S6 

6i85 

5989 

5894 

58ox 

57!0 

5621 

5534 

5449 

5365 

21 

6284 

6i83 

6084 

5987 

5892 

58oo 

5709 

5620 

5533 

5447 

5364 

5281 

21 

22 

6282 

6181 

6082 

5985 

5891 

5798 

5707 

56x8 

553  X 

5446 

5362 

5280 

22 

23 

62S1 

6179 

6081 

5984 

5SS9 

5796 

5706 

56x7 

553o 

5445 

536i 

5279 

23 

24 

25 

6279 

6178 

6079 

5982 

5888 

5795 

5704 

56x5 

5528 

5443 
5442 

535o 
5358 

5277 
5276 

24 

25 

6277 

6176 

6077 

5981 

5886 

5793 

5703 

56i4 

5527 

26 

6276 

6174 

6076 

5979 

5884 

5792 

5701 

56x3 

5526 

5440 

5357 

5275 

2/j 

27 

6274 

6173 

6074 

5977 

5883 

5790 

5700 

56ix 

5524 

5439 

5355 

5278 

•-7 

28 

6272 

6171 

6072 

59-b 

588 1 

5789 

5698 

56x0 

5523 

5437 

5354 

5272 

28 

29 

3o 

6271 

6169 

6071 

5974 

588o 

5787 

5697 

56o8 

5521 
5520 

5436 

5353 

D271 
5269 

29 
3o  ■ 

6269 

6168 

6069 

5973 

5878 

5786 

5695 

5607 

5435 

535x 

3i 

6267 

6166 

6067 

5971 

5877 

5784 

5694 

56o5 

55x8 

5433 

535o 

5268 

3i 

32 

6265 

6i65 

6066 

5969 

5875 

5783 

5692 

56o4 

55x7 

5432 

5348 

5266 

32 

33 

6264 

6i63 

6064 

5968 

5874 

5.78  X 

5691 

56o2 

55x6 

543o 

5347 

5265 

33 

34 
35 

6262 

6161 

6o63 

5966 

5S72 

5780 

5689 

56ox 

55x4 

5429 

5346 

5264 

34 
35 

6260 

6160 

6061 

5965 

5S70 

5778 

5688 

5599 

55x3 

5428 

5344 

5262 

36 

62  5g 

61 58 

6059 

5963 

5869 

5777 

5686 

5598 

55x1 

5426 

5343 

5261 

36 

37 

6257 

6i56 

6o58 

596  X 

5867 

5775 

5685 

5596 

55x0 

5425 

534  X 

5260 

37 

38 

6255 

6i55 

6o56 

5960 

5866 

5774 

5683 

5595 

55o8 

5423 

5340 

5258 

38 

39 
40 

6254 

6x53 

60  5  5 

5958 

5864 

5772 

5682 

5594 

55o7 

5422 

5339 

5257 

39 

40 

6252 

6i5i 

6o53 

5957 

5863 

5771 

568o 

5592 

55o6 

542  X 

5337 

5256 

4i 

6250 

6i5o 

6o5x 

5955 

586i 

5769 

5679 

559X 

55o4 

54x9 

5336 

5254 

4i 

42 

6248 

6i48 

6o5o 

5954 

586o 

5768 

5677 

5589 

55o3 

54x8 

5335 

5253 

42 

43 

6247 

6i46 

6o48 

5952 

5858 

5766 

5676 

5588 

55oi 

5416 

5333 

5252 

Ai 

45 

6245 

6145 

6046 

5950 

5856 

5765 

5674 

5586 

55oo 

54x5 
54x4 

5332 

525o 

AA 
45 

6243 

6143 

6045 

5949 

5855 

5763 

5673 

5585 

5498 

533i 

5249 

46 

6242 

6i4i 

6043 

5947 

5853 

5761 

5671 

5583 

5497 

54x2 

5329 

5248 

4b 

47 

6240 

6i4o 

6042 

5946 

5852 

5760 

5670 

5582 

5496 

54x1 

5328 

5246 

47 

48 

6233 

6x38 

6n4o 

5944 

585o 

5758 

5669 

558o 

5494 

5409 

5326 

5245 

48 

49 
5o 

6237 

6x36 

6o38 
6037 

5942 

5849 

5757 

5667 

5579 

5493 

5408 

5325 

5244 

49 

5o 

6235 

6x35 

5941 

5847 

5755 

5666 

5578 

5491 

5407 

5324 

5242 

5 1 

6233 

6x33 

6o35 

5939 

5846 

5754 

5664 

5576 

5490 

54o5 

5322 

524x 

5x 

52 

6232 

6x3x 

6o33 

593s 

5844 

5752 

5663 

5575 

5488 

54o4 

5321 

5240 

52 

53 

6230 

6i3o 

6o32 

5936 

5843 

575x 

566  X 

5573 

5487 

5402 

5320 

5238 

53 

54 
55 

6228 

6x28 

60  3o 

5935 

584 1 

5749 

566o 

5572 

5486 

540  X 
5400 

53x8 

5237 

54 
55 

6226 

6126 

6029 

5933 

5839 

5748 

5653 

5570 

5484 

53x7 

5235 

56 

6225 

6x25 

6027 

5931 

5838 

5746 

5657 

5569 

5483 

5398 

53x5  5234 

56 

57 

6223 

6x23 

6025 

5930 

5836 

5745 

5655 

5567 

5481 

5397 

53i4 

5233 

57 

53 

6221 

6l2X 

6024 

5928 

5835 

5743 

5654 

5566 

5480 

5395 

53x3 

523i 

58 

,  59 

S. 

6220 
0°42' 

6x20 

6022 

5927 

5833 

5742 

5652 

5564 

5478 

5394 

53x1 

523o 

S. 

0°43' 

0°44' 

0°  45' 

0°  4G'!0°  47' 

0°48' 

0°  4u'  0°  50' 

0°  51'|0°  52' 

0°53' 

TABLE  XXII.                [!'' 

.0.137 

Proportional  Logarithms. 

S. 

o 

h   m 

k   m 

/(  'm 

h    w 

ll    VI 

It  m 

h   m 

h    m  \  li   m 

h   m 

/i   m 

h    m 

0°54' 

0°55' 

0°56' 

0"57' 

4994 

0°58' 
4918 

O''  5!)' 

4844 

1°0' 

1°1' 

1^2' 

1°3' 

104/ 

1°5' 

S. 
0 

5229 

5i49 

5071 

4771 

4699 

4629 

4559 

4491 

4424 

I 

5227 

5i48 

5070 

4993 

4917 

4843 

4770 

4698 

4628 

4558 

4490 

4422 

I 

2 

5226 

5i46 

5o68 

4991 

491(3 

4842 

4769 

4697 

4626 

4557 

4489 

442  1 

2 

3 

5225 

5i45 

5067 

4990 

491^ 

484 1 

4768 

4696 

4625 

4556 

4488 

4420 

3 

4 
5 

5223 

5 144 

5ob6 

4989 

4913 

4839 

4766 

4695 

4624 

4555 

4486 

4419 

4 
5' 

5222 

5r43 

5o64 

4988 

4912 

4838 

4765 

4693 

4623 

4554 

4485 

4418 

6 

5221 

5i4i 

5o63 

4986 

491 1 

4837 

4764 

4692 

4622 

4552 

4484 

4417 

6 

7 

5219 

5i4o 

5062 

4985 

4910 

4836 

4763 

4691 

4C21 

455i 

4483 

4416 

7 

8 

5218 

5i39 

5o6i 

4984 

4908 

4834 

4762 

4690 

4619 

455o 

4482 

44i5 

8 

9 

lO 

5217 

5i37 

5o59 

49S3 

^907 

4833 

4760 

4689 

4618 

4549 
4548 

4481 

44j4 

10 

52i5 

5i36 

5o58 

4981 

49116 

4832 

4759  1  4688 

4617 

4480 

44^^- 

II 

52i4 

5i35 

5o57 

4980 

4905 

483 1 

4758 

4686 

46 1 6 

4547 

4479 

44 1 1 

II 

12 

52i3 

5 1 33 

bob:j 

4979 

4903 

48  3o 

4757 

4685 

46i5 

4546 

4477 

44io 

12 

iJ 

52II 

5i32 

5o54 

4977 

4902 

4828 

47^6 

4684 

46i4 

4544 

4476 

4409 

i3 

i4 
i5 

5210 

5i3i 

5o53 

4976 

4901 

4827 

4754 

4683 

4612 

4543 

4475 

44o8 

i4 
i5 

5209 

5129 

5o5i 

4975 

4900 

4826 

4753 

4682 

4611 

4542 

4474 

4407 

i6 

5207 

5 1 28 

5o5o 

4974 

4899 

4825 

47^2 

46So 

4610 

4541 

4473 

4406 

16 

17 

5206 

5i27 

5o49 

4972 

4897 

4823 

475i 

4679 

4609 

4540 

4472 

44o5 

17 

i8 

52o5 

5i25 

5o48 

4971 

4896 

4822 

475o 

4678 

4608 

4539 

4471 

4404 

18 

19 

20 

52o3 

5i24 

5o46 

4970 

4895 

4821 

4748 

4677 

4607 

4'3d\i 

4469 
4468 

44o2 
4401 

'9 

20 

5202 

5i23 

5o45 

4969 

4894 

4820 

4747 

4676 

4606 

4536 

21 

5201 

5l22 

5o44 

4967 

4892 

4819 

4746 

4675 

46o4 

4535 

4467 

44oo 

21 

22 

5.99 

5l20 

5o43 

4966 

4891 

4817 

4745 

4673 

46o3 

4534 

4466 

4399 

22 

23 

5198 

5ii9 

5o4i 

4965 

4890 

48i6 

4744 

4672 

4602 

4533 

4465 

4398 

2  3 

24 
25 

5i97 

5ri8 

5o4o 

4964 

4889 

48 1 5 

4742 

4671 

4601 

4532 
453i 

4464 
4463 

4397 

24 

25 

5195 

5ii6 

5o39 

4962 

4887 

48 1 4 

4741 

4670 

4600 

4396 

25 

5194 

5ii5 

5o37 

4961 

4886 

4812 

4740 

4669 

4599 

453o 

4462 

4395 

26 

27 

5193 

5ii4 

5o36 

4960 

4885 

4S11 

4739 

4668 

4597 

4528 

4460 

4394 

27 

28 

5191 

5i  12 

5o35 

4959 

4884 

48 10 

4738 

4666 

4596 

4527 

4459 

4393 

28 

29 

3o 

5190 

5iii 

5()34 

49!)7 

4882 

4809 

4736 

4665 

4595 

4526 

4458 

4391 

29 

3o 

5189 

5iio 

5o32 

4956 

488i 

4808 

4735 

4664 

4594 

4525 

4457 

4390 

3i 

6187- 

5io8 

5o3i 

49'j5 

488o 

4806 

4734 

4663 

4593 

4024 

4456 

4389 

3i 

32 

5i86 

5i07 

5o3o 

4954 

4S79 

48o5 

4733 

4662 

4592 

4523 

4455 

4388 

32 

33 

5i85 

5io6 

5028 

4952 

4877 

48o4 

4732 

4660 

4590 

4522 

4454 

4387 

33 

35 

5i83 

5io5 

5027 

4951 

4876 

48o3 

4730 

4659 

4589 

4520 

4453 

4386 

34 
35 

5182 

5io3 

5026 

4950 

4875 

4801 

4729 

4658 

4588 

45i9 

4452 

4385 

36 

5i8i 

5l02 

5o2  5 

4949 

4874 

4S00 

4728 

4657 

4587 

45i8 

445o 

4384 

36 

37 

5i79 

5ioi 

5o23 

4947 

4873 

4799 

4727 

4656 

4586 

45i7 

4449 

4383 

37 

38 

5178 

5099 

5022 

4946 

4871 

4798 

4726 

4655 

4585 

45i6 

4448 

438 1 

38 

39 

4o 

5i77 

5098 

5o2i 

4945 
4943 

4870 

4797 

4724 

4653 

4584 

45i5 
45i4 

4447 
4446 

438o 
43^ 

39 

40 

5175 

5097 

5019 

4869 

4795 

4723 

4652 

4582 

4i 

5i74 

5095 

5oi8 

4942 

4668 

4794 

4722 

465 1 

458i 

45i2 

4445 

4378 

4i 

42 

5.73 

5094 

5oi7 

4941 

4866 

4793 

4721 

465o 

458o 

45ii 

4444 

4377 

42 

43 

5172 

5093 

5oi6 

4940 

4865 

4792 

4720 

4649 

4579 

45io 

444'i 

4376 

43 

45 

5170 

5092 

5oi4 

4938 
4937 

4864 
4863 

4791 
4789 

4718 

4648 

4578 

4509 
45o8 

444 1 

4375 

44 
45 

5169 

5090 

5oi3 

4717 

4646 

4577 

4440 

4374 

46 

5 1 68 

50S9 

5oi2 

4936 

4861 

4788 

4716 

4645 

4575 

45o7 

4439 

4373 

46 

47 

5r66 

5o88 

5oii 

4935 

4860 

4787 

47i5 

4644 

4574 

45u6 

4438 

4372 

47 

48 

5 1 65 

5o86 

5009 

4933 

4859 

4786 

4714 

4643 

4573 

45o5 

4437 

4370 

48 

49 

5o 

5 1 64 

5o85 

5oo8 

4932 

4858 

4785 
4783 

4712 

4642 

4572 

45o3 

4436 

4369 
4368 

49 
5o 

5162 

5o84 

5007 

4931 

4856 

471 1 

464o 

4571 

45o2 

4435 

5i 

5i6i 

5082 

5oo5 

4930 

4855 

4782 

4710 

4639 

4570 

45oi 

4434 

4367 

5 1 

52 

5 1 60 

5o8i 

5oo4 

4928 

4854 

4781 

470Q 

4638 

4569 

45oo 

4433 

4366 

52 

53 

5i58 

5o8o 

5oo3 

4927 

4853 

4780 

4708 

4637 

4567 

4499 

443 1 

4365 

53 

54 
55 

5i57 

5079 

5o02 

4926 

4852 

4778 

4707 

4705 

4636 

4566 

4498 

443o 

4364 

54 
55 

5i56 

5o77 

5ooo 

4925 

485o 

4777 

4635 

4565 

4497 

4429 

4363 

56 

5i54 

5076 

4999 

4923 

4849 

4776 

4704 

4633 

A'M 

4495 

4428 

4362 

56 

57 

5i53 

50-75 

4998 

4922 

4848 

4775 

4703 

4632 

4563 

4494 

4427 

436i 

57 

58 

5 1 52 

5073 

4997 

4921 

4847 

4774 

4702 

463 1 

4562 

4493 

4426 

4359 

58 

59 
S. 

5i5o 

5072 

4995 

4920 

4845 

4772 

4701 

463o 

456o 

4492 

4425 

4358 

59 
S. 

0^^  54' 

0°  5.^' 

0°  56' 

0°  57' 

0°  58' 10°  59' 

1°0' 

1°I' 

1°2' 

1°3' 

1°4' 

i°5' 

IH 


r»g«i38j               TABLE  XXII. 

Proportional  Logarithms. 

S. 
o 

h  m 

k  m 

h  m 

h   m 

A  m 

h  m 

h  m 

h   m 

h  m 

h  m 

h  VI 

h  m 

F6' 

1°7' 

1°8' 

1°9' 

1°10' 

1°11' 

1°12' 

1°  13' 

1°14' 

1°15 

1°1G' 

V  IT 

S. 
0 

4357 

4292 

4228 

4i64 

4102 

4o4o 

3979 

3919 

386o 

38o2 

3745 

3688 

I 

4356 

4291 

4227 

4i63 

4ioi 

4039 

3978 

3919 

3859 

38oi 

3744 

8687 

I 

2 

4355 

4290 

4226 

4162 

4ioo 

4o38 

3977 

3918 

3858 

38oo 

3743 

3686 

2 

3 

4354 

4289 

4224 

4i6i 

4099 

4o37 

3976 

3917 

3857 

3799 

3742 

3685 

3 

4 
5 

4353 

4288 

4223 

4i6o 

4098 

4o36 

3975 

3916 

3856 

3798 

3741 

3684 

4 
5 

4352 

4287 

4222 

4i59 

4097 

4o35 

3974 

3915 

3856 

3797 

3740 

3683 

6 

435i 

4285 

4221 

4i58 

4096 

4o34 

3973 

3914 

3855 

3796 

3739 

8682 

6 

7 

435o 

4284 

4220 

4i57 

4095  '  4o33 

3972 

3913 

3854 

3795 

3788 

368 1 

7 

8 

4349 

4283 

4219 

4i56 

4093 

4o32 

3971 

3912 

3853 

3794 

3787 

368o 

8 

9 

lO 

4347 

4282 

4218 

4i55 

4092 

4o3i 

3970 

391 1 

3852 

3793 

3786 
8735 

8679 
8678 

9 

10 

4346 

4281 

4217 

4i54 

4091 

4o3o 

3969 

3910 

385i 

3792 

1 1 

AM^ 

4280 

4216 

4i53 

4090 

4029 

3968 

3909 

385o 

3792 

3734 

3677 

II 

12 

4344 

4279 

42i5 

4i52 

.4089 

4028 

3967 

3908 

3849 

3791 

8733 

8677 

12 

i3 

4343 

4278 

4214 

4i5i 

4088 

4027 

3966 

3907 

3848 

3790 

8782 

8676 

i3 

i4 
i5 

4342 

4277 

42i3 

4i5o 

4087 

4026 

3965 

3906 

3847 

3789 

8781 

8675 
8674 

i4 
i5 

4341 

4276 

4212 

4149 

4o86 

4025 

3964 

3905 

3846 

3788 

8780 

i6 

4340 

4275 

4211 

4i47 

408  5 

4024 

3963 

3904 

3845 

3787 

3729 

8673 

16 

17 

4339 

4274 

4210 

4i46 

4o84 

4o23 

3962 

3903 

3844 

3786 

8728 

8672 

17 

[8 

4338 

4273 

4209 

4i45 

4o83 

4o22 

3961 

3902 

3843 

3785 

8727 

8671 

18 

19 
20 

4336 

4271 

4207 

4i44 

4082 

402I 

3960 

3901 

3842 

3784 

8727 

8670 

19 
20 

4335 

4270 

4206 

4i43 

4081 

4020 

3959 

3900 

384 1 

3783 

8726 

8669 

21 

4334 

4269 

42o5 

4i42 

4080 

4019 

3958 

3899 

384o 

3782 

8725 

3668 

21 

22 

4333 

4268 

4204 

4i4i 

4079 

4018 

3957 

3898 

3839 

3781 

8724 

8667 

22 

23 

4332 

4267 

4203 

4i4o 

4078 

4017 

3956 

3897 

3838 

3780 

3728 

3666 

23 

24 

25 

433 1 

4266 

4202 
4201 

4i39 

4077 

4oi6 

3955 

3896 

3837 

3779 

8722 

8665 

24 

25 

433o 

4265 

4i38 

4076 

4oi5 

3954 

3895 

3336 

3778 

8721 

8664 

26 

4329 

4204 

4200 

4i37 

4075 

4oi4 

3953 

3894 

3835 

3777 

0720 

3663 

26 

27 

4328 

4263 

4199 

4i36 

4074 

4oi3 

3952 

3893 

3834 

3776 

8719 

3663 

27 

28 

4327 

4262 

4i9« 

4i35 

4073 

40I2 

3951 

3892 

3833 

3775 

8718 

3662 

28 

29 

3o 

4326 

4261 

4197 

4i34 

4072 

4oii 

3950 

3891 

3832 

3774 

8717 

8661 

29 

3o 

432  3 

4260 

4196 

4i33 

4071 

4oio 

3949 

3890 

383 1 

3773 

8716 

366o 

3i 

4323 

4259 

4195 

4i32 

4070 

4009 

3948 

3889 

383o 

3772 

3715 

3659 

3i 

32 

4322 

42  58 

4194 

4i3i 

4069 

4008 

3947 

3888 

3829 

3771 

3714 

3658 

32 

33 

4321 

4256 

4193 

4i3o 

4068 

4007 

3946 

3887 

3828 

3770 

8718 

3657 

3^ 

34 
35 

4320 
4319 

4255 

4192 

4129 

4067 

4oo6 

3945 

3886 

3827 

3769 
8768 

8712 
8711 

3656 

34 
35 

4254 

4191 

4128 

4066 

4oo5 

3944 

3885 

3826 

3655 

:ib 

43i8 

4253 

4189 

4127 

4o65 

4oo4 

3943 

3884 

382  5 

3768 

8710 

3654 

36 

37 

43i7 

425a 

4i88 

4126 

4o64 

4oo3 

3942 

3883 

3824 

3767 

8709 

3653 

37 

38 

43i6 

425i 

4187 

4i25 

4o63 

4002 

3941 

3882 

3823 

3766 

3709 

3652 

38 

39 
4o 

43i5 

42  5o 

4i86 
4i85 

4124 

4062 

4001 

3940 

388i 

3822 

3821 

3765 

8708 

365i 

39 

40 

43i4 

4249 

4l22 

4o6i 

4ooo 

3939 

388o 

3764 

8707 

365o 

41 

43i3 

4248 

4i84 

4l2I 

4o6o 

3999 

3938 

3879 

3820 

3763 

8706 

3649 

4i 

42 

43ii 

4247 

4i83 

4l20 

4059 

3998 

3937 

3878 

3820 

3762 

3705 

3649 

42 

43 

43io 

4246 

4182 

4II9 

4o58 

3997 

3936 

387^ 

3819 

3761 

3704 

3648 

Ai 

44 
45 

4309 

4245 

4i8i 

4ri8 

4o56 

3996 

3935 

3876 

38i8 

3760 

8708 

3647 

44 
45 

43o8 

4244 

4 1 80 

4117 

4o55 

3995 

3934 

3875 

3817 

3759 

3702 

3646 

4t) 

4to7 

4243 

4179 

4116 

4o54 

3993 

3933 

3874 

38i6 

3758 

8701 

3645 

46 

47 

43o6 

4241 

4i7« 

4ii5 

4o53 

3992 

3932 

3873 

38i5 

3757 

8700 

3644 

47 

48 

43o5 

4240 

4i77 

4ii4 

4o52 

3991 

3931 

3872 

38i4 

3756 

8699 

3643 

48 

49 
5o 

43o4 

4239 

4176 

4ii3 

4o5i 

3990 

3930 

3871 

38i3 

3755 

8698 

3642 

49 
5o 

43o3 

4238 

4175 

4lI2 

4o5o 

3989 

3920 

3870 

38i2 

3754 

8697 

364 1 

5i 

43o2 

4237 

4174 

4iii 

4o49 

3988 

3928 

3869 

38ii 

3753 

8696 

364o 

5i 

5 1 

43oi 

4236 

4173 

4iio 

4o48 

3987 

3927 

3868 

38io 

3752 

8695 

8689 

52 

53 

43cc  I  4235 

4172 

4109 

4o47 

3986 

3926 

3867 

3809 

375i 

8694 

8638 

53 

54 
55 

429S 

42J4 
4233 

4171 

4 1 08 

4o46 

3985 

3925 

3866 

38o8 

3750 

8698 

8687 

54 
55 

4297 

4169 

4107 

4045 

3984 

3924 

3865 

3807 

374Q 

8698 

3636 

56 

4296 

4232 

4168 

4 1 06 

4044 

39S3 

3923 

3864 

38o6 

3748 

8692 

3635 

56 

57 

4295 

423 1 

4167 

4io5 

4043 

3982 

3922 

3863 

38o5 

3747 

8691 

8635 

57 

58 

4294 

423o 

4i66 

4io4 

4042 

3981 

8921 

3862 

38o4 

3746 

8690 

3634 

58 

59 
S. 

4293 

4229 

4i65 

4io3 

4o4i 

3980 

3920 

386i 

38o3 

3746 

3689 

3638 

59 
S. 

1°(3' 

1°7' 

1°8' 

1°9' 

1°10' 

Pll' 

1°12' 

1°  13' 

1°14' 

1  15' 

1°  16'1°17'| 

TABLE  XXII.               iP^^ge  139 
Proportional  Logarithms. 

S. 

o 
I 

2 

3 
4 
5 
6 

7 
8 

9 

10 

1 1 

12 

i3 
i4 
i5 
i6 

17 
i8 

_L9_ 

20 
31 
22 
23 
24 
25 
26 

^7 
28 
29 

3o 
3i 

32 

33 

34 
35 
36 
37 
38 

39 
40 
4i 
42 
43 
44 
'  45 
46 

47 

48 
49 
5o 
5i 

52 

53 
54 
55 
56 

57 
58 

J9_ 

S. 

h    m 

P18' 

h    m 

1°19' 

h  m 
1°20' 

h  m 
1°21' 

h  m 
1°  22'' 

h  m 
1°23' 

h  vi 
I°24' 

h    m 
1°25' 

h  m 
1°26' 

h    m 
1°27' 

h    m 

1°28' 

h    m 
1°  29' 

S. 

0 
I 
2 
3 
4 
5 
6 

8 

10 
1 1 
12 

]3 
i4 
i5  . 
16 

'7 
18 

19 
20 
21 
22 

23 

24 

25 

26 
27 
28 
29 

3o 
3i 

32 

33 
34 
35^ 
36 

37 
38 
39 

40 
4i 
42 
43 

44 
45 
46 
47 
48 

49 
5o 
5i 

52 

53 

54 
55 
56 
57 
58 
69 

S. 

3632 
363 1 
363o 
3629 
362S 

3576 
3576 
3575 
3574 
3573 

3522 

3521 
3520 
3519 
35i8 

3468 
3467 
3466 
3465 
3464 

34i5 
34i4 
34i3 
3412 
3411 

3362 
336i 
336o 
3359 
3358 

33io 
3309 
33o8 
33o7 
33o6 
33o6 
33o5 
33o4 
33o3 
33o2 

3259 
3258 
3257 
3256 
3255 

3208 
3207 
3206 
32o5 
32o4 

3i58 
3,57 
3i56 
3 1 55 
3i54 
3i53 
3i53 
3i52 
3i5i 
3i5o 

3io8 
3i07 
3 106 
3io5 
3io5 

3io4 
3io3 
3 102 
3ioi 
3ioi 

3o59 
3o58 
3o57 
3o56 
3o56 

3o55 
3o54 
3o53 
3o52 
3o52 

3627 
3626 
3625 
3624 
3623 

3572 
3571 
3570 
3569 
3568 

35i7 
35i6 
35i5 
35i5 
35i4 

3463 
3463 
3462 
3461 
3460 

34io 
3409 
34  o8 
34d8 
3407 

3358 
3357 
3356 
3355 
3354 

3254 
3253 
3253 

3252 

325i 

32o4 
32o3 

3202 
3201 

3200 

3623 
3622 
3621 
3620 
3619 

3567 
3566 
3565 
3565 
3564 

35i3 
35i2 
35ii 
35io 
3509 

3459 
3458 
3457 
3456 
3455 

3406 
34o5 
3404 
34o3 
3402 

3353 
3352 
335i 
335i 
335o 

33oi 
33oo 
33oo 

325o 
3249 
3248 
3247 
3247 

3'99 
3198 
3198 
3197 
3196 

3i49 
3i48 
3i48 
3i47 
3 1 46 

3i45 
3i44 
3i43 
3i43 
3i42 

3 1 00 
3099 
3098 
3097 
3096 
3096 
3095 
3094 
3093 
3092 

3o5i 
3o5o 
3o49 
3o48 
3o47 

3o47 
3o46 
3  04  5 
3o44 
3o43 

36i8 
36i7 
36 16 
36i5 
36i4 

3563 
3562 
356 1 
356o 
3559 

35o8 
3507 
35o6 
35o6 
35o5 

3454 
3454 
3453 
3452 
345 1 

3401 
3400 
3400 
3399 
3398 

3349 
3348 
3347 
3346 
3345 

3297 
3296 
3295 
3294 
3294 

3246 
3245 
3244 
3243 
3242 

3195 
3194 
3193 
3193 
3192 

36i3 
36i2 
36ii 
36 10 
36 10 

3558 
3557 
3556 
3555 
3555 

35o4 
35o3 
35o2 
35oi 
35oo 

345o 
3449 
3448 
3447 
3446 

3397 
3396 
339J 
3394 
3393 

3345 
3344 
3343 
3342 
334i 

3293 
3292 
3291 
3290 
3289 

3288 
3288 
3287 
3286 

3285 

3242 
3241 
3240 
3239 
3238 

3237 
3236 
3236 
3235 
3234 

3191 
3190 
3189 
3i88 
3i88 

3187 
3i86 
3i85 
3]84 
3i83 

3i4i 
3i4o 
3i39 
3i38 
3i38 

8091 
3091 
3090 
3089 
3o88 

3o43 
3o42 
3o4i 
3o4o 
3089 

3609 
36o8 
3607 
36o6 
36o5 

3554 
3553 
3552 
355i 
355o 

3499 
3498 
3497 
3497 
3496 

3446 
3445 
3444 
3443 
3442 

3393 
3392 
3391 
3390 
3389 

3340 
3339 
3338 
3338 
3337 

3i37 
3i36 
3i35 
3i34 
3i33 

3i33 
3i32 
3i3i 
3i3o 
3129 

3087 
3087 
3o86 
3o85 
3o84 
3o83 
3082 
3082 
3o8i 
3o8o 

8089 
3o38 
3o37 
3o36 
3o35 
3o34 
3o34 
3o33 
3o32 
3o3i 

36o4 
36o3 
36o2 
36oi 
36oo 

3549 
3548 
3547 
3546 
3545 

3495 
3494 
3493 
3492 
3491 
3490 
3489 
3488 
3488 
3487 

3441 
3440 
3439 
3438 
3438 

3388 
3387 
3386 
3386 
3385 

3336 
3335 
3334 
3333 
3332 

3284 
3283 
3282 
3282 
3281 

3233 

3232 

323i 
323i 
323o 

3i83 
3i82 
3i8i 
3i8o 
3179 

3599 
359S 
3598 

3597 
3596 

3545 
3544 
3543 
3542 
3541 

3437 
3436 
3435 
3434 
3433 

3384 
3383 
3382 
338r 
338o 

3332 
333i 
333o 
3329 
3328 

3280 
3279 
3278 
3277 
3276 

3229 
3228 
3227 
3226 

32  25 

3178 
3178 
3i77 
3176 
3.75 

3129 
3i28 
3i27 
3i26 
3i25 

8079 
3078 
3078 
3077 
3076 

3o3o 
3o3o 
8029 
8028 
8027 

8026 
3o?6 
3o25 
3o24 
3o23 

3595 
3594 
3593 
3592 
3591 

3540 
3539 
3538 
3537 
3536 

3486 
3485 
348  i 
3483 
3482 

3432 
343 1 
343 1 
343o 
3429 

3379 
3379 
3378 

3377 
3376 

3327 
3396 
3325 
3325 
3324 

3276 
3275 
3274 
3273 
3272 

3225 

3224 

3223 
3222 
3221 

3i74 
3173 
3173 
3172 
3171 

3i24 
3i24 
3i23 

3l22 
3l2I 

3075 
3074 
3073 
3073 
3072 

3590 
3589 
3588 
3587 
3587 

3535 
3535 
3534 
3533 
3532 

3481 
3480 
3480 

3479 
3478 

3428 
3427 
3426 
3425 
3424 

3375 
3374 
3373 
3372 
3372 

3323 

3322 

3321 
3320 
3319 

3271 
3270 
3270 
3269 
3268 

3220 
3220 
3219 
3218 
3217 

3170 
3169 
3i68 
3 1 68 
3167 

3l20 

3i  19 
3119 

3II7 

3071 
8070 
3069 
8069 
3c68 

3022 
3022 
302I 
3020 
8019 

3586 
3585 
3584 
3583 
3582 

353i 
353o 
3529 
3528 
3527 

3477 
3476 
3475 
3474 
3473 

3423 
3423 
3422 
3421 
3420 

3371 
3370 
3369 
3368 
3367 

3319 
33i8 
33i7 
33i6 
33i5 

3267 
3266 
3265 
3265 
3264 

3216 

32i5 
32i4 
32i4 
32i3 

3 1 66 
3 1 65 
3 1 64 
3i63 
3 1 63 

3ii6 
3ii5 
3ii4 
3ii4 
3ii3 

8067 
3o66 
3o65 
3o65 
3o64 
3o63 
8062 
3o6i 
3o6o 
3oDo 

3oi8 
3oi8 
3oi7 
3oi6 
3(>i5 

358 1 
35So 

3579 
3578 
3^77 

3526 
3525 
3525 
3524 
3523 

3472 
3471 
3471 
3470 
3469 

3419 
3418 

3417 
3416 
34i5 

3366 
3365 
3365 
3364 
3363 

33i4 
33i3 
33i3 
33i2 
33 1 1 

3263 
3262 
3261 
3260 
3259 

3212 
32II 
3210 

3209 
3209 

3162 
3i6i 
3i6o 
3i59 
3i58 

3lI2 

3iii 
3iio 
3iio 
3 1 09 

3oi4 
3oi4 
3oi3 

3oi2 

3oii 

1°  J8'; 

1°  19' 

1°  20' 

l°2ri°22' 

r23' 

I°24' 

1°  25'il°  26'  1°  27';!"^  28'|1°  29^ 

Page  140]                   TABLE  XXII. 

Proportional  Logarithms. 

S. 

0 

I 

2 

3 
4 
5 
6 

7 
8 

9 

lO 

II 

12 

1 3 
i4 
i5 
i6 
17 
i8 

19 
20 
21 
22 

23 

24 

25 

26 

27 
28 
29 

3o 
3i 

32 

33 

.  34 

35 

36 

37 
38 
39 

4o 
4r 
42 

43 
44 
45 
46 
47 
48 

49 
5o 
5i 

52 

53 
54 
55 
56 
57 
58 
59 

S. 

h     m 

1°30 

h    m 

r  31' 

1°  32' 

k    m 
1°  33' 

h    VI 

1°34' 

h    m 
1°35' 

h    m 
1°36' 

h    m 
1°37' 

h    m 

1°38' 

h     m 
F39' 

h    m 

r40' 

li    m 
1°41' 

25lO 

2509 
25o8 
25o7 
25o7 

S. 

0 
I 
2 
3 
4 
5 
6 

7 
8 

9 

10 
II 
12 
i3 
i4 
i5 
16 

17 
18 

19 

3oio 
3009 
3009 
3oo8 
3oo7 

2962 
2962 
2961 
2960 
2909 

2915 
2914 
2913 
2912 
2912 

2868 
2867 
2866 
2866 
2865 

2821 
2821 
2820 
2819 
2818 

2775 
2775 
2774 
2773 
2772 

2730 
2729 
2729 
2728 
2727 

2685 
2684 
2684 
2683 
2682 

2640 
2640 
2639 
2638 
2638 

2596 
2596 
2595 
2594 
2593 

2553 

2552 

255i 
255i 
255o 

3  006 
3oo5 
3oo5 
3oo4 
3on3 

2958 
2958 
2957 
2955 
2955 

2911 
2910 
2909 
2909 
2908 

2864 
2863 
2862 
2862 
2861 

2818 
2817 
2816 
28[5 
2815 

2772 

2771 
2770 
2769 
2769 

2726 
2725 
2725 
2724 
2723 

2681 
2681 
2680 
2679 

2678 

2637 
2636 
2635 
2635 
2634 

2593 

2592 
2591 
2591 
2590 

2549 
2548 
2  548 
2547 
2  546 

2  5o6 
2  5o5 
2  5o4 
25o4 
25o3 

3oo2 
3ooi 
3ooi 
3ooo 
2999 

2954 
2954 
2953 
2952 
2951 

2907 
2906 
2905 
2905 
2904 

2860 
2859 
2859 
2858 
2857 

2814 
2813 
2812 
2811 
2811 

2768 
2767 
2766 
2766 
2765 

2722 
■i.-jii 
2721 
2720 
2719 

2678 
2677 
2676 
2675 
2675 

2633 

2632 
2632 

263 1 
263o 

2589 
2588 
2588 
2587 
2586 

2545 
2545 
2544 
2543 
2543 

25(12 
25(J2 
25oi 
2  5oC) 

2499 

299S 
2997 
2997 
2996 
2995 

2950 
2950 
2949 
2948 
2947 

2903 
2902 
2901 
2901 
2900 

2856 
2855 
2855 
2854 
2853 

2810 
2809 
2808 
2808 
2807 

2764 
2763 
2763 
2762 
2761 

2719 
2718 
2717 
2716 
2716 

2674 
2673 
2672 
2672 
2671 

2629 
2629 
2628 
2627 
2626 

2585 
2585 
2584 
2583 
2583 

2542 
254i 
2540 
2540 
2539 

2499 
2498 

2497 
2497 
2496 

2994 
2^93 
2993 
2992 
2991 

2946 
2946 
2945 
2944 
2943 

2898 
2897 
2896 

2852 
2852 

285i 
285o 
2849 

2806 
2805 
2805 
2804 
2803 

2760 
2760 
2759 

2758 
2757 

2715 
2714 
2713 
2713 
2712 

2670 
2669 
2669 
2668 
2667 

2626 
2625 
2624 
2624 
2623 

2582 

258 1 
258o 
258o 
2579 

2538 
2538 
2537 
2536 
2535 

2495 
2494 
2494 

2493 

2492 

20 
21 
22 

23 

24 

25 

26 

27 

28 

29 

3o 
3i 

32 

33 
34 

2990 
2989 
2989 
2988 
2987 

2942 
2942 
2941 
2940 
2939 

2895 
2894 
2894 
2893 
2892 

2848 
2848 
2847 
2846 
2845 

2802 
2801 
2801 
2800 
2799 

2756 
2756 
2755 
2754 
2753 

2711 
2710 
2710 
2709 
2708 

2666 
2666 
2665 
2664 
2663 

2622 
2621 
2621 
262c 
2610 

2578 
2577 
2577 
2576 
2575 

2535  1  2492 
2534  1  2491 
2533  1  2490 
2533  2489 
2532  1  2459 

2986 
29S5 
2985 
2984 
2983 

2939 
2938 
2937 
2936 
2935 

2891 
2891 
2890 
2889 
2888 

2845 
2844 
2843 
2842 
2842 

2798 
2798 
2797 
2796 
2795 

2753 
2752 
2751 
2750 
2750 

2707 
2707 
2706 
2705 
2704 

2663 
2662 
2661 
2660 
2660 

2618 
2618 
2617 
2616 
2615 

2574 
2574 
2573 
2572 
2572 

253i 
253o 
253o 
2529 
2528 

24S0 
24S7 
2487 
2486 
2485 

2982 
2981 
2981 
2980 
2979 

2935 
2934 
2933 
2932 
2931 

28S7 
2887 
28S6 
2885 
2884 

2841 
2840 
2839 
2838 
2838 

2795 
2794 
2793 
2792 
2792 

2749 
2748 
2747 
2747 
2746 

2704 
2703 
2702 
2701 
2701 

2659 
2658 
2657 
2657 
2656 

26i5 
2614 
2613 
2612 
2612 
261 1 
2610 
2610 
2609 
2608 

2571 
2570 
2569 
2569 
2568 

2527 
2527 
2526 

2525 
2525 

2485 
2484 
2483 
2482 
2482 

35 
36 

37 
38 

39 

40 
4i 
4-2 
43 
44 
45 
46 

47 
48 
49 
5o 
5i 

52 

53 
54 
55 
56 
57 
58 
59 

S. 

2978 
2977 
2977 
2976 
2975 

2931 
2930 
2929 
2928 
2927 

2883 
2883 
2882 
2881 
28S0 

2837 
2836 
2835 
2835 
2834 

2791 

2790 
2789 
2788 
2788 

2745 
2744 
2744 
2743 
2742 

2700 
2699 
2698 
2698 
2697 

2655 
2655 
2654 
2653 

2652 

2567 
2566 
2566 
2565 
2  564 

2524 

2523 
2522 
2522 
2521 

2481 
2480 
2480 
2479 
2478 

2974 
2973 
2973 
2972 
2971 

2927 
2926 
2925 
2924 
2924 

2880 
2879 
2878 
2877 
2876 

2833 

2832 

283  r 
2S3i 
283o 

2829 
2828 
2828 
2827 
2826 

2787 
2786 
2785 
2785 
2784 

2741 
2741 
2740 
2739 
2738 

2696 
2695 
2695 
2694 
2693 

2652 

265 1 
265o 
2649 
2649 

2607 
2607 
2606 
2605 
2604 

2  564 
2563 
2562 
256i 
256i 

2520 
252Q 
2519 
25l8 

25i7 

2477 
2477 
2476 
2475 
2475 

2970 
2969 
2969 
2968 
2967 

2923 
2922 
2921 
2920 
2920 

2876 
2875 
2S74 
2873 
2873 

2783 
2782 
2782 
2781 
2780 

2738 
2737 
2736 
2735 
2735 

2692 
2692 
2691 
2690 
2689 

2648 
2647 
2646 
2646 
2645 

2604 
2603 
2602 
2601 
2601 

256o 
2559 
2559 
2  558 
2557 

25i7 
25i6 
25i5 
25i5 
25i4 

2474 
2473 
2472 
2472 
2471 

2966 
2965 
2965 
2964 
2963 

2919 
2918 
2917 
2916 
2916 

2872 
2871 
2870 
28G9 
2869 

2825 
2825 
2824 
2823 
2822 

2779 

2779 
2778 

2777 
2776 

2734 
2733 
2732 
2732 
2731 

2689 
2688 
2687 
2687 
2686 

2644 
2643 
2643 
2642 
2641 

2600 
2599 

:|? 

2597 

2556 
2556 
2555 
2554 
2553 

25i3 

25l2 
25l2 
25ll 
25 1 C 

2470 
2470 
2469 
2468 
2467 

1°30' 

1°  31' 

1°  32' 

1°  33' 

r34' 

1°35' 

1°3G' 

1°37' 

1°38' 

1°39' 

1°40'1°41' 

TABLE  XXII. 

[I'afe^-  141 

Proportional  Logarithms. 

s. 

//.  m    h    m 

A   7ft 

h    VI 

h    m 

h    VI 

h    VI  \h    m 

h    m 

h    VI 

h    m 

h    VI 

1°  42'  1°  43 

'1°44 

1"45 

'1°4G 

'  1°  47' 

1°48 

1°49 

1°50' 

P51 

'1°52 

1°53' 

s. 

o 

2467  1  2424   2382 

2341 

23oo 

2259 

22X8 

2x78 

2x39 

2099 

2061 

2022 

0 

I 

2466 

2424  1  2382 

2340 

2299 

2258 

2218 

2x78 

2x38 

2099 

2060 

2021 

I 

2 

2455 

2423  1  238i 

2339 

2298 

2258 

23X7 

2x77 

2x37 

2098 

2059 

2021 

2 

3 

2465 

2422 

238o 

2339 

2298 

2257 

33X6 

2x76 

2i37 

2098 

2059 

2020 

3 

4 

2464 

2422 

338o 

2338 

2297 

3356 

33l6 

2x76 

2x36 

2097 

2C.58 

20x9 

4 

5 

2463 

2421 

2379 

2337 

2296 

3356 

33l5 

2175 

2x36 

2096 

2o57 

2019 

5 

6 

2462 

2420 

2378 

2337 

2296 

2255 

33X4 

2174 

2x35 

2096 

2057 

2018 

6 

7 

2462 

2419 

2378 

2336 

2295 

2254 

22X4 

2x74 

2x34 

2095 

2o56 

20x7 

7 

S 

2461 

2419 

3377 

2335 

2294 

2253 

22x3 

2x73 

2i34 

2094 

3o55 

20x7 

8 

9 

2460 

2418 

3376 

2335 

2294 

2253 

22X2 
23X2 

2x72 
2172 

2x33 

2X32 

2094 

3o55 

;oi6 

9 

lO 

2460 

2417 

2375 

2334 

2393 

2252 

2093 

3o54 

2016 

10 

II 

2459 

2417 

2375 

2333 

3392 

225l 

23X1 

217X 

2X32 

2092 

3o53 

20  X  5 

II 

12 

2458 

2416 

2374 

2333 

2291 

225x 

23(0 

2x70 

2x3l 

2092 

3o53 

20X4 

12 

i3 

2458 

24i5 

2373 

2332 

2291 

2250 

22X0 

2170 

2x3o 

2091 

2052 

20X4 

i3 

i4 

24'i7 

24 1 5 

3373 

233i 

2390 

2249 

2209 
23()5 

2 169 
2169 

2x3o 

2090 

3053 

20X3 

i4 
i5 

i5 

2456 

24 1 4 

2373 

233i 

2289 

2249 

2x29 

2090 

2o5x 

2012 

i6 

2455 

24i3 

2371 

233o 

2289 

3248 

2208 

2168 

2X38 

2089 

2o5o 

20X2 

16 

17 

2455 

2412 

2371 

2339 

2388 

3347 

2307 

3167 

3138 

2088 

2o5o 

301  1 

17 

i8 

2454 

2412 

3370 

3328 

2287 

2247 

2206 

2167 

2x27 

2088 

2049 

30X0 

18 

_i9_ 

20 

2453 

241 1 

2369 
2368 

2338 

2287 

2346 

3206 

2166 

2126 

2087 

2o48 

20X0 

19 

30 

2453 

3410 

3327 

2386 

3345 

32o5 

2x65 

2x26 

2086 

2048 

2009 

31 

2452 

2410 

2368 

3326 

2285 

3245 

3  2o4 

2x65 

2X25 

3086 

2047 

2009 

2X 

22 

2.45 1 

2409 

3367 

2326 

2285 

2344 

2204 

2x64 

2X34 

3o85 

2046 

3008 

33 

23 

245o 

2408 

3366 

2335 

2284 

2343 

2  2o3 

2x63 

3X34 

3o85 

2o46 

3007 

23 

24 
25 

2450 

241 .8 

3366 

3334 

3283 

3343 

2202 

2x63 

2123 

3o84 

2045 

2007 

34 
25' 

2449 

2407 

3365 

2334 

3283 

3342 

2202   2162 

2X22 

2  08  3 

2044 

2006 

26 

3448 

2406 

2364 

2323 

3283 

224x 

220X 

2x6x 

3X32 

3  08  3 

2044 

2oo5 

26 

27 

2448 

24o5 

2364 

2  33  2 

3381 

2241 

2200 

2x61 

3I2I 

208 '^ 

3043 

20o5 

27 

28 

2447 

24o5 

2363 

2333 

3381 

2240 

2200 

2160 

2120 

2f;8l  1  3042 

2004 

38 

29 

2446 

2404 

2362 

2321 

2280 

2239 

2199 

2x59 

3X20 

3081 

2043 

2oo3 

39 

3o 

3o 

244^ 

24o3 

2363 

2320 

2279 

2339 

2198 

2x59 

2x19 

3080 

3o4x 

2003 

3i 

2445 

24o3 

236i 

2  3  20 

2279 

2338 

219S 

3x58 

21X8 

2079 

304x 

2002 

3x 

32 

2444 

2402 

2  36o 

2319 

2278 

2237 

2x97 

2X37 

2I18 

2079 

2o4o 

2001 

32 

33 

2443 

2401 

2359 

23i8 

2277 

2237 

2196 

2x57 

.2117 

2078 

2039 

2001 

33 

34 
~35 

2443 

2401 

2359 

3358 

23i7 
33i7 

2277 

2236 

2196 

3 1 56 

21X6 

2077 

3039 

2000 

34 
35 

2442 

2400 

2376 

2235 

2195 

2x55 

21X6 

20"'7 

3o38 

2000 

36 

2441 

2399 

2357 

33i6 

2275 

2235 

2x94 

2x55 

21x5 

2076 

3o37 

'999 

36 

37 

244  r 

2398 

2357 

23i5 

2274 

2234 

2x94 

2x54 

2XX5 

2075 

2o37 

1998 

37 

38 

2440 

2398 

2356 

23x5 

2274 

2333 

2x93 

2x53 

2  1  14 

2075 

3o36 

1998 

38 

39 

2439 

2397 

2355 

23x4 

3373 

2233 

2192  2x53 

2Xl3 

3074 

3o35 

'997 

39 

4o 

40 

2438 

2396 

2355 

23x3 

3373 

2232 

2193  21 53 

2X  x3 

3073 

3o35 

iv,6 

4i 

2438 

2396 

2354 

33x3 

2272 

233l 

2191  2l5x 

2X12 

3073 

3o34 

10^6 

4i 

42 

2437  1  2395 

2353 

33X2 

227X 

223l 

2190 

3l5l 

2XXX 

3072 

3o33 

X995 

42 

43 

2436 

2394 

2353 

23X1 

2270 

333o 

2190 

3x5o 

2IX  I 

2072 

3o33 

'994 

43 

44 

2436 

3394 

2352 

23X1 

2270 

3339 

3x89 

2149 

21X0 

2071 

2032 

1994 
1993 

44 
45 

45 

3435 

3393 

235i 

23 10 

2369 

3229 

2188 

3x49 

2109 

2070 

3032 

46 

2434 

2392 

235o 

2309 

2268 

2228 

2x88 

3x48 

2x09 

3070 

2o3  X 

X993 

46 

47 

2433 

2391 

335o 

2309 

2268 

2337 

2x87 

3x47 

2x08 

3069 

2o3o 

1992 

47 

48 

2433 

2391 

3349 

23o8 

2367 

3227 

2x86 

2X47 

2x07 

3068 

3o3o 

199X 

48 

49 

2432 

2390 

3  348 

23o7 

2266 

2226 

2186 

2x46 

2107 

3068 

2029 

'99' 

49 

5o 

243i 

2389 

3  348 

23o7 

2266 

2235 

2x85 

2x45 

2106 

3067 

2028 

1990 

5o 

5i 

243i  2389  1 

3347 

23o6 

2265 

2225 

2184 

2145 

2I05 

2066 

2028 

X989 

5i 

52 

243o 

2388  1 

3346 

23o5 

2364 

2224 

2x84 

2:44 

2io5 

3066 

3037 

igfic, 

52 

53 

2439 

2387 

2  346 

2  3o4 

2264 

2223 

2x83 

2i43 

2104 

2o65 

3026 

19S8 

53 

54 

2429 
2428 

2387 

2345 

23o4 

2263 

2223 

2x82 

3i43 

2xo3 
2io3 

3  064 

3036 

19S7 

54 
55 

55 

3  386 

2344 

23o3 

2262 

2232 

2:82 

2l42 

3064  ■  203  5  1 

1987 

56 

2427 

3385 

2344 

23o2 

2262 

2221 

2181 

2X4I 

21C2 

2o63 

2025 

1986 

56 

57 

2426 

2384 

2343 

2302 

2261 

2220 

2x80 

2l4x 

2!0X 

2062 

2024 

1986 

57 

58 

2436 

2384 

2342 

23oi 

2260 

2220 

2180 

2l4o 

2IOI 

2062 

2023 

X985 

58 

59 
S. 

2425 

2383 

2342 

23oo 

2260  22x9  1 

2x79 

2x39 

2100 

2o6x 

2023 

1984 

D9 

1°42' 

1°  43' 

1°44' 

r  45' 

1°4G'|1°47'| 

r  J 8  1°  49'| 

1°50' 

Pol' 

1°52'1°53'| 

S.l 

P-^gei42]               TABLE  XXII. 

Proportional  Logarithms. 

S. 

o 

I 

2 

3 

4 
5 
6 

7 
8 

9 

lO 

1 1 

12 

i3 
i4 
i5 
i6 

17 
i8 

19 
20 
21 
22 

23 

24 

25 

26 

27 
28 
29 

3o 
3i 

32 

33 

34 

"35 

36 

37 
38 
39 

40 
4i 
41 
43 
44 
45 
46 
47 

48 
49 
5o 
5i 

52 

53 

54 
^5 
56 
f^i 
58 
59 

h   m 
1°54' 

h  m 
1°55' 

A  m 

1°56' 

h   m 
1°57' 

A  7n 

1°58' 

h  m 
1°59' 

h   m 
2°  0' 

k   m 
2°  V 

h   m 
2°  2' 

h   m 

2°   3' 

h   m 

2°  4' 

S. 

0 

I 
2 
3 
4 
5 
6 

7 
8 

9 
10 
II 
12 
i3 
i4 
[5 
16 

17 
18 

19 
20 
21 
22 

23 

24 

25 

26 

27 

28 

29 , 

3o  '( 
3i 

32 

33 
34 
35 
36 

37 
38 

39 
4o 
4i 
42 
43 
44 
45 
46 
47 
48 

49 
5o 
5i 

52 

53 
54 
55' 
56 

57 
58 

S. 

1984 
1983 
1982 
1982 
1981 

1946 
1945 
1944 
1944 
1943 

1908 
1908 
1907 
1906 
1906 

1871 
1870 
1870 
1869 
1868 

1 834 
i833 
i833 
i832 
i83i 

1797 
1797 
1796 
1795 
1795 

1761 
1760 
1760 
1759 
1759 

1725 
1724 
1724 
1723 
1722 

16S9 
1689 
1688 
16S7 
1687 

1 654 
i653 
i652 
i652 
i65i 

1619 
1618 
1617 
1617 
1616 

1981 
1980 
1979 
1979 
1978 

1943 
1942 
1941 
1941 
1940 

1905 
1904 
1904 
1903 
1903 

1868 
1867 
1867 
1866 
1 865 

i83i 
i83o 
i83o 
1829 
1828 

1794 
1794 
1793 
1792 
1792 

1758 
1757 
1757 
1756 
1755 

1722 
1721 
1721 
1720 
1719 

1686 
1686 
1 685 
1684 
1684 

i65i 
i65o 
i65o 
1649 
1648 

1616 
i6i5 
i6i4 
i6i4 
i6i3 

1977 

1975 
1975 

1939 
1939 
1938 
1938 
1937 

1902 
1901 
1 90 1 
1900 
1899 

1 865 
1 864 
1 863 
i863 
1862 

1828 
1827 
1827 
1826 
1825 

1791 
1791 
1790 
1789 
1789 

1755 
1754 
1754 
1753 
1752 

1719 
1718 
1718 

17.7 

i683 
1 683 
1682 
1681 
1681 

1648 
1647 
1647 
1 646 
1645 

i6i3 
1612 
1612 
1611 
1610 

1974 
1974 
1973 
1972 
1972 

1936 
1936 
1935 
1934 
1934 

1899 
1898 
1898 
1897 
1896 

r862 
1861 
i860 
i860 
1859 

1825 
1824 
1823 
1823 
1822 

1788 
1788 
1787 
1786 
1786 

1752 
1751 
1751 
1750 
1749 

1716 
1715 
1715 
1714 
1714 

1680 
1680 
1679 
1678 
1678 

1645 
1644 
1644 
1643 
1643 

1610 
1609 
1609 
1608 
1607 

1971 
1970 

1970 
19O9 
1068 

1933 
1933 
1932 
1 93 1 
1 93 1 

1896 
1895 
1894 
1894 
1893 

1859 
i858 
i857 
i857 
i856 

1822 
1821 
1820 
1820 
1819 

1785 
1785 
1784 
1783 
1783 

1749 
1748 
1748 
1747 
1746 

1713 
1712 
1712 
1711 
1711 

1677 
1677 
1676 
1676 
1675 

i642 
i64i 
i64i 
i64o 
i64o 

1607 
i6q6 
1606 
i6q5 
i6o5 

1968 
1967 
1967 
1966 
1965 

1930 
1929 
1929 
1928 
1928 

1893 
1892 
1891 
1891 
1890 

i855 
i855 
1 854 
i854 
i853 

1819 
1818 
1817 
1817 
1816 

1782 
1781 
1781 
1780 
1780 

1746 
1745 
1745 
1744 
1743 

1710 
1709 
1709 

1708 
1708 

1674 
1674 
1673 
1673 
1672 

1639 
i638 
i638 
1637 
i637 

1604 
i6o3 
i6o3 
1602 
1602 

1965 
1964 
1963 
1963 
1962 

1927 
1926 
1926 
1925 
1924 

1889 
1889 
1888 
1888 
1887 

i852 
i852 
i85i 
i85o 
i85o 

1816 
i8i5 
i8i4 
i8i4 
i8i3 

1779 

1778 

1778 
1777 
1777 

1743 
1742 
1742 
1741 
1740 

1707 
1706 
1706 
1705 
1705 

1671 
1671 
1670 
1670 
1669 

1 636 
i635 
i635 
1 634 
1 634 

1601 
1600 
1600 
1599 
1599 

1962 
1961 
19G0 
i960 
1959 

1924 
1923 
1923 
1922 
1921 

1886 
1886 
1 885 
1884 
1884 

1849 
1849 
1848 
1847 
1847 

1812 
1812 
1811 
1811 
1810 

1776 
1775 
1775 
1774 
1774 

1740 
1739 
1739 
-1738 
1737 

1704 
1703 
1703 
1702 
1702 

1668 
1668 
1667 
1667 
1666 

i633 
i633 
i632 
i63i 
i63i 

1598 
1598 
1597 
1596 
1596 

1958 
195s 

1956 
1956 

1919 
1919 
1918 

i883 
:88  3 
1SS2 
1881 
18S1 

1 846 
1845 
1845 
1844 
1844 

1809 
1809 
1808 
1808 
1807 

1773 
1772 
1772 
1771 
1771 

1737 
1736 
1736 
1735 
1734 

1701 
1700 
1700 
1699 
1699 

1 665 
1 665 
1 664 
1664 
1 663 

i63o 
i63o 
1629 
1628 
1628 
1627 
1627 
1626 
1626 
1625 

1595 
1595 
1594 
1593 
1593 

1592 
1592 
1591 
1591 
1590 

1955 
195D 
1954 
1953 
1953 

1952 
i95i 
1951 
1950 
1950 

1918 
1917 
1916 
1916 
1915 

1880 
1880 
1S79 
1878 
1878 

1843 
1843 
1842 
i84i 
i84i 

1806 
1806 
i8o5 
i8o5 
1804 

1770 
1769 
1769 
1768 
1768 

1734 
1733 
1733 
1732 
1 73 1 

1698 
1697 
1697 
1696 
1696 

1 663 
1662 
1661 
1661 
1660 

1914 
1914 
1913 
1913 
1912 

1877 
1876 
1876 
1875 
1875 

1840 
1839 
1839 
i838 
i838 

i8o3 
i8o3 
1802 
1802 
1801 

1767 
1766 
1766 
1765 
1765 

1 73 1 
1730 
1730 
1729 
1728 

1695 
1694 
1694 
1693 
1693 

1660 
1659 
i658 
i658 
1657 

1624 
1624 
1623 
1623 
1622 

1589 
1 589 
1 588 
1 588 
1 587 

1949 
1948 
1948 
19^7 
1946 

1911 
1911 
1910 

1909 
1909 

1874 
1873 
1873 
1872 
1871 

1837 
i836 
i836 
1 835 
i835 

1800 
1800 
1799 
1798 
1798 

1764 
1763 
1763 
1762 
1762 

1728 
1727 
1727 
1726 
1725 

1692 
1692 
1691 
1690 
1690 

1657 
1 656 
i655 
i655 
1 654 

1621 
1621 
1620 
1620 
1619 

1 587 
1 586 
1 585 
1 585 
1 584 

S. 

1°  54' 

l°55' 

rsG' 

1°57' 

l°o8' 

1°  59' 

2°  0' 

2°  1' 

2°  2' 

2°  3' 

2°  4' 

TABLE  XXII.                [^»seH3 
Proportional  Logarithms. 

S. 

o 

I 

2 

3 
4 
5 
(') 

7 
8 

9 

lO 

1 1 

12 

i3 

i4 
i5 
i6 

17 
i8 

19 
20 
21 
22 

23 

24 

35 

26 

27 
28 
29 

3o 
3i 

32 

33 

34 
35 
36 
37 
38 

39 

4o 
4i 
42 
43 
Ai 
■45 
46 

47 

48 
49 
5o 
5i 

52 

53 
54 
55 
56 

57 
58 
59 

S. 

h    m, 

2°  5' 

h    m 

2=  & 

/i  VI 

20  7, 

h    VI 
2°  8' 

h    m 

2°  9' 

k    m 
2°  10' 

h    VI 
2°  11' 

h    TO 
2°  12 

h    VI 
2°  13' 

h    VI 
2°  14' 

h    m 
2°  15' 

s. 

0 

I 

2 

3 

4 
5 
5 
7 
8 

_9_ 

10 
II 
12 
i3 
i4 
i5 
16 

17 
18 

19 
20 
21 
22 

23 

24 

25 

26 

27 

23 

29 

3o 
3i 
82 
83 
34 
35 
36 

37 
38 

39 
40 
4i 
42 
43 
44 
"45 
46 
47 
48 
49 
5o 
5i 

52 

53 
54 
55 
56 

57 
58 
59 

S. 

1 584 
i583 
i582 
i582 
i58i 

1 549 
1 548 
1 548 
1 547 
1 547 

i5i5 
i5i4 
i5i4 
i5i3 

l5l2 

i48i 
1 480 
1479 
1479 
1478 

1 447 
1 446 
1 446 
1445 
1445 

i4i3 
i4i3 
14:2 
1412 

i4ii 

i38o 
1879 
1879 
1878 
1378 

i347 
1 346 
1 346 
i345 
1 345 

i8i4 
i3i4 
i3i3 
i3i3 

l3l2 

1282 
1281 
1281 
1280 
1280 

1249 
1249 
1248 
1248 
1247 

i58i 
i58o 
!  5So 
i579 
1578 

1 546 
1 546 
i545 
1 544 
1 544 

l5l2 

i5ii 
i5ii 
i5io 
i5io 

1478 
i477 
1 477 
1476 
1476 

1 444 
1443 
1 443 
1442 
1442 

i4ii 
i4io 
1409 
1409 

i4o8 

1877 
1877 
1876 
1376 
1875 

1 344 
1 344 
1 343 
1 343 
1842 

i3ii 
i3ii 
i3io 
i3io 
1809 

1279 
1278 
1278 

1277 
1277 

1247 
1246 
1246 
1345 
1245 

1578 
1 577 
1 577 
1576 
1576 

1 543 
1 543 
1 542 
1 542 
i54i 

1 509 
i5o8 
i5o8 
i5o7 
1 507 

1475 
1474 
1474 
1473 
1473 

1472 
1472 
1471 
1470 
1470 

i44r 
i44i 
1 440 
1 440 
1439 
1 438 
i438 
1437 
1437 
i436 

1 408 
1407 
1 407 
i4o6 
i4o6 
i4o5 
i4o4 
i4o4 
i4o3 
i4o3 

1874 
1874 
1878 
1878 
1872 

1872 
1871 
1871 
1870 
1870 
1869 
1 368 
1 368 
1867 
1867 
1 366 
1 366 
i365 
1 865 
1864 

1342 
i34i 
1 340 
i34o 
1889 

1809 
i3o8 
1808 
1807 
1807 

1276 
1276 
1275 
1275 
1274 

1244 
1 24'' 
1243 
1242 
1242 

1575 
1 574 
1 574 
1S73 
1573 

1 540 
1 540 
1539 
1539 
1 538 

i5o6 
i5o6 
i5o5 
i5o4 
i5o4 

1339 
1 338 
i338 
i337 
i337 
i336 
i335 
i335 
1 334 
1 334 

i3o6 
i3o6 
i3o5 
i3o4 
i3o4 
i3o8 
i3o3 

l302 
l302 

i3oi 

1274 
1273 
1278 
1272 
1271 

1241 
1241 
1240 
1240 
1289 

1572 
1571 
1571 
1670 
1570 

i538 
1 537 
1 536 
1 536 
i535 

i5o3 
i5o3 

l502 
l502 

i5oi 

1469 
1469 
1 468 
1 468 
1467 

1 436 
1435 
i435 
1434 
i433 

l402 
l402 

i4oi 
i4oi 
i4oo 

1271 
1270 
1270 
1269 
1269 

1289 
1238 
1238 
1287 
1287 

1569 
1 569 
1 568 
1 567 
1 567 

i535 
1 534 
1 534 
1 533 
i532 

i5oo 
i5oo 
1 499 
1499 
1498 

1467 
1 466 
1 465 
1 465 
1 464 

1 43  3 
1432 
i432 
i43i 
i43i 

1899 

1399 
1398 
1398 
1897 

i338 
i833 
1882 
1882 
i33i 

i3oi 
i3oo 
i3oo 
1299 
1298 

1268 
1268 
1267 
1267 
1266 

1236 
1235 
1285 
1284 
1284 

1 566 
1 566 
1 565 
i565 
1 564 

i532 
i53i 
i53i 
i53o 
i53o 

1498 
1497 
1496 
1496 
1495 

1 464 
1 463 
1 463 
1462 
i46i 

i43o 
1429 
1429 
1428 
1428 

1897 
1396 
1396 
1895 
1894 

1 368 
1 363 
1862 
1862 
i36i 

i33i 
i33o 
1829 
1829 
1828 

1298 
1297 
1297 
1296 
1296 

1266 
1265 
1264 
1264 
1263 

12  83 
1233 
1282 

1232 
1281 

1 563 
i563 
1 562 
1 562 
i56i 

1529 
i528 
i528 
i527 
i527 

1495 
1494 
1494 
1493 
1493 

i46i 
1 460 
1 460 
1459 
1459 

1427 
1427 
1426 
1426 
1425 

1394 
1398 
1893 
1892 
1892 

i36i 
i36o 
i36o 
1859 
1359 

1828 
1827 
1827 
1826 
1826 

1295 
1295 
1294 
1294 
1298 

1263 
1262 
1262 
1261 
1261 

I23l 
1280 
1280 
1229 
1229 

i56i 
i56o 
1559 
1559 
1 558 

i526 
1 526 
i525 
i524 
i524 

1492 
1491 
1491 
1490 
1490 

i458 
i458 
1457 
i456 
i456 

1424 
1424 
142.3 
1423 
1422 

1891 
1391 
1 390 
i389 
1889 

i358 
i357 
i357 
i356 
1 356 

i355 
i355 
i354 
1 354 
i353 

i325 
1825 
1824 
1823 
i323 

1822 
1822 
1821 
1821 
1820 

1292 
1292 
1291 
1 291 
1290 

1290 
1289 
1289 
1288 
1288 

1260 
1260 
1259 
1259 
1258 

1228 
1227 
1227 
1226 
1226 

1 558 
1 557 
i556 
1 556 
i555 

i523 
i523 

l522 
l522 
1 52 1 

1489 
1489 
1 488 
1487 
1487 

i455 
i455 
1454 
1454 
i453 

1422 
1421 
1421 
1420 
1419 

1 388 
1 388 
1 387 
1887 
1 386 

1257 
1257 
1256 
1256 
1255 
1255 
1254 
1254 
1253 
1253 

1225 
1225 
122.4 
1224 
1228 

1228 
1222 
1222 
I22I 
I  221 

1 555 
1 554 
1 554 
i553 
i552 

I  520 
l520 

i5i9 
i5i9 
i5i8 

i486 
i486 
i485 
i485 
i484 

1452 
1452 
1 45 1 

i45i 
i45o 

1419 
i4i8 
i4i8 
1417 
i4i7 

1 386 
i385 
1 384 
i384 
i383 

i352 
i352 
i35i 
i35i 
i35o 

1820 
1819 
i3i9 
i3i8 

1817 

1287 
1287 
1286 
1285 
1285 

i552 
i55i 
i55i 
i55o 
i55o 

i5i8 
i5i7 
i5i6 
i5i6 
i5i5 

i483 
i483 
1482 
1482 
i48i 

i45o 
1449 
1449 
1 448 
1447 

i4i6 
i4i6 
i4i5 
i4i4 
i4i4 

1 383 
i382 
1 382 
i38i 
i38i 

i35o 
1 349 
1 349 
1 348 
1848 

1817 
i3i6 
i3i6 
i3i5 
i3i5 

1284 
1284 
1288 
1283 
1282 

1252 
1252 
T25l 
I250 
I25o 

X220 
I219 
1219 
1218 
13l8 

2°  5' 

2^^  0' 

2°  7' 

2°  8' 

2=  9' 

2°  10' 

2°  11' 

2^12' 

2"^  13' 

2M4' 

2°  15' 

Page  ]44] 

TABLE  XXII. 

Proportional  Logarithms. 

S. 

o 

I 

2 

3 
4 
5 
6 

7 
8 

9 

10 

1 1 

12 

i3 
i4 
i5 
i6 

17 
18 

19 
20 
21 
22 

23 

24 

25 

26 
27 
28 
29 

3(' 
3i 
3-2 
33 
34 
35 
36 

37 
38 
39 

4o 
4i 
42 
43 
44 
45 
46 
47 
48 
49 
5o 
5[ 

52 

53 
54 
55 
56 

57 
58 
59 

2°iG' 

k   m 
2°  17' 

h   m 

2°  18' 

h   m 
2°  19' 

h   m 
2°  20' 

h  m 
2°  21' 

h   m 
2°  22' 

h   m 
2°  23' 

h   m 

2°  24' 

h   m 
2°  25' 

h    m 

2°2G' 

S. 

0 

I 
2 
3 
4 
5 
6 

7 
8 

9 
10 
II 
12 
i3 
i4 
i5 
i6- 

17 
18 

19 
20 
21 
22 

23 

24 

25 

26 
27 
28 
29 

3o 
3i 

32 

33 

34 
35 
36 

37 
38 
39 

4o 
4i 
42 
43 
A^ 
45 
46 
47 
48 

49 
5o 
5i 

52 

53 
54 
55 
56 

57 
58 
59 

S. 

1217 
1217 
1216 
1216 

I2l5 
I2l5 

I2I4 

I2l4 
I2l3 
I2l3 

1 186 
ii85 
ii8'4 
1 184 
ii83 

ii54 
ii53 
ii53 

Il52 
Il52 

II23 

1122 
1122 
1121 
1 1 20 

1091 
1091 
1090 
1090 
1089 

1061 
1060 
1060 
1059 

io58 

1000 
1029 
1029 
1028 
1028 

0999 
0999 
0998 
0998 
0997 

0969 
0969 
0968 
0968 
0967 

0939 
0939 
oo38 
0938 
0937 

0909 
0909 
0908 
0908 
0907 

ii83 
1182 
1182 
1181 
1181 

ii5i 
ii5i 
ii5o 
ii5o 
1 149 

1120 
1119 
1119 
1118 
1118 

1089 
1088 
1088 
1087 
1087 

io58 
io57 
io57 
io56 
io56 

1027 
1027 
1026 
1026 

1025 

0997 
0996 
0996 
0995 
0995* 

0967 
0966 
0966 
0965 
0965 

0937 
0936 
0936 
0935 
0935 

0907 
0906 
0906 
0905 
0905 

I2I2 

I2II 
121  I 
I2I0 
I2I0 

II»0 

1 180 
1 179 
1179 
1 1 78 

1 149 
ii48 
I  [48 
ii47 
1 147 

1117 
1117 
1116 
1116 
iii5 

1086 
1086 
io85 
io85 
1084 

io55 
io55 
io54 
io54 
io53 

1025 

1024 
1024 

I023 

1023 
1022 

1022 
I02I 
I02I 
1020 

0994 
0994 
0993 
0993 
099" 
0992 
0991 
0991 
0990 
0990 

0964 
C964 
0963 
0963 
0962 

0962 
0961 
0961 
0960 
0960 

0934 
0934 
0933 
0933 
0932 

0904 
0904 
0903 
0903 
0902 

1209 
1209 
1208 
1208 
1207 

1178 

1177 
1177 
1 176 
1175 

1 1 46 
ii46 
1 145 
1145 
ii44 

iii5 
1114 
iii4 
iii3 
iii3 

1084 
io83 
io83 
1082 
1082 

io53 
io52 

I052 

io5i 
io5i 

0932 
0931 
0931 
0930 
0930 

0902 
0901 
0901 
0900 
0900 

1207 
1206 
1206 
I205 
I2o5 

1 175 
1 174 
1 174 
1173 
1 173 

1 143 
1 143 
1142 
1 142 
ii4i 

1112 
1112 
nil 
I  III 
II 10 

1081 
1081 
1080 
1080 
1079 

io5o 
io5o 
1049 
1049 
io48 

1020 
IOI9 
1019 
IO18 
IO18 

0989 
0989 
0988 
0988 
0987 

0959 
0969 
0958 
0958 
0957 

0929 
0929 
0928 
0928 
0927 

0899 
CS99 
0S98 
0898 
0897 

1204 
1204 
I203 
1202 
1202 

1172 
1172 
1171 
1 171 
1 170 

ii4i 
ii4o 
ii4o 
1 1 39 
1 1 39 

mo 
1 109 
II 09 
1 108 
1 108 

1079 
1078 
1078 
1077 
1076 

1048 
1047 
1047 
io46 
io46 

IOI7 
IOI7 
IO16 
IO16 

ioi5 

0987 
0986 
0986 
0985 
0985 

0957 
0956 
0956 
0955 
0955 

0927 
0926 
0926 
0925 
0925 

0897 
0896 
0896 
0895 
0895 

I20I 
I  201 
1200 
1200 
I  199 

1 170 
1 1 69 
1 169 
1 1 68 
1168 

ii38 
ii38 
ii37 
ii37 
ii36 

1 107 
1 106 
1 106 
iio5 
iio5 

1076 
1075 
1075 
1074 
1074 

1045 
1045 
1044 
1044 
1043 

ioi5 
ioi4 
.  ioi4 
10x3 
ioi3 

0984 
0984 
0983 
0983 
0982 

0954 
0954 
0953 
0953 
0952 

0924 
0924 
0923 
0923 
0922 

0894 
0894 
0893 
0893 
0892 

1199 
I  198 
II9S 
I  197 
IP97 

1167 
1 167 
1 166 
ii65 
ii65 

ii36 
ii35 
ii35 
ii34 
ii34 

1 104 
iio4 
iio3 
II  o3 
1102 

1073 
1073 
1072 
1072 
1071 

1043 
1042 
1042 
io4i 
io4t 

1012 
1012 

lOI  I 

ion 

lOIO 

0982 
0981 
0981 
0980 
0980 

0952 
0951 
0951 
0950 
0950 

0922 
0921 
0921 
0920 

0Q20 

0892 
0891 
0891 
0890 
0890 

I  I  96 
I  I  96 
I  195 
I  195 
I  194 
.193 
I  193 
I  I  92 
.1192 
II9I 

1 1 64 
ii64 
ii63 
1 1 63 
1 162 

ii33 

Il32 
Il32 

ii3i 
ii3r 

1102 

IIOI 
IIOI 

1 100 
1100 

1071 
1070 
1070 
1069 
1069 

io4o 
io4o 
1039 
1039 
io38 

1009 
1009 
1008 
100& 
1007 

0979 
0979 
0978 
0978 
0977 

0949 
0949 
0948 
0948 
0947 

0919 
0919 
0918 
0918 
0917 

0889 
0889 
0S88 
0888 
0887 

1 162 
1161 
1161 
1 1 60 
1 1 60 

ii3o 
ii3o 
1129 
1129 

1 1 28 

1099 
1099 
1098 
1098 
1097 

1068 
1068 
1067 
1067 

1066 

io37 
io37 
io36 
io36 
io35 

1007 
1006 
1006 
ioo5 
100  5 

0977 
0976 
0976 
0975 
0975 

0947 
0946 
0946 
0945 
0945 

0917 
0916 
0916 
0915 
0915 

0887 
0886 
08S6 
o885 
oS85 

H9I 

I  I  90 

:  190 
1.89 
1.89 

ii58 
ii58 
ii57 

1128 
1127 
1127 
1 1 26 

IT  26 

1097 
1096 
1096 
1095 
1095 

1066 
io65 
io65 
1064 
1064 

io35 
io34 
io34 
io33 
io33 

1004 
ioo4 
ioo3 
ioo3 
1002 

0974 
0974 
0973 
0973 
0972 

0944 
0944 
0943 
0943 
0942 

0914 
0914 
0913 
0913 
0912 

0884 
0884 
o883 
o883 
o883 

n88 
1 188 
1187 
1187 
1 186 

ii57 
ii56 
ii56 
ii55 
ii54 

II25 
II25 
I  I  24 
I  I  24 
II23 

1094 
1094 
1093 
1092 
1092 

io63 
io63 
1062 
1062 
1061 

io32 

1032 

io3i 
io3i 
io3o 

1002 

lOOI 

100 1 
1000 
1000 

0972 
0971 
0971 
0970 
0970 

0942 
0941 
0941 
0940 
0940 

0912 
09-1 
091 1 
0910 
0910 

0882 
0882 
0881 
0881 
0880 

9°  16' 

2°  17' 

2°  18' 

2°  19' 

2°  20' 

2°  21' 

2°  22' 

2°  23' 

2°  24' 

2=25' 

2°  96' 

1 

TABLE  XXII. 

[l-age  115  1 

Proportional  Lc 

(garilhms 

s 

o 

/*  7n 

h    m 

h    m 

h    VI 

k    m 

h    m 

A  in 

A  m 

h    m 

A  m 

A  m 

2°  27' 

2°  28' 

2°  29' 

2^30' 

2°  31' 

2'' 32' 

2°  33' 

2°  34' 

2°  35' 

2°  3G' 
0621 

2°  37' 

0594 

S. 
0 

0880 

08  5o 

0821 

0792 

07()3 

0734 

0706 

0678 

0649 

I 

0879 

o85o 

0820 

0791 

07G2 

0734 

0705 

0677 

0649 

0621 

0593 

I 

2 

0879 

0S49 

0820 

0791 

0762 

0733 

0705 

0677 

0648 

0621 

0593 

2 

3 

0878 

0849 

0819 

0790 

0762 

0733 

0704 

0676 

0648 

0620 

0592 

3 

4 
5 

0878 

o848 

0819 

0790 

07()i 

0732 

0704 

0676 

0648 

0620 

0592 
0591 

4 
5 

0877 

o848 

0818 

0789 

0761 

0732 

0703 

0675 

0647 

0619 

6 

0877 

0847 

0818 

0789 

0760 

0731 

0703 

0675 

0G47 

0619 

0591 

6 

7 

0876 

0847 

0817 

0788 

0760 

0731 

0703 

0674 

o646 

0618 

0591 

7 

8 

0876 

o846 

0817 

0788 

0759 

0730 

0702 

0674 

0646 

0618 

0590 

8 

9 

10 

0875 

.0846 

0816 

0787 

0759 

0730 

0702 

0673 

0645 

0617 
0617 

0590 

0589 

9 
10 

0875 

0845 

0816 

0787 

0758 

0730 

0701 

0673 

0645 

II 

0874 

0845 

0816 

0787 

0758 

0729 

0701 

0672 

0644 

0616 

0589 

II 

12 

0874 

0844 

08 1 5 

0786 

0757 

0729 

0700 

0672 

0644 

0616 

(,588 

12 

i3 

0873 

0844 

o8[5 

0786 

0757 

07^8 

0700 

0671 

0643 

061 5 

o588 

i3 

i4 
i5 

0873 

0843 

o8i4 

0785 

0756 

0728 

0699 
0699 

0671 

0643 

06 1 5 
i;6T5~ 

0587 
o587 

i4 
i5 

0872 

0843 

0814 

0785 

0756 

0727 

0670 

0642 

i6 

0872 

0842 

o8i3 

0784 

0755 

0727 

0698 

0670 

0642 

06 1 4 

c586 

16 

17 

0871 

0842 

o8i3 

0784 

0755 

0726 

0698 

0670 

064 1 

o6i4 

o586 

17 

i8 

0871 

0841 

0812 

0783 

0754 

0726 

0697 

o6()9 

064 1 

06 1 3 

o585 

18 

19 
20 

0870 

084 1 

0812 

0783 

0754 

0725 

0697 

0669 

oG4i 

061 3 

(.585 

19 
20 

0870 

0840 

081 1 

0782 

0753 

0725 

0696 

0668 

o64o 

0612 

o585 

21 

0869 

0840 

081 1 

0782 

0753 

0724 

0696 

0668 

0640 

0612 

o584 

21 

22 

0869 

0809 

0810 

07S1 

0752 

0724 

0695 

0667 

0639 

06 1 1 

o584 

22 

2j 

o8()8 

0839 

0810 

0781 

0752 

0723 

0695 

0667 

0639 

061 1 

o583 

23 

24 
25 

(.868 

o838 

0809 

0780 
07S0 

0751 

0723 

0694 

0666 

o638 

0610 

o583 
o582 

24 

25 

0867 

()838 

0809 

0751 

0722 

0694 

0666 

o638 

0610 

26 

0S67 

0837 

0808 

0779 

0751 

0722 

0694 

o665 

0637 

0609 

(>582 

26 

27 

oSfifi 

0S37 

0808 

0779 

0750 

0721 

0693 

o665 

0637 

0609 

o58i 

27 

28 

0S66 

08  3G 

0807 

0778 

0760 

0721 

0693 

0664 

o636 

0609 

()58i 

28 

29 

3o 

o865 

08  36 
o835 

0807 

0778 

0749 

0721 

0692 

0692 

0664 

o636 

0608 
0608 

o58o 
o58o 

29 

3o' 

o865 

0806 

0777 

0749 

0720 

oG63 

0635 

3i 

0HG4 

08  3  5 

0806 

0777 

0748 

0720 

0691 

o663 

o635 

0607 

0579 

3i 

32 

o8(34 

o834 

o8o5 

0776 

0748 

0719 

0691 

o663 

o634 

0607 

0579 

32 

33 

o863 

o834 

o8o5 

0776 

0747 

0719 

0690 

0662 

o634 

0606 

o579 

33 

34 
35 

o863 

oS34 

0804 

0775 

0747 

0718 

0690 

0DD2 

o634 

0606 

0578 

M 
35 

0862 

o833 

0804 

0775 

0746 

0718 

0689 

0661 

o633 

06'  >5 

0578 

36 

0862 

0833 

080  3 

0774 

0746 

0717 

0689  0661 

o633 

oGm5 

o577 

36 

37 

0861 

C.832 

o8o3 

0774 

0745 

0717 

0688  '  0660 

o632 

0604 

o577 

37 

38 

0861 

o832 

0802 

0774 

0745 

0716 

0688  0660 

o632 

0604 

0576 

38 

39 
4o 

0860 

o83i 

0802 

0773 

0744 

0716 

0687 

0669 

o63i 

060  3 

0576 

39 

40 

0860 

o83i 

0801 

0773 

0744 

0715 

0687 

0659 

o63i 

060  3 

o575 

4i 

0859 

o83() 

0801 

0772 

0743 

0715 

0686 

o658 

oG3o 

0602 

0575 

4i 

42 

0859 

o83c) 

0801 

0772 

0743 

0714 

0686 

06'")  8 

o63o 

0602 

0574 

42 

43 

08  58 

0829 

0800 

0771 

0742 

0714 

0686 

0()*17 

0629 

0602 

o574 

43 

44 
45 

C.858 

0829 

0800 

0771 

0742 

0713 

068  5 
068  5 

0657 

0629 

0601 

0573 

44 
45 

08  57 

0828 

0799 

0770 

0741 

0713 

0628 

060 1 

0573 

46 

08  57 

0828 

0799 

0770 

0741 

0712 

0684 

o656 

0628 

o6uo 

0573 

46 

47 

08  56 

0827 

0798 

0769 

0740 

0712 

0684 

06  5  5 

0628 

0600 

0572 

47 

48 

08  56 

0827 

0798 

0769 

0740 

071  I 

o683 

(,655 

0627 

0599 

o5-'2 

48 

49 
5o 

o855 

0826 

0797 

0768 

0740 

07  II 

o683 
0682 

o655 

"c.654" 

0627 
0G26 

0599 

0571 

49 
5o 

(xS55 

0826 

0797 

076S 

0739 

07  I  I 

0598 

(.571 

5i 

oS55 

0825 

0796 

0767 

0739 

0710 

0682 

o654 

C626 

0598 

0570 

5i 

52 

()854 

0825 

0796 

0767 

0738 

0710 

0681 

06  5  3 

0625 

0597 

0570 

b2 

53 

08  54 

0824 

0795 

0766 

0738 

0709 

0(58 1 

o()53 

0625 

0597 

o569 

53 

54 
55 

08  5  3 

0824 

0795 

0766 

0737 

0709 

0680 

0652 
~c65"i' 

0624 
0624 

0596 
0596 

0569 
o5fi8 

54 
■55" 

oS53 

0823 

0794 

0765 

0737 

0708 

0680 

56 

08  5  2 

0823 

0794 

0765 

0736 

0708 

0679 

on5 1 

0623 

0596 

o568 

56 

57 

o852 

0822 

0793 

0764 

0736 

0707 

0679 

o65i 

0G23 

0595 

o568 

57 

58 

o85i 

0822 

0793 

0764 

0735 

0707 

0678 

o65() 

0622 

0595 

o567 

58 

.A9_ 
S 

o85i 

0831 

0792 

0763 

0735 

0706 

0678 
2°  33' 

06  5o 
2°"ji4' 

0622 
2^^  35' 

0594 

o5G7 

59 
S. 

2°  27' 

2°  28' 

9=29' 

2°  30' 

2°  31' 

2°  33' 

2°3()' 

2°  37' 

19 


^''seJ^G]               TABLE  XXII. 

Proportional  Logaritbms. 

S. 

o 
I 

2 

3 

4 
5 
6 
7 
8 

9 

10 

II 

12 

i3 

i4 
i5 
i6 

I? 
i8 

19 
20 

31 
2  2 
23 
24 
25 
26 

27 
28 
29 

3o 
81 

32 

33 
34 
35 
36 

37 
38 
39 

40 
4t 
42 
43 
44 
45 
46 
47 
48 
49 
5o 
5i 

52 

53 
54 
55 

56 

57 
58 
59 

S. 

h    m 

2°  38' 

h    m 
2^39^ 

h    m 
2°  40' 

li    m 
2°  41' 

/i  m 

2°  42' 

h    VI 

2°  43' 

h    m 

2°  44' 

h    m 
2°  45' 

h    m 
2°  46' 

h  m 

h   m 

2°  48' 

S. 

0 
1 
2 
3 
4 
5 
6 

7 
8 

9 

10 
1 1 
12 
i3 
i4 
i5 
16 

17 
18 

19 
20 
21 
22 

23 

24 

25 

26 

27 
28 
29 

80 

3i 
82 
33 
34 
35 
86 

37 
38 

39 
4o 
4i 
42 
43 
44 
45 
46 
47 
48 
49 
5o 
5i 

52 

53 
54 
55 
56 
57 
58 
59 

S. 

o566 
o566 
o565 
o565 
o564 

0589 
o538 
o538 
o587 
0537 

o5i2 
o5n 
o5ii 
o5io 
o5io 

o484 
o484 
o484 
o483 
o4S8 

o458 
0457 
0457 
04  56 
0456 

043 1 
o43o 
0480 
o48o 
0429 

o4o4 
o4o4 
o4o3 
o4o3 
o4o8 

0878 
0877 

0377 
0877 
0876 

o852 
o35i 
o85i 
o35o 
o35o 

0826 
0825 
0825 
0824 
0824 

o3oo 
0299 
0299 
0298 
0298 

o564 
o568 
o563 
o562 
o562 

o586 
o536 
o586 
o585 
o535 

o5o9 
o5o9 
o5o8 
o5o8 
o5o7 

0482 
0482 
0481 
048  i 
0480 

0455 
0455 
0454 
0454 
0454 

0429 
0428 
0428 
0427 
0427 

o4o2 
o4o2 
o4oi 
o4oi 
o4oo 

0876 
0875 
0875 
0874 
0874 

0849 
0349 
0849 
o348 
0848 

0828 
0828 
0828 

0322 
0822 

0297 
0297 
0297 
0296 
0296 

o562 
o56i 
o56i 
o56o 
o56o 

o534 
0534 
o533 
o533 
o532 

o5o7 
o5o7 
o5o6 
o5o6 
o5o5 

0480 
o48o 
0479 

0479 
0478 

0453 
0458 
0452 
0452 
o45i 

0426 
0426 
0426 
0425 
0425 

o4oo 
0399 
0899 
0899 
0898 

0874 
0878 
0873 
0872 
0872 

o347 
o347 
0846 
o346 
o346 

o32i 

0821 
0820 
0820 
0819 

0295 
0295 
0294 
0294 
0294 

0559 
0559 
o558 
o558 
0557 

o532 
o53i 
o58i 
o53i 
o53o 

o5o5 
o5o4 
o5o4 
o5o3 
o5o3 

0478 
0477 
0477 
0476 
0476 

o45i 
o45o 
o45o 
o45o 
0449 

0424 
0424 
0428 
0428 
0422 

0898 
0897 
0897 
0896 
0896 
0895 
0895 
0895 
0894 
0894 

0871 
0871 
0870 
0870 
0870 

0845 
0845 
o344 
o344 
o348 

0819 
0819 

o3i8 
o3i8 
0817 

0298 
0293 
0292 
0292 
0291 

0557 
o557 
o556 
o556 
o555 

o53o 
0529 
0529 
0528 
0528 

0502 

o5o2 
o5o2 
o5oi 
o5oi 

0475 
0475 
0475 
0474 
0474 

0449 
o448 
0448 
o44i 
o44i 

0422 
0422 
0421 
0421 
0420 

0869 
0369 
o368 
0868 
0867 

o343 
0842 
0842 
0842 
0841 

0817 
o3i6 
0816 
o3i6 
o8t5 

0291 
0291 
0290 
0290 
0289 

o555 
0554 
o554 
o553 
o553 

0527 
0527 
0526 
o526 
0526 

o5oo 
o5oo 
0499 
0499 
0498 

0473 
0473 
0472 
0472 
0471 

o446 
o446 
o446 
0445 
0445 

0420 
0419 
0419 
o4i8 
o4i8 

0898 
0893 
0892 
0892 
0892 

0867 
0866 
o366 
o366 
o365 

0841 
o34o 
o34o 
0889 
0889 

o8i5 
o8r4 
o8i4 
08 1 3 
o3i3 

0289 
0288 
0288 
0288 
0287 

o552 
o552 
o552 
o55i 
o55i 

o525 
o525 
o524 
o524 
o523 

0498 
0498 
0497 
0497 
0496 

0471 
0471 
0470 
0470 
0469 

0444 
0444 
0443 
0443 
0442 

o4i8 
0417 
0417 
o4i6 
o4i6 

0891 
0891 
0890 
0890 
0889 

o865 
0864 
o364 
o368 
o863 

0339 
o838 
0338 
0887 
0887 

0818 
0812 

03l2 

o3ii 
o3ii 

0287 
0286 
0286 
0285 
0285 

o55o 
o55o 
0549 
0549 
o548 

o523 

o522 
0522 
052I 
052I 

0496 
0495 
0495 
0494 
0494 

0469 
0468 
o468 
0467 
0467 

0.442 
0442 
044 1 
044 1 
o44o 

o4i5 
•o4i5 
o4i4 
o4i4 
o4i4 

0889 
o388 
o388 
o388 
0887 

o863 
0862 
0862 
0861 
o36i 

o386 
0886 
0886 
o335 
o335 

o3io 
o3io 
0810 
0809 
0809 

0285 
0284 
0284 
0283 
0288 

o548 
o547 
o547 
o546 
o546 

052I 
o520 
o520 

o5i9 
o5i9 

0498 
0493 
0498 
0492 
0492 

0467 
0466 
0466 
o465 
o465 

o44o 
0439 
0489 
0438 
0488 

o4i8 
o4i3 

04l2 
04l2  • 

o4ii 

0887 
0886 
0886 
o385 
o385 

o36o 
o36o 
0359 
0359 
0359 

o334 
0884 
o833 
o883 
o388 

0808 
0808 
0807 
0807 
0807 

0282 
0282 
0282 
0281 
0281 

o546 
o545 
0545 
o544 
o544 

o5i8 
o5i8 
o5i7 
o5i7 
o5i7 

0491 
0491 
0490 
0490 
0489 

oi64 
0464 
0468 
o463 
0462 

o488 
0437 
0437 
o436 
o436 

o4ii 
o4io 
o4io 
o4io 
0409 

o384 
o384 
0884 
o383 
o383 

o358 
o358 
0857 
0857 
o356 

o332 
0882 
o33i 
o38i 
o33o 

0806 
o3o6 
o3o5 
o8o5 
0804 

0280 
0280 
0279 
0279 
0279 

o543 
o543 
0542 
0542 
o54i 

o5i6 
o5i6 
o5i5 
o5i5 
o5i4 

0489 
0489 
0488 
o488 
0487 

0462 
0462 
0461 
o46i 
0460 

0485 
0435 
0434 
0434 
0484 

0409 
o4o8 
o4o8 
0407 
0407 

o382 
0882 
0881 
0881 
o38i 

08  56 
o356 
o355 
o855 
o354 

o38o 
0829 
0829 
0829 
0828 

o3o4 
0804 
o8o3 
o8o3 
o3o2 

0278 
0278 
0277 
0277 
0276 

o54i 
o54i 
o54o 
o54o 
0539 

o5i4 
o5i3 
o5i3 
o5r2 
o5r2 

0487 
o486 
0486 
o485 
0485 

o46o 
0459 
0459 
o458 
o458 

o438 
0488 
0432 
0482 
0481 

o4o6 
o4o6 
o4o6 
o4o5 
o4o5 

0880 
o38o 
0879 
0879 
0878 

o854 
o858 
o353 
o353 
o852 

0828 
o327 
0827 
0826 
0826 

0802 
o3oi 
o3oi 
o3oo 
0800 

0276 
0276 
0275 
0275 
0274 

2°  38' 

2°  39' 

2M0' 

2°  41' 

2°  42' 

2°  43' 

2°  44' 

2°  45' 

2°  46' 

2°  47' 

2°  48' 

' 

TABLE  XXIL 

[Page  147 

Proportional  Logarithms. 

A  771 

h   VI 

h   m 

Ii   in 

h  in 

A  7/1 

h    m 

h   m 

h   m 

h   m 

h   m 

s. 

o 

2°  49' 

2°  50 

2"  51 

2^52 

2°  53' 

2=54 

2=55 

2°  50 

2=5? 

2=58' 

2°  59' 

S. 

0274 

0248 

0223 

0197 

0172 

0147 

0122 

0098 

0G73 

0049 

0024 

0 

I 

0273 

0248 

0222 

0197 

0172 

0147 

0122 

0097 

0073 

0048 

0024 

I 

2 

0273 

0247 

0222 

0197 

0171 

CI  46 

0122 

0097 

0072 

0048 

0023 

2 

3 

0273 

0247 

0221 

0196 

0171 

oi46 

0121 

0096 

0072 

0047 

0023 

3 

_4_ 
5 

0272 

0247 

0221 

0196 

0171 

oi46 

0121 

0096 

0071 

0047 

0023 

4 
5 

0272 

0246 

0221 

0195 

0170 

0145 

0120 

0096 

0071 

0046 

0022 

6 

0271 

0246 

0220 

0193 

0170 

0145 

0120 

0095 

0071 

0046 

0022 

6 

7 

0271 

0245 

0220 

0194 

0169 

0144 

0U9 

0095 

0070 

0046 

0021 

7 

8 

0270 

0245 

0219 

0194 

0169 

oi44 

0119 

0094 

0070 

0045 

0021 

8 

9 

0270 

0244 

0219 

0194 

0169 

0143 

0119 

0094 

0069 

0045 

002f 

_9_ 

10 

lO 

0270 

0244 

0219 

0193 

oi63 

0143 

0118 

0093 

0069 

0044 

0020 

ti 

0269 

0244 

02I& 

0193 

0168 

0143 

0118 

C093 

0068 

oo44 

0020 

1 1 

12 

0269 

0243 

0218 

0192 

0167 

0(42 

0117 

0093 

0068 

0044 

0019 

12 

i3 

0268 

0243 

0217 

0192 

0167 

0142 

0117 

0092 

0068 

0043 

0019 

i3 

i4 

0268 

0242 

0217 

0192 

0166 

oi4i 

0117 

0092 

0067 

0043 

0019 

i4 
i5 

i5 

0267 

0242 

0216 

0191 

0166 

oi4i 

0116 

0091 

0067 

0042 

0018 

i6 

0267 

0241 

0216 

0191 

0166 

oi4i 

0116 

0091 

0066 

0042 

0018 

16 

17 

0267 

0241 

0216 

0190 

oi65 

oi4o 

oii5 

0091 

0066 

0042 

0017 

17 

i8 

0266 

0241 

02l5 

0190 

oi65 

oi4o 

oii5 

0090 

0066 

oo4i 

0017 

18 

'9 

0266 

0240 

02l5 

0189 

0164 

ot39 

oii4 

0090 

oo65 

004 1 

0017 

19 

20 

0265 

0240 

02l4 

0189 

0164 

0139 

oii4 

0089 

006  5 

oo4o 

0016 

20 

21 

0265 

0239 

02l4 

0189 

oi63 

0139 

oii4 

0089 

0064 

oo4o 

0016 

21 

22 

0264 

0239 

02l3 

0188 

oi63 

oi38 

oii3 

0089 

0064 

oo4o 

ooi5 

22 

23 

0264 

0258 

02l3 

0188 

oi63 

oi38 

oii3 

0088 

0064 

0039 

001 5 

23 

24 

0264 

0238 

02l3 

0187 

0162 

oi37 

01 12 

0088 

006  3 

0039 

ooi5 

24 

25 

0263 

0238 

0212 

0187 

0162 

01 37 

0112 

0087 

oo63 

oo38 

00 14 

25 

26 

0263 

0237 

0212 

0187 

0161 

01 36 

01 1 2 

0087 

0062 

oo38 

00 1 4 

26 

27 

0262 

0237 

02  I  I 

o}86 

0161 

01 36 

OIII 

0087 

0062 

oo38 

001 3 

27 

28 

0262 

0236 

02II 

0186 

0161 

oi36 

OIII 

0086 

0062 

0037 

001 3 

28 

29 

3o 

0261 

0236 

02  11 

oi85 

0160 

oi35 

Olio 

0086 

0061 

oo37 

0012 

29 

0261 

0235 

0210 

oi85 

0160 

oi35 

OHO 

oo85 

0061 

oo36 

0012 

3o 

3i 

0261 

0235 

0210 

0184 

0159 

oi34 

OHO 

oo85 

0060 

oo36 

0012 

3i 

32 

0260 

0235 

0209 

0184 

0159 

oi34 

0109 

0084 

0060 

oo36 

001 1 

32 

33 

0260 

0234 

0209 

0184 

01 58 

oi34 

0109 

0084 

0060 

oo35 

00 1 1 

33 

34 
35 

0259 

0234 

0208 

oi83 

oi58 

oi33 

0J08 

0084 

0059 

oo35 

0010 

34 
35 

0259 

.0233 

0208 

oi83 

01 58 

oi33 

0108 

oo83 

0059 

oo34 

0010 

36 

0258 

0233 

0203 

0182 

oi57 

Ol32 

0107 

oo83 

oo5S 

oo34 

0010 

36 

37 

0258 

0233 

0207 

0182 

oi57 

Ol32 

0107 

0082 

oo58 

oo34 

0009 

37 

38 

0258 

0232 

0207 

0181 

oi56 

oi3i 

0107 

0082 

oo57 

oo33 

0009 

38 

39 

0257 

0232 

0206 

0181 

oi56 

oi3i 

0106 

0082 

co57 

oo33 

0008 

J9_ 

4o 

40 

0257 

023l 

0206 

0181 

oi56 

oi3i 

0106 

0081 

0057 

oo32 

0008 

4i 

0256 

023l 

020D 

0180 

oi55 

oi3o 

oio5 

0081 

oo56 

00  3  2 

0008 

4i 

42 

0256 

023o 

0205 

0180 

oi55 

oi3o 

oio5 

0080 

00  5  6 

oo3i 

0G07 

42 

43 

o2d:) 

023o 

0205 

0179 

oi54 

0129 

oio5 

0080 

oo55 

oo3i 

0007 

43 

44 

0255 

023o 

0204 

0179 

oi54 

0129 

oio4 

0060 

oo55 

oo3i 

0006 

44 
45 

45 

0255 

0229 

0204 

0179 

oi53 

0129 

0104 

0079 

oo55 

oo3o 

0006 

48 

0254 

0229 

O203 

0178 

oi53 

0128 

oio3 

0079 

oo54 

oo3o 

0006 

46 

47 

0254 

0228 

0203 

0178 

oi53 

0128 

oio3 

0078 

oo54 

0029 

ooo5 

47 

48 

0253 

0228 

0202 

0177 

Ol52 

0127 

oio3 

0078 

oo53 

0029 

ooo5 

48 

49 

■50 

0253 

0227 

0202 

0177 

Ol52 

0127 

0102 

0077 

oo53 

0029 

ooo4 

49 
5o 

0252 

0227 

0202 

0176 

Gl5l 

0126 

0102 

0077 

oo53 

0028 

0004 

61 

0252 

0227 

0201 

0176 

oi5i 

0126 

OIOI 

0077 

0052 

0028 

0G04 

5i 

52 

0252 

0226 

0201 

0176 

oi5i 

0126 

OIOI 

0076 

oo52 

0027 

ooo3 

52 

53 

025i 

0226 

0200 

0175 

01 5o 

OI25 

0100 

0076 

oo5i 

0027 

ooo3 

53 

54 
55' 

025l 

0225 

0200 

0175 

oi5o 

0125 

0100 

0075 

oo5i 

0027 

0002 

54 
'55 

D25o 

0225 

0200 

0174 

0149 

0124 

0100 

0075 

oo5i  • 

OC26 

0G02 

56 

0260 

0224 

0199 

0174 

0149 

0124 

0099 

0075 

00  5o 

0026  i 

0002 

56 

5)7 

035o 

0224 

0199 

0:74 

oi48 

0124 

0099 

0074 

oo5o 

0025 

0001 

57 

dS 

0249 

0224 

0198 

0173 

oi48 

OI23 

0098 

0074 

0049 

0025 

0001 

58 

59 

0249 

0223 

0198 

0173 

oi48 

0123 

0098 

0073 

0049 

0025 

0000 
2°59^ 

59 
S. 

s. 

2°  49' 

2^50' 

i 

2°5V 

2°  52' 

2^53 

2°  54' 

2°  55' 

2=5G' 

2°  57' 

f)0  KOI 

Page  148]  TABLE    XXIII 

To  find  the  Latitude  by  two  Altitudes  of  the  Sun. 


HALF  ELAPSED  TIME. 


MIDDLE  TLME. 


0  Hour. 


0  Hour. 


M. 


Iniinite. 

3. 

2. 

2.36oi8 
2.05916 
I . 

i.883o7 
* .75814 

1 .60125 
I .5820S 
I .5i5i5 
I. 45718 
I .4o6o5 


23 

li. 

25 

26 
27 
28 

3o 
3i 

32 

33 
3, 

~3T" 
36 
3- 
38 

39 
40 
4i 
42 
43 
J4 
45' 
46 
47 
48 

i9_ 
5o 
5i 

52 

53 

55" 

j6 

5? 

58 

59 


I .36o32 
I .31896 
I .28120 
I .24647 
t .21432 

I .18440 
I .i5642 
I .i3oi3 
1 . io536 
I .08193 


56ii 
o35i5 
o  1 5 1 6 
99606 

97777: 


I .05970 

I.03857 
I .01843 
0.99918 

0.98077 

0.96310396023 
0.9.(614194338 
0.92982  92716 
0.91411  91 154 
0.89094189647 


10" 


13833 

29324 
02440 

85959 
74042 


20" 


04701 
5701S 
50494 
•  823 
398. 

353 1 5 
31243 
27522 
24095 
20919 

1 796 1 
5192 
12590 
1 01 36 
07814 


8373o 
23525 

99221 

83732 
723^9 
63322 
5586i 
49496 
43946 
39027 


30" 


40" 


66121 153627 


50" 


43936 


34609 

3o6oo 

6931 

3549 

204 1 2 


7487 
14748 
12171 
09740 
07439 

o5254 
o3i75 
01 192 
99296 

9748c 


o.8843o 
87015 
0.8 5644 
0.84317 
).83o3o 


0.81780 
0.80567 
793S7 
0.78239 

0-77I22 

0.76033 

0.74972 

0.73937 
0.72927 
0-7194') 
o . 70976 
0.70034 
0.691 13:68962 
0.68212  6806^1 
0.6733067185 

o.i:;6l')66'66324 
o.  65620  6548 
o.6479ij64655 
o. 6397816 38 - 
o.63iSi  63(yjo 

0.62400 

(ii632 

60879 
o . 60 1 4o 


a8i9i 
86783 
85420 
84 1 00 
82819 

87576 
8o368 

79193 
7805 1 
7693s 

75854 
74797 
73767 
72760 
7^778 

70818 
5988 


95738 
94o63 
92452 
90899 
89401 


8-953 
86553 
85 197 
83884 
83609 

81372 
80170 
9001 
77863 
76756 


18409  1 3834  0969! 

962 2  5 '9342 2  90790 
81 613^79593  77663 
7070069121J67597 
61986,60690 
54733:53634 
4852047566 
43(_i86'42243 
38258  375o3 

33915 33I3T 


29967 
26349 
23oio 
19910 

17018 
14307 
11757 
09348 
07067 


04901 
02838 
00870 
98988 
97184 


95454 
93790 


29342 
25774 
22478 
19415 


16554 
13872 
1 1 346 
08960 
06699 

o455o 
o2  5o4 
oo55o 
98682 
96891 


95172 
93519' 


92i89'9i928 
90646:9039. 
80156:88913 


59431 
5256i 
46632 

41417 
36762 


32558 
28727 
25207 
21952 
8925 


0" 


16096 
1 3440 
0939 

08574 
6333 


04202 

0217 

00233 

98378 

96600 


75676 

74624 

3597 


87717,87481 
86324:86096 
84976184755 
8366983455 
82401 182193 
81 169,80967 
7997379777 
78809178618 
7767777491 
76574176393 

7549975323 
7445 1  74279 


734 


0.59414 


02271 

6i5o(i 
60755 
600 1 8 
59294 


72595:7243 

71616171455 

70660  7o5o3 

69725169571 

''-'■'°--   68660 


68811 
67916 
67040 

66782 
65342 

6  ;5l9 

63711 
62919 


62142 
61 J80 
606  3 1 

59897 
59175 


67769. 
66896 

6604 1 
653o4 
64383 
63578 
62789 

620 1"4  6788 

61254 

6o5o8 


73261 

72266 

71295 

70346 

69418 

685  TO 

6762' 

66752 

65900 

65o66 

64248 

63/t45 

62659 


94892 
93250 
91669 
90143 
88671 


87247 
85870 
84535 
83242 
986 


80767 
79581 
78428 
773c6 
76212 


59775 
59056 


01  129 
6o3S5 
59654 
5893- 


75 1 47 
74107 
73093 
72103 
7 1 1 36 


Jnf.lNeg 
2. 

2.94085 
3. 

3.24187 
3..  4 1 796 
3.54289 

3.63978 
3.71895 
3.78588 
3.84385 
3.89498 

3.94071 

3.98207 

4- 

4.01983 

4 -05456 

4.08671 


7019c 

69265 

6836 

67476 

66609 


65760 
6  -(928 
64ii3 
633 1 3 
62529 

61759 
61004 
60262 
59534 
588i8j 


23 
25 

26 
27 
28 

?9 
3o 
3i 

32 

33 

35 
36 

37 
38 
39_ 

4o 
4i 
42 
43 
_44_ 
45 
46 

47 

48 
49_ 
5o 
5i 

52 

53 

i^i 
55 
56 
'J7 
58 
59 


11663 
4-i446i 

17090 
4-19567 
4.21910 

4-24i33 
4-2624( 
4.28260 
4.3oi85 

4-32026 


_10" 

16270 

00779 
27663 
44 1 44 
56061 


654o2 
78085 
79609 
85380 
90294 


20" 


46373 

06578 
3oS82 
46371 
57764 


947S8 
98860 

0258i 
06008 
09 1 84 


12142 
14911 
i75i3 
996 
22289 


4-3:^793 
4.35489 
4.37121 
4.38692 
40209 


4.41673 
43o88 
4.44459 
4.45786 
4.47073 
4-48323 
4-49536 
4.50716 
4.5i864 
4.52981 


24492 
26588 
28587 
3o497 
32026 


06781 
74242 
80607 
86157 
91076 

95494 
995o3 

o3i72 
06554 
09691 


30"  40"  50" 


63982 

11694 

33878 
48490 
09408 
6877^ 

75370 

8i583 
87017 
91845 
96188 


76476 

16269 

36681 
5o5io 
60982 


0013600761 
03754  04329 


694 1 3 
76469 
82537 
S7860 
9260c 
96872 


86167 

2o4o8 
39313 

02440 
o2  5o6 


12616 
15355 
17932 
2o363 
22664 


34080 
35765 
37387 
38949 
4o456 

47^ 
43320 
44683 
46oo3 
7284 


48527 
49735 
50910 

52052 

53i65 


4.54070I54249 


4.55i3i 
4.56166 
4.57176 
4.58i63 


4.59127 
4 . 60069 
4  •  6(  1990 
4.61891 
4.62773 


53o6 
56336 
57343 
58325 


59285 
60223 
6ii4i 
62039 
62918 


63779 
64622 
65448 
662  58 
67053 


4.63637 
.64483 
4.65312 
4.66125 
4.66922 

4.67703  67832 

4.6847if6S597 

'1.69224 

4.69963 

4.70689 


24849 
26928 
28911 
30807 
32623 


34365 
36o4o 
37651 
39204 
40702 

42i5o 
4355o 
44906 
46219 
47494 


8731 

9933 

5i  102 

52240 

53347 


07093 
10193 

73oS5 
1 5706 
1 8346 
20755 
2  3o36 


25202 
27265 
29233 
3iii5 
32919 


34649 

363x3 

37914 

39457 

40947 

42386 

43779 

45127 

4643 

47702 


07625 
10688 


13549 
1623 
1S757 
21 143 
23404 
553 
27599 
29553 
3i42i 

332  12 


54427 
55479 
565o6 
57508 
58487 


59443 
60878 
61292 
62187 
63o63 


63921 

6476 1 

65584 

66392 

67184 

G7961I 

6S723: 

6934869472} 

70085  70206; 

70809I709281 


48934 
5oi3o 
5129.' 
52426 
53529 

54604 
55652 
56674 
57673 
58648 

59600 
6o532 
61443 
62334 
63207 


64062 
64899 
65720 
66525 
67314 


34931 
36584 
38175 
39709 
41190 
42622 
44007 
45348 
46648 
47910 


70672 
77542 
83471 
8686 
93341 


97545 
01376 


081 5 1 
178 


14007 
i6663 
19164 
21529 
23770 


5901 
27931 
29870 
31725 
335o3 


49186 
50826 
5i485 
52612 
53710 


352II 

36853 

38434 

39960 

1432 


2856 
44233 
45568 
46861 
481 17 


547S0 
55824 
56842 

57837I58000 
58So8  58967 


49336 

5o522 

1675 
52797 
53891 
54956 

55996 
57010 


59757 

60681 
61593 
62481 

6335 1 


68089 
68849 
69595 
70338 
71047 


6420  3 
65o37 
65855 
66658 
C->7444 


6821668344 


59913 

6o838 
6 1 742 
62627 
63494 


64343 
65175 
65990 
66790 
67574 


6S974 
69718 
70449 
7 1 1 66 


19099 


70569 
71286 


TABLE  XXIII               [Page  149 
To  find  the  Latitude  by  two  Altitudes  of  the  Sun. 

HALF  ELAPSED  TIME. 

MIDDLE  TIME. 

1  Houii. 

1  Hour. 

M. 

o 

I 

2 

3 
4 

5 
6 

7 
8 

9 

10 

II 

12 

i3 
i4 
i5 
i6 

17 
i8 

19 

20 
21 
22 
23 
24 
25 
26 

27 
28 

29 

3o 
3i 

32 

33 

34 
35 
36 
37 
38 
39 
40 
4i 
42 
43 
44 
45 
46 

47 

48 
49 
5o 
5i 

52 

53 
54 
55 
56 
57 
58 
59 

0" 

]0' 

20" 

30" 

40" 

58232 
57539 
56857 
56187 
55528 
54880 
54242 
536 1 4 
52995 
52387 

51787 
5.197 
5o6i5 
5oo42 

49477 

50" 

58ii5 
57424 
56745 
56076 
55419 

54773 
541 36 
535IO 
52893 
52286 
5 1688 
51099 
5o5i9 

49947 
49383 

M. 

0 
I 
2 
3 
4 
5 
6 

7 
8 

9 
10 
II 
12 
i3 
14 

0" 

10" 

20" 

30" 

40" 

50" 

0. 58700 
0.57999 
0.57310 
o.5ti633 
0.55966 

58583 
57884 
57196 
56521 
55856 
552o3 
54559 
53926 
533o3 
52690 

5208b 
51491 
50905 
5o327 
49758 

58465 
57768 
57083 
56409 
55747 
55095 
54453 
53822 
53x00 
52589 
5 1 986 
51393 
5o8o8 
5o232 
49664 

58348 
57653 
56970 
56298 
55637 

4.71403 
4.72104 
4.72793 
4.73470 
4.74137 

7i52o 
72219 

72907 
-3582 
74247 

71638 
72335 
73020 
73694 
74356 

71755 
72450 
73i33 
738o5 
74466 

7.871 
72564 
73246 
73916 

74575 

71988 
72679 
73358 
74027 
74684 
75330 

75967 
76593 
77210 
77817 
7841 5 
79004 
79584 
801 56 
80720 

81275 
81823 
82  363 
82896 
83421 

83940 
8445i 
84956 
85454 
85945 

8643o 
86909 
87382 
87849 
883ii 

88766 
89216 
89661 
90100 
90534 
90963 
91387 
91806 
92221 
92630 

o.553i I 
0. 54666 
0 . 54o3 1 
0.53406 
0.52791 

54987 
54347 
53718 
53098 
52487 

4.74792 
4.75437 
4.76072 
4.76697 
4.77312 

74900 
75544 
76177 
76800 
7741 3 

75008 
75650 
76281 
76903 
775i4 

751 16 
75756 
76385 
77005 
77616 

78217 
78809 
79392 
799()6 
8o533 

81091 
8i64i 
82184 
82719 
83247 
83768 
84281 
84788 
85288 
85782 

86269 
86750 
87225 
87694 
88 1 58 

75223 
75861 
76489 
77108 
77716 
7S3i6 
78906 
79488 
8006. 
80626 
8ii83 
81732 
82274 
82808 
83334 

83854 
84366 
84872 
85371 
85864 
86350 
8683o 
87304 
87772 
88234 

0.52 1 86 
0.51589 
0.5 1 002 
o.5o42  3 
0.49852 

5 1 886 
51294 
5071 1 
5oi37 
49570 

4.779'7 
4.78514 
4.79101 
4.796S0 
4.8o25i 

78017 
78612 
79198 
79776 
80345 

78117 
78710 
79295 
7987 1 
80439 

0.49290 
0.48736 
0.48189 
0.47650 
0.47119 

49197 

48644 
48099 
47561 
47o3i 

49104 
48553 
48009 
47473 
46944 
46422 
45907 
45399 
44898 
444o3 

49012 

4846a 

47919 
47384 
46856 

48920 

48371 
47829 
47295 
46769 

46249 
45737 
4523i 
44732 
44239 

43753 
43273 
42799 
4233i 
41869 

48828 
48280 
47740 
47207 
46682 

461 63 
45652 
45 1 47 
44649 
44 1 58 

43673 
43194 
42721 
42254 
41792 
4i337 
40887 
40442 
4ooo3 
39569 
39140 
38716 
38297 
37882 
37473 
37068 
36668 
36272 
35881 
35494 
35111 
34732 
34357 
33987 
33620 

33257 
32899 
32543 
32192 
3 1844 
3 1 5oo 
3ii59 
30822 
3o48S 
3oi58 

i5 
16 

17 
18 

'9 
20 
21 
22 

23 

24 

25 

26 
27 
28 
29 

3o 
3i 

32 

33 

34 
35 
36 
37 
38 
39 

40 
4i 
42 
43 
U 
45 
46 
47 
48 
49 
5o 
5i 

52 

53 

54 
55 
56 
57 
58 
59 

4.80813 
4.81367 
4.81914 
4.82453 
4.82984 
4.835o8 
4.84025 
4.84536 
4.85o39 
4.85536 

80906 
81459 
82004 
82542 
83072 

83595 
84iii 
84620 
85 1 22 
856i8 

80999 
8i55o 
S2094 
82630 
83i59 
83681 
84196 
84704 
85205 
85700 

0.46595 
0.46078 
0.45567 
0.45064 
0.44567 

465oS 
45992 
45483 
44981 
44485 

46335 
45822 
453i5 
448 1 5 
4432  1 

43834 
43353 
42878 
42409 
41945 

0.44077 
0.43592 
o.43ii4 
0.42642 
0.42176 

43995 
435i2 
43o35 
42564 
42099 

43915 
43432 
42956 
42486 
42022 

4.86026 
4.865n 
4.86989 
4.87461 
4.87927 

86108 
86591 
87068 
87539 
88004 
88463 
88917 
89365 
89808 
90246 

86188 
86671 

87147 
87617 
8S081 

88539 
88992 
89439 
89881 
903 1 8 

0.41716 
0.41261 
0.40812 
o.4o368 
0.39930 

4i64o 
41186 
40738 
40295 
39857 
39425 
38998 
38575 
38 1 58 
37745 

4i564 
4ii  II 
40664 
40222 
39785 

39354 
38927 
385o6 
380S9 
37677 
37270 
36867 
36469 
36076 
35687 

4 1 488 
4io36 
40590 
40149 
39713 

4i4i2 
4096 1 
4o5i6 
40076 
39641 

4.88387 
4.88842 
4.89291 
4.89735 
4.90173 

8861 5 
89067 
89513 
89954 
90390 

90821 

91247 
91667 
92083 
92494 

88691 
89.42 
89587 
90027 
90462 
90892 
9i3i7 
91737 
92152 
92562 

0.39497 
0.39069 
0.38646 
0.38227 
0.37814 

39282 
38856 
38436 
38o2o 
37609 

37203 
36801 
364o3 
36oi  I 
3562  2 

3921 1 

38786 
38366 
37951 
37541 
37135 
36734 
36338 
35946 
35558 

35174 
34795 
34420 
340 -[S 
33681 

333i8 
32958 
32602 

3225o 

31902 

3 1 2 1 6 
30878 
3o544 
3o2i3 

4 . 90606 
4.91034 
4.91457 
4.91S76 
4.92289 

90678 
91  io5 
91528 
91945 
92358 

90749 
91176 
91597 
92014 
92426 

0.37405 
0.37001 
0.36602 
0.36206 
o.358r6 

37338 
36934 
36535 
36i4i 
35751 

4.92698 
4.93102 
4.93501 
4.93897 
4.94287 

92765 
93169 
93568 
93962 
94352 

92833 
93236 
93634 
94027 
94416 

92900 
93302 
93700 
94092 
94481 

92968 
93369 
93765 
94 1 57 
94545 

93o35 
93435 
9383 1 
94222 
94609 

0.35429 
o.35o47 
0 . 34669 
0.34295 
0.33925 

35365 
34984 
34607 
34233 
33864 

35302 
34921 
34544 
34172 
338o3 

33438 
33078 
32720 
32367 
32018 

35238 
34858 
34482 
341 10 
33742 
33378 
3^018 
32661 
32309 
31960 

3i6i4 
31272 
30934 
3o599 
3026S 

4.94674 
4.95o56 
4.95434 
4.95808 
4.96178 

94738 
95119 
95496 
95870 
96239 

94801 
95182 
95559 
95931 
96300 

94865 
95245 
95621 
95993 
96361 

94929 
95308 
95683 
96055 
9642? 

94992 
95371 
95746 
96116 
96483 
96846 
97204 
97560 
9:911 
98259 

98603 
98944 
99281 
99615 
99945 

0.33559 
0.33197 
0.32839 
0.32485 
0.32 1 34 

33499 
33i37 
32780 
32426 
32076 

4.96544 
4 . 96906 
4.97264 
4.97618 
4.97969 

96604 
96966 
97323 

97677 
98027 

98374 
98717 
99057 
99393 
99725 

966()5 
97025 
97383 
97736 
98085 

98437 
98774 
991 1 3 
99448 
99780 

96725 
97085 
97442 

9779^ 
98143 

98489 
98831 
99169 
99504 
95835 

96785 
97145 
97501 
97853 
98201 

98546 
98887 
99225 
99559 
99S90 

0.31787 

o.3r443 
o.3iio3 
0.30766 
o.3o433 

3i729'3i672 
3i386i3i329 
3io46l3o99o 
30710  3o655 
3o378[3o323 

4.98316 
4.98660 
4.99000 
4.99337 
4.99670 

^^seim                                    TABLE  XXIII. 

To  find  the  Latitude  by  two  Altitudes  of  the  Sun. 

HALF  ELAPSED  TIME. 

MIDDLE  TIME. 

2  Hours. 

2  Hours. 

31. 

o 
I 

2 

3 
4 
5 
6 

7 
8 

9 

lO 

II 

12 

i3 
i4 
i5 
i6 

17 
iS 

19 

20 
21 
22 
23 

24 
25 

26 
27 
28 
29 

3o 
3i 

32 

33 
34 
35 
36 

37 
38 
39 

4o 
4i 
42 
43 
44 
45 
46 
47 
48 
49 
5o 
5i 

52 

53 
54 
55 
56 
57 
58 
59 

0" 

10" 

20" 

30" 

40" 

50" 

31. 

0" 

10" 

20" 

30" 

40" 

50'' 

D.3oio3 
0.29776 
0.29453 
0.29133 
0.28816 

3oo48 
29722 
29400 
29080 
28764 

29994 
29068 
29346 
29027 
2871 1 

29939 
29614 
29293 
28974 
28659 

28346 
28037 
27731 
27428 
27127 

29885 
29561 
29239 
28921 
28607 

29831 
29507 
29186 
28869 
28554 

0 
I 
2 
3 
4 
5 
6 
7 
8 

9 
10 
II 
12 
i3 
14 
i5 
16 

17 
18 

19 
20 
21 
22 

23 

24 

25 

26 

27 
28 
29 

3o 
3i 

32 

33 
34 
35 
36 

37 
38 
39 

40 
41 
42 
43 
44 
45 
46 
47 
48 

49 
5o 
5i 

52 

53 
54 
55 
56 

57 
58 
59 

5 . ooooo 
5.00327 
5.oo65o 
5.00970 
5.01287 

ooo55 
oo38i 
00703 

0IO23 

f)i339 

00109 
00435 
00757 
01076 
01892 

01705 
02014 
02821 
02625 
02926 

00164 
00489 
00810 
01129 
01444 

00218 
00  54  2 
00864 
01 182 
01496 

00272 
00596 
009 1 7 
01284 
01549 

0.28502 
0.28191 

0.27884 
0.27579 
0.27277 

28450 
28140 
27833 
27529 
27227 

269-'9 
26633 
26341 
2605 1 
25763 

28398 
28089 
07782 
27478 
27177 

26879 
26584 
26292 
26003 
25716 
25432 
25i5o 
24872 
24595 
24322 

28295 
27986 
27680 
27378 
27078 

28243 
27935 
27630 
27327 
27028 
2673, 
26438 
26147 
25859 
25573 

5.01601 
5.01912 
5.02219 
5.02524 
5.02826 

5.o3i25 
5.03421 
5.03714 
5 . o4oo4 
5.04292 

5.04577 
5.04859 
5.o5i39 
5.o54i6 
5.o569Ci 

5.05962 
5.06232 
5.06498 
5.06763 
5.07025 

oi653 
01963 
02270 
02574 
02876 

01757 
02066 
02872 
02<375 
02976 

01808 
021 17 
02423 
02725 
08025 

01860 
02168 
02473 
02776 
08075 
08872 
o3665 
08956 
t)4244 
o453o 

0.26978 
0.26682 
0.26389 
0.26099 
0.258(1 

2683o 
26535 
26244 
25955 
25668 

25385 
25io4 
24825 
24550 
24276 

26781 
36487 
26195 
25907 
25621 

o3i74 
03470 
08762 
o4o52 
04340 

08224 
o35i9 
o38ii 
o4ioo 

04387 

08278 
o3568 
08859 
o4i48 
04435 

08822 
o36i6 
08908 
04196 
04482 

0.25526 
0.25244 
0 . 24964 
0.24687 
0.2441 3 

25479 
25197 
24918 
24641 
24367 

25338 
25o57 
24779 
24504 
24231 

25291 

250Ii 

24733 

24458 
24186 

04624 
04906 
o5i85 
05462 
o5736 

04671 

04953 

o523i 
o55o8 
05781 

04718 
04999 
05278 
o5553 
05827 

04765 
o5o46 
o5324 
05599 
05872 

04812 
(J5092 
05370 
o5645 
05917 

0.24141 
0.23871 
o.236o5 
0.23340 
0.23078 

24096 
23827 
2356o 
23296 
23o35 
22775 
22519 
22264 
22012 
21762 

24o5i 
23782 
235i6 
23253 
22991 

22732 
22476 
22222 
21970 
21720 

24006 
23738 
23472 
23209 
22948 

23961 
23693 
23428 
23i65 
22905 

23916 
23649 
23384 

23l22 
22862 

06007 
06276 
06543 
06807 
07068 

o6o52 
06821 
06587 
o685o 
07 1 1 2 

06097 
06365 
o663i 
06894 
07155 

06142  06187 
0641006454 
0667506719 
06938106981 
0719SI07241 

0.22819 
0.22561 

0.223o6 
0.22054 

o.2i8o3 

22690 
22433 
22180 
21928 
21679 

22647 
22391 

22138 

21887 

21 638 

22604 
22349 
22096 

21 845 
21596 

5.07284 
5.07542 
5.07797 
5 . 08049 
5 .o83oo 

07328 
07584 
07839 
(j8o9i 
0834 1 

07871 
07627 
078S1 
081 33 
08383 
oS63o 
08875 
091 18 
09359 
09597 

07418 
07670 
07928 
08175 
08424 

07456 
07712 
07965 
08216 
o8465 

07499 
07754 
08007 
(.82  58 
o85o7 

08753 
08997 
09289 
09478 
09716 

09951 
10184 
io4i6 
10645 
10872 

1  1097 
1 1820 
1 1542 
1 1761 
11978 
12194 
12407 
1 26 19 
12829 
i3(.37 
i3243 

.3447 
13650 
i3S5i 
i4n5o 

0.21555 
0.2 1 309 
0.21 066 
0.20824 
0.20585 

2i5i4 
21269 
21025 
20784 
20545 

21473 
21228 
20985 
20744 
2o5o6 

21432 
21187 
20945 
20704 
2o466 

2023o 
19996 
19764 
19534 
19306 

19081 
18857 

1 8635 
184 1 5 
18197 
17981 
17767 
17554 
17344 
17135 

21391 

21147 
20905 
2o665 
20427 

2i35o 
21 106 
20864 
20625 
2o387 

5.08548 
5.08794 
5.09037 
5.09279 
5.09518 

085S9 
08834 
09078 
09319 
09558 

08671 
08916 
09158 
09899 
09687 

09878 
1 0 1 07 
10889 
10569 
10797 

11022 
1 1 246 
1 1 468 
11 688 
1 1 906 

(JS7 1 2 
08956 
09198 
09438 
09676 
099 1 2 
ioi46 
10877 
10607 
io834 
1 1  o6<j 
1 1 283 
ii5o5 
1 1725 
11942 

0.20348 

0.20Il3 

0. 19S80 
O.I  96  ■(9 
0. 19420 

2o3o9 
20074 
19841 
1961 1 
19382 

20269 
2oo35 
1 9803 
19572 
1 9344 

2tll91 

19957 
19726 
19496 
19269 

19043 
18820 
18598 
•  8378 
i8i6r 

17945 
17731 
17519 
17309 
1 7 1 0 1 

201 52 

19919 
19687 
19458 
19231 
1 9006 
18783 
i856i 
18342 
18125 
17909 
17696 
17484 
17274 
17066 

5.09755 
5.09990 
5. 10223 
5.10454 
5.10683 

09794 
10029 
10262 
10492 
10721 

09834 
10068 
io3oo 
io53i 
10759 

0. 19193 
0 . 1 8968 
0.18746 
0.18535 
0 . 1 83o6 

19156 
1S931 
1S709 
18488 
18269 

i8o53 
17838 
17625 
17414 
17205 

19118 
18894 
18672 
i845i 
18233 
1 80 1 7 
17802 
17590 
17379 
17170 

5.10910 
5.III35 

5.II357 
5.11578 
5. 1 1797 

10947 
11172 
1 1394 
ii6i5 
1 1 834 

10985 
1 1209 
ii43i 
1 1652 
11870 

0. 18089 
0.17S74 
0 . r  7660 
0 . 1 7449 
0. 17239 

5. i2oi4 
5. 12229 
5.12443 
5.12654 
5.12864 

12o5o 

12265 

12478 
12689 
12898 

1 2086 
12801 
i25i3 
12724 
12933 

12122 
i233( 
12549 
12759 
12968 

i3i75 
i338o 
i3583 
18784 
18984 

I2i58 
12872 
12584 
12794 
i3oo2 

18209 
i34i3 
i36i6 
i38i8 
14017 

0. 17032 
0.16826 
0. 16622 
0 . 1 64 1 9 
0. 16219 

0. 16020 
U.I5823 
0. 15627 
0.15434 
0. 15242 

16997 
16792 
i6588 
16386 
16186 

16963 
16758 
16554 
x6352 
i6i52 

16928 
16723 
1 6520 
i63i9 
161 19 
1 592 1 
15725 
i553o 
i5338 
i5i46 

16894 
1 6690 
16487 
162S5 
16086 

i5888 
15692 
15498 
i53o6 
i5ii5 

16860 
i6656 
16453 
16252 
i6o53 

T5856 
i566n 
15-166 
i52-4 
i5o83 

5.18071 
5.13277 
5.i348i 
5.13684 
5.13884 
5.i4o83 
5.14280 
5.14476 
5.146G9 
5.14861 

i3io6 
i33ii 
i35i5 
18717 
13917 

1 3 1 4o 
1 3345 
13549 
i375i 
18951 

15987 
15790 
1 5595 
1 5402 

l52lO 

15954 
15758 
i5563 
15370 
15178 

i4i  16 
i43i3 
i45o8 
14701 
14893 

i4i49 
14345 
14540 
14733 
1492.5 

14182 
14378 
14573 
14765 
14957 

i42i5 
i44ii 
1 460  5 

14797 
149S8 

14247 
14443 
14637 
T4S29 
1 5o2o 

TABLE 

XXIIL                     [P^Se  151 

To  find  the  Latitude  by  two  Altitudes  of  the  Sun. 

HALF  elapsp:d  time. 

MIDDLE  TIME. 

3  Hours. 

3  Hours. 

M. 
o 

0" 

10" 

20" 

30" 

40" 

50" 

31. 
0 

0" 

10" 

i5o83 

20" 
i5i  i5 

30" 

40" 

50" 

o.i5o5i 

1 5o2o 

14988 

14957 

14956 

14894 

5.i5o5i 

i5i46  15177I 

15209 

I 

O.I4863 

i4832 

i48oo 

14769 

14738 

14707 

I 

5.i524o 

15271 

i53o3 

i5334i5365| 

15396 

2 

0.14676 

1 4645 

i46i4 

14583 

1455? 

i452i 

2 

5.15427 

15458 

1 5489 

i552o 

i555i 

i5582 

3 

0.14490 

1 4460 

14429 

14398 

1 4368 

14337 

3 

5.i56i3 

15643 

15674 

i57o5 

15735 

15766 

4 
5 

0.14307 

14276 

14246 

i42i5 

i4i85 

i4i55 

4 

5.15796 

15827 

i5857 

i5888 

15918 

15948 

o.i4i24 

14094 

1 4064 

i4o34 

i4oo4 

13974 

5 

5.15979 

1 6009 

16039 

16069 

16099 

16129 

6 

0. 13944 

13914 

i3884 

i3854 

i3S24 

1 3794 

6 

5.16159 

16189 

16219 

16249I16279I 

i63o9 

0.13765 

13735 

i37o5 

13676 

i3646 

1 36 1 7 

7 

5.16338 

16368 

16398 

16427 

16457 

16486 

8 

0.13587 

i3558 

i352S 

13499 

13470 

1 344 1 

8 

5.i65i6 

16545 

16575 

16604 

i6633 

16662 

9 

10 

0.1 34 1 1 

i33S2 

i3353 

13324 

13295 

i3266 

9 

10 

5.16692 

16721 

16750 
16924 

16779 
16953 

16808 
16982 

16837 
17010 

o.i3.>3- 

i32o8 

i3i79 

i3i5o 

l3l2I 

1 3093 

5.16866 

16S95 

II 

0. i3u64 

i3o35 

i3oo7 

12978 

12950 

1 292 1 

1 1 

5. 17039 

17068 

17096 

17125 

17 1 53 

17182 

12 

0.12S93 

12864 

12836 

12808 

12779 

12731 

12 

5.17210 

17239 

17267 

17295 

17324 

17352 

i3 

0. 12723 

12695 

12666 

12638 

I26IO 

12582 

i3 

5.17380 

17408 

17437 

17465 

17493 

17521 

i4 
i5 

0. 12  554 

12526 

12499 

12332 

1 247 1 

12443 

i24i5 

i4 

5.17549 

17577 

17604 

17632 

1 7660 

176S8 

0.12387 

i236o 

i23o5 

12277 

12249 

i5 

5.17716 

17743 

17771 

17798 

17826 

17854 

i6 

0. 12222 

12195 

12167 

I2l40 

I21l3 

i2o85 

16 

5.17881 

17908 

17936 

17963 

17990 

18018 

17 

O.I2058 

12o3l 

12004 

1 1977 

1 1949 

11922 

17 

5.18045 

18072 

18099 

18126 

i8i54 

18181 

i8 

0.11S95 

11868 

11842 

ii8i5 

11788 

11761 

18 

5.18208 

18235 

18261 

18288 

i83i5 

■  8342 

19 

20 

0 . 1 1 734 

11708 

11681 

ii654 

11628 

1 1601 

'9 

5.18369 

18395 
18555 

18422 
i858i 

18449 
i86(j8 

18475 

i85o2 

0. 1 1 575 

11 548 

1]522 

1 1495 

1 1469 

11443 

20 

5.18528 

18634 

18660 

21 

0. ii4[6 

1 1390 

1 1 364 

11 338 

Il3l2 

11285 

21 

5.186S7 

18713 

18739 

18765 

18791 

18818 

22 

0. 1 1259 

11233 

1 1207 

11181 

iii56 

1 1 1 3o 

22 

5.18844 

18870 

18896 

18922 

18947 

18973 

23 

0. II 104 

1 1078 

I1052 

11027 

1 1 00 1 

10975 

23 

5.18999 

19025 

19051 

19076 

19102 

19128 

24 
25 

0.10950 
0. 10797 

10924 
10772 

10899 
10746 

10873 
10721 

10848 
1 0696 

10822 
10671 

24 

25 

5.19,53 

19179 

19204 

i923t 

19255 

19281 

5.19306 

19331 

19357 

19382 

19407 

19432 

26 

0 . 1 0646 

10620 

10595 

105/0 

10545 

10520 

26 

i>.  19457 

1948^ 

19508 

19533 

19558 

19583 

27 

0.10496 

10471 

io446 

10421 

10396 

10371 

27 

5 . 1 9607 

19632 

19657 

19682 

19707 

19732 

28 

0.10347 

Io322 

10298 

10273 

10248 

10224 

28 

5.19756 

19781 

19805 

19830 

19855 

19879 

29 

3o 

0.1 01 99 

IOI75 

ioi5i 

10126 

10102 

10078 

29 

5 . 1 9904 

19928 

19952 

19977 

20001 

20025 

0. ioo53 

10029 

iooo5 

09981 

09957 

09933 

3o 

5.2oo5o 

20074 

20098 

20122 

20 1 46 

20170 

3i 

0.09909 

09885 

09861 

09837 

09813 

09789 

3i 

5.20194 

20218 

20242 

20266 

20290 

2o3i4 

3;! 

0.0976509741 

09718 

09694 

09670 

09647 

32 

5.20338 

2o362 

2o385 

20409 

20433 

2o456 

33 

0.0962309599 

09576 

09552 

09529 

09506 

33 

5.20480 

2o5o4 

20527 

2o55i 

20574 

20597 

34 
35 

0.094s 2 

09459 

09435 

09412 

o9389[o9366 

34 

5.20621 

20644 

20668 

20691 

20714 

20737 
20876 

0.09343 

093  I  9 

09296 

09273 

09250  09227 

35 

5.20760 

20784 

20807 

2o83o 

20853 

36 

0.09204 

09181 

09158 

09136 

091 1 3  09090 

36 

5.20899 

20922 

20945 

20967 

20990 

210l3 

37 

0 . 09067 

09044 

09022 

08999 

08977  0S954 

37 

5.2io36 

21059 

21081 

21 104 

21126 

21149 

38 

0.08931 

08909 

08886 

08S64 

0884508819 

38 

5.21172 

21194 

21217 

21239 

21261 

21284 

39 

40 

0.08-97 

08775 
0864  I 

08752 
08619 

08730 
08597 

08708 

08686 

39 

40 

5.21 3o6 

21328 

2i35i 

21373 

21395 

21417 

0.08664 

08575 

08553 

5.21 439 

21462 

21 484 

2i5o6 

21528 

2i55o 

4i 

0.0853 1 

o85io 

08488 

08466 

08444 

08422 

4i 

5.21572 

21593 

2i6i5 

216J7 

21659 

21681 

42 

0 . o84o 1 

o837Q 

08357 

08336 

o83i4 

08293 

42 

5.21 702 

21724 

2  1746 

21767 

21789 

21810 

43 

0.08271 

0S250 

0S228 

oSao7 

081 85 

08164 

43 

5.2x832 2i853 

21875 

2189C 

21918 

21939 

45 

0.08143 

08121 

08100 

08079 

o8o58 

o8o36 

45 

5.21960  21982 

220o3 
22l3o 

22024 

22045 

22067 

o.oSoi5 

07994I07973 

07952 

07931 

07910 

5.22088  22109 

22l5l 

22172 

22193 

46 

0.07889 

07868 

07848 

07827 

07806 

07785 

46 

5.222l4|22235 

22255 

2227C 

22297 

223l8 

47 

0.07765 

07744 

07723 

0770307682 

07661 

47 

5.22338  22359|2238o 

224OG 

22421 

22442 

48 

0 .  0764 1 

07620 

07600 

0757907559 

07539 

48 

5.22462  22483 

2  2  5o3 

22524 

2  2544 

22  564 

49 
5o 

0.07518 

07498 

07478 

07458  0-437 

07417 

49 

5.22585 22605 

22625 

22645 
22766 

22666 
22786 

22686 
22806 

0.07397 

07377 

07357 

07337  07317 

0729- 

5o 

5.22706J22726 

22746 

5i 

0.07277 

07257 

07237 

0721707197 

07178 

5i 

5.2282622846 

22866 

22886 

22906 

22925 

52 

f).  07 1 58 

07 1 38 

07119 

07099  07079 

O7o6f) 

52 

5.22945  22965 

22984 

23oo4 

23o24 

23o43 

53 

0 . 07040 

07021 

07001 

06982  06962 

06943 

53 

5.23o63  23o82 

23l02 

23121 

23i4i 

23 160 

54 
55 

0.06923 

06904 

06885 

06866  06846 
06751  06731 

06827 
067 1 2 

54 

55 

5.23i8o  23199 

5.23595J23314 
5.23410  23429 

23218 

23333 

23237 

23257 

23276 
23391 

0.06808 

06789 

06770 

23352 

23372 

56 

0 . 0(5693 

06674 

o6656 

0663706618 

06599 

56 

23447 

23466 

23485  235o4 

57 

o.o658o 

o656i 

o6543 

o6524  o65o5 

06487 

57 

5.23523  23542 

2356o 

23579 

23598  236i6 

58 

0 . 06468 

06449 

0643 1 

064 12  06394 

06375 

58 

5.23635123654 

23672 

23691 

2370923728 

59 

0.06357 

o6338 

o632o 

o63o2  06283 

06265 

59 

5.2374623765 

2J783 

238oiJ2382o|23838| 

"^'^^J               TABLE  XXIII 

To  find  the  Latitude  by  two  Altitudes  of  the  Sun. 

HALF  ELAPSED  TIME. 

MIDDLE  TIME. 

4  Hours. 

4  Hours. 

r.i. 

o 

1 

2 

3 
4 
5 
6 

7 
8 

9 

lO 

II 

12 

i3 

i4 
i5 
i6 

17 
i8 

19 

20 
21 
22 
23 

24 
25 
26 
27 
28 
29 

3o 
3i 

32 

33 

34 
35 
36 
37 
38 

39 

4o 
4i 
42 
43 
4^ 
45 
46 
47 
48 
49 
5o 
5i 

52 

53 

54 
55 
56 

57 
58 
59 

0"  1  10" 

20" 

30" 
06192 
06084 
05977 
05871 
05766 
o5662 
05559 
05457 
05356 
05257 
o5i58 
o5o6o 
04964 
04868 
04774 
04680 
04588 
04496 
o44o6 
043 17 

40" 

50" 

M. 

0 
I 
2 
3 
4 
5 
6 

7 
8 

9 
10 
II 
12 
i3 
i4 
i5 
16 

17 
18 

19 

0" 

10" 

2387'4 
23983 
24091 
24197 
24302. 

20" 

23892 
24001 
24108 
24215 
24320 

30" 

40" 

50" 

0.06247 
0.061 3b 
0 . o6o3o 
1;. 05924 
o.o58i8 

06229 
06120 
06012 
05906 
o58oi 

06211 

(j6i02 

05995 
o5888 
05783 

06174 
06066 
05959 
o5853 
05748 

J5645 
05542 
o544o 
o534o 
o524o 
o5i42 
o5o44 
04948 
04852 
04758 

0466  5 
04573 
04481 
04391 
o43o2 

061 56 
06048 
05941 
05836 
o573i 
05627 
05525 
05423 
o5323 
05224 

o5i25 
o5o28 
04932 
04837 
04743 

04649 
04557 
04466 
04376 
04287 

5.23856 
5.23965 
5.24073 
5.24179 
5.24285 

2391 1 
24019 
2412O 
24232 
24337 
24441 
24544 
24646 
24747 
24846 

24945 
25o43 
25i39 
25235 
25329 

28929 
24037 
24144 
24251-) 
24355 
24458 
24561 
24663 
24763 
24863 
24961 
25o59 
25i55 

2525l 

25345 

23947 
24o55 
24 1 62 
24267 
24372 

24476 
24578 
24680 
24780 
24879 

24978 
25075 
25171 
25266 
2  5360 

0.05714 
o.oSoio 
o.o55o& 
0  o54o7 
o.o53o6 

o56y6 
05593 
05491 
05390 
05290 

05679 
05576 
05474 
05373 
05273 

o5i74 
o5o77 
04980 
04884 
04789 

5.24389 
5.24493 
5.24595 
5.24696 
5.24797 

24407 
24510 
24612 
24713 
2481 3 

24424 
24527 
24629 
24730 
2483o 

0.0020/ 

o.o5io9 
o.o5oi2 
0.04916 
o.(j482i 

05191 
05093 
04996 
04900 
o48o5 

5.24896 
5 . 24994 
5.25091 
5.25187 
5.25282 

5.25376 
5.25469 
5.25561 
5.25652 
5.25742 

24912 
25oio 
25107 

252o3 

25298 

24929 
25026 

25l23 

25219 
253i4 

'0. 04727 
0.04634 
0.04542 
o.o445i 
0.04361 

04711 
04619 
04527 
04436 
04346 

04696 
o46o3 
045 1 2 
04421 
04332 

25392 
25484 
25576 
25667 
25757 
25845 
25933 
26020 
26105 
26190 

25407 
255oo 
25591 
25682 
25771 
25860 
25947 
26034 
26120 
26204 

25423 
255i5 
25607 
25697 
25786 

25875 
25962 
26048 
26134 
26218 

25438 
2553o 
25622 
25712 
258oi 

25454 
25546 
25637 
25727 
258i6 

0.04275 
0.041 85 
0.04098 
0 . o4o 1 2 
0.03927 

04258 
04170 
o4o83 
03998 
03913 

04243 
o4i56 
04069 
03983 
03899 

04228 
o4i4i 
o4o55 
03969 
o3bS5 
o38o2 
03719 
03638 
o355^ 
03478 

03399 

o3322 

o3245 
o3i7o 
03095 

04214 
04127 
o4o4o 
03955 
03871 

037S8 
03706 
o3624 
o3544 
o3465 

o3386 
o33o9 
o3233 
o3i57 
o3o83 

04199 
04112 
04026 
03941 
o3857 

03774 
03692 
o36ii 
o353i 
03452 

03373 
03296 

03220 

o3i45 
o3o70 

20 
21 
22 

23 

24 

25 

26 
27 
28 
29 

3o 
3i 

32 

33 

34 

5.25831 
5.25918 
5.26005 
5.26091 
5.26176 
5.26260 
5.26343 
5.26425 
5.2C5o6 
5.26586 

5.26665 
5.26743 
5.26820 
5.26896 
5.26971 

25889 
25976 
26063 
26148 
26232 

25904 
25991 
26077 
26162 
26246 

o.o3843 
0.037G0 
0.03678 
0.03597 
o.o35i7 

o.o3438 
o.o336o 
o.o3>83 
o.o3;o7 
o.o3i32 

03829 
03747 
o3665 
o3584 
o35o4 
o3425 
03348 
03271 
03195 
o3i2o 

o38i5 
03733 
o365 1 
03571 
03491 
o34i2 
03335 
o3258 
o3i82 
o3i07 

26274 
26356 
26438 
26519 
26599 
26678 
26755 
26832 
26908 
26983 

26288 
26370 
26452 
26532 
26612 

26691 
26768 
26845 
26921 
26996 

263oi 
26384 
26465 
26546 
26625 

263 1 5 
26397 

26479 
26559 
2D638 

26829 
26411 
26492 
26572 
2665 1 

26704 
26781 
26858 
26933 
27008 

26717 
26794 
26870 
26946 
27020 

26780 
26807 
26883 
26958 
27033 

o.o3o5S 
0.029S5 
0.02913 
0.02841 
0.02771 

o3o46 
02973 
02901 
02829 
02759 

o3o34 
02961 
02889 
02818 
0274s 

o3o2i 
02949 
02877 
02806 
02736 

o3oo9 
02937 
02865 
02794 
02724 

02997 
02925 
02853 
02783 
02713 

35 
36 

37 
38 

39 

40 
4i 
42 
43 

45 
46 
47 
48 

49 

5.27045 
5.27118 
5.27190 
5.27262 
5.27332 

27057 
271 3o 
27202 
27274 
27344 

27069 
27142 
27214 
27285 
27355 

27082 
27154 
27226 
27297 
27367 

27436 
27504 
27571 
27637 
27703 

27094 
27166 
27288 
27309 
27379 

27447 
275i5 
27582 
27648 
27713 

2710b 
2717S 
27250 
27820 
27890 

27459 
27526 
27593 
27659 
27724 

0.02701 
0.02633 
0.02565 
0.02499 
0.02433 

02690 
02622 
02554 
02488 
02422 

02357 
02294 

o223l 
02169 
02108 

02678 
02610 
02543 
02477 
0241 1 
02347 

02283 

02221 
02159 

02098 

O2o38 

01979 

01921 

0 1 864 
01808 

01752 
01698 
0 1 644 
01591 
01 540 

02667 
02599 
02532 
02466 
o»4oo 

o233() 
02273 
02210 
02149 
020S8 

02656 
025S8 
02521 
02455 
02390 

02326 
02262 
02200 
02139 
02078 

02644 
02577 

025lO 

02444 
02379 

023 1 5 

02  252 

02190 
02128 
02068 

5.27402 
5 . 27470 
5.27538 
5.27604 
5.27670 

274i3 
27481 
27549 
27615 
27681 

27425 
27493 
27560 
27626 
27692 

0.02 368 

O.02  3o4 

0.02241 
0.02179 
0.021 18 

5.27735 
5.27799 
5.27862 
5.27924 
5.27985 

27746 
27809 
27872 
27934 
27995 

27756 
27820 
27882 
27944 
28005 

28065 
28124 
28182 
28239 
28295 
2835T 
284o5 
28459 

285l2 

28563 



27767 
27830 
27893 
27954 
28015 

28075 
28134 
28191 
28249 
283o5 

27777 
27841 
27903 
27964 
28025 

28085 
28143 
28201 
28258 
283i4 

27788 
27851 
27918 
27975 
28035 

28094 
28153 
28211 
28267 
28823 

28878 
28432 
28485 
28538 
28589 

o.o2o58 
0.01999 
0 . 0 1 940 
O.OI883 
0.01826 

02048 
01989 
01931 
01S73 
01817 

OI761 
01707 

01 653 
0 1 600 
o:548 

02028 
01969 
01912 
01 854 
01798 

02018 
01960 
01902 
01845 
01789 

02009 
01950 
01892 
01 836 
01780 

5d 
5i 

52 

53 
54 
55 
56 

57 
58 
59 

5.28045 
5.28104 
5.28163 
5.28220 
5.28277 

28055 
28114 
28172 
28230 
28286 
2S342 
2S396 
28450 
285o3 
28555 

0.01771 
0.01716 
0.01662 
0.01609 
0.01557 

01743 
0 1 6S9 
oi635 
01 583 
oi53i 

01734 
0 1 680 
01627 
01574 
oi523 

01725 
01671 
01618 
01 565 
oi5i4 

5.28332 
5.28387 
5.28441 
5 . 28494 
5.28546 

2836o 
28414 
28468 

28520 

28572 

28369 
28423 
28476 
28529 
28580 

TABLE  XXIJI. 

To  find  the  Latitude  by  two  Altitudes  of  the  Sun. 


[I'jge  153 


'1 


HALF  ELAPSED  TLME. 


5  HouHs. 


MIDDLE  TIME. 


5  Hours. 


i3 
i4 
i5 
i6 

17 
i8 

19 


23 
24 
25 

26 

27 
28 

29 
3o 
3i 

33 

33 
34 
35 

36 
37 
38 

4o 
4i 
42 
43 
44 
45 
46 
47 
48 

49 
5o 
5i 

52 

53 
54 
55 
56 
57 
58 
59 


0" 


. o I 5o6 
.01455 
.oi4o6 
.01357 
.oi3io 


10' 


.01263 
.01217 
.01172 

01  I28|0I  120 
.0108401077 


01497 
01447 
01398 
01349 
Ol302 

OI255 
01209 
01 164 


01489 
01439 
01390 
01 34 1 
01294 


20' 


o  1 247 
01202 
OII57 
oiii3 
01070 


.0104201035 
.0100000993 
.0096000953 
.oo92o]oo9i3 
.00881  00874 


.ou843bo836 
.oo8o5  00799 
.00769  00763 
.0073300728 
.  0069900693 


01028 
00987 
00946 
00907 
00868 
oo83o 
00793 
00757 
00722 
00687 


.00665  00659 
.00632:00626 


. 00600 
.oo568 
.00538 


0059 

oo563 

oo533 


.oo5oS 
.00480 
.00452 
.00425 
.00399 


oo5o4 


oo475bo47o 


.00373 
.00349 
.0032.5 
.oo3o2 
.00280 


.00259 
.00239 
.00219 


10654 
00621 
00589 
oo558 
00528 


00499 


00447 
00420 
00394 
00369 
oo345 
oo32i 
00298 
00276 


00443 
oo4i6 
00390 


00255 
00235 
00216 


oo365 
oo34i 
oo3i7 
00295 
00273 


.0020000197 
.00 1 83  00180 

001 63 
00147 

00l32 

00117 
00104 


.00166 
.00149 
.00 1 34 
.00120 
. 00 1 06 


. 00093 
.ooobi 
.00070 
.00060 
.ooo5o 


. ooo4 I 
.ooo33 
.00026 
.00020 


252 
00232 
002  1  3 

00 1 94 
00177 


00160 
001 44 
00129 
001 1 5 

00102 


00091 
00079 
00068 
ooo58 
00049 


ooo4o 

00032 

00025 
00019 


0.0001 5  000 1 4 


.00010 
. 00007 
.00004 
.00002 
.00000 


00010 
00006 
oooo3 
0000 1 
00000 


00089 
00077 
00066 
ooo56 
00047 


00039 
ooo3i 
00024 
00018 
(joo  1 3 


00009 
00006 
0000  3 
oooo'i 
00000 


30" 

01480 
oi43o 
oi38i 
oi333 
01286 
01240 
01194 
o  1 1 5o 
01 106 
oio63 
01021 
00980 
00940 
C0900 
00862 

00824 
00787 
00751 
007 1 6 
00682 
00648 
00616 
00584' 
oo553 
oo523 

00494 
00466 
00438 
oo4i2 
oo386 

oo36i 
00337 
oo3i3 
00291 
00269 
00249 
002  29 
00210 
00191 
00 1 74 
001 57 
00142 
00127 
001 1 3 
00099 


40"  50" 


01472 
01422 
01373 
o  1 3  2  5 
0127S 


01232 
01187 
01 142 
o  1 099 
oio56 


oioi4 
00973 
00933 
00894 
00855 
00S18 
00781. 
00745 
007 1  o 
006-6 


00643 
006 1  o 
00579 
oo548 
oo5i8^ 


01464 
oi4i4 
01 365 
oi3i7 
01271 

01224 
01 179 
01 135 
o  1 09 1 
01049 


01007 
00966 
00926 
00887 
00849 
0081 1 
00775 
00739 
00704 
00670 

00637 
oo6o5 
00574 
00543 
oo5i3 


00489 
oo46 1 
00434 
00407 
oo382 

oo357 
oo333 
oo3 1  o 


00484 
oo456 
00429 
oo4o3 
oo377 


oo353 
00329 
oo3o6 


0028700284 
0026600262 

00245  00242 
00225  00222 

00207  00203 

0018800185 


00171 

001 55 
00 1 39 
00 1 24 
00 1 1  o 
00097 


00087 
00075 
0006  5 
ooo55 
ooo46] 

000  3  7 
ooo3o 
00023 
00017 
000 1 3 


00008 
0000  5 
oooo3 
00001 
00000 


ooo85 
00074 
ooo63 
ooo53 
o'  )o44 
ooo36 
00029 
00022 
00017 
0001 2 


00008 

0000  5 
00002 

0000 1 
00000 


00168 
001 52 
00 1 37 
00122 
00  [  08 
00095 


00083 
00072 
00061 
000  5  2 
00043 

ooo35 
00028 
0002 1 
00016 
3001 1 


00007 
00004 
00002 

00(J0 

oofioo 


M. 


0" 


28597 
28648 
28697 
28746 
28793 


10" 


28840 
28S86 

2S931 

28975 

290 1 9 

29061 
29103 
29143 
29183 
29222 
29260 
29298 
29334 


28606 
28656 
28705 
28754 
28801 

28848 
28894 
28939 
28983 
29026 


29068 
291 10 
29150 
2919c 
29229 


20" 


28614 
28664 
28713 
28762 
28809 

28856 
28901 
28946 
28990 
29033 


29267 
29304 
29340 
29370129375 
29404  29410 


2943829444 
29471  29477 
29503  29509 
29535  29540 


29565 


29595 
29623 
29651 
29678 
29704 
29730 
29754 
29778 
29801 
29823 

29844 
29864 
29884 
29903 
59920 


29937 

29954 
29969 
29983 
29997 


, 3oo I o 

.3o02  2 

,3oo33 
.3oo43 
.  3oo53 


3oo62 
30070 
3oo77 
3oo83 

3ooi 


30093 
30096 
30099 
3o  I  o  I 


29570 

29599 

29628 

29656 

29683J29687 

29709  29713 

29734:29738 


29075 
291 16 
29157 
29196 
29235 
29273 
29310 
29346 
29381 
29416 

29449 
29482 
29514 
29545 
29575 

29604 
29633 
29660 


29758 
29782 
29805 


29848 
29868 
29887 
29906 
29923 


29940 
29956 
29971 
29986 
29999 

3ooi2 
3oo24 


29762 
29786 
29808 
29830 

I985T 
29S71 
•'9890 
29909 
29926 


29943 
29959 
29974 
29988 
3  0001 


3ooi4 
3oo26 
3oo35j3oo37 
3oo45j3oo47 
3oo54  3oo56 

3oo63i3oo64 


30071 
30078 
3oo84 


30072 
3oo79 
3oo85 


30089130090 

3oo93j3oo94 
30097  30097 
3oioo  3oioo 
3oio2i3oi02 


3oio3|3oio3i3oio3 


30"  40" 

28623  2863i 
28673  2S681 
28722  28730 
2877028778 
2881728825 
28863  28871 
2890928916 
2895328961 
28997  29004 
29040J  29047 
9082  29089 


29123 
29163 
29203 
29241 
29279 
29316 
29352 
29387 
29421 
29455 
29487 
29519 
29550 
29580 


29130 
29170 
29209 
29248 

29285 
29322 
29358 
29393 
29427 


29609 

29637 

29665 

29691 

29717 

29742 

29766 

29790 

2981 

208; 


29854 
29874 
29893 
29912 
29929 

29946 

2996 

29976 

29990 

3ooo4 


2946. 

29493 

29524 

29555 

29585 


29614 


19L 

28639 
28689 
8738 

878e 

28833 
28879 
28924 
28968 
29012 
29054 
29096 
29137 
29177 
29216 
29254 
29292 
29328 
29364 
29399 
29433 

29466 
29498 
29529 
29560 
29590 
29619 
29642129647 
29669  29674 
29696  29700 
29721  29726 
29746  29750 
977029774 


29793 
29816 
29837 


29858 


>.99i5 
!9932 


29797 
29819 
29841 
2^ 
29881 
29900 
29918 
29935 
29951 
29966 
29981 


3ooi6  3ooi8 
30028  30029 
3oo38|3oo4o 
3oo48i3oo5o 
3oo57  3oo59 
3oo66  30067 
3oo73i3oo74 
3oo8o  3oo8 


29948 
29964 
29979 : 
29993129995 
3ooo6i3ooo£ 

30020 


3oo3i 
3oo42 
3oo5i 
3oo6o 


3oo68 

30075 

30082 

3oo86|3oo87 

30091  30092 

30095130095  30096 
30098I30098 


3oo86 
30090 


3o  1 00 

3oi02 

3oio3 


3oioi 

3oi02 

3oio3 


30099 
3oioi 

30102 

3oio3 


20 


Page  154]  TABLE    XXIII. 

To  find  the  Latitude  by  two  Altitudes  of  the  Sun. 


LOG.    RISING   OR    VERSED    SINE 


0  Hour. 


4o 
4 1 
4a 
43 
44 

"45 
4G 

4i 
48 

il 
5o 
5i 

52 

53 
_5_4 
55" 
56 

57 

58 
59 


0" 


Inf.Neg 


9 . 97860 

o. 

o . 58o66 

0.93284 

I . 

[ .  18271 


I .37053 
I. 53488 
I .66877 
1.78474 
1.88703 

1.97554 
2 . 

2.061 3 1 
2 . [ 3687 
2. 2 06 38 
2.27073 


2.33o63 
2.38667 
2.43930 
2.48893 
2.5:^586 

2.58o39 
2.62274 
2.663i2 
2.701 70 
2.73863 


2.774o5 
2.80809 
2.84o83 
•2.87238 
2 .90282 


2.93223 

2 .96067 

2 .98820 

3. 

3. 01 488 

3.04077 

3.06590 
3.09032 
3 . 1 1 406 
3.13718 
3 . 15969 


10"  20" 


42230 


r  i25o 
65oi9 
97980 

21S17 


4o5oi 
55868 
68920 
80265 
90297 


02,436 

22848 
71455 

02435 
25224 


99289 

07437 
14885 
2 1 744 
28100 


34023 
39567 

44777 
49693 
54344 


58759 
62960 
66967 
^0796 

74464 


59474 
63641 
67617 
71418 
75o6o 


779S2 
81 363 
84617 
87753 
90779 


93703 
96532 
99270 

01925 

o45oi 


3.18162 
3 . 2()3oi 
3.223S9 
3.24427J24762 
3.264:8  26745 


07001 
09432 
1 1 796 
1 4097 
i6338 


l8522 

2o653 
22732 


3.28363 
3 .3<j^fJ6 
3.32128 
3.3395 
3.35734 


37482 
3.39195 

40875 
3.42522 
3.441 38 


3.45724 
3.47282 
3.48811 
3.5o3i4 
51791 


43258 
58 184 
70917 
82019 
91862 


00701 
08723 
16066 
2  2836 
291 16 


30"  40' 


34972 
40457 
1616 
5o486 
55096 


78555 
81914 
85i48 
88265 
9'273 
94 1 81 
96994 
997  >  9 

.»236o 

)492  2 


0741 1 
19831 
12184 
14475 
16706 


8881 
2ioo3 
2  3., 73 
25095 

27071 


28683:29002 
3o579 30891 


3243 
3425f 
36028 
37770 

39477 
4 1 152 
42794 
4440  5 


459S6 
47539 
49064 
5o562 
52o35 


32739 
34549 
"632  1 


38o57 
39759 
41427 
43o64 
44670 

46247 

47795 
493 1 5 
50809 
52278 


37654 

33079 
77448 

06673 

28502 

45^ 
6o44o 
72S69 
83739 
93399 

02091 
09991 
17202 
23915 

3oi20 

35910 
41339 

46447 
51271 
55841 
6m82 
643 16 
68262 
72o36 
75652 


62642 

42  23o 

83o54 


1071 
3 1 660 


79124 
82461 
85675 
88773 
91765 


94656 
97454 

00164 
02792 
05342 


07819 
10227 
12570 
i485o 
17072 


48524 
62639 
74778 
85426 
94909 


o3456 
1 1 240 
i8382 
24980 
3i  1 12 


50" 


S2024 

5o5o9 
88019 

4575 
34708 


5io4i 
64784 
76646 
7080 
96394 


1  Hour. 


o48o5 
12472 
19517 
26(j33 
32093 
37758 
43075 
4S6S5 
52821 


36839 
42211 

47270 

52o5o 

5658o|573i3 

6o8S5  67582 

6498765652 

68903  69538 

72649 


76241 
79689 
S3oo5 
86199 
89279 
92254 


19238 
2i35i 
234 1 4 
25428 
27396 


29320 
3 1202 
33o44 

34847 
366 1 3 


38343 
4oo39 
41702 
43334 
44935 


46507 
48o5r, 
49566 
5io56 

52520 


95129 

97912 

00608 

0322 

05760 

08225 
10622 
12954 

l5225 

1743; 


19^94 
21699 
23753 
25759 
i-j-jic 

29637 

3i5i2 

333 

35i44 

36903 


38628 
4o3i9 

4i97fi 

36o3 

45199 


46766 
483o5 
49816 
5i3oi 
52761 


73258 
76825 
8025T 
83546 
86720 
S9782 
92739 


i3 


0" 


3.53243 
3 . 5.4670 
3.56074 
3.57455 
3.58814 


3 . 60 1 5  2 
3.61469 
3.62766 
3.64043 
3.653o2 


95599 
9836 

01049 
o365i 
06 1 76 


08629 
1  ioi5 
i3337 
i559' 


9949 

22tj4^ 
2409c 
26089 
28042 


9952 
31820 

33649 
35439 
37193 


38912 
40597 

4225(1 

i387i 
45462 


47024 
48558 
5oo6fi 
5 1 547 
53oo2 


23 
_2_4 

2  5" 

26 

27 
28 

='9 

3o 
3i 

32 

33 

34 


3.66542 
3.67765 
3.68969 
3.70158 
3.71329 


3.72485 
3.73625 
3.74750 
3.75860 
3.76955 
3 .78037 
3.79105 
3.80159 
3.81201 
3.82230 


3.83246 
3.&4250 
3.85242 
3.86223 
3.87192 

3.88i5( 
3 .89097 
3 . 90034 
3 .  90( 
3.91876 


10" 


53482 
54905 
563o6 
57683 

59038 

60373 
61686 
62980 
64254 
655io 


66747 
67967 
69169 
70354 
7i523 


72676 
738 1 3 
74936 
76043 
77 '37 
78216 
79282 
80334 
81373 
82400 


20" 
53721 

55i4o 

56537 

57910 

59262 
60593 
61903 
63 194 
64465 
65717 


66952 
68168 
69367 
7o55o 
71716 


72S67 
74ooi 

75l2I 

76227 

77318 
7S395 

79458 
8o5o8 
8 1 545 
82570 


30" 

53959 

55375 

56767 

58i37 

59486 
t)o8 1 3 
62120 
63407 
64675 
65924 


67156 
68369 
69566 
70745 
71909 


834:4  83582 
8441684582 


6 

86385 
87352 


,92782 

,9367 

9456 

95443 

963 


97170 
98021 
98862 
99696 


.o53o4 
,  06074 
.06838 
.07095 
.08344 


S83o9 
S9254 
90189 
91114 
92028 


85570 
86547 
S7513 


92933 

9382 

94712 

95588 

06455 


973 1 3 
98162 
9900 
99834 

00657 
01473 

)228o 

o3o8o 
o387 
o4656 
^5433 
06202 
)6965 
)77lo 
08468 


09087 
09823 
io552 
1 1 275 
11992 


88467 
894 II 
90344 
91267 
92179 

93082 
93975 
94859 
95733 
96599 


97455 
98302 
99141 
99972 

00794 


70057 
74189 
75307 
76409 
77498 
78573 
79634 
80682 

8.717 
82739 


83749 
84748 
85734 


40" 


J4197 
556o8 
56997 
58363 


50" 


54434 
5584 1 
57226 
58589 


59708  59930 


6io32i6i  25i 

62336!6255i 

o362o!6383 

64885i65o94 

66i3ij66337 

67359^67562 
68570J68770 
69763169961 
094071 1 35 
72101  72293 


73247  73436 
7437674563 
5491  75676 
659276774 
77678 177858 
78750J78928 
'09I79985 
8o855!8io28 
81 888  j8  2059 
32908183077 


83917:84083 
849i385o78 
85897  86060 
86709j8(.)87o|87o3i 
87672i87S32!8799i 

8S625  88783188940 
89567  89723189879 
90498  91-653 '90807 
91420  91  572  91724 
92331  924S2  92632 


09210 

09945 

10673 

1 1395  I  i5i5 

121 1 1  12229 


o  1 608 
02414 

o32I2 

o4oo3 
04786 
("^67 
o633o 
1709  T 
07845 
08592 

09333 
1 0067 
0794 


93232 
94123 
95oo5 
95878 
96742, 

97597 
98443 
99280 

00109 

0093 

01743 
02547 
03344 
o4i34 
049 1 6 
05690 
06457 
07217 


93381I93530 
94271 J94418 
95i52  95297 
9602396167 
96885197028 


97738 
98583 
99419 

00247 
01066 


97880 
98723 
99557 

oo384 
01202I 


01877 
02681 
03477 
04265 
o5o45 


o58i8 

<>6584 

07343 
07970I08095 
08716108840 

09456109578 
101 88|  io3 10 
io9i5!i  io35 
ii634  1 1754  11873! 
1 2348]  12466  125841 


02814 
(j36o8 
04395 
o5i75 
05946 
067 1 1 
07469 
08220 
8q64 


09701 
1 043 1 
1 1 155 


TABLE  XXIII.  Li-a.-eiss 

To  find  the  Latitude  by  two  Altitudes  of  tiie  Sun. 

LOG.    RISING    OR    VERSED    SINE. 


2  lloLliS. 


0" 


23 

24 
25 
26 

27 

28 

^9 
3o 
3i 

32 

33 
34 
35 
36 

37 
38 

4o 
4i 
42 
43 
J_4_ 
45 
46 
47 
48 
49 
5o 
5i 

52 

53 

54 
55 
56 
57 
58 

^"9 


12702 
i34o6 
i4io4 
14797 
1 5483 


,i6i63 
i6S3S 
17507 
18171 
18829 


.1948 
.20129 
,20771 
,  2 1 409 

, 2Jo4l 


,22668 

,23290 
2390 
24520 
25128 


25731 
2633o 
26924 
27514 
28099 


.28681 
.29257 
.29S30 
.30398 
.30963 


.3.523 
.32079 
.32631 
.33i8o 
.33724 
.34265 
.34802 
.35335 
.35865 
.36391 


10" 

12820 
i3523 
1 42  20 
14911 
15597 

16276 

1695 

17618 

18281 

18938 


1959. 

20236 

20878 
2i5i4 
22 1 46 


22772 
23393 
24010 
24622 
25229 


83i 
26429 
27023 
276)  2 
28197 


28777 
29353 
9925 
30493 
3io56 


3i6i6 
32171 
32723 
33271 
338i5 


34355 
34891 
35424 
35953 
36478 


.36913 
.37432 
.37948 
. 3846n 
.38968 


.39473 
.39975 
.40474 
. 40969 
.41461 


37000 
3751S 
38o33 
38545 
39052 


39557 
4oo58 
4o556 
4io5i 
4i543 
42o3 1 
42  5 16' 
^2998 


:.0" 
1 2938 
i3ri4o 
i4336 
i5o26 
i57io 


16389 
17062 
17729 
18391 
19047 


1 9698 
2o344 
20984 
21620 

2225o 
22S76 
23496 
241  12 
24723 

2533o 


25931 
26529 
27121 
27710 
38294 

288^3 
29449 

3o020 

3o587 
3 1 1 5o 


31709 
32264 
328i5 
33362 
33905 


34444 
34980 
355i2 
36o4i 
36565 


37087 
37604 
38119 
38629 
39137 

39641 
40142 
40639 
4ii33 
41624 


4195" 

,42435 

,42918 

,43398143477 

,4387443953 

74434844426 
.448!  8144896 
.45286J45363'4544i 
.45750J45S27  45905 
.  46 -M  2 146289  4^365 


421 12 
42597 
43078 
43557 
44o32 

445o5 


44974 


_3t> 

73o55 

i3756 

i445 

i5i4o 

i5824 

i65oi 

17173 

17840 

i85oc 

19156 

1 9806 

2045 1 

21091 

21725 

22355 

22980 

23599 

24214 

24825 

2543o 

2603 1 
26628 
27220 
27807 
28391 


28969 
29544 
3oi  i5 
3o68i 
3i243 


3i8oi 
32356 
32906 
33453 
33995 

34534 
35069 
35601 
3612S 
36653 


37173 
37690 
38204 
38714 
39221 

39725 

4o2  25 

40722 
4i2i5 
41706 

42193 

42677 

43i58 

43636 

44i 

44583 

45n52 

455 18 

45982 

.-16442 


40"  I  50 ' 


i3i72 
13S72 
14567 
i5255 
15937 

76674 
17285 
1 7950 
18610 
19265 


19914 
20558 
21197 
2i83i 
22459 

23f783 
28702 
243 16 
24926 
2553i 


26i3i 

26727 
27318 
27905 
28487 


13289 
13989 
14682 
15369 
i6o5o 

16726 
17396 
18060 
18719 
19373 


20022 
20665 
2 1 3o3 
2  1036 
22064 
23'i"87 
238o5 
244 1 8 
25027 
2  563 1 


26231 
26826 
27416 
28002 
28584 


29066 
29639 
30209 
30775' 
3i337 

37894 
32448 
32997 
33543 
3408  5 


34623 
35i58 
35689 
36216 
3674( 
'37^ 
37776 
.3S289 
38799 
393o5 


29161 
29735 
3o3o4 
30869 
3i43o 

37^7 
32540 

33089 

33634 
341^5 
34713 
352.47 
35777 
363o4 
36827 


39808 

4o3o8 

4080. 

41297 

4r7_82 

42274 

42758 

43238 

43716 

44190 

44667 

45i3o 

45596 

46o5 

465 1 8 


37346 
37S62 
38374 
38884 
39389 

39892 
40391 
4c)887 
4 1 379 
4r868 


2355 
42838 
433 18 
43795 
44269 


3  Hours. 


44740 
4520S 
45673 
461 35 
46595 


U" 


4. 


i:jOU7i 
47127 
47580 
48o3i 
48479 
48924 
49366 
49806 
50243 
50677 


. 5i 109 
.5i539 
. 5 1 966 
.52390 
.52812 

753717 
. 53648 
.54u63 
.54475 
.54885 

.55293 
.55698 
.56101 
. 565oi 
.56900 


,57296 
.57690 
,58082 
,58471 
,,58859 


59244 
59627 
60008 
6o388 
60765 
6 1 1 39 


JO'' 


46747 
47203 
47656 
48 1 06 
48553 

48998 
49440 
49879 
5o3i6 
507  5o 

577s7 

5i6io 
52o37 
52461 
52882 


533oi 
53718 
54 1  32 
54544 
549_53 

5536o 
55765 
56 168 
56568 
56966 
57862 
57755 
58 1 47 
58536 
58923 

59308 
59691 
60072 
6o45o 


:^U'' 


4662  3 
47278 
47731 
48180 
48627 

49071 
4951 3 
49952 
5o388 
50822 
5i253 
5i68i 
52107 
5253i 
52952 

5T3"77 

53787 

54201 

546  I  2 

55o2i 

55428 
55832 
56235 
56635 
57032 

57428 
57821 

58212 

58601 

58988 


59372 
59755 
60 1 35 
6o5i3 


60827  608  9(. 
61202  I 


,6i5i 2  61 574 
.6188361945 
.62252 
.62619 


62680 

,62984631745 

,6334763407 

.63708 

.64068 

.64425 

.64780 

.65 1 34 


61264 

6 1 636 

62006 

6231362375 


62741 


63io5 
63468 


4.65486 
4.65836 
4.66184 


,6653o 
.66875 
.672 17 
,67558 
.67S97 


6376863828 
6412764187 
64484,64544 
64839I64898 
65193:65251 
65544'656o3 
65S94J65952 
66242  66299 

(76588:66645 
6693266989 
67274|6733i 
67615^67672 
67954,68010 

682^68347 
6S627'68682 


68235 

68571 

6890516896069016 

69-.'3-,69>92  69348 

69568169623,69678 


'.iU'     40"  50" 

46899146975:47051 
47354J4743o475o5 
478o()  47881  47956 
48255  4833ol484o4 
4S701  4877648850 

491454921949293 
495864966049733 
5ooj5  50098150170 


5o46 

50894 

57374 

51753 

52178 

52601 

53o22 


53440 

5385(i 
54269 
54680 
55089 

55496 
55900 
563oi 
56701 
57098 


5o533 
50966 
5'7396 
51824 
52249 
52672 
53092 

sTFio 
53925 

54338 

54749 
55i57 

53563 
55967 
56368 
56767 
57^64 
57559 
57951 
58342 
58730 
59116 

59500 
59882 


57493 
57886 
58277 
586t,5 
59052 

5.9436 

598 1 8 

60198160261 

6o576|6o639 

60952161015 

67r26!{77388 

6169S61760 

62068J62129 

62436I6249- 

62802  62S63 

6TiW> 
635*8 
63888 
64246 
646o3 


5o6o 
5io38 


51467 
51895 
52319 
52742 
53162 


53579 
53994 
54407 
54817 
55225 


55630 
56o34 
,56435 
56834 
57230 

57625 
58017 
584o7 
58794 
59180 

59564 
59945 
6o324 
60701 
61077 


6i45o 

61822 

62 191 

62558 

63923 

632  2.6  63287 

63588  63648 

6394864008 

6.'f3o6l64365 

64662164721 

649576501665075 

6531016536965427 

656616571965777 

6601066068 

6635766415 

66702  6676( 


67046167103 
67388J67445 
67728:67785 
68066I68123 


66126 
66472 

668 1 7 
67  T  60 
67502 
67841 
68179 


685 1 5 


684o3T)8459 
68738|68794'68849 
69071169127  69182 
69403  69458  69513 
:69733i69788l69842 


''^'^ei5G]  TABLli  XXllI. 

To  find  the  Latitude  by  two  Altitudes  of  the  Sun. 


LOG.    RISING    OR    VERSED    SINE. 


4  Hours. 


5  Hours. 


Al. 


i3 
i4 
i5 
i6 

17 
18 

19 
20 
21 
22 

23 

24 

25 

26 

27 

23 

29 

3o 
3i 

32 

33 
34 
35 
36 
37 
58 
_39_ 

4o 
41 
42 
43 
44 
45 
46 
47 
48 
_49. 
5o 
5i 

52 

53 

54 
55 
56 

57 
58 
59 


0^^   10"  20 

:6^^ 
,70224 
.70550 
.70874 
.71197 


,7i5iS 
,71837 
,72155 
,72471 
,72785 


71571 
71890 
72208 
72523 
72838 


76196 
76492 
76787 
77081 
77373 

064 
77954 
78242 
78529 
78814 
79098 

: 79381 
1 79662 

.  .  79942 
.8017580221 


69952 
70279 
70604 
70928 

7i25u 


73i5i 
73462 
73772 
74oS<j 
74386 
74692 
74995 
75298 
75599 
5898 


,80452  S049S 
,80729180775 
,8ioo4j8io49 
8127781323 
81595 

81866 
82i36 
824o5 
82672 
8293s 


4.83i59 


832o3 
,83423j83467 
,83685  83729 
,8394  ■- 


84207 
84466 
84724 
84981 
85236 
8549c 

85^4 
8599(1 
86247 
86496 


8399" 
842  5o 

S4509 

847ti7 
85o23 
85278 
85533 

85786 
86o3S 
86288 
86538 


70006 
70333 
7o658 
70982 
7i3o4 
71624 
71943 
72260 
72576 
72890 

732o3 
7351 4 
73823 
74i3i 
74437 
74742 
75o46 
75348 
75649 
759-I8 


76245 
76542 
76836 
77i3o 
77422 

77713 
78002 
78290 
78576 
78861 


79145 
79428 
79709 
79989 
80267 


80545 
80820 
81095 
81 368 
8i64i 
81911 
82181 
82449 
82716 
8298 


4.86745186786, 


83247 
835io 
83773 
84o34 
84293 

84552 
84810 
85o66 
85321 
85^5 

85828 
86079 
86330 
86579 
86828 


30'' 


7006 1 
70387 
70712 
7io36 
71357 


701 15 
70442 
70766 
71089 
71411 


7167S 
71996 
723 1 3 
72628 
72942 

78254 
73565 
78874 
74182 

74488 


71781 
72049 
72366 
72681 
72994 
733o6 
73617 
75926 
74233  7428, 


7479 

75096 

75395 

75699 

75997 

76295 
7()59i 
76885 

77179 
77470 


77761 

78050 

78338 

78624 

78909 

7919 

79475 

79756 

8oo35 

8o3i 


80591 
80866 
8 II 4 1 
8i4i4 
81686 

81956 
82226 
82 '194 
S2761 
88026 
832^7 
83554 
838 1 6 
84077 
84337 

84^95 
84852 
85io8 
85363 
85617 
85870 
86 1 2 1 
86372 
8662 1 
86869 


40" 


50" 


M. 


70170 
70496 
70820 
71143 

ri_464 
71784 
7210V 
72418 
72733 

73o46 
73"358 
78668 
73977 


74539 
74844 
75i47 
75448 
5748 
76047, 

7(3344 
766-io 
6934 
77227 
77519 
77809 
78098 
78385 
78671 
78956 

79240 
79522 
79802 
8oi_)82 
8o36o 


74591. 


74894 
75197 
75498 
75798 
76097 


76894 
76689 
76983 
77276 
77567 


77S57 
78146 
78433 
78719 
79004 

79287 
9568 
79849 
80128 
8o4o6 


80687 
80912 
81186 
81459 
81781 

82001 
82271 
82538 
82805 
88071 


8o683 

80958 

8128 

8i5o5 

81776 


83335 
835^)8 
8386o 
84 120 
84380 


84638 
84895 
85i5i 
854o6 
85659 


85912 
86i63 
864 1 3 
86662 
86910 


82046 
823i5 
82583 
8285o 
83ii5 


83379 
83642 
88903 
84 164 
84423 


8468 1 
84988 
85194 
85448 
85701 

85954 
86205 
8645 
86704 
8695 1 


0" 


86992 
87289 
87484 
87728 
8797' 
88213 
88454 
8S694 
88933 
89171 


8703, 

87280 

87525 

87769 

880 


88254 


,89407 
,89643 
.89877 
,9011 1 
.90343 

.90575 
.90805 
.91084 
.91268 
.91490 


88734 
88973 
89210 

89447 


89916 
9oi5o 
90882 


91716 

91942 

92166 

92891 

9261 


92833 
98054 
98278 
98492 
98709 


.98926 
.94141 
.94356 
.94570 
.94782 


.94994 
.95205 
.95415 
.95624 
.95882 

.96040 
.96246 
.96451 
.96656 
.96860 


10"  20 


87075 
87321 
87516 
87809 
88o52 


90618 
90843 
91078 
91801 
91528 


91754 
91979 
92208 
92427 
92649 


88294 
88534 
88774 
89012 
89250 

89486 
89721 
89955 
90188 
90421 


87116 
87362 
87606 
87850 
88093 

88334 
88574 
814 
89052 
89289 

89"525 
.B9760 
89994 
90227 
90459 


92870 
98090 
93310 
93528 
98745 


90652 
908S2 
9111 1 
91889 
91 566 
91792 
92017 
92241 
92464 
92686 

92907 
98127 
93346 
93564 
98781 


98962 

94177 
94392 
94605 


98998 
94213 
94427 
94641 


9481894853 


95029  95065 
95240,95275 
9545095485 


95659 
95867 


.97062 
.97264 
.97465 
.97665 
.97865 


98068 
98261 
98457 
,98653 
,98848 


,99042 
,99235 
99428 


96074 
96280 
96486 
96690 
9689, 


97096 
97298 

97499 
97699 
97898 


<;8o96 
98293 
98490 
98686 
98880 


99074 
99267 
99460 


.9961999651 
.  99810I99842 


95694 
95902 

96 1  Oi 

9681: 

96520 

96724 

96927 


97 1 3o 

9733 

97582 

97782 

97981 


98129 
98826 
98528 
98718 
98918 


99107 
99800 
99492 
99688 
,99873 


30'  40"  50 


87157 
87402 

S7647 
87890 
88188 


88874 
88614 
88653 
89091 
89328 


87198 
87443 
87688 
87981 
88178 
884T4 
88654 
88898 
89181 
89868 

89604 
89888 
90072 


89864 
89799 
90033 

90266'9o3o5 
90536 


90498 
90728 

90920  90958 

91149 

91877 

91608 


91187 
9i4i4 
91641 


91829 
92o5-) 
92278 
92601 
92728' 

92944 
98164 
93382 
93600 
98817 

94084 
94249 
94463 
94676 
94888 
95 1 00 
95810 
95520 
95728 
95935 


91867 
92092 
92815 
92538 
92760 


92980 


90767 
90996 
91225 
91452 
9 '679 
;9i9o4 
92129 
92352 
92575 
92796 
98017 


98201)  9828 

9841993455 

93687  93673 

98854193890 

94o69'94io5 

94284J94320 

94498,94534 

19471294747 

94924:94959 


95135I95170 
95345|9538o 
95555I95589 


95768 
9597 


96143196177 
9634996383 
9655496588 
96758  9679 
9696 1 


95798 
96005 


97163 
97365 
97565 
97765 
97964 
98162 
98359 
98555 
98751 
98945 
99189 
9933'^ 
9952- 


96212 

96417 

96622 

96826 

9699')'97029 

9719797281 
97898197432 
9759997682 


97790 
97997 
98195 
98892 
98588 


97832 
98030 


98228 
9S425 
98620 
98788198816 
98977J99010 
99171 J99203 
99364199896 
99556:99587 
997J5|99747|99778 
99905 ,99987 199968 


TABLE  XXIII.                    [Page  157 

To  find  the  Latitude  by  two  Altitudes  of  the  Sun. 

LOG.  RISING  OR  VERSED  SINE. 

G  Hours. 

7  Hours. 

M. 
o 

0" 

10" 

20" 

30" 

40" 

50 

M. 

0-" 

10" 

20" 

30" 

40" 

50" 

5.00000 

ooo32 

ooo63 

00095 

00126 

001 58 

0 

5.09996 

1002I 

10045 

10069 

10093 

10117 

I 

5.00189 

002  21 

00252 

o<i283 

oo3i5 

oo346 

I 

5.10141 

10166 

10190 

10214  10238 

10262 

2 

5  .(.iu377 

00409 

00440 

00471 

oo5o2 

oo534 

2 

5.10286 

io3io 

io334 

io358io382 

io4o6 

3 

5.oo565  00590 

00627 

oo658 

00C89 

00720 

3 

5.  io43o|io454 

10477 

io5oi 

io525 

10549 

4 
5 

5.00751 

00782 

oo8i3 

00844 

00S75 

00906 
01091 

4 
5 

5.10573 

10597 

10620 

10644 
10786 

10668 
10810 

10691 
10833 

5.00937 

00968 

00999 

oio3o 

01061 

5.10715 

10739 

10763 

D 

5.01 122 

oiibJ 

01184 

01214 

01245 

0127b 

6 

5.10857 

10881 

1 0904 

10928 

10951 

10975 

7 

5.oi3o6 

oi337 

01 368 

01398 

01429 

01459 

7 

5.10998 

I  I022|llo45 

1 1069 

1 1092 

iiii5 

8 

5.01490 

Ol520 

oi55i 

oi58i 

01612 

of642 

8 

5.11139 

tl  162 

11185 

11209 

1  1232 

II255 

9 

10 

5.01672 

01703 

01733 

01763 

01794 

01824 

9 

10 

5.11279 

1  l3o2 

ii325 

11 348 

1  1372 

11395 

5.oib54 

U1884 

01915 

01945 

01975 

02uo5 

5.ii4i8 

ii44i 

11464 

11487 

ii5io 

11 533 

1 1 

5.o2o35 

02065 

02095 

02  125 

02 1 55 

021 85 

1 1 

5.1:557 

ii58o 

1  i6o3 

11626 

1 1649 

11672 

12 

5. 022 1 5 

0224D 

02275 

023o5 

02335 

02  365 

12 

5.11695 

11717 

11740 

11763 

11786 

1 1809 

i3 

5.0^395 

02425 

02455 

02484 

025i4 

02544 

i3 

5.11832 

11855 

11878 

1 1 900 

1 1923 

11946 

i4 
i5 

5.02574 

02603 

02633 

02663 

02692 

02722 

i4 

5.11 969 

11991 

12014 

12037 

1 2059 

12082 

5.02751 

02781 

0281 1 

02840 

02870 

02899 

i5 

5.12105 

12127 

12l5o 

12173 

12I95[l22l8| 

i6 

5.o^9.-'8 

02958 

02987 

o3o  1 7 

o3o46 

o3o75 

16 

5.I224o 

12263 

12285 

i2  3o8 

123jo 

12353 

17 

5.ij3io5 

o3i34 

o3i63 

03193 

o3222  0325l 

17 

5.12375 

12397 

12420 

12442 

1 2465 

12487 

18 

5  .o3?8o 

o33io 

03339 

o3368 

03397,03426 

18 

5.12509 

12532 

12554 

12576 

12598 

12621 

_i9_ 
20 

5.o3-i55 
5.o3i)29 

03484 
o3658 

o35i3 
03687 

o3542 
03716 

03571  o36oo 

'9 

20 

5 . 1 2643 

12665 

12687 

£2709 
12842 

12732 
1 2864 

12754 
128S6 

03745 

03774 

5.12776 

12798 

12820 

21 

5.o38o-' 

o383i 

o386o 

03889 

03918 

03940 

21 

5.12908 

12930 

12952 

12974 

12996 

i3oi8 

22 

5.03975 

o4oo4  o4o32 

o4o6i 

04090 

o4i  18 

22 

5. i3o4o 

i3o62 

i3o84 

1 3 1 06 

i3i28 

i3i49 

23 

5.o4i47 

04175 

o42o4 

04232 

04261  042S9 

23 

5.i3i7i 

13193 

i32i5 

13237 

i3258 

13280 

24 

25 

5. 043 18 
5.oi4»8 

(.4346 
045 16 

04375 
04545 

o44o3 

o443 1 

o446o 

24 

25 

5.i33o2 
5.13432 

i3323 
13453 

13345 
13475 

13367 
13496 

13388 
73578 

i34io 

04573 

04601 

04629 

13539 

26 

5.o4657 

o4686 

o4ii4 

04742 

0477^ 

04798 

26 

5.i356i 

i3582 

i36o4 

i362  5 

13647 

13668 

27 

5 .04826 

04854 

048S2 

049 1 0 

04938 

04966 

27 

5 . 1 3690 

1 37 11 

13732 

13754 

13775 

13797 

28 

5.0^994 

o5o22 

o5o5o 

o5o78 

o5io6o5i3.^ 

28 

5.i38i8 

13839 

i386o 

13882 

1 3903 

13924 

29 

3o 

5.o5i6.< 
5.05328 

o5i89 
o5356 

o52i7 
o5383 

05245 
05411 

05273  o53oo 
05439  o5466 

29 

3o 

5.13945 
5.14072 

13967 
14093 

13988 
T4'i74 

1 4009 

i4o3o 

i4o5i 

i4i36 

i4i57 

14178 

3i 

5.05494 

o552i 

05549 

05577 

o56o4o5632 

3i 

5.14199 

14220 

14241 

14262 

14282 

i43o3 

32 

5.05659 

o56S6 

05714 

0574 1 

05769  05796 

32 

5.14324 

14345 

14366 

14387 

1 44o8 

14429 

33 

5.o5S23 

o585i 

05878 

05905 

05933  05960 

33 

5.14449 

14470 

14491 

i45i2 

14^33 

14553 

34 
35 

5.05987 
5.o6i5o 

06014 
06177 

0604 1 
06204 

06069 
06231 

0609606123 
0625s  06285 

34 
35 

5.14574 

14595 

i46i5 

1 4636 

14657 

14677 
14801 

5.14698 

14719 

14760  14780 

36 

5.o63i2 

06339 

06366 

06393 

06420,06447 

36 

5.14821 

14842 

14862 

i4883j  14903 

14924 

37 

5.06474 

o65oo 

06527 

o6554 

o658i 

06608 

37 

5.14944 

14964 

i49«5 

i5oo5  i5o26 

i5o46 

33 

5.06634 

06661 

06688 

06714 

06741 

06768 

38 

5.i5o66 

i5o87 

i5io7 

i5i27 

i5i47 

i5i68 

39 
4o 

5.06794 

06821 

o6848 

06874 

06901 

06927 

39 
40 

5.i5i88 

i52o8 

15228 
15349 

i5248 
T5369 

15269 
15389 

15289 

1 5409 

5.06954 

06980 

07007 

07033 

07060 

07086 

5. i53o9 

15329 

4i 

5.071 12 

07139 

07165 

07192 

07218 

07244 

4i 

5.15429 

15449 

15469 

15489 

i5bo9 

15520 

42 

■J. 07270 

07297 

07323 

07349 

07375 

07401 

42 

5.15549 

15569 

15589 

i56o9 

15629 

1 5649 

43 

5.o74?8 

07454 

07480 

07506 

07532 

07558 

43 

5.15668 

i5688 

i57o8 

15728 

1 5748 

15767 

44 
45 

5.07584 
5.07740 

07610 

07636 

07662 

07686 

07714 

44 
45 

5.15787 
5.15905 

i58o7 
15925 

i5827 
15944 

i5846 
15964 

i5866 
15984 

15886 
i6oo3 

07766 

07792 

07818 

07844 

07869 

46 

5.07895 

07921 

07947 

07973 

07998 

08024 

46 

5.16023 

16042 

16062 

16081 

16101 

16120 

47 

5 .o8o5o 

0S075 

08101 

08127 

c8i52 

08178 

47 

5.16140 

16159 

16179 

16196 

16217 

16237 

48 

5.08204 

08229 

08255 

08280 

o83o6 

o833i 

48 

5.16256 

16276 

16295 

i63i4 

i6333 

16353 

49 
5o 

5.08357 

o83S2 

oS4o8 

08433 
o8585 

o8458 
08610 

o8484 
08636 

49 
5o 

5.16372 

16391 

16410 
16526 

i643o 

16449 

16468 

5.o85(i9 

08534 

o856o 

5.16487 

i65o6 

i6545  16564 

16583 

5i 

5.08661 

08686 

0871 1 

08736 

08762 

08787 

5, 

5.16602 

16621 

i664o 

16659  16678 

16697 

52 

5.08812 

08837 

08S62 

0S887 

08912 

08937 

52 

5.16716 

16735 

16754 

16773  16792 

168  n 

53 

5.08962 

0898709012 

09037 

09062 

09087 

53 

5.]6S3o 

16849 

16867 

16886  16905 

16924 

54 
55 

5.091 12 

09137  09162 

09187 
09335 

092 1 1 
09360 

09236 
09  38  5 

54 
55 

5.16943  16961 

16980 

16999  17018 

i7o36 

5.09261 

0928609311 

5.17055 

17074 

17093 

1 7 1 1 1  1 7 1 3o 

17148 

56 

5.09^09 

0943409459 

09483 

0950S 

09533 

56 

5.17167 

17186 

17204 

17223  17241 

17260 

57 

5.09557 

09582  09606 

09631 

09655 

09680 

57 

5.17278 

17297 

i73i5 

17334  17352 

17371 

58 

5.09704 

0972909753 

09777 

09S02 

09826 

5H 

5.17389 

17408 

17436 

17444  17463 

17481 

59 

5.09851 

09875I09899 

09924 

09948 

09972 

59 

5.17499 

17518 

17336 

17554  17573 

17591 

i'=isei58]               TABLE  XXIII. 

To  find  the  Latitude  by  two  Altitudes  of  the  Sun. 

LOG.  RISING  OR  VERSED  SINE. 

8  Hours. 

9  Hours. 

M. 

o 
I 

2 

3 
4 
5 
6 

7 
8 

9 

10 

II 

12 

i3 
i4 
i5 
i6 

17 
i8 

19 

20 
21 
22 
23 
24 
25 
26 
27 
28 
29 

3o 
3i 

32 

33 
34 
35 
36 

37 
38 

39 
4o 
4i 
.42 
43 
M 
45 
46 
47 
48 

49 
5o 
5i 

52 

53 
54 
55 
56 

57 
58 
59 

0" 

10" 

20" 

30" 

17664 
17773 
17881 
17989 
18096 

18203 
i83o9 
i84i4 
i85i9 
18624 
18728 
i883i 
18934 
19036 
19188 

19240 
19340 
19441 
19540 
19689 

40" 

50" 

M. 

0 
I 
2 
3 
4 
5 
6 
7 
8 

9 
10 
11 
12 
i3 
i4 
i5 
16 
17 
18 

19 

0" 

10" 

20" 

30" 

40" 

50' 

5.17609 
5.17718 
5.17827 
5.17935 
5.18042 

17627 
17736 
17845 
17953 
18060 

17646 
17755 
17863 
17971 
18078 

17682 
1 779 1 
17899 
18007 
18114 
18220 
18826 
18432 
18537 
1864 1 
18745 
18848 
18951 
19053 
19155 

19256 
19357 
19457 
19557 
19656 

17700 
17809 

17917 
18024 
18182 

18288 
i8344 
18449 
i8554 
18059 
18762 
18866 
18968 
19070 
19172 

19273 
19874 
19474 
19573 
19672 

5.23226 
5.2  33o4 
5.28882 
5.23459 
5.23536 

28289 
23317 
28895 
23472 
23549 
23625 
28701 
23776 
2385i 
28925 

23252 
28880 
23408 

23485 
23562 

23638 
28714 
28789 
23863 
28988 

28265 
23348 
23421 
23498 
23574 

28278 
23356 
23434 
235ii 
23587 

28291 
28869 
23447 
23523 
28600 

5.18149 
5.18256 
5.18362 
5.18467 
5.18572 

J.  16676 
5.18780 
5.18883 
5.18985 
5.19087 

18167 
18273 
18379 
1S484 
18589 

18185 

I829I 

18397 

i85o2 

18606 

18710 

18814 
18917 

19019 
19I2J 

19223 
19324 
19424 
19524 
19623 
19722 
19820 
19918 

200l5 

20111 

5.28612 
5.23688 
5.28764 
5.23839 
5.28918 

2365o 
28726 
28801 
28876 
28950 

28663 
28789 
238i4 
28888 
28962 

28676 
28751 
23826 
28901 
28975 

18693 
18797 
18900 
19002 
19104 
19206 
19307 
19407 
19507 
19606 

19705 
1 9804 
19901 
19999 
20095 

5.28987 
5.24060 
5.24i33 
5.24206 
5.24278 

28999 
24078 
24145 
24218 
24290 

24011 
24085 
24i58 
24280 
24802 

24024 
24097 
24170 
24242 
243 14 

24086 
24109 
24182 
24254 
24326 

24048 
24121 
24194 
24266 
24338 

5.19189 
5.19290 
5.19390 
5.19490 
5.19590 

5.24349 
5.24421 
5.24491 
5.24561 
5.24681 

24861 
24432 
245o3 
24573 
24643 

24378 

245 1 5 
24585 
24654 

24385 
24456 
24526 
24596 
24666 
24735 
248o3 
24871 
24989 
25oo6 

24397 

24468 
24538 
24608 
24677 

24409 
24479 
24550 
24619 
24689 

5.19689 
5.19787 
5.19885 
5.19982 
5.20079 

19788 
19886 
19934 

2003l 

20127 

19754 
19852 
19950 
20047 
20143 

19771 
19869 
19966 
20o63 
20159 

20 
21 
22 

23 

24 

5.24700 
5.24769 
5.24887 
5.24905 
5.24972 

5.25089 
5. 25 106 
5.25172 
5.25237 
5.25302 

24712 
24780 
24849 
24916 
249S3 

25o5o 
25ii7 
25182 
25248 
253i3 

24728 
24792 
24860 
24927 
24995 
25o6i 
25128 
25193 
25259 
25324 
25388 
25452 
255i5 
25578 
2  564 1 
25708 
25765 
25826 
25887 
25947 
26007 
26066 
26125 
26184 
26242 

26299 
26356 
264:3 
26469 
26525 
2658o 
26635 
26689 
26743 
26797 
26850 
26908 
26955 
27007 
27058 

24746 
24814 
24882 
24950 
25oi7 

2.4757 
24826 
24894 
24961 
25028 

5.20175 
5.20271 
5.20366 
5.20461 
5.20555 

20191 

20287 

20382 

20477 
20571 

20207 
2o3o3 
20398 
20493 
20587 

20680 
20773 
20866 
20958 
21049 

2Il40 
21281 
2l32I 
2l4lO 
21499 

21588 
21676 
21763 
2i85o 
21986 

202'23 
20819 

2o4i4 
2o5o8 
20602 

20696 
20789 

20881. 

20978 
2io65 
IIT55 

2 1  2/j6 

2 1 336 
21425 
2i5i4 

20289 
20335 
2o43o 

20524 

20618 

2071 1 
20804 
20897 
20988 
21080 

20255 

2o35i 

20445 
2o54o 
2o634 
20727 
20820 
20912 
21004 
21095 

25 

26 
27 
28 
29 

3o 
3i 

32 

33 
34 
35 
36 

37 
38 
39 

25072 
45189 
25204 
25270 
25334 

25o84 
25i5o 

252l5 

25280 
25345 

25095 
25i6i 
25226 
25291 
25856 

5.20649 
5.20742 
5.20835 
5.20927 
5.21019 

2o665 
20758 
2o85o 
20943 
2io34 

2II25 

2I2I6 
2i3o6 
21395 
2 1 484 
21573 
21661 
21748 
21835 
21922 

5.25367 
5.25431 
5.25494 
5.25557 
5.25620 

25377 
25441 
255o5 
25568 
2563i 

25899 
25463 
25526 
25589 
2565i 

25409 

25473 

25536 
25599 
25662 

25420 
25484 
25547 
256io 
25672 

25734 
25795 
25856 
25917 
25977 

5.21110 
5. 2 1 20 1 
5. 2 1 291 
5.2i38o 
5.21470 
5.21558 
5.21646 
5.21734 
5.21821 
5.21 908 

21170 
21261 
2i35i 
21 440 
21529 

21186 
21276 
2 1 366 
21455 
21543 

5.2  5682 
5.25744 
5.258o6 
5.25866 
5.25927 

25698 
25755 
258i6 
25877 
25987 

25997 
26o56 
26115 
26174 
26282 

25713 
25775 
25836 
25897 
25957 

25724 
25785 
25846 
25907 
25967 

21602 
21690 
21777 
21864 
21951 

21617 
21705 
21792 
21870 
21965 

2l632 

21719 

21806 

21893 

2r979 

22065 
22l5o 
22235 

22819 

2  24o3 

40 
4i 
42 
43 

45 
46 
47 
48 

49 

5.25987 
5.26046 
5.26105 
5.26164 
5.26222 

26017 
2^076 
26135 
26198 
26251 
26809 
26366 
26422 
2G478 
26534 
26589 
26644 
26698 
26752 
26806 

26859 

2691  T 
26968 
27015 
27066 

26027 
26086 
26145 
26208 
26261 

26087 
26096 
26154 
26218 

26270 

5.21994 
5.22079 
5.22164 
5.22249 
5.22333 

22008 
22094 
22179 
22263 
22347 

22022 
22108 
22193 

22277 

2  236  1 

22087 
22122 
22207 
22291 
22875 

2205l 
22l36 
22221 
223o5 
22889 

5.26280 
5.26337 
5.26394 
5.26450 
5.265o6 

26290 
26347 
26403 
26460 
265i6 

26818 
26875 
26432 
26488 
26543 
2659S 
26653 
26707 
26761 
268 1 5 
2686S 
26920 
26972 
27024 
27075 

26828 
26385 
26441 
26497 
26553 

26608 
26662 
26716 
26770 
26823 

26876 
26929 
26981 
27082 
27083 

5.22417 
5.225oo 
5.22583 
5.22665 
5.22746 

22431 

225l4 

22596 

22678 

22760 

22445 
22528 
22610 
22692 

22773 

22458 

22541 
22624 
22706 

22787 

22472 

22555 
22687 

22719 

22801 

22486 
22569 

2265l 

22733 

22814 
22895 

22975 
23o55 
28184 
28218 

5o 
5i 

52 

53 
54 
55 
56 
57 
58 

59 

5.26562 
5.26617 
5.26671 
5.26725 
5.26779 
5.26882 
5.26885 
5.26937 
5.26989 
5.27041 

26571 
26626 
26680 
26734 
26788 

2684T 
26894 
26946 
26998 
27049 

5.22828 
5. 2 2 90S 
5.2298S 
5.23o68 
5.23i47 

22841 

22922 

23002 

23o8i 
23 160 

22854 
22935 
23oi5 
23095 
23 1 74 

22868 

22948 
23028 

28108 
28187 

22881 

22962 
28042 
28121 
28200 

TABLE  XXIII.                     [Page  159 

To  find  the  I^atitude  by  two  Altitudes  of  the  Sun. 

LOG.  RISING  OR  VERSED  SLNE. 

10  Hours. 

11  Hours.          j 

M. 

o 
I 

2 

3 
4 
5 
6 

7 
8 

9 

lO 

II 

12 

1.3 
i4 
i5 
i6 

17 
i8 

19 

20 
21 
2  2 
23 
24 
25 
26 
27 

28 
29 

3o 
3i 

32 

33 
34 
35 
36 

37 
38 
39 

40 
4i 
42 
43 
44 
45 
46 
47 
48 
49 
5o 
5i 

52 

53 
54 
55 
56 
57 
58 
59 

0" 

10" 

20" 

27109 
27159 
27209 
27259 
2'j3o8 

27356 
27404 
27452 
27500 
27546 
27593 
27639 
27684 
27730 
27774 

27819 
27862 
27906 

27949 
27991 

28033 
28075 
28116 
28157 
28197 

28237 
28277 
283i6 
28354 
28393 

30" 

40" 

50" 

M. 

0 
I 
2 

3 
4 
5 
6 
7 
8 

9 

10 
11 
12 
i3 
i4 
i5 
16 
•7 
iS 

'9 
20 
21 
22 

23 

24 

25 

26 
27 
28 
29 

3o 
3i 

32 

33 
34 
35 
36 
37 
38 
39 

40 
4i 
42 

43 
44 
45 
46 

47 
48 

49 

0" 

10" 

29361 
29386 
29410 
29434 
29457 

29480 
29503 
29525 
29546 
39568 

20" 

29365 
29390 
29414 
29438 
29461 

29484 
29506 
29528 
29550 
29571 

30" 

40"' 

50"- 

5.27092 
5.27142 
5.27192 
5.27242 
5.27291 

5.27340 
5.27388 
5.27436 
5.27484 
5.27531 

5.27577 
5.27624 
5.27669 
5.27715 
5.27759 

5.27804 
5.27848 
5.27891 
5.27934 
5.27977 

5.28019 
5.28061 
5.28102 
5.28143 
5.28184 

27100 
27i5i 
27201 
27250 
27299 

27348 
27396 
27444 
27492 
27539 

27535 
27631 
27677 
27722 
27767 
27811 
27855 
27899 
27942 
27984 
28026 
2806& 
28109 
28i5o 
28191 

27117 
27167 
27217 
27267 
27316 

27126 
27176 
27226 
27275 
27324 

27134 
27184 
27234 
27283 
27332 

5.29357 
5.29381 
5.29406 
5 . 29430 
5.29453 

5.39476 
5.29499 
5.29521 
5.29543 
5.29564 

29369 
29394 
29418 
29441 
29465 

29373 
29390 
29422 
29445 
29469 

29377 
2940J 
29426 
29449 
29472 

27364 
27412 
27460 
27507 
37554 

27372 
27420 
27468 
27515 
27562 

27380 
27428 
27476 
27523 
27570 

29488 
29510 
29532 
29554 
29575 

29491 
29514 
29536 
29557 
29578 

29495 
29517 
29539 
2956 1 
39582 

27601 

27647 
27692 
27737 
277S2 

27608 
27654 
27700 
27745 
27789 

27616 
27662 
27707 
27752 
27796 

5.29585 
5.29606 
5.29626 
5.29646 
5.29665 

29589 
29609 
29629 
29649 
29668 

296S7 
29705 
29723 
29741 
29758 

29592 
29612 
29632 
29652 
29671 
29690 
29708 
29726 
29744 
29761 

29596 
29616 
29636 
29655 
29674 

29599 
29619 
29639 
29658 
29677 

29602 
29623 
29642 
29662 
29681 

29699 
29717 
29735 
29752 
39769 

29786 
29802 
29817 
29832 
29847 
29S61 
29875 
29889 
29902 
29915 

27826 
27870 
27913 
27956 
27998 

28040 
28082 
28123 
28164 
28204 
28244 
28283 

28322 

2836i 
28399 

28437 
28474 
285ii 
23547 
28583 

28619 
28654 
28688 
28723 
28757 

27833 
27877 
27990 
27963 
28005 

28047 
28089 
28i3o 
28170 
28211 

28250 
28290 
28329 
28367 
284o5 

28443 
28480 
285i7 
28553 
28589 

2S624 
28660 
28694 
28728 
28762 

27840 
27884 
27927 
27970 
28012 

28054 
28096 
28137 
2S177 
28217 

28257 
28296 
28335 
28374 
2841 1 

28449 
28486 
28523 
28559 
28595 
28637; 
28665 
28700 
28734 
2876S 

5.29684 
5.29702 
5.29720 
5.29738 
5.29755 

29693 
29711 
29729 
29747 
29764 
29780 
29796 
29812 
29827 
29842 

29857 
29871 
29884 
29898 
2991 1 

29696 
29714 
29732 
29749 
29766 

29783 
29799 
29815 
29830 
29845 

29859 
29873 
29887 
29900 
29913 

5.29772 
5.29788 
5.29804 
5.29820 
5.29835 

29775 
29791 
29807 
29822 
29S37 
29852 
29866 
29880 
29893 
29906 

29777 
29794 
29809 
29S25 
29840 

29854 
29868 
29882 
29896 
29908 
2992 1 
29933 
29945 
39956 
29967 

5.28224 
5.28264 
5.2S3o3 
5.28342 
5.28380 

28231 
28270 
283o9 
28348 
28386 

5.29850 
5.29864 
5.29S78 
5.29S91 
5 . 29904 

5.28418 
5.28455 
5.28492 
5.2S529 
5.28565 

28424 
2S461 
2S498 
28535 
28571 

28430 
2  8468 
285o5 
2S541 
28577 

5.29917 
5.29929 
■5.29941 
5.29952 
5 . 29963 

29919 
29931 
29943 
29954 
29965 

29923 
29935 

29947 
29958 
29969 

29925 

29937 
29948 
29960 
29970 

29937 
29939 
29950 
29961 
39972 

5.28601 
5.28636 
5.28671 
5.28706 
5.28740 

28607 
28642 
28677 
2871 1 
28745 

28613 
28648 
28683 

28717 
28751 

5.29974 
5.29984 
5.29994 
5.3ooo3 
5.3ooi2 

29975 
29986 
29995 
3ooo4 
3oo  1 3 

29977 
29987 
29997 
3ooo6 
3ooi5 

29979 

29989 
29998 
30007 
3ooi6 

29981 
39990 
3oooo 
30009 
3ooi8 

29982 
29992 

3oooi 
3ooio 
3oo  1 9 

5.28773 
5.28806 
5.28839 
5.28872 
5.28904 

28779 
28812 
28845 
28877 
28909 

28784 
28«i7 
3885o 
28882 
28914 

28790 
28823 
28855 
28SSS 
28919 

28795 
28828 
28861 
28893 
28925 

2895^ 
289S7 
29017 
29047 
29076 

28801 
28834 
2S866 
28898 
28930 

28961 
28992 
29022 
29052 
29081 

5.3oo2o 
5.30028 
5.3oo36 
5 . 3oo43 
5.3oo5o 

3o02  2 

3oo3o 
3oo37 
3oo44 
3oo5i 

3oo2  3 
3oo3i 
3oo38 
3oo46 
3oo52 

3oo59 
3oo64 
30070 
30075 
3oo79 

3o^4 
30087 
30091 
30094 
30096 

30098 
3oioo 

30102 
30I02 

3oio3 

30024 
3oo32 
3oo4o 
3oo47 
3oo53 

30026 
3oo34 
3oo4 1 
3oo48 
3oo54 

30027 
3oo35 
3oo42 
3  0049 
3oo55 

5.28935 
5.28966 
5.28997 
5.29027 
5.29057 

28940 
28971 
29002 
29032 
29062 

28945 
28976 
29007 
29037 
29067 

28951 
28981 
29012 
29042 
29072 

5.3oo56 
5.30062 
5.3oo68 
5.30073 
5.30078 
5.30082 
5 . 3oo86 
5.30090 
5.30093 
5 . 30096 

3oo58 
3oo63 
30069 
30074 
30079 

3oo83 
30087 
30090 
30093 
30096 

3oo6o 
3oo65 
3co7i 
30076 
3oo8o 

3oo84 
3oo88 
30091 
30094 
30097 

3oo6i 
3oo66 
30072 
30076 
3oo8i 

30062 
30067 
30072 
30077 
30082 
3oo86 
30089 
30092 
2CC95 
3C097 

5.29086 
5.291 i5 
5.29144 
5.29172 
5.29199 

J.9091 
29120 
29148 
29176 
29204 

29096 
29125 
29153 
29181 
29209 

29101 
29129 
29158 
29186 
29213 

29240 
29267 
29293 
29319 
29344 

29106 
29134 
29162 
29190 
29218 

29245 
29271 
2929T 
29323 
2934s 

291 10 
29139 
29167 
29195 
29222 

29249 
39276 
29302 
29337 
29353 

5o 
5i 

52 

53 
54 
55 
56 

57 
58 
59 

3oo85 
30089 
3C091 
30095 
30097 

5.29227 
5.29254 
5.29280 
5.29306 
5.29332 

29231 
2925s 
29284 
29310 
29336 

29236 
29262 
29289 
29315 
29340 

5 . 30098 
5.3oioo 
5.3oioi 
5.3oio2 
5.3oio3 

30098 
3oioo 
3o  1 0 1 
3o  1 02 
3oio3 

30099 
3oioo 

30I02 

3oio3 
3oio3 

30099 
3cioi 

3oi02 

3oio3 
3oio3 

]^'?9 

30102 

3oio3 
3oio3 

!•»««  1601               TABLE  XXIV. 

Of  Natural  Sines. 

Prop, 
pans 

29 
o 

M 

0 

0° 

1° 

2° 

3° 

4° 

60 

Prop. 

parta 

2 

2 

N.  sine. 

N.  COS. 

N.  sine. 

N.  COS. 

N.  sine. 

N.  COS. 

N.  siiic- 

N.  COS. 

N.  sine. 

N.  COS 

00000 

I 00000 

01745 

99985 

03490 

99939 

05234 

99863 

06976 

99756 

0 

I 

00029 

1 00000 

01774 

99984 

o35i9 

99938 

o5263 

99861 

07005 

99754 

5g 

2 

I 

2 

ooo58 

1 00000 

oi8o3 

999S4 

o3548 

99937 

05292 

99860 

07034 

99752 

58 

2 

I 

3 

00087 

100000 

oi832 

99983 

03577 

99936 

o532i 

99858 

07063 

99750 

57 

2 

2 

4 

001 16 

1 00000 

01862 

99983 

o36o6 

99935 

o535o 

99857 

07092 

99748 

56 

2 

2 

b 

00145 

I 00000 

01 89 1 

99982 

03635 

99934 

053-9 

99855 

07121 

99746 

55 

2 

3 

~3' 

b 

7 

00175 

I 00000 

01920 

99982 

03664 
03693 

99933 

o54o8 

99854 

i07i5o 

99744 

54 
53 

2 
2 

00204 

I 00000 

01949 

99981 

99932 

05437 

99852 

07179 

99742 

A 

8 

00233 

1 00000 

01978 

99980 

03723 

99931 

o5466 

99S51 

0720S 

99740 

52 

2 

A 

9 

00262 

1 00000 

02007 

99980 

03752 

99930 

05495 

99849 

07287 

99738 

5i 

2 

b 

10 

00291 

1 00000 

02o36 

99979 

00781 

99929 

o5524 

99847 

07266 

99786 

5o 

2 

5 

11 

oo320 

99999 

02o65 

99979 

o3Sio 

99927 

o5553 

99846 

07295 

99734 

49 

2 

b 
6 

12 

00349 

99999 

02094 

99978 
99977 

o3839 
o3868 

99926 
99925 

o5582 

99844 

07824 

99731 

48 
47 

2 
2 

00378 

99999 

02123 

o56ii 

99842 

07353 

99729 

7 

1 4 

00407 

99999 

02l52 

99977 

03897 

99924 

o564o 

99841 

07382 

99727 

46 

2 

7 

lb 

oo436 

99999 

02I8I 

99976 

03926 

99923 

05669 

99839 

07411 

99725 

45 

2 

8 

i-b 

oo465 

99999 

022II 

99976 

03955 

99922 

05698 

99838 

07440 

99723 

AA 

8 

17 

00495 

99999 

02240 

99975 

03984 

99921 

0D727 

9983b 

07469 

99721 

l\i 

9 
9 

18 
19 

00524 

99999 

02269 

99974 

o4oi3 

99919 

05756 
05785 

99834 

07498 

99719 

42 
4i 

oo553 

99998 

02298 

99974 

04042 

99918 

99833 

07527 

99716 

10 

20 

00582 

99998 

02327 

99973 

04071 

99917 

o58i4 

99831 

07556 

99714 

4o 

10 

21 

0061 1 

99998 

02356 

99972 

o4ioo 

99916 

o5844 

99829 

07  58  5 

99712 

39 

1 1 

22 

oo64o 

99998 

02385 

99972 

04129 

99915 

05873 

99827 

07614 

99710 

38 

II 

23 

00669 

99998 

02414 

99971 

o4i59 

99918 

05902 

99826 

07643 

99708 

37 

12 
12 

24 

25 

00698 

99998 

02443 

99970 

o4i88 

99912 

05931 
05960 

99824 

07672 

99705 

3b 
"35" 

00727 

99997 

02472 

99969 

04217 

9991 1 

99822 

07701 

99708 

l3 

26 

00756 

99997 

025oi 

99969 

04246 

99910 

05989 

99821 

07780 

99701 

iA 

I  J 

27 

00785 

99997 

o2  53o 

99968 

04275 

99909 

060 1 8 

99819 

07759 

99699 

S6 

i4 

28 

008 1 4 

99997 

o256o 

99967 

o43o4 

99907 

06047 

90817 

07780 

99696 

32 

14 

29 

00844 

99996 

02589 

99966 

04333 

99906 

06076 

9981b 

07817 

99694 

3i 

lb 

i5 

3o 
IT 

00873 

99996 

02618 

99966 

04362 

99905 

o6io5 

99813 

07846 

99692 

3o 

T' 

00902 

99996 

02647 

99965 

04391 

99904 

06 1 34 

99812 

07875 

99689 

lb 

32 

00931 

99996 

02676 

99964 

04420 

99902 

061 63 

99810 

07904 

99687 

28 

i6 

33 

00960 

99995 

02705 

99963 

04449 

99901 

06 1 92 

99808 

07933 

99685 

27 

lb 

34 

00989 

99995 

P2734 

99963 

04478 

99900 

06221 

99806 

07962 

99683 

2b 

17 

35 

01018 

99995 

02763 

99962 

o45o7 

99898 

06250 

99804 

07991 

99680 

2b 

17 
i8 

36 
37 

01047 

99995 

02792 
02821 

99961 

04536 

99897 

06279 

99803 

08020 
08049 

99678 

24 

23 

-J- 

01076 

99994 

99960 

04565 

99896 

o63o8 

99801 

99676 

i8 

38 

oiio5 

99994 

o285o 

99959 

04594 

99894 

06337 

99799 

08078 

996-3 

22 

'9 

39 

01 1 34 

99994 

02879 

99959 

04623 

99893 

06366 

99797 

08107 

99671 

31 

19 

4o 

01164 

99993 

029(^8 

99958 

04653 

99802 

06395 

99795 

08 1 36 

99668 

20 

20 

4i 

01 193 

99993 

02938 

99957 

04682 

99S90 

06424 

99793 

08 1 65 

99666 

19 

20 
21 

42 
43 

01222 

99993 

02967 

99956 

0471 1 

99889 

06453 

99792 

08194 

99664 

18 
17 

— 

oi25i 

99992 

02996 

99955 

04740 

99S88 

06482 

99790 

08223 

99661 

21 

AA 

01280 

99992 

o3o2  5 

99954 

04769 

99886 

o65 1 1 

99788 

08252 

99659 

16 

22 

45 

0 1 309 

99991 

o3o54 

99953 

04798 

99S85 

o654o 

99786 

0828  F 

99657 

lb 

2  2 

46 

oi338 

99991 

o3oS3 

99952 

04827 

99S83 

06569 

99784 

o83io 

99654 

i4 

0 

23 

47 

0 1 367 

99991 

o3lI2 

99952 

o4856 

09882 

0659S 

99782 

08339 

99652 

i3 

0 

23 
"?4 

48 
49 

01396 

99990 

o3i4i 

99951 

0480  5 
04914 

99881 
99879 

06627 
o6656 

997S0 
99778 

08368 

99649 

12 

11 

0 
0 

01425 

99990 

o3i7o 

99900 

08397 

99647 

24 

5o 

01454 

99989 

o3i99 

99949 

04943 

99878 

06685 

99776 

08426 

99644 

10 

0 

25 

5i 

oi483 

99989 

o322S 

99948 

04972 

99876 

06714 

99774 

08455 

99642 

9 

0 

25 

52 

oi5i3 

99989 

o3257 

99947 

o5ooi 

99875 

06743 

99772 

08484 

99639 

8 

0 

26 

53 

oi542 

99988 

03286 

99946 

o5o3o 

99873 

06773 

99770 

o85i3 

99637 

7 

0 

26 

27 

54 
55 

01571 

99988 

o33i6 

99945 

o5o59 

99872 

06802 

99768 

08542 

99635 

6 

0 
0 

01600 

99987 

03345 

99944 

o5o88 

99870 

o683 1 

99766 

08571 

99632 

27 

56 

01629 

99987 

03374 

99943 

o5ii7 

99869 

06860 

99764 

08600 

99630 

4 

0 

28 

57 

01 658 

99986 

o34o3 

99942 

o5i46 

99867 

06889 

99762 

08629 

99627 

6 

0 

28 

58 

01687 

99986 

o3432 

99941 

o5i75 

99866 

069 1 8 

99760 

08658 

99635 

2 

c- 

2Q 

5q 

01716 

99985 

o346i 

99940 

o52o5 

99S64 

06947 

997b8 

0S687 

99622 

I 

0 

29 

60 

01745 

99985 

03490 

99939 

o5234 

99863 

06976 

99756 

08716 

99619 

0 

0 

N.  COS. 

N.  sine. 

N.  COS. 

N.sine. 

N.  COS. 

N.sine. 

N.  COS.  N.  sine. 

N.  COS. 

N.  sine. 

8 

9° 

88° 

87° 

80° 

85° 

I 


TABLE  XXIV.               [^^soiGi 
Of  Natural  Sines. 

Prop, 
parts 

29 

c 

0 

I 
I 

2 
2 
3 

3 

4 
4 
5 
5 
6 

6 
7 
7 
8 
8 
9 
9 

ID 
10 
II 
II 

12 

12 
l3 

i3 
i4 
i4 
i5 

i5 
i5 
i6 
i6 
17 
17 
1 8" 
i8 
19 
19 
20 
20 

21 
21 
22 
22 

23 
23 

24 
24 

25 
25 

26 
26 
27 
27 
28 
28 
29 

i?. 

M 

c 

T 
2 

3 
4 
5 
6 

7 
8 

.:; 
.. 

12 

T3 

i4 
i5 
16 
17 
►  8 

'9 
20 
21 
22 

2  3 

24 

25 

26 

27 

28 

? 

00 

TT 

32 

33 

34 
35 
36 

"3^ 
38 

3? 
4o 
4i 
42 

43' 
44 
45 
46 

47 
48 

5o 
5i 

52 

53 
54 
'55 
56 

57 

58 
60 

5' 

6° 

70 

8° 

90 

60 

59 
58 

■57 
56 
55 
54 
53 

52 

5i 
5o 

49 
48 

47 

46 
45 
44 
43 
42 

41 
4o 
39 
38 
37 
36 

~35 
34 
33 

32 

3i 
3o 

27 
26 

25 

24 

23 
22 
21 
20 

'9 

18 

7r7 
16 
i5 
1 4 
i3 
12 

II 
10 

9 
8 

7 
6 

~5 
A 
3 
2 
I 
0 

M 

Prop. 

paru 

4 

4 
4 
4 
4 
4 
4 
4 

4 
3 
3 
3 
3 
■  3 

3 
3 
3 
3 
3 
3 

3 
0 

3 
3 
2 
2 
2 
2 
2 
1 
2 
2 
2 
2 
2 
2 
2 
2 
2 

0 
0 

0 
0 
0 
0 
0 
0 

N.  sine. 

N.  COS. 

N.  sine. 

N.  COS. 

N.  sine. 

N.  COS. 

N.  sine. 

N.  COS. 

N.  sine. 
15643 
15672 
15701 
i573o 
1 5758 
i57tS7 
i58i6 

N.  COS. 
98769 
98764 
98760 
98755 
98751 
98746 
98741 

08716 
08745 
08774 
o88o3 
o883i 
08860 
08889 

99619 
99617 
99614 
99612 
99609 
99607 
99604 

10453 
10482 
io5ii 
io54o 
10569 
10597 
10626 

99452 
99449 
99446 
99443 
99440 
99437 
99434 

12187 
12216 

12245 
12274 

I23o2 

I233I 
i2  36o 

99255 
99251 
99248 
99244 
99240 
99237 
99233 

13917 
13946 
13975 

i4oo4 
i4o33 
1 406 1 
14090 

14119 
i4i48 
14177 
i42o5 
14234 
14263 

99027 
99023 

99019 
99015 

99011 
99006 
99002 

98998 
98994 
98990 
98986 

98982 

98978 

08918 
08947 
08976 
09005 
09034 
09063 

99602 

9yJ99 

99596 
99594 
99591 
99588 

io655 
10684 
10713 
10742 
1 077 1 
10800 

99431 
99428 
99424 
99421 
99418 
9941 5 

12389 
12418 
12447 
12476 
i25o4 
12  533 

99230 
99226 
99222 
99219 
99215 
992 1 1 

i5845 
15873 
15902 
15931 
15959 
15988 

98737 
98732 
98728. 
98723 
98718 
98714 

09092 
09121 
09150 
09179 
09208 
09237 

99586 
99583 
99580 
99578 
99575 
99572 

10829 
10858 
10887 
10916 
10945 
10973 

99412 
99409 
99406 
99402 
99399 
99396 

12562 
12591 
1 2620 
1 2649 
12678 
12706 

99208 
99204 
99200 

99' 97 
99193 
99189 

14292 
14320 

14349 
14378 
14407 
14436 

14464 
14493 
14522 
i455i 
i458o 
14608 

98973 

98969 
98965 
98961 
98957 
98953 

98948' 
98944 
98940 
98936 

98931 

98927 

16017 
16046 
<f  '74 
1 6 1  o3 
;6i32 
1 6 1 60 
16189 
16218 
16246 
16275 
i63o4 
16333 

i636i 
1 6390 
16419 
16447 
16476 
i65o5 

16533 
1 6562 
16591 
16620 
16648 
16677 

16706 
16734 
16763 
16792 
16820 
,16849 
16878 
16906 
16935 
16964 
16992 
1 702 1 

98709 
98704 
98700 
98695 
98690 
98686 
98681 
98676 
9S671 
98667 
98662 
98657 
98652 
98648 
98643 
98638 
98633 
98629 

98624 
98619 
98614 
08609 
98604 
98600 

09266 
09295 
09324 
09353 
09382 
09411 

99570 
99567 
99564 
99562 
99559 
99556 

II 002 
iio3i 
1 1 060 
110S9 
11118 
11147 

99393 

99386 
99383 
99380 
99377 

12735 
12764 
12793 
12822 
i285i 
12880 

99186 
99182 
99178 
99175 
99171 
99167 

09440 
09469 
09498 
09527 
09556 
09585 

99553 
99551 
99548 
99545 
99542 
99540 

11176 

I  I205 

II234 
1 1 263 
11291 

II  320 

99374 
99370 
99367 
99364 
99360 
99357 

12908 
12937 
12966 
1 2995 
i3o24 
i3o53 

99163 
99160 
991 56 
99152 
99i'48 
99144 

14637 
14666 
14695 
14723 
14752 
14781 
14810 
1 4838 
14867 
14896 
14925 
14954 

98923 
98919 
98914 

98910 
98906 
98902 

98897 
98893 
98889 
98884 
98880 
98876 

09614 
09642 
0967 1 
09700 
09729 
09758 

99537 
99534 
99501 
99528 
99526 
99523 

1 1 349 
1 1 378 
1 1 407 
II436 
1 1 465 
1 1494 

99354 
99351 
99347 
99344 
99341 
99337 

i3o8i 
i3ii0 
i3i39 
i3i68 
i3i97 
13226 

99141 
99^37 
99133 
99129 
99123 
99122 

097^^7 
098 1 6 
09845 
09874 
09903 
09932 

99520 
99517 
99514 
995 1 1 
99500 
99506 

1 1523 
ii552 
ii58o 
1 1 609 
ii638 
1 1667 

99-34 
99331 
99327 
9932.4 
99320 
99317 

i3254 
1 3283 
i33i2 
1 334 1 
13370 
13399 

991 18 
991 14 
991 10 
99106 
99102 
99098 

14982 
i5oi  1 
i5o4o 
15069 
1 5097 
i5i26 

98871 
98867 
98863 
98S58 
9S854 
98849 

9S595 
98590 
98585 
98580 
98575 
98570 

98565 
98561 
98556 
98551 
98546 
98541 

09961 
09990 
10019 
10048 
10077 
10106 

995o3 
99500 
99497 
99494 
99491 
99488 

11696 
11725 
1 1754 
1 1783 
11812 
ii84o 

993 1 4 
99310 
99307 
9930  3 
99300 
99297 

13427 
i3456 
1.3485 
i35i4 
1 3543 
13572 

99f>94 
99091 
99087 
99083 
99079 
99075 

i5i55 
i5i84 

l5212 

i524i 
15270 

15299 

98845 

98841 
98836 
9S832 
98827 
98823 

ioi35 
10164 
10.192 
10221 

I025o 

10279 

99485 
99482 

99479 
99476 
99473 
99470 

1 1869 
11898 
1 1927 
1 1 956 
1 1985 

I20l4 

99293 
99290 
99286 
99283 
99279 
99276 

1 36oo 
13629 
1.3658 
i36S7 
13716 
13744 

99071 
99067 
99063 
99059 
99055 
9905 1 

15327 
1 5356 
1 5385 
i54i4 
1 5442 
1 547 1 

98818 
98814 
98809 

98Sf.5 
98800 
98796 

1 7o5o 
17078 
17107 
17.36 
17164 
17193 

98536 
98531 
98526 
98521 
98516 
9851 1 

io3o8 
io337 
10  366 
10395 
10424 
10453 

99467 
99464 
99461 
99458 
99455 
99452 

12045 
I  207  I 
12100 
12129 

i2i53 
12187 

99272 
99269 
99265 
99262 
99258 
99255 

13773 
t38o2 
1 383 1 
i385o 
i.3t889 
13917- 

99047 
99043 
99039 
99035 
9903 1 
99027 

i55oo 
15529 
15557 
1 5586 
i56i5 
1 5643 

98791 
98787 
98782 
98778 
98773 
98769 

N.sine. 

17222 
17250 

17279 
17308 
17336 
17365 

98506 
98501 
98,^96 
98491 
984S6 
98481 

N.  COS. 

N.  sine. 

N.  COS. 

N.  sine. 

N.  COS. 

N.sine. 

N.  COS. 

N.  COS. 

N.  sine. 

84° 

83° 

82° 

8P 

80° 

Page  162^ 

TABLE  XXIV. 

Of  Natural  Sines. 

Prop, 
parts 

28 
o 

M 

0 

10° 

11° 

12° 

13° 

14° 

60 

Prop. 
p.%rta 

6 

6 

N.  sine. 

N.  COS. 

lV.  sine. 

N.  COS. 

N.  sine 

N.  COS. 

N.  sinc.|N.  COS. 

N.  sine. 

N.  COS. 

I73b5 

98481 

190S1 

98168 

20791 

97S15 

22495 

97487 

24192 

97o3o 

o 

1 

17393 

98476 

19109 

98ib7 

20820 

97809 

22528 

97480 

24220 

97028 

59 

6 

I 

2 

17422 

98471 

19138 

98152 

20848 

97808 

22552 

97424 

24249 

97015 

b8 

6 

I 

3 

17451 

98466 

19167 

98 1 46 

20877 

97797 

22580 

97417 

24277 

97008 

57 

6 

2 

4 

17479 

98461 

19195 

98140 

20905 

97791 

22608 

974 1 1 

24805 

97001 

b6 

6 

2 

5 

17508 

98455 

19224 

98185 

20988 

97784 

22687 

97404 

24333 

96994 

bb 

6 

3 
3 

(i 

7 

17W7 

98450 

19252 

98129 

20962 

97778 

22665 

97898 

24862 

96987 

54 
53 

5 

5 

17565 

98445 

19281 

98124 

20990 

97772 

22698 

97891 

24890 

96980 

4 

8 

17594 

98440 

19809 

98118 

21019 

97766 

22722 

97384 

24418 

96978 

b2 

5 

4 

9 

17623 

98435 

19338 

981 12 

21047 

97760 

22750 

97378 

24446 

96966 

bi 

5 

5 

10 

1 765 1 

98430 

19366 

98107 

2 1 0-6 

97754 

22778 

97871 

24474 

96959 

bo 

5 

b 

II 

17680 

98425 

19895 

98101 

21 104 

9774s 

22807 

97865 

245o3 

96952 

49 

5 

6 
d 

12 

17708 

98420 

19423 

98096 

21182 

97742 

22885 

97858 

24581 

96945 

48 
47" 

5 
5 

17737 

98414 

19452 

98090 

21161 

97735 

22863 

97351 

24559 

96987 

7 

i4 

17766 

98409 

1 948 1 

98084 

21189 

97729 

22892 

9734b 

24587 

96980 

4b 

5 

7 

lb 

17794 

98404 

19509 

98079 

21218 

97728 

22920 

97888 

246 1 5 

96928 

4  b 

5 

7 

lb 

17823 

98399 

19538 

98073 

21246 

97717 

22948 

9733 1 

24644 

96916 

44 

4 

8 

17 

17852 

98394 

19566 

98067 

21275 

97711 

22977 

97825 

24672 

96909 

4d 

4 

8 
9 

18 
"19" 

17880 

983.89- 

19595 

98061 

2i3o8 

97705 

23oo5 

97818 
97811 

24700 

96902 

42 
4 1 

4 

17909 

98383 

19623 

98056 

2i33i 

97698 

28088 

24728 

96894 

9 

20 

17937 

98378 

19652 

9S050 

21860 

97692 

28062 

97804 

24756 

96887 

40 

4 

lO 

21 

17966 

98373 

19680 

9S044 

21888 

976S6 

28090 

97298 

24784 

96880 

39 

4 

lO 

22 

17995 

98368 

19709 

98089 

2l4l7 

97680 

28118 

97291 

24818 

96878 

38 

4 

1 1 

23 

18023 

98862 

19787 

98088 

21445 

97678 

28146 

97284 

24841 

96866 

37 

4 

II 

12 

24 
T5" 

i8o52 

98357 

1 9766 

98027 

21474 

97667 

23175 

97278 
97271 

24869 

9685s 

36 
35 

4 
4 

180S1 

9S352 

19794 

98021 

2l5o2 

97661 

28208 

24897 

96851 

12 

2b 

18109 

98347 

19828 

9S016 

2i53o 

97655 

28281 

97264 

24925 

96844 

M 

3 

l3 

27 

i8i38 

9834: 

19851 

9S010 

21559 

97648 

28260 

97257 

24954 

96887 

3d 

3 

i3 

28 

18166 

98336 

19880 

98004 

21587 

97642 

28288 

97251 

24982 

96829 

3-2 

3 

1 4 

29 

18195 

98331 

19908 

97998 

2I6I6 

97686 

28816 

97244 

25oio 

96822 

3i 

3 

i4 
f4 

3o 
17 

18224 

98325 

19987 

97992 

21644 

97680 

28345 

97237 

25o88 

96815 

3o 
"29 

3 
3 

18252 

98320 

19965 

97987 

21672 

97628 

28878 

97280 

25o66 

96S07 

lb 

32 

1 828 1 

983 1 5 

19994 

97981 

21701 

97617 

28401 

97228 

25094 

96800 

28 

3 

lb 

:i6 

i83o9 

98310 

20022 

97975 

21729 

9761 1 

28429 

97217 

25l22 

96798 

27 

3 

1 5 

M 

i8338 

98004 

2O05l 

97969 

21758 

97604 

23458 

97210 

25i5i 

96786 

2b 

3 

i6 

3b 

18367 

98299 

20079 

97963 

21786 

97598 

23486 

97208 

25179 

96778 

2b 

3 

17 

17 

3b 

37 

18395 

98294 

20108 

97958 

21814 

97592 

235i4 

97196 

25207 

96771 

24 
23 

2 
2 

18424 

98288 

2oi36 

97952 

21843 

97585 

23542 

97189 

25235 

96764 

18 

38 

18452 

98283 

2oi65 

97946 

21871 

97379 

28571 

97182 

25268 

96756 

2  2 

2 

18 

39 

1 848 1 

98277 

20198 

97940 

21899 

97573 

23599 

97176 

25291 

96749 

21 

2 

19 

4o 

18509 

98272 

20222 

97934 

21928 

97566 

28627 

97169 

25820 

96742 

20 

2 

19 

4i 

i8538 

98267 

20250 

97928 

21956 

97560 

28656 

97162 

25848 

96784 

19 

2 

20 
20 

42 
43 

18567 

98261 

20279 

97922 

21985 

97bb3 

23684 

97155 

25376 

96727 

18 
17 

2 
2 

18595 

98256 

20807 

97916 

22018 

97547 

23712 

97148 

25404 

96719 

21 

44 

18624 

98250 

20886 

97910 

2204l 

97541 

28740 

97141 

25482 

967 1 2 

lb 

2 

21 

4b 

i8652 

98245 

20864 

97905 

22070 

97534 

28769 

97134 

25460 

96705 

lb 

2 

21 

4b 

18681 

98240 

20893 

97899 

22098 

97528 

28797 

97127 

25488 

96697 

i4 

22 

47 

18710 

98234 

20421 

97893 

22126 

97521 

23825 

97120 

255i6 

96690 

i3 

2  2 
l3 

48 
49" 

18738 

98229 

2o45o 

97887 

22l55 

97b  lb 

28858 

97113 

25545 

96682 

12 
1 1 

18767 

98223 

20478 

97881 

221S8 

97508 

28882 

97106 

25578 

96675 

23 

bo 

18795 

98218 

20D07 

97875 

22212 

97502 

28910 

97100 

256oi 

96667 

10 

24 

bi 

18824 

98212 

2o535 

97869 

22240 

97496 

28988 

97093 

25629 

96660 

9 

24 

b2 

iS852 

98207 

2o563 

97868 

22268 

97489 

28966 

97086 

25657 

96653 

8 

25 

bi 

18S81 

98201 

20592 

97857 

22297 

97483 

28995 

97079 

25685 

96645 

7 

2D 
26 

b4 
55 

18910 

98196 

20620 

97851 

22825 

97476 

24028 

97072 

25718 

96688 

b 

T 

-^ 

18938 

98190 

20649 

97845 

22353 

97470 

24o5i 

97065 

25741 

96630 

26 

bb 

18967 

98185 

20677 

97889 

22882 

97463 

24079 

97o58 

25769 

96628 

4 

0 

27 

^7 

18995 

98179 

20706 

97883 

22410 

97457 

24108 

9705 1 

25798 

96615 

3 

0 

27 

b8 

19024 

98174 

20784 

97827 

22438 

9745o 

24 1 36 

97044 

25826 

96608 

2 

0 

28 

b9 

19052 

98 1 68 

20763 

97S21 

22467 

97444 

24164 

97087 

25854 

96600   I  1 

0 

28 

()0 

1 908 1 

98163 

20791 

97815 

22495 

%-437 

24192 

97080 

2588?. 

96598 

0 

0 

N.  COS.  N.  sine. 

N.  COS. 

^.  sine. 

N.  COS. 

\.  sine. 

N.  COS.  N.  sine. 

N.  COS.  N.  sine. 

79° 

78° 

77° 

7C° 

75» 

TABLE  XXIV.                       [Page  163 

Of  Natural  Sines. 

Pttip 

pans 

27 

o 

M 

0 

15° 

10° 

17° 

18° 

19° 

60 

Prop. 

parU 

9 

9 

N.  sine 

.  N.  fos 
96593 

N.  sine 

.  N.  cos 

N.  sine 

.  N.  COS. 

N.  sine 

N.  cos 

N.  sine 

.  N.  cos 

25882 

27564 

96126 

29237 

9563o 

30902 

95106 

32557 

94552 

o 

I 

25910 

96585 

27592 

961 18 

29265 

95622 

30929 

95097 

32,584 

94542 

5q 

9 

I 

2 

2b938 

96578 

27620 

96110 

29293 

956 1 3 

30957 

95088 

32612 

94533 

58 

9 

I 

3 

2  59bb 

96570 

27648 

96102 

29321 

95605 

30985 

9D079 

32639 

94523 

57 

9 

2 

4 

25994 

96562 

2767b 

96094 

29348 

95596 

3l012 

95070 

32667 

945 14 

5n 

8 

2 

b 

2b022 

96555 

27704 

96086 

29376 

95588 

3io4o 

95061 

32694 

945o4 

55 

8 

3 

6 

7 

26o5o 

96347 

27731 

96078 

29404 

95579 

3i.ob8 

95o52 

32722 

94495 

54 
53 

8 

8 

2bo79 

96540 

27759 

96070 

29432 

95571 

31095 

95043 

32749 

94485 

4 

8 

26107 

96532 

27787 

96062 

29460 

95562 

3ii23 

95o33 

32777 

94476 

52 

8 

4 

9 

26135 

96524 

27815 

96054 

294S7 

95554 

3ii5i 

95024 

32804 

94466 

5i 

8 

b 

lO 

26163 

96517 

27843 

9604b 

29515 

95545 

31178 

95oi5 

32832 

94457 

5o 

8 

6 

u 

2bi9i 

96509 

27871 

96037 

29543 

95536 

3 1 206 

95qo6 

32859 

94447 

49 

~6" 

12 

71 

26219 

96502 

27899 

96029 

29571 

95528 
95519 

3i233 
31261 

94997 
94988 

32887 

94438 

48 
47 

7 

20247 

96494 

27927 

96021 

29599 

32914 

94428 

b 

i4 

26275 

96486 

279:0 

96013 

2962b 

95511 

31289 

94979 

32942 

94418 

46 

7 

7 

lb 

2b3o3 

96479 

279S3 

96005 

29654 

95502 

3i3ib 

94970 

32969 

94409 

45 

7 

7 

lb 

2633i 

96471 

280 1 1 

9^997 

29b82 

95493 

3 1 344 

94961 

32997 

94399 

AA 

7 

8 

17 

26359 

96463 

28039 

959S9 

29710 

95485 

3i372 

94952 

33o24 

94390 

43 

6 

8 
9 

■18 

'9 

26387 

96456 

28067 

95981 

29737 

95476 

3 1399 
31427 

94943 
94933 

33o5i 

94380 

42 
4i 

6 
6 

2b4i5 

9644s 

28095 

95972 

29765 

95467 

33079 

94370 

9 

20 

2b443 

96440 

28123 

95964 

29793 

95459 

3 1454 

94924 

33io6 

94361 

4o 

6 

9 

21 

26471 

96433 

28i5o 

95956 

29821 

95450 

3i482 

94915 

33 1 34 

94351 

39 

6 

lO 

22 

265oo 

96425 

2817S 

95948 

29849 

95441 

3i5io 

94906 

33i6i 

94342 

38 

6 

lO 

23 

26528 

96417 

28206 

95940 

29876 

95433 

3i537 

94S97 

33189 

94332 

37 

6 

1 1 
II 

24 

25 

26556 

96410 

28234 

95931 

29904 

95424 

3 1 565 
31593 

94888 
94878 

33216 

94322 

36 
35 

5 
5 

26584 

96402 

28262 

95923 

29932 

95415 

33244 

943 1 3 

12 

2b 

26612 

96394 

28290 

95915 

29960 

95407 

31620 

94869 

33271 

943o3 

M 

5 

12 

27 

26640 

96386 

283i8 

95907 

29987 

95398 

3 1 648 

94860 

33298 

94293 

33 

5 

iJ 

28 

2666S 

96379 

28346 

95898 

3ooi5 

95389 

31675 

9485 1 

33326 

94284 

32 

5 

1 3 

29 

26696 

96371 

28374 

95890 

3oo43 

953S0 

3 1 703 

94842 

33353 

94274 

3i 

5 

t4 

3o 
3i 

26724 

96363 

28402 

95882 

30071 

95372 

3 1730 

94S32 

3338 1 

94264 

3o 
29 

5 
4 

26752 

96355 

28429 

95874 

30098 

95363 

3i758 

94S23 

33408 

94254 

i4 

32 

26780 

96347 

284  D7 

95865 

3oi26 

95354 

3178b 

o48i4 

ZM^6 

94245 

28 

4 

ID 

33 

26S08 

96340 

28485 

95857 

3oi54 

95345 

3i8i3 

94So5 

33463 

94235 

27 

4 

l5 

34 

26836 

96332 

285 1 3 

9'iS49 

3oi82 

95337 

3i84i 

94795 

33490 

94225 

26 

4 

lb 

35 

26S64 

96324 

28541 

95841 

30209 

9532S 

3 1868 

94786 

335i8 

94215 

25 

4 

lb 
I? 

2Ci 
37 

26892 

96316 

28569 

95832 

3o237 

95319 

31896  94777 

33545 

94206 

24 

23 

4 
3 

26920 

96308 

28597 

95824 

3o265 

95310 

31923 

94768 

33573 

94196 

17 

38 

26948 

96301 

28625 

95816 

30292 

95301 

3i95i 

94758 

336oo 

94186 

22 

3 

i8 

39 

26976 

96293 

2S652 

95807 

3o32o 

95293 

31979 

94749 

33627 

94176 

21 

3 

i8 

40 

27004 

96285 

286S0 

95799 

3o348 

95284 

32006 

94740 

33655 

94167 

20 

3 

i8 

4i 

27032 

96277 

28708 

95791 

30376 

95275 

32o34 

94730 

33682 

94157 

19 

3 

19 

42 
43 

27060 

96269 

2S736 

95782 

3o4f)3 

95266 

32061 
32089 

94721 

33710 

94147 

18 

17 

3 
3 

270S8 

96261 

28764 

9^774 

3o43i 

95257 

94712 

33737 

94137 

20 

U 

27116 

96253 

28792 

95766 

30459 

9524s 

32116 

94702 

33764 

94127 

i6 

2 

20 

45 

27144 

96246 

28820 

95757 

3o48b 

95240 

32144 

94693 

33792 

94118 

1 5 

2 

21 

46 

27172 

96238 

28847 

95749 

3o5i4 

9523i 

32171 

94604 

33819 

94108 

i4 

2 

21 

47 

27200 

96230 

28S75 

95740 

3o542 

95222 

32199 

946-4 

33846 

94098 

i3 

2 

2  2 
22 

48 
49 

27228 

96222 

28903 

95732 

3o570 

95213 

32227 

94665 

94656 

33874 
33901 

94088 

12 
1 1 

2 
2 

27256 

96214 

28931 

95724 

3o597 

95204 

32254 

94078 

2J 

5o 

27284 

96206 

28959 

95715 

30625 

95195 

32282 

94646 

33929 

94068 

10 

2 

23 

5i 

27312 

96198 

28987 

95707 

3o653 

95186 

3^309 

94637 

3395b 

940  5  8 

9 

23 

52 

27340 

96190 

29015 

95698 

3o68o 

95177 

32337 

94627 

33983 

94049 

8 

24 

53 

27368 

96182 

29042 

95690 

30708 

95168 

32364 

94618 

34011 

94039 

7 

24 
25 

_5£ 
55 

27396 

96174 

29070 

95681 

3o736 

95159 

32392 

94609 

34o38 

94029 

6 

— 

27424 

96166 

2909S 

95673 

3(^763 

95i5o 

32419 

94599 

34o65 

94019 

2b 

56 

27452 

96158 

29126 

95664 

30791 

95142 

32447 

94590 

34093 

94009 

4 

2b 

57 

27480 

96150 

29154 

95656 

3o8 1 9 

95i33 

32474 

94580 

34120 

5,3999 

3 

0 

2b 

58 

27508 

96142 

29182  95647 

3o84(i 

95124 

32502 

94571 

34147 

93989 

2 

0 

27 

■jq 

27536 

96134 

29209  95639 

30874 

95ii5 

02529 

94561 

34175 

93979 

I 

0 

^ 

bo 

27564 

96126 

29237  9563o 

30902 

95106 

32557 

94552 

34202 

93969 

0 

0 

N.  COS. 

V.  sine. 

N.  COS.  N..sine. 

\.  co^.N.sino. 

N.  COS. 

V.  sine. 

V.  COS.  N.  sine. 

7-i 

" 

78^ 

72° 

7P    1 

70° 

Page  104]               TABLE  XXIV. 

Of  Natural  Sines. 

Prop, 
paru 

27 

o 

M 
0 

20° 

21° 

22° 

23° 

24° 

60 

Prop. 
p:ins 

11 

II 

N.  sine 

N.  COS. 

N.  sine 

N.  COS. 

N.  sine 

N.  COS 

N.  sine 

N.  COS. 

i\.  sine 

N.  COS. 

34202 

93969 

35837 

93358 

37461 

92718 

39073 

92o5o 

40674 

91355 

o 

I 

34229 

93959 

35864 

93348 

37488 

92707 

39100 

92089 

40700 

91343 

59 

II 

1 

2 

34257 

93949 

35891 

93337 

375x5 

92697 

39127 

92028 

40727 

9i33i 

58 

tl 

I 

3 

34284 

93939 

35918 

93327 

37542 

92686 

39153 

92016 

40753 

91819 

57 

10 

■2 

4 

343 1 1 

93929 

35945 

93316 

37569 

92675 

89180 

92005 

40780 

91807 

56 

[O 

2 

b 

34339 

93919 

35973 

93306 

37595 

92664 

39207 

91994 

40806 

91295 

55 

It; 

3 

b 

7 

34366 

93909 

36ooo 

93295 

37622 

93653 

39234 

919S2 

4o833 

91283 

54 
53" 

10 
10 

34393 

93899 

36027 

93285 

37649 

92642 

39260 

91971 

40860 

91272 

4 

8 

34421 

93S89 

36o54 

93274 

37676 

92631 

39387 

91959 

40886 

9 1 260 

53 

10 

4 

9 

34448 

9J879 

36o8r 

93264 

37703 

92620 

39814 

91948 

40918 

91248 

5 1 

9 

5 

10 

34475 

93869 

36 1 08 

93353 

37730 

92609 

39341 

91936 

40939 

91286 

5o 

9 

5 

II 

345o3 

93359 

36i35 

93343 

37757 

92598 

39367 

91925 

40966 

91224 

49 

9 

5 

6 

12 

34530 

93849 

36162 
36190 

93232 

37784 

92587 

39894 

91914 

40992 

91212 

48 
47 

9 

34557 

93839 

93222 

37811 

92576 

8942 1 

91902 

41019 

91200 

0 

14 

34584 

93829 

36217 

93211 

37838 

92565 

39448 

91S91 

41045 

91188 

46 

8 

7 

lb 

346 1 2 

93819 

36244 

93201 

37865 

92554 

39474 

91879 

41072 

91176 

45 

8 

7 

lb 

34639 

93809 

36271 

93190 

37S92 

92543 

39501 

9.868 

41098 

91164 

44 

8 

8 

17 

34666 

93799 

36298 

93180 

37919 

92532 

39528 

91855 

4ii35 

91 152 

43 

8 

8 
9 

18 

34694 

93789 

3632  5 

93169 

37946 

92521 

39555 
39581 

91845 
91833 

4ii5i 

91140 

42 
4i 

8 
8 

3472 1 

93779 

36352 

93 1 59 

37973 

92510 

41178 

91128 

9 

20 

34748 

93769 

36379 

93i48 

37999 

92499 

39608 

91822 

4i2o4 

91 116 

4o 

7 

9 

21 

34775 

93759 

364o6 

93.37 

38036 

92488 

39635 

91810 

4i23i 

91104 

39 

7 

lO 

22 

348o3 

93748 

36434 

93127 

38o53 

92477 

89661 

91799 

41257 

91092 

38 

7 

lO 

23 

3483o 

93738 

3646 1 

93t  16 

38o8o 

92466 

39688 

917S7 

41284 

91080 

37 

7 

II 
II 

24 

25 

34857 
34884 

93728 

36438 

93106 

38107 

92455 

39715 

91775 

4i3io 

91068 

36 
"3c 

7_ 
d 

93718 

365 1 5 

93095 

38 1 34 

92444 

39741 

91764 

4i337 

91056 

12 

2b 

34912 

93708 

36542 

93084 

38i6i 

92432 

39768 

91752 

4i363 

91044 

34 

b 

12 

27 

34939 

93698 

36569 

93074 

38 188 

92421 

39795 

91741 

41890 

91082 

33 

() 

l3 

28 

34966 

93688 

36596 

93o63 

382i5 

92410 

39822 

91729 

4i4i6 

9102c 

32 

b 

i3 

29 

34993 

93677 

36623 

93o52 

38241 

92399 

39S48 

91718 

41443 

9100B 

3i 

6 

i4 
i4 

3o 

17 

35o2i 
35o48 

93667 

36650 

93042 

38268 

92388 

39875 

91706 

41469 
41496 

90996 

90984 

3^ 
29 

6 
5 

93«7 

36677 

93o3i 

38295 

92377 

89902 

91694 

14 

32 

35075 

93647 

36704 

93020 

38322 

92866 

39928 

91683 

4 1 52 2  1  90972 

28 

5 

lb 

33 

35i02 

93637 

36731 

93010 

38349 

92355 

39955 

91671 

4;  1 54  J  9J980 
415.75  90948 

37 

5 

i5 

34 

35i3o 

93626 

36758 

92999 

38376 

92343 

89982 

91660 

26 

5 

i6 

3b 

35 1 57 

93616 

36785 

929SS 

384o3 

92332 

40008 

91648 

4i6o'j  1  90986 

25 

5 

i6 

17 

3b 

T7" 

35i84 

93606 

368 1 2 

92978 
92967 

38430 

92821 

4oo35 

9i636 

41628  90924 

24 

23 

4 

4 

352II 

93596 

36839 

38456 

92810 

40062 

91625 

4i655 

90911 

17 

38 

35239 

93585 

36867 

92956 

38483 

92299 

40088 

91618 

4s68i 

90899 

22 

4 

18 

39 

35266 

93575 

36894 

92945 

3S5io 

92287 

4oii5 

91601 

41707 

90887 

21 

4 

18 

4K 

35293 

93565 

3692 1 

92935 

38537 

92276 

4oi4i 

91590 

41734 

90875 

20 

4 

18 

41 

35320 

93555 

36948 

92934 

38564 

92265 

40168 

91 578 

41760 

90868 

'9 

3 

J2. 
19 

42 
43 

35347 

93544 

36975 

92913 

38591 

92254 

40195 
40221 

9:566 
91555 

41787 

9085 1 

18 
17 

3 
3 

35375 

93534 

37002 

92902 

38617 

92243 

4i8i3 

90889 

20 

^4 

35402 

93524 

37029 

92893 

38644 

92281 

40248 

01 543 

4 1 840 

90S  2  6 

16 

3 

20 

43 

35429 

935i4 

37o56 

92881 

38671 

92220 

40275  91 53 1 

41866 

90814 

i5 

3 

21 

46 

35456 

935o3 

37083 

92870 

38698 

92209 

4o3o  I 

91519 

41892 

90802 

i4 

3 

21 

47 

35484 

93493 

371 10 

93859 

38725 

92198 

40828 

9i5o8 

41919 

90790 

.3 

2 

22 
22 

48 
49' 

355 II 

93483 

37137 
37164 

92849 
92838 

38752 

92 1 86 

4o355 

91496 

41945 

90778 

12 
11 

2 
2 

35538 

93472 

38778 

92175 

4o38i 

91484 

41972 

90766 

23 

bf) 

35565 

93462 

37191 

92827 

3SSo5 

92164 

4o4o8 

91472 

41998 

90753 

10 

2 

23 

5i 

yynp 

93402 

37218 

92816 

38832 

92152 

40434 

91461 

42024 

90741 

9 

2 

23 

12 

35619 

93441 

37245 

93805 

38859 

92141 

4o46i 

91449 

42o5i 

90729 

8 

24 

bi 

35647 

93431 

37273 

92794 

38886 

92i3o 

4o488 

91437 

42077 

90717 

7 

24 

25 

b4 
55 

35674 

93420 

37299 

92784 

38oi2 

92119 

4o5i4 
4o54i 

91425 

42104 

90704 

6 
5 

35701 

93410 

37326 

92773 

38939 

92107 

91414 

42180 

90692 

25 

5f) 

35728 

93400 

37353 

92762 

38966 

92096 

40567 

91402 

421 56 

90680 

4 

2b 

1"" 

35755 

93389 

37380 

92751 

38993 

92085 

40594 

9 1 390 

42183 

90668 

3 

26 

5K 

35782 

93379 

37407 

92740 

39020 

92073 

40621 

91378 

42209 

90655 

2 

0 

27 

,.!■; 

358io 

93368 

37434 

92729 

39046 

92062 

40647 

91866 

42235 

90643 

I 

0 

27 

bo 

35837 

93358 

37461 

92718 

39073 

92o5o 

40674 
N.  COS. 

91355 
N.  sine. 

42262 

9063 1 

0 

0 
; 

N.  COS. 

V.  sine. 

N.  COS. 

N.  sine. 

N.  COS. 

\.  sine. 

M.  cos. 

N.  sine. 

69° 

68°       G7° 

66° 

65° 

TABLE  XXIV.               [rage  les 
Of  Natural  Sines. 

Prop, 
parts 

26 

o 

o 

I 
1 

2 
2 
3 

3 
3 

i 
4 
5 
5 
6 
6 
7 
7 
7 
8 

8 
9 
9 

lO 
10 
10 

1 1 
II 

12 
12 
l3 

i3 
71 
i4 
i4 
i5 
i5 
i6 

i6 
i6 
17 
17 
i8 
i8 

19 

19 
20 
20 
20 
21 

21 
22 
22 

23 
23 

53 

24 
24 

25 
25 

26 
26 

M 

0 
I 
2 
3 
4 
5 
6 

7 
8 

9 
:o 
u 
12 

7T 
14 

i5 
16 

17 
18 

'9 
20 
21 
22 

23 

M_ 
25 
26 

27 
28 

=9 
3o 

TT 

32 

33 
34 
35 
36 

37 
38 
39 
40 
4i 
42 

43 
44 

45 
46 
4i 
48 

49 
5o 
5i 

52 

53 
54 
55 
56 

57 
58 

60 

25= 

26° 

27° 

28° 

29° 

60 

59 

S8 

57 
56 
55 
54 
53 

52 

5i 
5o 
49 
48 

47 
46 
45 
44 
43 
42 

4i 
4o 
39 
38 

37 
36 

35 
34 
33 

32 

3i 

3o 

29 
28 

27 
26 

25 

24 

23 
22 
21 
20 

19 
18 

"17" 
16 

i5 
i4 
i3 
12 

1 1 

10 

9 

8 

7 
6 

5 
4 
3 
2 

I 
0 

Prop. 

14 

~T4 
i4 
i4 
i3 
i3 
i3 
i3 
12 
12 
12 
12 
11 
1 1 
11 
II 
II 
10 
10 
10 

10 
9 
9 
9 
9 
8 

~8' 

8 

8 

7 

7 
__7_ 

7 

7 

6 

6 

6 

6 

5 

5 

5 

5 

4 

4 

4" 
4 
4 
3 
3 
3 

~T 
2 
2 
2 
■k 
1 

I 
I 

I 
0 
0 
0 

N.  sine 

N.  COS. 

N.  sine 

N.  COS. 

N.  sine 

N.  COS. 

N.  sine 

N.  COS. 

N.  sine 

N.  COS. 

42262 
42288 
423i5 
42341 
42367 
42394 
42420 

91163 1 
906 1 8 
90606 
90594 
90582 
90569 
90557 

43837 
43863 
438S9 
43916 
43942 
43968 
43994 

89879 
89867 
89854 
89841 
89828 
89S16 
89803 

45399 
45425 
4545i 

45477 
455o3 
45529 
45554 

89101 
89087 
89074 
89061 
89048 
89035 
89021 

46947 
46973 
46999 
47024 
47o5o 
47076 
47101 
47127 
47153 
47178 
47204 
47229 
47255 

88295 
88281 
88267 
88254 
88240 
88226 
88213 

48481 
485o6 
48532 
48557 
48583 
^8608 
48634 

87462 
87448 
87434 
87420 
87406 
87391 
87377 

42446 
42473 
42^99 

42525 

42552 

4257S 

90545 
90532 
90520 
9u5o7 
90495 
90483 

44020 
44o46 
44072 
44098 
44124 
.44i5i 

89790 
89777 
89764 
89752 
S9739 
89726 

45580 
45606 
45632 
45658 
45684 
45710 

89008 
88995 
88981 
8896S 
88955 
88942 

88199 
88i85 
88172 
88 1 58 
88144 
88i3o 

48659 
48684 
48710 
48735 
48761 
48786 

87363 
87349 
87335 
87321 
87306 
87292 

42604 

4263 1 
42657 

42683 

42709 
42736 

90470 
90458 
90446 
90433 
90421 
9040S 

44177 
44203 
44229 
44255 
44281 
44307 

89713 
89700 
89687 
89674 
89662 
89649 

45736 
45762 
45787 
458 1 3 
45839 
45863 

88928 
88915 
88902 
88888 
88875 
88852 

47281 
473o6 
47332 
47358 
47383 
47409 

47434 
4746)0 
47486 
475ii 
47537 
47562 

47588 
47614 
47639 
47665 
47690 
47716 

88117 
88io3 
88089 
88075 
88062 
S804S 

88o34 
88020 
S8006 
-87993 
87979 
87965 

48811 
48837 
48862 
48888 
48913 
48938 

87278 
87264 
87250 
87235 
87221 
87207 

42762 
4278S 
42815 
42841 
42S67 
42894 

90396 
90383 
90371 
90358 
90346 
90334 

44333 
44359 
44385 
444 1 1 
444'^! 
44464 

89636 
89633 
89610 
89597 
89584 
89571 

45891 
45917 
45942 
45968 
45994 
46020 

88848 
88835 
88822 
88808 
88795 
88782 

48964 
48989 
49014 
49040 
49065 
49090 

49116 
49141 
49166 
49192 
49217 
49242 

87193 
87178 
87164 
87150 
87136 
87121 

87107 
87093 
87079 
87064 
87050 
87036 

42920 
42946 
42972 
42999 
43o25 
43o5i 

90321 
90309 
90296 
90284 
90271 
90259 

44490 
445 16 
44542 
44568 
44594 
44620 

89558 

89545 

89532 

89519. 

89506 

89493 

46o46 
46072 
46097 
46123 

46149 
46175 

88768 
88755 
88741 
88728 
88715 
88701 

87951 
87937 
87923 
87909 
87896 
87882 

43077 
43io4 
43i3o 
43 1 56 
43182 
43209 

90246 
90233 
90221 
90208 
90196 
90163 

44646 
44672 
44698 
44724 
4475o 
44776 

89480 
89467 
89454 
89441 
89428 
89415 

46201 
46226 
46252 
46278 
463o4 
4633o 

88688 
88674 
8S661 
88647 
88634 
88620 

47741 
47767 
47793 
47818 
47844 
47869 

47895 
47920 
47946 
47971 
47997 
48022 

87868 
87854 
87840 
87826 
87812 
87798 

49268 
49293 
49318 
49344 
49369 
49394 

87021 
87007 
86993 
86978 
86964 
86949 

43235 
43261 
43287 
433 1 3 
43340 
43366 

90171 
901 58 
90146 
90133 
90120 
9010S 

44802 
44828 
44854 
44880 
44906 
44932 
44958 
44984 
45oio 
45o36 
45062 
45o88 

89402 
89389 
89376 
89363 
89350 
89337 

46355 
4638 1 
46407 
46433 
46458 
4<^4H 

88607 
88593 
8858o 
88566 
88553 
88539 

87784 
87770 
87756 
87743 
87729 
87715 

49419 
49445 
49470 
49495 
49521 
49546 
49571 
49596 
49622 

49647 
49672 
49697 

86935 
86921 
86906 
86892 
86878 
80863 

86849 
86834 
86820 
868o5 
86791 
86777 

43392 
43418 
43445 
43471 
43497 
43523 

90095 
90082 
90070 
90057 
90045 
90032 

89324 
8931 1 
89298 
89285 
89272 
89259 

465 10 
46536 
4656 1 
46587 
4661 3 
46639 

88526 
885 1 2 
88499 
88485 
88472 
88458 

48048 
48073 
48099 
48124 
48i5o 
48175 

87701 
87687 
87673 
87659 
87645 
87631 

43549 
43575 
43602 
43628 
43654 
436So 

90019 

90007 
89994 
89981. 
89968 
89956 

45ii4 
45i4o 
45i66 
45192 
45218 
45243 

89245 
89232 
89219 
89206 
89193 
89180 

46664 
46690 
46716 
46742 
46767 
46793 

88445 
8843 1 
88417 
8S4o4 
88390 
88377 

48201 
48226 
48252 
48277 
483o3 
48328 

87617 
87603 
87589 
87575 
87561 
87546 

49723 
49748 
49773 
49798 
49824 
49849 

86762 
86748 
86733 
8O719 
86704 
86690 

86675 
86661 
86646 
86632 
86617 
8660; 

43706 
43733 
43759 
43785 
438 1 1 
43837 

89943 
89930 
89918 
89905 
89892 
89879 

45269 
45295 
45321 
45347 
45373 
45399 

89167 
89153 
89140 
89127 
89114 
89101 

46819 
4(iU4 
46870 
46896 
46921 
46947 

88363 
88349 
88336 
88322 
883o8 
88295 

48354 
48379 
484o5 
4843o 
48456 
48481 

87532 
87518 
87504 
87490 
87476 
87462 

49874 
49899 
49924 
49950 
49975 
5  0000 

N.  COS. 

.\.  sine. 

N.  COS. 

\.  sine. 

N.  co^.  N.  sine. 

N.  COS.  N.  s'iie. 

N.  COS. 

N.  sine. 

04°     1 

6.3=     1 

G2° 

61° 

60°    1 

Page  166]                       TABLE  XXIV. 

Of  Natural  Sines. 

Prop, 
par'.s 

16 

16" 

Prop, 
pans 

25 

o 

M 

o 

30° 

31° 

32° 

33° 

34° 

60 

N.  sine. 

N.  fos. 

N.  sine.  N.  COS. 

N.  sine. 

N.  COS. 

N.  sine. 

N.  COS. 

N.  sine. 

N.  COS. 

5oooo 

866o3 

5i5o4  85717 

52992 

848o5 

54464 

83867 

55919 

82904 

0 

I 

50025 

86588 

5i529 

85702 

53017 

84789 

54488 

8385 1 

55943 

82887 

59 

16 

I 

2 

5oo5o 

86573 

5i554 

85687 

53o4i 

84774 

545 1 3 

83835 

55968 

82871 

58 

i5 

I 

3 

50076 

86559 

bib79 

85672 

53o66 

84759 

54537 

838i9 

55992 

82855 

57 

i5 

2 

4 

5oioi 

86544 

5i6o4 

85657 

53091 

84743 

54561 

838o4 

56oi6 

82839 

56 

i5 

2 

b 

50126 

8653o 

51628 

85642 

53ii5 

84728 

54586 

83788 

56o4o 

82822 

55 

i5 

3 
3 

6 

7 

,5oi5i 

865 1 5 

5i653 

85627 

53i4o 

84712 

54610 
54635 

83772 

56o64 

82806 

54 
53' 

i4 

50176 

865oi 

51678 

856i2 

53 1 64 

840  n 
8468 1 

83756 

56oS8 

82790 

3 

8 

5020I 

86486 

5 1 703 

85597 

53189 

54659 

83740 

56ii2 

82773 

52 

1 4 

4 

9 

50227 

86471 

51758 

85582 

53214 

84666 

54683 

83724 

56 1 36 

82757 

5i- 

1 4 

4 

10 

50252 

86457 

51753 

85567 

53238 

8465o 

54708 

83708 

56 1 60 

82741 

5() 

i3 

5 

II 

50277 

86442 

51778 

8555i 

53263 

84635 

54732 

83692 

56i84 

82724 

4o 

i3 

5 
'  5 

12 

l3 

5o3o2 

86427 

5i8o3 

85536 

53288 

84619 

54756 

83676 

56208 

82708 

48 

47 

i3 

5o327 

8G4i3 

51828 

85521 

533 1 2 

846o4 

54781 

83660 

56232 

82692 

6 

i4 

5o352 

86398 

5i852 

855o6 

53337 

84588 

548o5 

83645 

56256 

82675 

46 

12 

6 

lb 

5o377 

86384 

bi877 

85491 

5336i 

84573 

54829 

83629 

56280 

82659 

45 

12' 

7 

i6 

5o4o3 

86369 

51902 

85476 

53386 

84557 

54854 

836 1 3 

563o5 

82643 

44 

12 

7 

17 

50428 

86354 

51927 

85461 

53411 

84542 

54S78 

83597 

56329 

82626 

43 

I  I 

8 
8 

i8 
19 

5o453 

86340 
86325 

51952 

85446 

53435 

84526 
845 1 1 

54902 

83581 

56353 

82610 

42 
4i 

I  1 
I  1 

50478 

51977 

85431 

53460 

54927 

83565 

56377 

82593 

8 

20 

5o5o3 

863 10 

52002 

85416 

53484 

84495 

54q5i 

83549 

564oi 

82577 

40 

I  I 

9 

21 

5o528 

86295 

52026 

854oi 

53509 

84480 

54975 

83533 

56425 

82561 

39 

TO 

9 

22 

5o553 

86281 

52o5i 

853S5 

53534 

84464 

54999 

83517 

56449 

82544 

38 

10 

10 

23 

50578 

86266 

52076 

S5370 

53558 

84448 

55o24 

83501 

56473 

82528 

37 

10 

10 
ID 

1± 
25 

5o6o3 

S6251 

52IOI 

85355 

53583 

84433 

55o48 

83485 

56497 

82511 

36 
IT 

10 

9 

50628 

86237 

52126 

85340 

53607 

84417 

55072 

83469 

56521 

82495 

II 

26 

5o654 

86222 

52i5i 

85325 

53632 

84402 

55097 

83453 

56545 

82478 

34 

9 

II 

27 

50679 

86207 

52175 

853io 

53656 

84386 

55i2i 

33437 

56569 

82.^62 

33 

9 

12 

28 

50704 

86192 

52200 

85294 

53681 

84370 

55i45 

83421 

56593 

82446 

32 

r3 

29 

50720 

86178 

52225 

85279 

53705 

84355 

55169 

834o5 

56617 

82429 

3i 

8 

i3 
i3 

3o 
3i 

50754 

86 1 63 

5225o 

85264 

53730 

84339 

55194 
55218 

83389 

56641 

82413 

3o 
29 

8 
~~8' 

50779 

86:48 

52275 

85249 

53754 

84324 

83373 

56665 

82396 

i3 

32 

5o8o4 

86i33 

52299 

85234 

53779 

843o8 

55242 

83356 

56689 

82380 

28 

7 

i4 

33 

50829 

86119 

52324 

852i8 

538o4 

84292 

55266 

8334o 

56713 

82363 

27 

7 

i4 

34 

5o854 

86104 

52349 

852o3 

53828 

84277 

55291 

83324 

56736 

82347 

26 

7 

i5 

35 

50879 

860S9 

52374 

85i88 

53853 

84261 

553i5 

833o8 

56760 

8233o 

25 

7 

i5 

1 5" 

36 

17 

50904 

86074 

52399 

85i73 

53877 

84245 

55339 

83292 

567S4 
568o8 

823i4 

24 
23 

6 
~6' 

50929 

86059 

52423 

85i57 

53902 

8423o 

55363 

88276 

82297 

i6 

38 

50954 

86045 

52448 

85i42 

53926 

84214 

55388 

83260 

56832 

82281 

22 

6 

i6 

39 

50979 

86o3o 

52473 

85i27 

53951 

84198 

55412 

83244 

56856 

82264 

21 

6 

17 

4o 

5ioo4 

860 1 5 

5249S 

85ii2 

53975 

84182 

55436 

83228 

56880 

82248 

20 

5 

17 

4i 

51029 

86000 

52522 

85096 

54000 

84167 

55460 

83212 

56904 

8223l 

10 

5 

i8 

42 

43 

5io54 

85985 

52547 

85o8i 

54024 

84i5i 

55484 

83195 

56928 

82214 

18 

17 

5 
5 

51079 

85970 

52572 

85o66 

54049 

84i35 

55509 

83 1 79 

56952 

82198 

i8 

44 

5iio4 

85956 

52597 

85o5i 

54073 

84120 

55533 

83i63 

56976 

82181 

16 

4 

19 

45 

51129 

85941 

52621 

85o35 

54097 

84 104 

55557 

83i47 

57000 

82165 

i5 

4 

!9 

46 

5ii54 

85926 

52646 

85o2o 

54122 

84088 

55581 

83i3i 

57024 

b2i48 

i4 

4 

20 

47 

5x179 

85911 

52671 

85oo5 

54 1 46 

84072 

556o5 

83n5 

57047 

82132 

i3 

3 

20 
20 

48 
49 

5i2o4 

85896 

52696 

849S9 

54171 

84o57 

5563b 

83098 

57071 

8211b 

82098 

12 
1 1 

3 

51229 

8588i 

52720 

84974 

54195 

84o4i 

55654 

83o82 

57095 

21 

bo 

bi254 

85866 

52745 

84959 

54220 

84025 

55678 

83o66 

57119 

82082 

10 

3 

21 

5i 

51279 

8585i 

52770 

84943 

54244 

84009 

55702 

83o5o 

57143 

82065 

9 

2 

22 

52 

5i3o4 

85836 

52794 

84928 

54269 

83994 

55726 

83o34 

57167 

82048 

8 

2 

2  2 

53 

5i329 

85821 

52819 

84913 

54293 

83978 

55750 

83oi7 

57191 

82032 

-7 

2 

23 
2  3 

_54_ 
55 

5i354 

858o6 

52844 
52869 

84897 
84882 

54317 

83962 

55775 

83ooi 
82985 

57215 

82015 

6 
5 

2 
1 

5i379 

85792 

54342 

83946 

55799 

57238 

81999 

23 

56 

5i4o4 

8b777 

52S93 

84866 

54366 

83900 

55823 

82969 

57262 

81982 

4 

1 

24 

57 

51429 

85762 

52918 

8485 1 

54391 

8391 5 

55847 

82953 

57286 

81965 

3 

I 

24 

58 

5i454 

85747 

52943 

84836 

544 1 5 

83899 

5587T 

82936 

5^310 

81949 

2 

I 

25 

5q 

5i479 

85732 

52967 

84820 

54440 

83SS3 

55895 

82920 

57334 

81932 

I 

0 

25 

6o 

5i5o4 

85717 

52992 

848o5 

54464 

83867  55919 

82904 

57358 

81915 

0 

IT 

0 

N.  co.s.|N.  sine. 

N.  COS. 

N.  sine. 

N.  COS. 

N.  sine. 

N.  COS. 

N.  sine. 

N.  COS. 

N.  sine. 

59° 

58° 

57° 

50° 

55° 

TABLE  XXIV.                [i'age 

167 

Of  Natural  Sines. 

Prop. 

23 

o 

M 

0 

35° 

36° 

37° 

38° 

39° 

60 

Prop. 
Dartj 

18 

:8 

N.siiie. 

N.  COS. 

N.  sine. 

i\.  COS. 

N.  sine. 

N.  COS. 

N.  sine. 

N.  COS. 

N.  sine.|N.  cos. 

57358 

81915 

58779 

80902 

60182 

79864 

61666 

78801 

62932 

777 1 5 

c 

r 

57381 

81899 

588o2 

8088  5 

60206 

79846 

616S9 

7S783 

62955 

77696 

59 

18 

I 

2 

574o5 

81882 

58826 

80867 

60228 

79829 

61612 

78765 

62977 

77678 

58 

17 

r 

3 

5742Q 

8 1 865 

58849 

8o85o 

60261 

798 1 1 

6i635 

78747 

63ooo 

77660 

57 

17 

2 

4 

57453 

81848 

58873 

8o833 

60274 

79793 

61668 

78729 

63022 

77641 

56 

17 

2 

5 

57477 

8i832 

58896 

80816 

60298 

79776 

61681 

7871 1 

63o45 

77623 

56 

17 

2 

"3" 

6 

7 

57501 

8i8i5 

58920 

80799 

6o32i 

79768 

61704 

78694 

63o68 

77606 

54 
53 

16 
"16" 

57524 

8.7Q8 

58943 

80782 

60344 

79741 

61726 

78676 

63090 

77686 

3 

8 

57548 

81782 

58967 

80765 

6o367 

79723 

61749 

78668 

63ii3 

77668 

52 

16 

3 

q 

57572 

81765 

58990 

80748 

60390 

79706 

61772 

78640 

63i35 

77660 

bi 

i5 

4 

10 

57396 

81743 

59014 

80730 

6o4i4 

79688 

61796 

78622 

63i58 

7753. 

bo 

i5 

4 

n 

57619 

81731 

59037 

80713 

60437 

79671 

61818 

78604 

63 1 80 

77bi3 

49 

i5 

b 
5 

12 
7T 

57643 

81714 

59061 

80696 

60460 

79663 

6i84i 

78686 

632o3 

77494 

48 
47 

i4 
i4 

57667 

81698 

59084 

80679 

6o483 

79635 

61864 

78568 

63226 

77476 

5 

14 

57691 

816S1 

59108 

80662 

60606 

79618 

61887 

78660 

63248 

77458 

46 

i4 

6 

rj 

57715 

81664 

59131 

80644 

60629 

79600 

61909 

78532 

63271 

77439 

4b 

i4 

6 

16 

57738 

81647 

59154 

80627 

6o553 

79583 

61932 

78614 

63293 

77421 

44 

i3 

7 

17 

57762 

8i63i 

59.78 

80610 

6o5v6 

79665 

61955 

78496 

633 16 

77402 

43 

i3 

7 
7 

18 
19 

57786 

81614 

59201 

80693 

60699 

79547 

61978 
62001 

78478 
78460 

63338 
63361 

77384 
77366 

42 
4i 

i3 
12 

57810 

81597 

59225 

80676 

60622 

79630 

« 

20 

57S33 

8i58o 

59248 

8o568 

60645 

79612 

62024 

78442 

63383 

77347 

4o 

12 

8 

21 

57857 

8 1 563 

59272 

8o54i 

60668 

79494 

62046 

78424 

634o6 

77329 

39 

12 

8 

22 

57881 

8 1 546 

59295 

80624 

60691 

79477 

62069 

78405 

63428 

77310 

38 

II 

9 

23 

57904 

8i53o 

59318 

80607 

60714 

79469 

62092 

7S387 

6345 1 

77292 

37 

1 1 

_9 

10 

24 

25 

57928 

8i5i3 

59342 

80489 

60738 

79441 

62116 

78369 

63473 

77273 

36 
35 

II 
II 

57952 

81496 

59365 

S0472 

60761 

79424 

62 1 38 

78351 

63496 

77255 

10 

26 

57976 

81479 

59389 

80455 

60784 

79406 

62160 

78333 

635i8 

77236 

34 

10 

ID 

27 

57999 

81462 

59412 

8o438 

60807 

79388 

62183 

783 1 5 

6354o 

77218 

.U 

10 

II 

28 

58o23 

81445 

59436 

80420 

608  3o 

79371 

62206 

78297 

63563 

77199 

32 

10 

II 

29 

58o47 

81428 

59459 

8o4o3 

6o853 

79353 

62229 

78279 

63585 

77181 

3i 

9 

12 
12 

3o 
3i 

58070 

8i4i2 

59482 

8o386 

60876 

79335 

62261 

78261 

636o8 

77162 

3o 

9 

58094 

81395 

59606 

8o368 

60899 

79318 

62274 

78243 

6363o 

77144 

12 

32 

58ii8 

81378 

bQb29 

8o35i 

60922 

79300 

62297 

78225 

63653 

77126 

28 

8 

l3 

33 

58i4i 

8i36i 

59552 

80334 

60945 

79282 

62320 

78206 

63675 

77107 

27 

8 

l3 

■M 

58i65 

8 1 344 

59576 

8o3 1 6 

60968 

79264 

62342 

78188 

63698 

77088 

26 

8 

i3 

35 

58189 

81327 

59599 

80299 

60991 

79247 

62365 

78170 

63720 

77070 

2b 

8 

i4 

i4 

36 
37 

582 1 2 

8i3io 

59622 

80282 

6ioi5 
6io38 

79229 

62388 

78162 

63742 

77o5i 

24 
23 

7 
7 

582  36 

81293 

59646 

So  2  64 

792 II 

6241 1 

78134 

63766 

77033 

lb 

38 

58260 

81276 

59669 

8o2.'[7 

61061 

79193 

62433 

78116 

63787 

77014 

22 

7 

lb 

39 

58283 

81259 

59693 

8o23o 

61084 

79176 

62466 

78098 

638 10 

76996 

21 

6 

lb 

4o 

583.17 

81242 

59716 

80212 

61 107 

79168 

62479 

78079 

63832 

76977 

20 

6 

i6 

4i 

58330 

81225 

59739 

80195 

6ii3o 

79140 

62602 

78061 

63854 

76969 

19 

6 

lb 
76" 

42 
T3" 

58354 

81208 

59763 

80178 

6ii63 

79122 

62624 

78043 

53877 

76940 
76921 

18 

17 

5 
5 

58378 

81191 

59786 

80160 

61176 

79106 

62647 

78026 

63899 

17 

44 

584oi 

81174 

59809 

80143 

61 199 

79087 

62670 

78007 

63922 

76903 

lb 

5 

17 

4  b 

58425 

81157 

59832 

80126 

61222 

79069 

62692 

77988 

63944 

76884 

lb 

■  b 

i8 

40 

58449 

8ii4o 

59856 

80108 

61245 

79061 

62616 

77970 

63966 

76866 

i4 

4 

i8 

47 

58472 

81123 

59879 

80091 

61268 

79033 

62638 

77952 

63989 

76847 

i3 

4 

i8 
19 

49 

58496 
585 19 

81106 

59902 

80073 

61291 

79016 

62660 

77934 

64o  1 1 

76828 

12 
1 1 

4 
3 

81089 

59926 

800  56 

6i3i4 

78998 

62683 

77916 

64o33 

76810 

19 

5o 

58543 

81072 

59949 

8oo38 

61337 

78980 

62706 

77S97 

64o56 

76791 

10 

3 

20 

bi 

58567 

8io55 

59972 

80021 

6i36o 

78962 

62728 

77879 

64078 

76772 

9 

3 

70 

52 

58590 

8io3S 

59995 

8ooo3 

6i383 

78944 

62761 

77861 

64100 

76754 

8 

2 

uo 

53 

586 1 4 

81021 

60019 

79986 

6i4o6 

78926 

62774 

77S43 

64123 

76735 

7 

2 

21 
21 

54 
"55" 

58637 
5866 1 

81004 
80987 

60042 

79968 

61429 

78908 

62796 

77824 

64 1 45 

76717 

6 
5 

2 
2 

60065 

79961 

6i45i 

78S91 

62819 

77806 

64167 

76698 

21 

bb 

58684 

80970 

60089 

79934 

61474 

788-73 

62842 

77788 

64190 

76679 

4 

1 

22 

b7 

58708 

80953 

60112 

79916 

61497 

78855 

()2864 

77769 

64212 

76661 

3 

I 

22 

58 

58731 

80936 

601 35 

79899 

61620 

78837 

62887 

77761 

64234 

76642 

2 

I 

2i 

b9 

58755 

80919 

601 58 

79S81 

61643 

78819 

62909 

77733 

64256 

76623 

I 

0 

23 

bo 

58779 

80902 

60182 

79864 

61666 

78801 

62932 

77716 

64279 

76604 

0 

IT 

0 

N.  COS. 

N.  sine. 

N.  COS. 

N.  sine. 

N.  COS. 

N.  sine. 

X.  COS.  JV.  sine. 

N.  COS. 

N.  sine. 

54= 

5; 

B° 

52° 

51° 

50° 

Pagei&s]                TABLE  XXIV. 

Of  Natural  Sines. 

Prop, 
pana 

22 

o 

M 
0 

40° 

41° 

42° 

43°    1 

44° 

60 

Prop. 
part5 

19 

19 

N.  sine. 

N.  COS. 

N.  sine. 

V.  COS. 

N.  sine. 

N.  COS. 

N.  sine. 

N.  cos. 

N.  sine. 

N.  cos. 

64279 

76604 

656o6 

75471 

66913 

743 14 

6O200 

73i35 

69466 

71934 

o 

I 

643oi 

76586 

65628 

75452 

66935 

74295 

68221 

73ii6 

69487 

71914 

59 

19 

I 

2 

64323 

76567 

6565o 

75433 

66956 

74276 

68242 

73096 

69508 

71894 

58 

18 

I 

3 

64346  76548 

65672 

75414 

66978 

74256 

68264 

73076 

69529 

71873 

57 

18 

I 

4 

64368  7653o 

65694 

7b39b 

66999 

74237 

6S285 

73o56 

69549 

71853 

55 

18 

2 

5 

64390 

765 1 1 

65716 

75375 

67021 

74217 

683o6 

73o36 

69570 

71833 

55 

17 

2 
~3 

b 

7 

644 1 2 

76492 

65738 

75356 

67043 

74198 

68327 

73016 

69591 

7i8i3 

54 
53 

17 
17 

64435 

76473 

65759 

75337 

67064 

74178 

68349 

72996 

69612 

71792 

3 

8 

64457 

76455 

65781 

753i8 

67086 

74159 

68370 

72976 

69533 

71772 

52 

lb 

6 

9 

64479 

76436 

658o3 

75299 

67107 

74139 

68391 

72957 

69654 

71752 

Dl 

lb 

4 

10 

6/:)5oi 

76417 

65825 

75280 

67129 

74 1 20 

68412 

72937 

69675 

71732 

bo 

lb 

4 

II 

64524 

76398 

65847 

75261 

671 5i 

74100 

68434 

72917 

69696 

71711 

49 

lb 

4 

5 

1 2 

64546 
64568 

7.6380 
76361 

65869 

75241 

67172 

74080 

68455 

72897 
72877 

69717 

71691 

48 

47 

lb 
i5 

65891 

75222 

67194 

74061 

68476 

6973,7 

71671 

1j 

i4 

64590 

76342 

65913 

75203 

67215 

74o4i 

68497 

72857 

69758 

7i65o 

46 

lb 

b 

lb 

646 1 2 

76323 

65935 

75 1 84 

67237 

74022 

685i8 

72837 

69779 

7i63o 

45 

14 

b 

lb 

64635 

763o4 

65956 

75i65 

67258 

74002 

68539 

7.2817 

69800 

71610 

44 

i4 

b 

17 

64657 

762S6 

65978 

75i46 

67280 

73^83 

68  56 1 

72797 

69821 

71590 

43 

i4 

7 
7 

18 

64679 

76267 

66000 

75126 

67301 

73953 

68582 

72777 

69842 

71569 

42 
'4V 

i3 
i3 

64701 

76248 

66022 

75107 

67323 

73944 

686o3 

72757 

69862 

71549 

7 

2(J 

64723 

76229 

66044 

75088 

67344 

73924 

68624 

72737 

69883 

71529 

4o- 

i3 

a 

21 

64746 

76210 

66066 

75069 

67366 

73904 

68545 

72717 

69904 

7i5o8 

39 

12 

8 

22 

64768 

76192 

66088 

75o5o 

67387 

73885 

68666 

72697 

69925 

71488 

38 

12 

8 

23 

64790 

76173 

66109 

75o3o 

67409 

73865 

68688 

72677 

69946 

71468 

37 

12 

9 
9 

24 
25 

648x2 

76154 

G6i3i 

7301 1 

67430 

73846 

68709 

72657 

69966 

71447 

3b 
I5" 

11 
II 

64834 

76135 

66 1 53 

74992 

67452 

73826 

68730 

72637 

69987 

71427 

lO 

2b 

64856 

761 16 

66175 

74973 

67473 

73806 

68751 

72617 

70008 

71407 

34 

II 

lO 

27 

64878 

7O0Q7 

66197 

74953 

G7495 

73787 

68772 

72597 

70029 

71386 

33 

10 

lO 

28 

64901 

76078 

66218 

74934 

67516 

73767 

68793 

72577 

70049 

7 1 366 

32 

10 

II 

29 

64923 

76059 

66240 

74915 

67538 

73747 

68814 

72557 

70070 

71345 

3i 

10 

f  I 
1 1 

Jo 

yr 

64945 

76041 

66262 

74896 

67559 
675S0 

73728 

68835 

72537 

70091 

71825 

3o 
29 

10 
9 

64967 

76022 

66284 

74876 

73708 

68857 

72517 

70112 

7i3o5 

12 

32 

649S9 

76003 

663o6 

74857 

67602 

73688 

68878 

72497 

70132 

71284 

28 

9 

12 

6:i 

65oii 

75984 

66327 

74838 

67623 

73669 

68899 

72477 

7013J 

71264 

27 

9 

12 

M 

65o33 

75965 

66349 

74818 

67645 

73649 

68920 

72457 

70174 

71243 

2b 

8 

iJ 

3d 

65o55 

75946 

66371 

74799 

67666 

73629 

68941 

72437 

70195 

71223 

2D 

8 

i4 

65o77 

75927 

66393 
664 1 4 

74780 
74760 

67688 

736'io 

68962 

72417 

70215 

71203 

24 
23 

8 
7 

65 1 00 

75908 

67709 

73590 

68983 

72397 

70235 

71182 

'4 

38 

65l22 

75889 

66436 

74741 

67730 

73570 

69004 

72377 

70257 

71162 

22 

7 

i-l 

39 

65 1 44 

75870 

66458 

74722 

67752 

73551 

69025 

72357 

70277 

71141 

21 

7 

i5 

4<' 

65 1 66 

7585i 

66480 

74703 

67773 

7353i 

69046 

72337 

70298 

71121 

20 

b 

1 5 

4>. 

65iS8 

75832 

665oi 

74683 

67795 

73511 

69067 

72317 

7o3i9 

71100 

19 

b 

lb 

42 
43 

65210 

758i3 

66523 

74664 

67816 
67837 

73491 
73472 

69088 

72297 

70339 

71080 

18 

17 

b 
5 

65232 

75794 

66545 

74644 

69109 

72277 

7o36o 

71059 

lb 

44 

65254 

75775 

66566 

74625 

67859 

73452 

69130 

72257 

7o38i 

71039 

lb 

b 

17 

43 

65276 

75756 

66588 

74606 

67880 

73432 

691 5 1 

72235 

70401 

71019 

lb 

b 

17 

4b 

65298 

75738 

66610 

74586 

67901 

73413 

69172 

72216 

70422 

70998 

i4 

4 

17 

47 

65320 

75719 

66632 

74567 

67923 

73393 

69193 

72196 

70443 

70978 

i3 

4 

i8 
i8 

48 
49 

65342 

75700 
75680 

66653 
66675 

74548 
74528 

67944 

73373 

69214 

72176 

70453 

70957 

12 
11 

4 
3 

65364 

67965 

73353 

69235 

72 1 56 

70484 

70907 

i8 

bo 

65386 

75661 

66697 

74509 

67987 

73333 

69255 

72 1 36 

7o5o5 

70916 

10 

3 

19 

bi 

654o8 

75642 

66718 

74489 

68008 

733i4 

69277 

■72116 

7o525 

70896 

9 

3 

'9 

32 

65430 

75623 

66740 

74470 

68029 

73294 

69298 

72095 

70546 

70875 

8 

3 

19 

bi 

65452 

756o4 

66762 

744  5 1 

68o5i 

73274 

69319 

72075 

70557 

70855 

7 

2 

20 
20 

!>4 
55 

65474 
65496 

75585 
75566 

667S3 

7443 1 

68072 

73254 

69340 

72055 

70587 

70834 

b 
"5 

2 
2 

668o5 

74412 

68093 

73234 

69361 

72035 

70608 

70813 

21 

bb 

655i8 

7iM7 

668  2  7 

74392 

6Sii5 

732 1 5 

693S2 

72015 

70628 

70793 

4 

I 

21 

b7 

6554o 

75528 

66848 

74373 

68 1 36 

73195 

69403 

71995 

70649 

70772 

3 

I 

21 

b8 

65562 

75509 

66870 

74353 

68157 

73175 

69424 

71974 

70670 

70752 

2 

I 

22 

b9 

65584 

75490 

G6891 

74334 

68179 

73i55- 

69445 

71954 

70690 

70731 

I 

0 

22 

b(j 

656o6 

75471 

66913 

743 1 4 

68200 

73i35 

69466  1  7 1 934 

7071 1 

707 II 

0 

0 

N.  cos.|N.  sine. 

i\.  ros. 

N.  sine. 

N.  COS. 

N.  sine. 

N.  ros.(N.  sine. 

.\.  cos. 

N.  sine. 

4 

9° 

48° 

47° 

4G° 

45° 

[Page  ]C9 

TABLE  XXV. 

Of  Logarithmic  Sines,  Tangents,  and  Secants  to  every  Point  and  Quarter 

Point  of  the  Compass. 

Points. 

Sine. 

Co-sine. 

Tangent. 

Co-tang. 

Secant. 

Co-sccant. 

o 

Lif.  neg. 

10.00000 

Lif.  neg. 

Infinite. 

10.00000 

Infinite. 

8 

oi 

8.69080 

9-99948 

8.69132 

ii.3o868 

IO.O0052 

II  .30920 

7^ 

o  h 

8.99130        9-99790 

8.99340 

11.00660 

10.00210 

II  .00870 

7  h 

o.i 

9.16652    :    9.99527 

9.17125 

10.82875 

10.00473 

10.83348 

7  i 

I 

9.29024 

9.99157 

9.29866 

10.70134 

10.00843 

ic. 70976 

7 

I  i 

9.38557 

9.98679 

9.39879 

10.601 2 1 

IO.Ol32I 

10.61443 

Gil 

I  4 

9.46282 

9.98088 

9.48194 

io.5i8o6 

IO.OI9I2 

10.53718  '       6  4       1 

I  % 

9.52749 

9-97384 

9.55365 

10. 44635 

10.02616 

10.47251 

6  i 

2 

9.5S284 

9.96562 

9.61722 

10.38278 

I0.03438 

10.41716 

6 

2    i 

9.63099 

9.95616 

9.67483 

io.325i7 

10.04384 

10.36901 

5  -I 

2    h 

9.67339 

9.94543 

9.72796 

10.27204 

10.05457 

10.82661 

5  h 

2    5 

9.71105 

9.93335 

9.77770 

I0.2223o 

io.o6665 

10.28895 

U 

3 

9-74474 

9.91985 

9.82489 

IO.I75II 

io.o8oi5 

10.25526 

5 

3  i 

9.77503 

9.90483 

9.87020 

10.12980 

10.09517 

10.22497 

4i 

3  h 

9.80236 

9. 888 1 9 

9.91417 

10. 08583 

10. 11 181 

10.19764 

4  h 

3i 

9.8270S 

9.86979 

9.95729 

10.04271 

10. l302I 

10.17292 

4  i 

4 

9.84949 

9-84949 

10.00000 

10.00000 

io.i5o5i 

io.i5o5i 

4 

Co-sine.             Sine. 

Co-tang. 

Tangent. 

Co-secant. 

Secant. 

Points. 

TABLE  XXVL 

Logarithms  of  Numbers. 

No.  1 100.                                                                   Log.  0.00000 2.00000. 

No. 

Log;. 

No. 

Log. 

No. 

Log. 

No.  j       Log. 

No. 

Log. 

1 

O . 00000 

21 

1.32222 

4i 

1. 61278 

61 

1.78533 

81 

I . 90849 

2 

o.3<)io3 

22 

I .34242 

42 

1.62325 

62 

1.79239 

82 

1.91381 

3 

0.47712 

23 

I. 36173 

43 

1.63347 

63 

1.79934 

83 

I .91908 

4 

0.6020G 

24 

i.38o2i 

44 

I. 64345 

64 

1.80618 

84 

I .92428 

5 

0.69897 

P.5 

1.39794 

45 

I .65321 

6 

5 

1.81291 

85 

1.92942 

6 

0.77815 

26 

I. 41497 

46 

I .66276 

6 

^ 

I. 81954 

86 

I . 93450 

7 

o.845io 

27 

i.43i36 

4i 

I .67210 

67 

1.82607 

87 

I .93952 

8 

0.90309 

28 

I. 44716 

48 

I. 68124 

68 

I.8325I 

88 

1.94448 

9 

0.95424 

29 

I .46240 

49 

I .69020 

69 

1.83885 

89 

1.94939 

lO 

I .00000 

3o 

I. 47712 

5o 

I .69897 

7 

0 

i.845io 

90 

1.95424 

II 

1.04139 

3i 

I .49136 

•  5i 

1.70757 

7 

I 

I.85I26 

91 

I .95904 

12 

I .07918 

32 

i.5o5i5 

52 

I .71600 

72 

I. 85733 

92 

I .96370 

1 3 

I . II 394 

33 

i.5fc85i 

53 

I .72428 

73 

I. 86332 

93 

1.96848 

•  i4 

I . i4Gi3 

M 

i.53i48 

54 

I .73239 

74 

I .86923 

94 

1 . 973 1 3 

i5 

1 . 1 7609 

35 

1.54407 

55 

I . 74o36 

.75 

i.875o( 

5 

95 

1.97772 

i6 

I .2o4l2 

36 

I.55630 

56 

I. 74819 

76 

1.8808 

96 

1.98227 

17 

I .2  3o45 

37 

1.56820 

57 

1.75587 

77 

I .88649 

97 

1.98677 

i8 

I .25527 

38 

1.57978 

58 

1.76343 

78 

1 .89209 

98 

1.99/23 

-9 

1.27875 

39 

1 .59106 

59 

1.77085 

79 

1.89763 

99 

I . 99564 

20 

I .3oio3 

40 

I .60206 

60 

1. 77815 

80 

I .90309 

100 

2.00000. 

2 

2 

Page  170] 


TABLE  XXVI. 

Logarithms  of  Numbers. 


No.  100- 


-1600. 


Loff.  00000- 


-20412. 


No. 


I02 

io3 

io4_ 
io5 
io6 
107 
io8 


:i3 


[2.3 

[_24_ 

125 
[26 

[27 
[28 

t3o 
[3i 

l32 

i33 
1 34 
l3:3 
[36 
i37 
1 38 
i39_ 

i4o 
i4i 

I  42 

13 
i44 
1 45' 

'(7 


[5i 

[55 

[53 
[_54 
[55" 
1 56 
[57 
58 

No. 


0 


GOOOO 

oo432 
00860 

01284 

01703 


021 19 
0253i 
02q38 
03342 
03743 


o4i39 
04532 
04922 
o53u8 
05690 


06070 
06446 
068 1 9 
071 88 
07555 


079(8 
S279 
8636 

08991 
9342 


9691 
oo37 
o38o 
079.1 
[059 


1394 

1727 
2o57 
2385 
2710 


3o33 
3354 
3672 
3988 
43oi 


46i3 
4922 
5229 
5534 
5836 

(■ii37 
6435 
6732 
7026 
7319 


7609 
7898 
8(84 
8469 
8752 

9rr3"3 

93  I  2 

9590 
9866 

2()l4o 

0 


1 


00043 
00475 
00903 
01326 
01745 


02160 
02572 

02979 

o3j83 
03782 


04179 
04571 
04961 
o5346 
05729 


06108 
o6483 
o6856 
07225 
07591 


07954 
oS3i4 
0S672 
09026 
09377 


09726 
0072 
04 1 5 
0755 
1093 


1428 
1760 
20Q0 
2418 
2743 


3(j66 
3386 
3704 
4019 
4333 


4644 
4953 
5259 
5564 
5866 


6167 
6465 
6761 
70  56 
7348 


7638 
7926 
82(3 
8498 
8780 

9061 
9340 
9618 
9893 
20167 


00087 
oo5i8 
00945 
01 368 
01787 


02202 
cfc6i2 
o3oi9 
03423 

03822 


04218 
o46io 
04999 
o53S5 
05767 


06145 
o652i 
06893 
07262 
07628 


07990 
o835o 
08707 
0906 1 
09412 

09760 
0106 
0449 
0789 
1 1 26 


i46[ 
1793 

2123 
2450 
2775 


3098 

34i8 
3735 
4(i5i 
4364 


4675 
49S3 
5290 
5594 
5897 


6197 
6495 
6791 
7085 

7377 


7667 
7955 
8241 
8526 
S80S 


9089 
9368 
9645 
9921 
20194 


ooi3o 
oo56i 
00988 
oi4io 
01828 


02243 
02653 
o3o6o 
o3463 
o3862 


04258 
o465o 
o5o38 
o5423 
o58o5 


o6i83 
06558 
06930 
07298 
07664 


08027 
08386 
08743 
09096 
09447 

09795 
oi4o 
o483 
0823 
1 160 


1494 
1826 
2 1 56 
2483 
2808 


3i3o 
345o 

3767 
4082 
4395 


4706 
5oi4 
5320 
5625 
5927 


6227 
6524 
6820 
7114 
7406 


7696 


8270 
8554 
8S37 


939^ 

9673 

9948 

20222 


00173 
00604 
oio3o 
01452 
01870 


02284 
02694 
o3ioo 
o35o3 
03902 


04297 
04689 
o5o77 
o546r 
o5843 


06221 
06595 


07335 
07700 


oSo63 
08422 
08778 
09132 
09482 


09830 
0175 
o5i7 
0857 
1 193 


1528 
i860 
2189 
25i6 
2840 


8162 
3481 
3799 
4ii4 
4426 


4737 
5o45 
535i 
5655 
5957 


6256 
6554 
685o 
7143 
7435 


7725 
801 3 
8298 
8583 
8865 


00217 
00647 
01072 
01494 
01Q12 


02325 

02735 
o3i4i 
03543 
03941 


04336 
04727 
o5ii5 
o55oo 
o588i 


06258 
06633 
07004 
07372 
07737 


08099 
08458 
08814 
09167 
09517 


9S64 
0209 
o55i 


1227 


ij5i 
1893 

2252 

2548 
2872 


3194 
35x3 
383o 
4i45 
4457 


7754 
8o4i 
8327 
861 1 
8S93 
9173 
945 1 
9728 

20003 

20776 


6 


00260 
00689 
oiiiS 
oi536 
01953 


02366 
02776 
o3i8i 
03583 
03981 


04376 
04766 
o5i54 
05538 
05918 


06296 
06670 
07041 
07408 
07773 


081 35 
08493 
08849 
09202 
09552 


10243 
10585 
[0924 
[1 261 


i63i6 
1 66  [3 
16909 
17202 

7492 
[7782 
18070 
[8355 
18639 

3921 


oo3o3 
00732 
01157 
01578 
01995 


02407 
02816 

03222 

o3623 

o4o21 


044 1 5 
o48o5 
05192 
05576 
05956 


06333 
06707 
07078 
07445 
07809 


08171 

08529 

08884 

19237 

'9587 


09934 
0278 
0619 
0958 

;i294 


[1628 
[1959 
[2287 
[26 1 3 
[2937 

i3258 
13577 
1 3S93 
14208 
14520 


14829 
[5i37 
[5442 
[5746 
16047 


1 6346 
16643 
16938 
[7231 
[7522 

1 78 II 
18099 
1 8384 
18667 
18949 


9229 

9507 

9783 

2oo58 

2o33o 


oo346 
00775 
01199 
01620 
o2o36 


02449 
02857 
03262 
03663 
o4o6o 


04454 
04844 
o523i 
o56i4 
05994 


06371 
06744 
071 15 
07482 
07846 


08207 
08565 
08920 
09272 
09621 


09968 

03l2 

o653 
10992 

;i327 


1661 
[1992 

[2320 

[2646 
2969 


16376 
16673 
16967 
17260 

17551 


17840 
I8I27 

i84i2 
18696 


19257 
19535 


9 


00889 
00817 
01242 
01662 
02078 


02490 
02898 
o33o2 
03703 
o4ioo 


04493 
04883 
05269 
o5652 
o6o32 


06408 
06781 
071 5 1 
07518 
07882 


08243 
08600 
08955 
09307 
09656 


[1025 

i36i 


1694 

[2024 
[2352 

[2678 

i3oo[ 


[3322 

1 364o 
[8956 
142-0 
[4582 

i489[ 
[5198 
[55o3 
j8u6 
16107 

16406 
16702 
16997 
[7289 
(7580 


201 12 

2o385 


43 

I 

4 

2 

9 

0 

i3 

4 

17 

5 

22 

6 

26 

7 

3o 

8 

M 

9 

39 

41 

40 

I 

4 

4 

2 

8 

8 

3 

12 

12 

4 

16 

16 

5 

21 

20 

6 

25 

24 

7 

29 

28 

8 

33 

33 

9 

J7 

36 

39 

38 

1 

4 

A 

2 

8 

8 

3 

12 

1 1 

4 

16 

i5 

5 

20 

If; 

6 

23 

23 

7 

27 

27 

8 

3i 

3o 

9 

35 

34 

37 

3i; 

I 

4 

4 

2 

7 

7 

3 

II 

II 

4 

i5 

i4 

5 

19 

18 

() 

22 

22 

7 

26 

a5 

8 

3o 

20 

9 

33 

32 

35 

34 

I 

4 

3 

2 

7 

7 

3 

II 

10 

4 

i4 

i4 

5 

18 

17 

6 

21 

20 

7 

25 

24 

8 

28 

27 

9 

32 

3i 

33 

1 

3 

2 

7 

i 

10 

4 

i3 

5 

17 

6 

20 

7 

2J 

8 

26 

9. 

3o 

32 
~3 
6 
10 
i3 
16 

19 
22 
26 

19 


TABLE  XXVI. 

Logarithms  of  Numbers. 


[Page  171 


No.  1600- 


-2200. 


hocr.  20412- 


-34242. 


:6o 

r6i 

162 

r63 

i64_ 

i65 

166 

167 

168 

K59_ 

[70 

171 

[72 
[73 

[75 

.76 

177 

178 

'79 
180 
181 
182 
[83 
'4 


i82_ 

190 
191 
[92 
93 

_9l 

.95 

196 

■97 
198 

199 
200 
201 
202 

2o3 
204 
205 

206 
207 

208 

209 
210 

211 
212 
2l3 
2l4 
2l5 

216 

217 

218 

219 


204l2 

2o6S3 
20952 
2I2I9 

21484 


21748 
2201 1 

22272 

2253l 

227S9 


23o45 

233oo 

23553 
23So5 
24u55 


243o4 
2455i 
24797 

25o42 

25285 


3(jio3 
3o3:>o 
3o535 
3o75o 
30963 


20439 
20710 
20978 
21245 

2l5l  I 


21775 
22037 
22298 
22557 
22814 
23070 
23325 
23578 
2383o 
24080 


24329 
24576 
24822 
25o66 
253io 


2555i 
25792 
2603 1 
26269 
265o5 


26741 
26975 
27207 
27439 
27669 


27898 
28126 
28353 
28578 
28803 


29026 
29248 
29469 
29G88 
29907 

3oi25 
3o34i 
3o557 
30771 
30984 


20466 
20737 

2I005 

21272 
2x537 


2 1 80 1 

2  2o63 

22824 
22583 
22840 


23096 
2335o 
236o3 
23855 
24 1  o5 


24353 
2460 1 
24846 
25091 
25334 


25575 
258i6 
26055 
26293 
26529 


26764 
26998 
27231 
27462 
27692 


27921 
28149 
28375 
28601 
28825 


29048 
29270 
29491 
29710 
29929 


3or46 
3o363 
30578 
30792 
3 1  oo() 


3i2i8 
31429 
3 1 639 
3 1 848 
32o56 


32263 
32469 
32675 
32879 
33082 


20493 
20763 

2Io32 
21299 

2 1 564 


21027 
22089 

2235o 

22608 

22866 


28121 
23376 
28629 
23880 
24i3o 


24378 
24625 
24871 
25i  i5 
25358 


256oo 
2584o 
26079 
263i6 
26553 


267S8 
27021 
27254 
27485 
27715 

27944 
28171 
28398 
28623 
28847 


29070 
29292 
29513 
29732 
29951 


3oi68 
3o384 
3o6oo 
3o8i4 
3  [027 


33284 
33486 
33686 
33885 
34084 


31239 
3i45o 
3 1 660 
31869 
32077 
32  2.84 
32490 
32695 
32899 
33io2 


333o4 
335o6 
33706 
33905 
34 1 04 


23i47 
23401 
23654 
23905 
24 1 55 


244o3 
2465o 
24S95 
25139 
25382 


25624 
2  5864 
26102 
26340 
26576 


26S 1 1 
27045 
27277 
27508 
27738 


27967 
28194 
28421 
28646 
28870 


29002 
29314 
29535 
29754 
29973 


3o  1 90 
3o4o6 
3062 1 
3o835 
3io48 


3 1 260 

3i47i 
3i68i 
81890 
32098 


323o5 
325ro 
32715 
32919 
33 1 22 


33325 
33526 
33726 
33925 
3412,4 


2o548 
20817 
2io85 

2l352 

21617 


24428 
24674 
24920 
25i64 
25406 


25648 
2  5888 
26126 
26364 
26600 


26834 
27068 
27800 
27531 
27761 


27989 
28217 
28443 
28668 
28892 


291 1 5 
29886 
29557 
29776 
29994 


32825 
3253i 
82786 
32940 
33i43 


33345 
33546 
33746 
33945 
34143 


20576 
20844 
21112 
21378 
21643 


21906 
22167 

22427 
22686 
22943 


28198 
23452 
28704 
28955 
24204 


24452 
24699 
24944 
25i88 
25481 


26672 
26912 
26160 
26887 
26623 


26858 
27091 
27828 
27664 
27784 
28012 
28240 
28466 
28691 
28914 


29187 
29358 
29679 
29798 
3ooi6 


3o233 
80449 
3o664 
80878 
3 1 09 1 


3i3o2 
3i6i3 
81723 
81931 
82189 


82846 
82662 
32766 
82960 
83i63 

333$5 
33666 
33766 
33965 
34i63 

6 


20602 
20871 

21130 

2i4o5 
21669 


21982 
22194 

22453 
22712 


28228 

23477 
28729 
28980 
24229 


24477 
24724 
24969 
26212 
26465 


28086 
28262 
28488 
28718 
28937 


29169 
29380 
29601 
29820 
3oo38 


80255 
80471 
3o6S6 
80899 

3X112 


3x323 
3x534 
3x744 
3x962 
82160 

32366 
82672 
82777 
32980 
33x83 


33386 
33586 
33786 
33986 
34 1 83 


20629 


21 166 
2i43x 
21696 


21968 
22220 
22479 
22737 


28249 
28602 
28764 
24006 
24264 


24502 
24748 
24998 
26237 
26479 


26720 
26969 
26198 
26435 
26670 


26906 
27188 
27370 
27600 
27880 


28068 
28286 
286x1 
28735 
28969 


29X8X 
29408 
29628 
29842 
80060 


30276 
80492 
80707 
30920 
8ii33 


3x345 
3i555 
3x766 
3x978 
32i8r 
32  387 
82693 
82797 
3  3  00 1 
332o3 


834o5 
336n6 
338o6 
34006 
34208 


7  I   8 


9 


2o656 
20926 
2X  X92 
2x468 
2x722 


21986 
22246 
22606 
22768 
28019 


28274 
23528 
28779 
24080 
24279 


24527 
24773 
260x8 
2526X 
26603 


26744 
26988 
262  2  X 
26458 
26694 


26928 
27161 
27898 
27623 
27862 


28081 
28307 
28533 
28-58 
289SX 


29208 
29426 
29645 
29868 
3()o8 1 


80298 
3o6i4 
30728 
30942 
3x164 


3 1 366 
8x676 
8x786 
81994 

3220X 

32408 
326x3 
32818 
33o2i 
33224 


33426 
33626 
33826 
34026 
34223 

9 


31 

30 

I 

3 

3 

2 

6 

6 

3 

9 

9 

4 

12 

12 

5 

16 

i5 

6 

19 

18 

7 

22 

21 

8 

26 

24 

9 

28 

27 

29 

28 

I 

3 

3 

2 

6 

G 

3 

9 

8 

4 

12 

II 

5 

i5 

14 

6 

17 

17 

7 

20 

20 

8 

20 

22 

9 

26 

26 

27 

2G 

I 

3 

3 

2 

5 

5 

3 

8 

8 

4 

11 

10 

c 

i4 

i3 

6 

16 

16 

7 

19 

18 

8 

22 

21 

9 

24 

23 

25 

24 

I 

3 

2 

2 

5 

5 

3 

8 

7 

4 

10 

10 

5 

i3 

xa 

6 

i5 

i4 

7 

18 

17 

8 

20 

19 

9 

23 

22 

2:3  22 


12 


i4  i3 
71  x6i i5 
8    i8'i8 


21 

I 

2 

2 

4 

3 

6 

4 

8 

5 

X  X 

6 

i3 

7 

16 

8 

17 

9 

19 

Page  172]                 TABLE  XXVI. 

Logarithms  of  Numbers. 

No.  2200 2800.                       Log.  34242 44710. 

No. 

0  1     1 

2 

3 

4 

5 

6 

7 

8 

34400 
34596 
34792 
34986 
35 1 80 

9 

34420 
34616 
34811 
35oo5 
35199 

220 
221 
222 
223 
224 
225 
226 
227 
228 
229 

23o 

23l 
232 

233 

234 

34242 
34439 
34635 
3483o 
35o25 

34262 
34459 
34655 
34850 
35o44 

34282 

34479 
34674 
34869 
35o64 

34301 
34498 
34694 
34889 
35o83 

34321 
34518 
34718 
34908 
35io2 

35295 
35488 
35679 
35870 
86059 

34341 
34537 
34733 
34928 

35l22 

353i5 
35507 
35698 
35889 
86078 

34361 
34557 
34753 
34947 
j5i4i 

84380 
34577 
34772 
34967 
35i6o 

35353 
35545 
35736 
35927 
36ii6 

I 
2 
3 
4 
5 
6 

7 
8 

9. 

2 
4 
6 
8 

35218 
354 II 
356o3 
35793 
35984 

35238 
35430 
35622 
358i3 
36oo3 

35257 
35449 
35641 
35832 
36o2i 

35276 
35468 
3566o 
3585i 
36o4o 

35334 
35526 

35717 
3.'59o8 
36097 

35372 
35564 
35755 
35946 
36i35 

35393 
35563 
35774 
35965 
36 1 54 

ic 
12 
i4 
16 
18 

36 1 73 
3636i 
36549 
36736 
36922 

36192 
3638o 
36568 
36754 
36940 

36211 
86899 
36586 
86778 
86959 

86229 
364 1 8 
366o5 
86791 
36977 

86248 
36436 
86624 
368 10 
36996 

86267 
36455 
36642 
86829 
87014 

86286 
36474 
3666 1 
36847 
87083 

363o5 
36493 
36680 
36866 
87051 

36324 
365 1 1 
86698 
36884 
87070 

36342 
3653o 
86717 
86903 
3-u88 

19 

I 
2 
3 
4 
5 
6 

7 
8 

9 

2 

4 
6 

8 

235 
236 
237 
238 
239 

37107 
37291 
37475 
37658 
37840 

37125 
37310 
37493 
37676 
37858 

37144 
87828 
37511 
37694 
37876 

38o57 
88288 
384 1 7 
88596 
88775 

37162 
87846 
37530 
877x2 
37894 

87181 
37365 
87548 
37781 
87912 

88098 
38274 
38453 
38632 
388io 

87199 
87883 
37566 
87749 
37981 

38ii2 
38292 
38471 
38650 
38828 

87218 
37401 
37585 
87767 
37949 

87236 
87420 
87608 
37785 
87967 

87254 
87488 
37621 
87808 
87985 
38 1 66 
38346 
38525 
88708 
3888 1 

89058 
89235 
89410 
89585 
89759 

87278 
37457 
87689 
87822 
38(H,3 

38 184 
38364 
38543 
38721 
38899 

89076 
89252 
89428 
89602 
39777 
39950 
40128 
40295 
40466 
4o637 

10 
1 1 

1 3 
i5 

17 

240 
241 
242 
243 

244 

38o2i 

38202 

38382 
38  56 1 
38-39 

38917 
39094 
39270 
39445 
39620 

88089 
38220 
38399 
38578 
38757 

88075 
38256 
38435 
386i4 
88792 

38 1 3o 
383 10 
88489 
38668 
38846 

38i48 
38328 
385o7 
38686 
38863 

18 

I 
2 
3 
4 
5 
6 

7 
8 

9_ 

1 

2 

4 
5 

245 
246 

247 
248 
249 
25o 

25l 
252 

253 

254 

38934 
89111 
39287 
39463 
89687 

88952 
39129 
39805 
89480 
39655 

88970 
89146 
39322 
89498 
39672 

38987 
89164 
89840 
39515 
89690 

89005 
89182 
89358 
89533 
89707 

89028 
39199 
89875 
89550 
89724 

39041 
892 1 7 
39893 

395es 

89742 

7 

9 

II 
i3 
14 
16 

7 

39794 
39967 
4oi4o 
4o3r2 
4o483 

39811 
39985 
4oi57 
40829 
4o5oo 

89829 
40002 
40175' 
4o346 
4o5i8 

40688 
4(>858 
41027 
4i  196 
4 1 363 

39846 
40019 
40192 
4o364 
4o535 

40705 
40875 
4io44 

4l2I2 

4i38o 

89868 
40087 
40209 
4o38i 
4o552 

89881 
4oo54 
40226 
40898 
40569 

89898 
40071 
40243 
4o4i5 
4o586 

89915 
400S8 
40261 
4o432 
4o6o3 

89933 
40106 
40278 
40449 
40620 

1 
2 
3 
4 
5 
6 
7 
8 

9 

2 
3 

5 

255 
256 
257 
258 
259 

4o654 
40824 
40993 
41162 
4i33o 

40671- 
4o84i 
4ioio 

41179 
4 1 347 

40722 
40892 
4io6i 
41229 
41897 

40739 
40909 
41078 
41246 
4i4i4 

40756 
40926 
41095 
41268 
4i43o 

40773 
40943 
4i  II I 
41280 
4 1 447 

40790 
40960 
41128 
41296 
4 1 464 

40807 
40976 
4ii45 
4i3i3 
4i4Si 

7 

9 

10 
12 

i4 

260 
261 
262 
263 

264 
265 
266 
267 
268 
269 
270 
271 
272 
273 
274 

41497 
4 1 664 
4i83o 
41996 
42160 

4i5i4 
41681 
41847 
42012 
42177 

4 1 53 1 
41697 
4 1 863 
42029 
42193 

4 1 547 
41714 
41880 
42045 
42210 

4 1 564 
41781 
41896 
42062 
42226 

4i58i 

41747 
41918 
42078 
42243 

41597 
41764 
41929 
42095 
42259 

4i6i4 
41780 
41946 
42111 
42275 

4i63i 

41797 
41968 
42127 
42292 

41647 
4i8i4 
41979 
42144 
42808 

i5 

1 

I 
2 
3 
4 
5 
6 

7 
8 

9_ 

1 

G 

2 
3 

5 
6 
8 
10 
II 
1 3 

42325 
42488 
4265 1 
42813 
42975 

42341 
42504 
42667 
42880 
42991 

43i52 
433 1 3 
43473 
43632 
43791 

42357 
42521 

42846 
43oo8 

42374 
42537 
42700 
42862 
48024 

42890 
42553 
42716 
42878 
43o4o 

42406 
42570 
42732 
42894 
43o56 

42428 
42586 
42749 
42911 
48072 

42439 
42602 
42765 
42927 
43o8S 
43249 
43409 
43569 
48727 
43886 

42455 
42619 
42781 
42943 
43 104 

42472 
42635 
42797 
42959 
43i2o 

43 1 36 
43297 
43457 
436i6 
43775 

43169 
43329 
43489 
43648 
43807 

43i85 
43345 
435o5 
43664 
48828 

43201 
43361 
43521 
4368o 
43838 

48217 
43377 
43537 
43696 
43854 

43233 
43398 
43553 
43712 
43870 

43265 
43425 
43584 
43743 
48902 

43281 
43441 
43600 
4.3759 
43917 

_i4 
5 

1 
2 
3 

5 
6 

7 
8 

9 

2 
3 

275 
276 
277 
278 
279 

43933 
44091 
44248 
44404 
44560 

43949 
44107 
44264 
44420 
44576 

1 

43965 

44l22 

44279 
44436 
44592 

43981 
44 1 38 

44295 
44451 
44607 

43996 
44 1 54 
443 1 1 
44467 
44628 

44012 
44170 
44826 
44483 
44638 

44028 
44i85 
44342 
44498 
44654 

44044 
44201 
44358 
445i4 
44669 

44059 
44217 
44373 
44529 
44680 

44075 
44282 
44389 
44545 
44700. 

5 
6 
8 

9 
11 

No. 

0 

0 

3 

4 

5 

G 

7 

8    i) 

12 
i4 

TABLE  XXVI.             [Pagans 
Logarithms  of  Numbers. 

No.  2300 ^3400.                       Log.  4471G 53148. 

No. 

0 

1 

2 

3 

1  4  1  5 

44778  1 44793 

6 

7 

8 

9 

280 
281 
2S2 
283 
2S4 

286 

287 
288 
289 

290 

291 

292 
293 
294 

295 
296 
297 

29S 

299 

3(jo 
3oi 

3f)2 

3o3 
3o4 
3o5 
3o6 
307 
3o8 
309 

3 10 
3ri 

3l2 

3i3 
3i4 
3i5 
3(6 

3,7 
3!8 
3,9 
320 
3?i 

322 

323 
324 
32  5 
326 
327 
328 
329 
"337." 
33i 
332 
333 
334 
335 
336 
337 
338 
33y 

No. 

447 1 6 
44871 
45o25 

45179 
45332 

44731 
448S6 
45o4o 
45194 
45347 

44747 
44902 
45o56 
45209 
45362 

44762 

44917 
45071 
45225 
45378 

44809 
44963 
45117 
45271 
45423 

44S24 

44979 
45 1 33 
45286 
45439 

44840 
44994 
45i48 
453oi 
45454 

44865 
46010 
45i63 
453i7 
45469 

1(3 

44932 
45o86 
45240 
45393 

44948 
45 102 
45255 
45408 

4556i 
45712 
45864 
4601 5 
461 65 

I 
2 
3 
4 
5 
6 

7 
8 

9 

2 
3 
5 
6 

45484 
45637 
45788 
45939 
46090 

45500 
45652 
458o3 
45954 
46io5 

455i5 
45667 
458i8 
45969 
46 1 20 

45530 
45682 
45834 
45984 
46i35 

45545 
45897 
45849 
46000 
46 .5o 

45576 
45728 
45879 
46o3o 
46180 

4633o 
46479 
46627 
46776 
46923 

45591 
45743 
45894 
46045 
46196 

46345 
46494 
46642 
46790 
46938 

45606 
46768 
46909 
46060 
46210 

45621 
45773 
46924 
46076 
46226 

8 
10 
II 
i3 
i4 

46240 
46389 
46538 
466S7 
46835 
46982 
47129 
47276 
47422 
47567 

46255 
464o4 
46553 
46702 
468  5o 

46997 
47144 
47290 
47436 
47582 

46270 
46419 
46568 
46716 
46864 

46285 
46434 
46583 
46731 
46879 

463oo 

46449 
46598 
46746 
46894 

463.5 
46464 
4661 3 
46761 
46909 

46359 
46609 
4(>657 
46806 
46953 

46374 
46523 
46672 
46820 
46967 

1 

I 
2 
3 
4 
5 
6 

7 
8 

9 

5 

2 

3 

c 

e 

8 

9 
1 1 

47012 
47159 
473o5 
4745 1 
47596 

47026 
47173 
47319 
47465 
4761 1 

47041 
47188 
47334 
47480 
47625 

47o56 
47202 
47349 
47494 
47640 

47()70 

4^.17 

47363 
47609 
47654 

47086 
47232 
47378 
47524 
47669 

47100 
47246 
47392 
47538 
47583 

471 14 
47261 
47407 
47563 
47698 

47712 
47857 
4S001 
48.44 
48287 

47727 
47871 
4801 5 
48i59 
483o2 

4774 1 
47885 
48029 
48.73 
483 1 6 

47756 
47900 
.48044 
48187 
4833o 

47770 
47914 
48o58 
48202 
48344 

47784 
47929 
48073 
48216 
48359 

47799 
47943 
48087 
4823o 
48373 

47813 
47958 
48.01 
48244 
48387 

47828 
47972 
48116 
48269 
484oi 

48544 
48686 
48827 
48968 
49108 

47842 
47986 
48i3o 
48273 
48416 
48658 
48700 
4884 1 
48982 
49122 

12 
14 

14 

4843o 
48572 
48714 
48855 
4899G 

48444 
48586 
48728 
48869 
49010 

48458 
48601 
48742 
48883 
49024 

48473 
486 1 5 
48756 
4SS97 
49038 

48487 
48629 
48770 
4891 1 
49052 

485oi 
4S643 
48785 
48926 
49066 

485.5 
48657 

4S799 
409110 
49080 

48530 
48671 
488 1 3 
48954 
49094 

I 
2 
3 
4 
5 
6 
7 
8 

9_ 

1 
3 

4 
6 

491 36 
49276 
4v4i5 
49554 
49693 

49i5o 
49290 
49429 
495(38 
49707 

49164 
49304 
49443 
49582 
49721 

49178 
49318 
49457 
49596 
49734 

49192 
49332 

49471 
49610 
49748 

49206 
49346 
49485 
49624 
497G2 

49220 
49360 
49499 
4963s 
49776 

49234 
49374 
49613 
49661 
49790 

49248 
49388 
49627 
49666 
49803 

49262 
49402 
49541 
49679 

49817 
49966 
60092 
60229 
6o365 
5o5oi 

7 

8 

10 

II 

i3 

49831 
49969 
5>ii.,6 
5oa43 
5o379 

49845 
49982 

5ol20 

5o256 
50393 

49859 
49996 
5oi33 
50270 
5o4o6 

49872 
5ooio 
5oi47 
50284 
5o42o 

49886 
50024 
5oi6i 
50297 
5o433 

49900 
5oo37 
50174 
5o3ii 
5o447 

49914 
5t)o5i 
5oi88 
5o325 
5o46i 

49927 
60066 

6(;202 

6o338 
5o474 
5o6io 
60745 
60880 
5ioi4 
5ii48 

61282 
5i4i5 
61648 
5i6So 
5i8i2 

61943 
62076 
62206 
52336 
62466 

49941 
60079 
6021 5 
5o352 
5o488 

IS 

1 

2 
3 
4 
5 
6 

7 
8 

9 

I 
3 
4 
5 

7 
8 

9 
10 
12 

5.,5;5 
5(i(i5i 
50786 
"^0920 
5io55 
5i  18S 
5i322 
5 1455 
5 1 587 
51720 

5i85i 
5.9S3 
5pii4 

5-2  244 

52375 

5o529 
5o664 
50799 
50934 
5 1 068 

5 1 202 
5i335 
5 1 468 
5 1 60 1 
51733 

5.865 
5 1 996 
52127 
52257 
52388 

5o542 
50678 
5o8i3 
50947 
5 1 08 1 

5o556 
50691 
50826 
50961 
51095 

5o56y 
5070D 
5oS4o 
50974 
5iio8 

5o583 
50718 
5o853 
50987 
5 1 1 2 1 

50696 
607.32. 
60866 
5iooi 
5ii36 
61268 
5i4o2 
616.34 
61667 
61799 

60623 
60769 
60893 
51028 
61162 

5o637 
60772 
60907 
6k)4i 
61176 

5i3o8 
6i44i 
5.674 
6 1 706 
5 1 838 

5i2i5 
5.34s 
5i48i 
5i6.4 
51746 

51228 
5 1 362 
51495 
51627 
51759 

51242 
5i375 
5i5o8 
5 1640 

51772 

5i255 
5i388 
5i52i 
5 1 654 
51786 

61295 
61428 
61661 
61693 
51826 

51878 
52009 
52i4o 
52270 
52401 

51891 
52022 
52  153 
52284 
52414 

5 1 904 
52f)35 
52.66 
52297 
52427 

51917 
52048 
52179 
523 10 
52440 

61930 
62061 
62192 
52323 
62453 

61967 
52o«8 
62218 
52349 
62479 

61970 
62101 
62231 
52362 
62492 

1 

I 
2 
3 
4 
5 
6 

7 
8 

9 

2 

I 
2 
4 
5 

52  5o4 
52634 
52763 
52S92 
53o2o 

525i7 
52647 
52776 
52905 
53o33 

5253o 
52660 
52789 
52917 
53<>46 

52543 
52673 
52802 
52930 
53u58 

52  556 
52(xS6 
528.5 
52943 
53071 

52569 
62699 
52827 
5^956 
53o84 

62682 
62711 
62840 
62969 
53097 

62696 
62724 
52853 
62982 
53iio 

62608 
52737 
62866 
62994 

53l22 

62621 
62760 
62879 
53007 
53i35 

5 

7 

8 

10 

II 

0    1 

2 

3 

4 

5 

6    7  1 

8 

9 

Page  174]            TABLE  XXVI. 

Logarithms  of  Numbers. 

Nn 

Q-ioo    Mon 

0.                     Log.  53148-    60206. 

No. 

34o 
34 1 
342 
343 
M4 
345 
346 

347 
348 

349 
.350 
35. 
352 
353 
354 

0 

1 

53i6i 
53288 
534 1 5 
53542 
53668 

2 

3 

4 

5 

6 

7 

8 

9 

53i48 
53275 
534o3 
53529 
53656 

53173 
53301 
53428 
53555 
53681 

53 186 
533i4 
53441 
53567 
53694 

53199 
53326 
53453 
5358o 
53706 

53212 

53339 

53466 
53593 
53719 

53224 
53352 

63479 
536o5 
53782 

63;  87 
53364 
53491 
536i8 
63744 

53260 
53377 
535o4 
6363 1 
53767 

53263 
53390 
68617 
63643 
68769 

13 

I 
2 
3 

4 
5 
6 

7 
8 

9 

I 

3 
4 

5 

53782 
53908 
54o33 
54 1 58 
54283 

53794 
53920 
54045 
54170 
54295 

53807 
53933 
54o58 
54i83 
54307 

53820 
53945 
54070 
54195 
54320 

53832 
53958 
54o83 
54208 
54332 

53845 
53970 
54095 
54220 
54345 

53857 
53983 
64108 
64233 
64357 

53870 
53995 
54120 
54245 
54370 

63882 
54008 
64i33 
64268 
54382 

53896 
64020 
54i46 
54270 
54394 

7 
8 

9 

10 
12 

54407 
■14531 
54654 
54777 
54900 

54419 
54543 
54667 
54790 
54913 

54432 
54555 
54679 
54802 
54925 

5UM 
54568 
54691 
548 1 4 
54937 

54456 
54580 
54-704 
54827 
54949 

54469 
54593 
54716 
54889 
54962 

5440  7. 
5460L 
54728 
5485 1 
54974 

54494 
54617 
54741 
54864 
54986 

645o6 
6463o 
64768 
54876 
54998 

64618 
54642 
54765 
54888 
66011 

355 
356 
357 
358 
359 
36o 
36 1 
362 
363 
364 
365 
366 
367 
368 
369 
370 
371 
372 
373 
374 
375 
376 
377 
378 
379 

38o 
38 1 
382 
383 
384 
385 
386 
387 
388 
389 

390 
391 
392 
393 
394 
395 
396 
397 
398 

1  399 

55023 
55i45 
55267 
55388 
55509 

55o35 
55i57 
55279 
55400 
55522 

55642 
55763 
55883 
56oo3 

56l22 

55o47 
55169 
55291 
554 1 3 
55534 

55q6o 
55f?2 
553o3 
,55425 
55546 

55072 
55194 
553i5 
55437 
55558 

55o84 
55206 
55328 
55449 
55570 

66096 
66218 
66340 
55461 
66682 

55708 
66823 
66943 
66062 
66182 

55io8 
55280 
55362 
55473 
55694 
55716 
66835 
55955 
66074 
56194 
663 1 2 
5643 1 
56549 
66667 
66786 

66121 
55242 
55364 
55485 
556o6 

65i33 
55265 
55376 

66497 
66618 

1 

I 
2 
3 
4 
6 
6 

7 
8 

_9_ 

2 

I 
i 

55630 
55751 
55871 
55991 
56iio 

55654 
55775 
55895 
56oi5 
56 1 34 

55666 
55787 
55907 
56027 
56i46 

56265 

56384 

565o2, 

56620 

56788 

55678 
55799 
55919 
56o38 
56i58 

55691 
558ii 
55981 
56o5o 
56170 

66727 
55847 
55967 
66086 
66206 

66324 
56443 
66661 
66679 
66797 

66739 
66869 
55979 
66098 
56217 

56336 
56466 
66673 
66691 
568o8 

A 
5 
6 

7 
8 

56229 
56348 
56467 
56585 
56703 

56241 
5636o 
56478 
56597 
56714 

56253 
56372 
56490 
566o8 
56726 

56844 
56961 
57078 
57194 
57810 

56277 
56896 
565i4 
56632 
56750 

56289 
56407 
56526 
56644 
56761 

663oi 
66419 
66538 
56656 
66778 

ic 
II 

56820 
56937 
57054 
571 71 
57287 

56832 
56949 
57066 
57183 
57299 

56855 
56972 
57089 
57206 
57822 

56867 
56984 
57101 
57217 
57334 

56879 
56996 
57118 
57229 
57345 

66891 
57008 
67124 
57241 
67357 

67473 
67688 
67703 
67818 
67933 

56902 
67019 
67186 
67262 
67868 

66914 
67081 
57148 
67264 
67380 

66926 
67043 
67169 
67276 
67892 

11 

574o3 
57519 
57634 
57749 
57864 

574 1 5 
57530 
57646 
57761 
57875 

57426 
57542 
57657 

57772 
57887 

57438 
57553 
57669 
57784 
57898 

57449 
57565 
57680 
57795 
57910 

57461 
57576 
57692 
57807 
57921 

57484 
67600 
67715 
67880 
67944. 

57496 
67611 
67726 
67841 
67955 

67607 
67623 
67738 
67862 
67967 

I 

2 
3 
4 
5 
6 

7 
8 

9^ 

I 
2 

3 

4 
6 

57978 
58092 
58206 
58320 
58433 

57990 
58 104 
58218 
5833 1 
58444 
58557 
58670 
58782 
58894 
59006 

58ooi 
58ii5 
58229 
58343 
58456 

58ot3 
58127 
58240 
58354 
58467 

58024 
58i38 
58252 
58365 
58478 

58o35 
58i49 
58263 
58377 
58490 

58o47 
68161 
68274 
68388 
685oi 

68o58 
68172 
582S6 
68899 
68612 

68070 
58 1 84 
68297 
58410 
58524 

68081 
68195 
68809 
6842^ 
58536 

I 

9 
10 

58546 
58659 

58771 
58883 
58995 

58569 
5868: 
58794 
58906 
59017 

58580 
58692 
588o5 
58917 
59028 

58591 
58704 
588 1 6 
58928 
59040 

58602 
58715 
58827 
58989 
59051 

586i4 
68726 
58838 
68960 
69062 

58626 
68787 
58850 
68961 
69078 

58636 
68749 
58861 
68978 
69084 

58647 
68760 
68872 
68984 
69096 

59106 
59218 
59329 
59439 
59550 

59118 
59229 
59340 
59450 
69561 
59671 
59780 
59890 
59999 
60108 

59129 
59240 
59351 
59461 
59572 

59140 
59251 
59362 
59472 
59583 

59i5i 
59262 
59878 
59483 
59594 

59162 
69278 
69884 
59494 
69606 

69178 
69284 
69895 
69606 
69616 

59184 
69296 
69406 
69617 
69627 

69196 
69806 
69417 
59628 
69688 

69207 
69818 
69428 
69539 
69649 

1( 

I 
2 
3 
4 
5 
6 

7 
8 

_9. 

) 

I 
1 

59660 
59770 
59879 
599S8 
60097 

59682 
59791 
59901 
60010 
60 1 1 9 

59693 
59802 
59912 
60021 
6oi3o 

59704 
59813 
59928 
6oo32 
6oi4i 

69715 
69824 
59934 
60043 
60162 

69726 
69835 
59945 
6oo54 
6oi63 

59787 
69846 
69966 
60066 
60178 

59748 
69867 
69966 
60076 
60184 

69769 
69868 

69977 
60086 
60196 

6 
4 
5 
6 

7 
8 

9 

1  No. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

1                            ■■  —                '        ■■ 

TABLE  XXVI.            [Page  175 
Logarithms  of  Numbers. 

1 

No.  4000     4600.                 hog.  60206     G6276. 

No. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

4oo 
4oi 
4o2 
4o3 
4o4 

60206 
6o3i4 
60423 
6o53i 
6o633 

60217 
6o32  5 
60433 
6o54i 
60649 

60228 
6o336 
60444 
6o552 
60660 

60239 
60347 
60455 
6o563 
60670 

60249 
6o358 
60466 
60574 
60681 

60260 
60869 
60477 
6o584 
60692 

60271 
60879 
60487 
60595 
60708 

60282 
60890 
60498 
60606 
60713 

60298 
60401 
60509 
60617 
60724 

60881 
60938 
61045 
6ii5i 
61257 

60804 
604 1 2 
6o52o 
60627 
60735 

] 

I 
2 

3 
4 
5 
6 

7 
8 

9 

1] 

I 
2 
3 
4 

4o5 
4o6 
407 
408 
409 

4io 
4ii 

4l2 

4i3 
4i4 

60746 
6o853 
60959 
61066 
61172 

60756 
6o863 
60970 
61077 
6ii83 

60767 
60874 
60981 
61087 
61 194 

60778 
6o885 
60991 
61098 
61204 

60788 
60895 
61002 
61109 
6i2i5 

60799 
60906 
6ioi3 
61119 
61225 

60810 
60917 
61028 
6ii3o 
6i236 

60821 
60927 
6io34 
6ii4o 
61247 

60842 
60949 
6io55 
61162 
61268 

6 

7 
8 

9 

ID 

61278 
61 384 
61490 
61595 
61700 

61289 
61395 
6i5oo 
61606 
61711 

6i3oo 
6i4o5 
6i5ii 
61616 
61721 

6i3io 
6i4i6 
6i52i 
61627 
61731 

61821 
61426 
6i532 
61687 
61742 

6i33i 
61437 
61542 
61648 
61752 

61342 
61448 
6i553 
6i658 
61768 

6i352 
6j458 
6 1 563 
61669 
61773 

6 1 863 
61469 
61574 
61679 
61784 

61874 
61479 
6 1 584 
61690 
61794 

4i5 

4i6 
417 
4i8 

419 

6i8o5 
61909 
62014 
62118 
62221 

6i8i5 
61920 
62024 
62128 
62232 

61S26 
61930 
62034 
62i38 
62242 

6i836 
61941 
62045 
62149 
62252 

61847 
6 1 95 1 
62055 
62159 
6226J 

61857 
61962 
62066 
62170 
62278 

61868 
61972 
62076 
62180 
62284 

61878 
61982 
620S6 
62190 
62294 

61888 
61993 
62097 
62201 
62804 

61899 
6200,3 
62107 
62211 
62815 

420 
421 
422 
423 
424 
425 
426 
427 
428 
429 
43o 
43t 
432 
433 
434 
435 
436 
437 
438 
439 
440 
44 1 
442 
443 
444 
445 
446 
447 
448 
449 
45o 
45i 
452 
453 
454 

62325 
62428 
6253i 
62634 
62737 

62335 
62439 
62542 
62644 
62747 

62346 
62449 
62552 
62655 
62757 

62356 
62459 
62562 
62665 
62767 

62366 
62469 
62572 
62675 
62778 

62877 
62480 
62583 
62685 
62788 

62887 
62490 
62593 
62696 
62798 

62897 
62500 
62608 
62706 
62808 

62408 
625ii 
62613 
62716 
62818 

62418 
62521 
62624 
62726 
62829 

62839 
62941 
63o43 
63 1 44 
63246 

62849 
62951 
63o53 
63 1 55 
63256 

62S59 
62961 
63o63 
63 1 65 
63266 

62870 
62972 
63073 
63 1 75 
63276 

62880 
62982 
63o83 
63i85 
68286 

62S90 
62992 
68094 
68195 
68296 

62900 
68002 
63io4 
632o5 
633o6 

62910 
63oi2 
63ii4 
632i5 
68817 

62921 
68022 
68124 
63225 
68827 

63428 
63528 
68629 
68729 
68829 

62981 
63o33 
63 1 34 
68286 
63337 

1 

I 
2 
3 
4 
5 
6 

7 
8 

9 

u 

1 

2 

3 
4 

63347 
63448 
63548 
63649 
63749 

63357 
63458 
63558 
63659 
63759 

63367 
63468 
63568 
63669 
63769 

63377 
63478 
63579 
63679 
68779 

63387 
63488 
63589 
63689 
63789 

63397 
63498 
63599 
68699 
68799 

63407 
63  5o8 
68609 
68709 
63809 

63417 
685:8 
636 19 
63719 
63819 

63438 
63538 
68689 
68789 
63839 

5 
6 
7 
8 

9 

63849 
63949 
64o48 
64 1 47 
64246 

63859 
63959 
64o58 
64i57 
64256 

63869 
63969 
64068 
64167 
64266 

64365 
64464 
64562 
64660 
64758 

63879 
68979 
64078 

64177 
64276 

64375 
64473 
64572 
64670 
64768 

68889 
68988 
64088 
64187 
64286 

68899 
68998 
64098 

64197 
64296 

68909 
64008 
64 1 08 
64207 
643o6 

68919 
64018 
64ii8 
64217 
643 1 6 

68929 
64028 
64128 
64227 
64326 

68939 
64o38 
64187 
64287 
64335 

I- 

64345 
64444 
.64542 
6464o 
64738 

64355 
64454 
64552 
6465o 
64748 

64385 
64483 
64582 
64680 
64777 

64395 
64493 
64591 
64689 
64787 

644o4 
645o3 
64601 
64699 
64797 

644 1 4 
645 1 3 
646 II 
C4709 
64807 

64424 
64523 
64621 
64719 
64816 

64484 
64532 
6463 1 
64729 
648  2  6 

J' 

T 

64836 
64933 
65o3i 
65i28 
65225 

64846 
64943 
65o4o 
65i37 
65234 

64856 
64953 
65o5o 
65i47 
65244 

64865 
64963 
65o6o 
65 1 57 
65254 

64875 
64972 
65070 
65i67 
65263 

64885 
64982 
65o79 
65176 
65273 

64S95 
64992 
65089 
65 1 86 
65283 

64904 
65oo2 
65o99 
65196 
65292 

64914 
65oii 
65 1 08 
65205 
65302 

64924 
65o2i 
65ii8 
652 1 5 
653 1 2 

65321 
654 1 8 
655i4 
656io 

65706 

6533i 
65427 
65523 
65619 
65715 

65341 
65437 
65533 
65629 

65725 

6535o 

65447 
65543 
65639 
65734 

65360 
65456 
65552 
65648 
65744 

65369 
65466 
65562 
65658 
6:,;^3 

65379 
65475 
65571 
65667 
65763 

65389 
65485 
65581 

65677 
63772 

65398 
65495 
65591 
65686 
65782 

654o8 
655o4 
656oo 
65696 
65792 

9 

2 :  -J 

3  3 
4\4 

455 
456 
457 
458 
459 

658oi 
65896 
65992 
66087 
66181 

658 11 
65906 
66001 
66096 
66191 

65820 
65916 
66011 
66106 

66200 

65830 
65925 
66020 
66ii5 
66210 

65839 
65935 
66o3o 
66124 
66219 

65849 
65944 
66089 
66134 
66229 

65858 
65954 
66049 
66143 
66288 

65868 
65968 
66o58 
661 53 

66247 

65877 
65973 
66068 
66162 
66257 

65887 
65982 
66077 
66172 
66266 

5, 
6 

7 
8 

9 

3 

5 
6 

7 
8 

No. 

0 

1 

2 

3 

4    5  1 

6 

7 

8 

9 

k 


Page 

1761            TABLE  XXVI. 

Logarithms  of  Numbers. 

No.  4000 5200.                   Log.  C6276 71600. 

No. 

0 

1 

2 

3 

4 

5 

6 

66332 
66427 
66621 
66614 
66708 

7 
66342 
66436 
66530 
66624 
66717 

8 

9 

46o 
46 1 
462 
463 
464 
465 
466 
467 
468 
469 

470 
471 
472 

473 

474 

475' 

476 

477 

47S 

479 

480 

481 

482 

483 

484 

485 

486 

487 

488 

489 

490 

491 

492 

493 

494 

495 

496 

497 

498 

499 

600 

5oi 

502 

5o3 
5o4 
5o6 
5o6 
607 
5oS 
609 

610 
5ii 
612 
5i3 
5i4 
5i5 
5x6 
617 
5i8 
619 

No. 

66276 
66370 
66454 
66668 
66662 

66286 
66380 
66474 
66667 
66661 

66296 
66389 
66483 
66577 
66671 

663o4 
66398 
66492 
66586 
66680 

663 1 4 
66408 
66602 
66696 
66689 

66323 
66417 
66611 
66606 
66699 

6636 1 
66445 
66539 
66633 
66727 

66361 
66455 
66649 
66642 
66736 

1 

I 
2 
3 

4 
5 
6 

7 
8 

9 

I 
2 
3 
4 

66745 
66839 
66932 
67026 
67117 

66755 
66848 
66941 
67034 
67127 

66764 
66867 
66960 
67043 
67136 

66773 
66S67 
66960 
67062 
67145 

66783 
66876 
66969 
67062 
67164 

66792 
66886 
66978 
67071 
67164 

66801 
66894 
669S7 
67080 
67173 

66811 
66904 
66997 
67089 
67182 

66820 
66913 
67006 
67099 
67191 

66829 
66922 
67016 
67108 
67201 

5 

6 

7 
8 

7 

67210 
67302 
67394 
674S6 
6767S 

67219 
67311 
67403 
67495 
67687 

67228 
67321 
67413 
67604 
67696 

67237 
67330 
67422 
67614 
67606 

67247 
67339 
67431 
67623 
67614 

67266 
67348 
67440 
67532 
67624 

67265 
67367 
67449 
67641 
67633 

67274 
67367 
67469 
67660 
67642 

67284 
67376 
67468 
67660 
67661 

67293 
67335 

67477 
67669 
67660 

67669 
67761 
67862 
67943 
68034 

68124 
68216 
683o5 
68396 
68485 

67679 
67770 
67861 
67962 
68043 

68 1 33 
68224 
683i4 
684o4 
68494 

676S8 

67779 
67870 
67961 
68062 

67697 
67788 
67879 
67970 
68061 

67706 
67797 
67888 
67979 
68070 

67715 
67806 
67897 
67988 
68079 

67724 
67816 
67906 
67997 
68088 

67733 
67826 
67916 
68006 
68097 

67742 
6-834 
67926 
68016 
68106 

67762 
67843 
67934 
68024 
68116 

68142 
68233 
68323 
684 1 3 
68602 

68161 
68242 
68332 
68422 
68611 

68160 
68261 
68341 
68431 
68620 

68169 
68260 
68350 
68440 
68629 

68178 
68269 
68359 

68449 
68538 

68187 
68278 
68368 
68458 
68647 

68196 
6S287 
68377 
68467 
68666 

68205 
68296 
68386 
68476 
68565 

68674 
68664 
68763 
68842 
68931 

68583 
68673 
68762 
6885 1 
68940 

68692 
68681 
68771 
68860 
68949 

6S601 
68690 
68780 
6S869 
68968 

68610 
68699 
68789 
68878 
68966 

68619 
68708 
68797 
68886 
68976 

68628 
6S717 
68806 
68896 
68984 

68637 
68726 
688 1 5 
68904 
68993 

68646 
68735 
68824 
68913 
69002 

68665 
68744 
68833 
68922 
69011 

I 
2 
3 
4 
5 
6 

7 
8 

9 

I 
2 
3 
4 

69020 
69108 
69197 
69285 
69373 

69028 
69117 
69206 
69294 
69381 

69037 
69126 
69214 
69302 
69390 

69046 
69135 
69223 
69311 
69399 

69066 
69144 
69232 
69320 
69408 

69064 
69162 
69241 
69329 
69417 

69073 
69161 
69249 
69338 
69426 

69082 
69170 
69268 
69346 
69434 

69090 
69179 
69267 
69355 
69443 

69099 
691S8 
69276 
69364 
69452 

69539 
69627 
69714 
69801 
69888 

69976 
70062 
70148 
70234 
70321 

5 
5 
6 
7 
f 

69461 
69548 
69636 
69723 
69810 

69469 
69667 
69644 
69732 
69819 

69478 
69666 
69663 
69740 
69827 

69487 
69674 
69662 
69749 
69836 

69496 
69683 
69671 
69768 
69846 

69604 
69692 
69679 
69767 
69854 

69613 
69601 
69688 
69776 
69S62 

69622 
69609 
69697 
69784 
69871 

69631 
69618 
69706 
69793 
69S80 

69966 
70063 
70140 
70226 
7o3i-2 

69897 
69984 
70070 
70167 
70243 

69906 
69992 
70079 
70166 
70262 

69914 
70001 
70088 
70174 
70260 

69923 

70010 
70096 
70183 
70269 

69932 
70018 
70106 
70191 
70278 

7o364 
70449 
70636 
7062 1 
70706 

69940 
70027 
701 14 
70200 
70286 

69949 
7oo36 
70122 
70209 
70296 

69968 
70044 
70i3i 
70217 
7o3o3 
70389 
70476 
70661 
70646 
70731 

70329 

704 1 5 
70601 
70686 
70672 

7o338 
70424 
70609 
70695 
70680 

70346 
70432 
70618 
70603 
70689 

7o365 
70441 
70626 
70612 
70697 

70372 
70458 
70644 
70629 
70714 

7o3Si 
70467 
70662 
7o638 
70723 

70398 
70484 
70669 
70666 
70740 

70826 
70910 
70995 
71079 
71164 
71248 
7i332 
71416 
71600 
71584 

70406 
70492 
70678 
70663 
70749 

70834 
70919 
7ioo3 
71088 
71172 

70767 
70842 
70927 
71012 
71096 

70766 
70861 
70935 
71020 
71106 

70774 
70869 
70944 
71029 
71113 

70783 
70868 
70962 
71037 
71122 

70791 
70876 
70961 
71046 
7ii3o 

70800 
70885 
70969 
71064 
71139 

70808 
70893 
70978 
71063 
71 147 
7i23i 
7i3i5 
71399 
71483 
71667 

70817 
70902 
70986 
71071 
71166 

I 
2 
3 
4 
5 
6 
7 
8 

9 

I 
2 
2 
3 

71 181 
71265 
71349 
71433 
71617 

71189 

71273 
71357 
71441 
71626 

71 198 
71282 
7 1 366 
7i45o 
71533 

71206 
71290 
71374 
71458 
71642 

71214 
71299 
71383 
71466 
71660 

71223 
7 1 307 
71391 

71475 
71669 

71240 
71324 
71408 
71492 
71676 

71267 
7i34i 
71425 
71608 
71692 

4 
5 
6 
6 

■7 

0    1 

(> 

3 

4    5  i 

G 

7 

8    9 

-  I 

TABLE  XXVI.                   [Page  177 

Logarithms  of  Numbers. 

No 

c;ooo     SsiOC 

).                       Log.  71600 76343. 

No. 

520 
521 
522 

523 
524 
525 
526 
527 
528 
529 

0 

1 

2 

3 

4 

5 

G 

7 

8 

9 

71600 

71684 
71767 

7i85o 
7193^ 

71609 
71692 

71775 
71858 
71941 

71617 
71700 
71784 
71867 
71950 

71625 
71709 
71792 
71875 
7.958 

71634 
71717 
71800 
71883 
71966 

71642 
71725 
71809 
71892 

71975 

7i65o 
71734 
71817 
71900 
71988 

71659 
71742 
71825 
71908 
71991 

71667 
7.750 
7.884 
7.9.7 
7.999 

7.675 
7.759 
71842 
7.925 
72008 

72016 
72099 
72181 
72263 
72346 

72024 
72107 
72189 
72272 
72354 

72082 
72115 
72198 
72280 
72862 

72041 
72123 
72206 
7  2  288 
72870 

72049 
72182 
72214 
72296 
72878 

72057 
72140 
72222 
72804 
72887 

72066 
72148 
72280 
72818 
72895 

72074 
72 1 56 
72289 
72821 
72403 

72082 
72.65 

72247 
72829 
724.1 

72090 
72178 
72255 
72887 
72419 

53o 
53i 
532 
533 
534 

72428 
72509 
72591 
73673 
72754 

7^835 
72916 

72997 
73078 
73159 

72436 
72518 
72599 
72681 
72762 

72843 
72925 
73006 
78086 
73167 

72444 
72526 
72607 
72689 
72770 

72452 
72534 
72616 
72697 
72779 

72460 
72542 
72624 
72705 
72787 
72868 
72949 
78080 
78111 
78191 

78272 
73852 
78482 
73512 
73592 

72469 
72550 
72682 
72718 
72795 
72876 
72957 
78088 
78119 
78199 
78280 
78860 
73440 
73520 
78600 

72477 
72558 
-2640 
72722 
72808 

72485 
72567 
72648 
72780 
72811 

72493 
72575 
72656 
72788 
72819 

725ui 
72583 
72665 
72746 
72827 

535 
536 
537 
538 
L)39 

54o 
54 1 
542 

543 
544 
545 
546 

547 
548 

549 
55o 
55i 
552 
553 
554 

72852 
72988 
78014 
78094 
78175 
78255 
78886 
78416 
78496 
73576 

72860 
72941 
78022 
78102 
78188 

72884 
72965 
78046 
78127 
78207 

72892 
72978 
78054 
78.35 
78215 

72900 
72981 
78062 
78143 
78328 

72908 
72989 
78070 
78.5. 
7828. 

73239 
73320 
73400 
73480 
73560 

73247 
73328 
73408 
73488 
73568 

78268 
73344 
73424 
73504 
73584 

78288 
78868 
73448 
73528 
78608 

78296 
78876 
73456 
73536 
78616 

78804 
73384 
78464 
73544 
78624 

78812 
78892 
78472 
78552 
78682 

7364o 
73719 
73799 
73878 
73957 

73648 
73727 
73807 
73886 
73965 

73656 
78785 
738 1 5 
73894 
78973 

73664 
73743 
78828 
78902 
78981 

78672 
78751 
73830 
78910 
78989 

78679 
78759 
78888 
78918 
73997 

78687 
78767 
78846 
78926 
74oo5 

78695 
73775 
78854 
78988 
740 1 3 

78708 
78788 
78S62 
7894. 
74020 

78711 
787^ 
78870 
78949 
74028 

74o36 
74u5 
74194 
74273 
7435i 

74o44 
74i23 
74202 
74280 
74359 

74o52 
74i3i 
74210 
74288 
74367 

74060 
74189 
74218 
74296 
74874 

74068 

74 1 47 
74225 

74304 

74382 

74076 
741 55 
74233 
74312 
74390 

74084 
74162 
74241 
74820 
74398 

74092 
74.70 
74249 
74337 
74406 

74'->99 
74178 
74257 
74335 
744.4 

74107 
74.86 
74265 
74348 
74421 

555 
556 

557 
558 
559 

74429 
74507 
74586 
74663 

74741 

74437 
745)5 
74593 
74671 
74749 

74445 
74523 
74601 
74679 
74757 

74458 
74581 
74609 
74687 
74764 

74461 
74539 
74617 
74695 
74772 

74468 

74547 
74624 
74702 
74780 

74476 
74554 
74682 
74710 
74788 

74484 
74562 
74640 
74718 

74796 

74492 
74570 
74648 
74726 
74S08 

745oo 
74578 
74656 
74788 
748 1 1 

56o 
56 1 
562 
563 
564 

74819 
74896 
74974 
75o5i 
75128 

74827 
74904 
74981 
75o59 
75i36 

74834 
74912 
74989 
75o66 
75i43 

74842 
74920 
74997 
75074 
75i5i 

7485o 
74927 
75oo5 
75082 
75i59 

74858 
74935 
75012 
75089 
75166 

74865 
74943 
75020 
75097 
75.74 

74873 
74950 
75028 
75io5 
75.82 

7488. 
74958 
75o35 
75ii3 
75.89 

74889 
74966 
75o48 
75 .  20 
75.97 

565 

566 
567 
568 
569 

75205 
75282 
75358 
75435 
755ii 

752i3 
75289 
75366 
75442 
75519 

75220 
75297 
75874 
75450 
75526 

75228 
753o5 
75881 
75458 
75534 

75286 
75312 
75389 
75465 
75542 

75248 
75820 
75397 
75473 
75549 

7525. 
75328 
75404 
7548. 
75557 

75259 
75385 
754.2 
75488 
75565 

75366 
75343 
75420 
75496 
75572 

75274 
75351 
75427 
75504 
75580 

75656 
75782 
75808 
75884 
75959 

570 
571 
572 
573 
574 
575 
576 
577 
578 

579 
No. 

75587 
75664 
73740 
758 1 5 
75891 

75595 
75671 
75747 
75823 
75899 

756o3 

7567? 
75755 
7583 1 
75906 

75610 
75686 
75762 
75888 
75914 

75618 
75694 
75770 
75846 
75921 

75626 
75702 
75778 
75853 
75929 

75633 
75709 
75785 
75861 
75987 

75641 

757.7 
75798 
75868 
75944 

75648 
75724 
75800 
75876 
75952 

75967 
76042 
761 18 
76193 
76268 

75974 
76o5o 
7-6125 
76200 
"6275 

75982 

76057 
76133 
76208 
76288 

75989 
76065 
76140 
76215 
76290 

75997 
76072 
76148 
76228 
76298 

76005 
76080 
76155 
76280 
76805 

76012 
76087 
76.68 
76288 
76818 

76020 
76095 

76.70 
76^45 
76820 

76027 
76 1  o3 
76178 
76253 
76828 

76085 
761 10 
76185 
76260 
76335 

0  1     1 

2 

3 

4 

5  1  6 

7 

8 

9 

3 
4 
4 
5 
6 
9'6 


23 


P^s«i78]             TABLE  XXVI. 

Logarithms  of  Numbers. 

No.  5800 6400.                       Log 

r  yrpi-ii    R 

0G18. 

No. 

58o 
58 1 
582 
583 
584 
585 
586 
587 
588 
589 
590 
591 
592 
593 
594 
595 
596 
597 
598 
599 
600 
601 
602 
6o3 
604 
6o5 
606 
607 
608 
609 

610 
611 
612 
6i3 
6i4 

0     1 

2 

3 

4 

5 

6 

7 

8 

9 

76343 
76418 
76492 
76567 
76641 
76716 
767-vo 
76864 
76938 
77012 

76350 
76425 
76^00 
76574 
76049 
76723 
76797 
76871 
76945 
77019 

76358 
76433 
76507 
76582 
76656 

76365 
76440 
765 1 5 
76589 
76664 

76738 
76812 
76886 
76960 
77034 

76373 
76448 
76522 
76597 
76671 

76880 
76455 
76530 
76604 
76678 

76388 
76462 
76537 
76612 
76686 

76395 
76470 
76545 
76619 
76693 

76403 

76477 
76662 
76626 
76701 

76410 
76485 
76669 
76634 
76708 

( 

I 
2 
3 
4 
5 
6 

7 
8 

9 

4 

■3 

76730 
76805 

76879 
76953 
77026 

76745 
76819 
76893 
76967 
77041 

76753 
76827 
76901 
76975 
77048 

76760 
76834 
76908 
76982 
77o56 

76768 
76842 
76916 
76989 
77063 

76775 
76849 
76923 

76997 
77070 

76782 
76866 
76930 
77004 
77078 

4 
5 
6 
6 

7 

77085 
77159 
77232 
773o5 
77379 

77093 
77166 
77240 
773 1 3 
77386 

77100 
77173 

77247 
77320 
77393 

77107 
77181 
77254 
77327 
77401 

77115 
77188 
77262 
77335 
77408 

77122 
77195 
77269 
77342 
774 1 5 

77129 
77203 
77276 
77349 
77422 

77187 
77210 
77283 
77357 
7743o 

77144 
77217 
77291 
77364 
77437 

77i5i 
77226 
77298 
77371 
77444 

77452 
77525 
77597 
77670 
77743 

77459 
77532 
77605 
77677 
77750 

77466 
77539 
77612 
77685 
77757 

77474 
77546 
77619 
77692 
77764 

77481 
77554 
77627 
77699 

77772 

77488 
77561 
77634 
77706 

77779 

77495 
77568 
77641 
77714 
77786 

77608 
77676 
77648 
77721 
77793 

77610 
77583 
77666 
77728 
77801 

77617 
77690 
77663 
77735 
77808 

77815 
77887 
77960 
78032 
78104 

77S22 
77895 

77967 
78039 
781 II 

77830 
77902 
77974 
78046 
78118 

77837 
77909 
77981 
78053 
78125 

77844 
77916 
77988 
78061 
78132 

77851 
77924 

77996 
78068 
78140 

77859 
77931 
78003 
78075 
78147 

77866 
77938 
78010 
78082 
78164 

77873 
77945 
78017 
78089 
78161 

77880 
77962 
78025 
78097 
78168 

7S176 
70247 
78319 
78390 
78462 

78183 
78254 
78326 
78398 
78469 

78190 
78262 
78333 
78405 
78476 

78197 
78269 
78340 
78412 

78483 

78204 
78276 
78347 
78419 
78490 

7821 1 
78283 
78355 
78426 
78497 

78219 
78290 
7S362 
78433 
78504 

78226 
78297 
78369 
78440 
78612 

78233 
78306 
78376 

78447 
78619 

78240 
78312 
78383 
78455 
78626 

7 

I 
2 
3 
4 
5 
6 

7 
8 

9 

I 
1 
2 
3 

78533 
7S604 
78675 
7S746 
78817 

78540 
78611 
78682 
78753 
78824 

78547 
78618 
786S9 
78760 
7883i 

7S554 
78625 
78696 
78767 
78838 

78561 
78633 
78704 
78774 
78845 

78669 
78640 
7871 1 
78781 
78852 

78576 
78647 
78718 
787S9 
78859 

78683 
78664 
78726 
78796 
78866 

78690 
78661 
78732 
78803 
78873 

78697 
78668 
78739 
78810 
7S880 

4 
4 
5 
6 
6 

6i5 
616 
617 
6r8 
619 

78888 
78958 
79029 
79099 
79169 

78895 
78965 
79o36 
79106 
79176 

78902 
78972 
79043 
79113 
79183 

78909 
78979 
79o5o 
79120 
79190 

78916 
78986 
79057 
79127 
79^97 

78923 
78993 
79064 
79134 
79204 

78930 
79000 
79071 
79141 
792 1 1 

78937 
79007 
79078 
79148 
79218 

78944 
79014 
79086 
79166 
79226 

78961 
79021 
79092 
79162 
79232 

620 
621 
622 
623 
624 

79239 
79309 
79379 
79449 
79518 

79246 
79316 
79386 
79456 
79525 
79S95 
79664 
79734 
79803 
•79872 

79941 
80010 
80079 
80147 
80216 

79253 
79323 
79393 
79463 
79532 

79260 
79330 
79400 
79470 
79539 

79267 
79337 
79407 

79477 
79546 

79274 
79344 
79414 
79484 
79553 

79281 
79351 
79421 
79491 
79660 

79288 
79358 
79428 

79498 
79667 

79295 
79366 
79436 
79606 
79574 

79302 
79872 
79442 
79611 
79681 

625 
626 
627 
628 
629 

63o 
63i 
632 
633 
634 
635 
636 
637 
638 
639 

No. 

79588 
79657 
79727 
79796 
79865 

79934 
8ooo3 
80072 
8oi4o 
80209 

79602 
79671 
79741 
79810 
79879 

79609 
79678 
79748 
79817 
79886 
79955 
80024 
80092 
80161 
80229 

79616 
79685 
79754 
79824 
79893 

79623 
79692 
79761 
79831 
79900 

79630 
79699 
79768 
79837 
79906 

79637 
79706 
79775 
79844 
79913 

79644 
79713 
79782 
79861 
79920 

79660 
79720 
79789 
79868 
79927 

79996 
8oo65 
8oi34 
80202 
80271 

79948 
80017 
8oo85 
801 54 
80223 

79962 
8oo3o 
80099 
80168 
8o236 

79969 
80037 
80106 
80175 
80243 

79975 
80044 
8oii3 
80182 
80260 

79982 
8oo5i 
80120 
80188 
80267 

79989 
8oo5S 
80127 
80195 
80264 

( 

I 

2 
3 
4 
5 
6 

7 
8 

9 

5 

I 
2 
1 

80277 
8o346 
804 1 4 
80482 
8o55o 

80.284 
8o353 
80421 
80489 
80557 

80291 
80359 
80428 
80496 
8o564 

80298 
8o366 
80434 
8o5o2 
80570 

8o3o5 
80373 
8o44i 
8o5o9 
80577 

8o3i2 
8o38o 
80448 
8o5i6 
8o584 

5 

8o3i8 
80387 
80455 
8o523 
80691 

80326 
80393 
80462 
8o53o 
80698 

8o332 
8o4oo 
80468 
8o536 
80604 

80339 
80407 
80475 
80543 
806 II 

3 
4 
4 
5 
5 

0 

1 

2 

3 

4 

6 

7 

8 

9 

TABLE  XXVI.             [vagem 

Logarithms  of  Numbers. 

No.  G400 7000.                      Log.  B0618 84510. 

No. 

64o 
64 1 
642 
643 
644 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

80618 
80686 
80754 
80821 
80889 

80625 
80693 
80760 
80828 
80895 

8o632 
80699 
80767 
8o835 
80902 

80688 
80706 
80774 
80841 
80909 

80645 
807 1 3 
80781 
80848 
80916 

8o652 
80720 
80787 
8o855 
80922 

80659 
80726 
80794 
80862 
80929 

8o665 
80788 
80801 
80868 
80986 

80672 
80740 
80808 
80875 
80943 

80679 
80747 
80814 
80S82 
80949 

4 

I 
2 
3 
4 
5 
6 

7 
8 

9 

I 

I 
a 
3 

645 
646 
647 
648 
649 

80956 
81023 
81090 
8ii58 
81224 

80963 
8io3o 
81097 
81164 
8i23i 

80969 
81037 
81104 
81171 
8i238 

80976 
81043 
81111 
81178 
81245 

80988 
8io5o 
81117 
81184 
8i25i 

810D7 
81124 
81191 
81258 

80996 
81064 
8ii3i 
81198 
81265 

81008 
81070 
81187 
81204 
81271 

81010 
81077 
81144 
81211 
81278 

81017 
81084 
8ii5i 
81218 
81285 

4 
4 
5 
6 
6 

65o 
65 1 
652 
653 
654 

81291 
8i358 
81425 
8 1 491 
8i558 

81298 
8i365 
8i43i 
81498 
81 564 

8i3o5 
81871 
81 438 
8i5o5 
81571 

8i3ii 
81878 
81445 
8i5ii 
81578 

8i3i8 
8i3S5 
8i45i 
8i5i8 
8 1 584 

8i325 
81891 
8i458 
8i525 
81591 

81881 
81398 
8i465 
8i53i 
81598 

81888 
8i4o5 
81471 
8i58S 
8 1 6o4 

81845 
8i4ii 
81478 
81544 
81611 

8i35i 
8i4i8 
81 485 
8i55i 
81617 

655 
656 
657 
658 
659 

81624 
81690 
81757 
81823 
81889 

8i63i 
81697 
81763 
81829 
81895 

81637 
81704 
81770 
8i836 
81902 

81644 
81710 
81776 
81842 
81908 

8i65i 
81717 
81788 
81849 
8191S 

81657 
81728 
81790 
8 1 856 
81921 

81664 
81730 
81796 
81862 
81928 

81671 
81787 
81808 
81869 
81985 

81677 
81748 
81809 
81875 
81941 

81684 
81750 
81816 
81882 
81948 

66o 
66 1 
662 
663 
664 

81954 
82020 
82086 
82i5i 
82217 

81961 
82027 
82092 
821 58 
82223 

81968 
82033 
82099 
82164 
82230 

81974 
82040 
82105 
82171 
82286 

81981 
82046 
82112 

82178 
82243 

82808 
82878 
82489 
82504 
82569 

81987 
82053 
821 19 
82184 
82249 

81994 
82060 
82125 
82191 
82256 

82000 
82066 
82182 
82197 
82263 

82007 
82078 
82i38 
82204 
82269 

82014 
82079 
82145 
82210 
82276 

665 
666 
667 
668 
669 

82282 
82347 
824i3 
82478 
82543 

82289 
82354 
82419 
82484 
82549 

82295 
82360 
82426 
82491 
82556 

82802 
82867 
82432 
82497 
82562 

82815 
82880 
82445 
82510 
82575 

82821 
82887 
82452 
82517 
82582 

82828 
82898 
82458 
82528 
82588 

8,2884 
82400 
82465 
82580 
82595 

82841 
82406 
82471 
82586 
82601 

670 
671 
672 
673 
674 
675 
676 
677 
678 

679 
680 
681 
682 
683 
684 

82607 
82672 
82737 
82802 
82866 

82614 
82679 
82743 
82808 
82872 

82620 
82685 
82750 
82814 
82879 

82627 
82692 
82756 
82821 
828S5 
82950 
88014 
83078 
83 1 42 
88206 

82638 
82698 
82763 
82827 
82892 

82640 
82705 
82769 
82884 
82898 

82646 
82711 
82776 
82840 
82905 

82653 
82718 
82782 
82847 
82911 

82659 
82724 
82789 
82853 
82918 

82666 
82780 
82795 
82860 
82924 

82930 
82995 
83o59 
83 1 23 
83187 

82937 
83ooi 
83o65 
83 1 29 
83193 

82943 
83oo8 
83o72 
83i36 
83200 

82956 
88020 
83o85 
88149 
832i3 

82968 
88027 
88091 
88i55 
88219 

82969 
88o83 
88097 
83i6i 
88225 

82975 
83o4o 
83 1 04 
88168 
88282 

82982 
83o46 
83iio 
88174 
88288 

82988 
88o52 
88117 
83i8i 
88245 

8325i 
833 1 5 
83378 
83442 
835o6 

83257 
83321 
83385 
83448 
83512 

83264 
83327 
83391 
834D5 
835i8 

88270 
88884 
88898 
88461 
83525 

88276 
83340 
834o4 
83467 
8353i 

88288 
83347 
83410 
88474 
83537 

88289 
83853 
83417 
83480 
83544 

88296 
88859 
88428 
83487 
83550 

88802 
83366 
88429 
83498 
83556 

833o8 
88872 
83436 
88499 
83563 

685 
686 
687 
688 
689 

83569 
83632 
83696 

83:59 
83822 

83575 
83639 
83702 
83765 
83828 

83582 
83645 
83708 
83771 
83835 

83588 
8365 1 
88715 
88778 
88841 

83594 
83658 
83721 
83784 
83847 

88601 
88664 
88727 
88790 
83858 

88607 
88670 
83734 
83797 
83S6o 

88618 
88677 
88740 
88808 
83866 

88620 
83683 
88746 
88809 
88872 

83626 
88689 
83753 
83Si6 
83879 

I 
2 
3 
4 
5 
6 
7 
8 

9 

0 

I 
I 
a 
2 

690 
691 
692 
693 
694 

83885 
88948 
84oii 
84073 
84 1 36 

83891 
83954 
84017 
84080 
84 1 42 

88897 
88960 
84028 
84o86 
84 1 48 

88904 
88967 
84029 
84092 
84 1 55 

88910 
83978 
84o36 
84098 
84i6i 

88916 
83979 
84042 
84io5 
84167 

88928 
88985 
84o48 
84111 
84173 
84236 
84298 
8436i 
84423 
84485 

88929 
88992 
84o55 

84117 
84 180 

88935 
8899S 
84061 
84123 
84186 

88942 
84oo4 
84067 
84i3o 
84192 

3 
4 
4 
5 
5 

695 
656 
697 
698 
699 

No. 

84198 
84261 
84323 
84386 
84448 

842o5 
84267 
8433o 
84392 
84454 

84211 
84273 
84336 
84398 
8446o 

84217 
84280 
84842 
844o4 
84466 

84228 
84286 
84848 
844 10 
84473 

84280 
84292 
84354 
84417 
84479 

84242 
843o5 
84867 
84429 
84491 

84248 
848 11 
84373 
84435 
84497 

84255 
843 1 7 

84379 
84442 
845o4 

0 

1 

2  1  3 

4 

5 

6 

7 

8 

9 

Page  180]                  TABLE  A 

Logaritlmiri  of 

AV'i. 

iN  umbers. 

Nn  70nf)      7(11)0                             I  nir  PiVAO 

CQnQI 

No. 

0 

1 

2 

3 

4 

5 

6  !  7 

8 

9 

700 
701 
702 
708 
704 

7o5 
706 
707 
708 

710 
711 
712 
7i3 
714 
'7!  5 
716 
717 
7.8 
719 
720 
721 
722 
728 
724 
725 
726 
727 
728 
729 

780 
73 1 
782 
788 
734 
735 
786 

737 
788 
789 

74o 
74 1 
742 
743 
744 

845io 
84572 
84634 
84696 
84757 
84819 
84SSo 
8^<942 

85o65 

845i6 

84578 
84640 
84702 
84768 

84522 
84584 
84646 
84708 
84770 

84528 
84590 
84652 
84714 
84776 

84535 
84597 
84658 
84720 
84782 

84541 
84608 
84665 
84726 
84788 

84547 
84609 
84671 
84733 
84794 
84856 
84917 
84979 
85o4o 
85ioi 

84558 
846 1 5 
84677 

84739 
84800 

84559 

8462  X 
84688 
84745 
84807 

84566 
84628 
84689 
8475 1 
848 1 3 

7 

I 
2 
3 

4 
5 
6 

7 
8 

9 

I 

2 
3 

84825 
84887 
84948 
85009 
85o7i 

84881 
84898 
84954 
85oi6 
85o77 

84887 
84899 
84960 

85o22 

85o88 

84844 
84905 
84967 
85o28 
85089 

84850 
8491 1 
84973 
85o84 
85095 

84862 
84924 
84985 
85o46 
85io7 

84868 
84980 
84991 
85o52 
85ii4 
85i75 
85236 
85297 
85858 
854i8 

84S74 
84986 
84997 
85o58 
85i2o 

85i8i 
85242 
853o8 
85364 
85425 

4 
4 
5 
6 
6 

85i26 
85187 
85248 
85309 
85370 

85i82 
85198 
85254 
858i5 
85376 

85i88 
85 1 99 
85260 
85821 
85382 

85i44 
85205 
85266 
85827 
85888 

85i5o 
852II 
85272 
85388 
S5394 

85i56 
85217 
85278 
85389 
854oo 

85i63 
85224 
85285 
85345 
854o6 

85169 
8523o 
85291 
85352 
85412 

8543 1 
85491 
85552 
856i2 
85673 

85487 
85497 
85558 
856i8 
85679 

85443 
855o8 
85564 
85625 
85685 

85449 
855o9 
85570 
8  568 1 
85691 

85455 
855i6 
85576 
85637 
85697 

8  546 1 
85522 
85582 
85648 
85708 

85467 
85528 
85588 
85649 
85709 

85473 
85534 
85594 
856d5 
857i5 

85479 
85540 
85600 
8566i 
85721 

85485 
85546 
856o6 
85667 
85727 

85788 
85794 
85854 
85914 
85974 

85789 
85800 
85860 
85920 
85980 

85745 
858o6 
85866 
85926 
85986 

8575i 
858i2 
85872 
85982 
85992 

85757 
858i8 
85878 
85988 
85998 

85768 
85824 
85884 
85944 
86004 

85769 
8583o 
85890 
85950 
86010 

85775 
85836 
85896 
85956 
86016 

85781 
85842 
85902 
85962 
86022 

85788 
85848 
85908 
85968 
86028 

86084 
86094 
86i58 
86213 
86278 

86o4o 
86100 
86159 
86219 
86279 

86o46 
86106 
86i65 
86225 
86285 

86o52 
86112 
86171 
86231 
86291 

86o58 
86118 
86177 
86287 
86297 

86856 
864 1 5 
86475 
86584 
86598 

86064 
86124 
86188 
86243 
86808 

86070 
86i3o 
86189 
86249 
86808 

86076 
86186 
86195 
86255 
86814 

86874 
86483 
86498 
86552 
8661 1 

86082 
86i4i 
86201 
86261 
86820 

S6088 
86147 
86207 
86267 
86826 

r 

I 
2 
3 

4 
5 

6 

7 
8 

9 

1 

1 
I 
2 
2 

86332 
86892 
8645 1 
865 10 
86570 

86838 
86898 
86457 
865 1 6 
86576 

86344 
86404 
86468 
86522 
8658i 

8664 1 
86700 
86759 
86817 
86876 

8685o 
864 10 
86469 
86528 
86587 

86862 
86421 
8648 1 
86540 
86599 

86368 
86427 
864S7 
86546 
866o5 

86880 
86439 
86499 
86558 
86617 

86386 
86445 
865o4,- 
86564 
86628 

86682 
86741 
86S00 
86859 
86917 

3 
4 
4 
5 
5 

86629 
86688 

86747 
86806 
86864 

86685 
86694 
86753 
86812 
86870 

86646 
86705 
86764 
86823 
86882 

86652 
8671 1 
86770 
86829 
86888 

86658 
86717 
86776 
86885 
S6894 

86664 
86728 
86782 
8684 1 
86900 

86670 
86729 
86788 
86847 
86906 

86676 
86735 
86794 
86858 
8691 1 

86928 
86982 
87040 
87099 
87.57 

86929 
86988 
87046 
87105 
87168 

86935 
86994 
87052 
87111 
87169 

86941 
86999 
87058 
87116 
87175 

86947 
87005 
87064 
87122 
87181 

86953 
8701 1 
87070 
87128 
87186 

86958 
87017 
87075 
87184 
87192 

86964 
87028 
87081 
87140 
87198 

86970 
87029 
87087 
87146 
87204 

86976 
87035 
87098 
87151 
87210 

745 
746 
747 
748 
749 
75o 
75i 
752 
753 
754 

87216 
87274 
87882 
87890 
87448 

87221 
87280 
87888 
87896 
87454 

87227 
87286 
87844 
87402 
87460 

87233 
87291 
87849 
87408 
87466 

87289 
87297 
87355 
87418 
87471 
87529 
87587 
87645 
87708 
87760 

87245 
87808 
87861 
87419 
87477 
87535 
87598 
87651 
87708 
87766 

87251 
87809 
87867 
87425 
87488 

87256 
87815 
87878 
87431 
87489 

87262 
87820 
8787-9 
87437 
87495 

87268 
87826 
87884 
87442 
87500 

87506 
87564 
87622 
87679 
87787 

87512 
87570 
87628 
87685 
87743 

87518 
87576 
87688 
87691 
87749 

87528 
87581 
87689 
87697 
87754 

87541 

87599 
87656 

87714 
87772 

87547 
87604 
87662 
S7720 

87777 

87552 
87610 
87668 
87726 
877S8 

87558 
87616 
87674 
87731 
87789 

I 

I 
2 
3 
4 
5 
6 

7 
8 

9 

) 

X 

I 
2 
2 

755 
756 

757 
758 

759 

87795 
878^2 
87910 
8^967 
8S024 

87800 
87858 
87915 
87973 
88o3o 

87S06 
87864 
87921 
87978 
88o36 

87812 
87869 
87027 
87984 
8804 1 

87818 
87875 
87988 
87990 
88047 

87828 
87881 
87988 
87996 
88o53 

87829 
87887 
87944 
88001 
88o5S 

87885 
87802 
87950 
88007 
88064 

87841 
87898 
87955 
88018 
88070 

87846 
87904 
87961 
88018 
88076 

3 
3 
4 
4 
5 

No. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

TABLE  XXVI.            [P^g«  '81 

Logarithms  of  Numbers. 

i\o.  7(Ji)0— 

8200.                      Log.  88081     91381. 

No. 

0 

1 

2 

3 

4 

5 

G 

7 

8 

9 

760 

88081  88087 

88093 

88098 

88104 

88110 

88116 

88i2i 

815127 

88i33 

761 

88 I 38  88144 

88i5o 

881 56 

88161 

88167 

88173 

88178 

88184 

88190 

762 

88195  88201 

8S207 

88213 

88218 

88224 

88230 

88235 

88241 

8S247 

763 

88252  88258 

88264 

88270 

88275 

88281 

88287 

88292 

88298 

88355 
88412 

8S3o4 

764 

883091  883 1 5 

88321 

88326 

88332 

88338 

88343 

88349 

8836o 

7()5 

88366 

8S372 

88377 

88383 

88389 

88395 

88400 

884o6 

88417 

766 

88423 

S8429 

88434 

88440 

88446 

8845i 

88457 

88463 

88468 

88474 

767 

88480 

884-85 

88491 

88497 

885o2 

8S5o8 

885 1 3 

88519 

88525 

88530 

768 

88536 

88542 

88547 

88553 

88559 

88564 

88570 

885-6 

8858i 

88587 

769 

88093 

8S598 

8S604 

88610 

886i5 

88621 

88627 

88632 

88638 

88643 

770 

88649 

88655 

88660 

88666 

88672 

8S677 

88683 

886S9 

88694 

88700 

771 

8S705 

8871 1 

88717 

88722 

88728 

88734 

88739 

88745 

88750 

88756 

772 

88762 

88767 

88773 

88779 

88784 

88790 

88795 

88801 

88807 

88812 

773 

88818 

8S824 

88829 

88835 

88840 

88846 

88852 

88857 

88863 

88868 

774 

88874 

88880 

88885 

88891 

88897 

88902 

88908 

88913 

88919 

88925 

775 

88930 

88936 

88941 

88947 

88953 

88958 

88964 

88969 

88975 

88981 

776 

88986 

88992 

88997 

89003 

89009 

89014 

89020 

8qo25 

89031 

89087 

777 

89042 

89048 

89053 

89059 

89064 

89070 

89076 

89081 

89087 

89092 

77S 

89098 

89104 

89109 

89115 

89120 

89126 

89131 

89137 

89143 

89148 

779 

89154 

89159 

89165 

89170 

89176 

89182 

89,87 

89193 
89248 

89198 

89204 
89260 

780 

89209 

89215 

89221 

89226 

89232 

89237 

89243 

89254 

781 

89265 

89271 

89276 

89282 

89287 

89293 

89298 

89304 

89310 

89315 

782 

89321 

89326 

89332 

89337 

89343 

89348 

89354 

89360 

89365 

89371 

783 

89376 

89382 

893S7 

89393 

89398 

89404 

89409 

89415 

80421 

89426 

7S4 

89432 

89437 

89443 

89448 

89454 

89459. 
89515 

. 89465 

89470 

89476 
89531 

89481 

785 

89487 

89492 

89498 

89504 

89509 

89520 

89526 

89537 

786 

89542 

89548 

89553 

89559 

89564 

89570 

89575 

89581 

89586 

89592 

787 

89597 

89603 

89609 

89614 

89620 

89625 

89631 

89636 

89642 

89647 

788 

89653 

89658 

89664 

89669 

89675 

89680 

89686 

89691 

89697 

89702 

789 
790 

89708 

89713 

89719 

89724 

89730 

8973:) 

89741 

89746 

89752 

89757 
89812 

89763 

89768 

89774 

89779 

89785 

89790 

89796 

89801 

89807 

791 

89S18 

89823 

89829 

89834 

89840 

89845 

S9851 

89856 

89862 

89867 

792 

89873 

89878 

89883 

89889 

89894 

89900 

89905 

89911 

89916 

89922 

79:^ 

89927 

89933 

89938 

89944 

89949 

89955 

89960 

89966 

89971 

89977 

794 
795 

89982 

89988 

89993 

89998 

90004 

90009 

90015 

90020 

90026 

90081 

90037 

90042 

90048 

90053 

90059 

90064 

90069 

90075 

90080 

90086 

796 

90091 

90097 

90102 

90108 

90113 

90J19 

90124 

90129 

90135 

90140 

797 

90146 

901 5i 

90157 

90162 

90168 

90173 

90179 

90184 

90189 

90195 

798 

90200 

90206 

902 II 

90217 

90222 

90227 

90233 

90238 

90244 

90249 

799 

90255 

90260 

90266 

90271 

90276 

90282 

90287 

90293 

90298 
90352 

90804 

800 

90309 

903 1 4 

90320 

90325 

9033 1 

903  36 

90342 

90347 

908  58 

801 

9o363 

90369 

90374 

90380 

9o385 

90390 

90396 

90401 

90407 

904 1 2 

802 

90417 

90423 

90428 

90434 

90439 

90445 

90450 

90455 

9046 1 

90466 

8o3 

90472 

90477 

90482 

90488 

90493 

90499 

9o5o4 

90509 

905 1 5 

90520 

804 
8o5 

90526 

9053 1 

90536 

90542 

90547 

90553 

90558 

9o563 

90569 

90574 

9o58o 

9o585 

90590 

90596 

90601 

90607 

90612 

90617 

90623 

90628 

806 

90634 

90639 

90644 

90650 

90655 

90660 

90666 

90671 

90677 

90682 

807 

90687 

90693 

90698 

90703 

90709 

90714 

90720 

90725 

90780 

90786 

808 

90741 

90747 

90752 

90757 

90763 

90768 

90773 

90779 

90784 

90789 

809 
810 

90795 

90800 

90806 

908 II 

90816 

90S  2  2 

90827 

90832 

90838 

90843 

90849 

90854 

90859 

90865 

90870 

90875 

9088 1 

90886 

90891 

90897 

8n 

90902 

90907 

90913 

90918 

90924 

90929 

90934 

90940 

90945 

90950 

812 

90956 

90961 

90966 

90972 

9«977 

90982 

90988 

90993 

90998 

91004 

8i3 

91009 

91014 

91020 

91025 

9io3o 

9io36 

91041 

9 1  o46 

91052 

91057 

8i4 

91062 

91068 

91073 

91078 

91084 

91089 

91094 

91100 

91105 

91110 

8i5 

911 16 

91121 

91126 

91132 

91137 

91142 

91148 

91153 

91 158  '  91 164  1 

816 

91 169 

91 174 

91 180 

91185 

91190 

91 196 

91201 

91206 

91212 

91217 

bl7 

91222 

91228 

91233 

91238 

91243 

91249 

91254 

91259 

91265 

91270 

818 

91275 

91281 

91286 

91291 

91297 

9i3o2 

91307 

9i3i2 

9i3i8 

91828 

819 

9132S 

91334 

91339 

91344 

9i35o 

91355 

9i36o 

91365 

91371 

91876 

No. 

b 

1 

2 

3 

4 

5 

6 

7 

8 

9 

2  1 

8  2 

4  2 

5  3 

6  3 

7  4 

8  4 

9  5 


Page  1 

82]             TABLE  XXVi. 

Logarithms  of  Numbers. 

No. 

'^"On     880( 

).                      Log.  91 381- 

0 

1448. 

No. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

820 
821 
822 
823 
824 

9i38i 
91434 
91487 
91540 
91593 

91387 
91440 
91492 
91545 
91598 

91392 
91445 
91498 
9i55i 
91603 

91397 
91450 
9i5o3 
91556 
91609 

9i4o3 
91455 
9i5o8 
9i56i 
91614 

91408 
91461 
9i5i4 
91 566 
91619 

91413 
91466 
91519 
91572 
91624 

91418 
91471 
91524 
9x577 
9i63o 

9x424 

91477 
91529 
9x582 
91635 

9x429 
91482 
9x535 
91587 
9x640 

( 

1 
2 
3 
4 
5 
6 

7 
8 

9 

I 
1 
2 
2 

825 
826 
827 
828 
829 

91645 
91698 
91751 
91803 
91855 

9i65i 

91703 
91756 
91808 
91861 

9i656 
91709 
91761 
91814 
91866 

91661 

91714 
91766 
91819 
91871 

91666 
91719 

91772 
91824 
91876 

91672 
91724 

91777 
91829 
91882 

91677 
91730 
91782 
91834 
91887 

91939 
91991 
92044 
92096 
92148 

91682 
91735 
91787 
91840 
91892 

91944 
91997 
92049 
9210X 
92x53 

91687 
91740 
9x793 
91845 
91897 

91693 
91745 
91798 
9x85o 
91903 

3 

4 
4 
5 
5 

83o 
83i 
832 
833 
834 

91908 
91960 
92012 
92065 
92117 

9' 9'^ 
91965 

92018 

92070 

92122 

91918 
91971 
92023 
92075 
92127 

91924 
91976 
92028 
92080 
92132 

91929 
91981 
92033 
920S5 
92137 

91934 
91986 
92o38 
92091 
92143 

91950 
92002 
92054 
92106 
92x58 

91955 
92007 
92059 
92 1 II 
92x63 

835 
836 

837 
838 
839 

84o 
84 1 
842 
843 
844 

92169 
92221 

92273 
92324 
92376 

92174 
92226 
92278 
92330 
92381 

92179 
92231 
92283 
92335 
92387 

92184 
92236 
92288 
92340 
92392 

92443 
92495 
92547 
92598 
92650 

92189 
92241 
92293 
92345 
92397 

92195 
92247 
92298 
92350 
92402 

92200 
92252 
92304 
92355 
92407 

92205 
92257 
92309 
9236x 
92412 

922x0 
92262 
923x4 
92366 
92418 

92215 
92267 
92319 
92371 
92423 

92428 
92480 
92531 
92583 
92634 

92433 
92485 
92536 
92588 
92639 

92438 
92490 
92542 
92593 
92645 

92449 
92500 
92552 
92603 
92655 

92454 
92  5o5 
92557 
92609 
92660 

92459 
92511 
92562 
92614 
92665 

92464 
92516 
92567 
92619 
92670 

92469 
9252X 
92572 
92624 
92675 

92474 
92526 
92578 
92629 
92681 

845 
846 

847 
848 

849 
85o 
85i 
852 
853 
854 
855 
856 
857 
858 
859 

92686 
92737 
92788 
92840 
92891 

92691 
92742 
92793 
92845 
92896 

92696 

92747 
92799 
92850 
92901 

92701 
92752 
92804 
92855 
92906 

92957 
93008 
93059 
93x10 
93161 

92706 
92758 
92809 
92860 
92911 

92711 
92763 
92814 
92865 
92916 

92716 
92768 
92819 
92870 
92921 

92722 
9*773 
92824 
92875 
92927 

92727 
92778 
92829 
9288  X 
92932 

92732 
92783 
92834 
92886 
92937 

X 

2 
3 

4 
5 
6 
7 
8 

9 

X 

2 

9294^ 
92993 
93044 
93095 
93 1 46 

92947 
92998 
93049 
93100 
93i5i 

92932 
93oo3 
93o54 
93io5 
93 1 56 

92962 
93oi3 
93064 
931 1 5 
93166 

92967 
93018 
93069 
93120 
93171 

92973 
93024 
93075 
93x25 
93176 

92978 
93029 
93080 
931 3 1 
93181 

92983 
93o34 
93o85 
93x36 
93x86 

92988 
93039 
93090 
93i4i 
93192 

3 
3 
4 
4 
5 

93197 

93247 
93298 
93349 
93399 

93202 
93252 
933o3 
93354 
93404 

93207 
93258 
93308 
93359 
93409 

93212 
93263 
933i3 
93364 
93414 

93217 
93268 
93318 
93369 
93420 

93222 

93273 
93323 
93374 
93425 

93227 
93278 
93328 

93379 
93430 

93232 
93283 
93334 
93384 
93435 

93237 
93288 
93339 
93389 
93440 

93242 
93293 
93344 
93394 
93445 

860 
861 
862 
863 
864 
865 
866 
867 
868 
869 
870 
871 
872 
873 
874 

93450 
935f)o 
93551 
93601 
9365i 

93455 
935o5 
93556 
93606 
93656 

93460 
93510 
93561 
9361 1 
93661 

93465 
935i5 
93566 
93616 
93666 

93470 
93520 
93571 
93621 
93671 

93475 
93526 
93576 
93626 
93676 

93480 
93531 
93581 
9363 1 
93682 

93435 
93536 
93586 
93636 
93687 

93490 
9354X 

93591 
9364  X 
93692 

93495 
93546 
93596 
93646 
93697 

93-02 
93752 
93802 
93852 
93902 

93707 

93757 
93807 
93857 
93907 

93712 
93762 
93812 
93862 
93912 

93717 
93767 
93817 
93867 
93917 

93722 
93772 
93822 
93872 
93922 

93727 

93777 
93827 

93S77 
93927- 

93732 
93782 
93832 
93882 
93932 

93737 
93787 
93837 
93887 
93937 

93742 
93792 
93842 
93892 
93942 

93747 
93797 
93847 
93897 

93947 

93952 

94002 
94o52 
94101 
94i5i 

93957 
94007 
94057 
94 1 06 
94 1 56 

93962 
94012 
94062 
941 1 1 
94161 

93967 
94017 
94067 
941 16 
94166 

93972 
94022 
94072 
941 2 1 
94171 

93977 
94027 
94077 
94 1 26 
94176 

93982 
94o32 
94082 
94i3i 
94181 

93987 
94037 
94086 
94 1 36 
94186 

93992 
94042 
94091 
94i4i 
9/' 1 91 

93997 
94047 
94096 
94x46 
94196 

i 

I 
2 
3 
4 

5 
6 

7 
8 
9 

1 

0 
I 
I 

875 
876 
877 
878 
879 

No. 

94201 
94250 
94300 
94349 
94399 

0 

94206 
94255 
943o5 
94354 
944o4 

1 

942 1 1 
94260 
94310 
94359 
94409 

94216 
94265 
943 1 5 
94364 
94414 

94221 
94270 
94320 
94369 
94419 

94226 
94275 
94325 
94374 
94424 

94231 
94280 
94330 
94379 
94429 

6 

94236 
94285 
94335 
94384 
94433 

y4240 
94290 
94340 
94389 

94438 

94245 
94295 
94345 
94394 
94443 

2 

2 
3 
3 
4 

2 

3 

4 

5 

7 

8 

9 

TABLE  XXVI.            [P'^ge  183 

Logarithms  of  Numbers. 

• 

Nn 

Q^no        CfAQ 

0.                     Log.  9444.8 97313. 

No. 

880 
88  r 
882 
883 
884 
885 
886 
887 
883 
889 

890 
891 
892 
893 
894 
895 
896 
897 
898 

899 
900 
901 
902 
903 
904 
905 
906 
907 
908 
909 
910 
911 
912 
9i3 
914 
915 
916 
917 
918 
919 

920 
921 
922 
923 
924 
926 
926 
927 
928 
929 
980 

932 
933 
934 
935 
936 
937 
938 
939 
No. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

94488 
94537 
94586 
94635 
94685 

9 

94448 
94498 
94547 
94596 
94646 

94453 

945o3 
94662 
94601 
94660 

94468 
94507 
94557 
94606 
94666 

94463 
94612 
94662 
94611 
94660 

94468 
94517 
94567 
94616 
94665 

94473 
94622 
94571 
94621 
94670 

94478 
94627 
94676 
94626 
94676 

94483 
94632 
94581 
9463o 
94680 

94493 
94542 
94591 
94640 
94689 

t 

1 
2 
3 
4 
5 
6 

7 
8 

9 

) 

I 
I 
2 
•> 

94694 
94743 
94792 
94841 
94890 

94699 
94748 
94797 
94846 
94895 

94704 
94753 
94802 
9485: 
94900 

94709 
94768 
94807 
94866 
94906 

94714 
94763 
94812 
94861 
94910 

94719 
94768 

94817 
94866 
94916 

94724 
94773 
94822 
94871 
94919 
94968 
96017 
96066 
96114 
96163 

94729 
94778 
94827 
94S76 
94924 

94973 
96022 
96071 
96119 
96168 

94734 
94783 
94832 
94880 
94929 

94733 

94787 
94836 
94885 
94934 

3 
3 

4 
4 
5 

94939 
94988 
96036 
96085 
96134 

94944 
94993 
96041 
96090 
96139 

94949 
94998 
96046 
96096 
96143 

94954 
96002 
96061 
96100 
96148 

94969 
96007 
96066 
96106 
96163 

94963 
96012 
96061 
96109 
96168 

94978 
96027 
96076 
96124 
96173 

94983 
96032 
96080 
96129 
96177 

96182 
9623  [ 
96279 
95328 
95376 

95424 
95472 
96621 
96669 
96617 

96187 
96236 
96284 
95332 
96381 

96192 
96240 
96289 
95337 
95386 

96197 
96245 
96294 
95342 
95390 

96202 
96260 
96299 

95347 
96396 

96207 
96266 
963o3 
96352 

96400 

96211 
96260 
96308 
95357 
96406 

96216 
96266 
963i3 
96361 
96410 

96221 
96270 
95318 
96366 
96416 

96226 
96274 
96323 
95371 
95419 

95429 

95477 
95526 
96674 
96622 

95434 
95482 
95530 
96678 
96626 

96439 
95487 
95535 
96683 
9563 1 

96444 
96492 
96640 
96688 
95636 

95448 
95497 
95546 
96693 
96641 

95453 
96601 
96660 
96698 
96646 

96458 
96606 
96554 
96602 
96660 

96463 
96611 
95569 
96607 
95666 

96468 
96616 
96664 
96612 
96660 

96666 
96713 
96761 
96809 
96866 

96670 
96718 
96766 
96813 
96861 

96674 
96722 
96770 
96818 
96866 

96679 
96727 
96776 
96823 
96871 

96684 
96732 
96780 
96828 
96876 

96689 
96737 
96786 
96832 
96880 

96694 
96742 
96789 
96837 
96886 

96698 
96746 
96794 
96842 
96890 

96703 
96761 
96799 
96847 
96896 

96708 
96766 
96804 
96862 
96899 

96904 
96962 
96999 
96047 
96096 

96909 
96967 
96004 
96062 
96099 

96914 
95961 
96009 
96067 
96104 

96918 
96966 
96014 
96061 
96109 

96923 
96971 
96019 
96066 
96114 

96928 
96976 
96023 
96071 
961 18 

96933 
96980 
96028 
96076 
96123 

96933 
96986 
96033 
96080 
96128 

96942 
96990 
96033 
96086 
96133 

96947 
96996 
96042 
96090 
96137 

96142 
96190 

96237 
96284 
96332 

96379" 
96426 
96473 
96620 
96667 

96147 
96194 
96242 
96289 
96336 

96384 
96431 
96478 
96626 
96672 

96162 
96199 
96246 
96294 
96341 

96166 
96204 
96261 
96298 
96346 

96161 
96209 
96266 
96303 
96350 

96166 
96213 
96261 
96308 
96365 

96171 
96218 
96266 
963 1 3 
96360 

96176 
96223 
96270 
96317 
96365 

96180 
96227 
96275 
96322 
96369 

96186 
96232 
96280 
96327 
96374 

96388 
0643  6 
96483 
96530 
96677 

96393 
96440 
96487 
96534 
96681 

96628 
96676 
96722 
96769 
96816 

96398 
96445 
96492 
96539 
96686 

96402 
96460 
96497 
96644 
96691 

96407 
96454 
96601 
96648 
96696 

96412 
96459 
96606 
96663 
96600 

96417 
96464 
96611 
96668 
96606 

96421 
96468 
96616 
96662 
96609 

96614 
9666  r 
96708 
96765 
96802 

96619 
96666 
96713 
96769 
96806 

96624 
96670 
96717 
96764 
96811 

96633 
96680 

96727 
96774 
96820 

96638 
96686 
96731 
96778 
96825 

96642 
96689 
96736 
96783 
96830 

96876 
96923 
96970 
97016 
97063 

96647 
96694 
96741 
96788 
96834 

96662 
96699 
96745 
96792 
96839 

96666 
96703 
96760 
96797 
96844 

I 
2 
3 
4 
5 
5 
7 
8 

9 

i 

0 
I 
I 
7 

96848 
96896 
96942 
96988 
97035 

96S53 
96900 
96946 
96993 
97039 

96868 
96904 
96961 
96997 
97044 
97090 
97137 
97183 
97230 
97276 

96862 
96909 
96966 
97002 
97049 

96867 
96914 
96960 
97007 
97063 

96S72 
96918 
96966 
97011 
97068 

96881 
96928 
96974 
97021 
97067 

96886 
96932 
96979 
97025 
97072 

9fj890 
96937 
96984 
97o3o 
97077 

2 
2 
3 
3 
4 

9708 1 
97128 
97174 
97220 
97267 

97086 
97132 

97179 
97225 
97271 

97096 
97142 
97188 
97234 
97280 

97100 
97146 
97192 
97239 
97286 

97104 
97161 
97197 
97243 
97290 

97109 
97155 
97202 
97248 
97294 

971 14 
97160 
97206 
97253 
97299 

971 18 
97165 
97211 
97267 
973o4 

97123 
97169 
97216 
97262 
97308 

0 

1 

2 

3 

4 

5 

G 

7 

8 

9 

^^sem                                 TABLE  XXVI. 

Logarithms  of  Numbers. 

IVf,  Oinri      lOflOO                          T.n"'  Q7"^ir?- 

0 

J996. 

No. 
940 
941 
942 
943 
944 
940 
946 
947 
948 

949 
950 
95 1 
952 
953 
954 
955 
956 
957 
958 
959 
960 
961 
962 
963 
964 
965 
966 
967 
968 
969 
970 
97 1 
972 
973 
974 

975 
976 

977 
978 

979 
980 
981 
982 
983 
984 
985 
986 
987 
988 
989 

990 
991 
992 
993 
994 
995 
996 

997 
998 

999 
No. 

0 

1 

2 

3 

4 

5 

6 

7    8 

9 

973 1 3 
C7359 
974o5 
974  5 1 
97497 

97317 
97364 
97410 
97456 
97502 

97322 
97368 
974 1 4 
97460 
97506 

97327 
97373 
97419 
97465 
975 1 1 

97331 

97377 
97424 
97470 
97516 

97336 
97382 
97428 
97474 
975- 

97340 
97387 
97433 
97479 
97525 

97345 
97391 
97437 
97483 
97529 

97350 
97396 
97442 
97488 
97534 

97354 
97400 
97447 
97493 
97539 

5 

1 
2 
3 

4 
5 
6 

7 
8 

9 

I 
I 
2 
2 

97543 
97589 
976,35 
97681 
97727 

97548 
97594 
97640 
97685 
9773: 

97552 
97598 
97644 
97690 
97736 

97557 
97603 
97649 
97695 
97740 

97562 
97607 
97653 

97699 
97745 

97566  ■ 

97612 

97658 

97704 

97749 

97571 
97617 
97663 
97708 
97754 

97575 
97621 
97667 
97713 

97759 

97580 
97626 
97672 

97717 
97763 

97585 
97630 
97676 
97722 
97768 

3 
3 

4 
4 
5 

97772 
97818 
97864 
97909 
97955 

97777 
97823 
9786^1 
97914 
97959 

97782 
97827 
97873 
97918 
97964 

97786 
97832 
97877 
97923 
97968 

97791 
97836 
97882 
97928 
97973 

97795 
97841 
978S6 
97932 
97978 

97800 
97845 
97891 

97937 
97982 

97804 
97850 
97896 

97941 
97987 

97809 
97855 
97900 
97946 
97991 

97813 
97859 

97903 
97950 
97996 

98000 
98046 
98091 
98137 
98182 

98005 
98050 
98096 
98141 
98186 

98009 
98055 
98100 
98146 
98191 

98014 
9S059 
98105 
98150 
98195 

98')i9 
98  064 
98109 
98155 
98200 

98023 
98068 
98114 
98 1 59 
98204 

98028 
98073 
98118 
98164 
98209 

98032 
9S078 
98123 
98168 
98214 

98037 
98082 
98127 
98173 
98218 

98041 
98087 
98132 
98177 
98223 

98227 
98272 
983:8 
98363 
9840S 
98453 
98498 
9S543 
98588 
98632 

98232 
98277 
98322 
98367 
98412 

98457 
98502 

98547 
98592 
98637 

98236 
9S2S1 
9S327 
98372 
98417 

98241 
98286 
9833 1 
98376 
98421 

98245 
98290 
98336 
98381 
98426 

98250 
98295 
98340 
98385 
98430 

98254 
98299 
98345 
98390 
98435 

98259 
98304 
98349 
98394 
98439 

98263 
9830S 
98354 
98399 
98444 

98268 
983 1 3 
98353 
98403 
98448 

98462 
98507 
98552 
98597 
98641 

98466 
98511 
98556 
98601 
98646 

98471 
98516 
98561 
98605 
98650 

98475 
98520 
98565 
98610 
98655 

98480 
98525 
98570 
98614 
98659 

984S4 
98529 
98574 
986:9 
98664 

98489 
98534 
98579 
98623 
98668 

98493 
98538 
98583 
9S628 
9S673 

98677 
9S722 
98767 
9S811 
98856 

986S2 
98726 
98771 
98816 
98860 

9S686 
98731 
9S776 
98820 
98865 

98691 
98735 
98780 
98825 
98869 

98695 
98740 
98784 
98829 
98874 

98700 
98744 
98789 
98834 
98878 

98704 
98749 
98793 
98838 
98883 

98709 
98753 
98798 
98843 
98887 

987:3 
98758 
98802 
98847 
98892 

98717 
98762 
98807 
98851 
98896 

98900 
9S945 
989S9 
99034 
99078 

98905 

98949 
98994 
990  3  8 
99083 

98909 
98954 
98998 
99043 
99087 

98914 
98958 
99003 
99047 
99092 

98918 
9S963 
99007 
99052 
99096 

98923 
98967 
99012 
99056 
99100 

98927 
98972 
99016 
9906 1 
99105 

9S932 
98976 
99021 
99065 
99:09 

98936 
98981 
99025 
99069 
991 14 
99 1 58 
99202 
99247 

99291 
99335 

98941 
98985 
99029 
99074 
99118 

99123 
99167 
99211 
99f5 
99300 

99127 
991 7 1 

992  ID 
99260 
99304 

99i3i 
99176 
99220 
99264 
99308 

99 1 36 
99180 
99224 
99269 
993 1 3 

99140 
99185 
99229 
99273 
99317 

99145 
99189 
99233 
99277 
99322 

99149 
99193 
99238 
99282 
99326 

99154 
99T98 
99242 
99286 
99330 

99162 
99207 
99251 
99295 
99339 

99344 
99388 
99432 
99476 
99520 

99348 
99392 

99436 
99480 
99524 

99352 
99396 
99441 
99484 
99528 

99357 
99401 
99445 
99489 
99533 

99361 
9940  5 
99449 
99493 
99537 

99366 
99410 
99454 
99498 
99542 

99370 
994:4 
99458 
99502 
99546 

99374 
994:9 
99463 
99506 
99550 

99379 
99423 
99467 
995 1 1 
99555 

99383 

99427 

99471 
995:5 

99559 

L 

I 
2 

3 
4 
5 
6 

7 
8 

_9_ 

0 

I 
I 

99554 
99607 
99651 
99695 
99739 

99568 
99612 
99656 
99699 
99743 

99572 
99616 
99660 
99704 
99747 

99577 
99621 
99664 
99708 
99752 

99581 
99625 
99669 
99712 
99756 

99585 
99629 
99673 
99717 
99760 

99590 
99634 
99677 
99721 
99765 

99594 
99638 
99682 
99726 
99769 

99599 
99642 
99686 
99730 

99774 

99603 

99647 
99691 
99734 
99778 

•1 
1 
0 
3 
3 
4 

99782 
99S26 
99870 
99913 
99957 

99787 
99830 
99874 
99917 
99961 

99791 
99835 
99878 
99922 
99965 

99795 
99839 
99883 
99926 
99970 

99S00 
99843 

99887 
99930 
99974 

99804 
99848 
99891 
99935 
99978 

99808 
99852 
99896 
99939 
99983 

99813 
99856 
99900 
99944 
99987 

99817 
99861 
99904 
99948 
99991 

99822 
99865 
99909 
99952 
99996 

0 

1 

2 

3 

4 

0 

6 

7 

8 

9 

TABLE  XXVIL 

[Page  185 

Log.  Sines,  Tangents,  and  Secants. 

M 

o 

V 

179° 

Hour  A.nr 

Hour  P.M. 

Sine. 

Diir.i 

Cosecnul. 

Tang'cnt. 

Diff.r 

Cotangent 

Secant. 

Cosine. 

M 

60 

12  00 

000 

Inf.  Neg 

Iiifiniie. 

Inf.  Nog. 

Iiifinitc. 

10.00000 

10.00000 

I 

II  59  52 

0   8 

6.46373 

3oio3 

13.53627 

6.46373 

3oio3 

13.53627 

00000 

00000 

59 

2 

59  44 

0  16 

7647C 

17609 

23524 

76476 

17609 

23524 

00000 

00000 

58 

3 

59  36 

0  24 

9408  5 

12494 

05915 

94o85 

1 2494 

05915 

00000 

00000 

57 

4 
^5 

59  28 

0  32 

7.06579 

9691 

12.93421 

7.06579 

9691 

12.93421 

00000 

00000 

56 
55 

II  59  20 

0  0  4o 

7.1627c 

7918 

12.83730 

7.16270 

7918 

12.83730 

1 0 . 00000 

10.00000 

6 

59  12 

0  48 

24188 

6694 

75812 

24188 

6694 

75812 

00000 

00000 

54 

7 

59  4 

0  56 

30882 

58oo 

69118 

30882 

58oo 

69118 

00000 

00000 

53 

8 

58  56 

I  4 

36682 

5ii5 

633 18 

36682 

5ii5 

633 1 8 

00000 

00000 

52 

_9 

lO 

58  48 

I  12 

41797 

4576 

58203 

41797 

4576 

58203 

OOOOCi 

00000 

5i 

5o 

II  58  4» 

0  I  20 

7.46373 

4i39 

12.53627 

7.46373 

4i39 

12.53627 

1 0 . 00000 

10.00000 

1 1 

58  32 

I  28 

5o5i2 

3779 

494SS 

5o5i2 

3779 

49488 

00000 

00000 

49 

12 

58  24 

I  36 

54291 

3476 

45709 

54291 

3476 

45709 

00000 

00000 

48 

i3 

58  16 

I  44 

57767 

3218 

42233 

57767 

3219 

42233 

00000 

00000 

47 

i4 
i5 

58  8 

I   52 

60985 

2997 

39015 

60986 

2996 

39014 

00000 

00000 

4b 
45 

II  58  0 

020 

7.639S2 

2802 

1 2 . 36o 1 8 

7.63982 

2803 

12.36018 

10.00000 

ic .00000 

i6 

57  52 

2  8 

66784 

2633 

33216 

66785 

2633 

332 1 5 

00000 

00000 

44 

17 

57  44 

2  16 

69417 

2483 

3o583 

69418 

2482 

3o582 

0000 i|  9.99999 

4i 

i8 

57  36 

2  24 

7 1 900 

2348 

28 1 00 

71900 

2348 

28  [  00 

00001 

99999 

42 

!9 
so 

57  28 

2  32 

74248 

2227 

25752 

74248 

2228 

25752 

00001 

99999 

4i 
4o 

II  57  20 

0  2  4o 

7.76475 

21 19 

12.23525 

7.76476 

2119 

12.23524 

10.00001 

9.99999 

21 

57  12 

2  48 

7S594 

2021 

2 1 4o6 

78595 

2020 

2i4o5 

00001 

99999 

39 

22 

57  4 

2  56 

So6i5 

1930 

19385 

806 1 5 

1931 

19385 

0000 1 

99999 

38 

23 

56  56 

3  4 

82545 

1 848 

17455 

82546 

1 848 

17454 

0000 1 

99999. 

37 

24 
25 

56  48 

3  12 

84393 

1773 

1 5607 

84394 

1773 

i56o6 

0000 1 

99999 

3b 
35 

II  56  4o 

0  3   2() 

7.861G6 

1704 

12. 1 3834 

7.86167 

1704 

I2.I3833 

10.00001 

9.99999 

26 

56  32 

3  28 

87870 

1639 

I2l3o 

87871 

1639 

12129 

0000 1 

99999 

M 

27 

56  24 

3  36 

89509 

1579 

1 049 1 

89510 

1579 

10490 

0000 1 

99999 

^.i 

2S 

56  16 

3  44 

91088 

i524 

08912 

91089 

i524 

0891 1 

0000 1 

99999 

32 

29 

3o 

56  8 

3  53 

92612 

1472 

07388 

92613 

1473 

07887 

00002 

99998 

3i 
3^ 

II  56  0 

0  4  " 

7.940.84 

1424 

12.05916 

7.94086 

1424 

12.05914 

10.00002 

9.99998 

3i 

55  52 

4  8 

95508 

1 379 

04492 

95510 

1879 

04490 

00002 

99998 

29 

32 

55  44 

4     16 

96S87 

1 336 

o3i  i3 

96889 

1 336 

o3iii 

00002 

99998 

28 

33 

55  36 

4  24 

98223 

1297 

01777 

98225 

1297 

01775 

00002 

99998 

27 

34 
35 

55  28 

4  32 

99520 

1259 

00480 

99522 

1259 

00478 

00002 

99998 

26 

25 

II  55  20 

0  4  4o 

8.00779 

1223 

II .99221 

8.00781 

1223 

II  .99219 

10.00002 

9.99998 

36 

55  12 

4  48 

02002 

I  1 90 

97998 

02004 

I  190 

97996 

00002 

99998 

24 

^7 

55  4 

4  56 

03192 

ii58 

96808 

o3i94 

1 1  59 

96806 

oooo3 

99997 

23 

38 

54  56 

5  4 

043  5o 

1128 

9565o 

04353 

I  I  28 

95647 

oooo3 

99997 

22 

39 
4o 

54  48 

5  12 

05478 

1 100 

94522 

o548i 

I  100 

94519 

oooo3 

99997 

21 

20 

II  54  4o 

0  5  20 

8.06578 

1072 

11.93422 

8.o658i 

1072 

II. 93419 

io.oooo3 

9.99997 

4i 

54  32 

5  28 

07650 

1046 

92350 

07653 

1047 

92347 

oooo3 

99997 

19 

42 

54  24 

5  36 

0S696 

1022 

9 1 3o4 

08700 

1022 

9i3oo 

oooo3 

99997 

18 

4i 

54  16 

5  44 

09718 

999 

90282 

09722 

998 

90278 

oooo3 

99997 

17 

44 

45 

54  8 

5  5?. 

10717 

976 

89283 

10720 

976 
955 

89280 
ii.883o4 

ooco4 

99996 

lb 
75 

II  54  0 

060 

3.11693 

954 

1 1.88 307 

8. 1 1696 

10.00004 

9.99996 

40 

53  52 

6  8 

1 2647 

934 

87353 

i265i 

934 

87349 

00004 

99996 

i4 

47 

53  44 

6  16 

i358i 

914 

86419 

i3585 

9.5 

864 1 5 

00004 

99996 

i3 

48 

53  36 

6  24 

14495 

89G 

855o5 

i45oo 

895 

855oo 

00004 

99996 

12 

49 
5o 

53  28 

6  32 

15391 

877 

84609 
II .83732 

15395 

878 

846o5 

00004 

99996 

1 1 
10 

II  53  20 

0  6  4o 

8.16268 

860 

8.16273 

860 

11.83727 

io.oooo5 

9.99995 

5i 

53  12 

6  48 

17128 

843 

82872 

17133 

843 

82867 

oooo5 

99995 

9 

52 

53  4 

6  56 

1 797 1 

827 

82029 

17976 

828 

82024 

oooo5 

99995 

8 

53 

52  56 

7  4 

18798 

812 

81202 

18804 

812 

81196 

oooo5 

99995 

7 

54 
55 

52  48 

7  12 

1 96 10 

797 

80390 

19616 

797 

8o384 

oooo5 

99995 

6 
5 

II  52  4o 

0  7  20 

8.20407 

782 

11.79593 

8.2o4i3 

782 

11.79587 

1 0 . 00006 

9.99994 

56 

52  32 

7  28 

21189 

769 

78811 

21 195 

769 

78805 

00006 

99994 

4 

57 

52  ■j4 

7  36 

21958 

755 

78042 

21964 

750 

78036 

00006 

99994 

3 

58 

52  16 

7  44 

22713 

743 

77287 

22720 

742 

77280 

00006 

99994 

2 

59 

52  8 

7  52 

23456 

73o 

76544 

23462 

73o 

76538 

00006 

99994 

I 

60 

52   0 

8   0 

24186 

717 

758 1 4 

24192 

718 

75808 

00007 

99993 

0 
M 

Hour  P.M. 

Hour  A.M. 

Cosine. 

Diff.l'  Secant.  | 

Cotangent 

Diff.r 

Tangent. 

Cosecant. 

Sine. 

90° 


80^ 


24 


Pa 

;e  18G] 

TABLE 

XXVIL 

1° 

Log.  Sines,  Tang( 

mts,  and  Secants. 

178° 

M 

o 

Hour  A.M.] 

Hour  P.M. 

Sine.  Diir.l'j 

Cosecant. 

11.75814 

Tangent.  iDiir.l'l 

Jotangent 

Secant. 

Cosine. 

M 

II  52  0; 

080 

8.24186  717 

8.24192 

718 

11.75808 

10.00007 

?• 99993 

I 

5i  521 

8  8 

24903  706 

75097 

24910 

706 

75090 

00007 

99993 

59 

2 

5i  44 

8  16 

25609 

695 

74391 

256i6 

696 

74384 

00007 

99993 

58 

3 

5i  36 

8  24 

263o4 

684 

73696 

263 12 

684 

73688 

00007 

99993 

37 

4 
~5 

5i  28 

8  3^ 

26988 

673 

73012 

26996 

678 

78004 

00008 

99992 

56 
55 

It  5 1  20 

0  8  40 

8.27661 

663 

II  .72339 

8 .  27669 

663 

II  .72331 

1 0 . 00008 

Q. 99992 

6 

5r  12 

8  48 

28324 

653 

71676 

28332 

654 

71668 

OOG08   99992 1 

54 

7 

5i  4 

8  56 

28977 

644 

71023 

28986 

643 

71014 

00008 

99992 

53 

8 

5o  56 

9  4 

29621 

bM 

70379 

29629 

634 

70871 

00008 

99992 

52 

_9 

lO 

5t)  48 

9  12 

30255 

624 

69745 

3o263 

625 

69787 

000C9 

99991 

5i 
5o 

1 1  5o  4o 

0  9  20 

8.30879 

616 

II .69121 

8.30S80 

617 

II .691 12 

10.000099.99991 1 

1 1 

5o  32 

9  28 

31495 

608 

685o5 

3i5o5 

607 

68495 

00009 

99991 

49 

12 

5o  24 

9  36 

32io3 

599 

67897 

321 12 

599 

67888 

00010 

99990 

48 

i3 

5o  16 

9  44 

32702 

590 

67298 

8271 1 

59c 

67289 

000 10 

99990 

47 

i4 
i5 

5o  8 

9  52 

0  10  0 

33292 

583 
575 

66708 

333o2 

584 

66698 

00010 

99990 

46 

45 

II  5o  0 

8.33875 

II .66125 

8.33886 

575 

1 1 . 66 1 1 4 

10.00010 

9.99990 

i6 

49  52 

10  8 

34450 

568 

65550 

34461 

568 

65539 

0001 1 

99989 

44 

17 

49  44 

10  16 

35oi8 

56o 

64982 

35029 

56 1 

64971 

0001 1 

99989 

43 

i8 

49  38 

10  24 

35578 

553 

64422 

35590 

553 

644 '0 

000 1 1 

99989 

42 

1 

20 

49  28 

10  32 

36i3i 

547 
539 

68869 
11 .63322 

36i43 

546 

63857 

0001 1 

99989 

4i 
4o 

II  49  20 

0  10  4o 

8.36678 

8.36689 

54o 

II .633ii 

10.00012 

9.99988 

21 

49  12 

10  48 

37217 

533 

62783 

87229 

533 

62771 

00012 

99988 

39 

22 

49  4 

10  56 

37750 

526 

62250 

37762 

527 

62288 

00012 

99988 

38 

23 

48  56 

II  4 

38276 

520 

61724 

38289 

520 

61711 

000 1 3 

99987 

37 

24 

25 

48  48 

II  12 

38796 

5i4 

61204 

38809 

5i4 

61191 

000 1 3 

99987 

3ti" 
35 

:i  48  4o 

0  1 1  20 

8.39310 

5o8 

1 1 .60690 

8.39323 

509 

II .60677 

io.oooi3 

9.99987 

26 

48  32 

II  28 

39818 

5o2 

60182 

39882 

502 

60168 

000 1 4 

99986 

34 

27 

48  24 

II  36 

4o320 

496 

59680 

40.334 

49^ 

59666 

00014 

99986 

33 

28 

48  16 

II  44 

40816 

491 

59184 

4o83o 

491 

59170 

000 1 4 

999S6 

32 

29 

3o 

43  8 

II  52 

4i3o7 

485 

58693 

4i32i 

486 
480 

58679 

000 1 5 
io.oooi5 

99985 

3i 

3^ 

(1  48  0 

0  12  0 

8.41792 

480 

1 1 .58208 

8.41807 

11.58193 

9.999S5 

3i 

47  52 

12  8 

42272 

474 

57728 

4:2287 

475 

57713 

000 1 5 

99985 

29 

32 

47  44 

12  16 

42746 

470 

57?54 

42762 

470 

57288 

00016 

99984 

28 

33 

47  36 

12  24 

43216 

464 

56784 

43282 

464 

56768 

00016 

99984 

27 

34 
35 

47  28 

12  32 

4368o 

459 

56320 

43696 

46o 

563o4 

00016 

999S4 

26 

II  47  20 

0  12  4<i 

8.44139 

455 

11.55861 

8.44i56 

455 

11.55844 

10.00017 

9.99983 

36 

47  '2 

12  48 

44594 

45o 

55406 

4461 1 

45o 

55389 

00017 

99983 

24 

37 

47  4 

12  56 

45o44 

445 

54956 

45061 

446 

54939 

00017 

99983 

23 

38 

46   56 

i3  4 

45489 

441 

545ii 

45507 

44 1 

54493 

00018 

99982 

22 

09 

40 

46  48 

i3  12 

45930 

436 

54070 
11.53634 

45948 
8.46385 

437 

54o52 

00018 

99982 
9.99982 

21 

20 

1 1  46  4" 

0  i3  20 

8.46366 

433 

432 

ii.536i5 

10.00018 

4i 

46  32 

i3  28 

46799 

427 

53201 

46817 

428 

53i83 

00019 

99981 

19 

42 

46  24 

1 3  36 

47226 

424 

52774 

47245 

424 

52755 

00019 

99981 

18 

43 

46  ]6 

i3  44 

47650'  419 

52350 

47669 

420 

52331 

00019 

99981 

17 

44 
45 

/i6  8 

i3  52 

48069 

4i6 
"4m~ 

51931 

48089 

4i6 

51911 

00020 

99980 

lb 
75 

1 1  46  0 

0  i4  0 

8.48485 

ii.5i5i5 

8.485o5 

4l2 

11 .51495 

10.00020 

9.99980 

40 

45  52 

i4  8 

4S896 

4o8 

5i  io4 

48917 

408 

5io83 

00021 

99979 

14 

47 

45  44 

i4  16 

49304 

4o4 

50696 

49825 

4o4 

50675 

00021 

99979 

i3 

.48 

45  36 

1 4  24 

49708 

4oo 

50292 

49729 

40 1 

60271 

00021 

99979 

12 

49 

5c 

45  28 

l4  32 

5oio8 
8 . 5o5o4 

396 
393 

49892 

5oi3o 

397 

49870 

00022 

99978 

II 

10 

II  45  20 

0  1 4  4<i 

11.49496 

8.5o527 

393 

11.49473 

10. 00022 

9.99978 

5i 

45  12 

i4  48 

50897 

890 

49105 

50920 

390 

49080 

00023 

99977 

9 

52 

45  4 

i4  56 

51287 

386 

48713 

5i3in 

386 

48690 

0002  3 

99977 

8 

53 

44  56 

i5  4 

51673 

382 

48327 

51696 

383 

483o4 

00023 

99977 

7 

54 
55 

44  48 

i5  12 

52o55 

379 

47945 

52079 

38o 

47921 

00024 

99976 

6 

'5 

11  44  4c 

0  1 5  20 

8.52434 

376 

11.47566 

8.52459 

376 

II. 4754 1 

10.0002^ 

9.99976 

56 

44  32 

1 5  28 

52810 

373 

47190 

52835 

373 

47165 

00025 

99975 

4 

57 

44  24 

i5  36 

53i83 

369 

46817 

5320& 

370 

46792 

00025 

99973 

3 

58 

44  if 

1 5  44 

53552 

367 

46448 

5357S 

367 

4642  2 

000  2C 

99974 

2 

5( 

44  8 

i5  52 

53919 

363 

4608 1 

53945 

363 

46o55 

00026 

99974 

1 

6c. 
M 

44    c 

16  0 

54282 

36o 

45718 

543o& 

36 1 

45692 

0002C 

99974 

0 
M 

Hour  p. M 

Hour.\.!M 

Cosine.  JDiir.l 

Secant. 

Cotangeii 

Difl'.l 

Tangent. 

Cosecant 

Siiio. 

9V 


88» 


TABLE 

XXVIL 

[Page  187 

Log.  Sines,  Tangents,  and  Secants. 

2° 

177° 

M 
o 

Hour  A.M. 

Hourp 

M. 

Sine.  |Diir.l'< 

ZJosecant. 

Tangent. 

DitT.l'l 

^lotangent 

Secant.  1  Cosine. 

6^ 

II  44  0 

0  16 

- 

8.54282  36o 

II. 45718 

8.54308 

36 1 

11.45692 

10.00026^9.99974 

I 

43  52 

16 

8 

54642 

357 

45358 

54669 

358 

45331 

ooo27|  99973 

59 

2 

43  44 

16 

16 

54999 

355 

45ooi 

55027 

355 

44973 

00027 

99973 

58 

3 

43  36 

16 

24 

55354 

35i 

44646 

55382 

352 

44618 

00028 

99972 

57 

4 
5 

43  28 

16  32 

55705 

349 

44295 

55734 

349 

44266 

00028 

99972 

55 

II  43  20 

0  16  4o 

8.56o54 

346 

11 .43946 

8.56o83 

346 

11.43917 

10.00029 

9.99971 

6 

43  12 

lO 

48 

564oo 

343 

43600 

56429 

344 

43571 

00029 

9997' 

54 

-^ 

43  4 

16 

56 

56743 

34 1 

43257 

56773 

341 

43227 

ooo3c) 

99970 

53 

8 

42  56 

17 

4 

57084 

337 

42916 

57114 

338 

42886 

ooo3o 

9997" 

52 

42  ^8 

17 

12 

57421 

336 

42579 

57452 

336 

42548 

ooo3i 

99969 

5i 

5^ 

1 1  42  40 

0  17 

20 

8.57757 

332 

II  .42243 

8.57788 

333 

1 1 .42212 

io.ooo3i 

9.99969 

1  1 

42  32 

17 

28 

58089 

33o 

41911 

58i2i 

33o 

41879 

ooo32 

99968 

49 

1  2 

42  24 

17 

36 

58419 

328 

4i58i 

5845i 

328 

4i549 

ooo32 

99968 

48 

i3 

42  16 

17 

44 

58747 

325 

41253 

58779 

326 

41221 

ooo33 

99967 

47 

i4 
i5 

42  8 

17 

52 

69072 

323 

40928 

59105 

323 

40895 

ooo33 

99967 

4b 
45 

1 1  42  0 

0  18 

0 

8.59395 

320 

1 1 .4060 5 

8.59428 

321 

1 1 .40572 

io.ooo33 

9.99967 

i6 

4i  52 

18 

8 

59715 

3i8 

40285 

59749 

319 

4o2  5i 

ooo34 

99966 

44 

17 

4 1  44 

18 

16 

6oo33 

3i6 

39967 

60068 

3i6 

39932 

ooo34 

99966 

43 

lb 

4i  36 

18 

24 

60349 

3i3 

39651 

6o384 

3i4 

396 1 6 

000  3  5 

99965 

42 

!9 

20 

4i  28 

18 

32 

60662 

3ii 

39338 

60698 

3ii 

39302 

ooo36 

99964 

41 
4o 

II  4 1  20 

0  18  4o\ 

8.60973 

309 

1 1 .39027 

8.61009 

5/0 

1 1 .38991 

10.000369.99964 

21 

4i  12 

18 

48 

61282 

307 

38718 

6i3i9 

3o7 

3868 1 

ooo37 

99963 

^2 

2  2 

4i  4 

18 

56 

61589 

3o5 

384 1 1 

61626 

3o5 

38374 

00037 

99963 

38 

23 

4o  56 

19 

4 

61894 

302 

38 1 06 

61931 

3o3 

3S069 

ooo38 

99962 

37 

24 
25 

4o  48 

19 

12 

62196 

3oi 

.  37804 

62234 

3oi 

37766 

ooo38 

99962 

3b 
35 

11  40  4o 

0  19 

20 

8.624^7 

298 

II .375o3 

8.62535 

299 

11.37465 

10.00039 

9.99961 

2b 

4o  32 

19 

28 

62795 

296 

37205 

62834 

297 

37166 

00039 

99961 

34 

27 

4o  24 

19 

36 

63091 

294 

36909 

63i3i 

295 

36869 

00040 

99960 

6i 

28 

40  16 

19 

44 

63385 

293 

366 1 5 

63426 

292 

36574 

ooo4o 

99960 

32 

29 

3o 

40  8 

19 

52 

6367S 

290 

3632  2 

63718 

291 

36282 

ooo4i 

99959 

3i 

33 

1 1  4o  0 

0  20 

0 

8.63968 

288 

11 .36o32 

8.64009 

289 

1 1 .35991 

1 0.0004 1 

9.99959 

3i 

39  52 

20 

8 

64256 

287 

35744 

64298 

287 

35702 

00042 

99958 

29 

32 

39  44 

20 

16 

64543 

284 

35457 

64585 

285 

354i5 

00042 

99958 

28 

33 

39  36 

20 

24 

64827 

283 

35173 

64870 

284 

35i3o 

00043 

99957 

27 

34 
35 

39  28 

20 

32 

65iio 

281 

34890 

65 1 54 

281 

34846 

00044 
10.00044 

99956 

2b 

II  39  20 

0  20 

4o 

8.65391 

279 

II  .34609 

8.65435 

280 

11.34565 

9.99956 

3b 

39  12 

20 

48 

65670 

277 

34330 

657 1 5 

278 

34285 

00045 

99955 

24 

37 

39  4 

20 

56 

65947 

276 

34o53 

65993 

276 

34007 

00045 

99955 

23 

3« 

38  56 

21 

4 

66223 

274 

33777 

66269 

274 

33731 

00046 

99954 

22 

39 

4f. 

38  48 

21 

12 

66497 
8.66769 

272 

335o3 

66543 

273 

33457 

00046 

99954 

21 

20 

II  38  4o 

0  21 

20 

270 

II  .3323i 

8.66816 

271 

ii.33i84 

10.00047 

9.99953 

4i 

38  32 

21 

28 

67039 

269 

32961 

67CJ87 

269 

32913 

ooo4S 

99952 

19 

42 

38  24 

21 

36 

67308 

267 

32692 

67356 

268 

32644 

00048 

99952 

18 

43 

38  16 

21 

44 

67575 

266 

32425 

67624 

266 

32376 

00049 

99951 

17 

44 
45 

38  8 

21 

52 

67841 

263 

32159 

67890 

264 

321 10 

00049 

99951 

lb 
75 

II  38  0 

0  22 

0 

8.68104 

263 

1 1. 31896 

8.68154 

263 

ii.5i846 

io.ooo5o 

9.99950 

4f) 

37  52 

22 

8 

68367 

260 

3 1 633 

68417 

261 

3 1 583 

ooo5i 

99949 

i4 

4? 

37  44 

22 

16 

68627 

269 

3 1 373 

68678 

260 

3l322 

ooo5i 

99949 

i3 

4B 

37  36 

22 

24 

68886 

258 

3i  ii4 

68938 

258 

31062 

000  5  2 

99948 

12 

49 

5o 

37  28 

22 

32 

69144 

256 

3o856 

69196 

357 

3o8o4 

000  5  2 

99948 

11 

10 

1 1  37  20 

0  22 

40 

8.69400 

254 

1 1 .3o6oo 

8.69453 

255 

1 1 .3o547 

10.00053I9.99947 

61 

37  12 

22 

48 

69654 

253 

3o346 

69708 

2  54 

30292 

00054'  99946 

9 

62 

37  4 

22 

56 

69907 

252 

30093 

69962 

252 

3oo38 

ooo54 

99946 

8 

53 

36  56 

23 

4 

70159 

25o 

29841 

70214 

25l 

29786 

ooo55 

99945 

7 

54 
55 

36  48 

23 

12 

70409 

249 

29391 

7o465 

249 

29535 

ooo56 

99944 

b 
~5 

n  36  4o 

0  23 

20 

8.70658 

247 

II  .29342 

8.70714 

248 

11 .29286 

10.000569.99944 

5b 

36  32 

23 

28 

70905 

246 

29095 

70962 

246 

29038 

00057  99943 

4 

57 

36  24 

23 

36 

7ii5i 

244 

28849 

71208 

245 

28792 

ooo58  99942 

3 

5b 

36  16 

23  44 

71395 

243 

28605 

71453 

244 

28547 

ooo58  99942 

2 

59 

36  8 

23 

52 

7i638 

242 

28362 

71697 

243 

283o3 

000591  99941 

I 

bo 
M 

36  0 

24 

0 

71880 

240 

28120 

71940 

241 

28060 

0006c  99940 

0 
M 

Hour  P.M. 

Hour  A 

.M. 

Cosino.  iDifT.l' 

Secant. 

CotangentlDiff.l 

Tangent. 

Cosecant. '  Sine. 

92^ 


87« 


Pa 

ge  183] 

TABLE 

XXVIL 

• 

Log.  Sines,  Tangents,  and  Secants, 

3° 

M 

0 

176° 

Hoar  A.M. 
1 1  36  o 

Hour  P.M. 
0  24  0 

Sine. 

Difif.l' 

Cosecant. 

Jangent. 

Diff.l' 

Cotangent 

Secant. 

Cosine. 

31 

5o 

8.71880 

240 

II  .28120 

8.71940 

24 1 

1 1 . 28060 

10.00060 

9.99940 

I 

35  52 

24  8 

72120 

239 

27880 

72181 

239 

27819 

00060 

99940 

5q 

2 

35  44 

24  16 

72359 

238 

27641 

72420 

239 

27580 

00061 

99939 

58 

<J 

3>5  36 

24  24 

72597 

237 

2-j4o3 

72659 

237 

27341 

00062 

99938 

57 

4 
"5 

35  28 

24  32 

72834 

235 

27166 

72896 

236 

27104 

00062 

99958 

56 
55 

1 1  35  20 

0  24  4o 

8.73069 

234 

II  .26931 

8.73i32 

234 

11.26868 

10.00063 

9.99937 

ti 

35  12 

24  48 

733o3 

232 

26697 

73366 

234 

26634 

00064 

99936 

54 

■  7 

35  4 

24  56 

73535 

232 

26465 

73600 

232 

26400 

00064 

99936 

53 

i " 

34  56 

25  4 

73767 

23o 

26233 

73832 

23l 

26168 

ooo65 

99935 

52 

9 

!0 

34  48 

25  12 

73997 
8.74226 

229 

26003 

74o63 

229 

25937 

00066 

99934 

5i 
56 

1 1  34  4o 

0  2  5  20 

228 

II .25774 

8.74292 

229 

II .25708 

10.00066 

9.99934 

1  I 

34   32 

25  28 

74454 

226 

25546 

7452  1 

227 

25479 

00067 

99933 

49 

12 

34  24 

25  36 

74680 

226 

25320 

74748 

226 

25252 

0006S 

99932 

48 

i3 

34  16 

25  44 

74906 

224 

25094 

74974 

225 

25026 

00068 

99932 

47 

i4 
i5 

34    8 

25  52 

75i3o 

223 

24870 

75199 

224 

24801 

00069 

99931 

46 
45 

II  34  0 

0  26  0 

8.75353 

222 

1 1 . 24647 

8.75423 

222 

11.24577 

10.000709.99930 

iti 

33  52 

26  8 

75575 

220 

24425 

75645 

222 

24355 

00071 

99929 

44 

17 

33  44 

26  16 

75795 

220 

24205 

75867 

220 

24i33 

00071 

99929 

43 

i8 

33  36 

26  24 

76015 

219 

23985 

76087 

219 

23913 

00072 

99928 

42 

19 

20 

33  28 

26  32 

76234 

217 

23766 

76306 

2iy 

23694 

00073 

99927 

4i 
40 

II  33  20 

0  26  4o 

8.76451 

216 

II  .23549 

8.76525 

217 

11.23475 

10.00074 

9.99926 

21 

33  12 

26  48 

76667 

216 

23333 

76742 

216 

23258 

00074 

99926 

39 

22 

33    4 

26  56 

768S3 

2l4 

23l  17 

76958 

2l5 

2  3o42 

00075 

99925 

38 

2j 

32  56 

27  4 

77097 

2l3 

22900 

77173 

2l4 

22827 

00076 

99924 

37 

24 
25 

32  48 

27  12 
0  27  20 

773 10 

212 

22690 

7738-j 

2l3 

22613 

00077 

99923 

36 
35 

1 1  32  40 

8.77522 

21  I 

1 1 .22478 

8 . 77600 

211 

1 1 .22400 

10.00077 

9.99923 

2fa 

32  32 

27  28 

77733 

210 

22267 

77811 

211 

22189 

00078 

99922 

34 

27 

32  24 

27  36 

77943 

209 

22057 

78022 

210 

21978 

00079 

99921 

33 

28 

32  16 

27  44 

78152 

208 

2184s 

78232 

209 

21768 

00080 

99920 

32 

'9 
3o 

32  8 

27  52 
0  28  0 

78360 

208 

21640 

78441 

208 

21559 

00080 

99920 

3i 
3o 

II  32   0 

8.78568 

206 

II  .21432 

8.78649 

206 

II .2i35i 

10.00081 

9.99919 

3 1 

3i  52 

28  8 

78774 

2o5 

21226 

78855 

206 

21145 

00082 

99918 

29 

32 

3 1  44 

28  16 

78979 

2o4 

2I02I 

79061 

205 

20939 

oooS3 

99917 

28 

33 

3 1  36 

28  24 

79183 

203 

20817 

79266 

204 

20734 

ooo83 

99917 

27 

35 

3i  28 

28  32 

79386 
8.79588 

202 
201 

20614 
II .20412 

79470 

203 

2o53o 

00084 

99916 

36 

25 

II  3 1  20 

0  28  40 

8.79673 

202 

II .20327 

1O.O0O85 

9.99915 

36 

3i  12 

28  48 

79789 

201 

202  I  I 

79875 

201 

20I2D 

00086 

99914 

24 

37 

3i  4 

28  56 

79990 

199 

20010 

80076 

201 

19924 

00087 

99913 

23 

38 

3o  56 

29  4 

80189 

199 

198  I  I 

80277 

199 

19723 

00087 

999x3 

22 

39 
4o 

3o  48 

29  12 

8o388 

197 

I  96  I  2 

80476 

198 

19524 

00088 

99912 

21 
20 

n  3o  4o 

0  29  20 

8.8o585 

197 

II .19415 

8.80674 

198 

II .19326 

10.00089 

9.99911 

4i 

3o  32 

29  28 

80782 

196 

1 92 1 8 

80872 

196 

I9128 

00090 

99910 

19 

42 

3o  24 

29  36 

80978 

195 

19022 

81068 

196 

18932 

00091 

99909 

18 

43 

3o  16 

29  44 

81173 

194 

18827 

81264 

195 

18736 

00091 

99909 

"7 

44 

45 

3o  8 

29  52 
0  3o  0 

8 1 367 

193 

i8533 

81459 

194 

i854i 

00092 

999f)S 

16 

75 

11  3o  0 

8.8i56o 

192 

II .18440 

8.8i653 

193 

11.18347 

10.00093 

9.99907 

4b 

29  52 

3o  8 

81752 

192 

18248 

81846 

192 

i8i54 

00094 

99906 

i4 

47 

29  44 

3o  16 

81944 

190 

i8o56 

82o38 

192 

17962 

00095 

99905 

i3 

48 

29  36 

3o  24 

82134 

190 

17866 

82230 

190 

17770 

00096 

99904 

12 

49 
5o 

29  28 

3o  32 

82324 
8.825i3 

189 
188 

17676 

82420 

190 

17580 

00(^96 

99904 

1 1 

10 

II  29  20 

0  3o  4o 

1 1. 17487 

8.82610 

189 

II .17390 

10.00097 

9.99903 

3[ 

29  12 

3o  48 

82701 

187 

17299 

82799 

188 

17201 

00098 

99902 

9 

52 

29  4 

3o  56 

82888 

187 

17112 

82987 

188 

17013 

00099 

99901 

8 

53 

28  56 

3i  4 

83075 

186 

16925 

83i75 

186 

16825 

00 1 00 

99900 

7 

54 
55 

28  48 

3i  12 

83261 

8.83446 

i85 
184 

16739 
1 1. 16554 

8336i 

186 

16639 

OOIOI 

99899 
9.99898 

6 
~5 

II  28  4o 

0  3i  20 

8.83547 

i85 

1 1. 16453 

10.00102 

56 

28  32 

3i  28 

8363o 

]83 

16370 

83732 

1 84 

16268 

00102 

99898 

4 

57 

28  24 

3i  36 

838 1 3 

1 83 

16187 

83916 

1 84 

16084 

ooio3 

99S97 

3 

58 

28  16 

3i  44 

83996 

!8l 

1 600  4 

84 100 

182 

15900 

00104 

99S96 

2 

59 

28  8 

3i  52 

84177 

181 

i5823 

84282 

182 

15718 

ooio5 

99S95 

I 

6o 
M 

28  0 

32   0 

84358 
Cosine. 

181 
DiffT' 

1 5642 
Secant. 

84464 

182 

15535 

00106 

99894 

0 
M 

Hour  P.M. 

Hour  A.M. 

Cotangent 

Diff.l' 

Tangent. 

Cosecant. 
,, .,   

Sine. 

93° 


86" 


TABLE  XXVIL 

[Page  189 

Log.  Sines,  Tangents,  and  Secants. 

4= 

'•* 

175" 

31 

0 

Hour  a.:m 

Flour  P.M. 

Sine. 

Difi-.  1 

Cosecant. 

Tang'ciit. 
8.8446/ 

Diff.  1 

Cotanj^cnt 

Secant. 

Cosine. 

M 

II  28  ( 

0  32  0 

8.84358 

181 

n .  1 5642 

182 

1 1. 1 5536 

10.00106 

9.99894 

I 

27  5i 

32  8 

84535 

179 

1 5461 

8464G 

180 

1 5354 

00107 

99893 

5q 

2 

27  4 

32  16 

84718 

179 

15282 

'  8482G 

180 

i5i74 

00108 

99892 

58 

3 

27  3c 

32  24 

84897 

178 

i5io3 

85oo6 

179 

14994 

00109 

99891 

57 

4 
5 

27  7b 

32  32 

85()75 

177 

14925 

85i85 

178 

i48i5 

00109 

99891 

56 
55 

I  I  27  2t 

0  32  4o 

8.85252 

•77 

1 1. 14748 

8.85363 

177 

11 .14637 

10.001 1( 

9.99890 

6 

27  I? 

32  48 

85429 

17b 

14571 

85540 

177 

1 4460 

001 II 

1   99''^89 

54 

7 

27  4 

32  56 

856o5 

175 

14395 

85717 

176 

14283 

001 12 

1   998S8 

53 

8 

26  5( 

33  4 

85780 

175 

14220 

85393 

176 

14107 

001 1 3 

1   99887 

52 

_9 

lO 

26  48 

33  12 

85955 

173 

i4o45 

86069 

174 

1 3931 

001 14 

998S6 

5i 
5^ 

II  26  4< 

0  33  20 

8.86128 

173 

II . 13872 

8.86243 

174 

li .13757 

lo.ooiiS 

9.99885 

1 1 

26  3? 

33  28 

86301 

173 

13699 

86417 

174 

13583 

00116 

99884 

49 

12 

26  24 

33  36 

86474 

171 

13526 

86591 

172 

13409 

001 17 

998S3 

48 

i3 

26  i() 

33  44 

86645 

171 

i3355 

86763 

172 

i3237 

001  ife 

99882 

47 

i4 
i5 

26  8 

33  52 

86816 

171 

i3i84 

86935 

171 

i3o65 

00119 

99881 

46 
45 

I  I  26  c 

0  34  0 

8.86987 

169 

II  .i3oi3 

8.87106 

171 

1 1. 1 2894 

10. 0012c 

9.99880 

[6 

2  5  52 

34  8 

87156 

169 

12844 

87277 

170 

12723 

00121 

99S79 

44 

17 

2  5  44 

34  16 

87325 

169 

12675 

87447 

169 

12553 

00121 

99879 

43 

iB 

2  5  3f) 

34  2/( 

87494 

ib7 

i2  5o6 

87616 

169 

12384 

00122 

99878 

42 

19 

20 

95  28 

34  32 

87661 

168 
166 

12339 
11 .12171 

87785 

168 

1 22 1 5 

00123 

99877 

4i 

4o 

II  2D  2u 

0  34  4o 

8.87829 

8.87953 

167 

1 1 . 1 2047 

10.00124 

9.99876 

21 

25  12 

34  48 

87995 

166 

1 2  00  5 

88120 

167 

1 1 880 

OOI25 

99875 

39 

22 

25  4 

34   56 

88161 

165 

1 1 839 

88287 

166 

11713 

00126 

99874 

38 

23 

24  56 

35  4 

88326 

164 

11674 

88453 

1 65 

1 1 547 

00127 

99873 

37 

24 
25 

24  48 

35  12 

88490 

164 

ii5io 

88618 

i65 

ii382 

00128 

99872 

36 
35 

II  24  4o 

0  35  2u 

8.8S654 

i63 

11.11 346 

8.88783 

i65 

II  .11217 

10.00129 

9.99871 

26 

24  32 

35  28 

888 1 7 

1 63 

iii83 

88948 

i63 

I1052 

ooi3o 

99870 

34 

27 

24  24 

35  36 

88980 

162 

11020 

891 1 1 

1 63 

10889 

ooi3i 

99869 

33 

28 

24  16 

35  44 

89142 

162 

io858 

89274 

ib3 

10726 

OOl32 

99868 

32 

29 

3o 

24  8 

35  52 

89304 

160 

1 0696 

89437 

ibi 

io563 

00 1 33 

99867 

3i 
3I; 

II  24  0 

0  36  0 

8.89464 

161 

II .io536 

8.8959S 

162 

II. 10402 

10.00134 

9 . 99866 

3i 

23  52 

36  8 

■89625 

159 

10375 

89760 

160 

10240 

001 35 

99865 

29 

32 

23  44 

36  16 

89784 

1 59 

10216 

89920 

160 

1 0080 

00 1 36 

99864 

26 

33 

2  3  36 

36  24 

89943 

159 

10057 

90080 

160 

09920 

00137 

99863 

27 

34 
35 

23  28 

36  32 

90102 

1 58 

0989S 

90240 

.59 

09760 

00 1 38 

99862 

26 

25 

I  I  23  20 

0  36  40 

8.9026(j 

1 57 

1 1 . 09740 

8.90391^ 

1 58 

1 1 . 0960 1 

10.00139 

9.99861 

3b 

23  12 

36  48 

90417 

1 57 

09383 

90557 

1 58 

09443 

ooi4o 

99860 

24 

37 

23  4 

36  56 

90574 

i5b 

09426 

90715 

i57 

09285 

ooi4i 

99859 

23 

38 

22  5(j 

37  4 

90730 

i55 

09270 

90872 

i57 

09128 

00142 

99858 

22 

39 
4u 

22  48 

37  12 

90885 

i55 

091 1 5 

91029 

1 5b 

08971 

00143 

99857 

21 

20 

It  22  4o 

0  37  20 

8.91 o4o 

i55 

1 1 .08960 

8.91 i85 

i55 

II .oS8i5 

10.00144 

9.99856 

4r 

22  32 

37  28 

91 195 

1 54 

oS8o5 

91340 

i55 

0S660 

00145 

99855 

iq 

42 

22  24 

37  36 

91349 

i5J 

o865i 

91495 

1 55 

o85o5 

00 1 46 

99854 

18 

43 

22  lb 

37  44 

91502 

i53 

08498 

91650 

133 

o835o 

00147 

99853 

17 

44 
45 

22   8 

37  52 

91655 

l52 

08345 

91S03 

1 54 

0S197 

00148 

99S52 

iG 
75 

U  22   0 

0  38  0 

8.91807 

l52 

11 .0S193 

8.91957 

i53 

1 1 .08043 

10.00149 

9.99851 

4b 

2[  52 

38  8 

91959 

i5i 

o£o4i 

921 10 

l52 

07890 

ooi5o 

99850 

1 4 

47 

2  1  44 

38  16 

921  10 

i5i 

07890 

92262 

l52 

07738 

OOl52 

99848 

i3 

48 

21  36 

38  24 

92261 

i5o 

07739 

92414 

i5i 

07586 

001 53 

99847 

12 

49 
5o 

21  28 

38  32 

92411 

i5o 

07589 

92565 

i5i 

07435 

001 54 

99846 

1 1 

10 

11  21  2U 

0  38  4o 

8.92561 

1 49 

11.07439 

8.92716 

i5o 

II .07284 

io.ooi55 

9.99845 

bi 

21  12 

38  48 

92710 

149 

07290 

92866 

i5o 

07134 

00 1 56 

99844 

9 

52 

21   4 

38  56 

92859 

1 48 

07141 

93016 

149 

06984 

00157 

99843 

8 

53 

20  56 

39  4 

93007 

147 

06993 

93i65 

1 48 

o6835 

001 58 

99842 

7 

54 
55 

20  48 

39  12 

93 1 54 

147 

06846 

933i3 

.49 

06687 

00159 

99841 

6 
5 

II  20  4o 

c  39  20 

8.93301 

i47 

1 1 . 06699 

8.93462 

147 

1 1. 06538 

10.00160 

9.99840 

56 

20  32 

39  28 

9344s 

1 46 

o6552 

93609 

'47 

06391 

00161 

99839 

4 

57 

20  24 

39  36 

93594 

i4b 

o64o6 

9I756 

147 

06244 

00162 

99838 

3 

58 

20  16 

39  44 

93740 

i45 

06260 

9  '903 

1 46 

06097 

00163 

99837 

2 

59- 

20  8 

39  52 

93885 

i45 

o6ii5 

94049 

1 4b 

05951 

00164 

99S36 

1 

60 

20  0 

40  0 

94o3o 

144 

0^970 

94195 

145 

o58o5 

00166 

99S34 

0 
M 

Hour  P.M. 

[lour  A.M. 

Cosine. 

DiiT.l' 

Secant. 

Cotangent 

Diff.  1' 

Tangent.  | 

Cosecant. 

h'iiic. 

w 


85" 


p 

ge  1901 

TABLE  XXVII. 

S' 

Log 

.  Sines,  Tangents,  and  Secants. 

G'. 

5= 

A 

A 

B 

B 

C        C  174° 

0 

HourA.M. 

Hour  P.M. 

Sine, 

Diif.  Cosecant. 

Tangent. 

Diff. 

Cotangent 

Secant. 

Diff. 

Cosine. 

M 

60 

II  20  00 

0  4o  0 

8 . 94o3o 

0  II  .05970 

8.94195 

0 

ii.o58o5 

10. 00166 

0 

9.99834 

I 

19  52 

4o  8 

94174 

2    05826 

94340 

2 

o5o6o 

00167 

0 

99833 

59 

2 

19  44 

4o  16 

94317 

4 

05683 

94485 

4 

o55i5 

00168 

0 

99832 

58 

J 

19  36 

4o  24 

94461 

7 

05539 

9463o 

7 

05370 

00169 

0 

9983 1 

57 

4 
5 

19  28 

4o  32 

94603 

9 

II 

05397 

94773 

9 

05227 

00170 

0 

99830 

56 
55 

II  19  20 

0  4o  4o 

8.94746 

ii.o5254 

8.94917 

II 

ii.o5o83 

10.00171 

0 

9.99829 

b 

19  12 

4o  48 

94887 

i3 

o5ii3 

95060 

i3 

04940 

00172 

0 

99828 

54 

7 

19/ 

4o  56 

95029 

i5 

04971 

95202 

i5 

04798 

00173 

0 

99827 

53 

8 

18  56 

4i  4 

95170 

18 

o483o 

95344 

18 

04656 

00175 

0 

99825 

52 

_9 

10 

18  48 

4i  12 

95310 

20 

04690 

95486 

20 

o45i4 

00176 

0 

99824 

5i 

5o 

II  18  4o 

0  4i  20 

8.95450 

32 

ii.o455o 

8.95627 

22 

11 .04373 

10.00177 

0 

9.99823 

II 

18  32 

4i  28 

95589 

24 

044 1 1 

95767 

24 

04233 

00178 

0 

99822 

49 

12 

ifl  ■^i 

4i  36 

95728 

2b 

04272 

95908 

27 

04092 

00179 

0 

99821 

48 

Ij 

18  lb 

4i  44 

95867 

29 

o4i33 

96047 

29 

03953 

00180 

0 

•  99820 

47 

i4 
i5 

18  8 

4i   52 

96005 

3i 

03995 

96187 

3i 

o38i3 

00181 

0 

99819 

46 
45 

II  18  0 

0  42  0 

8.96143 

33 

1 1. 03857 

8.96325 

33 

II .03675 

10.00x83 

0 

9.99817 

i6 

17  62 

42  8 

96280 

35 

03720 

96464 

35 

o3536 

00184 

0 

99816 

44 

17 

17  44 

42  16 

96417 

37 

03583 

96602 

38 

03398 

00 1 85 

0 

99815 

43 

i8 

IT  36 

42  24 

96553 

39 

03447 

96739 

40 

o326i 

00186 

0 

99814 

42 

£9 

20 

17  28 

42  32 

96689 

42 

o33ii 

96877 

42 

o3i23 

00187 

0 

99813 

4i 
40 

II  17  20 

0  42  4o 

8.96825 

44 

II  .o3i75 

8.97013 

44 

I"  02987 

10.00188 

0 

9.99812 

21 

17  12 

42  48 

96960 

4b 

o3o4o 

97i5o 

4b 

0285o 

00190 

0 

99810 

39 

2  2 

17  4 

42  56 

97095 

48 

02905 

97285 

49 

02715 

00191 

0 

99809 

38 

2J 

16  56 

43  4 

97229 

bo 

02771 

97421 

5i 

02579 

00 1 92 

0 

9980S 

37 

24 
25 

16  48 

43  12 

97363 

53 

02637 

97556 

53 

02444 

00193 

0 

99807 

36 
35 

II  16  4o 

0  43  20 

8.97496 

55 

II .025o4 

8.97691 

55 

II  .02309 

10.00194 

9.99806 

2b 

16  32 

43   28 

97629 

57 

02371 

97825 

58 

02175 

00196 

99804 

34 

27 

16  24 

43  36 

97762 

59 

02238 

97959 

bo 

0204 1 

00197 

99803 

33 

28 

16  16 

43  44 

97894 

bi 

02106 

98092 

b2 

01908 

00198 

99802 

32 

29 

3o 

16  8 

43   52 

98026 

b4 

01974 

98225 

b4 
"66 

01775 

00199 

99801 

3i 

3^ 

II  16  0 

0  44  0 

8.9815"^ 

66 

II. 01843 

8.98358 

II  .01642 

lo. 00200 

9.99800 

Ji 

i5  52 

44    8 

98288 

68 

OI7I2 

98490 

69 

oi5io 

00202 

99798 

29 

J2 

i5  44 

44   16 

98419 

70 

oi58i 

98622 

71 

01378 

002o3 

99797 

28 

JJ 

i5  36 

44  14 

98549 

72 

oi45i 

98753 

73 

01247 

00204 

99796 

27 

J4 
35 

i5  28 

44  32 

98679 

75 

Ol32I 

98884 

75 

01116 

002o5 

99795 

26 

25 

II  1 5  20 

0  44  4o 

8.98808 

77 

II  .01192 

8.99015 

77 

II  .00985 

10.00207 

9.99793 

db 

i5  12 

44  48 

98937 

79 

oio63 

99145 

80 

oo855 

00208 

99792 

24 

^7 

i5  4 

44   56 

99066 

81 

00934 

99275 

82 

00725 

00209 

99791 

23 

J8 

i4  56 

45  4 

99194 

83 

00806 

9940  5 

84 

00595 

00210 

99790 

22 

39 

40 

i4  48 

45  12 

99322 

8b 

00678 

99534 

8b 

00466 

00212 

99788 

21 
20 

II  i4  4o 

0  45  20 

8.99450 

88 

II .oo55o 

8.99662 

89 

II .oo338 

I0.002l3 

9.99787 

4i 

i4  32 

45  28 

99377 

90 

00423 

99791 

9' 

00209 

00214 

99786 

19 

42 

1 4  24 

45  36 

99704 

92 

00296 

99919 

93 

00081 

002l5 

99785 

18 

43 

i4  16 

45  44 

99S30 

94 

00170 

9.00046 

95 

10.99954 

00217 

99783 

17 

44 
45 

i4  8 

45  52 

99956 

9b 

00044 

00174 

97 

99826 

00218 

99782 

16 
l5 

II  i4  0 

0  46  0 

9.00082 

99 

10.99918 

9.oo3oi 

100 

10.99699 

10.00219 

9.99781 

4b 

i3  52 

46  8 

00207 

lOI 

99793 

00427 

102 

99573 

00220 

99780 

i4 

47 

i3  44 

46  16 

oo332 

io3 

99668 

oo553 

io4 

99447 

00222 

99778 

i3 

48 

i3  36 

46  24 

oo456 

io5 

99544 

00679 

106 

99321 

00223 

99777 

12 

49 
5o 

i3  28 

46  32 

oo58i 

107 

99419 

oo8o5 

108 

99795 

00224 

99776 

II 
10 

II  i3  20 

0  46  40 

9 . 00704 

no 

10.99296 

9.00930 

III 

10.99070 

10.00225 

9.99775 

5i 

i3  12 

46  48 

00S28 

112 

99172 

oio55 

ii3 

98945 

00227 

99773 

9 

52 

i3  4 

46  56 

00951 

ii4 

99049 

01179 

ii5   98821 

00228 

99772 

8 

5J 

12  56 

47  4 

01074 

lib 

98926 

oi3o3 

1 1 7   98697 

00229 

99771 

7 

t>4 
55 

12  48 

47  12 

01 196 

118 

98804 

01427 

120 
122 

98573 

0023  1 

99769 

6 
5 

II  12  4o 

0  47  20 

9.oi3i8 

121 

10.98682 

9.oi55o 

10.98450 

I0.00232 

I  9.99768 

5b 

12  32 

47  28 

oi44o 

123 

98560 

01673 

124 

98327 

00233 

I    99767 

4 

'i- 

12  24 

47  36 

oi56i 

125 

98439 

01796 

126 

98204 

0O235 

I    99765 

3 

58 

12  16 

47  44 

01682 

127 

98318 

01918 

128 

98082 

00  2  36 

I    99764 

2 

59 

12   8 

4i   52 

oi8o3 

129 

98197 

02040 

i3i 

97960 

00237 

I    99763 

I 

00 

12  0 

48  0 

01923, 

l32 

98077 

02162 

i33 

97838 

00239 

I    99761 

0 
M 

M 

Hour  P.M. 

HourA.M. 

Cosino.  ' 

Diff. 

Secant. 

Cotangent 

Difl-. 

Tangent. 

Cosecant. 

Diff.l  Sine. 

95° 


Seconds  of  time 

1» 

23 

3' 

4. 

5' 

6' 

7, 

Prop,  parts  of  cols.  <  B 

(c 

16 

17 
0 

33 
33 
0 

49 
5o 
0 

66 
66 

I 

82 
83 

I 

99 

100 

I 

ii5 
116 

I 

TABLE  XXVII. 

■ 

[Fiigc  191 

S' 

Log.  Sines,  Tan 

gents,  and  Secants. 

G'. 

G° 

A        A 

B         B 

C       C  ITS-^ 

31 

o 

Hour  A.ai. 
II  12  0 

Hour  p. .M. 

0  48  0 

Sine. 
9.01923 

Diir. 
0 

Cosecant. 

Tangent. 

Diff. 

Cotangent 

Secant. 

Diff.  Co.siue. 

M 

(k) 

10.98077 

9.02162 

0 

10.97838 

10.00239 

0  9.99761 

I 

II  52 

48  8 

o3o43 

2 

97957 

02283 

2 

97717 

00240 

0 

997D0 

J9 

2 

II  44 

48  16 

02i63 

4 

97837 

02404 

4 

97596 

00241 

0 

99759 

58 

3 

1 1  36 

48  24 

o;.283 

6 

97717 

02525 

6 

97475 

00243 

0 

99757 

57 

4 
5 

II  28 

48  32 

02402 

7 

97598 

02645 

8 

97355 

00244 

0 

99756 

56 
55 

II  II  20 

0  48  4o 

9.02520 

9 

1 0 . 97480 

9.02766 

9 

10.97234 

10.00245 

0 

9.99755 

6 

II  12 

48  48 

02639 

1 1 

97361 

02885 

1 1 

97115 

00247 

0 

99750 

54 

7 

!I   4 

48  56 

02757 

i3 

97243 

o3oo5 

i3 

96995 

00248 

0 

99-52 

53 

8 

10  56 

49  4 

02874 

i5 

97126 

o3i24 

i5 

96876 

00249 

0 

99751 

52 

_9 

10 

10  48 

49  12 

02992 

17 

97008 

03242 

17 

96758 

0025l 

0 

99749 

5i 
5^ 

11  10  4« 

0  49  20 

9.03109 

19 

10.96891 

9.03361 

19 

10.96639 

10.00252 

0 

9.99748 

1 1 

10  32 

49  28 

03226 

20 

96774 

o3479 

21 

96521 

00253 

0 

99747 

^9 

12 

10  24 

49  36 

03342 

22 

96658 

03597 

23 

96403 

00255 

0 

99745 

48 

i3 

10  16 

49  44 

o3458 

24 

96542 

03714 

24 

96286 

00256 

0 

99744 

47 

i4 
i5 

10  8 

49  52 

03574 

26 

96426 

o3832 

26 

96168 

00258 

0 

99742 

46 
45 

1 1  10  0 

0  5o  0 

9.03690 

28 

10.96310 

9.03948 

28 

10.96052 

10.00259 

0 

9.99741 

i6 

9  52 

5o  8 

o38o5 

3o 

96195 

o4o65 

3o 

95935 

00260 

0 

99740 

44 

17 

9  44 

5o  16 

03920 

3i 

96080 

o4i8i 

32 

95819 

00262 

0 

99738 

43 

i8 

9  36 

5o  24 

o4o34 

33 

95966 

04297 

34 

95703 

00263 

0 

99737 

42 

!9 

20 

9  28 

5o  32 

o4i49 

35 

9585i 

o44i3 

36 

95587 

00264 

0 

99736 

41 

4o 

II  9  20 

0  5o  40 

9.04262 

37 

10.95738 

9.04528 

38 

10.95472 

10.00266 

0 

9.99734 

21 

Q  12 

5o  48 

04376 

3o 

95624 

04643 

39 

95357 

00267 

99733 

39 

22 

9  4 

5o  56 

04490 

4i 

95510 

04758 

41 

95242 

00269 

99731 

3b 

23 

8  56 

5r  4 

o46o3 

43 

95397 

04873 

43 

95127 

00270 

99730 

37 

24 
25 

8  48 

5i  12 

047 1 5 

44 

95285 

04987 

45 

950 1 3 

00272 

99728 

3t) 
35 

II  8  4o 

0  5i  20 

9.04828 

46 

10.95172 

9.o5ioi 

47 

10.94899 

10.00273 

9.99727 

26 

8  32 

5i  28 

04940 

48 

95060 

o52i4 

49 

94786 

00274 

99726 

34 

27 

8  24 

5i  36 

o5o52 

5o 

94948 

05328 

01 

94672 

00276 

99724 

6i 

28 

8  16 

5i  44 

o5i64 

5? 

94836 

o544i 

53 

94559 

00277 

99723 

32 

29 

3o 

8  8 

5i  52 

05275 

54 

94725 

05553 

54 

94447 

00279 

99721 

3i 
3^ 

II  8  0 

0  52  0 

9.05386 

56 

1 0 . 946 1 4 

9.05666 

56 

10.94334 

10.00280 

9.99720 

3i 

7  52 

52  8 

05497 

57 

945o3 

05778 

58i   94222 

00282 

99718 

29 

32 

7  44 

52  16 

o56o7 

59 

94393 

05890 

60 

941 10 

00283 

997 '7 

2h 

33 

7  36 

52  24 

05717 

61 

94283 

06002 

62 

93998 

00284 

99716 

27 

3i 
35 

7  28 

52  32 

05827 

63 

94173 

061 13 

64 

93887 

00286 

997 '4 

2b 

II  7  20 

0  52  4o 

9.05937 

65 

10.94063 

9.06224 

66 

10.93776 

10.00287 

9.99713 

36 

7  12 

52  48 

o6o46 

67 

93954 

o6335 

68 

93665 

00289 

99711 

24 

37 

7  4 

52  56 

06 1 55 

69 

93845 

06445 

69 

93555 

00290 

99710 

23 

38 

6  56 

53  4 

06264 

70 

93736 

06556 

71 

93444 

00292 

99708 

22 

39 
4o 

6  48 

53  12 

06372 

72 

93628 

06666 

73 

93334 

00293 

99707 

21 
20 

II  6  4o 

0  53  20 

9.06481 

74 

10.93519 

9.06775 

75 

10.93225 

10.00295 

9.99705 

4i 

6  32 

53  28 

06589 

76 

934 1 1 

06885 

77 

931 15 

00296 

99704 

'9 

42 

6  24 

53  36 

06696 

78 

93304 

06994 

79 

93006 

00298 

99702 

18 

43 

6  16 

53  44 

06804 

8c 

93196 

07103 

81 

92897 

00299 

99;o> 

17 

44 
45 

6  8 

53  52 

0691 1 

81 

93089 

07211 

83 

92789 

oo3oi 

99699 

lb 

II  60 

0  54  0 

9.07018 

83 

10.92982 

9.07320 

84 

1 0 .  92^)80 

io.oo3o2 

9.99698 

40 

5  52 

54  8 

07124 

85 

92876 

07428 

86 

92572 

oo3o4 

99(596 

14 

47 

5  44 

54  16 

07231 

87 

92769 

07536 

88 

92464 

oo3o5 

99695 

i3 

48 

5  36 

54  24 

07337 

89 

92663 

07643 

90 

92357 

oo3o7 

99693 

12 

49 
5o 

5  28 

54  32 

07442 

91 

92558 

07751 

92 

92249 

oo3o8 

99692 

1 1 
10 

II  5  20 

0  54  40 

9.07548 

93 

10.92452 

9.07858 

94 

10.92142 

io.oo3ic 

9.99690 

5i 

5  12 

54  48 

07653 

94 

92347 

07964 

96 

92036 

oo3ii 

99689 

9 

52 

5  4 

54  56 

07758 

96 

92242 

08071 

98 

91929 

oo3i3 

99687 

8 

53 

4  56 

55  4 

07863 

98 

92137 

08177 

99 

91823 

oo3i4 

99686 

7 

54 
55 

4  48 

55  12 

07968 

100 

92032 

08283 

lOI 

9'7i7 

oo3i6 

99684 

6 

"5 

1 1  4  4o 

0  55  20 

9.08072 

102 

10.91928 

9.08389 

io3 

10.9161 1 

io.oo3i7 

9.99683 

56 

4  32 

55  28 

08 1 76 

io4 

91824 

08495 

io5 

9i5o5 

oo3i9 

9968 1 

4 

57 

4  24 

55  36 

08280 

loC 

91720 

08600 

107 

91400 

00320 

99680 

3 

58 

4  16 

55  44 

08383 

107 

91617 

08705 

loq 

91295 

oo32? 

99678 

2 

5q 

4  8 

55  52 

0848G 

IOC 

9i5i4 

08S10 

II I 

91190 

oo323 

95677 

I 

60 
M 

4  0 

56  0 

08589 

I II 

91411 

08914 

ii3   91086 

oo325 

99675 

0 

Hourp.M 

HonrA.Ai 

Cosine. 

Diff 

Secant. 

Cotangent 

Diff.  Tangent. 

Cosecant. iDiff 

Sine 

.% 

0 

A 

A 

B 

B 

C 

C 

83= 

Seconds  of  time 

1' 

2^ 

3» 

4s 

5» 

G' 

83 
84 

I 

7' 

97 
98 

I 

Prop,  parts  ot  cols.  <  B 

(c 

i4 
i4 
0 

28 
28 
0 

42 
42 

I 

56 
56 
I 

69 
70 

I 

Page  192] 

TABLE  XXVIL 

S 

Log.  S] 

nes,  Tangents,  and  Secants. 

G'. 

7° 

A 

A 

B 

B 

C        C  172° 

o 

H cur  A.M. 

Hour  P.M. 

Sine. 

Diff. 

Coserant. 

Tangent.  |Diff. 

Cotangent 

Secant. 

DifT. 

Cosine. 

M 

63 

II  40 

0  56  0 

9.08589 

0 

10.91411 

9.08914 

0 

10.91086 

io.oo325 

0 

9.99675 

I 

3  52 

56  8 

08692 

2 

9i3o8 

09019 

2 

90981 

00326 

0 

99674 

59 

2 

3  44 

S6  16 

08795 

3 

91205 

09120 

3 

90877 

00328 

0 

00672 

58 

J 

3  36 

56  24 

08897 

5 

91103 

09227 

5 

90773 

oo33o 

c  '  99670 

57 

4 
5 

3  28 

56  32 

08999 

6 

91001 

09330 

7 

90670 

oo33i 

0   99669 

56 
55 

II  3  so 

0  56  4o 

9.09101 

8 

10.90899 

9.09434 

8 

10.90566 

10.00333 

0 

9.99667 

t) 

3  12 

56  48 

09202 

10 

90798 

09537 

10 

Qo463 

00334 

0 

99666 

54 

7 

3  4 

56  56 

09304 

II 

90696 

09640 

n 

90360 

oo336 

0 

99664 

53 

8 

2  56 

57  4 

09405 

i3 

90595 

09742 

i3 

90258 

00337 

0 

99663 

52 

_9 

10 

2  48 

57  12 

09506 

i4 

90494 

09845 

i5 

90155 

00339 

0 

99661 

5i 

5o 

11  2  40 

0  57  20 

9 .  09606 

16 

10.90394 

9.09947 

16 

1 0 . 90053 

io.oo34i 

0 

9.99659 

11 

2  32 

57  28 

09707 

18 

90293 

10049 

18 

89951 

00342 

0  j  99658 

49 

12 

2  24 

57  36 

09807 

19 

90193 

ioi5o 

20 

89850 

oo344 

0 

99656 

48 

iJ 

2  16 

57  44 

09907 

21 

90093 

I0252 

21 

89748 

00345 

0 

99655 

47 

i4 
i5 

2  8 

57  52 

10006 

22 

89994 

io353 

23 

89647 

oo347 

0 

99653 

46 

45 

II  20 

0  58  0 

9.10106 

M 

10.89894 

9.10454 

24 

10.89546 

10.00349 

0 

9.99(151 

lb 

I  52 

58  8 

I0205 

2b 

89795 

io555 

2b 

89445 

oo35o 

0 

99650 

44 

17 

I  44 

58  16 

io3o4 

27 

89696 

io656 

28 

89344 

oo352 

0 

99648 

43 

i8 

I  36 

58  24 

I0402 

29 

89598 

10756 

29 

89244 

00353 

99647 

42 

19 

20 

I  28 

58  32 

io5oi 

3o 

89499 

108  56 

3i 

89144 

00355 

99645 

4 1 

4<. 

II  I  20 

0  58  40 

9.10599 

32 

10.89401 

9. 10956 

33 

10.89044 

10.00357 

9.99643 

21 

I  12 

58  48 

10697 

34 

89303 

iio56 

34 

88944 

oo358 

99642 

39 

22 

I  4 

58  56 

10795 

35 

89205 

iii55 

36 

88845 

oo36o 

I 

99640 

38 

2j 

0  56 

59  4 

10893 

37 

89107 

1 1254 

37 

88746 

oo362 

99638 

37 

24 
25 

0  48 

59  12 

10990 

38 

890 1 0 

ii353 

39 

88647 

oo363 

99637 

36 
35 

II  0  40 

0  59  20 

9. II 087 

4o 

10.88913 

9.11452 

4i 

10.88548 

io.oo365 

9.99635 

2b 

0  32 

59  28 

1 1 184 

42 

88816 

ii55i 

42 

88449 

00367 

99633 

34 

27 

0  24 

59  36 

11281 

43 

88719 

1 1 649 

44 

8835, 

oo368 

99632 

33 

2b 

0  16 

59  44 

1,377 

45 

88623 

1 1747 

46 

88253 

00370 

99630 

32 

29 

3o 

0  8 

59  52 

1 1474 

46 

88526 

11845 

47 

88 1 55 

00371 

99629 

3i 
3o 

II  00 

I  0  0 

9.11570 

48 

I0.88430 

9.11943 

49 

10.88057 

10.00373 

9.99627 

61 

10  59  5? 

0  8 

1 1666 

5o 

88334 

1204o 

5i 

87960 

00375 

99625 

29 

62 

59  44 

0  16 

11761 

5i 

88239 

i2i38 

52 

87862 

00376 

99624 

28 

dJ 

59  36 

0  24 

11857 

53 

88143 

12235 

54 

87765 

00378 

99622 

27 

34 
35 

59  28 

0  32 

1 1952 

54 

88o48 

12332 

55 

87668 

00380- 

99620 

26 

10  59  20 

I  0  4o 

9.12047 

56 

10.87953 

9. 12428 

57 

10.87572 

10.00382 

9.99618 

3b 

59  12 

0  48 

12142 

58 

87858 

12525 

59 

87475 

00383 

97617 

24 

^7 

59  ,4 

0  56 

12  236 

59 

87764 

I262I 

60 

87379 

oo385 

1 

996 1 5 

33 

38 

58  56 

I  4 

i233i 

61 

87669 

I27I7 

62 

87283 

oo387 

I 

99613 

22 

39 

4o 

58  48 

I  12 

12425 

62 

87575 

I28I3 

64 

87187 

00388 

996 1 2 

21 
20 

10  58  4o 

I  I  20 

9. 12519 

64 

10.87481 

9. 1  2909 

65 

10.87091 

10.00390 

9.99610 

4i 

58  32 

I  28 

12612 

66 

87388 

i3oo4 

67 

86996 

00392 

99608 

19 

42 

58  24 

I  36 

12706 

67 

87294 

i3o99 

68 

86901 

00393 

99607 

18 

43 

58  16 

I  44 

12799 

69 

87201 

i3i94 

70 

86806 

00395 

99605 

'7 

44 
45 

58  8 

I  52 

12892 

70 

87108 

13289 

72 

86711 
10.86616 

00397 

99603 

lb 
75 

10  58  0 

I  2  0 

9.12985 

72 

10.87015 

9.13384 

73 

10.00399 

9 . 9960 1 

4b 

67  52 

2  8 

13078 

74 

86922 

13478 

7^ 

86522 

oo4oo 

99600 

i4 

47 

57  44 

2  16 

i3i7i 

75 

86829 

13573 

77 

86427 

00402 

99598 

1 3 

48 

57  3(i 

2  24 

1 3263 

77 

86737 

1 3667 

78 

86333 

oo4o4 

99596 

12 

49 
5o 

57  28 

2  32 

13355 

7a 

86645 

1 376 1 

80 

86239 

oo4o5 

99595 

I, 

10 

10  57  20 

I   2  4o 

9.13447 

80 

10.86553' 

9-13854 

81 

10.86146 

10.00407 

9  99^*93 

5i 

57  12 

2  48 

13539 

82 

8646 1 

13948 

83 

86o52 

00409 

99591 

9 

52 

57  4 

2  55 

i363o 

83 

86370 

i4o4i 

85 

85959 

004 11 

99589 

8 

53 

56  56 

3  4 

13722 

85 

86278 

i4i34 

86 

85866 

004 1 2 

99588 

7 

54 
55 

56  48 

3  12 

i38i3 

87 

86187 

14327 

88 

85773 

oo4i4 

2 

99586 

6 

'~5 

ID  56  40 

I  3  20 

9.13904 

88 

1 0 . 86096 

9. 14320 

90 

io.8568o 

io.oo4i6 

2 

9.99584 

5b 

56  32 

3  28 

13994 

90 

86006 

1 44 1 2 

91 

85588 

004 1 8 

2 

99582 

4 

57 

56  24 

3  36 

i4o85 

91 

85915 

l45n4 

93 

85496 

00419 

2 

9958 1 

3 

58 

56  16 

3  44 

14175 

93 

85825 

14597 

95 

854o3 

00421 

2 

99579 

2 

59 

56  8 

3  52 

14366 

95 

85734 

1 4688 

96 

853i2 

00423 

2 

99577 

I 

bo 
M 

56  0 

4  0 

14356 

96 

85644 

14780 

98 

85220 

00425 

2 

99575 

0 
M 

lI(,urp.M. 

IIOUIA.M. 

Cosine. 

Difl-. 

Secant. 

Cotangent 

Dill 

Tangent. 

Cosecant. 

DilT. 

S-ine. 

U7° 

A 

A 

B 

B 

C 

c 

82" 

Seconds  of  time 

1- 

2' 

3' 

4. 

5« 

6» 

7. 

84 
86 

I 

Prop,  parts  of  cols.  J  B 

12 
12 
0 

24 

24 

0 

36 
37 

I 

48 

I 

60 
61 

I 

72 

73 

I 

TABLE  XXVIL 

fPage  193 

s 

Log.  Sines,  Tangents,  and  Secants. 

G 

8° 

A 

A 

B 

B 

c 

C  171° 

M 

0 

Hour  A.M. 

Hourp.M. 

Sine. 

Diff. 

Cosecant. 

Tangent. 

Diff. 

Cotangent 

Secant. 

Diff.  Cosine. 

M 

6^ 

10  56  0 

I  4  0 

9-14356 

0 

10. 85644 

9.147^^0 

0 

I0.85220 

10.00425 

0 

9.99575 

I 

55  52 

4  8 

14445 

1 

85555 

14872 

1 

85i28 

00426 

0 

99574 

59 

2 

55  44 

4  16 

14535 

3 

85465 

14963 

3 

85o37 

00428 

0 

99572 

58 

3 

55  36 

4  24 

14624 

4 

85376 

i5o54 

4 

84946 

oo43o 

0 

99570 

57 

4 
5 

55  28 

4  32 

i47i4 

6 

85286 

i5i45 

b 

84855 

00432 

0 

99568 

56 
55 

10  55  20 

I  4  40 

9. i48o3 

7 

10.85197 

9.15236 

7 

10.84764 

10.00434 

0 

9.99566 

6 

55  12 

4  48 

14891 

8 

85io9 

15327 

9 

84673 

00435 

0 

99565 

54 

7 

55  4 

4  56 

14980 

10 

85o2o 

i54i7 

10 

84583 

00437 

0 

99563 

53 

8 

54  56 

5  4 

15069 

II 

84931 

i55o8 

12 

84492 

00439 

0 

99561 

52 

_9 

10 

54  48 

5  12 

i5i57 

i3 

84843 

15598 

i3 

84402 

0044 1 

0 

99559 

5i 

5o 

10  54  4o 

I  5  20 

9.15245 

i4 

10.84755 

9.15688 

i4 

10.84312 

10.00443 

0 

9.99557 

II 

54  32 

5  28 

15333 

16 

84667 

15777 

16 

84223 

00444J 

0 

09556 

49 

12 

54  24 

5  36 

1 542 1 

17 

84579 

15867 

17 

84r33 

00446 

0 

99554 

48 

i3 

54  16 

5  44 

i55o8 

18 

84492 

15956 

19 

84044 

00448 

0 

99552 

47 

i4 
i5 

54  8 

5  52 

15596 

20 

844o4 

16046 

20 

83954 

oo45o 

0 

99550 

46 
45 

10  54  0 

I  6  0 

9.15683 

21 

10.84317 

9.16135 

22 

10.83865 

10.00452 

0 

9.99548 

i6 

53  52 

6  8 

15770 

23 

84230 

16224 

23 

83776 

00454 

99546 

44 

'7 

53  44 

6  16 

1 5857  24 

84i43 

i63i2 

25 

83688 

00455 

99545 

43 

i8 

53  36 

6  24 

15944 

25 

84o56 

i64oi 

26 

83599 

00457 

99'>43 

42 

19 

20 

53  28 

6  32 

i6o3o 

27 
28 

83970 

164S9 

27 

835ii 

00459 

99541 

4i 
4o 

10  53  20 

I  6  4o 

9.16116 

10. 83884 

9.16577 

29 

10.83423 

10.00461 

9.99539 

21 

53  12 

6  48 

16203 

3o 

83797 

16665 

3o 

83335 

oo463 

99537 

39 

22 

53  4 

6  56 

162891 

3i 

83711 

16753 

32 

83247 

oo465 

99535 

38 

23 

52  56 

7  4 

16374 

32 

83626 

i684r 

33 

83 1 59 

00467 

99533 

37 

24 

25 

52  48 

7  12 

1 6460 

34 
35 

83540 
10.83455 

16928 

35 

83072 

00468 

99532 

36 
35 

10  52  4o 

I  7  20 

9.16545 

9.17016 

36 

10.82984 

10.004-0 

9.99530 

26 

52  32 

7  28 

1 6631 

37 

83369 

17103 

37 

82897 

00472 

99528 

34 

27 

52  24 

7  36 

16716 

38 

83284 

17190 

39 

82810 

0047-' 

99526 

33 

28 

52  16 

7  44 

1 680 1 

39 

83i99 

17277 

40 

82723 

00476 

99524 

32 

29 

3o 

52  8 

7  52 

16886 

4i 

83ii4 

17363 

42 

82637 

00478 

99522 

3i 
3o 

10  52  0 

X  8  0 

9.16970 

42 

io.83o3o 

9.17450 

43 

10.82550 

1 0.00480 

9.99520 

3i 

5i  52 

8  8 

17055 

44 

82945 

17536 

45 

82464 

00482 

99518 

29 

32 

5t  A4 

8  16 

17139 

45 

82861 

17622 

46 

82378 

00483 

99317 

28 

33 

5i  36 

8  24 

17223 

47 

82777 

17708 

48 

82292 

0048  5 

995 1 5 

27 

34 
35 

5 1  28 

8  32 

17307 

48 

82693 

17794 

49 

82206 

00487 

995 1 3 

26 

25 

10  5i  20 

I  8  4o 

9.17I391 

49 

10.82609 

9. 17880 

5o 

10.82120 

10.00489 

9.9951 1 

3o 

5[  12 

8  48 

17474 

5i 

82526 

17965 

52 

82035 

00491 

99309 

24 

37 

5i  4 

8  56 

17558 

52 

82442 

i8o5i 

53 

81949 

00493 

99507 

23 

38 

5o  56 

9  4 

17641 

54 

82359 

i8i36 

55 

81864 

00495 

995o5 

22 

39 

5o  48 

9  12 

17724 

55 

82276 

18221 

5b 

81779 

00497 

995o3 

21 

20 

4o 

10  5o  4o 

I  9  20 

9.17807  56 

10.82193 

9.18306 

53 

10.81694 

10.00499 

9.99501 

41 

5o  32 

9  28 

17890 

58 

82110 

18391 

59 

81609 

oo5or 

99499 

19 

42 

5o  24 

9  36 

17973 

59 

82027 

18475 

bi 

8i525 

oo5o3 

99497 

18 

43 

5o  16 

9  44 

i8o55 

61 

8.945 

i856o 

62 

8i44o 

oo5o5 

99495 

17 

44 
45 

5o  8 

9  52 

i8i37 

62 

8 1 863 

18644 

63 

81 356 

oo5o6 

99494 

16 

75 

10  5o  11 

I  10  0 

9. 1S220 

63 

10.81780 

9.18728 

65 

10.81272 

io.oo5o8 

9 • 99492 

46 

49  52 

10  8 

i83o2 

65 

81698 

18812 

66 

81188 

oo5io 

99490 

i4 

47 

49  44 

10  16 

18383 

66 

81617 

18896 

68 

81104 

■ 005l2 

99488 

i3 

48 

49  36 

10  24 

18465 

68 

8i535 

18979 

69 

81021 

oo5i4 

2 

99486 

12 

49 
5o 

49  28 
10  49  20 

10  32 

18547 

69 

81453 

19063 

71 

80937 

oo5i6 

2 

99484 

11 

10 

I  10  40 

9.18628 

71 

10.81372 

9.19146 

72 

10.80854 

io.oo5i8 

2 

9.99482 

5i 

49  12 

10  48 

18709 

72 

81291 

19229 

74 

80771 

00520 

2 

99480 

9 

52 

49  4 

10  56 

18790 

73 

81210 

19312 

75 

80688 

Oo52  2 

2 

99478 

8 

53 

48  56 

II  4 

1 887 1 

75 

81 129 

19395 

76 

8o6o5 

oo524 

2 

99476 

7 

.'>4 
5"5 

48  48 

1 1  12 

18952 

76 

81048 

19478 

7B 

8o52  2 

00526 

2 

99474 

6 

10  48  40 

I  11  20 

9. 19033 

78 

10.80967 

9. 19561 

79 

10.80439 

10.00528 

2 

9.99472 

56 

48  32 

■  II  28 

191  r3 

79 

80887 

19643 

81 

80357 

oo53o 

2 

99470 

4 

!)7 

48  24 

II  36 

19193 

80 

80807 

19725 

82 

80275 

oo532 

2 

99468 

3 

58 

48  16 

II  44 

19273 

82 

80727 

19807 

84 

80193 

oo534 

2 

99466 

9 

59 

48  8 

II  52 

19353 

83 

80647 

19889 

85 

801 11 

oo536 

2 

99464 

I 

60 
M 

48  0 

12  0 

19433 

85 

80567 

1 997 1 

87 

80029 

oo538 

2 

99462 

Hour  P.M. 

Hour  A.M. 

Cosine. 

Diff. 

Secant. 

Cotann-entlDlff. 

Tangent. 

Cosecant. 

Diff.   Sine. 

9t>"^ 


C      81°' 


Seconds  of  time 

1- 

2' 

3" 

32 
32 

I 

4- 

42 
43 

I 

5' 

53 
5.^ 

I 

63 
65 

I 

7" 

74 
76 
a 

Prop  parts  of  cols  <  B 
f  C 

II 
II 
0 

21 
22 
0 

25 


Paye  191T 

TABLE 

.  XXVIL 

S'. 

Log.  Sines,  Tan 

gents,  and  Secants. 

QK 

9° 

A 

A 

B 

B 

C 

C  170° 

M 

0 

Hour  A.M. 

Hour  P.M. 

Sine. 

Diff. 

Cosecant. 

Tang'cnt. 

Uiff. 

Cotangent 

Secant. 

Uiff.  Cosine. 

M 

60 

10  48  0 

1  12  0 

9.19433 

0 

10.80567 

9.19971 

0 

10.80029 

io.oo538 

0  9.99462 

I 

47  52 

12  8 

19513 

I 

80487 

2oo53 

I 

799.17 

oo54o 

0 

99460 

59 

2 

47  44 

12  16 

19592 

3 

8o4o8 

20 1 34 

3 

79866 

oo542 

0 

99458 

58 

3 

Ai  36 

12  24 

19672 

4 

80328 

20216 

4 

79784 

oo544 

0 

99456 

57 

4 
5 

47  28 

12  32 

1 975 1 

5 

80249 

20297 

5 

79708 

oo546 

0 

99454 

56 
55 

10  47  20 

I  12  4o 

9.19830 

6 

10.80170 

9.20378 

6 

10.79622 

io.oo54S 

0 

9.99452 

6 

47  12 

12  48 

19909 

8 

80091 

20459 

8 

79541 

oo55o 

0 

99450 

54 

7 

47  4 

12  56 

19988 

9 

80012 

20  540 

9 

79460 

oo552 

0 

99448 

53 

8 

A^   56 

i3  4 

20067 

10 

79933 

20621 

10 

79879 

oo554 

0 

99446 

52 

_9 

10 

46  48 

i3  12 

20145 

II 

79855 

20701 

12 

79299 

oo556 

0 

99444 

5i 
5^ 

10  46  4<J 

1  1 3  20 

9.20223 

i3 

10.79777 

9.20782 

i3 

10.79210 

io.oo558 

0 

9.99442 

II 

46  32 

i3  28 

2o3o2 

lA 

79698 

20862 

14 

79188 

oo56o 

0 

99440 

49 

12 

46  24 

i3  36 

2o38o 

i5 

79620 

20942 

16 

79o58 

oo56:> 

0 

99438 

48 

i3 

46  16 

i3  AA 

20458 

16 

79543 

21022 

17 

78978 

oo564 

0 

99436 

47 

i4 
i5 

46  8 

i3  52 

2o535 

18 

79465 

21102 

18 

78S98 

oo566 

0 

99434 

46 
45 

10  4G  0 

I  i4  0 

9.20613 

19 

10.79387 

9.21182 

19 

10.78818 

10.00568 

9.99482 

i6 

45  52 

i4  8 

20691 

20 

79309 

2 1 261 

21 

78789 

00571 

99429 

AA 

17 

45  A4 

i4  16 

2076S 

21 

79232 

2i34i 

22 

78659 

00578 

99427 

Ai 

i8 

45  36 

i4  24 

20845 

23 

79155 

21420 

23 

78580 

00575 

99425 

42 

£9 

20 

45  28 

i4  32 

20922 

24 

79078 

21499 

25 

78501 

00577 

99428 

Ai 
4() 

10  45  20 

I  1 4  40 

9.20999 

25 

10.79001 

9.21578 

26 

10.78422 

10.00579 

9.99421 

21 

45  12 

i4  48 

21076 

26 

78924 

21657 

27 

78343 

oo5Si 

99419 

39 

22 

45  4 

i4  56 

2ii53 

28 

78847 

21736 

28 

78264 

oo583 

99417 

38 

23 

AA   56 

i5  4 

21229 

29 

78771 

21814 

3o 

78186 

oo585 

994 1 5 

37 

24 
25 

AA  48 

i5  12 

2i3o6 

3o 

78694 

21893 

3i 

78107 

00587 

994 1 3 

36 
35 

10  AA  40| 

I  i5  20 

9.21382 

3i 

10.78618 

9.21971 

32 

10.78029 

10.00589 

9.9941 1 

26 

AA  32 

i5  28 

2i458 

33 

78542 

22049 

34 

77951 

00591 

99409 

M 

27 

44  24 

i5  36 

2 1 534 

34 

78466 

22127 

35 

77873 

00593 

99407 

66 

28 

AA   16 

i5  44 

21610 

3b 

78390 

222o5 

36 

77795 

00596 

99404 

32 

29 

3o 

44  8 

i5  52 

2 1 635 

37 

783i5 

22283 

38 

77717 

00598 

99402 

3i 
3o 

10  44  0 

I  16  0 

9.21761 

38 

10.78289 

9.22361 

39 

10.77689 

10.00600 

9.99400 

3i 

43  52 

•  16  8 

2 1 836 

39 

78 164 

22433 

40 

77562 

00602 

99898 

29 

32 

43  44 

16  16 

21912 

4o 

7808S 

225l6 

4i 

77484 

00604 

99896 

28 

33 

43  36 

16  24 

21987 

42 

78013 

22593 

43 

77407 

00606 

99894 

27 

34 
35 

43  28 

16  32 

22062 

43 

77938 

22670 

44 

77880 

00608 

99J92 

26 

25 

10  43  20 

I  16  4<J 

9.22137 

AA 

10.77863 

9.22747 

45 

10.77258 

10.00610 

9.99890 

36 

43  12 

16  48 

22211 

45 

77789 

22824 

47 

77176 

00612 

99888 

24 

37 

43  4 

16  56 

22286 

47 

77714 

22901 

48 

77099 

006 1 5 

99385 

23 

38 

42  56 

17  4 

22361 

48 

77639 

22977 

49 

77028 

00617 

99888 

22 

39 

4o 

42  43 

17  12 

22435 

49 

77565 

23o54 

bo 

76946 

006 1 9 

99881 

21 
20 

10  42  4o 

I  17  20 

9.22509 

5o 

10.77491 

9.23i3o 

52 

10.76870 

10.00621 

9.99879 

4i 

42  32 

17  28 

22583 

52 

77417 

23206 

53 

76794 

00628 

99377 

19 

42 

42  24 

17  36 

22657 

53 

77343 

23283 

54 

76717 

00626 

99875 

18 

43 

42  16 

17  AA 

22731 

54 

77269 

23359 

56 

76641 

00628 

2 

99872 

17 

44 
45 

42  8 

17  52 

22805 

55 

77195 

23435 

57 

76565 

00680 

2 

99870 

16 

Is 

10  42  0 

1  18  0 

9.22878 

57 

10.77122 

9.23510 

58 

10.76490 

10.00682 

2 

9.99868 

46 

4i  5a 

18  8 

22952 

58 

77048 

23586 

60 

76414 

00684 

2 

99866 

i4 

47 

4i  AA 

18  16 

23o25 

59 

76975 

2366 1 

61 

76339 

00686 

2 

99864 

i3 

48 

Ai   36 

18  24 

23098 

60 

76902 

23737 

62 

76268 

00688 

2 

99862 

12 

49 
5o 

4i  28 

18  32 

23171 

9.23244 

62 

76829 

238l2 

63 

76188 

00641 

2 

99359 

II 
10 

10  4i  20 

I  18  4o 

63 

10.76756 

9.23887 

65 

10.76113 

10.00643 

2 

9.99857 

5i 

Ai   12 

18  48 

233i7 

64 

76683 

28962 

66 

76088 

00645 

2 

99355 

9 

52 

4i  4 

18  56 

23390 

65 

76610 

24087 

67 

75968 

00647 

2 

99853 

b 

53 

40  56 

19  4 

23462 

67 

76538 

24  I  I  2 

69 

75888 

00649 

2 

99831 

7 

54 
55 

4o  48 

19  12 

23535 

68 

76465 

24186 

70 

758 1 4 

oo652 

2 

99348 

6 

10  4o  4o 

I  19  20 

9.23607 

69 

10.76393 

9.24261 

71 

10.75789 

10.00654 

2 

9.99846 

56 

4o  32 

19  28 

23679 

71 

76321 

24335 

73 

75665 

oo656 

2 

99844 

4 

!)7 

4o  24 

19  36 

23752 

72 

76248 

24410 

74 

75590 

00658 

2 

99342 

3 

58 

4o  16 

19  AA 

238231  73 

76177 

24484 

75 

75516 

00660 

2 

99840 

2 

59 

40  8 

19  52 

23895  74 

76105 

24558 

76 

75442 

oo663 

2 

99337 

I 

60 
M 

4o  0 

20  0 

23967  76 

76033 

24632 

78 

75368 

oo665 

2 

99835 

0 

Hour  P.M. 

Hour  A.M. 

Cosine.  iDiff. 

Secant. 

Colansfcnt 

DilT 

Tangent. 

Cosecant. 

Diff 

Sine. 

OD" 


A 

A 

B 

B 

C 

1= 

2= 

3' 

4= 

38 
39 

I 

5^ 

47 

49 
I 

6= 

57 
58 
3 

7" 

"66~ 
68 
1 

Prop,  parts  of  cols. 

!■ 

9 
10 
0 

19 
19 

1 

28 
29 
I 

C       80=' 


TABLE  XXVII. 

[Page  195 

s 

/_ 

Log.  Smes,  Tangents,  and  Secants, 

G'. 

10 

0 

A 

A 

B 

E 

C 

C   169° 

M 

0 

Hour  A.M. 

Hourp.M. 

Sine. 

Diff. 

Cosecant. 

Tang'ent. 

Diff. 

Cotangent 

Secant. 

Diff. 

Cosine. 

31 

6^ 

10  4o  0 

I  20  0 

9.23967 

0 

10.76033 

9.24682 

0 

10.75368 

10.00665 

0 

9.99335 

I 

39  52 

20  8 

24039 

I 

75961 

24706 

1 

75294 

00667 

0 

99833 

59 

2 

39  44 

20  16 

24110 

a 

75890 

24779 

2 

75221 

00669 

0 

99331 

58 

3 

39  36 

20  24 

24181 

3 

75819 

24853 

4 

75i47 

00672 

0 

99828 

57 

4 
5 

39  28 

20  32 

24253 

5 

75747 

24926 

5 

75074 

00674 

0 

99826 

56 
55 

10  39  20 

I  20  40 

9.24324 

6 

10.75676 

9.25000 

6 

10.75000 

10.00676 

0 

9.99824 

6 

39  12 

20  48 

24395 

7 

756o5 

25078 

7 

749^7 

00678 

0 

99822 

54 

7 

39  4 

20  56 

24466 

8 

75534 

25i46 

8 

74854 

0068 1 

0 

99819 

53 

8 

38  56 

21  4 

24536 

9 

75464 

25219 

9 

74781 

00683 

0 

99817 

52 

10 

38  48 

21  12 

24607 

10 

75393 

25292 

11 

74708 

0068  5 

0 

9931 5 

5i 

5o 

10  38  4o 

I  21  20 

9.24677 

1 1 

10  75323 

9.25365 

12 

10.74635 

10.00687 

0 

9.99813 

II 

38  32 

21  28 

24748 

i3 

75252 

25437 

1 3 

74563 

00690 

0 

99810 

49 

12 

38  24 

21  36 

24818 

i4 

75182 

255io 

i4 

74490 

00692 

0 

99308 

48 

i3 

38  16 

21  44 

24888 

i5 

75i  12 

25582 

i5 

744.8 

00694 

99806 

47 

i4 
i5 

38  8 

21  52 

24958 

16 

75o42 

25655 

16 

74345 

00696 

I 

99804 

46 
45 

10  38  0 

I  22  0 

9.2502S 

17 

10.74972 

9.25727 

18 

10.74273 

10.00699 

9.99801 

i6 

37  52 

22  8 

25098 

18 

74902 

25799 

19 

74201 

0070 1 

99299 

44 

17 

37  44 

22  16 

25i68 

19 

74832 

25871 

20 

74129 

00703 

99297 

43 

i8 

37  36 

22  24 

25237 

20 

74763 

25943 

21 

74o57 

00706 

99294 

42 

!9 

20 

37  28 

22  32 

25307 

22 

74693 

26015 

22 

73985 

00708 

99292 

4i 
4o 

10  37  20 

I  22  40 

9.25376 

23 

10.74624 

9.26086 

24 

10.78914 

10.00710 

9.99290 

21 

37  12 

22  48 

25445 

24 

74555 

26i5S 

25 

78842 

00712 

99288 

39 

22 

37  4 

22  56 

255i4 

25 

74486 

26229 

26 

73771 

00715 

99285 

38 

23 

36  56 

23  4 

25583 

26 

74417 

26801 

27 

73699 

00717 

99283 

37 

24 
25 

36  48 

23  12 

25652 

27 

74348 

26872 

28 

73628 

00719 

99281 

36 
35 

10  36  4o 

I  23  20 

9.25721 

28 

10.74279 

9.26443 

29 

10.73557 

10.00722 

9.99278 

26 

36  32 

23  28 

25790 

3o 

74210 

265 1 4 

3. 

73486 

00724 

99276 

34 

27 

36  24 

23  36 

25858 

3i 

74142 

26585 

32 

7341 5 

00726 

99274 

33 

28 

36  16 

23  44 

22927 

32 

74073 

26555 

33 

73345 

00729 

99271 

32 

29 

3o 

36  8 

23  52 

25995 
9.26063 

33 

74oo5 

26726 

34 

73274 

0073  i 

99269 

3i 

3o 

10  36  0 

I  24  0 

34 

10.78937 

9.26797 

35 

10.73208 

10.00733 

k 

9.99267 

3i 

35  52 

24  8 

26i3i 

35 

73869 

26S67 

36 

73i33 

00736 

99264 

29 

32 

35  44 

24  16 

26199 

i5 

73801 

26987 

38 

73o63 

00788 

99262 

28 

33 

35  36 

24  24 

26267 

38 

73733 

27008 

39 

72992 

00740 

99260 

27 

35 

35  28 

24  32 

26335 

39 

73665 

27078 

40 

72922 

00743 

99257 

26 

ID  35  20 

I  24  4o 

9.26403 

40 

10.73597 

9.27148 

4i 

10.72852 

10.00745 

9.99255 

36 

35  12 

24  48 

26470 

4i 

73530 

27218 

42 

72782 

00748 

99252 

24 

37 

35  4 

24  56 

26538 

42 

73462 

27288 

44 

72712 

00750 

99250 

23 

38 

34  56 

25  4 

26605 

43 

73395 

27357 

45 

72643 

00752 

99248 

22 

39 

4o 

34  48 

25  12 

266-2 

44 

73328 

27427 

46 

72573 

00755 

1 

99245 

21 
20 

10  34  4o 

I  25  20 

9.26739 

45 

10.78261 

9.27496 

47 

10.72504 

10.00757 

1 

9.99243 

4i 

34   32 

25  28 

26S06 

47 

73194 

27566 

48 

72434 

00759 

2 

99241 

19 

42 

■  34  24 

25  36 

26873 

48 

78127 

27685 

49 

72365 

00762 

2 

99288 

18 

43 

34   16 

25  44 

26940 

49 

78060 

27704 

5i 

72296 

00764 

2 

99236 

17 

44 
45 

34  8 

25  52 

27007 

5o 

72998 

27773 

52 

72227 

00767 

2 

99233 

16 

75 

10  34  0 

I  26  0 

9.27C7J 

5. 

10.72927 

9.27842 

53 

10.72153 

10.00769 

2 

9.99231 

46 

33  52 

26  8 

27140 

52 

72860 

2791 1 

64 

72089 

00771 

2 

99229 

i4 

47 

33  44 

26  16 

27206 

53 

72794 

27980 

55 

72020 

00774 

2 

99226 

i3 

48 

33  36 

26  24 

27273 

55 

72727 

28049 

56 

71951 

00776 

2 

99224 

12 

49 
5o 

33  28 

26  32 

27339 

56 

72661 

28117 
9.28186 

53 

71888 

00779 

2 

99221 

II 
10 

10  33  20 

I  26  4o 

9.27405 

57 

10.72595 

59 

10.71814 

10.00781 

2 

9.99219 

5[ 

33  12 

26  48 

27471 

58 

72529 

28254 

60 

71746 

00783 

2' 

99217 

9 

52 

33  4 

26  56 

27537 

59 

72463 

28828 

61 

71677 

00786 

2 

99214 

8 

53 

32  56 

27  4 

27602 

()0 

72398 

28891 

62 

71609 

00788 

2 

99212 

7 

55 

32  48 

27  12 

27668 

61 

72882 

28459 

63 

71541 

00791 

2 

99209 

6 

5 

10  32  4o 

I  27  20 

9.27734 

63 

10.72266 

9.2S527 

65 

10.71473 

10.00793 

2 

9.99207 

5b 

32  32 

27  28 

27799 

64 

72201 

28595 

66 

7i4o5 

00796 

2 

99204 

4 

57 

32  24 

27  36 

27864 

65 

72186 

28662 

67 

7i338 

00798 

2 

99202 

3 

58 

32  16 

27  44 

27930 

66 

72070 

28780 

68 

71270 

00800 

2 

99200 

2 

59 

32  8 

27  62 

27995 

67 

72og5 

28798 

69 

71202 

oo8o3 

2 

99197 

I 

60 

32   0 

28  0 

28060 

68 

71940 
Secant. 

28865 

71 

7ii35 

oo8o5 

2 

99195 

0 
M 

I  [our  P.M. 

Hour  A.M. 

Cosine. 

Din: 

Cotangent 

Dift: 

Tangent. 

Cosecant. 

Difi-. 

Sine. 

100° 


79° 


Seconds  of  time 

1' 

2" 

3^ 

4s 

34 
35 

I 

5^ 

43 
44 

I 

6^ 
5i 
53 
2 

7.1 

60  j 
62 
2 

(A 
Prop,  parts  of  cols.  ^   B 

9 
9 
0 

17 
18 

1 

26 
26 

I 

Page  19G] 

TABLE  XXVIL 

S' 

Log.  Sines,  Tancrents,  and  Secants. 

G'. 

11 

D 

A 

A 

B 

B 

C 

C  168° 

0 

Hour  A.M. 

Hourp.M. 
I  28  0 

Sine. 

Diff. 

Cosecant. 

Tangent. 

Diff. 

Cotangent 

Secant. 

Diff. 

Cosine. 

M 

60 

10  32   0 

9.28060 

0 

10.71940 

9.28865 

0 

10.71135 

io.oo8o5 

0 

9.99195 

1 

3l  52 

28  8 

28125 

I 

71875 

28933 

I 

71067 

00808 

0 

99192 

59 

2 

3i  4i 

28  16 

28190 

2 

71810 

29000 

2 

71000 

00810 

0 

99190 

08 

3 

3i  36 

28  24 

28254 

3 

71746 

29067 

3 

70933 

008 1 3 

0 

99187 

57 

4 
5 

3i  28 

28  32 

28319 

4 
5 

71681 

29134 

4 

70866 

0081 5 

0 

99185 

5b 

55 

10  3i  20 

1  28  4o 

9.28384 

10.71616 

9.29201 

5 

10.70799 

10.00818 

0 

9.99182 

b 

3i  12 

28  48 

28448 

b 

71552 

29268 

6 

70732 

00820 

0 

99180 

54 

7 

3i  4 

28  56 

285l2 

7 

71488 

29335 

8 

70665 

00823 

0 

99 '77 

53 

8 

3o  56 

29  4 

28577 

8 

71423 

29402 

9 

70598 

00825 

0 

99I7D 

52 

_9 

10 

3o  48 
10  3o  4o 

29  12 
I  29  20 

28641 
9.28705 

9 

71359 

29468 

10 

70532 

00828 

0 

99172 

5i 
5o 

10 

10.71295 

9.29535 

11 

10.70465 

io.oo83o 

0 

9.99170 

II 

3o  32 

29  28 

28769 

II 

7i23i 

29601 

12 

70399 

00833 

0 

99167 

49 

12 

3o  24 

29  36 

28833 

12 

71167 

29668 

i3 

70332 

00835 

99165 

48 

iJ 

3o  16 

29  44 

28896 

i3 

71104 

29734 

i4 

70266 

oo838 

99162 

47 

i4 
i5 

3o  8 

29  52 

28960 

i4 

71040 

29800 

i5 

70200 

00840 

99160 

46 
45 

10  3o  0 

I  3o  0 

9.29024 

16 

10.70976 

9.29866 

16 

10.70134 

10.00843 

9.99157 

lb 

29  52 

3o  8 

29087 

17 

70913 

29932 

17 

70068 

00845 

99155 

44 

17 

29  44 

3o  16 

29150 

18 

7o85o 

29998 

18 

70002 

00848 

99152 

43 

i8 

29  36 

3o  24 

29214 

19 

70786 

3oo64 

19 

69936 

oo85o 

99i5o 

42 

19, 

20 

29  28 

3o  32 

29277 

20 

70723 

3oi3o 

20 

69S70 

00853 

99 '47 

4i 
4o 

10  29  20 

I  3o  4() 

9.29340 

21 

10.70660 

9.30195 

22 

10.69805 

10.00855 

9.99145 

21 

29  12 

3o  48 

29403 

22 

70597 

30261 

23 

69739 

oo858 

99142 

39 

22 

29  4 

3o  56 

29466 

23 

70534 

3o326 

24 

69674 

00860 

99140 

38 

2j 

28  56 

3i  4 

29529 

24 

70471 

3o39i 

25 

69609 

00863 

99137 

37 

24 
25 

28  48 

3l  12 

29591 

25 

70409 

30457 

26 

69543 

00865 

99135 

36 
35 

10  28  4o 

I  3i  20 

9.29654 

26 

10.70346 

9.3o522 

27 

10.69478 

10.00868 

9.99132 

26 

28  32 

3i  28 

29716 

27 

70284 

3o587 

28 

69413 

00870 

99i3o 

34 

27 

28  24 

3i  36 

29779 

28 

70221 

3o652 

29 

69348 

00873 

99127 

33 

28 

28  16 

3i  44 

.  29841 

29 

70159 

30717 

3o 

69283 

00876 

99124 

32 

29 

3o 

28  8 

3i  52 

29903 

3o 

70097 

30782 

3i 

69218 

00878 

99122 

3i 

3o 

lo  28  0 

I  32   0 

9.29966 

3i 

10.70034 

9.30846 

32 

10.69154 

10.008S1 

9.99119 

3i 

27  52 

32  8 

30028 

32 

69972 

30911 

33 

69089 

oo883 

991 17 

29 

32 

27  44 

32  16 

30090 

33 

69910 

30975 

35 

69025 

00886 

99114 

28 

33 

27  36 

32  24 

3oi5i 

34 

69849 

3io4o 

36 

68960 

00888 

99112 

27 

34 

35 

27  28 

32  32 

3o2i3 

35 

69787 

3i  io4 

37 

68896 

00891 

99109 

26 

25 

10  27  20 

I  32  40 

9.30275 

36 

10.69725 

9.31168 

38 

10.68832 

10.00894 

2 

9.99106 

36 

27  12 

32  48 

3o336 

37 

69664 

31233 

39 

68767 

00896 

2 

99104 

24 

37 

27  4 

32  56 

30398 

38 

69602 

31297 

4o 

68703 

00899 

2 

99101 

23 

38 

26  56 

33  4 

30459 

39 

69541 

3i36i 

4i 

68639 

00901 

2 

99099 

22 

39 
4o 

26  48 

33  12 

3o52i 

40 

69479 

3i425 

42 

68575 

00904 

2 

99096 

21 
20 

10  26  4o 

I  33  20 

9.3o582 

4i 

10.69418 

9.31489 

43 

io.6S5ii 

10.00907 

2 

9.99093 

4i 

26  32 

33  28 

3o643 

42 

69357 

3i552 

44 

68448 

00909 

2 

99091 

19 

42 

26  24 

33  36 

30704 

43 

69296 

3i6i6 

45 

68384 

00912 

2 

99088 

18 

43 

26  16 

33  44 

30765 

45 

69235 

31679 

46 

68321 

00914 

2 

99086 

17 

44 
45 

26  8 

33  52 

30826 

46 

69174 

31743 

47 

68257 

00917 

2 

99083 

16 

75 

10  26  0 

I  34  0 

9.308S7 

47 

10.69113 

9.31806 

49 

10.68194 

10.00920 

2 

9.99080 

46 

25  52 

34  8 

30947 

48 

69053 

31870 

5o 

68i3o 

00922 

2 

99078 

14 

47 

25  44 

34  16 

3 1008 

49 

68992 

31933 

5i 

G8067 

00925 

2 

99075 

i3 

48 

25  36 

34  24 

3 1 068 

5o 

68932 

31996 

52 

68004 

00928 

2 

99072 

12 

49 
5o 

25  28 

34   32 

31129 

5i 

68871 

32059 

53 

67941 

00930 

2 

99070 

II 

10 

10  25  20 

I  34  4o 

9.31189 

52 

10.68811 

9.32122 

54 

10.67878 

10.00933 

2 

9.99067 

5i 

25  12 

34  48 

3i25o 

53 

68750 

32185 

55 

67815 

00936 

2 

99064 

9 

52 

25  4 

34   56 

3i3io 

54 

68690 

32248 

56 

67752 

00938 

2 

99062 

8 

53 

24  56 

35  4 

3 1 370 

55 

6863o 

323ii 

57 

67689 

00941 

2 

99059 

7 

54 
55 

24  48 

35  12 

3i43o 

56 

68570 

32373 

58 

67627 

00944 

2 

99056 

b 

5 

10  24  4o 

I  35  20 

9.31490 

57 

1 0.685 10 

9.32436 

59 

10.67564 

1 0 . 00946 

2 

9.99054 

56 

24  32 

35  28 

3i549 

58 

6845 1 

32498 

60 

67502 

00949 

2 

9905 1 

4 

57 

24  24 

35  36 

3 1 609 

59 

6S391 

3256i 

61 

67439 

00952 

2 

99048 

3 

58 

24  16 

35  44 

31669 

60 

68331 

32623 

63 

67377 

00954 

2 

99046 

2 

5q 

24  8 

35  52 

31728 

61 

68272 

32685 

64 

67315 

00957 

3 

99043 

I 

60 
M 

24  0 

36  0 

31788 

62 

68212 

32747 

65 

67253 

00960 

3 

99040 

0 
M 

Hour  P.M. 

Hour  A.M. 

Cosine. 

Diff. 

Secant. 

Cotangent 

Diff. 

Tangent. 

Cosecant. 

Diff. 

Sine. 

101° 


C      78^ 


Seconds  of  time 

1* 

2" 

3' 

4« 

5- 

6' 

7' 

Prop,  parts  »>f  cois.  <  Q 

1                         (c 

8 
8 
0 

16 
16 

! 

23 

24 

I 

3i 

?2 

I 

39 
40 

2 

47 
4o 

3 

54 
57 
a 

"~ 

TABLE  XXVIL 

[Page  107 

SI. 

Log 

.  Sines,  Tan 

gents,  and  Secants. 

at. 

12= 

A 

A 

B 

B 

C 

C  167° 

o 

Hour  A.M. 

Hour  P.M. 

Sine. 

Diff. 

Cosecant. 

Tangent. 

Diff. 

Cotaiij^enl 

Secant. 

Difl.. 

Cosine. 

M 

60 

to  24  0 

I  36  0 

9.31788 

0 

10.68212 

9.32747 

0 

10.67253 

10.00960 

0 

9 . 99040 

I 

23  52 

36  8 

3 1847 

I 

68 1 53 

32810 

I 

67 1 90 

00962 

0 

99o38 

r)9 

? 

23  4i 

36  16 

31907 

2 

68093 

32872 

2 

67128 

00965 

0 

99035 

58 

3 

23  36 

36  24 

31966 

3 

68034 

32933 

3 

67067 

00968 

0 

99032 

57 

4 
5 

23  28 

36  32 

32025 

4 

67975 

32995 

4 

67005 

00970 

0 

99o3o 

06 

55 

10  23  20 

I  36  40 

9.320S4 

5 

10.67916 

9.33057 

5 

10.66943 

10.00973 

0 

9.99027 

6 

23  12 

36  48 

32143 

6 

67857 

33119 

6 

66881 

o<r976 

0 

99024 

54 

7 

23  4 

36  56 

32202 

7 

67798 

33 1 80 

7 

66820 

00978 

0 

99022 

53 

s 

22  56 

37  4 

32261 

8 

67739 

33242 

8 

66758 

0098 1 

0 

99019 

52 

_9 

10 

2  2  48 

37  12 

32319 

9 

67681 

333o3 

9 

66697 
10. 666 3 5 

00984 

0 

990 1 6 

5i 

5^ 

10  22  4l) 

I  37  20 

9.3^378 

10 

10.67622 

9.33365 

10 

10.00987 

0 

9.99013 

1 1 

22  32 

37  28 

32437 

10 

67553 

33426 

11 

66574 

00989 

990 1 1 

49 

[3 

22  24 

37  36 

32495 

II 

67505 

33487 

12 

665 1 3 

00992 

99008 

48 

i3 

23  16 

37  44 

32553 

12 

67447 

33548 

i3 

66452 

00995 

99005 

47 

i4 
i5 

22   8 

37  52 

32612 

i3 

67388 

33609 

i4 

66391 

00998 

99002 

46 

45 

10  22   0 

I  38  0 

9.32670 

i4 

10.67330 

9.33670 

i5 

io.6633o 

10.01000 

9 . 99000 

i6 

21  52 

38  8 

32728 

i5 

67272 

33731 

16 

66269 

oioo3 

98997 

44 

17 

21  44 

38  16 

32786 

16 

67214 

33792 

17 

66208 

0 1 006 

98994 

43 

i8 

21  36 

38  24 

32844 

17 

67156 

33853 

18 

66147 

01009 

98991 

42 

12 

20 

21  28 

38  32 

32902 

18 

67098 

33913 

'9 

66087 

0101 1 

98989 

4i 
4o 

10  21  20 

I  38  40 

9.32960 

19 

10.67040 

9.33974 

20 

10.66026 

10.01014 

9.98986 

2r 

21  12 

38  48 

33oi8 

20 

66982 

34o34 

21 

65966 

01017 

98983 

39 

22 

21   4 

38  56 

33075 

21 

66925 

34095 

22 

65905 

01020 

98980 

38 

23 

20  56 

39  4 

33i33 

22 

66867 

34i55 

23 

65845 

01022 

98978 

37 

24 
25 

20  48 

39  12 

33190 

23 

66810 

34215 

24 

65785 

01025 

98975 

3b 
35 

10  20  4o 

I  3q  20 

9.33248 

24 

10.66752 

9.34276 

25 

10.65724 

10.01028 

9.98972 

26 

20  32 

39  28 

333o5 

25 

66695 

34336 

26 

65664 

oio3i 

98969 

34 

27 

20  24 

39  36 

33362 

26 

66638 

34396 

27 

656o4 

oio33 

98967 

dS 

28 

20  16 

39  44 

33420 

27 

66580 

34456 

28 

65544 

oio36 

98964 

32 

29 

3o 

20  8 

39  52 

33477 

28 

66523 

34516 

29 

65484 

01039 

98961 

3i 
3^ 

10  20  0 

I  4o  0 

9.33534 

29 

10.66466 

9.34576 

3o 

10.65424 

10.01042 

9.98958 

3i 

19  52 

4o  8 

33591 

29 

66409 

34635 

3i 

65365 

01045 

98955 

29 

32 

19  44 

4o  16 

33647 

3o 

66353 

34695 

32 

653o5 

01047 

98953 

28 

33 

19  36 

4o  24 

33704 

3i 

66296 

34755 

33 

65245 

oio5o 

2 

98950 

27 

34 
35 

19  28 

4o  32 

33761 

32 

"33 

66239 
10.66182 

34814 

34 

65 186 

01053 

2 

98947 

2b 

10  19  20 

I  4o  4o 

9.33818 

9.34874 

35 

io.65i26 

io.oio56 

2 

9.98944 

36 

19  12 

40  48 

33874 

34 

66126 

34933 

36 

65067 

01059 

2 

98941 

24 

37 

19  4 

40  56 

33931 

35 

66069 

34992 

•37 

65oo8 

0 1 062 

2 

98938 

23 

38 

iS  56 

4i  4 

33987 

36 

660 1 3 

35o5i 

38 

64949 

01064 

2 

98936 

22 

39 

40 

18  48 

4i  12 

34043 

37 

65957 

35iii 

39 

64889 

0 1 067 

2 

98933 

21 

20 

10  18  4o 

I  4 1   20 

9.34100 

38 

10.65900 

9.35170 

40 

io.6483o 

io.ou>7o 

2 

9.98930 

4i 

18  32 

4i  28 

341 56 

39 

65844 

35229 

4i 

64771 

01073  2 

98927 

19 

42 

18  54 

41  36 

34212 

4o 

65788 

35288 

42 

64712 

01076  2 

98924 

18 

43 

18  16 

4 1  44 

34268 

4. 

65732 

35347 

4-i 

64653 

01079 

2 

98921 

17 

44 
45 

]8  8 

4i  52 

34324 

42 

65676 

354o5 

44 

64595 

01 08 1 

2 

98919 

lb 
l5 

n  18  0 

1  42  0 

9.34380 

43 

10.65620 

9.35464 

45 

10.64536 

10.01084 

2 

9.98916 

46 

17  52 

42  8 

34436 

44 

65564 

35523 

46 

64477 

01087 

2 

98913 

14 

47 

17  44 

42  j6 

34491 

45 

655o9 

35581 

47 

64419 

01090 

2 

98910 

i3 

48 

17  36 

42  24 

34547 

46 

65453 

35640 

48 

64360 

01093 

2 

98907 

12 

49 

5ci 

17  28 

42  32 

34602 
9.34658 

47 

6539S 

35698 

49 

t)0 

643o2 

0 1 096 

2  ■ 

98904 

11 
10 

uj  17  20 

I  42  4o 

48 

10.65342 

9.35757 

10.64243 

10.01099 

2 

9.98901 

5i 

17  12 

42  48 

3471 3 

48 

65287 

358i5 

5i 

64i85 

01 102 

2 

98898 

9 

52 

17  4 

42  56 

34769 

49 

6523i 

35873 

52 

64127 

01  io4 

2 

98896 

8 

53 

16  56 

43  4 

34824 

5o 

65176 

35931 

53 

64069 

01 107 

2 

98893 

7 

54 
55 

16  48 

43  12 

34879 

5i 

65i2i 

35989 

54 

64oi  1 

OHIO 

3 

98890 

b 
~5 

10  16  4o 

I  43  20 

9.3io3/ 

52 

io.65o66 

9.36047 

55 

10.63953 

10.01 1 i3 

3 

9.98S87 

56 

16  32 

43  28 

34989 

53 

65oii 

36io5 

56 

63895 

01 1 16 

3 

98884 

4 

57 

16  24 

43  36 

35o4 

54 

64956 

36i63 

57 

63837 

0 1 1 1 9 

3 

98881 

3 

58 

16  16 

43  44 

3509c 

55 

64901 

36221 

58 

63779 

01122 

3 

98878 

2 

59 

16  8 

43  52 

35i54 

1  56 

64846 

36279 

59 

63721 

01  125 

3 

98875 

I 

60 

16  0 

44  0 

35209 

1  57 

64791 

36336 

60 

63664 

01 128 

3 

98872 

c 
M 

Hour  p.M 

Hour  .\.:m. 

Cosine. 

iDiff 

.Secant. 

Colans^ent 

Diff 

Tansronl. 

Cosecant.  |Diff. 

Sine. 

102= 


A 

A 

B 

B 

C 

Seconds  of  time 

!• 

2' 

3» 

4* 

5' 

6' 

7- 

Prop,  parts  of  cols. 

{  C 

7 
0 

i4 
i5 
1 

21 
22 
I 

29 
3o 

I 

36 
37 
2 

43 
45 

2 

52 

3 

C      77« 


Page  198] 

TABLE  XXVIL 

5' 

Log.  Sines,  Tangents,  and  Secants. 

G 

13 

0 

A 

A 

B 

B 

C 

C  166° 

M 

o 

Hour  A.M. 

Hour  P.M. 

Sine. 

Diff. 

Cosecant. 

Tangent. 

Dlff. 

Cotangcn-. 

Secant. 

Diff. 

Cosine. 

M 

6^ 

10  16  0 

I  4i    0 

9.35209 

0 

10.64791 

9.36336 

0 

io.63f'd4 

10. 0x128 

0 

9.98872 

I 

i5  52 

44    8 

35263 

I 

64737 

36394 

I 

636o6 

oxi3i 

0 

98869 

59 

2 

i5  44 

44   16 

353i8 

2 

64682 

36452 

2 

63548 

01x33 

0 

98867 

58 

3 

i5  36 

44  24 

35373 

3 

64627 

365o9 

3 

63491 

oxi36 

0 

98864 

57 

4 
5 

i5  28 

44  32 

35427 

4 

64573 

36566 

4 

63434 

01 139 

0 

98861 

56 
55 

10  i5  20 

I  44  4o 

9.35481 

4 

10.64519 

9.36624 

5 

10.63376 

10.01142 

0 

9.9885s 

6 

i5  12 

44  48 

35536 

5 

64464 

3668 1 

6 

633 1 9 

01145 

0 

98S55 

54 

7 

i5  4 

44  56 

35590 

6 

644  K) 

36738 

6 

63262 

oii48 

0 

98852 

53 

8 

i4  56 

45  4 

35644 

7 

64356 

36795 

7 

632o5 

01X01 

0 

98849 

52 

_9 

lO 

i4  48 

45  12 

35698 

8 

643o2 
10.64248 

36852 

8 

63 1 48 

01x54 

0 

98846 

5i 

5^ 

10  I 4  40 

I  45  20 

9.35752 

9 

9.36909 

9 

10.63091 

X0.01157 

9.98843 

II 

i4  32 

45  28 

358o6 

10 

64194 

36966 

10 

63o34 

01 160 

98S40 

49 

12 

i4  24 

45  36 

35860 

II 

64i4o 

37023 

ir 

62977 

oii63 

98837 

48 

i3 

i4  16 

45  44 

35914 

11 

64086 

37080 

12 

62920 

01 166 

98834 

47 

i4 
i5 

i4  8 

45  52 

35968 

12 

64o32 

37137 

i3 

62863 

01169 

98831 
9.98828 

46 
45 

10  i4  0 

I  46  0 

9.36022 

i3 

10.63978 

9.37193 

i4 

10.62807 

X0.01172 

i6 

i3  52 

46  8 

36075 

i4 

63925 

i5 

62750 

01x75 

98825 

44 

17 

i3  44 

46  16 

36129 

i5 

63871 

37306 

lO 

62694 

01178 

98822 

43 

i8 

i3  36 

46  24 

36182 

16 

638 1 8 

37553'  17 

62637 

0X181 

988x9 

42 

12 

20 

i3  28 

46  32 

36236 
9.36289 

17 
18" 

63764 
10.63711 

37419 

18 

62581 

on  84 

98816 

4i 
4o 

10  I 3  20 

I  46  4o 

9.37476 

IQ 

10.62524 

10.01187 

9.9S813 

21 

i3  12 

46  48 

36342 

18 

63658 

37532 

IQ 

62468 

01190 

988x0 

39 

22 

i3  4 

46  56 

36395 

19 

636o5 

37588 

20 

62412 

01193 

98807 

38 

23 

12  56 

47  4 

36449 

20 

6355i 

37644 

21 

62356 

01 196 

9S804 

37 

24 
25 

12  48 

4?  12 

365o2 

21 

63498 

37700 

22 

62300 

01199 

98S01 

36 
35 

10  12  4o 

I  47  20 

9.36555 

22 

10.63445 

9.37756 

23 

10.62244 

I0.012U2 

9.98798 

26 

12  32 

47  28 

366o8 

23 

63302 

37812 

24 

62188 

0X205 

9S795 

34 

27 

12  24 

47  36 

36660 

24 

63340 

37868 

25 

62X32 

OI20S 

98792 

33 

28 

12  16 

47  44 

367x3 

25 

63287 

37924 

26 

62076 

012X  X 

98789 

32 

29 

3o 

12  8 

4i  52 

36766 

25 

63234 
io.63i8i 

37980 

27 

62020 

0 1  2  1 4 

987S6 

3i 

3^ 

10  12  0 

I  48  0 

9.36819 

9.38o35 

28 

10.61965 

XO.OI217 

2 

9.98783 

3i 

II  52 

48  8 

36871 

27 

63i29 

38091 

29 

61909 

01220 

2 

98780 

29 

32 

II  44 

48  16 

36924 

28 

63076 

38i47 

3o 

6x853 

0X223 

2 

98777 

28 

33 

II  36 

48  24 

36976 

29 

63024 

38202 

3i 

61798 

01226 

2 

98774 

27 

34 
35 

II  28 

48  3  a 

37028 

3o 

62972 

38257 

32 
32 

61743 

0x229 

2 

98771 

26 

25 

10  II  20 

I  48  4o 

9.37081 

3. 

10.62919 

9.383x3 

10.61687 

XO. 01232 

2 

9.98768 

36 

II  12 

48  48 

37133 

32 

62867 

38368 

33 

6x632 

0X235 

2 

98765 

24 

37 

II  4 

48  56 

37185 

32 

62815 

38423 

34 

6x577 

OI238 

2 

98762 

23 

38 

10  56 

49  4 

37237 

ii 

62763 

38479 

35 

6x521 

0x241 

2 

98759 

22 

39 
40 

10  48 

49  12 

37289 
9.3734. 

34 

62711 

38534 

36 

6x466 

0x244 

2 

98756 

21 
20 

10  10  40 

I  49  2n 

35 

10.62659 

9.38589 

37 

io.6i4i I 

10.0x247 

2 

9.98753 

4i 

10  32 

49  28 

37393 

3b 

62607 

38644 

33 

6x356 

OX25o 

2 

9S750 

19 

42 

10  24 

49  36 

37445 

37 

62555 

38699 

39 

6x3oi 

01254 

2 

98746 

x8 

43 

10  16 

49  A4 

37497 

38 

625o3 

38754 

4o 

6x246 

0x257 

2 

98743 

17 

44 
45 

10  8 

49  52 

37549 

39 

62451 

388o8 

4i 

6x192 

01260 

2 

98740 

16 

i5 

ID    10   0 

I  5o  0 

9.37600 

39 

10.62400 

9.38863 

42 

10.61137 

10. 01263 

2 

9.98737 

46 

9  52 

5o  8 

37652 

40 

62348 

38918 

43 

6x082 

01266 

2 

98734 

i4 

47 

9  44 

5o  16 

37703 

4i 

62297 

38972 

44 

6x028 

01269 

2 

9S731 

i3 

48 

9  36 

5o  24 

37755 

42 

62245 

39027 

45 

60973 

0x272 

2 

98728 

12 

49 
5o 

9  28 

5o  32 

37806 

43 

62194 

39082 

45 
46 

609x8 
X 0.60864 

01275 

2 

98755 

1 1 
10 

10  9  20 

I  5o  4o 

9.37858 

44 

10.62142 

9.39136 

10.0x278 

3 

9.9S722 

5i 

9  12 

5o  48 

37909 

45 

62091 

39190 

47 

60810 

0x281 

3 

98719 

9 

52 

9  4 

5o  56 

3796<:) 

4<i 

62040 

39245 

48 

60755 

OI285 

3 

98715 

8 

53 

8  56 

5i  4 

38on 

47 

61989 

39299 

49 

60701 

0x288 

3 

98712 

7 

64 
55 

8  48 

5i  12 

38062 

47 

61938 

39353 

5o 

60647 

OI29I 

3 

98709 

6 

10  8  40 

I  5i  20 

9.381 i3 

48 

10.618S7 

9.39407 

5i 

X 0.60593 

10.01294 

3 

9.9S706 

56 

8  32 

5i  28 

38 1 64 

49 

6i836 

39461 

52 

60539 

0x297 

3 

98703 

^ 

57 

8  24 

5i  36 

382i5 

5o 

61785 

39515 

53 

604s  5 

oi3oo 

3 

90700 

3 

58 

8  16 

5i  44 

38266 

5r 

61734 

39569 

54 

604  3  X 

ox3o3 

3 

98697 

2 

59 

8  8 

5i  52 

383 1 7 

52 

6i683 

39623 

55 

60377 

ox3o6 

3 

98694 

1 

60 
M 

8  0 

52   0 

38368 

53 

6i632 

39677 

56 

6o323 

oi3io 

-J 

98690 

0 
M 

Hour  P.M. 

Hour  A.M. 

Cosine. 

Difif. 

Secant. 

Cotangent 

Diff. 

Tangent. 

Cosecant. 

Diff. 

Sine. 

103° 


C     76° 


Seconds  of  time 

1» 

2^ 

3^ 

4s 

5^ 

39 
42 
2 

7' 
46 
49 
3 

Prop,  parts  of  cols.  <  B 

(c 

7 
7 
0 

x3 

i4 

I 

20 
21 
I 

26 
28 
2 

33 
35 
2 

TABLE  XXVIL 

[  Page  199 

5. 

Log.  Sines,  Tangents,  and  Secants. 

G'. 

14 

3 

A 

A 

B 

B 

C 

C  165° 

M 

0 

Hour  4..M. 

Hoiirp.M. 

Sine. 
9.38368 

Dirt". 
0 

Cosecant. 

Taii2j.'-nl. 

DifT. 

Colanjcnt 

Secant. 

Difl'. 

Cosine. 

6S 

10  8 

0 

I  52  0 

io.6i63.> 

9.39677 

0 

io.6o323 

I0.oi3io 

0 

9.98690 

I 

7 

52 

52  8 

384 18 

I 

6i582 

39731 

1 

60269 

oi3i3 

0 

98687 

59 

2 

7 

44 

52  16 

33469 

2 

6i53i 

39785 

2 

6021 5 

oi3i6 

0 

98684 

58 

3 

7 

36 

5a  24 

385i9 

2 

61481 

3y838 

3 

60162 

oi3i9 

0 

9S881 

57 

4 
5 

7 

28 

52  32 

3S570 

3 

6i43o 
io.6i38u 

3989? 

3 

60108 

01822 

0 

98678 

5b 
55 

10  7 

20 

I  52  40 

9.38620 

4 

9.39945 

4 

io.6oo55 

10  01825 

0 

9.98675 

6 

7 

12 

52  48 

38670 

5 

6i33o 

39999 

5 

60001 

01829 

0 

9867 1 

54 

7 

7 

A 

52  56 

38721 

6 

61279 

4oo52 

6 

59948 

oi332 

0 

98668 

53 

8 

6 

56 

53  4 

38771 

7 

61229 

4o  1 06 

7 

59894 

01 335 

0 

98665 

52 

_9 

lO 

6 

48 

53  12 

3882  1 

7 

61179 

40159 

8 

59841 

oi338 

0 

98662 

5i 
5^ 

10  6 

4o 

I  53  20 

9.38871 

8 

10.61 129 

9.40212 

9 

10.5978S 

io.oi34i 

9.98659 

II 

6 

32 

53  28 

38921 

9 

61079 

40266 

10 

59734 

01 344 

98656 

49 

12 

6 

24 

53  36 

38971 

U) 

61029 

4o3i9 

10 

59681 

01348 

9S652 

48 

i3 

6 

16 

53  44 

39021 

11 

60979 

40372 

1 1 

59628 

oi35i 

98649 

47 

i4 
i5 

6 

8 

53  52 

39071 

1 1 

60929 

40425 

12 

59575 

oi354 
10.01357 

98646 

46 

45 

10  6 

0 

I  54  0 

9.39121 

12 

10.60879 

9.4047^ 

i3 

10.59522 

9.98643 

i6 

5 

52 

54  8 

39170 

i3 

6oS3o 

4o53i 

i4 

59469 

oi36o 

9S640 

44 

17 

5 

44 

54  i6 

39220 

.i4 

60780 

4o584 

i5 

59416 

01 364 

98686 

43 

i8 

5 

36 

54  24 

39270 

n 

60730 

4o636 

16 

59364 

oi367 

98633 

42 

19 

20 

5 

28 

54  32 

39319 

i5 
16 

60681 

40689 

17 

59311 

01870 

98630 

4i 
4o 

10  5 

20 

I  54  4o 

9.39369 

io.6o63i 

9.40742 

17 

10.59258 

10.01373 

9.98627 

21 

5 

12 

54  48 

39418 

17 

6o582 

40795 

18 

59205 

01877 

98623 

39 

22 

5 

4 

54  56 

39467 

18 

6o533 

40847 

19 

59153 

oi38o 

98620 

38 

23 

4 

56 

55  4 

39517 

>9 

6048  3 

40900 

20 

59100 

oi383 

98617 

37 

24 
25 

4 

48 

55  12 

39566 

20 

60434 

4095^ 

21 

59048 

01 386 

98614 

3b 
35 

10  4 

4o 

I  55  20 

9.39615 

20 

io.6o385 

9.4 lOOD 

22 

10.58995 

10.01390 

9.98610 

26 

4 

32 

55  28 

39664 

21 

6o336 

4io57 

23 

58943 

01893 

98607 

34 

27 

4 

24 

55  36 

39718 

22 

602S7 

41109 

23 

5889. 

01896 

98604 

6d 

28 

4 

16 

55  44 

39762 

23 

60288 

41161 

24 

58S39 

01899 

2 

9860 1 

32 

29 

3o 

4 

8 

55  52 

3981 1 

24 

60189 

4i2i4 

20 

58786 

oi4o3 

2 

q85j7 

3i 
3o 

10  4 

0 

I  56  0 

9.39860 

24 

io.6oi4o 

9.41266 

26 

10.5S734 

io.oi4o6 

2 

9.98594 

3i 

3 

52 

56  8 

39900 

25 

60091 

4i3i8 

27 

58682 

01409 

2 

98591 

29 

32 

3 

44 

56  16 

3995s 

26 

6oo4a 

41370 

28 

58630 

0l4l2 

2 

98588 

28 

33 

3 

36 

56  24 

40006 

27 

59994 

41422 

29 

58578 

oi4i6 

2 

98584 

27 

34 
35 

3 

28 

56  32 

4oo55 

28 

59945 

4i474 

3o 

58526 

01419 

2 

98581 

2b 

15 

10  3 

20 

I  5t)  4o 

9.40103 

29 

10.59897 

9.41526 

3u 

10.58474 

10.01422 

2 

9.9S578 

36 

3 

12 

56  48 

4oi52 

29 

59848 

4.578 

di 

5S422 

01426 

2 

98574 

24 

37 

3 

4 

56  56 

40200 

3o 

59800 

41629 

32 

58371 

01429 

2 

9857. 

23 

38 

2 

56 

57  4 

40249 

01 

59751 

41681 

di 

,   5S3i9 

01432 

2 

98568 

22 

39 
4o 

2 

48 

57  12 

40297 

32 

59703 

41733 

M 

58267 

01435 

2 

98565 

21 
20 

10  2 

4'> 

I  57  20 

9.40346 

33 

10.59654 

9.41784 

35 

10.58216 

10.01439 

2 

9.98561 

4i 

2 

32 

57  28 

4039^ 

33 

59606 

4i836 

30 

58 164 

01442 

2 

98558 

19 

42 

2 

2.4 

57  36 

40442 

34 

59558 

41887 

3b 

58ii3 

01445 

2 

98555 

18 

43 

2 

i6 

57  44 

40490 

35 

59510 

41939 

37 

58o6i 

01449 

2 

q855i 

17 

44 
45 

2 

8 

57  52 

4o538 

36 

59462 

41990 

3S 

5So  1 0 

01452 

2 

9S548 

lb 
i5 

10  2 

0 

I  58  0 

9.4o586 

37 

10.59414 

9.42041 

39 

10.57959 

10.01455 

2 

9.98545 

46 

52 

58  8 

4o634 

37 

59366 

42093 

40 

57907 

01459 

3 

9854. 

14 

47 

44 

58  16 

40682 

38 

59318 

42144 

4i 

57856 

01462 

3 

98538 

i3 

48 

36 

58  24 

40730 

3q 

59270 

42195 

42 

57805 

01465 

3 

98535 

12 

49 
5o 

28 

58  32 

40778 
9.40S25 

4o 

59222 

42246 

43 

57754 

01469 

3 

98531 

11 
10 

10  I 

20 

1  58  4o 

4i 

10.59175 

9.42297 

4i 

10.57703 

10.01472 

3 

9.98528 

5i 

12 

53  48 

40873 

42 

59127 

42348 

4  a 

57652 

01475 

3 

98525 

9 

52 

4 

58  56 

4092 1 

42 

59079 

42399 

45 

57601 

01479 

3 

98521 

8 

53 

0 

56 

59  4 

40968 

43 

59032 

42450 

46 

57550 

01482 

3 

98518 

7 

54 

55 

0 

48 

59  12 

4ioi6 

44 

58984 

42501 

47 

57499 

OI485 

3 

9851 5 

6 

5 

10  0 

4<> 

I  59  20 

9.41 o63 

45 

10.58937 

9.42552 

48 

10.57448 

10.01489 

3 

9.98511 

56 

0 

32 

59  28 

4i  1 1 1 

46 

58889 

42603 

49 

57397 

01492 

3 

98508 

4 

57 

0 

24 

59  36 

4ii58 

46 

58842 

42653 

bo 

57347 

01495 

3 

985o5 

3 

58 

0 

16 

59  44 

4i2o5 

47 

58795 

42704 

5o 

57296 

01499 

3 

98501 

2 

59 

0 

8 

59  52 

4l252 

48 

58748 

42755 

5i 

57245 

01302 

3 

98498 

I 

60 
M 

0 

0 

200 

4l3or 

49 

58700 
Secant. 

42805 

52 

57195 

0 1 5o6 

3 

98494 

0 

M 

Hour 

f.M. 

Hour  A.M. 

Cosine.  Dili'. 

Cotangent 

DitT. 

Tangent. 

Cosecant. 

DifT. 

Sine. 

104" 


75" 


Seconds  of  time 

1' 

2' 

3' 

18 
20 

I 

4» 

24 
26 

2 

5' 

3i 
33 
2 

6' 

37 
39 
2 

7- 

43 
46 
3 

Prop,  parts  of  cols.  ^  B 

(c 

6 

7 
0 

12 
i3 
I 

Page  200] 

TABLE  XXVII 

S' 

Log.  Sines,  Tangents,  and  Secants. 

G  . 

15° 

A 

A 

B 

B 

C        C  164° 

M  Hour  A. M 

.  Hour  p. M 
)  2  0   0 

Sine. 

iDiff 

Cosecant 

Tangent. 

Diff 

Cotangent 

Secant. 

Diff 

Cosine. 

M 

60 

0  10  o  c 

9.4i3oc 

0 

10.58700 

9.4280^ 

0 

10.57195 

10.01 5oe 

0 

9 . 98494 

I   9  59  5: 

0   8 

4i34' 

I 

58653 

4285e 

I 

57144 

01509'  0 

98491 

59 

2   59  4^ 

0  16 

4139/ 

2 

586o6 

42906 

2 

57094 

Ol5l2 

0 

98488 

58 

3    59  3t 

)    0  24 

4i44i 

2 

58559 

4295- 

2 

57043 

oi5ie 

0 

9S484 

57 

4  59  28|    0  32 

5  9  69  20|  3  0  4o 

4i48S 
9.41535 

3 

585x2 

43007 

3 

56993 

oi5i9 

0 

98481 

56 
55 

4 

10. 58465 

9.4305- 

4 

10.56943 

IO.OI523 

0 

9.98477 

6    59  12 

:    0  48 

4i582 

5 

584 1 8 

43 1 08 

5 

56892 

oi52G 

0 

98474 

54 

7    59  A 

1   0  56 

41628 

5 

58372 

43 1 58 

6 

56842 

01529 

0 

98471 

53 

8    58  5e 

I  4 

41675 

6 

58325 

4320& 

7 

56792 

oi533 

0 

98467 

52 

9    58  48 

I  12 

41722 

7 

58278 

43258 

7 

56742 

oi536 

I 

98464 

5i 
5o 

10  9  58  4t 

2  I  20 

9.41768 

8 

10.58232 

9.43308 

8 

10.56692 

io.oi54( 

I 

9.98460 

II    58  32 

I  28 

4i8i5 

8 

58 1 85 

43358 

9 

56642 

01543 

I 

98457 

49 

12    58  24 

I  36 

41861 

9 

58i39 

43408 

10 

56592 

01547 

I 

98453 

48 

i3    58  16 

I  44 

4190& 

10 

58092 

43458 

II 

56542 

oi55o-  I 

98450 

47 

i4    58  & 

I   52 

41954 

II 

58o46 

43508 

1 1 

56492 

oi553 

98447 

46 
45 

i5  9  58  0 

220 

9.42001 

II 

10.57999 

9.43558 

12 

10.56442 

10. 01557 

9.98443 

16    57  52 

2  8 

42047 

12 

57953 

43607 

i3 

56393 

oi56o 

98440 

44 

17   57  44 

2  16 

42093 

i3 

57907 

43657 

i4 

56343 

01 564 

98436 

43 

18    57  36 

2  24 

42140 

14 

57860 

43707 

i5 

56293 

01567 

98433 

41 

19   57  28 

2  32 

42186 

14 

57814 

43756 

16 

56244 

01571 

98429 

4i 
4o 

20  9  57  20 

2  2  4o 

9.42232 

i5 

10.57768 

9.43806 

16 

10.56194 

10.01574 

9.98426 

21    57  12 

2  48 

42278 

16 

57722 

43855 

17 

56i45 

01578 

98422 

3q 

22   57  4 

2  56 

42324 

17 

57676 

43905 

18 

56095 

oi58i 

98419 

38 

23    56  56 

3  4 

42370 

17 

57630 

43954 

19 

56046 

oi585 

98415 

37 

24    56  48 

3  12 

42416 

18 

57584 

44oo4 

20 

55996 

01 588 

98412 

36 
35 

25  9  56  4o 

2  3  20 

9.42461 

19 

10.57539 

9.44053 

20 

10.55947 

10.01591 

9.98409 

26   56  32 

3  28 

42507 

20 

57493 

44102 

21 

55898 

01595 

2 

98405 

M 

27    56  24 

3  36 

42553 

21 

57447 

44i5i 

22 

55849 

01598 

2 

98402 

33 

28    56  16 

3  44 

42599 

21 

57401 

44201 

23 

55799 

01602 

2 

98398 

32 

29   56  8 

3  52 

42644 

22 

57356 

44  2  5o 

24 

55750 

oi6o5 

2 

98395 

3i 

3^ 

3o  9  56  0 

240 

9.42690 

23 

10.57310 

9.44299 

25 

10.55701 

10.01609 

2 

9.98391 

3i    55  52 

4  8 

42735 

24 

57265 

44348 

25 

55652 

01612 

2 

98388 

29 

32    55  44 

4  16 

42781 

24 

57219 

44397 

26 

556o3 

01616 

2 

98384 

28 

33    55  36 

4  24 

42826 

25 

57174 

44446 

27 

55554 

0 1 6 1 9 

2 

98381 

27 

34   55  28 

4  32 

42872 

26 

57128 

44495 

28 

555o5 

01623 

2 

98377 

26 

35  9  55  20 

2  4  4(1 

9.42917 

27 

10.57083 

9.44544 

29 

10.55456 

10.01627 

2 

9.98373 

36   55  12 

4  48 

42962 

27 

57088 

44592 

29 

55408 

oi63o 

2 

98370 

24 

37    55  4 

4  56 

43008 

28 

56992 

44641 

3o 

55359 

01634 

2 

98366 

23 

38    54  56 

5  4 

43o53 

29 

56947 

44690 

3t 

553io 

01637 

2 

98363 

22 

39   54  48 

5  12 

43098 
9.43143 

3o 

56902 

44738 

32 

55262 

01641 

2 

98359 

21 

20 

4o  9  54  40 

2  5  20 

3o 

10.56857 

9-44787 

33 

I0.552I3 

10.01644 

2 

9.98356 

4i    54  32 

5  28 

43 1 88 

3i 

568 1 2 

44836 

34 

55 164 

oi648 

2 

98352 

19 

42   54  24 

5  36 

43233 

32 

56767 

44884 

M 

55ii6 

oi65i 

2 

98349 

18 

43   54  16 

5  44 

43278 

33 

56722 

44933 

35 

55067 

01655 

3 

98345 

17 

44        54  8 

5  52 

43323 

M 

56677 

44981 

36 

55019 

oi658 

3 

98342 

16 
i5 

45  9  54  0 

260 

9.43367 

34 

10. 56633 

9.45029 

37 

10.54971 

10.01662 

3 

9.98338 

46    53  52 

6  8 

43412 

35 

56588 

45078 

38 

54922 

01666 

3 

98334 

i4 

47   53  44 

6  16 

43457 

36 

56543 

45126 

38 

54874 

01669 

3 

98331 

i3 

48    53  36 

6  24 

435o2 

36 

56498 

45174 

39 

54826 

01673 

3 

98327 

12 

49   53  28 

6  32 

43546 

37 

56454 

45222 

4o 

54778 

0 1 676 

3 

98324 

II 

lO 

5o  9  53  20 

2  6  4o 

9.43591 

38 

10.56409 

9.45271 

4i 

10.54729 

10.01680 

3 

9.98320 

5i    53  12 

6  48 

43635 

39 

56365 

45319 

42 

54681 

01683 

3 

98817 

9 

52    53  4 

6  56 

4365o  39  1 

56320 

45367 

43 

54633 

01687 

3 

9831 3 

g 

53    52  56 

7  4 

43724 

4o 

56276 

454 1 5 

43 

54585 

01691 

3 

98309 

7 

54    52  48 

7  12 

43769 

4i 

5623 1 

45463 
9.45511 

44 
■45 

54537 

01694 

3 

98306 

6 

5 

55  9  52  40 

2  7  20 

9.438i3 

42 

10.56187 

10.54489 

10.01698 

3 

9.98302 

56   52  32 

7  28 

43857 

43 

56 1 43 

45559 

46 

54441 

01 70 1 

3 

98299 

4 

57    52  24 

7  36 

43901 

A3 

56099 

456o6 

An 

54394 

01705 

3 

9829^. 

3 

58   52  16 

7  44 

43946 

44 

56o54 

45654 

4i 

54346 

01709 

3 

98291 

2 

59   52  8 

7  52 

43990 

45 

56oio 

45702 

48 

54298 

01712 

3 

98288 

I 

60    52  0 

8  0 

44o34 

46 

55966 

45750 

49 

54250 

01716 

4 

98284 

0 
M 

M  Hour  P.M. 

Hour  A.M. 

Cosine. 

Difi-. 

Secant. 

Cotangent 

Diff. 

Tangent.  | 

Cosecant. 

)iff. 

Sine. 

1U5" 


A 

A 

B 

B 

C 

Seconds  of  time 

1» 

2» 

3' 

4. 

5' 

6» 

7' 

Frop.  parts  cf  cols. 

\  C 

6 
6 
0 

II 
12 

I 

17 
18 
I 

23 
25 
2 

28 
3i 
2 

34 

37 
3 

40 
43 
3 

C     74° 


TABLE  XXVIL 

[Page  yOl 

^ 

Log 

.  Sines,  Tangents,  and  Secants. 

G'. 

16-^ 

A 

A 

B 

B 

c 

C  163° 

M 

o 

HourA.JI. 

Hour  P.M. 

Sine. 

Ditr. 

Coseraiit. 

Tangent. 

Ditr. 

Colanf^cnt 

Secant. 

Ditr. 

Cosine. 

M 

60 

9  62  0 

280 

9.44034 

0 

10.55966 

9.45750 

0 

10.5425(1 

10.01716 

0 

9.98284 

I 

5i  52 

8  8 

44078 

I 

55922 

45797 

I 

54203 

01719 

0 

98281 

59 

2 

5i  44 

8  16 

44122 

I 

55878 

45845 

2 

54 1 55 

01723 

0 

98277 

58 

3 

5 1  36 

8  24 

44166 

2 

55834 

45892 

2 

54108 

01727 

0 

98273 

57 

4 
5 

5i  28 

8  32 

44210 

3 

55790 

45940 

3 

54060 

01730 

0 

98270 

56 
55 

9  5i  20 

2  8  4o 

9.44253 

4 

10.55747 

9.45987 

4 

io.54oi3 

10.01734 

0 

9.98266 

6 

5i  12 

8  48 

44297 

4 

55703 

46o35 

5 

53965 

01738 

0 

98262 

54 

7 

5r  4 

8  56 

44341 

5 

55659 

46082 

5 

53918 

01741 

0 

98259 

53 

8 

5o  56 

9  4 

44385 

6 

556 1 5 

46 1 3o 

b 

53870 

01745 

0 

98255 

52 

_? 

10 

5o  48 

9  12 

44428 

6 

55572 

46177 

7 

53823 

01749 

98251 

5i 
5o 

9  5o  4o 

2  9  20 

9.44472 

7 

10.55528 

9.46224 

8 

10.53776 

10.01752 

9.9824s 

1 1 

5o  32 

9  28 

445 16 

8 

55484 

46271 

9 

53729 

01756 

98244 

49 

12 

5o  24 

9  36 

44559 

9 

55441 

463 1 9 

9 

53681 

01760 

98240 

48 

i3 

5o  16 

9  44 

44602 

9 

5539S 

46366 

10 

53634 

01763 

98237 

47 

i4 
i5 

5o  8 

9  52 

44646 

10 

55354 

464 1 3 

II 

53587 

01767 

98233 

4b 
45 

9  5o  0 

2  10  0 

9.44689 

II 

io.553i 1 

9.46460 

12 

10.53540 

10.01771 

9.98229 

1 6 

49  52 

10  8 

44733 

II 

55267 

465o7 

12 

53493 

01774 

98226 

44 

I? 

49  44 

10  16 

44776 

12 

55224 

46554 

i3 

53446 

01778 

98223 

43 

i8 

49  36 

10  24 

44819 

i3 

55i8i 

46601 

i4 

53399 

01782 

98218 

42 

!9 

20 

49  28 

10  32 

44862 

i4 

55i38 

46648 

i5 

53352 

01785 

98215 

4i 
4o 

9  49  20 

2  10  40 

9.44905 

14 

10.55095 

9.46694 

i5 

io.533o6 

10.01789 

9.9S211 

21 

49  12 

10  48 

44948 

i5 

55o52 

46741 

lb 

53259 

01793 

98207 

39 

2  2 

49  4 

10  56 

44992 

16 

55oo8 

46788 

17 

532  12 

01796 

98204 

38 

23 

48  56 

II  4 

45o35 

16 

54965 

46835 

18 

53i65 

01800 

98200 

37 

24 
25 

48  48 

II  12 

45077 

17 

54923 

4688 1 

19 

53 II 9 

01804 

98196 

3b 
35 

9  48  4o 

2  II  20 

9.45120 

18 

10.54880 

9.46928 

19 

10.53072 

I 0.0 I 80S 

2 

9.98192 

26 

48  32 

II  28 

45i63 

18 

54837 

46975 

20 

53025 

01811 

2 

98189 

M 

27 

48  24 

II  36 

45206 

19 

5479i 

47021 

21 

52979 

oi8i5 

2 

98185 

33 

28 

48  16 

II  44 

45249 

20 

54751 

47068 

22 

5293: 

01819 

2 

98181 

32 

29 

3o 

48  8 

II  52 

45292 

21 

5470S 
10.54666 

47114 

22 

52886 

01823 

2 

98177 

3i 

3^ 

9  48  0 

2  12  0 

9.45334 

21 

9.47160 

23 

io.5284i> 

10.01826 

2 

9.98174 

3i 

47  52 

12  8 

45377 

22 

54623 

47207 

24 

52793 

oi83o 

2 

98170 

29 

32 

47  44 

12  16 

45419 

23 

54581 

47253 

25 

52747 

01834 

2 

98166 

28 

33 

47  36 

12  24 

45462 

23 

54538 

47299 

2fa 

52701 

oi838 

2 

98162 

27 

34 
35 

47  28 

12  32 

455o4 

24 

54496 

47346 

26 

52654 

01841 

2 

98 1  59 

2b 

9  47  20 

2  12  4o 

9.45547 

25 

10.54453 

9.47392 

27 

10.52608 

10.01843 

2 

9.98155 

36 

47  12 

12  48 

45589 

26 

544 1 1 

47438 

28 

52563 

01849 

2 

98151 

24 

37 

47  4 

12  56 

45632 

26 

54368 

47484 

29 

525i6 

01 853 

2 

98147 

23 

38 

46  56 

i3  4 

45674 

27 

54326 

47530 

29 

52470 

01856 

2 

98144 

22 

39 
4o 

46  48 

i3  12 

45716 

28 

54284 

47576 

3o 

52424 

0 1 860 

2 

98140 

21 

20 

9  46  4o 

2  i3  20 

9.45758 

28 

10.54242 

9.47622 

3i 

10.52378 

10.01864 

2 

9.98136 

4i 

46  32 

i3  28 

458oi 

29 

54199 

47668 

32 

52332 

01868 

3 

98132 

'9 

42 

46  24 

i3  36 

45843 

3o 

54157 

47714 

32 

52286 

01871 

3 

98129 

18 

43 

46  16 

i3  44 

45885 

3i 

54ii5 

47760 

33 

52240 

01875 

3 

98125 

17 

44 
45 

46  8 

i3  52 

45927 

3i 

54073 

47806 

34 

52194 

01879 

3 

98121 

16 
75 

9  46  0 

2  i4  " 

9.45969 

32 

io.54o3i 

9.47852 

35 

10.52148 

10.01883 

3 

9  98117 

46 

45  52 

i4  8 

4601 1 

33 

53989 

47897 

36 

52io3 

01887 

3 

98113 

i4 

47 

45  44 

i4  16 

46o53 

33 

53947 

47943 

36 

52057 

01890 

3 

98 1 1 0 

i3 

48 

45  36 

i4  24 

46095 

34 

53905 

47989 

37 

5201  1 

01894 

3 

98 1 06 

12 

49 
5o 

45  28 

i4  32 

461 36 

35 

53864 

48o35 

38 

51965 

01898 

3 

98102 

1 1 

10 

9  45  20 

2  i4  4o 

9.46178 

36 

10.53822 

9.48080 

39 

10.51920 

10.01902 

3 

9 . 98098 

5i 

45  12 

i4  48 

46220 

36 

53780 

48126 

39 

51874 

01906 

3 

98094 

9 

52 

45  4 

i4  56 

46262 

37 

53738 

48171 

40 

51829 

01910 

3 

98090 

8 

53 

44  56 

i5  4 

463o3 

38 

53697 

48217 

41 

51783 

01913 

3 

98087 

7 

54 
55 

44  48 

i5  12 

46345 

38 

53655 

48262 

42 

5 1 738 

01917 

3 

98083 

6 
5 

9  44  4o 

2  i5  20 

9.46386 

io.536i4 

9.48307 

43 

10.51693 

10.01921 

3 

9.98079 

56 

44  32 

i5  28 

46428 

4o 

53572 

48353 

43 

5 1 647 

01925 

3 

98075 

4 

^7 

44   24 

i5  36 

46469 

4i 

5353 1 

48398 

44 

5 1 602 

01929 

4 

9807; 

3 

58 

44   16 

i5  44 

465 1 1 

4i 

53489 

48443 

45 

5i557 

01933 

4 

98067 

2 

59 

44    8 

i5  52 

46552 

42 

53448 

48489 

46 

5i5ii 

01937 

4 

98063 

I 

60 
M 

44  0 

16  0 

46594 

43 

53406 

48534 

46 

5 1 466 

0 1 940 

4 

98060 

0 
M 

Hour  P.M. 

Hour  A.M. 

Cosino. 

Diff. 

Secant. 

Cotangent|Dlfl". 

Tan!:^cnt. 

Cosecant. 

Diff. 

Sine. 

106° 


A 

A 

B 

B 

C 

1* 

2" 

3' 

4. 

5" 

6' 

32 

35 
3 

7' 

37 
4i 
3 

Prop,  parts  oT  cols. 

(■ 

5 
6 
0 

11 
12 

16 

17 
1 

21 

23 

2 

27 
29 
2 

2G 


Page  202] 

TABLE  XXVIL 

S' 

Log.  S 

nes,  Tangents,  and  Secants. 

G'. 

17 

0 

A 

A 

B 

B 

c 

C  162° 

M 

0 

HOUFA.AI. 

Hour  P.M. 

Sine. 

Diff 

Cosecant. 

Tangent. 

Diff. 

Cotangent 
io.5i466 

Secant. 

Diff. 

Cosine. 

M 

60 

9  44  c 

2  16 

0 

9.46594 

0 

10.53406 

9-48534 

0 

10.01940 

0 

9 . 98060 

I 

43  52 

16 

8 

46635 

I 

53365 

48579 

I 

5i42i 

01944 

0 

9S056 

59 

2 

43  44 

16 

16 

46676 

I 

53324 

48624 

I 

51376 

01948 

0 

9S052 

58 

3 

43  36 

16 

24 

46717 

2 

53283 

48669 

2 

5i33i 

01932 

0 

98048 

57 

4 
5 

43   28 

16 

02 

46758 

3 

53242 

48714 

3 

51286 

01956 

0 

9S044 

56 
55 

9  43  20 

2  16  4o 

9.46800 

3 

10.53200 

9-48759 

4 

io.5i24i 

10.01960 

0 

9.980^0 

6 

43  12 

16 

48 

4684! 

4 

53 1 59 

48S04 

4 

51196 

01964 

0 

98036 

54 

7 

43  4 

16 

56 

46882 

5 

53ii8 

48849 

5 

5ii5i 

01968 

0 

9S032 

53 

b 

42  56 

17 

4 

46923 

b 

53077 

4S894 

6 

5 1 106 

01971 

98029 

52 

_9 

10 

42  48 

17 

12 

20 

46964 

b 

53o36 

48939 

7 

5 106 1 
io.5ioi6 

01975 

9S025 

5i 
5o 

9  42  4o 

2  17 

9.47005 

7 

10.52995 

9.48984 

7 

10.01979 

9.98021 

II 

42  32 

17 

28 

47045 

7 

52955 

49029 

8 

50971 

01983 

98017 

49 

12 

42  24 

J7 

36 

47086 

8 

52914 

49073 

9 

50927 

019S7 

98013 

48 

i3 

42  16 

17 

44 

47127 

9 

52873 

491 18 

10 

50882 

01991 

98000 

47 

i4 
i5 

42  8 

17 

D2 

47168 

9 

52832 

49163 

10 

5o837 

01995 

98005 

46 
45 

9  42  0 

2  18 

0 

9.47209 

10 

10.52791 

9.49207 

1 1 

10.50793 

10.01 999 

9.98001 

i6 

4i  52 

18 

8 

47249 

II 

52751 

49252 

12 

5074s 

02003 

97997 

44 

17 

4i  44 

i» 

16 

47290 

II 

52710 

49296 

12 

50704 

r20O7 

97993 

4i 

i8 

41  36 

18 

24 

47330 

12 

52670 

49341 

i3 

5o659 

02011 

97989 

42 

!9 

20 

4i  28 

18 

32 

47371 

i3 

52629 

49385 

i4 

5o6i5 
io.5o570 

020  (4 

97986 

4i 
40 

9  4i  20 

2  18 

40 

9.47411 

i3 

10.52589 

9.49430 

i5 

10.02018 

9.97982 

21 

4i  12 

18 

48 

47452 

i4 

52548 

49474 

i5 

5o526 

02022 

97978 

39 

22 

4i  4 

18 

56 

47492 

i5 

525o8 

■49519 

16 

5o48i 

02026 

97974 

38 

23 

40  56 

19 

4 

47533 

i5 

52467 

49563 

17 

50437 

02030 

2 

97970 

37 

24 
25 

4o  48 

19 

12 

47373 

16 

52427 

49607 
9.49652 

18 
18 

50393 
io.5o348 

O2o34 

2 

97966 
9.979(32 

36 
35 

9  4o  4o 

2  19 

20 

9.47613 

17 

10. 52387 

io.o2o38 

2 

26 

4o  32 

19 

28 

47654 

17 

52346 

49696 

19 

5o3o4 

02042 

2 

97958 

M 

27 

4o  24 

'9 

36 

47694 

18 

523o6 

49740 

20 

50260 

02046 

2 

97954 

36 

28 

4o  16 

19 

44 

47734 

19 

5226b 

49784 

21 

5o2i6 

02o5o 

2 

97950 

32 

29 

3o 

40  8 

19 

52 

47774 

19 

52226 

49828  21 

50172 

o2o54 

2 

979^6 

3t 
3o 

9  40  0 

2  20 

0 

9 . 478 1 4 

20 

10.52186 

9.49872 

22 

10.50128 

10.02058 

2 

9.97942 

3i 

39  52 

20 

8 

47854 

21 

52146 

49916 

23 

5oo84 

02062 

2 

97938 

29 

32 

39  44 

20 

16 

47894 

21 

52106 

49960 

24 

5oo4o 

02066 

2 

97934 

28 

33 

39  36 

20 

24 

47934 

22 

52066 

5ooo4 

24 

49996 

02070 

2 

97930 

27 

'I 
35 

39  28 

20 

32 

47974 

23 

52026 

5oo48 

25 

49952 

02074 

2 

97926 

2b 

9  39  20 

2  20 

4o 

9.48014 

23 

10.5198b 

9 . 50092 

26 

10.49908 

10.02078 

2 

9.97922 

'36 

39  12 

20 

48 

48o54 

24 

51940 

5oi36 

26 

49864 

02082 

2 

97918 

24 

^7 

39  4 

20 

5b 

48094 

25 

5 1 906 

5oi8o 

27 

49S20 

02086 

2 

979 '4 

23 

38 

38  56 

21 

4 

48 1 33 

25 

51867 

5o233 

28 

49777 

02090 

3 

97910 

22 

39 

4o 

38  48 

21 

(2 

48173 

26 

51827 

50267 

29 

29 

49733 
10.49689 

02094 

3 

97900 

21 

20 

9  38  4o 

2  21 

20 

9.48213 

27 

10.517S7 

9.5o3i 1 

10.02098 

3 

9.97902 

4i 

33  32 

21 

28 

48252 

27 

5 1 748 

5o355 

3o 

49645 

02102 

3 

97S98 

19 

4a 

33  24 

21 

36 

48292 

28 

51708 

50398 

3i 

49602 

02106 

3 

97894 

18 

43 

33  16 

21 

44 

48332 

29 

5 1 668 

5o442 

32 

49558 

02II0 

3 

97890 

17 

44 
45 

38  8 

21 

52 

4837i 

39 

51629 

5o485 

32 

495 1 5 

021 14 

3 
3 

97886 
9.97S82 

lb 

Is 

9  38  u 

2  22 

0 

9.4841 1 

3o 

io.5i589 

9.50529 

33 

10.49471 

10.02118 

46 

37  52 

22 

8 

4845o 

3i 

5i55o 

5o572 

34 

49428 

02122 

3 

97878 

i4 

47 

37  44 

22 

.6 

48490 

3. 

5i5io 

50616 

35 

49384 

02126 

3 

97874 

i3 

48 

37  36 

22 

24 

48529 

32 

5i47i 

5o659 

35 

49341 

02l3o 

3 

97870 

12 

49 
5o 

37  3-8 

22 

32 

48568 

33 

5i432 

5o7o3 

36 

49297 

02 1 34 

3 

97S66 

II 
10 

9  37  20 

2  22 

40 

9.48607 

33 

10.51393 

9 . 50746 

37 

10.49254 

10.02139 

3 

9.97861 

5i 

37  12 

22 

48 

48647 

34 

5i353 

50789 

37 

49211 

02143 

3 

97857 

9 

'j2 

37  4 

22 

56 

48686 

35 

5[3i4 

5o833 

38 

49167 

02147 

3 

97853 

8 

J3 

36  56 

23 

4 

48725 

35 

51275 

50876 

39 

49124 

021 5 1 

4 

9784Q 

7 

54 
55 

36  40 

23 

[2 

48764 

36 
37 

5 1 236 

50919 

40 

49081 
10.49038 

02i55 

4 

97845 

b 
"5 

9  36  4o 

2  23 

20 

9.48803 

1 0 . 5 1 1 97 

9.50962 

4o 

10.02159 

4 

9.97841 

5b 

36  32 

23 

28 

48842 

37 

5ii58 

5ioo5'  4i 

48995 

02i63 

4 

97837 

4 

57 

36  24 

23 

36 

48881 

38 

5i  1 19 

5io48 

42 

48952 

02167 

4 

97833 

3 

58 

36  16 

23  44 

48920 

39 

5 1 080 

51092 

43 

48908 

02171 

4 

97839 

2 

59 

36  8 

23 

52 

48959 

39 

5io4i 

5ii35 

43 

48865 

02175 

4 

97825 

I 

60 

36  0 

24 

0 

48998 

4o 

5 1 002 

51178 

4i 

48822 

02179 

4 

9782! 

0 

M 

flour  P.M. 

Hour  A 

..■\i. 

Cosine. 

Diir. 

Secant. 

Cotangent 

Din. 

Tangent. 

Cosecant. 

Diff. 

Sine. 

107° 


7'^ 


Seconds  of  time  . 

1» 

2» 

3^ 

4s 

5^ 

6» 

7* 

r 

IC 

5 

10 

i5 

20 

25 

5o 

35 

Prop,  pnrts  of  cols 

6 

11 

17 

22 

28 

33 

39 

0 

I 

I 

2 

2 

3 

3 

TABLE  XXVIL 

[Page  203 

S' 

Log.  Sines,  Tangents,  and  Secants. 

Gi. 

18 

0 

A 

A 

B 

B 

C 

C  161° 

M 

D 

Hour  A.M. 

Hour  I'.r.i. 

Slue. 
9.48998 

DiflT. 
0 

Ccsecaut. 

I0.5l002 

Tan°;oiit. 

Diir. 

Cotangent 

Secant. 

Diff. 

Cosine. 

9  36  0 

2  24  u 

9.51 178 

0 

10.48822 

10.02179 

0 

9.97S21 

I 

35  52 

24  8 

49037 

I 

50963 

5l22I 

I 

48779 

02i83 

0 

97S17 

5q 

2 

35  44 

24  16 

49076 

I 

50924 

5 1 264  1 

48736 

02188 

0 

97812 

58 

3 

35  3(1 

24  24 

491 15 

2 

5o885 

5i3o6i  2 

48694 

02192 

0 

97808 

57 

4 
5 

35  28 

24  32 

49153 

3 

5o847 

5 1 349 

3 

48651 

02196 

0 

97804 

56 
55 

9  35  20 

2  24  40 

9.49192 

3 

10.50808 

9.51 392 

3 

10.48608 

10.02200 

0 

9.97800 

b 

35  12 

24  48 

49231 

4 

50769 

5i435 

4 

48565 

02204 

0 

97796 

54 

7 

35  A 

24  56 

49269 

4 

5073 1 

51478 

5 

48522 

02208 

0 

9779' 

53 

8 

34   56 

25  4 

49308 

5 

50692 

5i52o 

b 

48480 

02212 

97788 

52 

_9 

10 

34  4« 

25  12 

49347 
9.49385 

6 
6 

5o653 
io.5o6i5 

5 1 563 

b 

48437 

02216 

97784 

5i 

5^ 

9  34  4o 

2  25  20 

9.51 606 

7 

10.48394 

10.02221 

9-97779 
97775 

II 

34  32 

25  28 

49424 

7 

50576 

5i648 

8 

48352 

02225 

49 

12 

34  24 

2  5  36 

49462 

8 

5o538 

51691 

8 

483o9 

02229 

97771 

48 

fci 

34  16 

25  44 

49500 

8 

5o5oo 

51734 

9 

48266 

02233 

97767 

47 

i4 
i5 

54    8 

25  52 

49539 

9 

5o46i 

51776 

10 

48224 

02237 

97763 

46 
45 

9  34  0 

2  26  0 

9.49577 

9 

io.5o42  3 

9.51819 

10 

10.48181 

10.02241 

9.97759 

lb 

33  52 

26  8 

49615 

10 

5o385 

5i86i 

1 1 

48139 

02246 

97754 

4i 

I? 

33  44 

26  16 

49654 

II 

5o346 

51903 

12 

48097 

0225o 

97750 

43 

i8 

33  36 

26  24 

49692 

II 

5o3o8 

51946 

i3 

48o54 

02254 

97746 

42 

£9 

20 

33  28 

26  32 

49730 

12 

50270 

51988 

i3 

48012 

02258 

97742 

4i 
4o 

9  33  20 

2  26  4" 

9.49768 

i3 

10.5o232 

9.52o3i 

■  4 

10.47969 

10.02262 

9-97738 

21 

33  12 

26  48 

49806 

i3 

50194 

52073 

i5 

47927 

02266 

97734 

39 

32 

33  4 

26  56 

49844 

i4 

5oi56 

52ii5 

i5 

47885 

G2271 

2 

97729 

38 

2j 

32  56 

27.  4 

49S82 

i4 

5oii8 

52157 

16 

47843 

02275 

2 

97725 

37 

24 
25 

32  48 

27  12 

49920 

i5 

5oo8o 

52200 

17 

47800 

02279 

2 

97721 

36 
35 

9  32  4o 

2  27  20 

9.49958 

16 

io.5oo42 

9.52242 

17 

10.47758 

10.02  283 

2 

9.97717 

2b 

32  32 

27  28 

49996 

lb 

5ooo4 

52284 

18 

47716 

02287 

2 

97713 

34 

27 

32  24 

27  36 

5oo34 

17 

49966 

52326 

19 

47674 

02292 

2 

97708 

33 

28 

32  16 

27  44 

50072 

18 

49928 

52368 

20 

47632 

02296 

2 

977"4 

32 

29 

3o 

32  8 

27  52 

5oiio 

18 

49890 

52410 

20 

47590 

023oo 

2 

97700 

3i 

3^ 

9  32  0 

2  28  0 

9.50148 

19 

10.49852 

9.52452 

21 

10.47548 

io.023o4 

2 

9.97696 

di 

3i  52 

28  8 

5oi85 

20 

49S15 

52494 

22 

475o6 

02309 

2 

97691 

29 

62 

3i  44 

•  28  16 

50223 

20 

49777 

52536 

22 

47464 

023i3 

2 

97687 

28 

di 

3 1  36 

28  74 

5o26i 

21 

49739 

52578 

23 

47422 

02317 

2 

97683 

27 

35 

3i  28 

28  32 

50298 

21 

49702 

52620 

24 

473S0 

02321 

2 

97679 

26 

25 

9  3i  20 

2  28  4" 

9.5o336 

22 

I 0 . 49664 

9.52661 

24 

10.47339 

10.02326 

2 

9.97674 

cib 

3i  12 

28  48 

5o374 

23 

49626 

52703 

25 

47297 

o233o 

3 

97670 

24 

^7 

3i  4 

28  56 

5o4i  1 

23 

495S9 

52745 

2b 

47255 

02334 

3 

97666 

23 

ciS 

3o  56 

29  4 

5o449 

24 

49551 

52787 

27 

472 1 3 

02338 

3 

97662 

22 

39 
4o 

3o  48 

29  12 

5o486 

25 
25 

49514 

52829 

27 

47171 

02343 

3 

97657 

?.I 
20 

9  3o  4^1 

2  29  20 

9.5o523 

10.49477 

9.52S70 

28 

.o.47i3o 

10.02347 

3 

9.97653 

4i 

3o  32 

29  28 

5o56i 

26 

49439 

52912 

29 

4708S 

0235i 

3 

97649 

19 

42 

3o  24 

29  36 

5059S 

26 

49402 

52953 

29 

47047 

02355 

3 

97645 

18 

43 

3o  16 

29  44 

5o635 

27 

49365 

52995 

3o 

470o5 

0236o 

3 

97640 

17 

44 
45 

3o  8 

29  52 

50673 

28 

49327 

53o37 

3i 

46963 

02364 

3 

97636 

16 

i5 

9  3o  0 

2  3o  0 

9.50710 

28 

10.49290 

9.53078 

3 1 

10.46922 

1 0.02 368 

3 

9-97632 

4b 

29  52 

3o  8 

50747 

29 

49253 

53 120 

32 

46880 

02372 

3 

976.^8 

i4 

47 

29  44 

3o  16 

50784 

3o 

49216 

53i6i 

.33 

46839 

02377 

3 

97623 

i3 

48 

29  36 

3o  24 

5o82i 

3o 

49179 

53202 

34 

46798 

0238i 

3 

97619 

12 

49 
5o 

29  28 

3o  32 

5o858 
9.50896 

3i 
3i 

49142 
10.49104 

53244 

-i-i 

46756 

02385 

3 

97615 

II 
10 

9  29  20 

2  3o  4" 

9.53285 

10.46715 

10.02390 

4 

9. 97(1 10 

5i 

29  12 

3o  48 

50933 

32 

49067 

53327 

i6 

46673 

02394 

4 

97606 

Q 

52 

29  4 

3o  56 

50970 

33 

49o3o 

53368 

i6 

46632 

0239S 

4 

976J2 

8 

53 

28  5(i 

3i  4 

51007 

3  J 

48993 

53409 

37 

46591 

o24o3 

4 

97597 

7 

64 
55 

28  48 

3(  12 

5 1043 

34 

48957 

5345o 

38 
38 

4655o 
io.465oS 

02407 

4 

97'J93 

S 

9  28  40 

2  3i  20 

9.510S0 

3') 

10.48920 

9.53492 

10.0241 1 

4 

9.97589 

5b 

28  32 

3i  28 

51117 

35 

48883 

53533 

39 

46467 

02416 

4 

97584 

4 

!)7 

28  24 

3i  36 

5ii54 

36 

48846 

53574 

40 

46426 

02420 

4 

97580 

3 

58 

28  16 

3i  44 

5i  191 

37 

4S809 

536i5 

4i 

46385 

02424 

4 

97576 

T 

59 

28  8 

3i   52 

5l227 

37 

4S773 

53656 

41 

46344 

02429 

4 

97571 

T 

bo 
M 

28  0 

32   c 

51264 

38 

48736 
Secant. 

53697 

42 

463o3 

02433 

4 

97567 

0 

M 

Hour  P.M. 

Hour  A.M. 

Cosine. 

Difr. 

Cotangeut 

DifT. 

Tangent. 

Cosecant. 

DilT. 

Sine. 

108° 


A 

A 

B 

B 

C 

Seconds  of  time  . 

1' 

2' 

3^ 

i4 
16 
2 

4s 

19 
21 
2 

5" 

24 
26 
3 

0= 

28 

3i 
3 

7" 
33 
37 
4 

Prop,  parts  of  cols 

5 
5 
I 

9 

10 

I 

71" 


Page  204] 

TABLE  XXVII 

SI 

• 

Log.  Sines,  Tangents,  and  Secants. 

G'. 

19 

° 

A 

A 

B 

B 

C 

C  160° 

M 

o 

Hour  A. M 

Hour  P.M. 

Sine. 

Dlfl- 

Cosecant. 

Tangent. 

Dirt'. 

Cotangent 

Secant.  [DifT. 

Cosine. 

9.97567 

M 
60 

9  28  c 

2  32  0 

9.5126.^ 

0 

10.48736 

9.53697 

0 

io.463o3 

10.02433 

0 

I 

27  52 

32  8 

5i3oi 

I 

48699 

53738 

I 

46262 

02437 

0 

97563 

5q 

2 

27  44 

32  61 

5i33S 

I 

48602 

53779 

I 

46221 

02442 

0 

97558 

58 

J 

27  3t 

32  24 

5 1 374 

2 

4S626 

53820 

2 

461S0, 

02446 

0 

97554 

57 

4 
5 

27  2& 

32  32 

5i4ii 

2 

48589 

5386i 

3 

46 1 39 

02450 

0 

97550 

56 
55 

9  27  2C 

2  32  4o 

9-5i447 

3 

10.48553 

9.53902 

3 

10*46098 

10.02455 

0 

9-97545 

b 

27  12 

32  48 

5 1 484 

4 

485i6 

53943 

4 

46o57 

02459 

0 

97541 

54 

7 

27  4 

32  56 

5i52c 

4 

48480 

53984 

5 

46016 

02464 

97536 

53 

8 

26  5(j 

33     4 

5i557 

5 

48443 

54o25 

5 

45975 

02468 

97532 

52 

_9 

10 

26  48 

33  12 

51593 

5 

48407 

54o65 

6 

45935 

02472 

97528 
9.97523 

5i 
5o 

9  26  4o 

2  33  20 

9.51629 

6 

10.48371 

9.54106 

7 

10.45S94 

10.02477 

II 

26  32 

33  28 

5 1 666 

7 

48334 

54i47 

7 

45853 

02481 

97519 

4q 

12 

26  24 

33  36 

51702 

7 

48298 

54187 

8 

458i3 

02485 

975 1 5 

48 

iJ 

26  16 

33  44 

5 1 738 

8 

48262 

54228 

9 

45772 

02490 

97510 

47 

i4 
i5 

26  8 

33  52 

51774 

8 

48226 

54269 

9 

45731 

02494 

97506 

46 

45 

926  0 

2  34  0 

9.51811 

9 

10.48189 

9.54309 

10 

10.45691 

10.02499 

9.97501 

lb 

25  52 

34  8 

5 1 847 

10 

48i53 

54350 

II 

45650 

025o3 

97497 

44 

17 

25  44 

34  16 

5i883 

10 

48117 

54390 

II 

45610 

025o8 

97492 

43 

i8 

25  36 

34  24 

51919 

II 

48081 

54431 

12 

45569 

025l2 

97488 

42 

12 

20 

25  28 

34  32 

51955 

II 

48045 

54471 

i3 

45529 

025i6 

97484 

4i 

4o 

9  25  20 

2  34  4o 

9.51991 

12 

10.48009 

9.54512 

i3 

10.45488 

10.02521 

9-97479 

21 

25  12 

34  48 

52027 

12 

47973 

54552 

i4 

45448 

02525 

2 

97475 

3q 

22 

25  4 

34  56 

52063 

i3 

47937 

54593 

i5 

45407 

0253o 

2 

97470 

38 

2j 

24  56 

35  4 

52099 

14 

4790 ' 

54633 

i5 

45367 

02534 

2 

97466 

37 

24 
25 

24  48 

35  12 

52x35 

14 

47865 

54673 

16 

45327 

02539 

2 

97461 

36 
35 

9  24  4o 

2  35  20 

9.52171!  i5 

10.47829 

9.54714 

17 

10.45286 

10.02543 

2 

9-9/457 

2b 

24  32 

35  28 

52207I  1 5 

47793 

54754 

17 

45246 

02547 

2 

97453 

34 

2? 

24  24 

35  36 

52242 

lb 

47758 

54794 

18 

45206 

02552 

2 

97448 

33 

28 

24  16 

35  44 

52278 

17 

47722 

54835 

19 

45i65 

02556 

2 

97444 

32 

29 

3o 

24  8 

35  52 

523i4 

17 

47686 

54875 

19 

45i25 

o256i 

2 

97439 

3i 
3^ 

9  24  0 

2  36  0 

9.52350 

18 

10.47650 

9.54915 

20 

io.45o85 

10.02565 

2 

9-97435 

Si 

23  52 

36  8 

52385 

18 

4-6 1 5 

54955 

21 

45o45 

02570 

2 

97430 

29 

J2 

23  44 

36  16 

52421 

19 

47579 

54995 

21 

45oo5 

02574 

2 

97426 

28 

JJ 

23  36 

36  24 

52456 

20 

47544 

55o35 

22 

44965 

02579 

2 

97421 

27 

34 

35 

23  28 

36  32 

52492 

20 

47508 

55075 

23 

44925 

02583 

3 

97417 

26 
l5 

9  23  20 

2  36  4o 

9.52527 

21 

10.47473 

9.551 i5 

23 

10. 44885 

10.02588 

3 

9.97412 

6b 

23  12 

36  48 

52563 

21 

47437 

55i55 

24 

44845 

02592 

3 

97408 

24 

^7 

23  4 

36  56 

52598 

22 

47402 

55195 

25 

448o5 

02597 

3 

974o3 

23 

38 

22  56 

37  4 

52634 

23 

47366 

55235  25 

44765 

02601 

3 

97399 

22 

39 

4o 

22  48 

37  12 

52669 

23 

47331 

55275  26 

44725 

02606 

3 

97394 

21 
20 

9  22  4o 

2  37  20 

9.52705 

24 

10.47295 

9.553i5 

27 

10. 44685 

10.02610 

3 

9.97390 

41 

22  32 

37  28 

52740 

24 

47260 

55355 

27 

44645 

02615 

3 

97385 

IQ 

42 

22  24 

37  36 

52775 

25 

47225 

55395 

28 

446o5 

02619 

3 

97381 

18 

43 

22  16 

37  44 

528 II 

2b 

47189 

55434 

29 

44566 

02624 

3 

97376 

17 

44 
45 

22   8 

37  52 

52846 

2b 

47154 

55474 

29 

44526 

02628 

3 

97372 

16 

i5 

9  22   0 

2  38  0 

9.52881 

27 

10.47119 

9.555i4 

3o 

10.44486 

10.02633 

3 

9.97367 

4b 

21  52 

38  8 

52916 

27 

47084 

55554 

3i 

44446 

02637 

3 

97363 

i4 

47 

21  44 

38  16 

52951 

28 

47049 

55593 

3i 

44407 

02642 

3 

97358 

i3 

4b 

21  36 

38  24 

52986 

29 

47014 

55633 

32 

44367 

02647 

4 

97353 

12 

49 
5o 

21  28 

38  32 

53o2i 

29 

46979 

55673 

33 

44327 

o265i 

4 
4 

97349 

II 

10 

9  21  20 

2  38  4o 

9.53o56 

3o 

10.46944 

9.55712 

33 

10.4428S 

10. 02656 

9-97344 

5i 

21  12 

38  48 

53092 

3o 

46908 

55752 

34 

44248 

02660 

4 

97340 

9 

52 

21  4 

58  56 

53126 

3i 

46874 

55791 

35 

44209 

02665 

4 

97335 

ti 

53 

20  56 

39  4 

53i6i 

32 

46839 

5583i 

35 

44169 

02669 

4 

97331 

7 

64 
55 

20  48 

39  12 

53196 

32 

46804 

55870 

36 

44  i3o 

02674 

4 

97326 

6 
5 

9  20  4o 

2  39  20 

9.53231 

33 

10.46769 

9.55910 

37 

10.44090 

10.02678 

4 

9.97322 

5b 

20  32 

39  28 

53266 

33 

46734 

55949 

37 

44o5i 

02683 

4 

97317 

4 

i>7 

20  24 

39  36 

533oi 

M 

46699 

55989 

38 

44011 

02688 

4 

97312 

3 

58 

20  16 

09  44 

53336 

M 

46664 

56028 

39 

43972 

02692 

4 

97308 

2 

59 

20  8 

39  52 

53370 

35 

4663o 

56067 

39 

43933 

02697 

4 

973o3 

I 

bo 

20  0 

4o  0 

534o5 

3b 

46595 

56107 

4o 

43893 

02701 

4 

97299 

0 
M 

M 

Hour  p.ir. 

HourA.M. 

Cosine. 

Ditr. 

Secant. 

Cotangent 

Dilf. 

Tangent. 

Cosecant. 

Ditr. 

Sine. 

109= 


A 

A 

B 

B 

C 

1' 

2» 

3' 

4. 

5» 

6' 

7- 

Prop,  parts  of  cols. 

(• 

4 
5 
I 

9 
10 

I 

i3 
i5 
2 

18 
20 
3 

22 

25 

3 

27 
3o 
3 

3i 
35 
4 

lOP 


^~"" 

TABLE  XXVII. 

[Page  205 

S'. 

Loot 

.  Sines,  Tangents,  and  Secants. 

Q'. 

20= 

A 

A 

B 

B 

C 

C  159° 

IM 

0 

IIourA.M. 

Hour  P.M. 

Sine. 

Diir. 

Cosecant. 

Tangent. 

DifT. 

Cotangent 

Secant.  DifT.; 

Cosine. 

M 

6^ 
59 

9  20   0 
19  52 

19  44 
19  36 

2  40  0 

9.53405 

G 

10.46595 

9.56107 

0 

10.43893 

10.02701 

0 

9.97299 

I 

4o  8 

53440 

I 

46560 

56i46 

I 

43854 

02706 

0 

97294 

9 

4o  16 

53475 

I 

46525 

56i85 

I 

438i5 

02711 

0 

97289 

58 

57 
56 

55 
54 

3 

40  24 

53509 

2 

46491 

56224 

2 

43776 

02715 

0 

97285 

4 
5 

19  28 

40  32 

53544 

2 

46456 

56264 

6 
3 

43736 
10.43697 

02720 

0 

972S0 

9  19  20 

2  4o  40 

9.53578 

3 

10.46422 

9.563o3 

10.02724 

0 

9.97276 

fi 

19  12 
ly  4 

40  48 

536i3 

3 

46387 

56342 

4 

43658 

02729 

0 

9771 

7 

40  56 

53647 

4 

46353 

5638 1 

4 

43619 

02734 

97266 

53 

52 

iS 

18  56 

4i  4 

53682 

5 

463 1 8 

56420 

5 

4358o 

02738 

97262 

_9 

10 

18  48 

4i  12 

53716 

5 

46284 

56459 

6 

43541 

02743 

97257 
9.97252 

5i 

5^ 

9  18  40 

2  4i  20 

9.53751 

6 

10.46240 

9.56498 

6 

10.43502 

10.02748 

1 1 

18  32 

4i  28 

53785 

6 

4621^ 

56537 

7 

43463 

02752 

97241 

49 

1  L> 

18  24 

4 1  36 

53819 

7 

46t8i 

56576 

8 

43424 

02757 

97243 

46 

i3 

18  16 

4i  44 

53854 

7 

46 1 46 

566 1 5 

8 

43385 

02762 

97238 

47 

i4 
i5 

18  8 

4t  52 

53888 

8 

461 12 

56654 

9 

43346 

02766 

97234 

46 
45 
44 
43 
42 

0  lb  0 

2  42  0 

9.53922 

8 

10.46078 

9.56693 

IQ 

10.43307 

10.02771 

9.97229 

iG 

17  52 

42  8 

53957 

9 

46043 

56732 

10 

43268 

02776 

97224 

17 

17  44 

42  16 

53991 

10 

46009 

56771 

II 

43229 

02780 

97220 

i8 

17  36 

42  24 

54025 

10 

45975 

568 10 

12 

43190 

02785 

97215 

12 

20 

17  28 

42  32 

54059 

II 

45941 

56849 

12 

43i5i 

02790 

97210 

41 
4o 
39 

9  17  20 

2  42  4» 

9.54093 

II 

10.45907 

9.56887 

i3 

io.43ii3 

10.02794 

2 

9.97206 

21 

17  12 

42  48 

54127 

12 

45873 

56926 

iJ 

43074 

02799 

2 

97201 

22 

17  4 

42  56 

54161 

12 

45830 

56965 

i4 

43o35 

02804 

3 

97196 

38 

23 

16  56 

43  4 

54195 

i3 

458o5 

57004 

lb 

4299^. 

02808 

2 

97192 

37 
36 

35 

24 
25 

16  48 

43  12 

54229 

i4 

45771 

57042 

i5 

42958 

02813 

2 

97187 

9  16  4o 

2  43  20 

9.54263 

i4 

10.45737 

9.57081 

16 

10.42919 

10.02818 

2 

9.97182 

26 

16  32 

43  28 

54297 

i5 

45703 

57120 

17 

42880 

02822 

2 

97178 

34 

27 

16  24 

43  36 

54331 

i5 

45669 

57 1 58 

17 

42842 

02827 

2 

97173 

33 

28 

16  16 

43  44 

54365 

16 

45635 

57197 

18 

42803 

02832 

2 

97168 

32 

29 

So 

16  8 

43  52 

54399 

16 

45601 

57235 

19 

42765 

02837 

2 

97163 

3i 
3^ 

9  16  0 

2  44  0 

9.54433 

17 

10.45567 

9.57274 

19 

10.42726 

10.02841 

2 

9.97159 

3i 

i5  52 

44    8 

54466 

17 

45534 

57312 

20 

42688 

02846 

2 

97154 

29 

32 

i5  44 

44   16 

54500 

18 

45500 

57351 

21 

42649 

o285i 

3 

97149 

28 

33 

i5  36 

44  24 

54534 

19 

45466 

57389 

21 

42611 

02855 

3 

97145 

27 

34 
35 

i5  28 

44  32 

54567 

'9 

45433 

57428 

22 

42572 

02860 

3 

97140 

26 

9  1 5  20 

2  44  4o 

9.54601 

20 

10.45399 

9.57466 

22 

10.42534 

10.02865 

3 

9.97135 

36 

i5  12 

44  48 

54635 

20 

45365 

57504 

23 

42496 

02870 

6 

97i3o 

24 

37 

i5  4 

44  56 

54668 

21 

45332 

57543 

24 

42457 

02874 

6 

97126 

23 

38 

i4  56 

45  4 

54702 

21 

45298 

57581 

24 

42419 

02879 

3 

97121 

22 

39 

4o 

i4  48 

45  12 

54735 

22 

45265 

57619 

25 

4238i 

02884 

3 

97116 

21 

20 

9  1 4  4o 

2  45  20 

9.54769 

23 

io.4523i 

9.57658 

26 

10.42342 

10.028S9 

3 

9.97111 

4i 

i4  32 

45  28 

54802 

23 

45198 

57696 

26 

423o4 

02893 

3 

97107 

ly 

42 

1 4  24 

45  36 

5483r 

24 

45i64 

57734 

27 

42266 

02898 

3 

97102 

lb 

43 

i4  16 

45  44 

54869 

24 

45i3i 

57772 

28. 

42228 

02903 

3 

97097 

17 

44 
45 

14  8 

45  52 

54903 

25 

45097 

57S10 

28 

43190 

02908 

3 

97092 

16 

75 

9  i4  0 

2  46  0 

9.549^6 

25 

10.45064 

9.57849 

29 

io.42i5i 

10.02913 

4 

9.97087 

46 

i3  52 

46  8 

54969 

26 

45o3i 

57887 

3o 

42ii3 

02917 

4 

97083 

14 

47 

1 3  44 

46  16 

55oo3 

26 

44997 

57925 

3o 

42075 

02922 

4 

97078 

i3 

48 

i3  36 

46  24 

55o36 

27 

44964 

57963 

3i 

42037 

02927 

4 

97073 

12 

49 
5o 

i3  28 

46  32 

55069 

28 

44931 

58ooi 

3i 

41999 

02932 

4 

97068 

1 1 
10 

9  i3  20 

2  46  4o 

9.55102 

28 

10.44898 

9.58039 

32 

10.41961 

10.02937 

4 

9.97063 

bi 

i3  12 

46  48 

55i3C 

29 

44864 

58077 

33 

41923 

02941 

4 

97059 

y 

52 

i3  4 

46  56 

55169 

29 

4483 1 

58ii5 

33 

4i885 

02946 

4 

97054 

8 

53 

12  56 

47  4 

55202 

3o 

44798 

58i53 

34 

4 1847 

02951 

4 

97049 

7 

54 

55 

12  48 

47  12 

55235 

3o 

44765 

58191 

35 

41809 

02956 

4 

97044 

~5 

9  12  4o 

2  47  20 

9.55268 

3i 

10.44732 

9.58229 

35 

10.4177' 

10.02961 

4 

9.97039 

56 

12  32 

47  28 

553oi 

32 

44699 

58267 

36 

41733 

02965 

4 

97035 

4 

^7 

12  24 

47  36 

5533.J 

32 

44666 

583o4 

3? 

41696 

02970 

4 

97o3o 

3 

58 

12  16 

47  44 

55367 

33 

44633 

58342 

37 

4i658 

02975 

5 

97025 

2 

^9 

12  8 

4i   52 

55400 

33 

44600 

58380 

38 

41620 

02980 

5 

97020 

I 

60 
M 

12  0 

48  0 

55433 

34 

44567 

584i8 

39 

4i582 

02985 

5 

97015 

0 
M 

Hour  p.M 

Hour  A.M. 

Cosinn. 

Diff. 

Secant. 

Cotangen 

Ditr 

Tangent. 

Cosecant.  jDifT. 

Sine. 

110° 


A 

A 

B 

B 

C 

1- 

2» 

3' 

4- 

5' 

6' 

25 

29 
4 

7* 
3o 
34 
4 

Prop,  parts  of  cols. 

(■ 

4 
5 
I 

8 
10 

X 

i3 

i4 

a 

17 

19 

a 

31 
24 

3 

C       09" 


Page  206] 

TABLE  XXVIL 

5 

Log.  Sines,  Tanaents,  and  Secants. 

G'. 

21° 

A 

A 

B 

B 

C        C  158° 

M 
o 

Hour  A. M 

Hour  P.M. 

Sine. 

Diff 

Cosecant. 

Tangent. 

Diff. 

Cotangent 

Secant. 

Diff. 

Cosine. 

M 

60 

9  12  0 

2  48  0 

9.55433 

0 

10.44567 

9.58418 

0 

io.4i582 

10.02985 

0 

9.97015 

I 

II  52 

48  8 

55466 

I 

44534 

58455 

I 

4i545 

03990 

0 

97010 

59 

2 

II  4^ 

48  16 

55499 

I 

445oi 

58493 

I 

4i5o7 

02995 

0 

97005 

58 

6 

II  36 

48  24 

55532 

2 

44468 

58531 

2 

41469 

02999 

0 

97001 

57 

4 
5 

II  28 

43  32 

55564 

2 

44436 

58569 

2 

4i43i 

o3()o4 

0 

96996 

56 
55 

9  1 1  20 

2  48  4o 

9.55597 

3 

io.444o3 

9.58606 

3 

10.41394 

io.o3oo9 

0 

9.96991 

b 

II  12 

48  48 

55630 

3 

44370 

58644 

4 

4i356 

o3oi4 

0 

96986 

54 

7 

II  4 

48  56 

55663 

4 

44337 

5868 1 

4 

4i3i9 

o3oi9 

96981 

53 

8 

10  56 

49  4 

55695 

4 

443o5 

58719 

5 

41281 

o3o24 

96976 

52 

_9 

10 

10  48 

49  12 

55728 

5 

44272 

58757 

6 

41243 

o3o29 

96971 

5i 

5o 

9  10  4o 

2  49  20 

9.55761 

5 

10.44239 

9.58794 

6 

10.41206 

io.o3o34 

9.96966 

II 

10  32 

49  28 

55793 

b 

44207 

58832 

7 

4116S 

o3o38 

96962 

'40 

12 

in  94 

49  36 

55826 

b 

44174 

58869 

7 

4ii3i 

o3o43 

96957 

48 

iJ 

1  u  i  'J 

49  44 

55858 

7 

44x42 

58907 

8 

41093 

o3o48 

96952 

47 

i4 
i5 

10  8 

49  52 

55891 
9.55923 

7 

44109 

58944 

9 

4io56 

o3o53 

96947 

46 
45 

9  1 0  0 

2  5o  0 

8 

10.44077 

9.58981 

9 

10.41019 

io.o3o58 

9.96942 

lb 

952 

5o  8 

55956 

9 

44044 

59019 

10 

40981 

o3o63 

96937 

44 

17 

9  44 

5o  16 

55988 

9 

44012 

59056 

10 

40944 

o3o68 

96932 

43 

i8 

9  36 

5o  24 

56o2i 

10 

43979 

59094 

1 1 

40906 

o3o73 

96927 

42 

19 

20 

9  28 

5o  32 

56o53 

10 

43947 

59131 

12 

40869 

03078 

2 

96922 

4i 
4o 

9  9  20 

2  5o  4o 

9.56085 

1 1 

10.43915 

9.59168 

12 

to.4oS32 

io.o3oS3 

2 

9.96917 

21 

9  12 

5o  48 

56ii8 

II 

43882 

59205 

i3 

40795 

o3o88 

2 

96912 

3q 

22 

9  4 

5o  56 

56i5o 

12 

43850 

59243 

i4 

40757 

03093 

2 

96907 

38 

2  J 

8  56 

5i  4 

56182 

12 

438x8 

592S0 

i4 

40720 

o3o97 

2 

96903 

37 

24 
25 

8  4^ 

5i  12 

562  15 

i3 

43785 

59317 

i5 

4o683 

o3io2 

2 

96S98 

36 

35 

9  8  40 

■2   5i  20 

9.56247 

i3 

10.43753 

9.59354 

i5 

10.40646 

io.o3!07 

2 

9.96893 

2b 

8  32 

5i  28 

56279 

i4 

43721 

59391 

16 

40609 

o3i  12 

2 

96888 

34 

27 

8  24 

5i  36 

563 II 

1 4 

43689 

59429 

17 

4o57i 

o3ii7 

2 

96883 

33 

28 

8  16 

5i  44 

56343 

13 

43657 

59466 

17 

4o534 

o3l22 

2 

96S78 

32 

29 

3o 

8  8 

5i  52 

56375 

lb 

43625 
10.43592 

5q5c3 

18 

40497 

o3i27 

2 

96873 

3i 
3^ 

980 

2  52  0 

0 . 564o8 

16 

0.59540 

19 

!0.4o46o 

io.o3i32 

2 

9 . 96S68 

Ji 

7  52 

52  8 

56440 

17 

43560 

5g577 

IQ 

40423 

o3i37 

3 

96863 

29 

J2 

7  44 

52  16 

56472 

17 

43528 

5o6r4 

20 

4o386 

o3i42 

3 

96858 

28 

33 

7  36 

52  24 

565o4 

18 

43496 

5q65i 

20 

40349 

o3i47 

3 

96853 

27 

34 
35 

7  28 

52  32 

56536 

18 

43464 

5r^S8 

21 

4o3i2 

o3i52 

3 

96848 

26 

9  7  20 

2  52  40 

9.56568 

19 

10.43432 

9.59725 

22 

10.40275 

io.o3i57 

3 

9.96843 

3b 

7  12 

52  48 

56599 

'9 

43401 

59762 

22 

40238 

o3i62 

J 

96838 

24 

^7 

7  4 

52  56 

5663 1 

20 

43369 

59799 

23- 

40201 

.o3 1 67 

3 

96833 

23 

38 

6  56 

53  4 

56663 

20 

43337 

59835 

23 

4oi65 

o3i72 

3 

96828 

22 

39 

4o 

6  48 

53  12 

56695 

21 

433o5 

59872 

24 

40128 

o3i77 

3 

96823 

21 

20 

9  6  4o 

2  53  20 

9.56727 

21 

10.43273 

9.59909 

25 

10.40091 

io.o3i82 

3 

9.96S18 

4i 

6  32 

53  28 

56759 

22 

43241 

59946 

25 

4oo54 

o3i87 

3 

96813 

19 

42 

6  24 

53  36 

56790 

22 

43210 

59983 

26 

40017 

03192 

3 

96808 

18 

43 

6  iG 

53  44 

5682  2 

23 

.  43178 

60019 

27 

39981 

o3i97 

4 

96803 

17 

44 
45 

6  8 

53  52 

56854 

24 

43 1 46 

Goo56 

27 

39944 

03202 

4 

96798 

16 
75 

960 

2  54  0 

9.56886 

24 

io.43ii4 

9.60093 

28 

10.39907 

10.03207 

4 

9.96793 

4b 

5  52 

54  8 

56917 

25 

43o83 

6oi3o 

28 

39870 

o32I2 

4 

96788 

i4 

47 

5  44 

54  16 

56949 

25 

43o5i 

60166 

29 

39834 

o32i7 

4 

96783 

i3 

48 

5  36 

54  24 

56980 

26 

43o2o 

6o2o3 

3o 

39797 

03222 

4 

96778 

12 

49 
5o 

5  28 

54  32 

57012 

26 

42988 

60240 

3o 

39760 

03228 

4 

9677? 

11 
10 

9  5  20 

2  54  40 

9.57044 

27 

10.42956 

9.60276 

3i 

10.39724 

10.03233 

4 

9.96767 

5i 

5  12 

54  48 

57075 

27 

42925 

6o3i3 

3i 

396S7 

o323S 

4 

96762 

9 

52 

5  4 

54  56 

57107 

28 

42893 

fo349 

32 

39651 

03243 

4 

96757 

« 

53 

4  56 

55  4 

57i38 

28 

42862 

6o386 

33 

39614 

o3248 

4 

96752 

7 

54 
55 

4  48 

55  12 

57169 

29 

4283i 

6i  422 

33 

39578 

o3253 

4 

96747 

b 

9  4  40 

2  55  20 

9.57201 

29 

10.42709 

9.60459 

34 

10.39541 

io.n3253 

5 

9.96742 

5b 

4  32 

55  28 

57232 

3o 

42768 

60/95 

J3 

39505 

o3263 

5 

96737 

4 

t)7 

4  24 

55  36 

57264 

3o 

42736 

6o532 

35 

39468 

03268 

5 

96732 

3 

58 

4  16 

55  44 

57295 

3[ 

42705 

6o568 

36 

39432 

03273 

5 

96727 

2 

59 

4  8 

55  52 

57326 

32 

42674 

6o6c5 

36 

39395 

03278 

5 

96722 

I 

bo 

4  0 

56  0 

57358 

32 

42642 

6064! 

37 

39359 

o3283 

5 

96717 

0 

Hour  p. 51. 

Hour  A.m. 

Cosine,  joiff. 

Secant. 

Cotangent 

DilT. 

Tangent. 

Cosecant. 

Diif. 

Si.:e. 

ur 


A 

A 

B 

B 

( 

*_/ 

Seconds  of  time 

V 

2' 

3» 

4» 

5» 

6' 

7' 

(^ 

4 

8 

12 

16 

20 

24 

28 

Prop,  parts  of  cols. 

^ 

5 

9 

i4 

19 

23 

28 

32 

f  C 

I 

2 

2  1  3 

4 

4 

68* 


TABLi:  XXVII. 

[Page  'J07 

S' 

Log.  Sines,  Tangents,  and  Secants. 

G'. 

22 

0 

A 

A 

B 

B 

C 

C  157° 

M 

0 

I 

2 

3 
4 
5 
6 

7 
8 

_9 

10 

II 

12 

i3 

i4 
i5 
1 6 

17 
i8 

12 

20 
21 
22 
23 
24 
25 
26 
27 
28 

29 

3o 
3i 

32 

33 
34 
35 
36 

37 
38 
39 

4o 
4i 
42 
43 
44 
45 
46 
47 
48 

49 
5o 
5i 

52 

53 
54 
55 
56 

57 
58 

60 
M 

Hour  A.M. 

Hour  P.M. 

Sine. 

DilT. 

Cosecant. 

Tangent. 

DifT. 

Cotangent 

Secant. 

Difl". 

Cosine. 

M 
6^ 
59 
58 

57 
56 

55 
54 
53 

52 

5i 

53 

49 
48 

47 
46 

45 
44 
43 
42 
4i 
40 

39 

38 

37 
36 

35 
34 
33 

32 

3i 

3o 
29 

28 

27 
26 

25 

24 

23 
22 
21 

20 

•9 
18 

17 
16 

75 
i4 
i3 
12 
1 1 
10 

9 

8 

7 
6 

5 
4 
3 
2 
I 
0 

.?. 

940 
3  52 
3  44 
3  36 
3  28 

2  56  0 
56  8 
56  16 
56  24 
56  32 

9.57358 
57389 
57420 
57451 
57482 

0 

I 
I 
2 
2 

10.42642 
4261 1 
42580 
42'}4<) 
425i8 

9 . 6064 1 
60677 
60714 
60750 
60786 

0 
I 

I 
2 
2 

10.39359 
39323 
39286 
39250 
39214 

io.o32S3 
08289 
08294 
08299 

o33o4 

0 
0 
0 
0 
0 

9.96717 
96711 
96706 
96701 
96696 

9  3  20 
3  12 

3  4 
2  56 
2  48 

2  56  40 

56  48 

56  56 

57  4 
57  12 

9.57514 
57545 
57576 
57607 
57638 

3 
3 

4 
4 
5 

to. 42486 

42455 
42424 
42393 

42362 

9.60823 
6u859 
60895 
60931 
60967 

3 

4 
4 
5 
.5 

10.39177 
39141 
39105 
39069 
89033 

10.08809 
o33i4 
o33i9 
08824 
o333o 

0 

9.96691 
96686 
9668 1 
96676 
96670 

9  2  4o 

2  32 

2  24 
2  16 
2  8 

2  57  20 
57  28 
57  36 
57  44 

57  52 

9.57669 
57700 
57731 
57762 
57793 

5 
6 
6 

7 

7 

10.42331 
423oo 
42269 

42238 

42207 

9.61004 
6io4o 
61076 
61 1 12 
6ii48 

6 

7 
7 
8 
8 

10.38996 
38960 
38924 
38888 
38852 

10. 03335 
o334o 
o3345 
o335o 
03355 

9.96G65 
96660 
96655 
96650 
96645 

920 

I  52 

I  44 
I   36 
I  28 

2  58  0 
58  8 
53  16 
58  24 
58  32 

9.57824 
57855 
57885 
57916 
57947 

8 
8 
9 
9 

U) 

10.42176 
42145 
42ii5 
42084 
42053 

9.61184 
61220 
6i256 
61292 
6i328 

9 
10 
10 
1 1 
1 1 

io.3S8i6 
38780 
38744 
38708 
38672 

io.o336o 
03366 
03371 
03376 
o338i 

2 
2 

9.96640 
96634 
96629 
96624 
966 1 9 

9 . 966 1 4 
96608 
96608 
96598 
96593 

9  I  20 
I  12 
I  4 
0  56 
0  48 

2  58  4o 
58  48 

58  56 

59  4 
69  12 

9.57978 
58oo8 
58o39 
5S070 
5Sioi 

10 
1 1 
II 
12 
12 

10.42022 
41992 
4 1961 
41930 
41899 

9.61864 
6i4oo 
6 1 436 
61472 
6i5o8 

12 

i3 
i3 
1 4 

i4 

10. 38636 
386oo 
38564 
38528 
38492 

io.o33S6 
03392 
08897 
o34o2 
08437 

2 
2 

2 

2 

9  0  4o 

0  32 

0  24 
0  16 
0  8 

2  59  20 
59  28 
59  36 
59  44 
59  52 

9.58i3i 
58162 
58192 
58223 
58253 

i3 

i3 
1 4 
1 4 
i5 

10.41869 
4i838 
41808 
41777 
41747 

9.61544 
61579 
6i6i5 
6i65i 
61687 

i5 
i5 
16 
17 
17 

10.38456 
38421 
38385 
38349 
383 1 3 

ro.o34i2 
o34i8 
03423 
08428 
03433 

2 
2 
2 
2 
3 

9.965S8 
96582 
96577 
96572 
96567 

900 

8  59  52 

59  44 

59  36 

59  28 

3  0  0 
0  8 
0  16 

0  24 

0  32 

9.58284 
583 1 4 
58345 
58375 
584o6 

i5 
16 
16 
17 
17 

:o.4i7i6 
4 1 686 
4i655 
41625 
41594 

9.61722 
61758 
61794 
6i83o 
6 1 865 

18 
18 

'9 

20 
20 

10.08278 
38242 
382*06 
38170 
38i35 

I0.03438 
03444 
03449 
03454 
03459 

3 

0 

3 
3 
3 

9.95562 
96556 
96551 
96546 
96541 

8  59  20 
59  12 
59  4 
58  56 
58  48 

3  0  4o 
0  48 

0  56 

1  4 
I  12 

9.58436 
58467 
58497 
58527 
58557 

18 
18 
19 
19 
20 

20 
21 
21 
22 
22 

1 0 . 4 1 564 
4i533 
4i5o3 
41473 
41443 

io.4i4i2 
4i382 
4i352 

4  I  322 

41291 

9.61901 
61936 
61972 
62008 
62043 

21 
21 
22 

23 
23 

1 0 . 3S099 
38o64 
38028 
37992 
37957 

io.o3465 
03470 
03475 
o348o 
o3486 

3 
3 
3 
3 
3 

9.96535 

96530 
96525 
96520 
96514 

8  58  4o 
58  32 
58  2.4 
58  16 
58  8 

3  I  20 
1  28 
I  36 
I  44 

I    52 

9.58588 
586 18 
58648 
5S678 
53709 

9.62079 
62114 
62i5o 
62185 

6-'2:>l 

24 

24 

25 

26 
26 

10.87921 
37886 
37850 
37815 
37779 

10.03491 
03496 
o35o2 
o35o7 
o35i2 

3 
4 
4 
4 
4 

9.96509 
96504 
96498 
96493 
9648S 

8  58  0 
57  52 

57  4^ 
57  36 
57  28 

320 
2  8 
2  16 
2  24 

2  32 

9.58739 
58769 
58799 
58829 
58859 

23 

23 

24 
24 

25 
25 

26 
26 
27 
27 

I0.4I26I 
4i23i 
41201 
41171 
4ii4i 

io.4ii 1 1 
410S1 
4io5i 

4l02[ 
40991 

9.62256 
62292 

62827 

62362 
62398 

27 
27 
28 
29 
29 

10.37744 
37708 
37673 
37688 
37602 

io.o35i7 
03523 
03528 
03533 
03539 

4 
4 
4 
4 
4 

9.96488 

96477 
96472 
96467 
96461 

8  57  20 
57  12 
57  4 
56  56 
56  48 

3  2  4o 
2  48 

2  56 

3  4 
3  12 

9.58889 
58919 
58949 
58979 
59009 

9.62433 
62468 
62504 
62539 
62574 

3o 
3o 
3i 

32 
32 

10.87567 
37532 
37496 
37461 
37426 

10.03544 
03549 
03555 
o856o 
o3565 

4 
4 
5 
5 
5 

9.96456 
96451 
96445 
96440 
96435 

8  56  4o 
56  32 
56  24 
56  16 
56  8 
56  0 

3  3  20 
3  28 
3  36 
3  44 

3  52 

4  0 

9.59039 
59069 
59098 
59128 
59158 
59188 

28 
28 
29 

19 
3o 
3i 

10.40961 
40931 
40902 
40872 
40S42 
4oSl2 

9.62609 
62645 
62680 
62715 
62750 
62785 

33 
33 
34 
35 
35 
36 

10.37391 
37355 
37820 
37285 
37250 
37215 

10.03571 
03576 
o358i 
o3587 
08592 
03597 

5 
5 
5 
5 
5 
5 

9.96429 
96424 
96419 
96418 
96408 
96408 

IIoiiri'.iM. 

Hour  A.M. 

Cosine. 

DilT. 

Secant. 

Cotangent 

Diff. 

Tangent. 

Cosecant. 

Diff. 

Sine. 

112° 


or 


Seconds  of  time 

1' 

2' 

3» 

4» 

5* 

6' 

•ys 

Prop,  parts  of  cols.  ^  B 

4 
4 

I 

8 

9 
I 

IX 

i3 
3 

i5 
18 
3 

19 
22 
3 

23 

27 
4 

27 
3i 

Page2)S] 

TABLE  XXVIL 

5' 

Log.  Sines,  Tangents,  and  Secants. 

(?'. 

2-4 

0 

A 

A 

B 

B 

C 

G  156° 

M 

o 

Hour  A. M 

Hour  P.M. 

Sine. 

Diff. 

Cosecant. 

Tang-ent. 

Ditr. 

Cotang'ent 

Secant. 

Diff. 

Cosine. 

M 
6^ 

8  56  0 

340 

9.59188 

0 

10.40812 

9.62785 

0 

10.37215 

10.03597 

0 

9 . 96403 

I 

55  52 

4  8 

59218 

0 

40782 

62820 

I 

37180 

o36o3 

0 

96397 

5q 

2 

55  44 

4  16 

59247 

I 

40753 

62855 

I 

37145 

o36o8 

0 

96392 

58 

3 

55  36 

4  24 

59277 

1 

40723 

62890 

2 

371 10 

o36i3 

0 

96387 

57 

4 
5 

55  28 

4  32 

59307 

2 

40693 

62926 

2 

37074 

o36i9 

0 

96381 

56' 
55 

8  55  20 

3  4  4o 

9.59336 

2 

10.40664 

9.62961 

3 

10.37039 

10.03624 

0 

9.963-6 

b 

55  12 

4  48 

59366 

3 

4o634 

62996 

3 

37004 

o363o 

96370 

'^■^ 

7 

55  4 

4  56 

59396 

3 

4o6o4 

63o3i 

4 

36969 

03635 

96365 

53 

8 

54  56 

5  4 

59425 

4 

40575 

63o66 

5 

36934 

o364o 

96360 

52 

_? 

lO 

54  48 

5  12 

59455 
9.59484 

4 

4o545 

63ioi 

5 

36899 

03646 

96354 

5i 
5o 

8  54  4^ 

3  5  20 

5 

io.4o5i6 

9.63i35 

6 

10.36865 

io.o365i 

9.96349 

II 

54  32 

5  28 

59514 

5 

40486 

63170 

6 

36830 

o3657 

96343 

4q 

12 

54  24 

5  36 

59543 

b 

40457 

63205 

7 

36795 

03662 

96338 

48 

i6 

54  lb 

5  44 

59573 

b 

40427 

63240 

7 

36760 

03667 

96333 

47 

i4 
i5 

54  8 

5  52 

59602 

7 

40398 

63275 

8 

36725 

03673 

96327 

46 

45 

8  54  0 

3  b  c 

9.59632 

7 

io.4o363 

9.63310 

9 

10.36690 

10.03678 

9.96322 

lb 

53  52 

6  8 

59661 

8 

40339 

63345 

9 

36655 

o36S4 

96316 

44 

17 

53  44 

6  16 

59690 

8 

4o3io 

63379 

10 

3662  1 

03689 

2 

963 1 1 

43 

i8 

53  36 

6  24 

59720 

9 

40280 

634 1 4 

10 

36586 

03695 

2 

963o5 

42 

!9 

20 

53  28 

6  32 

59749 

9 

4o25i 

63449 

1 1 

3655i 

03700 

2 

96300 

4i 
4o 

8  53  20 

3  6  40 

9.59778 

10 

10.40222 

9.63484 

12 

io.3b5i6 

10  03706 

2 

9.96294 

21 

53  12 

6  48 

59808 

10 

40192 

635i9 

12 

3648 1 

0371 1 

2 

96289 

3q 

22 

53  4 

6  56 

59S37 

II 

4oi63 

63553 

i3 

36447 

03716 

2 

0/^284 

38 

2j 

■   52  56 

7  4 

59S66 

II 

4oi34 

63588 

i3 

364 1 2 

03722 

2 

90278 

37 

24 
25 

52  48 

7  12 

59S95 

12 

4oio5 

63623 

i4 

36377 

03727 

2 

96273 

36 
35 

8  52  40 

3  7  20 

9.59924 

12 

10.40076 

9.63657 

i4 

10.36343 

10.03733 

2 

9.96267 

2b 

52  32 

7  28 

59954 

i3 

40046 

63692 

i5 

363o8 

03738 

2 

96262 

34 

27 

'n  14 

7  36 

59983 

i3 

40017 

63726 

lb 

36274 

03744 

2 

96256 

33 

28 

52  iG 

'  A' 

60012 

14 

3998S 

63761 

lb 

36239 

03749 

3 

96251 

32 

29 

3o 

52  8 

7  52 

Loc^. 

—  1   9>  ? 

63796 
9~&383o 

17^ 
17 

36204 
10.36170 

03755 

3 

96245 

3i 
3^ 

8  52  0 

3  8  u 

9.60070 

i5 

10.39930 

10.03760 

3 

9.96240 

3 1 

5i  52 

8  8 

60090 
60128 

i5 

30901 

63865 

18 

36i35 

03769 

3 

96234 

2Q 

J  2 

5i  44 

8  16 

i5 

39872 

63899 

18 

36ioi 

03771 

3 

96229 

28 

33 

5 1  36 

8  24 

60157 

lb 

39843 

63934 

19 

36o66 

03777 

3 

96223 

27 

34 
35 

5r  28 

8  32 

60186 

lb 

39814 

63968 

20 

36o32 

03782 

3 

96218 

26 
25 

8  5i  20 

3  8  4o 

9.60215 

17 

10.39785 

9.64oo3 

20 

10.35997 

10.03788 

3 

9.96212 

3b- 

5i  12 

8  48 

60244 

17 

39756 

64o37 

21 

35963 

03793 

3 

96207 

2.4 

^7 

5r  4 

8  56 

60273 

18 

39727 

64072 

21 

35928 

03799 

3 

96201 

23 

38 

5o  56 

9  4 

6o3o2 

18 

39698 

64106 

22 

35894 

o3So4 

3 

96196 

22 

39 

4o 

5o  48 

9  12 

6o33i 

19 

39669 

64 1 40 

22 

35S6o 

o38io 

4 

96190 

21 
20 

8  5o  40 

3  9  20 

9.60359 

19 

10.39641 

9.64175 

23 

10.35825 

io.o38i5 

4 

9. 96 1 85 

4i 

5o  32 

9  28 

6o388 

20 

39612 

64209 

24 

35791 

o382i 

4 

96179 

IQ 

42 

5o  24 

9  36 

60417 

20 

39583 

64243 

24 

35757 

03826 

4 

96174 

18 

43 

5o  16 

9  A4 

6o446 

21 

39554 

64278 

25 

35722 

o3832 

4 

96168 

17 

44 
45 

5o  8 

9  52 

60474 

21 

39526 

643 1 2 

25 

35688 

o3838 

4 

96162 

16 

75 

8  5o  0 

3  10  0 

9.6o5o3 

22 

10.39497 

9.64346 

26 

10.35654 

10.03843 

4 

9.96157 

4b 

49  52 

10  8 

6o532 

22 

39468 

6438 1 

26 

35619 

o3849 

4 

961 5 1 

i4 

47 

49  A4 

10  16 

6o56i 

23 

39439 

644 1 5 

27 

35585 

03854 

4 

96146 

i3 

48 

49  36 

10  24 

60589 

23 

394 II 

64449 

28 

3555i 

o386o 

4 

96140 

12 

49 
5o 

49  28 

10  32 

60618 

24 

39382 

64483 

28 

35517 

03865 

4 

96135 

II 

10 

8  49  20 

3  10  4o 

9.60646 

24 

10.39354 

9.64517 

29 

10.35483 

10.03871 

5 

9.96129 

5i 

49  12 

10  48 

60675 

25 

'  39325 

64552 

29 

35448 

03877 

5 

96123 

9 

52 

49  4 

10  56 

00704 

25 

39296 

64586 

3o 

35414 

03882 

5 

96118 

8 

53 

48  56 

II  4 

60732 

26 

39268 

64620 

3i 

3538o 

o3888 

5 

961 1 2 

7 

54 
55 

48  48 

11  12 

60761 

26 

39239 

64654 

3i 

35346 

03893 

5 

96107 

6 
5 

8  48  4o 

3  II  20 

9.6C789 

27 

10.39211 

9.64688 

32 

io.353i2 

10.03899 

5 

9.96101 

5b 

48  32 

II  28 

60818 

27 

39182 

64722 

32 

35278 

03905 

5 

96095 

4 

^7 

48  24 

II  36 

60846 

28 

39154 

64756 

33 

35244 

03910 

5 

96090 

3 

58 

48  16 

II  44 

60875 

28 

39125 

64790 

33 

35210 

03916 

5 

96084 

2 

59 

48  8 

II  52 

60903 

29 

39097 

64824 

34 

35176 

o3i^2I 

5 

96079 

I 

bo 
M 

48  0 

12  0 

60931 

29 

39069 

64858 

35 

35i42 

00927 

b 

96073 
Sine. 

0 

M 

Hour  P.M. 

[lour  A.M. 

Cosine. 

Diir. 

Secant. 

Cotangent 

Diff. 

Tangent. 

Cosecant. 

Diir. 

113° 


A 

A 

B 

B 

C 

Seconds  of  time 

V 

2' 

3» 

4' 

5» 

6' 

7' 

Prop,  parts  of  cols. 

I  C 

4 
4 
I 

7 

9 

I 

II 
i3 
9 

i5 

17 
3 

18 
11 
3 

22 
26 
4 

25 

3i 
5 

66^ 


TABLE  XXVIL 

fl';i!;e  209 

S' 

Log.  Sines,  Tar 

gents,  and  Secant?. 

G'. 

24° 

A 

A 

B 

B   . 

C 

C  155° 

o 

Hour  A. ill. 

Hour  P.M. 

Sine.  Dirt". 

Cosecaiil. 

Tangent. 

Difl'. 

Cotang-enl 

Secant. 

uiir. 

Cosine;. 
,9.96073 

65, 

5  48  0 

3  12  0 

9.60931,  0 

10.39069 

9.64858 

0 

io.35i42 

10.03927 

0 

1 

47  52 

12  8 

60960'  0 

39040 

64892 

1 

35io8 

03933 

0   96067 

■^9 

2 

47  44 

12  16 

609881   I 

39012 

64926 

I 

35074 

03938 

0 

96062 

58 

3 

47  3fi 

12  24 

61016.  I 

38984 

64960 

2 

35o4o 

03944 

0 

96056 

57 

4 
5 

47  ?B 

12  32 

61045   2 

38955 

•  64994 

2 

35oo6 

03950 

0 

96050 

55 

8  47  20 

3  12  4o 

9.61073 

2 

10.38927 

9.65028 

3 

10.34972 

10.03955 

0 

9.96045 

6 

47  12 

12  48 

61 101 

3 

38899 

65o62 

3 

34938 

03961 

9(1039 

54 

7 

47  4 

12  56 

61  129 

3 

38871 

65096 

4 

34904 

03966 

96034 

53 

8 

46  56 

i3  4 

6m58 

4 

3S842 

65i3o 

4 

34870 

03972 

9')i-»28 

52 

_9 

10 

46-48 

i3  12 

61 186 

4 

388 1 4 

65i64 

5 

34836 

03978 

9602  2 

5i 

5o 

8  46  4-0 

3  1 3  20 

9.61214 

5 

10.3S786 

9.65197 

6 

io.348o3 

10.03983 
03989 

I 

9 . 960 1 7 

II 

46  32 

i3  28 

61242 

5 

38758 

65231 

6 

34769 

c6<,i;  I 

49 

12 

45  24 

i3  36 

61270 

6 

38730 

65265 

7 

34735 

o3o95 

n6oo5 

48 

i3 

46  16 

i3  44 

61298 

f? 

38702 

65299 

7 

34701 

OiOOO 

96000 

4-' 

1 4 
i5 

46  8 

i3  52 

61326 

6 

38674 

65333 

8 

34667 

o4oo6 

' 

__?^2?4 
9.95988 

4b 
45 

8  46  0 

3  14  0 

9.61354 

7 

10. 38646 

9.65366 

8 

10. 34634 

I0.040I2 

i6 

45  52 

i4  8 

6i382 

7 

386 1 8 

654oo 

9 

34600 

o4oi8 

2 

95982 

44 

17 

45  44 

i4  16 

6i4'  1 

8 

38589 

65434 

9 

34566 

o4o2  3 

2 

9^977 

43 

i8 

45  36 

i4  24 

6i438 

8 

38562 

65467 

10 

34533 

04029 

2 

9597 1 

43 

19 

20 

45  28 

i4  32 

6i.'i66 
9.61494 

9 

38534 

655oi 

1 1 

34499 

o4o35 

2 

95965 

4i 
4o 

8  45  20 

3  14  4" 

9 

io.385o6 

9.65535 

11 

It).  34465 

1 0 . o4o4o 

2 

9.9596c. 

21 

45  12 

i4  48 

6  I  522 

10 

38478 

65568 

12 

34432 

o4o46 

2 

95954 

t2 

22 

45  4 

i4  56 

6i55o 

10 

38450 

656o2 

12 

34398 

o4<'52 

2 

95948 

38 

2j 

44   56 

i5  4 

61578 

II 

38422 

65636 

i3 

34364 

o4o5S 

2 

9594? 

3- 

24 
25 

44  48 

i5  12 

61606 

1 1 

■  38394 

65669 

i3 

3433i 

o4o63 

2 

95937 

30 
35 

8  44  40 

3  i5  20 

9.61654 

12 

10.38366 

9.65703 

14 

10.34297 

10.04069 

2 

9.95931 

2b 

44  32 

i5  28 

6i6()2 

12 

38338 

65736 

i5 

34264 

04075 

2 

95925 

M 

27 

44  24 

i5  36 

61689 

12 

383 1 1 

65770 

i5 

34230 

o4o8o 

3 

95920 

33 

28 

44   16 

i5  44 

61717 

i3 

3S283 

658o3 

16 

34197 

o4o86 

3 

95914 

32 

29 

3o 

44    8 

i5  52 

61745 
9.61773 

i3 

3S255 

65837 

16 

34 1 63 

04092 

0 

959018 
9.95902 

3i 
3o 

8  44  0 

3  16  0 

10.38227 

9.65870 

17 

1 0.34 1 3o 

10.04098 

3 

3i 

43  52 

16  8 

6i8<», 

i4 

38200 

65904 

17 

34096 

o4 1  o3 

3 

95897 

29 

32 

43  44 

16  16 

61828 

ID 

38172 

65937 

18 

34o63 

o4 1 09 

95891 

Ob 

a 

43  36 

16  24 

6 1 856 

i5 

38i44 

65971 

18 

34029 

o4i  i5 

3 

95885 

2-7 

M 
35 

43  28 

16  32 

6:883 

16 

38i  17 

66004 

19 

33996 

o4i2i 

3 

95879 

2b 

8  43  20 

3  16  4" 

9.6191 1 

16 

10.38089 

9.66o38 

20 

10.33962 

10.04127 

3 

9.95873 

36 

43  12 

16  48 

61939 

17 

38o6i 

66071 

20 

33959 

o4i32 

3 

95868 

24 

37 

43  4 

16  56 

61966 

17 

38o34 

66104 

21 

33S96 

04 1 38 

4 

9586? 

23 

38 

42   56 

17  4 

61994 

18 

38oo6 

66 1 38 

21 

33862 

o4i44 

4 

95856 

22 

39 

40 

42  48 

17  12 

6202 1 

18 

37979 

66171 

22 

33829 

o4i5o 

4 

9585o 

21 

20 

8  42  4o 

3  17  20 

9.62049 

18 

10.37951 

9.66204 

22 

10.33796 

io.o4i56 

4 

9-95844 

4i 

42  32 

17  28 

62076 

19 

37924 

66238 

23 

33762 

o4i6i 

4 

95839 

'9 

42 

42  24 

17  36 

62104 

•9 

37896 

66271 

23 

33729 

04167 

4 

95833 

18 

43 

42  16 

17  44 

62i3i 

20 

37869 

663o4 

24 

33696 

04173 

4 

95827 

'7 

44 
45 

.  42  8 

17  52 

3  18  .. 

62159 

20 

37841 

66337 

25 

33663 

o4 1 79 

4 

95821 

lb 

73 

8  42  0 

9.62186 

21 

10..  378 1 4 

9.66371 

25 

10.33629 

io.o4iS5 

4 

g.958i5 

J,0 

4 1  52 

i3  8 

62214 

21 

37786 

66404 

26 

33596 

o4 1 90 

4 

95810 

i4 

47 

4 1  44 

18  16 

62241 

22 

37759 

66437 

26 

33563 

04196 

5 

95804 

i3 

48 

4i  36 

18  24 

62268 

22 

37732 

66470 

27 

3353o 

04202 

5 

95798 

IS 

49 
5o 

4i  28 

18  32 

3  18  4" 

62296 

23 

37704 

665o3 

27 

33497 

04208 

D 

95792 

II 
10  - 

8  4i  20 

9.62323 

23 

10.37677 

9.66537 

28 

10. 33463 

10.04214 

5  9.95786 

5i 

4i  12 

18  48 

62350 

24 

37650 

66570 

28 

*  33430 

04220 

5 

95780 

9 

■32 

4i  4 

18  56 

62377 

24 

37623 

666o3 

29 

333(>7 

04225 

5 

95775 

53 

4o  56 

19  4 

62405 

24 

37595 

66636 

3o 

33364 

0423 1 

5 

95769 

7 

54 
55 

4o  48 
8  4o  4o 

19  12 

62432 
9»63459 

25 
25 

37568 
10.37541 

66669 

3o 

3333i 

04237 

5 

95763 

b 

3  19  20 

9.66702 

3i 

10.33298 

10.04243 

D 

9.95757 

5b 

4t>  32 

19  28 

62486 

26 

37514 

66735 

3 1 

33265 

04249 

5 

95751 

4 

■J7 

4o  24 

19  36 

625i3 

26 

37487 

66768 

32 

33232 

04255 

5 

95745 

3 

58 

4o  16 

19  44 

62541 

27 

37459 

66801 

32 

33199 

04261 

6 

95739 

2 

59 

4o  8 

19  52 

62  568 

27 

37432 

66834 

33 

33 166 

04267 

6 

95733 

1 

bo 

4o  0 

20  0 

62595 

28 

374o5 

66867 

33 

33i33 

04272 

6   95728 

0 
.»1 

Kour  p.;m. 

Hour  A.M. 

Cnsiii(>. 

Diff. 

Secant. 

Cotann^enl 

Difl-. 

Tangent. 

Cosecant 

Ditri  Sine. 

114° 


A 

A 

B 

B 

C 

P 

2' 

li' 

4. 

5- 

G' 

7* 

_  

f  ^ 

3 

7 

10 

i4 

17 

21 

24 

Prop,  parts  of  cols. 

^ 

4 

8 

i3 

17 

21 

25 

29 

(c 

I 

I 

2 

3 

4 

4 

5 

27 


Pa 

ge  210] 

TABLE  XXVIL 

S'. 

Loo 

'.  Sines,  Tangents,  and  Secants. 

G'. 

25 
M 

o 

5 

A 

A 

B 

B 

c 

C  154° 

Hour  A.M. 

Hour  P.M. 

Sine. 

Diff. 

Cosecant. 

Tangent. 

Ditr. 

Cotangent 

Secant.  Diff. 

Cosine. 

31 

6^ 

8  4o  0 

3  20  0 

9.62595 

0 

10.37405 

9.66867 

0 

10. 33 1 33 

10.04272  0 

9.95728 

I 

39  52 

20  8 

62622 

0 

37378 

66900 

I 

33 100 

04278  0 

9J722 

59 

2 

39  4i 

20  16 

62649 

I 

37351 

66933 

I 

33067 

04284  0 

95716 

58 

3 

39  36 

20  24 

62676 

I 

37324 

66966 

2 

33o34 

04290  0 

93710 

57 

4 
5 

39  28 

20  32 

62703 

2 

37297 

66999 

2 

33ooi 

04296 

0 

957C»4 

56 
55 

8  39  20 

3  20  4o 

9.62730 

2 

10.37270 

9.67032 

3 

10.3296S 

io.o43o2 

9.95698 

b 

39  12 

20  48 

62757 

3 

37243 

67065 

3 

32935 

o43o8 

95692 

54 

7 

39  4 

20  56 

62784 

3 

37216 

67098 

4 

32902 

043 1 4 

95686 

53 

8 

•38  56 

21  4 

62811 

4 

37189 

67131 

4 

32869 

04320 

95680 

52 

_9 

lO 

38  48 

21  12 

62838 

4 

37162 

67163 

5 

32837 

04326 

95674 

5i 
5o 

8  38  4" 

3  21  20 

9.62865 

4 

10.37135 

9.67196 

5 

1 0.32804 

10.04332 

9.95668 

II 

38  32 

21  28 

62892 

5 

37108 

67229 

6 

J2771 

04337 

9566? 

49 

12 

38  24 

21  36 

62918 

5 

370S2 

67262 

7 

32738 

04343 

95657 

48 

iJ 

38  16 

21  44 

62945 

6 

37055 

67295 

7 

32705 

04349 

95651 

47 

i4 
i5 

38  8 

21  52 

62972 

b 

37028 

67327 

8 

32673 
10.33640 

04355 

9564'j 

46 
45 

8  38  0 

3  22  0 

9.62999 

7 

10.37001 

9.67360 

8 

10. 04361 

2 

9.95639 

i6 

37  5-^ 

22  8 

63o26 

7 

36974 

67393 

9 

32607 

04367 

2 

95633 

44 

17 

37  44 

22  16 

63o52 

8 

36948 

67426 

9 

32574 

04373 

2 

95627 

43 

i8 

37  36 

22  24 

63o79 

8 

36921 

67458 

10 

32542 

04379 

2 

95621 

42 

19 

20 

37  28 

22  32 

63 1 06 

8 
9 

36894 
10.36867 

67491 

10 

32509 

o4385 

2 

956x5 

4i 
40 

8  37  20 

3  22  4o 

Q.63i33 

9.67624 

II 

10.32476 

10.04391 

2 

9.95609 

21 

37  12 

22  48 

63 1 59 

9 

3684 1 

67556 

11 

32444 

04397 

2 

956o3 

39 

22 

37  4 

22  55 

63 186 

10 

368 1 4 

67589 

12 

3241 1 

o44o3 

2 

95597 

38 

2J 

36  56 

23  4 

632 1 3 

10 

36787 

67622 

12 

32378 

04409 

2 

95591 

37 

24 
25 

36  48 

23  12 

63239 

1 1 

36761 

67654 

i3 

32  346 
io.323i3 

044 1 5 

2 

95585 

36 
35 

S  36  40 

3  23  20 

9.63266 

1 1 

10.36734 

9.67687 

i4 

10.04421 

3 

9.95579 

26 

36  32 

23  28 

63292 

1 1 

36708 

67719 

i4 

32281 

04427 

3 

95573 

34 

i^7 

36  24 

23  36 

63319 

12 

3668 1 

67752 

i5 

32248 

04433 

3 

95567 

33 

28 

36  16 

23  44 

63345 

12 

36655 

67785 

i5 

322l5 

04439 

3 

95561 

32 

29 

3o 

36    8 

23  52 

63372 

i3 

36628 

67817 

16 

32i83 
ro.32i5o 

04445 

3 

95555 

3i 

3^ 

8  36  0 

3  24  0 

9.63398 

i3 

1 0 . 366o2 

9.67850 

16 

io.o445i 

3 

9.95549 

3i 

35  52 

24  8 

63425 

i4 

36575 

67882 

17 

32118 

04457 

3 

95543 

29 

32 

35  44 

24  16 

6345 1 

i4 

36549 

67915 

17 

3  208  5 

04463 

3 

95537 

28 

33 

35  36 

24  24 

63478 

i5 

3652  2 

67947 

18 

32o53 

04469 

3 

95531 

27 

35 

35  28 

24  3a 

635o4 

i5 

36496 

67980 

18 

32020 

04475 

3 

95525 

26 
I5 

8  35  20 

3  24  40 

9.63531 

i5 

10.36469 

9.6S012 

19 

10.31988 

10.04481 

4 

9.95519 

36 

35  12 

24  48 

63557 

16 

36443 

68044 

20 

31956 

04487 

4 

955i3 

24 

37 

35  4 

24  56 

63583 

16 

36417 

68077 

20 

31923 

04493 

4 

95507 

23 

38 

34  56 

25  4 

636io 

17 

36390 

68109 

21 

31S91 

o45oo 

4 

95500 

22 

39 
40 

34  48 

25  12 

63636 
9.63662 

17 
18 

36364 

68142 

21 

3i858 

o45o6 

4 

95494 

21 
20 

8  34  40 

3  25  20 

10.36338 

9.68174 

22 

10.31826 

io.o45i2 

4 

9.95488 

4i 

34  32 

25  28 

63689 

18 

363 1 1 

68206 

22 

31794 

o45i8 

4 

95482 

19 

42 

34  24 

25  36 

637 1 5 

19 

36285 

68239 

23 

31761 

04524 

4 

95476 

18 

43 

34  16 

25  44 

63741 

'9 

36259 

68271 

23 

31729 

o453o 

4 

95470 

17 

44 
45 

34  8 

2  5  52 

63767 

19 

36233 

683o3 

24 

31697 

04536 

4 

95464 

16 
75 

8  34  0 

3  26  0 

9.63794 

20 

10.36206 

9.68336 

24 

io.3i664 

10.04542 

5 

9.95458 

46 

33  52 

26  8 

63820 

20 

36 1 80 

68368 

25 

3i632 

04548 

b 

9^452 

14 

47 

33  44 

26  16 

63846 

21 

36 1 54 

684oo 

25 

3 1 600 

04554 

5 

95446 

i3 

48 

33  36 

26  24 

63872 

21 

36128 

68432 

26 

3 1 568 

04560 

b 

95440 

12 

49 
5o 

33  28 

26  32 

6389S 
9.63924 

22 
22 

36io2 

68465 

27 

3i535 

04566 

b 

95434 

1 1 

10 

8  33  20 

3  26  4o 

10.36076 

9.68497 

27 

io.3i5o3 

10.04573 

5 

9.95427 

5i 

33  12 

26  48 

63950 

23 

36o5o 

68529 

28 

3 1471 

04579 

5 

95421 

9 

5  J 

33  4 

26  56 

63976 

23 

36024 

6S56i 

28 

31439 

o45S5 

b 

95ii5 

8 

53 

32  56 

27  4 

64002 

23 

35998 

68593 

29 

3 1 407 

04591 

b 

95409 

7 

54 
55 

32  48 

27  12 

64028 

24 

35972 

68626 

29 

3i374 

04597 

b 

95403 

0 
"5 

£  32  4o 

3  27  20 

9.64054 

24 

10.35946 

9.68658 

3o 

io.3i342 

io.o46o3 

6 

9.95397 

56 

32  32 

27  28 

64080 

25 

35920 

68690 

3o 

3i3io 

04609 

6 

95391 

4 

57 

32  24 

27  36 

64 1 06 

25 

35894 

68722 

3i 

31278 

04616 

6 

95384 

3 

58 

32  16 

27  44 

64 1 32 

26 

35868 

68754 

3i 

31246 

04622 

6 

95378 

2 

59 

32  8 

27  52 

64 1 58 

26 

35842 

68786 

32 

3i2i4 

04628 

6 

95372 

I 

60 

M 

32   0 

28  0 

64»84 

26 

358i6 

68818 

33 

31182 

o4634 

6 

95366 

0 
M 

Hour  P.M. 

Hour  A.M. 

Cosine. 

Din: 

Secant. 

• 

Cotangent 

Diir. 

Tangent. 

Cosecant. 

Diir. 

Sine. 

U5" 


C      64' 


Seconds  of  time 

V 

2= 

3^ 

4. 

5» 

20 
24 
5 

7, 

"23 

28 
5 

Prop,  parts  of  cols.  <  B 
I  C 

3 
4 

I 

7 
8 
2 

10 
12 
3 

i3 
16 
3 

17 
20 
4 

TABLE  XXVn. 

[Page  211 

S". 

Log 

.  Sines,  Tanrrents,  and  Secants. 

G'. 

26^ 

> 

A 

A 

B 

B 

C 

C    153° 

M 

o 

Hour  A.M. 

Hour  P.M. 

Sine. 

Diff.  Cosecant.] 

Tangent. 

Diff. 

Cotangent 

Secant. 

Diff. 

Cosine. 

M 

6?, 

8  32  0 

3  28  0 

9.64184 

0 

io.358i6 

9.6S818 

0 

io.3ii82 

10.04634 

0 

9.95366 

I 

3i  52 

28  8 

64210 

0 

35790 

6885o 

I 

3ii5o 

o464o 

0 

95360 

59 

2 

3i  44 

28  16 

64236 

I 

35764 

68882 

1 

3iii8 

04646 

0 

95354 

58 

3 

3i  36 

28  24 

6426? 

I 

35738 

68914 

2 

3 1 086 

o4652 

0 

95348 

57 

4 
5 

3i  28 

28  32 

64^88 

2 

35712 

68946 

2 

3io54 

04659 

0 

953.^1 

56 
55 

8  3r  20 

3  28  4o 

9.643i3 

2 

10.35687 

9.68978 

3 

10.3l022 

I0.04665 

I 

9.95335 

6 

3i  12 

28  48 

64339 

3 

3  566 1 

69010 

3 

30990 

04671 

I 

95329 

54 

7 

3i  4 

28  56 

64365 

3 

35635 

69042 

4 

30958 

04677 

I 

95323 

53 

8 

3o  56 

29  4 

64391 

3 

35609 

69074 

4 

30926 

04683  I 

95317 

52 

_9 

10 

3o  48 

29  12 

64417 

4 

35583 

69106 

5 

30894 

04690I  I 

95310 

5i 

5o 

8  3o  4o 

3  29  20 

9.64442 

4 

10.35558 

9.69138 

5 

10.30862 

10.04696 

9.95304 

II 

3o  32 

29  28 

64468 

5 

35532 

69170 

b 

3o83o 

04702 

95298 

49 

12 

3o  24 

29  36 

64494 

5 

355o6 

69202 

6 

30798 

64708 

95292 

48 

j3 

3o  16 

29  44 

64519 

5 

35481 

69234 

7 

30766 

04714 

95286 

47 

i4 
i5 

3o  8 

29  52 

64545 

6 

35455 

69266 

7 

30734 

04721 

95279 

46 
45 

8  3o  0 

3  3o  0 

9.64571 

6 

10.35429 

9.69298 

8 

10.30702 

"10.04727 

2 

9.95273 

i6 

29  52 

3o  8 

64596 

7 

35404 

69329 

8 

30671 

04733 

2 

95267 

44 

17 

29  44 

3o  16 

64622 

7 

35378 

69361 

9 

3o639 

04739 

2 

95261 

43 

lb 

29  36 

3o  24 

64647 

« 

35353 

69393 

9 

30607 

04746 

2 

95254 

42 

!9 

20 

29  28 

3o  32 

64673 

8 

8 

35327 

69425 

10 

3o575 

04752 

2 

95248 

4i 
4o 

8  29  20 

3  3o  4o 

9.64698 

io.353o2 

9.69457 

1 1 

io.3o543 

10.04758 

2 

9.95242 

21 

29  12 

3o  48 

64724 

9 

35276 

69488 

1 1 

3o5i2 

04764 

2 

95236 

39 

22 

29  4 

3o  56 

64749 

9 

3525i 

69520 

12 

3o48o 

04771 

2 

95229 

38 

23 

28  56 

31  4 

64775 

10 

3^225 

69552 

12 

3o44S 

04777 

2 

95223 

37. 

?.4 

25 

28  48 

3i  12 

64800 

10 

35200 

69584 

1 3 

3o4i6 

04783 

3 

95217 

36 
35 

8  28  4o 

3  3i  20 

9.64S26 

1 1 

10.35174 

9 . 696 1 5 

i3 

io.3o385 

10.04789 

3 

9.95211 

2b 

28  32 

31  28 

6485 1 

II 

35i49 

69647 

i4 

3o353 

04796 

3 

95204 

M 

27 

28  24 

3i  36 

64877 

1 1 

35i23 

69679 

i4 

3o32i 

04802 

3 

95198 

33 

28 

28  16 

3i  44 

64902 

12 

35098 

69710 

1 5 

30290 

04808 

3 

95192 

32 

29 

So 

28  8 

3i  52 

64927 

12 
i3 

35073 

69742 

i5 

3o258 

o48i5 

3 

95i85 

3i 
3o 

8  28  0 

3  32  0 

9.64953 

io.35o47 

9.69774 

16 

10.30226 

10.04S21 

3 

9.95179 

3i 

27  52 

32  8 

64978 

i3 

35o22 

69805 

lb 

3o  1 95 

04827 

3 

95173 

29 

32 

27  44 

32  16 

65oo3 

1 4 

34997 

69837 

17 

3oi63 

04833 

3 

96167 

28 

3i 

27  36 

32  24 

65029 

i4 

34971 

69868 

•7 

3oi32 

o484o 

3 

96160 

27 

34 
35 

27  28 

32  32 

65o54 

i4 

34946 

69900 

18 

3oioo 

04846 

4 

961 54 

26 

25 

8  27  20 

3  32  40 

9.65079 

i5 

10.34921 

9.69932 

18 

io.3oo68 

10.04852 

4 

9.96148 

3t) 

27  12 

32  48 

65io4 

i5 

34896 

69963 

'9 

3oo37 

04859 

4 

96141 

24 

^7 

27  4 

32  56 

65i3o 

lb 

34870 

69995 

20 

3ooo5 

04865 

4 

q6i35 

23 

38 

26  56 

33  4 

65i55 

16 

34845 

70026 

20 

29974 

04871 

4 

96129 

22 

3<; 
4o 

26  48 

33  12 

65 1  So 

16 

34820 
10.34795 

70o58 

21 

29942 

04878 

4 

96122 

21 

20 

8  26  4o 

3  33  20 

9.652o5 

17 

9 . 700S9 

21 

10.2991 1 

10.04884 

4 

9.96116 

4i 

26  32 

33  28 

65230 

17 

34770 

7012 1 

22 

29879 

04890 

4 

96110 

19 

42 

26  24 

33  36 

65255 

18 

34745 

701 52 

22 

29848 

04897 

4 

96103 

18 

ii 

26  .16 

33  44 

65281 

18 

347 '9 

70184 

23 

2^816 

04903 

5 

96097 

J7 

44 
45 

26  8 

33  52 

653o6 

•9 

34694 

70215 

23 

29785 

04910 

5 

96090 

16 
i5 

8  26  0 

3  34  0 

9.653ii 

19 

10.34669 

9.70247 

24 

10.29753 

10.04916 

5 

9.96084 

40 

25  52 

34  8 

65356 

19 

34644 

70278 

24 

29722 

04922 

5 

96078 

1 4 

47 

25  44 

34  16 

6538: 

20 

34619 

7o3o9 

25 

29691 

04929 

5 

9607 1 

1 3 

48 

25  36 

34  24 

654o6 

20 

34594 

70341 

25 

29659 

04935 

5 

9K)frj 

12 

49 
So 

25  28 

34  32 

6543 1 

21 

34569 

70372 

26 

29628 

04941 

5 

96069 

1 1 
10 

8  25  20 

3  34   40 

9.65456 

21 

10.34544 

9.70404 

26 

10.29596 

1C.04948 

5 

9.96062 

?' 

25  12 

34  48 

6548 1 

22 

34519 

70435 

27 

29565 

04954 

5 

96046 

9 

.S2 

25  4 

34  56 

655o6 

22 

34494 

70466 

27 

29534 

0496 1 

5 

96039 

8 

53 

24  56 

35  4 

6553i 

22 

34469 

70498 

28 

2^502 

04967 

6 

96033 

7 

54 
5'5 

24  48 

35  17 

65556 

23 

34444 

70529 

28 

29471 

04973 

6 

96027 

6 

8  24  4o 

3  35  20 

9.655SO 

23 

10.34420 

9.70560 

29 

10.29440 

10.04980 

6 

9.96020 

5 

50 

24  32 

35  28 

656o5 

24 

34395 

70592 

3o 

29408 

04986 

6 

96014 

4 

'•.7 

24  24 

35  36 

6563o 

24 

34370 

70623 

3o 

29377 

04993 

6 

96007 

3 

58 

24  16 

35  44 

65655 

25 

34345 

70654 

3i 

29346 

04999 

6 

96001 

2 

59 

24  8 

35  52 

6568o 

25 

34320 

70685 

3c 

29315 

o5oo5 

6 

94996 

I 

bo 
VI 

24  0 

36  0 

657o5 

23 

34295 

70717 

32 

29283 

o5oi2 

6 

9498S 

0 
M 

Hour  p. M 

Hour  A.M. 

Cosine. 

Diff. 

Secant. 

Cotangent 

Diff 

Tangent. 

Cosecant. 

Diff. 

Sine. 

116° 


»/ 


Seconds  of  time , 

1' 

2^ 

3' 

4- 

5' 

6» 

7' 

(^ 

3 

6 

10 

i3 

16 

19 

t2 

Prop,  parts  of  cols 

U 

4 

8 

:2 

16 

20 

24 

28 

(c 

I 

1 

2 

3 

4 

5 

Jij 

■'• 

.i:-\-] 

TABLE  XXVII. 

. 

S' 

hog.  S 

ines,  Tangents,  and  Secants. 

G'. 

•27 

i\ 

o 

A 

A      B 

B 

G 

C  152° 

HourA.M 

Hour  P.M. 

Sine. 

Dirt' 
0 

Cosecant. 

Tang-en  t. 

Diff. 

Cotangent 

Secant. 

Dift-. 

Cosine. 

M 

8  24  0 

3  36  0 

9.65705 

10.34295 

9.70717 

0 

10.29283 

io.o5oi2 

0 

9.94988 

I 

23  52 

36  8 

65729 

0 

34271 

70748 

I 

29252 

o5oi8 

0 

94982 

'^9 

2 

23  4^ 

36  16 

65754 

I 

34246 

70779 

1 

29221 

o5o25 

0 

94975 

58 

3 

23  3b 

36  24 

65779 

1 

34221 

70810 

2 

29190 

o5o3i 

0 

94969 

57 

4 
5 

23  28 

36  32 

658o4 

2 

34196 

70841 

2 

29159 
10.29127 

o5o38 

0 

94962 

56 
55 

8  23  20 

3  36  4<) 

9.65828 

2 

10.34172 

9.70873 

3 

io.o5o44 

9.94956 

b 

23  12 

36  48 

65853 

2 

34i47 

70904 

3 

29096 

o5o5i 

94949 

54 

7 

23  4 

36  56 

65878 

3 

34122 

70935 

4 

29065 

o5o57 

94943 

53 

« 

22  56 

37  4 

65902 

3 

34098 

70966 

4 

29034 

o5o64 

94936 

52 

J? 

lO 

22  48 

37  12 

65927 

4 

34073 

70997 

5 

29003 

o5o7o 

94'93o 

5i 
5o 

8  22  40 

3  37  20 

9.65952 

4 

10.34048 

9.71028 

5 

10.28972 

io.o5o77 

9.94923 

II 

22  32 

37  28. 

65976 

4 

34024 

71059 

6 

28941 

o5o83 

94917 

4q 

12 

22  24 

37  36 

66001 

5 

33999 

7 1 090 

b 

28910 

o5o89 

94911 

48 

IJ 

22  16 

37  44 

66025 

5 

33975 

71 121 

7 

28879 

05096 

94904 

47 

i4 

25 

22  8 

37  52 

63o5o 

b 

33950 

71 153 

7 

28847 

05l02 

2 

94898 

46 
45 

8  22  0 

3  38  0 

9.6()075 

6 

10.33025 

9.71184 

8 

10.28816 

io.o5i09 

2 

9.94891 

i6 

21  52 

38  8 

66099 

b 

3J701 

71215 

8 

28785 

o5i  i5 

2 

94885 

44 

17 

21  44 

33  16 

66124 

7 

33S76" 

71246 

9 

28754 

05l22 

2 

94878 

43 

i8 

21  36 

33  24 

66 1 48 

7 

33852 

71277 

9 

28723 

o5i29 

2 

94871 

42 

19 
so 

21  28 

38  32 

66173 

8 

33827 

7i3o8 

10 

28692 

o5i35 

2 

94865 

4i 
4o 

8  21  20 

3  38  4o 

9.66197 

8 

io.338o3 

9.71339 

10 

10.28661 

io.o5i42 

2 

9-94858 

21 

21  12 

38  48 

66221 

8 

33779 

71370 

1 1 

28630 

o5i48 

2 

94852 

3o 

22 

21  4 

38  56 

66246 

9 

33754 

7i4oi 

11 

28599 

o5i55 

2 

94845 

38 

23 

20  56 

39  4 

66270 

9 

33730 

7>43fc 

12 

28569 

o5i6i 

3 

94839 

37 

24 
25 

20  48 

39  12 

66295 

10 

33705 

71462 

12 

28538 

o5i68 

3 

94832 

36 
35 

8  20  4o 

3  39  20 

9.66319 

10 

10.33681 

9.71493 

i3 

10.28507 

io.o5i74 

3 

9.94826 

2b 

20  32 

39  28 

6634^ 

1 1 

33657 

71524 

i3 

28476 

o5i8i 

3 

94819 

M 

27 

20  24 

39  36 

66368 

II 

33632 

71555 

14 

28445 

05187 

3 

9481 3 

33 

28 

20  16 

39  44 

6639'^ 

II 

336o8 

71586 

i4 

28414 

05194 

3 

94S06 

32 

29 

3o 

20  8 

39  52 

66416 

12 

33584 

71617 

i5 

28383 

05201 

3 

94799 

3 1 

37; 

8  20  0 

3  4o  0 

9.66441 

12 

10.33559 

9.71648 

i5 

10.28352 

10.05207 

3 

9.94793 

3i 

19  52 

4o  8 

66465 

i3 

33535 

71679 

16 

28321 

o52i4 

3 

94786 

2Q 

32 

19  44 

4o  16 

66489 

i3 

335ii 

71709 

16 

28291 

05220 

4 

94780 

28 

33 

19  36 

4o  24 

665 1 3 

i3 

33487 

71740 

17 

28260 

05227 

4 

94773 

27 

35 

19  28 

4o  32 

66537 

i4 

33463 

7'77' 

17 

28229 

05233 

4 

947G7 

26 
25 

8  19  20 

3  4o  40 

9.66562 

14 

10. 33438 

9.71802 

iS 

10.28198 

io.o524o 

4 

9.94760 

36 

19  12 

4o  48 

66586 

i5 

334 1 4 

71833 

19 

28167 

05247 

4 

94753 

24 

^7 

19  4 

4o  56 

66610 

i5 

33390 

71863 

19 

28i3t 

o5253 

4 

94747 

23 

38 

18  56 

4i  4 

66634 

i5 

33366 

71894 

20 

28106 

05260 

4 

9474<-i 

22 

39 
4o 

18  48 

4r-  12 

66658 

16 

33342 

71925 

20 

28075 

o5266 

4 

94734 

21 
20 

8  18  40 

3  4i  20 

9.66682 

16 

io.333i8 

9.71955 

21 

10.28045 

10.05273 

4 

9.94727 

4i 

18  32 

4i  28 

66706 

17 

33294 

71986 

21 

28014 

o523o 

4 

94720 

•'9 

42 

18  24 

4i  36 

66731 

17 

33269 

72017 

22 

27983 

05286 

5 

94714 

18 

4S 

18  16 

4i  44 

66755 

17 

33245 

72048 

22 

27952 

05293 

5 

94707 

17 

44 
45 

18  8 

4i  52 

66779 
9.66803 

18 
18 

33221 

10.33197 

72078 

23 

27922 

o53oo 

5 

94700 

16 

75 

8  18  0 

3  42  0 

9.72109 

23 

10.27891 

io.o53o6 

5 

9.94694 

4b 

17  52 

42  8 

66827 

19 

33173 

72i4(j 

24 

27860 

o53i3 

5 

94687 

i4 

47 

17  44 

42  16 

66851 

19 

33 14<) 

72170 

24 

27830 

o532o 

5 

94680 

1 3 

48 

17  36 

42  24 

66875 

19 

33125 

72201 

25 

27799 

05326 

5 

94674' 

12 

49 
5o 

17  28 

42  32 

66899 

20 

33ioi 

72231 

25 

27769 

05333 

5 

94667 

1 1 

10 

8  17  20 

3  42  4o 

9.66922 

20 

10,330-8 

9.72262 

26 

10.27738 

io.o534o 

5 

9.94660 

5i 

17  12 

42  48 

66946 

21 

33o54 

72293 

26 

27707 

05346 

6 

9^654 

9 

52 

17  4 

42  56 

66970 

21 

33o3o 

72323 

27 

27677 

05353 

6 

9I647 

8 

t>i 

16  56 

43  4 

66994 

21 

33oo6 

72354 

27 

27646 

o536o 

6 

94640 

7 

54 
55 

16  48 

43  12 

67018 

22 

32983 
:o. 32958 

72384 

28 

27616 

o5366 

6 

94634 

6 
1> 

8  16  4o 

3  4^   20 

9.67042 

22 

9.72415 

28 

10.27585 

10.05373 

6 

9.94627 

5b 

16  32 

43  28 

67<)66 

23 

3?93i 

72445 

20 

27555 

o538o 

6 

94620 

4 

57 

16  24 

43  36 

67090 

23 

32910 

72476 

29 

27524 

05386 

6 

94614 

3 

58 

16  16 

43  44 

67113 

23 

32887 

725o6 

3o 

27494 

05393 

6 

91607 

2 

59 

16  8 

43  52 

67.37 

24 

32  863 

72537 

3o 

27463 

o54oo 

b 

94600 

1 

bo 
M 

16  0 

44    0 

67161 

24 

32839 

72567 

3i 

27433 

o54o7 

7 

94593 

0 

Hour  P.M. 

HourA.M. 

Cosine. 

Diif. 

Secant. 

Cotangent 

uiir. 

Tangent. 

Cosecant. 

Diff. 

Sine. 

117° 


A 


62' 


1' 

2' 

3^ 

9 

4' 

12 

5» 

6' 

iS 

7» 
21 

(^ 

3 

G 

i5 

I*'ro['.  parta  of  cols. 

N 

4 

8 

12 

i5 

19 

23 

27 

(c 

1 

2 

2 

3 

4 

5 

6 

TABLE  XXVIL 

[Page  21:) 

i" 

Log 

.  Sines,  Tan 

gents,  and  Secants. 

G'. 

28 
M 
c 

A 

A 

B 

B 

C 

C  151° 

Ilo'.ir  A.Bi.jHour  p.m. 

Sino. 

Diir. 

Cosscaut. 
3D.3a839 

Tangent. 

Diir.jCotaiigcnl 

Secant. 

DiiT.  Cosine. 

M 

6^ 

£  16  0 

3  44  0 

9.67161 

0 

9.72567 

0 

10.27433 

io.o54o7 

0  '9.94593 

1 

i5  52 

44    8 

67185 

0 

32815 

72598 

I 

27402 

o54i3 

0  1  94687 

r)9 

2 

1 5  44 

44   16 

67208 

I 

32792 

72628 

I 

27372 

05420 

0  ]  94680 

58 

3 

1 5  36 

44  24 

67232 

J 

32768 

72659 

2 

27341 

05427 

0 

94673 

^7 

4 
5 

1 5  28 
£  1 5  20 

44  32 
3  44  4" 

67256 

2 

32744 

72689 

2 

27311 

05433 

0 

94567 

Ob 
55 

9.67280 

2 

10.32720 

9.72720 

3 

10.27280 

io.o544o 

9.94560 

6 

[5  12 

44  48 

67303 

2 

32697 

72750 

3 

27250 

05447 

94553 

64 

7 

i5  4 

44  56 

67327 

3 

32673 

72780 

4 

27220 

05454 

94546 

63 

8 

i4  56 

45  4 

67359 

3 

3265o 

72811 

4 

27189 

o546o 

94540 

62 

ID 

i4  48 

45  12 

67374 

3 

32626 

72841 

5 

27159 

05467 

94533 

5i 
57. 

8  i4  4o 

3  45  20 

9.6739S 

4 

10.32602 

9.72072 

5 

10.27128 

10.05474 

9.94526 

1  I 

i4  32 

45  28 

6742  1 

4 

32579 

72902 

6 

27098 

o548i 

94519 

49 

12 

1 4  24 

45  36 

67445 

5 

32555 

72932 

6 

27068 

05487 

945 1 3 

48 

i3 

i4  16 

45  44 

67463 

5 

32532 

72963 

7 

27037 

05494 

94606 

47 

r4 
i5 

i4  8 

45  52 

67492 

5 

32  5o8 

72993 

7 

27007 

o55oi 

2 

94499 

4b 
46 

8  i4  0 

3  46  0 

9.67515 

6 

10.32485 

9.73023 

8 

10.26977 

io.o55o8 

2 

9.94492 

i6 

i3  52 

46  8 

67539 

6 

32461 

73o54 

8 

26946 

o55i5 

2 

94486 

44 

17 

i3  44 

46  16 

67562 

7 

32438 

73084 

9 

26916 

o552i 

2 

94479 

4^ 

18 

i3  36 

46  24 

67586 

7 

324i4 

73ii4 

9 

26886 

05528 

2 

94472 

42 

!9 

20 

i3  28 

46  32 

67609 
9.67633 

7 

32391 

73i44 

10 

26856 

05535 

2 

94465 

4i 

40 

8  i3  20 

3  46  40 

8 

10.32367 

9.73175 

10 

10.26825 

10.05542 

2 

9-94458 

21 

i3  12 

46  48 

67656 

8 

32344 

732o5 

II 

26795 

05549 

2 

94451 

39 

22 

i3  4 

46  56 

67680 

9 

32320 

73235 

II 

26765 

05555 

3 

94445 

38 

2j 

12  56 

47  4 

67703 

9 

32297 

73265 

12 

26735 

o5562 

3 

94438 

37 

24 
25 

12  48 

47  12 

67726 

9 

32274 

73295 

12 

26705 

05569 

3 

9443 1 

3b 
35 

8  12  40 

3  47  20 

9.67750 

10 

I0.3225o 

9.73326 

i3 

10.26674 

10.05576 

3 

9.94424 

2b 

12  32 

47  28 

G7773 

10 

32227 

73356 

i3 

26644 

o5583 

3 

94417 

M 

27 

12  24 

47  36 

67796 

10 

32204 

73386 

i4 

26614 

05590 

3 

94410 

.ii 

28 

12  16 

47  44 

6-7820 

II 

32180 

73416 

i4 

26584 

05596 

3 

94404 

32 

29 

3o 

12  8 

47  52 

67843 
9.67866 

1 1 

32157 

73446 

i5 

26554 

o56o3 

3 

94397 

3i 
3^ 

8  12  0 

3  48  0 

12 

io.32i34 

9-73476 

i5 

10.26524 

1 0 .  ()56 1 0 

3 

9.94390 

3i 

II  62 

48  8 

67890 

12 

32110 

73507 

16 

26493 

o56i7 

4 

94383 

29 

i> 

11  44 

48  16 

67913 

12 

32087 

73537 

16 

26463 

05624 

4 

94376 

28 

34 

II  36 

48  24 

67936 

i3 

32064 

73567 

17 

26433 

o563i 

4 

94369 

27 

34 
35 

II  28 

48  32 

67959 

i3 

32o4i 

73597 

17 

26403 

05638 

4 

94362 

2b 

8  1 1  20 

3  48  4o 

9.67982 

14 

10.32018 

9.73627 

18 

10.26373 

I0.05645 

4 

9.94355 

3b 

II  12 

48  48 

68006 

i4 

31994 

73657 

18 

26343 

o565i 

4 

94349 

24 

47 

11  4 

48  56 

68029 

i4 

31971 

73687 

'9 

263 1 3 

05658 

4 

94342 

23 

38 

10  56 

49  4 

68o52 

i5 

31948 

73717 

19 

26283 

05665 

4 

94335 

22 

39 

4<. 

10  48 

49  12 

68075 

i5 

31925 

73747 

20 

26253 
10.26223 

05672 

4 

94328 

21 

20 

8  10  4o 

3  49  20 

9.68098 

16 

10.31902 

9-73777 

20 

10.05679 

5 

9.94321 

4i 

10  32 

49  28 

68121 

16 

31879 

73807 

21 

26193 

o5686 

5 

943 1 4 

IQ 

42 

10  24 

49  36 

68144 

16 

3i856 

73837 

21 

26163 

05693 

6 

94307 

18 

43 

10  16 

49  44 

68167 

17 

3i833 

73S67 

22 

26133 

05700 

5 

94300 

'7 

44 

45 

10  8 

4g  52 

68190 

17 

3i8io 

73897 

22 

26103 

05707 

5 

94293 

16 

16 

8100 

3  5o  0 

9.68213 

17 

10.31787 

9.73927 

23 

10.26073 

10.05714 

5 

9.94286 

4b 

9  52 

5o  8 

6S237 

18 

31763 

73957 

23 

26043 

05721 

5 

94279 

i4 

47 

9  44 

5o  16 

68260 

18 

3 1740 

73987 

24 

26013 

06727 

5 

94273 

i3 

48 

9  36 

5o  24 

68283 

19 

31717 

74017 

24 

25983 

05734 

5 

94266 

12 

49 
5o 

9  ?8 

5o  32 

683o5 

'9 

31695 

74047 

25 

25953 

0574 1 

6 

94269 

1 1 
III 

S  9  20 

3  5o  40 

9.68328 

iQ 

10.31672 

9.74077 

25 

10.25923 

io.o5748 

6 

9.94262 

61 

9  12 

5o  48 

68351 

20 

.  3i649 

74107 

26 

25893 

05755 

6 

94246 

Q 

52 

9  4 

5o  56 

68374 

20 

31626 

74 1 37 

26 

25863 

06762 

6 

94238 

8 

53 

8  56 

5i  4 

68397 

21 

3i6o3 

74166 

27 

25834 

06769 

6 

9423 1 

7 

54 
55 

8  48 

5i  12 

68420 

21 

3i58o 

74196 

27 

258o4 

06776 

6 

94^24 

6 

~5 

8  8  40 

3  5i  20 

9.68443 

21 

ic.3i557 

9.74226 

28 

10.25774 

10.05783 

6 

9.94217 

56 

8  32 

5i  28 

68466 

22 

3i534 

74256 

28 

25744 

06790 

6 

94210 

4 

5t 
5S 

8  24 

5 1  36 

68489 

22 

3i5ii 

74286 

29 

25714 

06797 

7 

94203 

3 

8  16 

5i  44 

685 1 2 

22 

3 1 488 

743i6 

29 

2  5684 

o58o4 

7 

94196 

2 

59 

8  8 

5i  52 

68534 

23 

3 1 466 

74345 

3o 

25655 

o58ii 

7  ■ 

94189 

I 

8  0 

52   0 

68557 

23 

3i443 

74375 

3o 

25625 

o58i8 

94182 

0 

Hour  p.?,i.  Hour  a.m. 

Cosine. 

DifiT. 

Secant. 

Coiangcnt  DilT.| 

Tangent. 

Cosecant. 

Diir. 

Sine. 

M 

118° 


A 

A 

B 

B 

C 

Seconds  of  time  . . . 

..^ 

1' 

2» 

3" 

4- 

5« 

6' 

7* 

Prop,  parts  of  cols. 

I  C 

3 

4 

I 

6 
8 
2 

9 
II 
3 

12 

!5 

3 

i5 

19 
4 

17 

23 

5 

20 
26 
6 

GV 


P; 

Se  2141 

TABLE  XXVIL 

S' 

Log.  S 

nes,  Tancrents,  and  Secants. 

G'. 

29 

0 

A 

A 

B 

B 

C 

C  150° 

o 

Hour  A.M. 

Hour  P.M. 

Sine. 

DilT. 

Cosecant. 

Tangent. 
9-74375 

Ditr. 

0 

Cotangent 
lo. 25625 

Secant. 

Diff.  Cosine. 

M 

880 

3  52  0 

9.68557 

0 

I 0.3 1443 

io.o58i8 

0  9.94182 

60 

I 

7  52 

52  8 

6858o 

0 

3 1420 

744o5 

0 

25595 

o5825 

0   o4i75 

5o 

2 

7  44 

52  16 

6S6o3 

I 

3.397 

74435 

I 

25565 

05832 

0 

94168 

58 

3 

7  36 

52  24 

6S625 

1 

3 1 375 

74465 

I 

25535 

o5839 

0 

94161 

67 

4 
5 

7  28 

52  32 

68648 

I 

3i352 

74494 

2 

255o6 

o5846 

0 

94164 

56 

8  7  20 

3  52  40 

9.68671 

2 

io.3i329 

9.74524 

2 

10.25476 

io.o5P.:3 

: 

9.94147 

55 

b 

7  12 

52  48 

68694 

2 

3 1 3o6 

74554 

3 

25446 

o586o 

04 1 40 

54 

7 

7  4 

52  56 

68716 

0 

3 1 284 

74583 

3 

25417 

05867 

I  1  94i33 

63 

8 

6  56 

53  4 

68739 

3 

31261 

7461 3 

4 

25387 

05874 

I   q4i26 

52 

9 

10 

6  48 

53  12 

68762 

3 

3.238 

7 -{643 

4 

25357 
10.25327 

o588i 

94 119 

5. 

8  6  4o 

3  53  20 

9.68784 

4 

1 0 . 3 1 2 1  b 

9.74673 

5 

10.05888 

9.94112 

5o 

II 

6  32 

53  28 

6S807 

4 

31193 

74702 

5 

25298 

05895 

94io5 

4c) 

12 

6  24 

53  36 

68829 

4 

3117. 

74732 

6 

25268 

o5oo2 

9409S 

48 

i3 

6  16 

53  44 

68852 

5 

3ii48 

74762 

6 

25238 

06910 

2 

94090 

4- 

i4 
i5 

6  8 

53  52 

68875 

5 

3ii25 

7479' 

7 

25209 

05917 

2 

94o83 

46 

860 

3  54  0 

9.68897 

6 

io.3iio3 

9.74821 

7 

10.25179 

10.06924 

2 

9 . 94076 

46 

i6 

5  52 

54  8 

68920 

b 

3 1080 

7485 1 

8 

25i49 

06981 

2 

94069 

44 

17 

5  44 

54  16 

68942 

b 

3io58 

74880 

8 

25  120 

05938 

2- 

04062 

43 

i8 

5  36 

54  24 

68965 

7 

3io35 

74910 

9 

25090 

05945 

2   94o55 

42 

12 

20 

5  28 

54  32 

68987 

7 

3ioi3 

74939 

9 

25o6i 

06962 

2 

94048 

4i 

8  5  20 

3  54  4o 

9 . 690 1 0 

7 

1 0 . 30990 

9.74969 

10 

io.25o3i 

10.06959 

2 

9.9404. 

4o 

21 

5  12 

54  48 

69032 

8 

30968 

74998 

10 

250O2 

06966 

3 

94o34 

39 

22 

5  4 

54  56 

69055 

8 

30945 

75028 

1 1 

24972 

06978 

3 

94027 

38 

23 

4  56 

55  4 

69077 

9 

30923 

75o58 

1 1 

24942 

06980 

3 

94020 

37 

24 
25 

4  48 

55  12 

69100 

9 

30900 

75087 

12 

24913 

06988 

3 

94012 

36 

8  4  4o 

3  55  20 

9.69122 

9 

10.30878 

9.75117 

12 

10. 24883 

10.06996 

3 

9.94006 

35 

26 

4  32 

55  28 

69144 

lu 

3o856 

75i46 

i3 

24854 

06002 

3 

93998 

34 

27 

4  24 

55  36 

69167 

JO 

3o833 

75.76 

i3 

24824 

06009 

3 

93991 

33 

28 

4  16 

55  44 

69189 

10 

3o8ii 

752o5 

i4 

24795 

06016 

3 

93984 

32 

29 

3o 

4  8 

55  52 

69212 
9.69234 

II 
II 

30788 

75235 

14 

24765 

06028 

3 

93977 

3i 

8  4  0 

3  56  0 

10.30766 

9.75264 

i5 

10.24736 

io.o6o3o 

4 

9.98970 

3o 

3i 

3  52 

56  8 

69256 

12 

30744 

75294 

i5 

24706 

06087 

4 

93963 

29 

32 

3  44 

56  16 

69279 

12 

30721 

75323 

16 

24677 

06045 

4 

98955 

28 

33 

3  36 

56  24 

69301 

12 

30699 

75353 

16 

24647 

06062 

4 

93948 

27 

34 
35 

3  28 

56  32 

69323 

i3 

80677 

75382 

17 

24618 

06069 

4 

93941 

26 

8  3  20 

3  56  40 

9.69345 

i3 

io.3o655 

9.75411 

17 

zo. 24589 

10.06066 

4 

9.93934 

25 

36 

3  12 

56  48 

69368 

i3 

3u632 

7544  i 

18 

24559 

06078 

4 

93927 

24 

37 

3  4 

56  56 

69390 

i4 

3o6 1 0 

75470 

18 

2453u 

06080 

4 

98920 

23 

38 

2  56 

57  4 

69412 

i4 

3o58S 

75500 

19 

245ou 

06088 

5 

98912 

22 

39 
4o 

2  48 

57  12 

69434 
9.69456 

i5 
i5 

3o566 
io.3o544 

75529 

19 

24471 

06095 

5 

98906 

2. 

8  2  4o 

3  57  an 

9.75558 

20 

10.24442 

10.06102 

5 

9.98898 

20 

4i 

2  32 

57  28 

69479 

i5 

3o52i 

75588 

20 

24412 

06109 

5 

93891 

19 

42 

2  24 

57  36 

69501 

16 

3o499 

75617 

21 

24383 

061 16 

5 

98884 

18 

43 

2  16 

57  44 

69523 

lb 

3o477 

75647 

21 

24353 

06124 

5 

93876 

'7 

44 

45 

2   8 

57  52 

69545 

16 

3o455 

75676 

22 

24324 

oGi3i 

5 

98869 

lb 

820 

3  58  0 

9.69567 

17 

jo.3o433 

9.75705 

22 

10.24295 

io.o6i38 

5 

9.93862 

i5 

46 

I   52 

58  8 

69589 

17 

3o4ii 

75735 

23 

24265 

06145 

b 

93865 

i4 

47 

f  44 

58  16 

6961 1 

17 

3o389 

75764 

23 

24286 

061 53 

b 

93847 

18 

48 

I  36 

58  24 

69633 

18 

3o367 

75793 

24 

24207 

06160 

b 

98840 

12 

49 

5o 

I  28 

58  32 

69655 

18 

3o345 

75822 

24 

24178 

06167 

b 

93833 

IT 

8  1  20 

3  58  4o 

9.69677 

19 

■io.3o323 

9-75852 

25 

io.24i48 

10.06174 

6 

9.93826 

10 

5i 

1  12 

58  48 

69699 

19 

3o3o  1 

75881 

25 

241 19 

06181 

6 

93819 

9 

52 

.  4 

58  56 

69721 

19 

30279 

7D910 

26 

24090 

06189 

b 

98811 

b 

53 

0  56 

59  4 

69743 

2(J 

3o25^ 

75939 

26 

24061 

06196 

b 

93So4 

7 

54 
55 

0  48 

59  12 

69765 

20 

30235 

75969 

27 

24o3i 

06208 

6 

93797 

b 

8    0    Ao 

3  59  20 

9.69787 

20 

io.3o2i3 

9.75998 

27 

10.24002 

10.0621 1 

7 

9.93789 

5 

56 

0    32 

59  28 

69809 

21 

80191 

76027 

28 

28978 

06218 

7 

98782 

4 

57 

0  24 

59  36 

69831 

21 

80169 

76056 

28 

28944 

06226 

7 

98773 

3 

58 

0  16 

59  44 

69853 

22 

3oi47 

76086 

29 

28914 

06282 

7 

93768 

2 

5q 

0  8 

59  52 

69875 

22 

30125 

761 1 5 

29 

23885 

06240 

7 

93760 

I 

60 
M 

0  0 

4  0  0 

69897 

22 

3oio3 

76144 

29 

23856 

06247 

7 

93753 

0 

Hour  P.M. 

Hour  A.M. 

Cosine. 

Diff. 

Secant. 

Cotangent 

Diff. 

Tangent. 

Cosecant. 

Ditr. 

Sin<;. 

M 

119° 


A 

A 

B 

B 

( 

^ 

1' 

2^ 

3= 

4s 

.5^ 

6' 

7" 

Prop,  parts  of  cols. 

I  c 

3 

4 

6 

7 
a 

8 
11 

3 

i5 

4 

14 
iS 
i 

17 
22 
5 

20 
26 
6 

GO* 


TABLE  XXVIL 

• 

[Page  215 

S' 

Leg.  S 

ncs,  Tangents,  and  Secants. 

G'. 

30 

3 

A 

A 

B 

B 

C 

C  149° 

M 

0 

Hour  A.M. 

Hour  P.M. 

S'uio, 

Ditr. 

Coserant. 

Tangent. 

Diff. 

Colang-cnl 

Secant. 

Diir. 

Cosine. 

6^ 

800 

400 

9.69S97 

0 

io.3oio3 

9.76144 

0 

10.23856 

10.06247 

0 

9.93753 

1 

7  59  52 

0  8 

69919 

0 

3oo8i 

76173 

0 

23827 

06254 

0 

93746 

59 

2 

59  44 

0  16 

69941 

I 

3oo59 

76202 

1 

23798 

06262 

0 

93738 

58 

3 

59  36 

0  24 

69963 

I 

3oo37 

76231 

I 

23769 

06269 

0 

9373. 

57 

4 
5 

59  28 

0  32 

69984 

3ooi6 

76261 

2 

23739 

06276 

0 

93724 

56 

55 

7  59  20 

4  0  40 

9 . 70006 

2 

10.29994 

9.76290 

2 

10.23710 

10.06283 

9.937.7 

6 

59  12 

0  48 

70028 

2 

29972 

76319 

3 

23681 

06291 

93709 

54 

7 

59  4 

0  56 

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3 

29950 

76348 

3 

23652 

06298 

9370? 

53 

8 

58  56 

I  4 

70072 

3 

29928 

76377 

4 

23623 

o63o5 

936y5 

52 

_9 

lO 

58  48 

I  12 

70093 

3 

29907 

76406 

4 

23594 
10. 23565 

o63i3 

93687 

5i 
5c 

7  58  40 

4  I  20 

9.701 i5 

4 

10.29885 

9,76435 

5 

io.o632o 

9.936S0 

1 1 

58  32 

I  28 

70137 

4 

29863 

76464 

5 

23536 

06327 

93673 

49 

12 

58  24 

I  36 

70159 

4 

29841 

76493 

6 

23507 

06335 

93665 

48 

i3 

58  16 

I  44 

70180 

5 

29820 

76522  6 

23478 

06342 

2 

93658 

4- 

i4 
i5 

58  8 

I  52 

70202 

5 

29798 

76551 
9.76580 

7 
7 

23449 

o635o 

2 

9365o 

-16 

45 

7  58  0 

420 

9.70224 

5 

10.29776 

10.23420 

10.06357 

2 

9.93643 

i6 

57  52 

2  8 

70245 

b 

29755 

76609 

8 

23391 

06364 

2 

93636 

-n 

17 

57  44 

2  16 

70267 

b 

29733 

76639 

8 

2336i 

06372 

2 

93628 

43 

i8 

57  36 

2  24 

70288 

6 

29712 

76668 

9 

23332 

06379 

2 

9362. 

A-' 

!9 

20 

57  28 

2  32 

7o3io 

7 

29690 

76697 
9.76725 

_9. 
10 

233o3 

o6386 

2 

936.4 

4. 
4o 

7  57  20 

4  2  40 

9.70332 

7 

10.29668 

10.23275 

10.06394 

2 

9.93606 

21 

57  12 

2  48 

70353 

8 

29647 

76754 

10 

23246 

06401 

3 

93599 

3g 

22 

57  4 

2  56 

70375 

8 

29625 

76783 

1 1 

23217 

06409 

3 

93591 

38 

23 

56  56 

3  4 

70396 

8 

29604 

76812 

1 1 

23i88 

o64i6 

•  3 

93584 

37 

24 
25 

56  48 

3  12 

70418 

9 

29582 

76841 

12 

23i59 

06423 

3 

93577 

36 
35 

7  56  4o 

4  3  20 

9.70439 

9 

10.29561 

9.76870 

12 

io.23i3o 

10.06431 

3 

9.93569 

26 

56  32 

3  28 

70461 

9 

29539 

76899 

i3 

23l01 

06438 

3 

93562 

34 

27 

56  24 

3  36 

70482 

10 

29518 

76928 

i3 

23072 

06446 

3 

93554 

33 

28 

56  16 

3  44 

7o5o4 

10 

29496 

76957 

i3 

23o43 

06453 

3 

93547 

32 

29 

3o 

56  8 

3  52 

70525 

10 

29475 

76986 
9.77015 

i4 
1 4 

23oi4 
10.22985 

06461 
10.06468 

4 
4 

93539 
9.93532 

3i 

3^. 

7  56  0 

440 

9.70547 

II 

10.29453 

3i 

55  52 

4  8 

7o568 

1 1 

29432 

77044 

lb 

22956 

06475 

4 

93525 

20 

32 

55  44 

4  16 

70590 

11 

39410 

77073 

i5 

22927 

o6483 

4 

93517 

28 

33 

55  36 

4  24 

7061 1 

12 

29389 

77101 

lb 

22899 

00490 

4 

9351c 

27 

34 
35 

55  38 

4  3} 

70633 

12 

29367 

77i3o 

lb 

22870 

06^8 

4 

93502 

26 
I5 

7  55  20 

4  4  40 

9.70654 

i3 

10.29346 

9.77159 

17 

10.22841 

io.o65o5 

4 

9.93495 

3b 

55  12 

4  48 

70675 

i3 

29325 

77188 

17 

22812 

o65i3 

4 

93487 

■}4 

37 

55  4 

4  56 

.70697 

i3 

29303 

77217 

18 

22783 

06520 

5 

93480 

23 

3« 

54  56 

5  4 

70718 

i4 

29282 

77246 

18 

22754 

06528 

5 

93472 

2/ 

39 

4o 

54  48 

5  12 

70739 

i4 

29261 

77274 

'9 

22726 

o6535 

10.06543 

5 
~5~ 

.  93465 
9.93457 

21 
20 

7  54  4o 

4  5  20 

9.70761 

i4 

10.29239 

9.77303 

•9 

10.22697 

41 

54  32 

5  a8 

70783 

i5 

29318 

77332 

20 

22668 

o655o 

5 

93450 

19 

42 

54  24 

5  36 

70803 

i5 

29197 

7736. 

20 

22639 

06558 

5 

93442 

.8 

43 

54  16 

5  44 

70824 

i5 

29176 

7739(1 

21 

22610 

06565 

5 

93435 

17 

44 
45 

54  8 

5  J2 

70846 
9 . 70867 

16 

29154 

774.8 

21 

33582 

06573 

5 

93427 

16 

75 

7  54  0 

460 

16 

10.29133 

9-77447 

22 

10.22553 

io.o658o 

6 

9.93420 

4b 

53  52 

6  8 

70888 

lb 

29112 

77476 

22 

22524 

06588 

6 

q34i2 

i4 

47 

53  44 

6  16 

70909 

17 

29091 

775o5 

23 

22495 

06595 

6 

q34<>5 

i3 

4« 

53  36 

6  24 

70931 

17 

29069 

77533 

23 

22467 

o66o3 

6 

93397 

12 

49 

5o 

53  28 

6  32 

70952 

18 

29048 

77562 

24 

22438 
10.2  2409 

06610 
10.06618 

6 
IT 

93390 
9.93382 

1 1 

.0 

7  53  20 

4  6  40 

9.70973 

18 

10.29027 

9.7759. 

24 

5i 

53  12 

6  48 

70994 

18 

29006 

77619 

25 

22381 

06625 

6 

93375 

9 

52 

53  4 

6  56 

71015 

'9 

289S5 

77648 

25 

22352 

o6633 

6. 

93367 

8 

53 

52  56 

7  4 

7io36 

19 

28964 

77677 

26 

22333 

06640 

7 

93360 

7 

54 
55 

52  48 

7  12 

7io58 

'9 

28942 

77706 

26 

22294 

06648 

7 

93352 

6 
~5 

7  52  4o 

4  7  20 

9.71079 

20 

10.28921 

9.77734 

26 

10. 22266 

io.o6656 

7 

9  93344 

5b 

52  32 

7  28 

71100 

20 

28900 

777631  27 

22237 

06663 

7 

93337 

.'i 

^7 

52  24 

7  36 

71121 

20 

28879 

77791!  27 

22209 

0667 1 

7 

93329 

3 

M 

52  16 

7  44 

71 142 

21 

28858 

77820  28 

22180 

06678 

7 

93322 

2 

59 

52  8 

7  52 

71163 

21 

28837 

77849  28 

22l5l 

06686 

7 

933i4 

I 

bo 
M 

52  0 

8  0 

71184 

21 

28816 

778771  29 

22123 

06693 

7 

93307 

c 
M 

Hour  P.M. 

Hour  .\.m 

fosliip. 

Diflr. 

Secant. 

Cotang-enlDlflr. 

Taii£:ent. 

Cosccnnl. 

DifT. 

Sine. 

120= 


Seconds  of  time 

1 

2 

3 

4 

5 

6 

7 

Prop,  parts  of  cols.  <.   B 

(c 

3 
4 

I 

5 

7 
2 

8 
1 1 

3 

1 1 
i4 

4 

i3 

18 

16 
22 

6 

19 

25 

7 

Page2lG] 

• 

TABLE  XXVII. 

.5'. 

Lo£r.  Sines,  Tangents,  and  Secant.s. 

G' 

31 

3 

A 

A 

B 

B 

C 

C  148° 

o 

Hour  A.M. 

Hour  P.M. 

Sine. 

Ditr. 

0 

Cosecant. 
10.28816 

Tangent. 

Diff. 

Cotangent 

.Secant. 

Diir 

Cosine. 

60 

7  52  0 

480 

9.71184 

9.77877 

0 

10.22123 

10.06693 

0 

9.93307 

I 

5i  52 

8  8 

7[2o5 

0 

28795 

77906 

0 

2209^ 

0670 1 

0 

93299 

5q 

2 

5 1  44 

8  16 

71226 

I 

28774 

77935 

I 

22065 

06709 

0 

93291 

58 

3 

5r  36 

8  24 

71247 

I 

28753 

77963   1 

22037 

067 1 6 

0 

93284 

57 

4 
5 

5j  28 

8  32 

71268 

1 

28732 

77992 

2 

22008 
10.21980 

06724 

93276 

56 
55 

7  5 1  20 

4  8  40 

9.71289 

2 

10.2871 1 

9.78020 

2 

10.06731 

9.93269 

b 

5i  12 

8  48 

7i3io 

2 

28690 

78049 

3 

21951 

06739 

95261 

54 

7 

5i  4 

8  56 

7i33i 

2 

28669 

78077 

3 

21923 

06747 

93253 

53 

8 

5o  56 

9  4 

7i352 

3 

28648 

78106 

4 

21894 

06754 

93246 

52 

_9 

10 

5o  48 

9  12 

4  9  20 

71373 

3 

28627 

78135 

4 

2 1 865 

06762 

93238 

5i 
5o 

7  5o  40 

9.71393 

3 

10.28607 

9.78163 

5 

10.21837 

10.06770 

9.93230 

II 

5o  32 

9  28 

7i4i4 

4 

28586 

78 1 92 

5 

21808 

06777 

93223 

49 

12 

5o  24 

9  36 

71435 

4 

28565 

78220 

b 

21780 

06785 

2 

932  i5 

48 

i3 

5o  16 

9  44 

71456 

4 

28544 

78249 

6 

21751 

06793 

2 

93207 

47 

i4 
i5 

5o  8 

952 

7'477 

5 

28523 

78277 

7 

21723 

06800 

2 

93200 

46 
45 

7  5o  0 

4  10  0 

9.71498 

5 

10.2S502 

9.78306 

7 

10.21694 

io.o68r)9 

2 

9.93192 

lb 

49  52 

10  8 

7i5i9 

5 

28481 

78334 

8 

21666 

06816 

2 

93184 

44 

17 

49  44 

10  16 

71539 

6 

28461 

78363 

8 

21637 

06823 

2 

93 '77 

43 

18 

49  30 

10  24 

7i56o 

6 

28440 

78391 

9 

2 1 609 

o683 1 

2 

93169 

42 

!9 

20 

49  28 

10  32 

7i58i 

7 

28419 

78419 

9 

2i58i 

06839 

2 

93161 

4i 

4o 

7  49  20 

4  10  4o 

9.71602 

7 

10.28398 

9.78448 

9 

io.2i552 

10.06846 

3 

9.93154 

21 

49  12 

10  48 

7lb22 

7 

28378 

78476 

10 

2l524 

06854 

3 

93t46 

3q 

22 

49  4 

10  56 

71643 

8 

28357 

785o5 

10 

21495 

06862 

3 

93 1 38 

38 

aJ 

48  56 

•  II  4 

71664 

8 

28336 

78533 

11 

21467 

06S69 

3 

93i3i 

37 

24 

25 

48  48 

II  12 

71685 

8 

283i5 

78562 

II 

21438 

06877 

3 

93123 

36 
35 

7  48  4o 

4  II  20 

9.71705 

9 

10.28295 

9.78590 

12 

I0.2l4lO 

10.06885 

3 

9.93115 

3b 

48  32 

11  28 

71726 

9 

28274 

78618 

12 

2i382 

06892 

3 

93108 

M 

27 

48  24 

II  36 

71747 

9 

28253 

78647 

i3 

2i353 

06900 

3 

93100 

33 

28 

48  16 

II  44 

71767 

10 

28233 

78675 

i3 

2i325 

06908 

4 

93092 

32 

29 
.30 

48  8 

II  52 

71788 

10 

28212 

78704 

i4 

21296 

06916 

4 

93084 

3i 
3^ 

7  48  0 

4  12  0 

9.718U9 

10 

10.28191 

9.78732 

i4 

10.21268 

10.06923 

4 

9.93077 

3i 

47  52 

12  8 

71829 

II 

28171 

78760 

i5 

2 1 240 

06931 

4 

93069 

29 

3?. 

47  44 

12  16 

7i85o 

II 

28i5o 

78789 

i5 

2121 1 

06939 

4 

93061 

28 

3i 

4i   36 

12  24 

71870 

1 1 

28i3o 

78817 

lb 

21183 

06947 

4 

93o53 

27 

34 
35 

47  28 

I^  32 

71891 

12 

28109 

78845 

lb 

2ii55 

06954 

4 

93o46 

26 

25 

7  47  20 

4  12  4" 

9-71911 

12 

10.28089 

9.78874 

17 

10.21 126 

10.06962 

5 

9.93038 

Jb 

47  12 

12  48 

71932 

12 

28068 

78902 

17 

21098 

06970 

5 

93o3o 

24 

J7 

47  4 

12  56 

71952 

i3 

28048 

78930 

17 

21070 

06978 

5 

93022 

23 

38 

46  56 

i3  4 

71973 

i3 

28027 

78959 

18 

2 104 1 

06986 

5 

93oi4 

22 

39 
4o 

46  48 

i3  12 

71994 
9.72014 

i3 

"i4 

28006 
10.27986 

78987 

18 

2IOl3 

10.20985 

06993 

5 

93007 

21 

20 

7  46  40 

4  i3  Qo 

9.79015 

19 

1 0 . 0700 1 

5 

9.92999 

41 

46  3? 

i3  28 

72034 

i4 

27966 

79043 

19 

20957 

07009 

5 

92991 

19 

42 

46  24 

i3  36 

72055 

i4 

27945 

79072 

20 

20928 

07017 

5 

929S3 

18 

43 

46  16 

i3  44 

72075 

i5 

27925 

79100 

20 

20900 

07024 

6 

92976 

17 

44 
45 

46  8 

i3  52 

72096 

i5 

27904 

79128 

21 

20872 

07032 

6 

92968 

lb 
75 

7  46  0 

4  i4  0 

9.721 16 

i5 

10.27884 

9.79156 

21 

10.20844 

10.07040 

6 

9.92960 

4b 

45  52 

i4  8 

72137 

16 

27863 

79185 

22 

208 1 5 

07048 

6 

92952 

i4 

47 

45  44 

i4  16 

72157 

16 

27843 

79213 

22 

20787 

07056 

6 

92944 

i3 

48 

45  36 

i4  24 

72177 

16 

27823 

79241 

23 

20759 

07064 

5 

92936 

12 

49 
5o 

45  28 

i4  32 

73198 

17 

27802 
10.27782 

79269 

23 

20731 

07071 

b 

92929 

II 
10 

7  45  20 

4  i4  4o 

9.72218 

17 

9.79297 

24 

10.20703 

10.07079 

6 

9.92921 

bi 

45  12 

i4  48 

72238 

18 

27762 

79326 

24 

20D74 

07087 

7 

92913 

9 

.12 

45  4. 

i4  56 

72259 

18 

27741 

79354 

25 

2064b 

07095 

7 

92905 

8 

i>3 

44   56 

i5  4 

72279 

18 

27721 

79382 

25 

20618 

07103 

7 

92897 

7 

.'14 
55 

44  48 

i5  12 

72299 

'9 

27701 

79410 

2b 

20590 

071 1 1 

7 

92889 

6 
5 

7  44  4o 

4  1 5  20 

9.72320 

'9 

10.27680 

9.7943s 

26 

10.20562 

10.07119 

7 

9.92881 

5b 

44  32 

i5  26 

_  72340 

19 

27660 

79466 

26 

20534 

07126 

7 

92874 

4 

57 

44  24 

i5  36 

72360 

20 

27640 

79495 

27 

2o5o5 

07134 

7 

92866 

3 

58 

44   16 

i5  44 

72381 

20 

27619 

79523 

27 

20477 

07142 

7 

92858 

2 

59 

44  8 

i5  52 

72401 

20 

27599 

79551 

28 

20449 

O7i5o 

8 

92850 

I 

bii 
M 

44    0 

16  0 

72421 

21 

27579 

79579 

28 

20421 

07 1 58 

8 

92842 

0 
M 

Hour  P.M. 

Hour  A.M. 

Cosine. 

Difl-. 

Secant. 

Cotangent 

Diff. 

Tangent. 

Cosecant. 

Dift: 

Sine. 

12r 


A 

A 

B 

B 

C 

Seconds  of  time  . 

3 
4 
I 

2» 

5 
7 

St 

3' 

8 
II 
3 

4- 

10 
i4 
4 

5' 

i3 
18 
5 

6' 

i5 
21 
6 

7. 



18 

25 

1 

Prop,  parts  of  cols 

!■ 

C      58« 


TABLE  XXVIL 

-1 
[Page  217 

.v. 

Log.  Sines,  Tangents,  and  S 

ecants. 

G- 

33' 

M 

o 

A 

A 

B 

B 

c 

C  147° 

Hour  A.M. 

Hour  P.M. 

Sine.  |Difi'. 

Cosecant. 

Tang-ent. 

Diir. 

Cotangent 

Secant. 

Din: 

Cosine. 

M 

60 

7  44  0 

4  16  0 

9. 7242  J 

0 

10.27579 

9.79579 

0 

10.20421 

10.07158 

0 

9.92842 

I 

43  52 

16  8 

7244r 

0 

27559 

79607 

0 

20393 

07166 

0 

92834 

^9 

2 

43  44 

16  16 

72461 

I 

27539 

79635 

I 

2o365 

07174 

0 

92826 

58 

3 

43  36 

16  24 

72482 

I 

27518 

79663 

I 

20337 

07182 

0 

92818 

!57 

4 
5 

43  28 

16  32 

72502 

1 

27498 

79691 

2 

2o3o9 

07190 

92810 

'Jb 

55 

7  43  20 

4  16  4o 

9.72522 

2 

10.27478 

9.79719 

2 

10.20281 

10.07197 

9.92803 

6 

43  12 

16  48 

72542 

2 

2745? 

79747 

3 

20253 

07205 

92795 

54 

- 

43  4 

16 -56 

72562 

2 

27438 

79776 

3 

20224 

07213 

92787 

53 

8 

42  56 

17  4 

72582 

3 

27418 

79804 

4 

20 1 96 

07221 

92779 

52 

_9 

10 

42  48 

17  12 

72602 

3 

27398 

79832 

4 

20168 

07229 

92771 
0.93763 

5^ 

7  42  4" 

4  17  20 

9.72622 

3 

10.27378 

9.79860 

5 

10.20140 

10.07237 

1 1 

42  32 

17  28 

72643 

4 

27357 

79888 

5 

201 12 

07245 

92755 

^9 

13 

42  24 

17  36 

72663 

4 

27337 

79916 

6 

20084 

07253 

92747 

4  b 

i3 

42  16 

17  44 

72683 

4 

27317 

79944 

6 

2oo56 

07261 

2 

92739 

47 

i4 
i5 

42  8 

17  52 

72703 

5 

27297 

79972 

7 

20028 

07269 

2 

92731 

40 
45 

7  42  0 

4  18  0 

9.72723 

5 

10.27277 

9 . 80000 

7 

1 0 . 20000 

10.07277 

2 

9.92723 

1 6 

4 1  52 

18  8 

72743 

6 

27257 

80028 

7 

19972 

07285 

2 

92715 

44 

I- 

4i  4i 

18  16 

72763 

6 

27237 

8oo56 

8 

19944 

07293 

2 

92707 

43 

i8 

4 1  36 

18  24 

72783 

6 

27217 

80084 

8 

1 99 16 

07301 

2 

92699 

42 

!9 
ao 

4i  28 

18  32 

72803 

6 

27197 

801 12 

9 

1988S 

07309 
10.07317 

3 

92691 

9.92683 

41 
4o 

7  4i  20 

4  18  40 

9.72S23 

7 

10.27177 

9.80140 

9 

1 0 . 1 9860 

21 

4i  12 

18  48 

72843 

7 

27157 

80168 

10 

19832 

07325 

3 

92675 

39 

22 

4t  4 

18  56 

72S63 

7 

27137 

80195 

10 

19805 

07333 

3 

92667 

38 

23 

40  56 

19  4 

72883 

8 

27117 

80233 

1 1 

19777 

07341 

3 

92659 

37 

24 
25 

4o  48 

19  12 

72902 

8 

27098 

8025 1 

1 1 

19749 

07349 

3 

9265 1 

3b 
35 

7  4o  4»' 

4  19  20 

9.72922 

8 

10.27078 

9.80279 

12 

10. 19721 

ic. 07357 

3 

9.92643 

26 

40  32 

19  28 

72942 

9 

27058 

8o3o7 

12 

19693 

07365 

3 

92635 

34 

27 

40  24 

19  36 

72962 

9 

27038 

8o335 

i3 

19665 

07373 

4 

92627 

ii 

28 

4o  16 

19  44 

72982 

9 

27018 

8o363 

i3 

19637 

0738 1 

4 

92619 

32 

^9 
3o 

40  8 

19  52 

73002 

10 

26998 

80391 

i3 

1 9609 

07389 
10.07397 

4 

9261  1 

3i 

3^ 

7  4o  0 

4  20  0 

9.73«22 

10 

10.26978 

9.80419 

14 

10. 19581 

4 

9.92603 

3i 

39  52 

20  8 

73o4i 

10 

26959 

80447 

14 

19553 

074o5 

4 

92595 

29 

32 

39  44 

20  16 

73i)6 1 

II 

26939 

80474 

i5 

19526 

0741 3 

4 

92587 

2b 

3J 

39  36 

20  24 

73o8 1 

1 1 

26919 

8o5o2 

i5 

19498 

07421 

4 

92579 

27 

34 
35 

39  28 

20  32 

73 101 
9.73121 

1 1 

26899 

8o53o 

lb 

■  19470 

07429 

.5 

92571 

2b 
^5 

7  39  20 

4  20  4(1 

12 

10.26879 

9.8o558 

16 

10.19442 

10.07437 

5 

9.92563 

36 

39  12 

20  48 

73i4o 

12 

26860 

8o586 

17 

19414 

07445 

5 

92555 

24 

37 

39  4 

20  56 

73 160 

12 

26840 

80614 

17 

19386 

07454 

5 

92546 

23 

38 

38  56 

•  21  4 

73180 

i3 

26820 

80642 

18 

19358 

07462 

5 

92538 

22 

39 
40 

38  48 

21  12 

73200 

i3 

26800 

80669 

18 

1 9331 

07470 

5 

92530 

21 
20 

7  38  40 

4  21  20 

9.73219 

i3 

10.26781 

9.80697 

19 

10. 19303 

10.07478 

5 

9.93522 

4i 

38  32 

21  28 

73239 

i4 

26761 

80725 

'9 

19275 

07486 

6 

92514 

'9 

42 

38  24 

21  36 

73259 

i4 

26741 

80753 

20 

19247 

07494 

6 

92506 

18 

43 

38  16 

21  44 

73278 

14 

26722 

80781 

20 

19219 

07502 

6 

92498 

'7 

44 
4!^ 

38  8 

21  52 

73298 

i5 

26702 

80808 

20 

19192 

07510 

6 

92490 

lb 

75 

7  38  0 

4  22  0 

9.73318 

i5 

10.26682 

9.80836 

21 

10.19164 

10.07518 

6 

9.92482 

40 

37  52 

22  8 

73337 

i5 

26663 

S0864 

21 

19136 

07527 

6 

92473 

14 

47 

37  44 

22  16 

73357 

16 

26643 

80892 

22 

19108 

07535 

6 

92465 

i3 

48 

37  3() 

22  24 

73377 

16 

26623 

80919 

22 

1 908 1 

07543 

b 

93457 

12 

19 
5ci 

37  28 

22  32 

73396 

16 

26604 

80947 

23 

19053 

07551 

7 

93449 

1 1 
10 

7  37  2C 

4  22  4'-' 

9.73416 

17 

10. 26584 

9.80975 

23 

10.19025 

10.07559 

7 

9.93441 

DI 

37  12 

22  48 

73435 

17 

26565 

8ioo3 

24 

18997 

07567 

7 

93433 

9 

bi 

37  4 

22  00 

73455 

'7 

26545 

8io3o 

24 

18970 

07575 

7 

93425 

8 

53 

36  56 

23  4 

73474  18 

26526 

8io58 

25 

18942 

07584 

7 

93416 

7 

54 
55 

36  48 

23  12 

73494  I? 

265o6 

81086 

25 

18914 

07592 

7 

93408 

~5 

7  36  4o 

4  23  2C1 

9.73513  18 

io.26'i87 

9.81113 

26 

10.18887 

10.07600 

7 

9.92400 

d6 

36  32 

23  28 

73533  19 

26467 

8ii4i 

?6 

18859 

07608 

8 

92392 

4 

37 

36  24 

23  36 

73552 

19 

26448 

81169 

26 

i883i 

076 1 6 

8 

92  384 

3 

:)y 

36  16 

23  ^^ 

73572 

•9 

26428 

81196 

27 

18804 

07624 

8 

92376 

2 

^9 

36  8 

23  52 

73591 

20 

26409 

81224 

27 

18776 

07633 

8 

92367 

1 

0(1 

36  0 

24  0 

736 1 1 

20 

26389 

81252 

28 

1874s 

07641 

8 

.  9' ^^9 
Sntc. 

0 

[Tour  P.M. 

Hour  .\.M. 

Cosino.  DilT. 

Recant. 

Cotangent 

DilT. 

Tangent. 

Cosecant 

Diir. 

122=^ 


A 

A 

B 

B 

C 

Seconds  of  time 

!• 

2» 

3' 

4- 

5' 

•6' 

7' 

(^ 

2 

5 

7 

10 

12 

i5 

>7 

Prop,  ptirts  of  cols. 

" 

3 

7 

10 

i4 

17 

21 

24 

(c 

I 

2 

3 

4 

5 

6 

7 

C      57" 


'<J8 


Paae  218] 

TABLE  XXVIL 

.5  . 

Log.  Sines,  Tangents,  and  Secants. 

G 

•33° 

A 

A 

B 

B 

C        C  14G" 

^1 

o 

Hour  A.M. 

Hour  P.M. 

Siiio. 

uiir 

Cosecant. 

Tangent. 

Ditr. 

Cotangent 

Secant. 

Diir. 

Cosine. 

M 

6S 

7  36  0 

4  24  0 

9.7361 1 

0 

10.26389 

9.81262 

0 

10.18748 

10.07641 

0 

9.92359 

I 

35  52 

24  8 

7363o 

0 

26370 

81279 

0 

18721 

07649 

0 

92351 

5q 

2 

35  M 

24  16 

73650 

I 

2635o 

8i3o7 

I 

18693 

07667 

0 

92343 

58 

3 

35  36 

24  24 

73669 

I 

2633i 

8i335 

1 

18666 

07666 

0 

92335 

57 

A 
5 

35  28 

24  32 

73689 

I 

263 1 1 

8 1 362 

2 

i8638 

07674 

92326 

56 
66 

7  35  20 

4  24  40 

9.73708 

2 

10.26292 

9.81 390 

2 

10.18610 

10.07682 

9.92318 

b 

35  12 

24  48 

73727 

2 

26273 

8t4i8 

3 

18682 

07690 

92310 

64 

7 

35  4 

24  56 

73747 

2 

26253 

81445 

3 

i8565 

07698 

92302 

63 

8 

34  56 

25  4 

73766 

3 

26234 

81473 

4 

18627 

07707 

92293 

62 

_9 

10 

34  48 

25  12 
4  25  20 

73785 

3 

26215 

8i5oo 

4 

18600 

07716 

92286 

61 

5^ 

7  34  4" 

9.73805 

3 

10.26195 

9.81628 

6 

10.18472 

10.07723 

9.92277 

1 1 

34  32 

25  28 

73824 

3 

26176 

8i556 

6 

mA4 

07731 

2 

92269 

4o 

12 

M   24 

25  36 

73843 

4 

26157 

8i583 

5 

I84I7 

07740 

2 

92260 

48 

IJ 

34  16 

25  A4 

73863 

4 

26137 

81611 

6 

18389 

07748 

2 

92262 

47 

i4 
i5 

34  8 

25  52 

73882 

4 

26118 

8i638 

6 

i8362 

07766 

2 

92244 

46 
45 

7  34  0 

4  26  0 

9.73901 

5 

1 0 . 26099 

9.81666 

7 

10.18334 

10.07766 

2 

9.92235 

!b 

33  52 

26  8 

73921 

5 

26079 

81693 

7 

i83o7 

07773 

2 

92227 

44 

'7 

33  M 

26  16 

73940 

5 

26060 

81721 

8 

18279 

07781 

2 

92219 

43 

i8 

:^  36 

26  24 

73959 

b 

26041 

81748 

8 

18262 

07789 

3 

92211 

42 

12 

io 

33  28 

26  32 

73978 

b 

26022 

81776 

9 

18224 

07798 

3 

92202 

4i 

4o 

7  33  20 

4  26  40 

9.73997 

6 

10.26003 

9.81803 

9 

10.18197 

10.07806 

3 

9.92194 

21 

33  12 

26  48 

74017 

7 

26983 

8i83i 

10 

18169 

07814 

3 

92186 

3q 

22 

33  4 

26  56 

74o36 

7 

25964 

81868 

10 

18142 

07823 

3 

92177 

38 

20 

32  56 

27  4 

74o55 

7 

25945 

81886 

1 1 

18114 

07831 

3 

92169 

37 

24 
25 

32  48 

27  12 

74074 

8 

25926 

81913 

11 

18087 

07839 

3 

92  161 

36 
35 

7  32  4o 

4  27  20 

9.74093 

8 

10.25907 

9.S1941 

II 

10.18069 

10.07848 

3 

9.92162 

20 

32  32 

27  28 

74n3 

8 

2:)887 

81968 

12 

i8o32 

07866 

4 

92 1 4/\ 

M 

27 

32  24 

27  36 

74 1 32 

9 

2  5868 

81996 

12 

1 8oo4 

07864 

4 

92 1 36 

33 

28 

32  16 

27  M 

74i5i 

9 

25849 

82023 

i3 

17977 

07873 

4 

92127 

32 

29 

3o 

32  8 

27  52 

74170 

9 

2583o 

82061 

i3 

17949 

07881 

4 

92119 

3i 
3o 

7  32  0 

4  28  0 

9.74189 

10 

io.258ii 

9.82078 

14 

10.179^2 

10.07889 

4 

9.921 1 1 

3i 

3t  52 

28  8 

74208 

10 

26792 

82106 

i4 

17894 

07898 

4 

92102 

29 

32 

3t  44 

28  16 

74227 

10 

25773 

82133 

i5 

17867 

07906 

4 

92094 

28 

ii 

3i  '^6 

28  24 

74246 

10 

26754 

82161 

i5 

i783g 

07914 

5 

92086 

27 

34 

3 1  28 

28  32 

74265 

11 

26735 

82188 

16 

17812 

C7923 

5 

92077 

26 

I! 

3b 

7  3i  20 

4  28  40 

9.74284 

II 

10.26716 

9.82216 

16 

10.17786 

10.07931 

5 

9.92069 

3b 

3i  12 

28  48 

743o3 

1 1 

26697 

82243 

lb 

17757 

07940 

5 

92060 

24 

37 

3i  4 

28  56 

74322 

12 

26678 

82270 

17 

17730 

07948 

5 

92062 

23 

3y 

3o  56 

29  4 

74341 

12 

26669 

82298 

17 

17702 

07966 

•5 

92044 

22 

09 
40 

3u  48 

29  12 

74360 

12 

25640 

82326 

18 

17G75 

07966 

6 

92035 

21 

20 

7  3o  4o 

4  29  20 

9.74379 

i3 

10.26621 

9.82362 

18 

10.17648 

10.07973 

6 

9.92027 

41 

3o  32 

29  28 

7439S 

i3 

26602 

82380 

19 

17620 

07982 

6 

92018 

19 

42 

3o  24 

29  3o 

74417 

i3 

25683 

82407 

19 

17693 

07990 

6 

92010 

18 

4J 

3o  16 

29  44 

74436 

i4 

2  5564 

82435 

20 

17666 

07998 

6 

92002 

17 

44 
45 

3o  8 

29  52 

74455 

i4 

25545 

82462 

20 

17638 

08007 

6 

91993 

lb 
is 

7  3t)  0 

4  3o  0 

9-74474 

i4 

10,26626 

9.82489 

21 

10.1761 1 

10.08016 

6 

9.91985 

40 

29  52 

3o  8 

74493 

i5 

26607 

82617 

21 

17483 

08024 

6 

91976 

i4 

47 

29  U 

3o  16 

745 1 2 

i5 

25488 

82644 

22 

17466 

o8o32 

7 

9 1 968 

i3 

46 

29  36 

3o  24 

7453i 

i5 

26469 

82671 

22 

17429 

o8o4i 

7 

91969 

12 

49 
5o 

29  28 

3o  32 

74549 

lb 

25451 

82699 

22 

17401 

08049 

7 

7 

91961 

1 1 
10 

7  29  20 

4  3o  40 

9.74568 

16 

10.26432 

9.82626 

23 

10.17374 

10.08068 

9.91942 

5i 

29  ij 

3o  48 

74587 

lb 

254i3 

82653 

23 

17347 

08066  7 

91934 

9 

52 

29  4 

3o  56 

74606 

17 

25394 

82681 

24 

17319 

08076 

7 

91926 

•8 

53 

28  56 

3i  4 

74625 

17 

25376 

82708  24  1 

17292 

o8o83 

7 

91917 

7 

54 
55 

28  48 

3i  ,2 

74644 

17 

25366 

82735 

25 

17266 

08092 

8 

9 1 908 

b 

7  28  40 

4  3i  20 

9.74662 

17 

10. 25338 

9.82762 

26 

10.17238 

10.08100 

8 

9.91900 

5b 

28  32 

3i  28 

74681 

18 

•■25319 

82790 

26 

17210 

08109 

8 

91891 

4 

'>7 

28  -ji 

3i  36 

74700 

18 

2  53oo 

82817 

26 

17183 

08117 

8 

91883 

3 

58 

28  16 

3 1  44 

74719 

18 

26281 

82844 

27 

17166 

08126 

8 

91874 

2 

^9 

28  6 

3i  52 

74737 

19 

26263 

82871 

27 

17129 

08 1 34 

8 

91866 

I 

bo 
I\I 

28  o| 

32   0 

74756 

19 

26244 

8280Q 

27 

17101 

08143 

8 

91867 

0 

Uourr.M 

lour  A.M. 

Cosine. 

DiflT. 

Secant. 

Cotangent 

Diff. 

Tangent. 

Cosecant. 

DilT.  Sine.  | 

123° 


5e» 


P 

2'^ 

3' 

4' 

5' 

& 

'*'« 
r 

(^ 

2 

5 

7 

10 

12 

i4 

17 

Prop,  parts  of  cols. 

^ 

-) 

7 

10 

i4 

17 

21 

24 

(c 

I 

2 

3 

4 

5 

6 

1 

TABLE  XXVIJ                ['"='«« -19 

Log.  Sines,  Tangents,  and  Secants.              '^'• 
34^              A        A      B        B      C        C  345° 

o 
I 

2 

3 

4 
"5 
6 

8 
_9 

10 

1 1 

12 

i3 
i4 
i5 
i6 

17 
i8 

11 

20 
21 
22 
23 
24 
25 
26 
27 
28 

29 

3o 
3i 

32 

33 
34 
35 
36 
37 
38 
39 

40 
4i 
42 
43 
44 
45 
46 
47 
48 
49 
5o 
5. 

52 

53 
54 
55 
56 

57 
58 

60 
M 

Hour  A.M 

Houip.M. 

Sine. 

DiflT 

Cosecant. 

Tangent. 

Ditr. 

Cotang-ent 

Secant. 

Diir. 

Cosine. 

M 

60 
59 
58 

57 
56 

55 
54 
53 

52 

5. 
5o 

t 

4i 
46 

45 
44 
43 
42 
4i 
4o 

39 
38 

37 
36 

35 
34 
33 

32 

3. 

3Z 
29 

28 
27 
26 
J5 
24 

23 

22 
2. 

20 

19 

.8 

17 
.6 

75 
i4 
i3 
12 
11 
10 

9 

8 

7 
6 

5 
4 
3 
2 
I 
0 

7  28  0 
27  52 

27  4^' 
27  3t) 
27  28 

4  32  0 

32  8 
32  16 
32  24 

32  32 

9-74756 
74775 
74794 
748(2 
7483 1 

0 
0 

I 
I 
I 

10.25244 

25225 
25206 

25.88 
25.69 

9.82899 
82926 
82953 
82980 
83oo8 

0 
0 

I 
I 
2 

10. 17.01 
17074 
17047 
17020 
16992 

10.08.43 
08. 5. 
08160 
08.68 
08.77 

0 
0 
0 
0 

9.9.857 
9.849 
91840 
9.832 
91823 

7  27  20 
27  12 
27  4 
26  56 
26  48 

7  26  4'-> 

26  32 
26  24 
26  16 
26  8 

4  32  4u 
32  48 

32  56 

33  4 
33  12 

9-74850 
74868 
74887 
74906 
74924 

2 
2 

2 
2 

3 

io.25.5o 
25.32 
25ii3 
25094 
25076 

9.83o35 
83o62 
83089 
83.17 
83.44 

2 
3 

■  3 

4 
4 

10.16965 
.6938 
169. . 
16883 
1 6856 

10.08185 
08.94 
0S202 
0S2  1 . 
082.9 

.0. 08228 
08237 
08245 
08254 
08262 

9.9.8.5 
91806 

9' 798 
9.789 
9.78. 

4  33  20 
33  28 
33  36 
33  44 
33  52 

9.74943 
74961 
74980 

74999 
75017 

3 
3 
4 
4 
4 

10  25o57 
25o39 

25020 

25oo. 
24983 

9.83.7. 
83198 
83225 
83252 
83280 

5 
5 
5 
6 
6 

10.16829 
16802 
16775 
16748 
16720 

2 
2 
2 
2 

9.9.772 
9.763 
9.755 
91746 
91738 

7  26  0 

25  52 

25  44 
2  5  36 
25  28 

4  34  0 
34    8 
34  16 
34   24 
34  32 

9.75036 
75o54 
75073 
75091 
75 1 10 

5 
5 
5 
6 
6 

10.24964 
24946 
24927 
24909 
24890 

9-83307 
83334 
8336. 
83388 
834.5 

7 
7 
8 
8 
9 

10. 16693 
16666 
16639 
16612 
16585 

.0.0827] 
08280 
08288 
08297 
o83o5 

2 
2 
2 
3 
3 

9.91729 

91720 
917.2 
9.703 
91695 

7  25  20 
25  12 
25  4 
24  56 
24  48 

4  34  4o 
34  48 

34  56 

35  4 
35  12 

.9.75128 
75i47 
75i65 
75.84 
7520-;' 

6 
6 

7 
7 
7 

10.24872 
24853 
24835 
24816 
24798 

9-83442 
83470 
83497 
83524 
83551 

9 

9 

10 

10 

II 

10.16558 
i653o 
i65o3 
16476 
16449 

io'.o83.4 
o832  3 
o833i 
o834o 
08349 

3 
3 
3 
3 
3 

9.91686 
91677 
91669 
9 1 660 
9.65. 

7  24  4o 

24  32 
24  24 
24  16 
24  8 

4  35  20 
35  28 
35  36 
35  44 
35  52 

9.75221 
75239 
75258 
75276 
75294 

8 
8 
8 
9 
9 

9 
9 

.0 

.0 

10 

10.24779 
2476. 
24742 
24724 
24706 

9.83578 
836o5 
83632 
83659 
83686 

I . 

12 
12 
i3 
i3 

10.16422 
16395 
i636S 
i634i 
i63i4 

10.08357 
o8366 
08375 
08383 
08392 

4 
4 
4 
4 
4 

9.91643 
9.634 
91625 
916.7 
91608 

7  24  0 
23  52 
23  44 
23  36 
23  28 

4  36  0 
36  8 
36  16 
36  24 
36  32 

9.75313 
75331 
75350 
75368 
75386 

10.24(187 
24669 
2465o 
24632 
24614 

9.837.3 
83740 
83768 
83795 
83822 

i4 
14 
1 4 
i5 
i5 

ID. 16287 
16260 
16232 
16205 
16178 

10.08401 
08409 
084.8 
08427 
08435 

4 
4 
5 
5 
5 

9.9.599 
9.591 
9.582 
9'573 
9.565 

7  23  20 

23  12 

23  4 

22  56 

22  48 

4  36  40 
36  48 

36  56 

37  4 
37  12 

9.75405 
75423 
75441 
75459 
75478 

1 1 
II 
1 1 
12 
12 

10.24595 

24577 
24559 
24541 
24522 

9.83849 
83876 
83903 
83930 
83957 

16 
16 

'7 
17 
18 

10. i6i5i 
16124 
16097 
1 6070 
16043 

10.08444 
08453 
08462 
08470 
08479 

5 
5 
5 
5 
6 

9.9.  ::T 

91547 
9.538 
9i53o 
9.521 

7  22  4o 

22  32 
22  24 
22  16 
22   8 

4  37  an 
37  28 
37  36 
37  44 
3j   52 

9.75496 
755i4 
75533 
7555! 
75569 

.2 

.3 
i3 
i3 
i3 

io.245o4 
24486 
24467 
24449 
2443 1 

9.83984 
84oi  1 
84o38 
84o65 
84092 

18 
18 
'9 
■9 
20 

1 0 . 1 60 1 6 
15989 
15962 
15935 
1 5908 

.0.08488 
0S496 
oS5o5 
o85.4 
o8523 

6 
6 
6 
6 
6 

9.915.2 
9i5o4 
91495 
9.486 
91477 

7  22   0 
21  52 

21  44 
21  36 

21  28 

4  38  0 
38  8 
38  16 
38  24 
38  32 

9.75587 
756o5 
75624 
75642 
75660 

9.75678 
75696 
757.4 
75733 
75751 

14 
i4 
i4 
i5 
i5 
75 
16 
16 
16 
17 

io.244i3 
24395 
24376 
24358 
24340 

9.84 1 19 
84.46 
84.73 
84200 
84227 

20 
21 
21 
22 
22 

10..588I 
i5854 
i5827 
i58oo 
15773 

io.c)853i 
o854o 
08549 
o8558 
08567 

7 
7 
7 
7 
7 

9-91469 
9 1 460 
9145. 
9.442 
91433 

7  21  20 
21  12 
21   4 
20  56 
20  48 

4  38  4o 
38  48 

38  56 

39  4 
39  12 

10.24322 
24304 
24286 
24267 
24249 

9.84254 
84280 
84307 
84334 
8436. 

23 
23 
23 

24 
24 

10.15746 
15720 
15693 
15666 
1 5639 

10.08575 
08584 
08593 
08602 
08611 

7 
7 
8 
8 
8 

9.9.425 
9.416 
9.407 
9.398 
9.389 

7  20  4" 

2(.    32 
20  24 
20  16 
20   8 
20   0 

4  39  20 
39  28 
39  36 
39  44 

39  52 

40  0 

9.75769 
75787 
758o5 
75823 
7584. 
75859 

17 
17 
17 
18 
18 
18 

10.2423. 
24213 
24.95 

24.77 
24.59 
24.41 

9.84388 
844 1 5 
84442 
84469 
84496 
84523 

25 
25 

26 
26 

27 

27 

10. 1 56 12 
i5585 
1 5558 
i553. 
i55o4 
15477 

10.08619 
08628 
08637 
08646 
08655 
08664 

8 
8 
8 
8 
9 
9 

9.9.381 
91372 
91363 
9.354 
91345 
9.336 

IIourP.M.|HourA-Bi. 

Cosine. 

Diff. 

Secant. 

Co(angent|Difl'. 

Tangent. 

Cosecant. 

Diff 

Sinc- 

rM° 


Seconds  of  time 

1' 

2= 

3' 

4' 

9 

i4 

5' 

II 

17 
5 

6' 

i4 
20 

7 

7' 
16 
24 
8 

Prop,  parts  of  cols.  <  B 
(  C 

2 
3 
I 

5 

7 
2 

7 
10 
3 

1\ 

ISP  220] 

TABLE  XXVIL 

S'. 

Log 

^  Sines,  Tangents,  and  Secants. 

g: 

35 

0 

A 

A 

B 

B 

C 

C  144° 

Hoar  A.M. 

Hour  P.M. 

4  4o  0 

Sine. 

bifl-. 

Cosecant. 

Tang-ent. 

DilT. 

Colang^onl 

Secant. 

Diir. 

Cosint;. 

M 

60 

7  20  0 

9.75859 

0 

Io.24i4i 

9.84523 

0 

10. 1 5477 

10.08664 

0 

9.91336 

1 

19  52 

4o  8 

75877 

0 

24123 

8455o 

0 

i545o 

08672 

0 

91328 

5q 

2 

19  4  i 

4o  16 

75895 

I 

24io5 

84576 

I 

15424 

08681 

0 

91319 

58 

3 

19  36 

4o  24 

759.3 

I 

24087 

846o3 

1 

15397 

08690 

0 

9i3io 

57 

4 

5 

19  28 

4o  32 

4  4o  4o 

75931 

I 

24069 
io.24o5i 

8463o 

2 

15370 

08699 

I 

9i3oi 

56 
55 

7  19  20 

9.75949 

I 

9.84657 

2 

10.15343 

10.08708 

I 

9.91292 

0 

19  12 

4o  48 

75967 

2 

24o33 

84684 

3 

i53i6 

08717 

I 

91283 

54 

7 

.9  4 

40  56 

75985 

2 

340 1 5 

84711 

3 

15289 

08726 

I 

91274 

53 

« 

18  56 

4i  4 

76003 

3 

23997 

84738 

4 

15262 

08734 

I 

91266 

52 

10 

iS  4S 

4r  13 

7602  I 

3 

[0. 23961 

S4764 
9.84791 

4 
4 

15236 

08743 

I 

91257 
9.91248 

5i 

5o 

7  18  4o 

4  4i  20 

9.76039 

3 

10. l520Q 

10.08753 

2 

1 1 

18  32 

4 1  28 

76057 

3 

23943 

84818 

5 

■   i5i82 

08761 

2 

91239 

49 

12 

18  24 

4 1  36 

76075 

4 

23925 

84845 

5 

i5i55 

08770 

2 

9 1 2  3o 

48 

i3 

18  16 

4i  44 

■  76093 

4 

23907 

84872 

b 

i5i28 

08779 

2 

91221 

47 

i4 
i5 

18  8 

41  52 

761 1 1 

4 

23889 
10.23S71 

84899 

b 

i5ioi 

08788 

2 

91212 

46 
45 

7  iS  0 

4  42  0 

9.76129 

4 

9.84935 

7 

io.i5o75 

10.08797 

2 

9.91203 

i6 

17  52 

42  8 

76146 

5 

23854 

84952 

7 

i5o48 

08806 

2 

91 194 

44 

17 

17  44 

42  iG 

76164 

5 

23836 

84979 

8 

l502I 

08815 

3 

91185 

43 

iS 

17  36 

42  24 

76182 

b 

238i8 

85oo6 

8 

14994 

08824 

3 

91176 

42 

20 

17  28 

42  32 

4  42  4" 

76200 

6 

238oo 

85o33 

8 

14967 

08833 

3 

91 167 

4i 
40 

7  17  20 

9.76218 

6 

10.23782 

9.85039 

9 

10. 14941 

10:08842 

3 

9.91158 

21 

17  12 

42  48 

76236 

b 

23764 

85o86 

9 

I49I4 

o885i 

3 

91149 

3q 

22 

17  4 

42  56 

76253 

b 

23747 

85ii3 

10 

14887 

08859 

3 

91141 

38 

23 

16  56 

43  4 

76271 

7 

23729 

85i4o 

10 

14860 

08868 

^ 

91132 

37 

24 
25 

16  48 

43  12 

4  43  20 

76289 
9.76307 

7 

2371 1 

85 166 

1 1 

14834 

08877 

4 

91123 

36 
35 

7  16  4u 

7 

10.23693 

9.85193 

1 1 

10. 14807 

10.08886 

4 

9.91114 

3  b 

16  32 

43  28 

76324 

8 

23676 

852  2U 

12 

147S0 

08895 

4 

91  io5 

34 

27 

16  24 

43  36 

76342 

8 

23658 

85347 

12 

14753 

08904 

4 

91096 

33 

■>s 

16  16 

43  44 

76360 

8 

23640 

85273 

1  3- 

14727 

0S913 

4 

91087 

32 

29 

3o 

16  8 
7  16  0 

43  52 
4  44  0 

7637S 

9 

2  362  2 

853oo 

i3 

14700 

08922 

4 

91078 

3i 
3^ 

9.76395 

9 

io.236c5 

9.85327 

.3 

10.14673 

10.08931 

5 

9.91 069 

h 

1 5  52 

44  8 

764 1 3 

9 

23587 

85354 

I A 

14646 

08940 

5 

9 1 060 

29 

J  2 

1 5  44 

44   16 

76431 

9 

-N^t^l^O 

85330 

i4 

14620 

08949 

5 

9io5i 

28 

3J 

1 5  36 

44   24 

76448 

10 

23552 

85407 

1 5 

14593 

08958 

b 

91042 

27 

34 
35 

i5  28 

44  32 

76486 

10 

23534 
io.235i6 

85434 

i5 

14566 

08967 

b 

91033 

26 

25 

7  !5  20 

4  44  4o 

9.76464 

10 

9.85460 

16 

10.14540 

10.08977 

5 

9.91023 

3o 

i5  12 

44  48 

76501 

1 1 

23499 

85487 

16 

i45i3 

089S6 

5 

91014 

2  4 

J7 

i5  4 

44  56 

765  I  Q 

1 1 

23i8i 

855i4 

16 

14486 

0S995 

6 

91005 

2  3 

36 

1 4  56 

45  4 

76537 

1 1 

23463 

85540 

17 

1 4460 

-  09004 

6 

90996 

2  2 

4o 

i4  48 

45  12 
4  45  20 

76554 

12 
12 

23446 
10.23428 

85567 

1 7 

14433 

09013 

6 

909S7 

21 
20 

7  1 4  40 

9.76572 

9.85594 

18 

1 0 . 1 44i  16 

10.09023 

6 

9.9097S 

4i 

1 4  32 

45  28 

76590 

12 

•  23410 

85620 

18 

i43So 

09031 

b 

90969 

'9 

42 

i4  24 

45  36 

76607 

12 

33393 

85647 

19 

14353 

09040 

b 

9096(.; 

18 

43 

14  16 

45  44 

76625 

i3 

23375 

85674 

19 

14326 

09049 

6 

9095 1 

'7 

45 

i4  8 

45  52 

4  46  0 

76643 

i3 

23358 
10.23340 

85700 
9".85727 

20 
20 

1 43oo 

09058 

7 

90943 

lb 
75 

7  i4  0 

9 . 76660 

i3 

10. 14273 

10.09067 

7 

9.90933 

4b 

1 3  52 

46  8 

76677 

1 4 

23323 

85754 

20 

14246 

09076 

7 

90924 

14 

47 

1 3  44 

46  16 

76695 

i4 

233o5 

85780 

21 

14220 

09085 

7 

909 1 5 

i3 

4a 

1 3  36 

46   24 

76712 

14 

23288 

85807 

21 

14193 

09094 

7 

90906 

12 

49 
5o 

i3  28 
7  i3  20 

46  32 
4  46  40 

76730 

i4 

23270 

85834 

22 

i4i66 

09104 

7 

90S  96 

1 1 
10 

9.76747 

10 

10.23253 

9.85860 

22 

10. i4i4o 

10.091 i3 

8 

9.90887 

5i 

i3  12 

46  48 

76765 

i5 

23235 

85887 

23 

i4ii3 

09122 

8 

9087S 

9 

h2 

i3  4 

46  56 

76782 

i5 

23218 

85913 

23 

14087 

ogi3i 

8 

90S69 

8 

53 

12  56 

47  4 

76S0O 

16 

23200 

85940 

24 

i4o6o 

09140 

8 

90860 

7 

^1 

12  48 

47  12 
4  47  20 

768,7 

lb 
16 

23i83 
io.23i65 

85967 

24 

i4o33 

09149 

8 

9085 1 

6 

7  12  4<> 

9.76835 

9.85993 

24 

10. 14007 

10.09158 

8 

9.90843 

30 

12  32 

47  28 

76S52 

17 

23i48 

86020 

25 

13980 

09168 

8 

90833 

4 

!37 

12  24 

47  36 

76870 

17 

23i3o 

86o46 

25 

13954 

09177 

9 

90S  2  3 

3 

i8 

12  16 

47  44 

768S7 

17 

23  m  3 

86073 

26 

13927 

09 1 86 

9 

90814 

2 

59 

12  8 

47  52 

76904 

17 

23096 

86100 

26 

13900 

09195 

9 

90805 

1 

7l 

12  0 

48  0 

76922 

18 

23078 

86126 

27 

13874 

09204 

9 

90796 

0 

Hour  F.. 11. 

Hi)ur.\..M. 

Cosine. 

Diir. 

Secant. 

Colang-eiit 

Diir. 

Tangent. 

Cosecant. 

Difl". 

Sine. 

125= 


A 

A 

B 

B 

C 

. 

1' 

2» 

4 

7 
2 

7 
10 
■i 

4s 

9 
i3 
5 

5^ 

II 

17 
6 

6^ 
i3 

20 
7 

7' 
16 

23 

8 

Prop,  parts  of  cols. 

i 

!• 

2 

3 
I 

C      54« 


TABLE  XXVIL 

# 

■s-' 

Log.  S 

mes,  Tangents,  and  Secants. 

C. 

3G 

A 

A 

B 

B 

C        C  143° 

vr 

0 

Hour  A.M. 
7  12  0 

Hour  P.M. 

Sine. 

Diff. 

Cosct-nnl. 

Tang;ent. 

I)itr.|Cotangent 

Secant. 

Diff. 

Cosine. 

60 

4  48  0 

9.76922 

0 

10.2307S 

9.86126 

0 

10.13874 

10.09204 

0 

9,90796 

I 

II  D2 

48  8 

76939 

0 

23o6i 

86 1 53 

0 

1 3847 

09213 

0 

90787 

59 

2 

1 1  /\A 

48  16 

76957 

I 

23o43 

86179,  1 

1 382  1 

09223 

0 

90777 

58 

3 

1 1  26 

48  24 

76974 

I 

23o26 

86206 

I 

13794 

09232 

0 

90768 

57 

4 
5 

II  28 

48  32 

76991 

I 

23009 

86u32 
9.86259 

2 

13768 

09241 

90759 

56 
55 

7  1 1  20 

4  48  4o 

9.770U9 

I 

10.22991 

2 

10. 1 3-41 

10.09250 

9.90750 

6 

1 1  12 

48  48 

77026 

2 

22974 

86285 

3 

i37i5 

09259 

9074 1 

54 

7 

II  4 

48  56 

77043 

2 

22957 

863 12 

3 

1 3688 

09369 

90731 

53 

8 

10  5v> 

49  4 

77061 

2 

22939 

86338 

4 

1 3662 

09278 

90722 

52 

_9 

10 

10  48 

49.  '2 

77078 

3 

22922 

86365 

4 

13635 

09287 

907 1 3 

5i 

5^ 

7  10  4o 

4  49  20 

9.77095 

3 

10.22905 

9.86392 

4 

io.i36o8 

10.09296 

2 

9  90704 

1 1 

10  Sa 

49  28 

77112 

3 

22888 

864 18 

5 

1 3582 

09306 

2 

90694 

49 

15 

10  24 

49  36 

.77i3o 

3 

22870 

86445 

5 

i3555 

09315 

2 

906H5 

48 

iJ 

10  16 

49  ^^ 

77147 

4 

22853 

86471 

() 

13529 

0932.4 

2 

90676 

4- 

i4 
i5 

10  8 

49  52 

77164 

4 

22836 

8649^ 

6 

i35o2 

09333 

2 

90667 

46 
45 

7  10  0 

4  5o  0 

9.77181 

4 

10.22819 

9.86524 

7 

10. 13476 

10.09343 

2 

9.90657 

1 6 

952 

5o  8 

77199 

5 

22S01 

86551 

7 

13449 

09352 

2 

90648 

AA 

17 

9  44 

5o  16 

77216 

5 

22784 

86577 

7 

1342  3 

09361 

3 

90639 

43 

iS 

9  36 

5o  24 

77233 

5 

22767 

866o3 

8 

13397 

09370 

3 

90630 

42 

20 

9  28 

5o  32 

77250 

5 

22750 
10.22732 

8663o 

8 
9 

13370 

09380 

3 

90620 

4i 
4o 

7  9  20 

4  5o  4o 

9.7726S 

6 

9.86656 

(0.13344 

10.093S9 

3 

9 . 906 1 1 

21 

9  12 

5o  48 

77285 

b 

22715 

86683 

9 

i33i7 

09398 

3 

90602 

3q 

22 

9  4 

5o  56 

77302 

6 

22698 

86709 

10 

13291 

0940S 

3 

90592 

38 

23 

8  56 

5i  4 

77319 

7 

22681 

86736 

10 

1 3264 

09417 

4 

9o583 

37 

24 
25 

8  48 

5 1  12 

77336 

7 

22664 

86762 

I  1 

i3238 

09426 

4 

90574 

36 
35 

7  8  40 

4  5i  20 

9.77353 

7 

10.22647 

9.86789 

1 1 

10. l3211 

10.09435 

4 

9.90565 

2(3 

8  32 

5i  28 

■77370 

7 

2263o 

868 1 5 

1 1 

i3i85 

09445 

4 

90555 

34 

27 

8  24 

5i  36 

77387 

8 

22613 

86842 

12 

i3i58 

09454 

4 

90546 

33 

28 

8  16 

5i  A^ 

774o5 

8 

22595 

86868 

12 

i3i32 

09463 

4 

90537 

32 

29 

3o 

8  8 

5i  52 

77422 

« 

22578 

86894 

i3 

i3io6 

09473 

5 

90527 

3i 

3^ 

780 

4  ^2   0 

9.77439 

9 

I0.2256l 

9.86921 

i3 

10. i3o79 

10.09483 

5 

9.90518 

Ji 

7  52 

52  8 

77456 

9 

22544 

86947 

i4 

i3o53 

09491 

5 

90509 

29 

i> 

7  44 

52  !6 

77473 

9 

22527 

86974 

i4 

1 3026 

09501 

5 

90499 

28 

33 

7  36 

52  24 

77490 

9 

225l0 

87000 

lb 

i3ooo 

09510 

5 

90490 

27 

34 
35 

7  28 

52  32 

77507 

10 

22493 

87027 

i5 

12973 

09520 

5 

90480 

26 

7  7  20 

4  52  40 

9.77524 

10 

10.22476 

9.87053 

i5 

10.12947 

10.09529 

5 

9.90471 

3() 

7  12 

52  48 

77541 

10 

22459 

87079 

16 

1 292 1 

09538 

6 

90462 

24 

37 

7  4 

52  56 

77558 

1 1 

22442 

87106 

16 

12894 

09548 

6 

90452 

23 

38 

6  56 

53  4 

77575 

1 1 

22425 

87132 

'7 

12868 

09557 

6 

90.443 

2  2 

39 

40 

6  48 

53  12 

77592 

1 1 

22408 

87158 

17 

12842 

09566 

6 

90434 

2  1 

20 

7  6  4o 

4  53  an 

9.77609 

'  11 

10.22391 

9.87185 

18 

10. 12815 

10.09576 

6 

9.90424 

4i 

6  32 

53  28 

77626 

12 

22374 

8721 1 

18 

12789 

09585 

6 

904 1 5 

19 

4? 

6  24 

53  36 

77643 

12 

22357 

87238 

18 

12762 

09595 

7 

90405 

18 

43 

6  16 

53  A/i 

77660 

12 

22340 

87264 

19 

12736 

09604 

7 

90396 

17 

44 
45 

6  8 

53  52 

77677 

i3 

22323 

87290 

19 

12710 

09614 

7 

9o386 
9.90377 

16 

75 

760 

4  54  0 

9.77694 

i3 

10.223o6 

9.87317 

20 

10.12683 

10.09623 

7 

40 

5  52 

54  8 

777" 

i3 

22289 

87343 

20 

12657 

09632 

7 

90368 

i4 

47 

5  44 

54  16 

7772S 

i3 

22272 

87369 

21 

1 263 1 

09642 

7 

9035s 

.  0 

48 

5  36 

54  24 

77744 

14 

22256 

87396 

21 

12604 

09651 

7 

90349 

12 

49 

5o 

5  28 

54  32 

77761 

14 

22239 

87422 

22 

12578 

09661 
in. 09670 

8 

90339 

1 1 

10 

7  5  20 

4  54  4o 

9.77778 

i4 

10.22222 

9.87448 

22 

10.12552 

8 

9.90330 

5r 

5  12 

54  48 

77795 

i5 

22205 

87475 

22 

12525 

09680 

8 

90320 

9 

52 

5  4 

54  56 

77812 

i5 

22188 

87501 

2  3 

12499 

09689 

8 

9c:>i  1 

8 

53 

4  56 

55  4 

77829 

j5 

2217I 

87527 

2-3 

12473 

09699 

8 

9o3o  I 

7 

54 
55 

4  48 

55  12 

77846 

ij 

22l54 

87554 

24 

12446 

09708 

« 

90292 
9.90282 

b 
5 

7  4  4o 

4  55  20 

9.77862 

16 

10.221 38 

9.87580 

24 

10.12420 

10.09718 

9 

5b 

4  32 

55  28 

77879 

16 

22121 

87606 

25 

12394 

097271  9 

90273 

4 

57 

4  24 

55  36 

77896 

16 

22104 

87633;  25 

12367 

097371  9 

90263 

3 

58 

4  16 

55  AA 

77913 

lb 

22087 

87659,  26 

12341 

09746|  9 

90254 

2 

59 

4  8 

55  52 

77930 

17 

22070 

87685,  26 

i23i5 

09756  9 

90244 

I 

bo 
M 

4  0 

56  0 

77946 

17 

22054 

8771 1  26 

12209 

09765  9 

90235 
Sine. 

0 

Hour  P.M.  Ilor.r  A.M. 

Cosine. 

Difr. 

Secant. 

Cotangent  DifT. 

Tangent. 

Cosecant.  Dili. 

■26° 


53' 


Seconds  of  time 

1' 

2' 

3' 

4. 

5» 

6= 

7a 

Prop,  parts  of  cols.  <  B 

2 
3 
I 

4 

7 
2 

6 

10 
4 

9 

i3 
5 

II 

17 
6 

i3 

20 
7 

i5 

23 

8 

I' 

ige  9^21 

TABLE  XXVn. 

I 

6'- 

Log.  S 

ines,  Tangents,  and  Secants. 

G'. 

:J7 

0 

0 

Hour  A. M 

A 

A 

B 

B 

C 

C  142^ 

Hour  P.M. 

Sine. 

Diff 

Cosecant. 

Tangent. 

Diir 

Cotangent 

Secant. 

Diir. 

Cosine. 

M 

6^ 

7  4  c 

4  56  0 

9.77946 

0 

10  22054 

9.87711 

0 

10. 12289 

10.09765 

0 

9.90235 

I 

3  52 

56  8 

77963 

0 

22007 

87738 

0 

12262 

09775 

0 

90225 

59 

2 

3  44 

56  16 

7798c 

I 

22020 

87764 

I 

12236 

09784 

0 

90216 

58 

J 

3  36 

56  24 

77997 

1 

2  2003 

8779a 

1 

12210 

09794 

0 

90206 

57 

4 
'5 

3  28 

56  32 

7S013 

I 

21987 

87817I  2 

I2i83 

09803 

90197 

56 

55 

7  3  20 

4  56  40 

9.7803c 

I 

10.21970 

9.87843   2 

10.12157 

10.09813 

9.90187 

b 

3  12 

56  48 

78047 

2 

21953 

87869  3 
878951  3 

12l3l 

09822 

90178 

54 

7 

3  4 

56  56 

78063 

2 

21937 

12105 

09832 

90168 

53 

8 

2  56 

57  4 

78080 

2 

21920 

87922 

3 

I207S 

09841 

90159 

5? 

9 

lO 

2  48 

57  12 

78097 

2 

21903 

87948 

4 

I2o52 

09851 

90149 

5i 

7  2  4o 

4  57  20 

9.78113 

3 

10.21887 

9.87974 

4 

10.12026 

10. 09861 

2 

9.90189 

II 

2  32 

57  28 

78i3o 

3 

21870 

88000 

5 

12000 

09870 

2 

90180 

4o 

12 

2  24 

57  36 

78147 

3 

21853 

88027 

3 

11973 

09880 

2 

901  20 

/iS 

iJ 

2  iT 

57  44 

78163 

4 

2,1837 

88o53 

6 

II  947 

09889 

2 

901 1 1 

47 

i4 
i5 

2  8 

57  52 

78180 

4 

21820 

88079 

6 

II  921 

09899 

2 

90 1 0 1 

46 
45 

720 

4  58  0 

9.78197 

4 

io.2i8o3 

9.88105 

7 

10. 1 1895 

10.09909 

2 

9 . 9009 1 

lb 

I  5? 

58  8 

78213 

4 

21787 

88i3i 

7 

11869 

09910 

3 

90082 

44 

17 

I  44 

58  16 

78230 

5 

21770 

881 58 

7 

II842 

09928 

3 

90072 

43 

iS 

I  36 

58  24 

78246 

b 

21754 

88184 

8 

I18I6 

09937 

3 

90063 

42 

12 

20 

I  28 

58  32 

78263 

b 

21737 

88210 

8 

1 1790 

09947 

3 

90053 

4i 

4o 

7  I  20 

4  58  4o 

9.78280 

5 

10.21720 

9.88236 

9 

10. 1 1764 

10.09957 

3 

9.90043 

21 

I  12 

58  48 

78296 

b 

21704 

88262 

9 

11738 

09966 

3 

90034 

39 

22 

1  4 

58  56 

783 1 3 

b 

21687 

88289 

10 

1171 1 

09976 

4 

90024 

88 

2j 

0  56 

59  4 

78329 

b 

2 1 67 1 

883 1 5 

10 

11 685 

09986 

4 

90014 

J/ 

24 
25 

0  48 

59  12 

78346 

7 

2 1 654 

8834 1 

10 

1 1659 

09995 

4 

90005 

36 
35 

7  0  4o 

4  59  20 

9.78362 

7 

[0.21638 

9. 88367 

1 1 

10.11633 

10. iooo5 

4 

9.89995 

2b 

0  32 

59  28 

7S379 

7 

21621 

88393 

1 1 

1 1607 

iooi5 

4 

899S5 

84 

27 

0  24 

59  36 

78395 

7 

2i6o5 

88420 

12 

ii58o 

10024 

4 

89976 

33 

28 

0  16 

5944 

7S412 

8 

2 1 588 

88446 

12 

11554 

ioo34 

5 

89966 

3^ 

29 

3o 

0  8 

59  52 

78428 

8 

21572 

88472 

i3 

ii528 

10044 

5 

89956 

3i 

3(1 

700 

5  0  0 

9-78445 

8 

I0.2I555 

9.88498 

i3 

10. 1 l502 

10.  ioo5'3 

5 

9.89947 

:ii 

6  59  52 

0  8 

78461 

9 

2 1 539 

88524 

1 4 

1 1476 

ioo63 

b 

89987 

20 

32 

59  44 

0  16 

7847S 

9 

21  j22 

8855o 

14 

ii45o 

10073 

b 

89927 

28 

JJ 

59  3b 

0  24 

7S494 

9 

2i5o6 

88577 

14 

ii423 

10082 

5 

89918 

27 

34 
35 

59  28 

0  32 

785io 

9 

21490 

886o3 

i5 

II 397 

10092 

b 

89908 

26 

25 

6  59  20 

5  0  4o 

9.78527 

10 

10.21473 

9.8S629 

i5 

10.11371 

10. 10102 

6 

9.89898 

3b 

59  12 

0  48 

78  543 

10 

21457 

88655 

16 

ii345 

10112 

6 

89888 

24 

^7 

59  4 

0  56 

78560 

10 

2i44o 

88681 

16 

ii3i9 

10121 

6 

89879 

28 

38 

58  56 

I  4 

78576 

10 

21424 

88707 

17 

1 1293 

ioi3i 

6 

89869 

22 

39 
4o 

58  48 

I  12 

78592 

1 1 

2i4o8 

88733 

17 

1 1267 

ioi4i 

6 

89859 

21 

20 

6  58  4o 

5  I  20 

9.78609 

1 1 

10.21391 

9.88759 

17 

10. 11241 

io.]oi5i 

6 

9.89849 

41 

58  32 

I  28 

78625 

It 

21375 

8S786 

18 

11214 

10160 

7 

89840 

'9 

42 

58  24 

I  36 

78642 

12 

2i358 

88812 

18 

1 1 188 

10170 

7 

89880 

18 

43 

58  16 

I  44 

78658 

12 

21342 

88838 

19 

1 1 162 

10180 

7 

89820 

17 

44 
45 

58  8 

I  52 

7S674 

12 

2 1 326 

88864 

>9 

1 1 136 

10190 

7 

89S10 
9.89801 

16 

i5 

6  58  0 

320 

9 . 7S69 I 

12 

io.2i3o9 

9.88890 

20 

10.111 10 

10.10199 

7 

40 

57  52 

2  8 

78707 

i3 

21293 

88916 

20 

11084 

10209 

7 

89791 

i4 

47 

57  44 

2  16 

78723 

1 3 

21277 

8S942 

20 

11058 

10219 

8 

897S1 

i3 

48 

57  36 

2  24 

78739 

i3 

21261 

88968 

21 

II032 

10229 

8 

89771 

12 

49 
5o 

57  28 

2  32 

78756 

i3 

2 1 244 

88994 

21 

1 1006 

10239 

8 

89761 

1 1 

6  57  20 

5  2  40 

9.78772 

i4 

10.21228 

9.89020 

22 

10.10980 

10.10248 

8 

9.89752 

HI 

'31 

57  12 

2  48 

7878S 

14 

21212 

89046 

22 

10954 

10258 

8 

89742 

9 

■32 

57  4 

2  56 

78805 

i4 

21 195 

89073 

23 

10927 

10268 

8 

89782 

W 

56  56 

3  4 

78821 

i5 

21179 

89099 

23 

10901 

10278 

9 

89722 

7 

54 

55 

56  48 

3  12 

78837 

i5 

21 163 

89125 

24 

10875 

10288 

9 

89712 

() 
~5 

6  56  4o 

5  3  20 

9.78853 

i5 

10.21 147 

9.89151 

24 

10.10849 

10.10298 

9 

9.89702 

5b 

56  32 

3  28 

7S869 

ID 

21  i3i 

89177 

24 

10823 

io3o7 

9 

89698 

4 

57 

56  24 

3  36 

78886 

lb 

2 1 1 1 4 

89203 

25 

10797 

io3i7 

9 

89688 

3 

58 

56  16 

3  44 

78902 

16 

2109S 

89229 

.25 

10771 

10327 

9 

89678 

2 

59 

56  8 

3   52 

78918 

16 

21082 

89255 

26 

10745 

10337 

10 

89668 

I 

bo 

56  0 

4  0 

78934 

18 

21066 

89281 

26 

1G719 

io347 

10 

89658 

0 
M 

M 

riourp.M.j 

[lour  A.M.  Cosine. 

DifT. 

Secant. 

Cotangent 

Ditr. 

Tangent.  | 

Cosecant. 

Diff. 

Sine. 

127° 


rj2« 


1' 

2' 

3' 

4. 

8 
18 
5 

5- 

10 
i5 
6 

6= 
12 

20 
7 

7° 
i4 

23 

8 

Prop,  parts  of  cols 

2 

3 
1 

4 
7 
2 

6 
10 
4 

TABLE  XXVIL 

[I'n;;e223 

S'. 

Log 

Sines,  Tan 

gents,  and  S 

ccants 

''". 

38° 

A 

A 

B 

B 

c      c  l4i°| 

0 

Hour  A.M. 

Hour  I'.M. 

Slue. 

Diir. 

Cosecant. 

Tnngeiil. 

Diff. 

Cotangent 

Secant. 

Diir.  Cosine. 

M 

6^ 

6  56  0 

5  4  0 

9.78934 

0 

10.21066 

9.89281 

0 

10. 10719 

,  ).io347 

0  :9. 89663 

I 

55  52 

4  8 

78950 

0 

2io5o 

89307 

0 

10693 

io357 

0  ;  89643 

D9 

3 

55  AA 

4  16 

78967 

I 

2io33 

89333 

I 

10667 

1 0367 

0 

89633 

bS 

3 

55  36 

4  24 

78983 

I 

21017 

89359 

I 

1 0641 

10376 

89624 

37 

4 

55  28 

4  32 

78999 

I 

21001 

89385 

2 

io6i5 

io386 

89614 

^6 

6^5 

5 

6  55  20 

5  4  4o 

9 . 790 1 5 

1 

10.20985 

9.8941 1 

2 

10.10689 

10.10396 

9 . 89604 

6 

55  12 

4  48 

7903 1 

2 

20969 

89437 

3 

10563 

I  o4o6 

89694 

64 

n 

55  4 

4  56 

79047 

2 

20953 

89463 

3 

10537 

io4i6 

89604 

63 

8 

54  56 

5  4 

79063 

2 

20937 

89489 

3 

1061 1 

10426 

89674 

b2r 

9 

54  48 
6  54  4" 

5  12 

79079 
9.79095 

2 

tj   2092 1 

89515 

4 

10485 
10. 10469 

1 04  36 

89664 

bi 
5o 

to 

5  5  20 

3 

10.20905 

9.89541 

4 

1 0 . 1 0446 

2 

9.89664 

1 1 

54  32 

5  28 

791 1 1 

3 

20889 

89567 

b 

10433 

10456 

2 

89644 

49 

12 

54  24 

5  36 

79128 

3 

20872 

89593 

b 

10407 

io466 

2 

89634 

48 

l3 

54  16 

5  44 

79144 

3 

20856 

89619 

Jb 

io38i 

10476 

2 

89624 

■4  / 

'4 
i5 

54  8 
6  54  0 

5  52 
5  6  0 

79 1 60 

4 

20840 

89645 

6 

io355 

J  0486 

2 

89614 

40 
46 

9.79176 

4 

10.20824 

9.S9671 

6 

10. 10329 

10.10496 

3 

9.89604 

iG 

53  52 

6  8 

79192 

4 

20808 

89697 

7 

io3o3 

io5o6 

3 

89496 

44 

17 

53  44 

6  16 

79208 

5 

20792 

89723 

7 

10277 

io5i6 

3 

89486 

4^' 

18 

53  36 

6  24 

79224 

5 

20776 

89749 

8 

10261 

10626 

3 

89476 

r) 

!9 
20 

53  28 

6  32 

79240 
9.79256 

5 

20760 

89775 

8 

10226 

io636 

3 
~3 

89466 
9.89466 

41 
4<'. 

6  53  20 

5  6  40 

5 

10.20744 

9.89801 

9 

10. 10199 

10. 10646 

21 

53  12 

6  48 

79272 

6 

20728 

89827 

9 

10173 

10666 

4 

89445 

39 

22 

53  4 

6  56 

79288 

6 

20712 

89853 

10 

10147 

10666 

4 

89435 

38 

23 

52  56 

7  4 

79304 

6 

20696 

89879 

10 

10121 

10675 

4 

89426 

^7 

24 

25 

52  48 

7  12 

79319 

6 

20681 

89906 

10 

10096 

io585 

4 

89416 

35 

6  52  4o 

5  7  20 

9.79335 

7 

io.2o665 

9.89931 

II 

10. 10069 

10. 10696 

4 

9.89406 

2b 

52  32 

7  28 

79351 

7 

20649 

89957 

II 

10043 

10606 

4 

89396 

11 

27 

52  24 

7  36 

79367 

7 

20633 

89983 

12 

10017 

1 061 5 

b 

89386 

ii\ 

28 

52  16 

7  44 

79383 

7 

20617 

90009 

12 

09991 

10626 

b 

89376 

32 

29 

3o 

52  8 

7  52 

79^99 

8 

20601 

90035 

i3 

09966 

io636 

b 

89364 

3i 
3c 

6  52  0 

5  8  0 

9.79415 

8 

io.2o585 

9 . 9006 1 

i3 

I p. 09939 

10.10646 

5 

9.89354 

3i 

5 1  52 

8  8 

79431 

8 

20569 

90086 

i3 

09914 

10666 

5 

89344 

29 

32 

5 1  44 

8  16 

79447 

8 

20553 

901 1 2 

i4 

09888 

10666 

b 

89334 

28 

33 

5i  36 

8  24 

79463 

9 

20537 

90 1 38 

i4 

09862 

10676 

6 

89324 

27 

34 

5[  28 
6  5i  20 

8  32 

79478 

9 

20522 

90164 

lb 

09836 

1 06S6 

b 

8y3  1 4 

26 
26 

35 

5  8  4o 

9.79494 

9 

io.2o5o6 

9.90190 

i5 

10.09810 

10. 10696 

6  l9.';93o4 

30 

5r  12 

8  48 

79510 

10 

20490 

90216 

16 

09784 

10706 

ti  '  89594 

>A 

37 

5i  4 

8  56 

79526 

10 

20474 

90242 

16 

09768 

10716 

6  '  89284 

2  3 

38 

5o  56 

9  4 

79542 

10 

20458 

90268 

16 

09732 

10726 

6 

8^274 

22 

39 

40 

5o  48 

9  12 

79558 

10 

20442 

90294 

17 

09706 

10736 

7 

89264 

21 

20 

6  5o  ^0 

5  9  20 

9.79573 

II 

10.20427 

9.90320 

17 

10.09680 

10. 10746 

7  I9.S9264 

41 

5o  32 

9  28 

79589 

II 

2o4l  1 

90346 

18 

09654 

10766 

7 

89244 

'9 

42 

5o  24 

9  36 

79605 

1 1 

20395 

90371 

18 

09629 

10767 

7 

89233 

18 

43 

5o  16 

9  M 

79621 

1 1 

20379 

90397 

19 

09603 

10777 

7 

89223 

'7 

44 
45 

5o  8 

9  52 

79636 

12 

2o364 

90423 

'9 

09677 

10787 

7 

89213 

lb 
75 

6  5o  0 

5  10  0 

9.79652 

12 

10.20348 

9.90449 

19 

10.09661 

10. 10797 

8 

9.89203 

40 

49  52 

10  8 

79668 

12 

20332 

90475 

20 

09626 

10S07 

8 

89193 

i4 

47 

49  44 

.0  .6 

79684 

12 

2o3i6 

9o5oi 

20 

09499 

1 08 1 7 

8 

89183 

i3 

48 

49  3(i 

10  24 

79699 

i3 

2o3oi 

90527 

21 

09473 

10827 

8 

89173 

12 

49 
5o 
5. 

49  38 

10  32 

79715 

i3 

20285 

90553 

21 

09447 

io838 

8 

89162 

1  1 

10 

6  49  20 

5  10  40 

9.79731 

i3 

10.20269 

9.90678 

22 

10.09422 

10.1 084s 

8  9.89162 

49  12 

10  48 

79746 

i4 

20254 

90604 

22 

09396 

10858 

9   89142 

9 

52 

49  4 

10  56 

79762 

i4 

20238 

9o63o 

22 

09370 

10868 

9 

89132 

8 

53 

48  56 

II  4 

79778 

i4 

20222 

90666 

23 

09344 

10878 

9 

89122 

7 

54 
55 
56 
57 
58 

48  48 

II  12 

79793 

i4 

20207 

90682 

23 

09318 

10888 

9 

891 12 

6 
"5 

6  48  4o 

5  1 1  uo 

9.79809 

.5 

I0.20I9I 

9 . 90708 

24 

10.C9292 

10. 10899 

9  9.89101 

43  32 

11  a8 

79825 

lb 

20175 

90734 

24 

O920tj 

1 0909 

9 

89091 

4 

48  24 

II  36 

7984(J 

lb 

20160 

m? 

25 

09241 

10919 

10 

89081 

^ 

4*  16 

11  44 

79856 

lb 

20 1 44 

r^K 

002 1 5 

10929 

10 

89071 

2 

60 
M 

48  8 

11  52 

79872 

i6 

20128 

908 1 1 

26 

09189 

10940 

10 

89060 

1 

48  0 

12  0 

79887 

lb 

20Il3 

90837 

26 

09 1 63 

10960 

10 

89060 

"SX 

Hour  P.M. 

flour  A.M. 

Cosliio. 

DiiT. 

Scrniit. 

Cntnno^ent 

Diir. 

Tangent. 

Cosecant.  DilT. 

Sine. 

128° 


A 

A 

B 

B 

C 

P 

Os 

3' 

4' 

5» 

6^ 

7" 

f^ 

2 

4 

6 

8 

10 

12 

i4 

Prop  parts  of  cols.  < 

P 

3 

6 

10 

i3 

16 

19 

23 

fc 

I 

3 

4 

5 

6 

8 

9 

Page  224] 

TABLE  XXVIL 

5''. 

Log.  Sines,  Tangents,  and  Secpails. 

G'. 

39°   • 

A 

A     B 

B 

C 

C  140^ 

M 

0 

1 

Hour  A.M. 

Ilourr.M. 

Sine. 

Dinr. 

Cosecant. 

Tangent. 

Diir. 

Cotangent 

.Secant. 

Diff. 

Cosine. 

M 

60 
5g 

6  48  0 
47  52 

5  12  0 
12  8 

9.79887 
79903 

•0 
0 

10.201 i3 
20097 

9.90837 
90863 

0 
0 

10.09163 
09137 

10.10950 
1096',) 

0 
0 

9.89050 
S9040 

2 

47  44 

12  '.6 

79918 

I 

20082 

90889 

I 

091 1 1 

10970 

0 

89030 

58 

J 

47  36 

12  24 

79934 

1 

20066 

90914 

1 

09086 

10980 

80020 

57 

4 
5 

47  28 
6  47  20 

12  32 

79950 

1 

20o5o 

90940 

3 

09060 
10.09034 

1 099 1 

I  1  89009 

56 
55 

5  12  40 

9.79965 

I 

io.2oo35 

9 . 90966 

7 

10.11  0( )  1 

9.88999 

b 

47  12 

12  48 

79981 

2 

20019 

90992 

3 

09008 

1 1 0 II 

88989 

54 

7 

47  4 

12  56 

79996 

2 

200011 

91018 

3 

08982 

1 1022 

8S978 

53 

•  8 

46  56 

i3  4 

8001  2 

2 

19988 

9 1 04  3 

3 

08957 

Ilo3i 

8806S 

5^ 

_9 

io 

46  48 
6  /i&  4o 

i3.  12 

80027 

2 

19973 

91069 

_,  0893 1 

1 1042 

2  !  88o58 

5i 
r- 

5  i3  20 

9.80043 

3 

10. 19937 

9.91095 

'4 

10.08905 

10. 1 11)52 

2  9  .8E948 
2  ,  88937 

11 

46  32 

i3  28 

8oo58 

3 

19942 

91 121 

6 

08879 

1  io63 

40 

12 

46  24 

1 3  36 

80074 

3 

19926 

91147 

5 

08853 

11073 

2  1  88027 

48 

IJ 

AQ   16 

i3  AA 

800S9 

3 

.19911 

91172 

6 

0882S 

iio83 

a 

88917 
88906 

9  88896 

'17 

i4 
i5 

46  8 

i3  52 

8oio5 

4 

19895 

91 198 

6 

08802 

1 1094 

2 

46 
45 

6  46  0 

5  i4  0 

9.801 20 

4 

10.19880 

9.91224 

6 

10.08776 

1 0 . 1 1 1 04 

3 

lb 

45  52 

r4  8 

801 36 

4 

19864 

91250 

7 

08750 

1 1 1 1 4 

3 

88886 

44 

17 

45  44 

i4  16 

8oi5i 

4 

19849 

91276 

7 

08724 

III25 

3 

88S75 

43 

i8 

45  36 

i4  24 

80166 

5 

19834 

9i3oi 

8 

08609 

iii35 

3 

88865 

42 

!9 

20 

45  28 

i4  32 

80182 

5 

19818 

91327 

8 

08673 

1 1145 

3 

88855 

4i 
4o 

6  45  20 

5  i4  4o 

9.80197 

5 

10. 19803 

9.91353 

9 

10.0S647 

1 0 . 1 1 1 56 

3 

9.88844 

21 

45  12 

i4  48 

8021 3 

5 

19787 

.  91379 

9 

08621 

1 1 1 66 

4 

88834 

3q 

22 

45  4 

!4  56 

80228 

b 

19772 

91404 

9 

08596 

11176 

4 

8S824 

38 

2j 

44  5b 

■  5  4 

80244 

() 

19756 

9i43o 

10 

08570 

1 1 187 

4 

8881 3 

37 

24 
25 

M   48 

ID  12 

80259 

b 

19741 

91456J  10 

08544 
io.o85iS 

1 1 197 

4 

888o3 

36 
35 

6  Ai  40 

5  i5  2u 

9.80274 

6 

10. 19726 

9.91482 

11 

10.11207 

4 

9.88790 

2b 

44  32 

i5  28 

80290 

7 

19710 

91507 

II 

08493 

11218 

5 

88782 

34 

27 

A^   24 

i5  36 

8o3o5 

7 

19695 

91533 

12 

08467 

1 1228 

5 

88772 

33 

28 

AA   16 

1 5  44 

8o32o 

7 

19680 

91559 

12 

08441 

1 1239 

5 

88761 

32 

29 

3o 

AA    8 

1 5  52 

8o336 
9.80351 

7 
~8" 

19664 

9i585 

12 

084 1 5 

II 249 

5 

88751 

3i 

3«3 

6  44  0 

5  16  0 

10. 19649 

9.91610 

i3 

10.08390 

10,11259 

5 

9.88741 

Ji 

43  52 

16  8 

8o366 

8 

19634 

9 1 636 

i3 

08364 

1 1 270 

5 

88730 

29 

i2 

43  44 

16  16 

8o3S2 

8 

19618 

91662 

i4 

08338 

1 1280 

6 

88720 

28 

J  J 

43  36 

16  24 

80397 

8 

1 9603 

91 688 

i4 

o83i2 

11291 

6 

88709 

27 

M 
35 

43  28 
6  43  20 

16  32 

8o4 1 2 

9 

19588 

91713 

i5 

08287 

ii3oi 

6 

88699 

26 

25 

5  16  4<> 

9.80428 

10. 19572 

9.91739 

i5 

10.08261 

!0.1l3l2 

6 

9.88688 

3b 

43  12 

16  48 

80443 

9 

19557 

91765 

i5 

08235 

Il322 

6 

88678 

23 

37 

43  4 

16  56 

8o458 

9 

19542 

91791 

lb 

08209 

1 1 332 

6 

88668 

23 

3b 

42  56 

•7  4 

80473 

H) 

'.9527 

91816 

lb 

08184 

1 1 343 

7 

88657 

22 

39 
4o 

42  48 

17  12 

80489 

10 

1951 1 
1 0 . 1 9496 

91842 

17 

081 58 

11353 

7 

88647 
9.88636 

21 

20 

6  42  4" 

5  17  20 

9.8o5o4 

10 

9.91868 

17 

io.o8i32 

io.ii36-i 

7 

4i 

42  i-j 

,7  28 

8o5i9 

10 

1 948 1 

91893 

18 

08107 

1 1 374 

7 

88626 

19 

42 

42  24 

17  36 

8o534 

1  I 

19466 

91919 

18 

08081 

ii385 

7 

8861 5 

18 

4J 

42  16 

17  44 

SoS'o 

1  I 

19450 

91945 

18 

o8o55 

1 1395 

7 

88605 

17 

44 

45 

42  8 

17  52 

8o565 

1  1 

19435 

91971 

19 

08029 
10.08004 

1 14'')6 

8 

88594 

16 

i5 

6  42  0 

5  18  0 

9.8o58u 

12 

10. 19420 

9.91996 

19 

to. ii4i6 

8 

9.88584 

4b 

4i  52 

18  8 

80595 

12 

19405 

92022 

20 

07978 

1 1427 

8 

88573 

.4 

47 

4 1  44 

18  16 

8u6io 

12 

19390 

92048 

20 

07952 

1 1437 

8 

88563 

ij 

48 

4 1  36 

18  24 

80625 

12 

19375 

92073 

21 

07927 

1 1 448 

8 

88552 

12 

49 
5o 

4 1  28 

18  32 

8064 1 
9.80656 

l3 

19359 

92099 

21 

07901 

ii458 

9 

88542 

1 1 

11) 

6  4i  20 

5  18  40 

10.19344 

9.92125 

21 

10.07875 

10. 1 1469 

9 

■9.88531 

5i 

4i  12 

18  48 

8067 1 

1-! 

19329 

92 1 5o 

22 

07850 

11479 

9 

88521 

9 

h2 

4i  4 

18  56 

806S6 

1 3 

19314 

92176 

22 

07824 

1 1490 

9 

885 10 

8 

:)3 

4o  56 

19  4 

80701 

1 4 

19299 

92202 

23 

07798 

ii5oi 

9 

88499 

7 

;>4 
55 

4o  48 

19  12 

80716 

1 4 

19284 

92227 

23 

07773 
10.07747 

ii5i  1 

9 

8S4^i9 

b 

6  4o  4o 

5  19  20 

9.80731 

i4 

10.19269 

9  92253 

24 

10. 11 52 2 

10 

9.88478 

5b 

4o  32 

19  28 

80746 

14 

19254 

92279 

24 

07721 

1 1 532 

10 

88468 

4 

37 

4o  24 

19  36 

80762 

1 5 

19238 

92304 

24 

07696 

1 1 543 

10 

88457 

3 

38 

4o  16 

19  ^^ 

80777 

i5 

19223 

92330 

25 

07670 

ii553 

10   8i447 

2 

59 

4o  8 

19  52 

80793 

i5 

19208 

92356 

25 

07644 

1 1 561 

10  .     88436 

1 

bo 

4o  0 

20  0 

80807 

i5 

19193 

92381 

26 

07619 

1 1 575 

10  ;   8842  5 

0 

Hour  P.M. 

Hour  A.M. 

Cosine. 

DilT. 

Secant. 

Cotang-enl[Difir. 

Tangent. 

Cosecant. 

DiiT.  Sine. 

129" 


V.       5(/ 


Seconds  of  lime 

1' 

2» 

4 
6 

3 

3^ 

6 

10 

4 

4s 

8 
,3 

5 

5- 

10 

16 

12 

'9 
8 

7^1 
.  1 

Frop.  parts  of  cols.  I   B 
»  C 

2 

3 
I 

13 

23 

__9_| 

S'. 
40° 


TABLE   XXVII. 

Log.  Sines,  Tangents,  and  Secants. 
A  °      A      B        B 


i3 


IIourA.M  iHourp.M. 


23 
24 
25 
26 

27 
28 

2y 

So 
3i 

32 

33 
34 
35 
36 
37 
38 
39 

4o 
4i 

42 

43 
44 
45 
46 
47 
48 
49 
5o 
5i 

52 

53 
54 
55 
56 

57 
58 


6  4o  o 
39  52 

39  44 
39  36 
39  28 


6  39  M) 
39  12 

39  4 

3H  5(. 
38  48 


20  o 
20  8 
20  16 

20  24 

20  3:> 


6  38  4'J 
38  32 
38  24 
38  16 
38  8 


20  4ii 
20  48 

20  5() 

21  4 
21  12 


5  21 


20 

!  I  28 

M  36 
n  44 

52 


21 


6  38  o\ 
37  52 
37  44 
37  36 
37  28 

6  37  20 
37  J  2 
37  4 
36  56 
36  48 


5  22  o 

22  8 

32  16 

2  2  S4 

22  32 


6  36  40 
36  32 
36  24 
36  16 
36  8 


6  36  o 
35  52 
35  44 
35  3() 
35  28 


35  20 
35  12 
35  4 
34  56 
34  48 


34  4o 
34  32 

34  26 
34  16 
34    8 


5  22  4" 

22  48 

22  56 

23  4 

23    12 

5  23  20 
23  28 
2  3  36 
23  44 

23    52 


5  24  o 
24  8 
24  16 
24  24 
24   32 


Siiio 


9.80811" 
80S  2  2 
8:  .837 

808  5  2 

80867 
80882 

80897 

809  I  2 

80927 
80942 


Difl: 
o 


Cosecant. 


9.81254 
8 1  269 
81284 
81  299 
8i3i4 


24  4" 
24  48 

24  56 

25  4 
25  12 


34  o 
33  52 
33  44 
33  36 
33  2b 


33  20 
33  12 
33  4 
32  56 
32  4b 


32  4o 
32  3 
32  24 
32  16 

32  8 

32 


M   Hour  p.. ir.  Hour  A. M 


5  25  2fi 

25  28 
25  36 
25  44 

25  52 

5  26  o 

26  8 
26  16 
26  24 
26  32 


26  4" 
26  48 

26  56 

27  .4 
27  12 


10.18968 
18953 
18939 
18924 
18909 

10.18894 


9.81328 
81343 
81 358 
81372 
81387 

9.81402 
81417 
8i43i 
81446 
81461 


9.81475 
81/190 
8i5o5 
8i5i9 
8i53 


27  20 
27  28 
27  36 

27  44 

27  52 

28  o 


9.81549 
81 563 
81578 
81592 
81607 

9.81622 
8 1 636 
8i65i 
81 665 
81680 
81694 
Cosine 


18864 
18849 
1 883 


10.1S820 
i88o5 
18790 
18775 
18760 


10.18746 
18731 
18716 
18701 
18686 


Taiiffent.  Diir 


.92381 
92407 
92433 
93458 
9248 1 


.92510 
92535 
92  56 1 
925S7 
9261  2 


9.92638 
92663 
9.689 
927 
92740 


Cotanafcnl 


4 

5 
5 
6 
6 

9.92766'  6 

7 
7 
8 
8 


10.07619 
07593 
07567 
07542 
07516 


10.07490 
07465 
07439 
0741 3 
07388 


10.07362 
07337 
0731 1 
07285 
07260 


92792 
928 1 7 
9^843 
92868 

9.92894 
92920 
92945 
92971 
92996 


9.93022 
93048 
93073 
93099 
93124 


10.18672 
18657 
18642 
18628 
i86i3 


10.18598 
i8583 
18569 
18554 
18539 


DitT. 


10.18525 
i85io 
18495 
i848i 
18466 

10. i845i 
18437 
18422 
i84o8 
18393 

10.18378 
1 8  364 
18349 
18335 
i832o 
t83o6 


10.07234 
07208 
07183 
07157 
07132 


o . 07 1 06 
07080 
07055 
07029 
07004 


10.06978 
0695? 
06927 
06901 

06876 


Serant. 


,11575 
ii585 
1 1596 
1 1 606 
1 1617 


[Page  SW."! 
G  . 

_C_139° 

Diir?  Cosine   jTl 

6? 


10. 1 1628 
1 1 638 
1 1649 
1 1 660 
1 1670 


10. 1 1681 
1 1692 
1 1702 
11713 
11724 


10. 1 1734 
1 1745 
1 1756 
1 1766 

1177- 


10.11788 
1 1 799 
1 1 809 
1 1820 
ii83 


io.o6S5o 
06825 
06799 
06773 
06748 


9.93406 
93431 
93457 
93482 
93508 


9.93533 
93559 
93584 
93610 
93636 


9.93661 
93687 
93712 
93738 
93763 


Secant. 


9.93789 
93814 
93f 
93865 


93916 


10.06722 
06697 
0667 1 
06646 
06620 


10.06594 
06569 
06543 
o65i8 
06492 


1 o . 06467 

.  0644 1 

064 16 

06390 

o6364 


0.1 1842 
ii852 
1 1 863 
1 1874 
1 1 885 


9.88425 
884i5 
884o4 
88394 
88383 

9.88372 
88362 
8835 1 
88340 
8833o 

9  883 1 9 
883o8 
88298 
88287 
88276 


9.88266 
88255 
88244 
88234 
88223 


9.882 12 
882C1 
8819- 
8818^ 
88169 


9.88i58 
881 48 
88 1 37 
88126 
8811 5 


10. 1 1895 
1 1906 
11917 
1 1928 
11939 


[o.i 1949 
1 1 960 
1 197 1 
1 1982 
11993 


I o . 1 2004 
120 

12025 

1 2o36 
1 2047 


Cotansrent 


10.06339 
o63i3 
06288 
06262 
06237 


23 
23 

~^ 

24 
24 

25 
2  5 

26 

Difil".'  Tansjent. 


10.062  1 1 
06186 
06 1 60 
o6i35 
06109 
06084 


10. 1 2o58 
1 2069 
1 20S0 
1209 
12102 


9.88105 
88094 
8808  3 
88072 
88061 


9.8805 1 
88<->4o 
88039 
88018 
88007 

87996 
87985 
87975 
87964 
87953 


8  I9. 87942 

8  :  87931 
87920 
87909 
8789S 


9.S78S7 

87877 
87866 
87855 
8784. 


9.87833 
87822 
878 1 1 
87800 
87789 
87778 


Cosecant.  Dill'.     .Sine. 


130° 


A 


B 


B 


vy 


Seconds  of  time 

1' 

2^ 

3^ 

4^ 

7 
i3 
5 

5^ 

9 
16 

6' 
1 1 

'9 

7^ 

i3 

22 

I'rop.  parts  nf  cols.  <   B 

f  C 

2 

3 

! 

4 
6 
3 

6 
10 
A 

S'. 


TABLE   XXVII 

Log.  Sines,  Tangents,  and  Secanls. 


41 
M 
o 

0 

A 

A 

B 

B 

C 

C  138° 

Hour  A. M 

Hour  P.M. 

Sine. 

DiflT 

Cosecant. 

Tangent. 

Diir. 

Cotangent 

Secant. 

Difl- 

Cosine. 

60 

6  32  0 

5  a8  0 

9.81 694 

0 

I c . 1 83o6 

9.9391b 

0 

10.06084 

10. 12222 

0 

9.87778 

I 

3i  52 

28  8 

8 1 709 

0 

i8?9i 

93942 

0 

o6o58 

12233 

0 

87767 

5q 

2 

3i  44 

28  16 

8i7i>3 

0 

18277 

93967 

1 

o6o33 

12244 

0 

87756 

58 

3 

3 1  3b 

28  24 

8  i  73fe 

1 

18262 

93993 

I 

06007 

12255 

87745 

57 

4 
5 

3i  38 

28  32 

81752 

1 

18248 
ic. 18233 

94018 

2 

05983 

72266 

87734 

5o 
55 

6  3i  20 

5  28  40 

9.81767 

I 

9.94044 

2 

10.05956 

10.12277 

9.87723 

6 

3i  12 

28  48 

81781   I 

18219 

94069 

3 

05931 

12288 

87712 

54 

7 

3i  4i   28  56 

81796   2 

18204 

94095 

3 

05905 

12299 

87701 

53 

8 

3o  56-   29  4 

81810   2 

18190 

94 1 20 

3 

o588o 

i23io 

8^690 

53 

_? 

10 

3o  48 

29  12 

81825 

2 

18175 

94 1 46 

4 

o5854 

12321 

2 

87679 

5i 
5<. 

6  3o  4o 

5  29  20 

9.81839 

2 

10.18161 

9.94171 

4 

10.05829 

10.12332 

2 

9 .  87668 

1 1 

3o  32 

29  28 

8i854 

3 

i8i46 

94 '97 

5 

o58o3 

12343 

2 

87657 

4o 

12 

3o  24 

29  36 

81868 

3 

i8i32 

94222 

5 

05778 

12354 

2 

87646 

48 

i3 

3o  16 

29  44 

81882 

3 

18118 

94248 

b 

05753 

12365 

2 

87635 

47 

i4 
i5 

3o  8 

29  52 

81897 

3 

i8io3 

94273 

b 

05727 

12376 

3 

87624 

46 
45 

6  3o  V. 

5  3o  0 

9.81911 

4 

10.18089 

9.94299 

6 

10.05701 

10.12387 

3 

9.87613 

i6 

29  5? 

3o  8 

81926 

4 

18074 

94324 

7 

05676 

12399 

3 

87601 

44 

17 

29  44 

3o  16 

81940 

4 

18060 

94350 

7 

o565o 

I24IO 

3 

87590 

43 

i8 

29  36 

3o  24 

81955 

4 

18045 

94375 

8 

o5625 

1 2421 

3 

87579 

42 

19 

20 

29  28 

3o  32 

81969 

5 

i8o3i 

94401 

8 

05599 

12433 

4 

87568 

4i 

4o 

6  29  20 

5  3o  4o 

9.81983 

5 

10. 18017 

9.94426 

8 

10.05574 

10. 13443 

4 

9.87557 

21 

29  12 

3o  48 

8199S 

5 

18002 

94452 

9 

o554S 

12454 

4 

87546 

39 

22 

29  4 

3o  56 

82012 

5 

17988 

94477 

9 

o5523 

1 3465 

4 

87535 

38 

2j 

28  56 

3i  4 

82026 

5 

17974 

945o3 

10 

05497 

12476 

4 

87524 

37 

24 
25 

28  48 

3i  12 

82041 

6 

17959 

94528 

10 

05472 

1 2437 

4 

87513 

36 
35 

6  28  4o 

5  3i  20 

9.82055 

b 

10.17945 

9.94554 

1 1 

io.o5446 

10. 12499 

5 

9.87501 

26 

.28  32 

3i  28 

82069 

6 

17931 

94579 

1 1 

05421 

I25lO 

5 

87490 

34 

27 

28  24 

3i  36 

82084 

6 

17916 

94604 

1 1 

05396 

I252I 

^ 

87479 

33 

28 

28  16 

3i  44 

82098 

7 

17902 

9463o 

12 

05370 

12532 

5   87468 

32 

29 

28  8 

3i  52 

82112 

7 

17^88 

94655 

12 

05345 

12543 

5 

87457 

3i 
3o 

6  28  0 

5  32  0 

9.82126 

7 

10.17S74 

9.94681 

i3 

10.05319 

10. 13554 

6 

9-87446 

27  52 

32  8 

82141 

7 

17859 

94706 

i3 

05294 

12  566 

6 

87434 

29 

32 

27  44 

32  16 

82155 

8 

17845 

94732 

i4 

05368 

12577 

b 

87423 

28 

33 

27  36 

32  24 

82169 

8 

17831 

94757 

i4 

05243 

12588 

b 

87412 

27 

34 
35 

27  28 

32  32 

82184 

8 

17816 

947S3 

i4 

05217 

12599 

6 

87401 

26 
l5 

6  27  20 

5  32  40 

9.82198 

8 

10. 17802 

9.94808 

i5 

10.05193 

[o. 12610 

7 

9.87390 

3b 

27  12 

32  48 

82212 

9 

17788 

94834 

i5 

o5i66 

12622 

7 

87378 

24 

37 

27  4 

32  56 

82226 

9 

"7774 

94859 

16 

o5i4i 

12633 

7 

87367 

23 

38 

26  56 

33  4 

82240 

9 

17760 

94884 

16 

o5ii6 

12644 

7 

87356 

23 

39 

4o 

26  48 

33  12 

82355 

9 

17745 

94910 

17 

05090 

12655 

_7_ 

7 

87345 

21 

20 

6  26  4o 

5  33  20 

9.82269 

10 

10.17731 

9.94935 

17 

io.o5()65 

1 0 . 1 2666 

9.87334 

4 1 

26  32 

33  28 

82283 

10 

17717 

94961 

17 

o5o39 

12678 

8 

87322 

19 

42 

26  24 

33  36 

82297 

10 

17703 

94986 

18 

o5oi4 

12689 

8 

87311 

18 

43 

26  16 

33  44 

82311 

10 

17689 

95012 

18 

04988 

12700 

8 

87300 

17 

44 
45 

26  8 

33  52 

83326 

10 

17674 

95o37 

'9 

04963 

12712 

8 

87388 

lb 
i5 

6  26  0 

5  34  0 

9.82340 

II 

10.17660 

9.95062 

19 

10.04938 

10.12723 

8 

9.87377 

4b 

25  52 

34  8 

82354 

II 

17646 

95088 

20 

04912 

12734 

9 

87366 

14 

47 

25  44 

Z4   16 

82368 

1 1 

17632 

95ii3 

20 

04887 

12745 

9 

87255 

1 3 

48 

25  36 

34  24 

82382 

1 1 

17618 

95139 

20 

o486 1 

12757 

9 

87243 

13 

49 
5o 

25  28 

34  32 

82396 

13 

17604 

95164 

21 

o4836 

12768 

9 

87233 

1  1 

10 

6  25  20 

5  34  4o 

9.8241Q. 

12 

10.17590 

9 . 95 1 90 

21 

10.04810 

10.12779 

9 

9.87321 

bi 

25  12 

34  48 

82424 

12 

17576 

95215 

22 

04785 

12791 

10 

87209 

9 

b2 

25  4 

34  56 

82439 

12 

17561 

95240 

22 

0476a 

12802 

10 

87198 

8 

53 

24  56 

35  4 

82453 

l3 

17547 

95266 

22 

04734 

12813 

10 

87187 

7 

54 
55 

24  48 

35  12 

82467 

i3 

17533 
10.17519 

95291 

23 

04709 

12825 

10 

87175 

b 
■5 

6  24  4<> 

5  35  20 

9.8248] 

i3 

9.95317 

23 

I0.04683 

10. 12836 

10 

9.87164 

5b 

24  3i 

35  28 

82495 

1 3 

i75o5 

95343 

24 

04658 

12847 

10 

8-i53 

4 

57 

24  -M 

35  36 

82509 

i4 

17491 

95368 

24 

04632 

I28f9 

II 

87141 

3 

58 

14   16   35  44 

82523 

14 

17477 

95393 

25 

04607 

12870 

II 

87130 

2 

^9 

24  8   35  52 

82537 

i4 

17463 

95418 

25 

04582 

12881 

1 1 

87119 

I 

bo 
M 

24  0   36  0 

82551 

i4 

17449 

95444 

25 

04556 

12893 

11 

87107 

0 
M 

Hourp.M.  IIourA.M. 

Cosine. 

Diff. 

Secant. 

Cotangent 

Diff. 

Tangent. 

Cosecant. 

Diff. 

Sine. 

131' 


A 

A 

B 

B 

C 

Seconds  of  time 

V 

2' 

3^ 

4» 

5» 

6' 

7' 

Prop,  parts  of  eols. 

2 

3 

4 
6 
3 

5 

10 
4 

7 
i3 
6 

9 
16 

7 

1 1 

•9 
8 

12  1 
22  1 
m  : 

4S' 


TABLE    XXVII. 


40^ 


A 


Sines,  Tan 
A 


gents,  and  Secants. 
B  B 


[Page  2a7 
G\ 

C    137° 


si 

0 

IIourA.M.]HourP.M. 

Sine, 

DilT. 

Cosecant. 

Tangent. 

DifT.  Cotangent 

Secant. 

DifT. 

Cosine. 

60 

6  24  o|  5  36  0 

9.8255r 

0 

10.17449 

Q. 95444 

0 

10.04556 

10.12893 

0 

9.87107 

1 

23  52   36  8 

82S65 

0 

17435 

'  95469 

0 

0453 1 

12904 

0 

87096 

59 

2 

23  44   36  16 

82579 

0 

17421 

95495 

I 

o45o5 

12915 

0 

87085 

58 

3 

23  36   36  24 

82593 

I 

17407 

95520 

I 

o448o 

12927 

87073 

57 

4 
'5 

23  28   36  32 

82607 

I 

17393 

95545 

2 

04455 

12938 

I 

87062 

Dbt 
55 

8  23  20 

5  36  4o 

9.82621 

1 

10.17379 

9.95571 

2 

10.04429 

10. 12950 

J 

9.87060 

6 

23  12 

36  48 

82635 

I 

17365 

95596 

3 

o44o4 

12961 

87039 

54 

7 

23  4 

36  56 

82649 

2 

1 735 1 

95622 

3 

04378 

12972 

87028 

53 

s 

22  56 

37  4 

82663 

2 

17337 

95647 

3 

04353 

12984 

2 

87016 

52 

_9 
10 

22  48 

37.2 

82677 

2 

17323 

95672 

4 

04328 

12995 

2 

87005 

5i 
57) 

6  S2  4o;  5  37  20 

9.82691 

2 

10.17309 

9.95698 

4 

10.04302 

10. 1 3007 

2 

9.86993 

1 1 

22  32 

37  28 

82705 

3 

17295 

95723 

5 

04277 

i3oi8 

2 

86982 

49 

t2 

22  24 

37  3(i 

82719 

3 

17281 

95748 

5 

04252 

i3o3o 

2 

86970 

48 

i3 

22  16 

37  M 

82733 

3 

17267 

95774 

5 

04226 

i3o4i 

3 

86959 

47 

i4 
i5 

22  8 

37  52 

82747 

3 

17253 

95799 

6 

04201 

i3o53 

3 

86947 

4b 
45 

6  22  0 

5  38  0 

9.82761 

3 

10. 17239 

0.95825 

6 

10.04175 

io.i3o64 

3 

9.86936 

16 

21  62 

38  8 

82775 

4 

17225 

95850 

7 

o4i5o 

13076 

3 

86924 

44 

17 

21  44 

38  16 

82788 

4 

17212 

95875 

7 

o4i25 

1 3087 

3 

86913 

Ai 

18 

21  36 

38  24 

82802 

4 

17198 

95901 

8 

04099 

13098 

3 

86902 

42 

!9 

20 

21  28 

38  32 

82816 
9.82830 

4 

17184 

95926 

8 

04074 

i3iio 

4 

86890 

41 

40 

6  21  20 

5  38  40 

5 

1 0 . 1 7 1 70 

9.95952 

8 

1 0 . o4o48 

10. l3l2I 

4 

9.86879 

21 

21  12 

38  48 

82844 

5 

I7i56 

95977 

9 

.i4o23 

i3.33 

4 

86867 

39 

22 

21  4 

38  56 

82858 

5 

17142 

96002 

9 

03998 

i3i45 

4 

86855 

38 

23 

20  56 

39  4 

82872 

5 

17128 

96028 

10 

03972 

i3i56 

4 

86844 

il 

24 

25 

20  48 

39  12 

82885 

6 

17115 

96053 

10 

03947 

i3i68 

5 

86832 

3b 
35 

6  20  4o 

5  39  20 

9.82899 

6 

10. 17101 

9.96078 

II 

10.03922 

'io.i3i79 

5 

9.86821 

lb 

20  32 

39  28 

82913 

6 

17087 

96104 

II 

03896 

13191 

5 

86809 

34 

27 

20  24 

39  ZG 

82927 

6 

17073 

96129 

II 

0^871 

l3202 

5 

86798 

ii 

28 

20  16 

39  44 

82941 

b 

17059 

96155 

12 

03845 

i32i4 

5 

86786 

32 

29 
3o 

20  8 

39  52 

82955 

7 

17045 

96180 

12 

o382o 

l3225 

b 

86775 

3i 
3^ 

6  20  0 

5  4o  0 

9.82968 

7 

10. 17032 

9.96205 

>3 

10.03795 

io.i3237 

6 

9.86763 

U 

19  52 

40  8 

82982 

7 

4 1 70 1 8 

96231 

i3 

03769 

13248 

b 

86752 

29 

3.' 

.■9  44 

4o  16 

82996 

7 

1   1 7004 

96256 

i4 

03744 

13260 

b 

86740 

28 

33 

19  36 

4o  24 

83oio 

8 

1 6990 

96281 

14 

03719 

13272 

b 

86728 

27 

34 
35 

19  28 

4o  32 

83023 

8 

16977 

96307 

i4 

03693 

i3283 

7 

86717 

2b 

6  19  20 

5  4o  40 

9.83o37 

8 

10. 16963 

9.96332 

i5 

10.03668 

10.13295 

7 

9.86705 

3b 

19  12 

4o  48 

83o5i 

8 

16949 

96357 

i5 

03643 

i33o6 

7 

86694 

24 

3? 

19  4 

40  56 

83o65 

8 

16935 

96383 

16 

o36i7 

i33i8 

7 

86682 

23 

3S 

18  56 

4i  4 

83o7S 

Q 

16922 

96408 

16 

03592 

i333o 

7 

86670 

22 

39 
40 

18  48 

4i  12 

83092 

9 

16908 

98433 

16 

o3567 

i334i 

8 

86659 

21 
20 

6  18  4o 

5  4i  20 

9.83 106 

9 

10. 16894 

9.96459 

17 

10  o354i 

10. 13353 

8 

9. 86647 

4i 

i8  32 

4i  28 

83 1 20 

9 

16880 

96484 

17 

o35i6 

i3365 

8 

86635 

19 

42 

18  24 

4i  36 

83i33 

10 

16867 

96510 

18 

03490 

13376 

8 

86624 

18 

43 

18  16 

4i  44 

83.47 

10 

16853 

96535 

18 

o3465 

1 3388 

8 

86612 

17 

44 
45 

18  8 

4i  52 

83i6i 

10 

16839 

96560 

19 

o344o 

1 3400 

8 

86600 

16 

75 

6  18  0 

5  42  0 

9.83174 

10 

10. 16826 

9.96586 

19 

io.o34i4 

10. i34ii 

9 

9.86589 

4b 

17  52 

42  8 

83 188 

II 

16812 

96611 

19 

03389 

1 3423 

9 

86577 

i4 

47 

17  M 

42  16 

83202 

1 1 

16798' 

96636 

20 

o3364 

13435 

9 

86565 

i3 

48 

17  36 

42  24 

832i5 

1 1 

16785 

96662 

20 

033.38 

13446 

9 

86554 

12 

49 
5c. 

17  28 

42  32 

83229 

11 

16771 

96687 

21 

o33i3 

13458 

9 

86543 

11 

10 

6  17  20 

5  42  40 

9.83242 

1 1 

10.16758 

9.96712 

21 

10.03288 

10.13470 

10 

9.86530 

5i 

17  12 

42  48 

83?5G 

12 

16744 

96738 

22 

03262 

1 3483 

10 

865 18 

9 

52 

17  4 

42  56 

83270 

12 

16730 

96763 

22 

03237 

13493 

10 

86507 

8 

53 

16  56 

43  4 

83283 

12 

■16717 

96788 

22 

o32I2 

i35o5 

10 

86495 

7 

54 
55 

16  48 

43  12 

83297 

12 

16703 

96814 
9.96839 

23 
23 

o3i86 

i35i7 
10.13528 

10 

86483 

6 
'5 

6  16  4o 

5  43  20 

9-83310 

i3 

10.16690 

io.o3i6i 

II  9.86472 

5b 

16  32 

43  28 

83324 

i3 

16676 

96864 

24 

o3i36 

1 3540 

II 

86460 

4 

!)7 

16  24 

43  36 

83338 

i3 

16662 

96890 

24 

o3iio 

i3552 

II 

86448 

3 

;>« 

16  16 

43  M 

8335i 

i3 

16649 

96915 

25 

o3o85 

1 3  564 

1 1 

86436 

2 

^9 

16  8 

43  52 

83365 

i4 

1 663  5 

96940 

25 

o3o6o 

13575 

1 1 

8642  5 

I 

bo 

16  0 

4i    0 

83378 

i4 

16622 

96966 

25 

o3o34 

i3587 

12 

864 1 3 

Hour  P.M. 'Hour  A.M. 

Cosino. 

DifT. 

Secant. 

Cotanjjcnt 

DifT. 

Tangent. 

Cosecant. 

DifT. 

Sine. 

132° 


A 

A 

B 

B 

C 

Seconds  of  time 

1' 

2» 

3' 

4. 

5" 

6^ 

10 

7' 
12 

f^ 

2 

3 

5 

7 

9 

Prop,  parts  of  cols. 

r 

3 

6 

10 

i3 

16 

'9 

22 

(c 

> 

3 

4 

6 

7 

9 

10 

* 

■  ■ 

rage  228' 

TABLE  XXVIL 

SI 

Log.  S 

lies,  Tangents,  and  Secants. 

G'. 

43°              A 

A 

B 

B 

C 

C  136° 

M 

o 

Hour  A.M. 

Hour  P.M. 

Sine. 

Ditr. 

Cosecant. 

Tangent.  Diff. 

Cotangent 

Secant 

DiflT. 

Co.tnie. 
0786413 

M 

60 

6  16  0 

5  44    0 

9.83378 

0 

10. 16622 

9 . 96966   0 

io.o3o34 

10.13587 

0 

I 

i5  52 

44  8 

83392 

0 

16608 

9699 I 1  0 

o3oo9 

13599 

0   86401 

5g 

2 

i5  44 

44   16 

834o5 

0 

16595 

97016 

1 

02984 

i36i  I 

0   86389 

58 

6 

i5  36 

44   24 

83419 

I 

i658r 

97042 

1 

02q5S 

i3623 

86377 

57 

4 
5 

i5  28 

44  32 

83432 

J 

1 6568 

97067 

2 

02933 

i3634 

86366 

56 
55 

6  i5  20 

5  4440 

9  83446 

1 

10.16554 

9.97092 

2 

10.02908 

10. 1 3646 

9.86354 

b 

i5  12 

44  48 

83459 

I 

i654i 

97118 

3 

02882 

1 3658 

86342 

54 

7 

i5  4 

44  56 

83473 

2 

16527 

97143 

3 

02857 

1 3670 

86330 

53 

8 

r4  56 

45  4 

83486 

2 

i65i4 

97168 

3 

02832 

1 3682 

2 

863 1 8 

52 

_9 

lO 

i4  48 

45  12 

835oo 

2 

i65oo 

97193 

4 

02807 

13694 

2 

863o6 

5i 

5^ 

6  i4  4o 

5  45  20 

9.835i3 

2 

to. 16487 

9.97219 

4 

10.02781 

10. 1 3705 

2 

9.86^95 

II 

i4  32 

45  28 

83527 

2 

16473 

97244 

5 

02756 

:37i7 

2 

86283 

49 

12 

i4  24 

45  36 

83540 

3 

i646o 

97269 

5 

02731 

13729 

2 

86271 

48 

iJ 

i4  16 

45  44 

83554 

3 

t6446 

97295 

5 

02705 

i374. 

3 

86259 

47 

•  4 
i5 

i4  8 
6  i4  0 

45  52 
5  46  0 

83567 

3 

16433 

97320 

6 

02680 

13753 

3 

86247 

46 
45 

9-83581 

3 

10.16419 

9.97345 

6 

10.02655 

10.13765 

3 

9.86235 

lb 

i3  52 

46  8 

83594 

4 

I  64(j6 

97371 

7 

02629 

13777 

3 

86223 

4i 

J7 

i3  44 

46  16 

836o8 

4 

16392 

97396 

7 

02604 

13789 

3 

862 1 1 

43 

i8 

i3  36 

46  24 

8362  1 

4 

16379 

97421 

8 

02579 

i38oo 

4 

86200 

42 

!9 

20 

i3  28 

46  32 

83634 

4 

1 6366 

97447 

8 

02553 

i38i2 

4 

86188 

4i 
40 

6  i3  20 

5  46  4o 

9.83648 

4 

10. [6352 

9.97472 

8 

10.02528 

10.13824 

4 

9.86176 

21 

i3  12 

46  48 

8366 1 

5 

16339 

97497 

9 

095o3 

13836 

4 

86164 

39 

22 

i3  4 

46  56 

83674 

5 

16326 

97523 

9 

02477 

13848 

4 

.  86i52 

38 

2j 

1-2  56   47  4 

83638 

5 

i63i2 

97548 

10 

02452 

i386o 

5 

86 1 40 

37 

25 

12  48 

47  12 

83701 

5 

16299 

97573 

10 
II 

02427 

13872 

5 

86128 

36 
35 

6  12  4o 

5  47  20 

9.8^715 

6 

10.16285 

9-97598 

10.02402 

10. 13884 

5 

9.86116 

2b 

12  32 

47  28 

83728 

b 

16272 

97624 

II 

02376 

13S96 

5 

86104 

34 

2? 

12  24 

47  36 

83741 

6 

16259 

97649 

1 1 

o2  35i 

1 3908 

5 

86092 

33 

28 

12  16 

47  44 

83755 

'  6 

16245 

97674 

12 

02326 

13920 

6 

86080 

32 

29 

3o 

12  8 

47  52 

83768 
9.S3781 

6 

7 

16232 

97700 

12 

o2  3oo 

13932 

6 

86068 

3i 
3^ 

6  12  0 

5  48  0 

to. 16219 

9.97725 

i3 

10.02275 

10.13944 

6 

9.86056 

Ji 

II  52 

48  8 

83795 

7 

16205 

97750 

i3 

0225o 

13956 

6 

86044 

29 

J2 

II  44 

48  16 

838o8 

7 

16192 

97776 

1^ 

02224 

13968 

6 

86o32 

28 

JJ 

II  36 

48  24 

83821 

7 

16179 

97801 

i4 

•  02199 

13980 

7 

86020 

27 

M 
35 

II  28 

48  32 

83834 

8 
~8 

16166 

97826 

i4 

02174 
10.02149 

13992 

7 

86008 

2b 

6  II  20 

5  48  4o 

9.83848 

io.i6i52 

9.97851 

i5 

io.i4oo4 

7 

9.85996 

Jb 

11  12 

48  48 

83861 

8 

16139 

97877 

i5 

02123 

14016 

7 

85984 

24 

^7 

11  4 

48  56 

83874 

8 

161 26 

97902 

16 

02098 

14028 

7 

85972 

23 

J8 

10  56 

49  4 

83887 

8 

i6n3 

97927 

16 

02073 

i4o4'' 

8 

85960 

22 

39 

4o 

10  48 

49  12 

83901 
9.83914 

_9 
9 

1 6099 

97953 
9.97978 

16 

17 

02047 
10.02022 

i4o52 

8 

85948 

21 
20 

6  '0  40 

5  49  20 

1 0 . 1 60S6 

io.i4o64 

8 

9.8593b 

41 

to  32 

49  28 

83927 

9 

16073 

98003 

17 

01997 

1407G 

8 

85924 

'9 

42 

10  24 

49  36 

83940 

9 

1 6060 

98029 

18 

OI97I 

14088 

8 

85912 

18 

43 

10  16 

49  44 

83954 

10 

i6o46 

9S054 

18 

01946 

i4ioo 

9 

85900 

17 

44 
45 

10  8 

49   52 
5  5o  0 

83967 

10 

i6o33 

98079 

19 

01921 

l4lI2 

9 

85888 

lb 

73 

6  10  0 

9.839S0 

10 

10. 16020 

9.98104 

'9 

10.01 896 

IO.I4I24 

9 

9.85876 

4b 

952 

5o  8 

83993 

10 

16007 

98130 

19 

01870 

i4i36 

9 

85864 

i4 

47 

9  44 

5o  16 

84006 

10 

1 5994 

98155 

20 

01845 

i4i49 

9 

8585 1 

iJ 

48 

9  36 

5o  24 

84o2() 

1 1 

15980 

98180 

20 

01820 

i4i6i 

10 

85839 

12 

49 
5o 

9  28 

5o  32 

84o33 

1 1 

15967 
10.15954 

98  2  06 

21 

01794 

14173 

10 

85827 

1 1 

10 

6  9  20 

5  5o  40 

9.84046 

1 1 

9.98231 

21 

10.01769 

10. i4iS5 

10 

9.8^815 

5i 

9  12 

5o  48 

84059 

1 1 

15941 

98256 

22 

01744 

14197 

10 

85So3 

9 

h2 

9  4 

5o  56 

84072 

12 

15928 

9S281 

22 

01719 

14209 

10 

85791 

8 

53 

8  56 

5i  4 

84oS5 

12 

15915 

9S307  22 

01693 

14221 

11 

85779 

7 

54 
55 

8  48 

5i  12 

5  5i  20 

84098 

12 

15902 

98332 

2  3 

01668 

14234 

11 

85766 

6 
"5 

6  8  4o 

9.841 12 

12 

IO.I588S 

9.98357 

23 

;o. 01643 

10.14246 

11 

9.85754 

5b 

8  3i 

5i  28 

84i25 

12 

15875 

98383 

24 

01617 

14258 

II 

85743 

4 

^7 

8  24 

5 1  36 

84 1 38 

i3 

1 5862 

98408 

24 

01592 

14270 

11 

8573o 

3 

58 

8  16 

5i  44 

84i5i 

i3 

1 5849 

98433 

24 

01567 

14282 

12 

85718 

2 

59 

8  8 

5i  52 

84i64 

i3 

15836 

9S45S 

25 

01 542 

14294 

12 

85706 

1 

bo 
M 

8  0 

52   0 

84177 

Din; 

1 5S23 

Secant. 

984S4 

25 

oi5i6 

1 4307 

12 

85693 

0 

Hour  P.M. 

Hour  A.M. 

Cosine. 

Cotangent 

Difl-. 

Tangent. 

Cosecant. 

Difl".  Sine. 

M 

133' 


A 

A 

B 

B 

C 

y 

Seconds  of  tame 

1' 

2' 

3» 

4. 

5' 

'8~ 

6' 

10 

7' 
12 

(^ 

2 

3 

5 

7 

1  Prop,  parts  of  cols. 

l"" 

3 

6 

9 

i3 

16 

19 

22 

1 

Ic 

2 

3 

5 

6 

8 

9_ 

1 1 

C       4(7' 


J 

TABLE  XXVIL 

[  rage  2-39 

s 

Log.  Sines,  Tangents,  and  Secants, 

G'. 

14 

\i 

o 

3 

Hour  J 

.M. 

A 

A 

B 

B 

C 

C  135° 

Hour  P.M. 

Sine.  IDiff. 

Cosecant. 

IO.I5823 

Tangent. 

Diir. 

Cotangent 

Secant. 

Diff. 

Cosine. 

6  8 

0 

5  52  0 

9.84177 

0 

9-98484 

0 

io.oi5i6 

io.i43o7 

0 

9.85693 

I 

7 

52 

52  8 

84190 

0 

i58io 

98509 

0 

01491 

14319 

0 

8568i 

5.? 

2 

7 

44 

52  16 

84203 

0 

15797 

98534 

I 

oi466 

i433i 

0 

85669 

58 

3 

7 

36 

52  24 

84216 

1 

15784 

98560 

I 

oi44o 

14343 

85657 

57 

4 
5 

7 

28 

52  32 

5  52  4" 

84229 

1 

1 5771 

98585 
9 . 986 1  ( ) 

2 
2 

oi4i5 
10.01390 

14355 
10. 14368 

-j 

S5645 
9.85632 

56 

55 

6  7 

20 

9.8424.) 

I 

10. 1 5758 

6 

7 

12 

52  48 

84255 

I 

15745 

98635 

3 

01 365 

i438o 

80620 

54 

7 

7 

4 

52  56 

84269 

2 

1 5731 

98661 

3 

01339 

14392 

856oS 

53 

8 

6 

5() 

53  4 

8428J 

2 

15718 

98(i86 

3 

0 1 3 1 4 

i44o4 

2 

8^596 

52 

_? 

10 

6 

48 

53  12 

84295 
9.84308 

2 
2 

1 5705 

9R71. 

4 

01289 

14417 

2 

85583 

5i 

5o 

6  6 

4(. 

5  53  20 

10.  i5()92 

9.98737 

4 

10.01263 

10.14429 

2 

9  85571 

11 

6 

32 

53  28 

8432  1 

2' 

15679 

98762 

5 

01 238 

i444i 

2 

85559 

4q 

12 

6 

24 

53  36 

84334 

3 

1 5656 

98787 

5 

0I2l3 

14453 

2 

85547 

48 

i3 

6 

16 

53  44 

84347 

3 

1 5653 

98S12 

5 

OII88 

1 4466 

3 

85534 

47 

i4 
i5 

6 

8 

53  52 

8436o 
9.84373 

3 
~3 

1 5640 
10. 15627 

9883s 

6 

01 162 

14478 

3 

8552? 

46 

45 

6   6 

0 

5  54  0 

9.98863 

6 

10.01 1 37 

10.14490 

3 

9.85510 

i6 

5 

52 

54  8 

84385 

3 

i56i5 

•  9888S 

7 

01  I  12 

i45o3 

3 

85497 

44 

17 

5 

44 

54  16 

8439S 

4 

i56o2 

98913 

7 

0  I  087 

i45i5 

4 

85485 

43 

i8 

5 

36 

54  24 

844  m 

4 

15589 

98939 

8 

01061 

14527 

4 

85473 

42 

!9 

20 

5 

28 

54  32 

84424 

4 

15576 

98964 

8. 

oio36 

14540 

4 

85460 

4i 
40 

6  5 

20 

5  54  4" 

9-84437 

4 

10. .15563 

9.98989 

8 

lO.OIOI 1 

10.14552 

4 

9-85448 

21 

5 

12 

54  48 

84450 

5 

i555o 

990 1 5 

9 

00985 

14564 

4 

85436 

39 

22 

5 

4 

54  56 

84463 

5 

15537 

99040 

9 

00965) 

14577 

5 

85423 

38 

23 

4 

56 

55  4 

84476 

5 

i5524 

99065 

10 

00935 

14589 

5 

85411 

37 

24 
25 

4 

48 

55  12 

84489 

5 

i55i  1 

99090 

10 

0^9 1 0 

1 4601 

5 

85399 
9.85386 

36 
35 

6  4 

40 

5  55  20 

9.84502 

5 

10. 15498 

9.991 16 

1 1 

10.00884 

10. i46i4 

5 

26 

4 

32 

55  28 

845 1 5 

6 

15485 

99141 

1 1 

00859 

14626 

5 

85374 

34 

^7 

4 

24 

.  55  36 

84528 

6 

15472 

99166 

I' 

00834 

1 4639 

6 

85361 

33 

28 

4 

16 

55  44 

84540 

6 

1 5460 

99191 

12 

00809 

i465i 

6 

85349 

32 

29 

3o 

4 

8 

55  52 

84553 

6 
6 

1 5447 

99217 

12 

00783 

14663 

6 

.  85337 

3i 

3^) 

6  4 

0 

5  56  0 

9.84566 

10.15434 

9.99242 

i3 

10.00758 

10. 14676 

6 

9.85324 

3i 

3 

52 

56  8 

84579 

7 

15421 

99267 

i3 

00733 

1 4688 

6 

853i2 

29 

32 

3 

4i 

56  16 

84592 

7 

i54o8 

99293 

i3 

00707 

1 470 1 

7 

85299 

•28 

33 

3 

36 

56  24 

846o5 

7 

15395 

99318 

i4 

00682 

i47i3 

7 

85287 

27 

34 
35 

3 

28 

56  32 

84618 

7 

i5382 

99343 

i4 

00657 

14726 
10. 14738 

_7_ 
7 

80274 
9.85262 

26 
l5 

6  3 

20 

5  56  4" 

9;8463o 

8 

10.15370 

9.99368 

i5 

io.oo632 

36 

3 

12 

56  48 

84643 

8 

15357 

99394 

10 

00G06 

i475o 

7 

852  5o 

24 

37 

3 

4 

56  56 

84656 

8 

15344 

99419 

16 

oo58i 

14763 

8 

85237 

23 

38 

2 

56 

57  4 

84669 

8 

i533i 

99444 

16 

oo556 

14775 

8 

85225 

22 

39 

4o 

2 

48 

57  .2 

5  57  20 

84682 
9 . 84694 

8 

i53i8 

99469 

16 

oo53i 
io.oo5o5 

14788 

8 

85212 

21 

20 

6  2 

40 

9 

io.i53o6 

9.99495 

"7 

1 0 . 1 4800 

8 

9.85200 

4i 

2 

32 

57  28 

84707 

9 

15293 

.  99520 

17 

oo48o 

i48i3 

8 

85i87 

'9 

42 

2 

24 

57  36 

84720 

9 

15280 

99545 

18 

00455 

14825 

9 

85.75 

18 

4i 

2 

16 

57  44 

84733 

9 

15267 

99570 

18 

oo43o 

14838 

9 

85 162 

17 

44 
45 

2 

a 

57  52 

84745 
9.84758 

_9 
10 

i5255 
10.15242 

99596 

'9 

oo4o4 

i485o 

9 

85i5o 

lb 
75 

6  2 

0 

5  58  0 

9.99621 

19 

10.00379 

10. 14863 

9 

9. 85 1 37 

46 

52 

58  8 

8477' 

10 

15229 

99646 

19 

oo354 

14875 

85 125 

14 

47 

44 

58  16 

84784 

10 

i52i6 

99672 

20 

00328 

14888 

85 1 12 

i3 

48 

36 

56  24 

84796 

10 

1 5  204 

99697 

20 

oo3o3 

1 4900 

85ioo 

12 

^9 
5o 

28 

58  32 

84809 

1 1 

15191 

99722 

21 

00278 

14913 

S5087 

1  1 
10 

0  I 

20 

5  58  40 

9.84822 

II 

10.15178 

9-99747 

21 

10.00253 

10. 14926 

9.S5074 

5i 

12 

58  48 

84835 

II 

i5i65 

99773 

21 

00227 

14938 

8  5062 

9 

52 

4 

58  56 

84847 

1 1 

i5i53 

99798 

22 

00202 

14951 

80049 

8 

53 

0 

56 

59  4 

848Go 

II 

i5i4o 

99823 

22 

00177 

14963 

85o37 

.  7 

54 
55 

0 

48 

59  12 

84S73 

12 

i5i27 

99848 

23 

00l52 

14976 

iT 

85o24 
9 . 8  5o 1 2 

6 

6  0 

4o 

5  59  20 

9.84885 

12 

io.i5ii5 

9.99874 

23 

10.00126 

I c. 14988 

56 

0 

32 

.59  28 

84898 

12 

l5l02 

99899 

24 

00101 

i5ooi 

12 

8^999 

4 

^7 

0 

24 

59  36 

84911 

12 

i5cC9 

99924 

24 

00076 

i5oi4 

12 

84986 

3 

58 

0 

16 

59  44 

84923 

12 

1 5077 

99949 

24 

000  5 1 

1 5026 

12 

84974 

2 

59 

0 

8 

59  52 

84936 

i3 

i5o64 

99975 

25 

00025 

1 5o39 

12 

84961 

I 

60 
M 

0 

0 

5  0  0 

84949 

i3 

i5o5i 

10.00000 

25 

00000 

•i5o5i 

12 

84949 

0 

Hour  P.M. 

Houl  A  -M . 

Cosine. 

Di(r. 

Secant. 

Cotangent 

Ditr. 

Tangent. 

Cosecant. 

Ditr. 

Sine. 

134° 


45» 


Seconds  of  time 

V 

2' 

3' 

4= 

5' 

8 
16 
8 

6» 
to 
•9 
9_ 

7' 

1 1 
22 
11 

(A 

Prop,  parts  of  cols.  <  B 

ic 

2 

3 
2 

3 
6 
3 

5 

9 

5 

c 

i3 
.1 

^^s^  230]                              TABLES   XXVIII,  XXIX. 

TABLE   XXVIII. 

TABLE  XXIX. 

For  reducing  the  Time  of  the  Moon's  passage  over  the  Merid 

an  of 

Correction  of  Moon's 

Greenwich,  to  the  Time  of  its  passage  over  any  other  Meridian. 

altitude  for  Paral- 

The numbers  taken  from  this  Table  are  to  be  added  to  the  Time  at 

lax    and     Refrac- 

Greenwich in  West  Longitude,  but  subtracted  in  East. 

tion. 

Daily  Variation  of  the  Moon's  passing  the  Meridian. 

Dak. 
Deg. 

Corr. 
Min. 

Dalt. 
Deg. 

5i 

Uorr- 
Mm. 

35 

Ship's 

/ 

/ 

/ 

/ 

1 

/ 

/ 

/ 

/ 

/ 

/ 

/ 

/ 

/ 

Ship's 

10 

5i 

Lou. 

40 

42 

44 

46 

48 

50 

52 

54 

56 

58 

60 

62 

64 

66 

Loii. 

II 
12 

52 
52 

52 

53 

35 
34 

/ 

1 

1 

1 

1 

/ 

/ 

/ 

1 

/ 

/ 

/ 

1 

/ 

i3 

52 

54 

33 

o 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

14 

52 

55 

32 

5 

I 

1 

1 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

5 

i5 

52 

56 

32 

lO 

I 

I 

I 

I 

I 

I 

I 

I 

2 

2 

2 

2 

2 

2 

10 

16 

52 

5? 

3i 

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2 

2 

2 

2 

2 

2 

2 

2 

2 

2 

2 

3 

3 

3 

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17 

52 

58 

So 

20 

2 

2 

2 

3 

3 

3 

3 

3 

3 

3 

3 

3 

4 

4 

20 

18 

52 

59 
60 

29 

28 

25 

3 

3 

3 

•  3 

3 

3 

4 

4 

4 

4 

4 

4 

4 

5 

25 

19 
20 

52 

3o 

3 

3 

4 

4 

4 

4 

4 

4 

5 

5 

5 

5 

5 

5 

3o 

5i 

35 

T 

4 

4 

4 

5 

5' 

5 

5 

T 

6 

6 

~6" 

6 

6 

35 

21 

5i 

~67~ 

27 
26 

4o 

4 

5 

5 

5 

5 

6 

6 

6 

6 

6 

7 

7 

7 

7 

40 

22 

5i 

62 

45 

5 

5 

5 

6 

6 

6 

6 

7 

7 

7 

7 

8 

8 

8 

45 

23 

5i 

63 

26 

5o 

6 

6 

6 

6 

7 

7 

7 

7 

8 

8 

8 

9 

9 

9 

5o 

24 

5o 

64 

25 

55 

6 

7 

6 

7 

7 
7 

7 
8 

7 
8 

8 

8 

8 
9 

8 
9 

_9_ 
9 

_9_ 
10 

__9_ 

ID 

_9_ 
10 

10 
II 

10 
11 

55 

25 

26 

5o 
5o 

65 
66 

24 

23 

60 

60 

65 

7 

8 

8 

8 

9 

9 

9 

10 

10 

10 

I  I 

II 

12 

12 

65 

27 

49 

67 

22 

70 

8 

8 

9 

9 

9 

10 

10 

10 

1 1 

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12 

12 

12 

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70 

28 

49 

68 

21 

75 

8 

9 

9 

10 

10 

10 

II 

II 

12 

12 

12 

i3 

i3 

i4 

75 

29 

49 

69 

20 

80 

85 

•_9_ 
9 

_9_ 
10 

10 
10 

10 
II 

II 
II 

II 

I2t 

12 
1 2 

12 
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12 
73" 

i3 

i4 

I  3 

T4 

i4 
i5 

i4 
i5 

i5 
16 

80 

85 

00 

48 

70 

19 

48 

71 

i8 

90 

10 

10 

II 

1 1 

12 

12 

i3 

i3 

i4 

i4 

i5 

i5 

16 

16 

90 

32 

47 

72- 

17 

95 

1 1 

II 

12 

12 

i3 

l3 

i4 

14 

i5 

i5 

16 

16 

17 

17 

95 

33 

47 

73 

17 

100 

1 1 

12 

12 

i3 

i3 

i4 

i4 

i5 

16 

16 

17 

17 

18 

18 

100 

34 

46 

74 

16 

io5 

12 

12 

i3 

i3 

i4 

i5 

i5 

16 

16 

'7 

17 

18 

'9 

19 

io5 

35 

46 

75 

i5 

36 

37 
38 

39 

40 

45 
45 
44 
44 
43 

76 

i4 

1 10 

12 

73" 

73" 

T4" 

i5 

i5 

16 

16 

17 

18 

18 

'9 

20 

20 

1 10 

ii5 
120 

i3 
i3 

i3 

i4 

i4 
i5 

i5 
i5 

i5 
16 

16 

17 

17 
17 

17 
18 

18 

'9 

'9 
19 

19 

20 

20 
21 

20 
21 

21 
22 

1x5 
120 

77 
78 

i3 
12 

125 

i3o 

1 4 
1 4 

i5 
i5 

i5 
16 

16 

•7 

17 
17 

17 
18 

18 
_L?_ 

19 
20 

'9 
'9 

19 
20 

20 
21 

21 
22 

22 
22 

22 

23 

23 

24 

125 

i3o 

79 
80 

1 1 

ID 

i35 

i5 

16" 

'x'^ 

17 

Tb 

19 

20 

21 

22 

22 

77 

77 

25 

i35  . 

4i 

42 
42 
4i 
4o 
4o 
39 

81 
82 
83 
84 
85 
86 

9 

1 40 

16 

16 

17 

18 

'9 

19 

21 

22 

23 

23 

24 

25 

26 

i4o 

42 
43 
44 
45 
46 

8 

i45 

16 

17 

18 

'.9 

'9 

20 

21 

22 

*23 

23 

^4 

55 

26 

27 

i45 

7 
6 
5 
4 

i5o 

17 

17 

18 

'9 

20 

21 

22 

22 

23 

04 

25 

26 

27 

27 

i5o 

1 55 

17 

-L^_ 

_L9 

20 

21 

22 

32 

23 

24 

25 

26 

27 

28 

28 

i55 

160 

18 

19 

20 

20 

21 

22 

T3" 

Ta 

2  5' 

76 

27 

28" 

T8~ 

39 

160 

47 

38 

87 

3 

ifj-) 

18 

'9 

20 

21 

22 

23 

24 

25 

56 

27 

27 

28 

29 

3o 

i65 

48 

38 

88 

2 

170 

'9 

20 

21 

22 

23 

24 

25 

25 

26 

27 

28 

29 

3o 

3i 

170 

it 

37 

89 

I 

17') 

19 

20 

21 

22 

23 

24 

25 

26 

27 

28 

29 

3o 

3i 

32 

175 

36 

90 

0 

iSo 

5  0 

21 

22 

23 

H 

25 

26 

27 

28 

19 

3o 

3i 

32 

33 

180 

40' 

42' 

44' 

40' 

48' 

50' 

52' 

54" 

56' 

58']  GO' 

62' 

64' 

66' 

TABLE  XXX. 

[Page  231 

For  finding  the  Variation  of  ih 
of  the  ftlooii  s  Iliglit  Ascension 

eS 

un's  Right  Ascension,  of  the  Pec 

lination,  of  the  Equation  of  Time  or 

in 

an\'  number  of 

minutes  of  time 

,  the  Horary'  Motion  being  given  at 

the  top  of  the  page  in  seconds,  and  the  number  of  minutes  of  time  in  the  side-column  y —                           J 

Also,  for  finding  the  Variation 

of  the  Moon's 

Declination  in  seconds  of  time ;  the  motion  in  one  I 

minute  being  given  at  the  top,  and  the  numbers  in  the  side-column  being  taken  (or  seconds. 

Horai-y  Motion. 

M 

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21 

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23 

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16 

17 

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23 

23 

24 

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26 

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54 

55 

2 

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6 

6 

7 

8 

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23 

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12 

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20 

21 

22 

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25 

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27 

28 

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2 

3 

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6 

7 

8 

9 

10 

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12 

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16 

17 

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21 

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23 

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25 

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28 

29 

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59 

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2 

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16 

17 

18 

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20 

21 

22 

23 

24 

25 

26 

27 

28 

29130 

60 

1 
Page  232]                       TABLE  XXX. 

For  fiiKling  Ihe  Variation  of  the  Sun's  Right  Ascension,  of  the  Declination,  of  the  Equation  of  Time  or 

of  llie  Moon's  Rifrlil  Ascension,  in  any  number  of  minutes  of  time,  the  Horary  Motion  being  given  at 

the  top  of  llie  page  in  seconds,  and  the  number  of  minutes  of  time  in  the  side-column  ; — 

Also,  for  finding  the  Variation  of  tiic  ftloon's  Declination  in  seconds  of  time;  l!ie  motion  in  one 

minute  being  given  at  the  top,  and  llie  numbers  in  ihc  side-colunui  being  taken  for  seeomls. 

Horary  Motion. 

M 

'/ 1 "  1 

// 

// 

;/ 

// 

II 

'/  i 

// 

// 

/'  II   II 

II 

;/ 

II 

/; 

;/ 

// 

// 

// 

If 

II 

II 

II 

// 

// 

II 

II 

II 

M 

31 

32 

33 

34 

35 

36 

37 

33; 

39 

4U 

41 

42 

43 

44 

45 

4G 

47 

48 

49 

50 

51 

52 

53 

54 

55 

56 

57 

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59 

GO 

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I 
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I 

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1 

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42 

4344 

45 

46 

47 

48 

49 

5o 

5i 

53 

53 

54 

55 

56 

'^7 

58  59 

59 

60 

3i 

|3: 

33 

|34 

|35  3r 

137  38 

39 

4o 

4i 

42 

4-i 

44'45 

46 

47 

48 

49 

5o 

5i 

52 

53 

54 

55 

56 

57 

58 

59I60 

60 

TABLE  XXX.                [P^se^^3 

For  fiiuling  ihe  Variation  of  llie  Sun's  Rigiit  Ascension,  of  the  Declination,  of  the  [■^ciiuuion  of  Time  or 
of  tlie  iMoon  s  Rig;ht  Ascension,  in  any  number  of  minutes  of  time,  the  lloiary  Motion  bci.ig  given  at 

llie  top  of  tlie  page  in  seconds,  and  the  number  of  minutes  of  lime  in  the  side-column, — 

Also,  for  finding  the  Variation  of  the  Jloon's  Declination  in  seconds  of  time;  the  motion  in  one 

minute  being  given  at  the  top,  and  the  numbers  in  the  side-column  being  taken  for  seconds. 

Horary  Motion. 

M 

II 

// 

II 

/( 

// 

„ 

II 

\i 

// 

II 

,7 

;/ 

/; 

// 

// 

/'  1 II 

II 

// 

II 

// 

// 

// 

// 

n 

// 

II 

II 

II 

II 

M 

61 

(i2 

(i:3 

(i-J 

65 

m 

u 

6« 

6'J 

70 

71 

72 

7:3 

74 

75 

76 

77 

78 

7il 

80 

SJ 

82 

83 

84 

85 

86 

87 

8b 

8ii 

i)0 

I 

2 

1 
2 

2 

I 
2 

I 
2 

I 
2 

I 
2 

I 
2 

2 

1 
2 

I 
2 

1 
2 

I 

2 

I 
2 

I 
2 

I 
3 

I 
3 

1 
3 

3 

I 
3 

3 

I 

3 

I 

3 

I 
3 

3 

I 
3 

I 
3 

I 
3 

3 

I 
3 

2 

3 

I 
2 

3 

3 

3 

3 

3 

3 

3 

3 

3 

3 

4 

4 

4 

4 

4 

4 

4 

4 

4 

4 

4 

4 

4 

4 

4 

4 

4 

4 

4 

4 

5 

3 

4 

4 

4 

4 

4 

4 

4 

4 

5 

5 

5 

5 

5 

5 

5 

5 

5 

5 

5 

5 

5 

5 

5 

6 

6 

6 

6 

6 

6 

6 

6 

4 

5 

5 

5 

5 

5 

5 

6 

6 

6 

6 

6 

6 

6 

6 

6 

6 

6 

6 

7 

7 

7 

7 

7 

7 

7 

_7 

7 

7 

J 

_7 

8 

5 

6 

6 

() 

6 

6 

7 

7 

7 

7 

7 

7 

7 

7 

7 

7 

8 

~8 

"8 

"8 

"8 

8 

8 

'8 

8 

8 

9 

9 

9 

9 

9 

9 

6 

7 

7 

7 

7 

7 

8 

8 

8 

8 

8 

8 

8 

8 

9 

9 

9 

9 

9 

9 

9 

9 

9 

10 

10 

10 

10 

U) 

10 

K 

i(j 

1 1 

7 

8 

8 

8 

8 

9 

9 

9 

9 

9 

9 

9 

9 

10 

10 

10 

10 

10 

10 

10 

1 1 

1 1 

1 1 

1 1 

1 1 

1 1 

1 1 

1 1 

12 

12 

12 

12 

8 

9 

9 

9 

9 

10 

10 

10 

H) 

10 

10 

1 1 

1 1 

1 1 

1 1 

1 1 

II 

1 1 

12 

12 

12 

12 

12 

12 

12 

i3 

i3 

i3 

i3 

i3 

i3 

14 

9 

lO 

10 

lu 

1 1 

1 1 

1 1 

1 1 

I  I 

1 1 

12 

12 

12 

12 

12 

12 

i3 

i3 

i3 

i3 

i3 

i3 

14 

14 

i4 

i4 

i4 

:4 

i5 

i5 

i5 

i5 

lO 

1 1 

1 1 

1  1 

12 

12 

12 

12 

12 

12 

73 

73 

73 

73 

73 

74 

i4 

i4 

i4 

14 

74 

75 

i5 

i5 

i5 

i5 

16 

16 

76 

76 

16 

17 

1 1 

12 

12 

1  2 

i5 

i3 

i3 

i3 

\3 

i4 

1 4 

i4 

i4 

14 

i5 

i5 

1 5 

i5 

1 5 

16 

16 

16 

16 

16 

17 

■7 

17 

17 

■7 

18 

18 

18 

12 

i3 

i3 

i3 

i4 

i4 

i4 

i4 

i5 

i5 

1 5 

i5 

i5 

16 

16 

16 

16 

16 

17 

17 

17 

17 

18 

18 

18 

18 

18 

'9 

'9 

'9 

'9 

20 

i3 

i4 

i4 

i4 

i5 

i5 

i5 

i5 

16 

16 

16 

16 

17 

17 

17 

17 

18 

18 

18 

18 

18 

19 

19 

'9 

19 

20 

20 

2p 

2U 

21 

21 

21 

i4 

i5 

1 5 

16 

16 

16 

16 

17 

17 

17 

17 

18 

18 

18 

18 

12 

i? 

[9 

19 

2C1 

20 

20 

20 

21 

21 

21 

21 

2  2 

22 

22 

22 

23 

i5 

7(3 

76 

'7 

'7 

17 

'7 

18^ 

78 

18 

18 

19 

19 

^ 

'9 

20 

20 

20 

21 

21 

21 

21 

22 

22 

22 

22 

73 

73 

73 

73 

24 

?4 

16 

I? 

17 

18 

18 

18 

18 

'9 

'9 

'9 

20 

20 

20 

20 

21 

21 

21 

22 

22 

22 

22 

23 

23 

23 

24 

24 

24 

24 

25 

25 

25 

26 

17 

i8 

18 

19 

'9 

'9 

20 

20 

2CI 

20 

21 

21 

21 

22 

22 

22 

23 

23 

23 

23 

24 

24 

24 

25 

25 

2  5 

26 

26 

26 

26 

27 

27 

18 

'9 

'9 

.)(! 

2U 

20 

2  1 

21 

21 

22 

22 

22 

22 

23 

23 

23 

24 

24 

24 

25 

25 

2  5 

26 

26 

26 

27 

27 

27 

28 

28 

28 

29 

'9 

20 

20 

2  1 

2i 

21 

22 

22 

2  2 

23 

23 

23 

24 

24 

24 

2  5 

25 

25 

26 

26 

26 

27 

27 

27 

28 

28 

28 

29 

29 

29 

3o 

3o 

2(1 

21 

21 

22 

22 

22 

73 

73 

73 

74 

24 

75 

25 

25 

26 

76 

26 

27 

27 

27 

78 

28 

28 

29 

29 

29 

37 

3o 

3o 

3i 

37 

37 

21 

22 

22 

23 

23 

23 

24 

24 

25 

25 

25 

26 

26 

26 

27 

27 

28 

28 

28 

29 

29 

29 

3o 

3o 

3i 

3i 

3i 

32 

32 

32 

33 

33 

22 

23 

23 

24 

24 

25 

25 

25 

26 

26 

26 

27 

27 

28 

28 

28 

29 

29 

3o 

3o 

3(; 

3i 

3i 

3i 

32 

32 

33 

33 

33 

34 

34 

35 

23 

24 

24 

25 

2  5 

26 

26 

26 

27 

27 

28 

28 

28 

29 

29 

3o 

3o 

3o 

3i 

3i 

32 

32 

32 

33 

33 

34 

34 

34 

34 

35 

36 

36 

24 

25 

25 

2() 

26 

27 

27 

28 

28 

28 

20 

29 

3o 

3o 

3o 

3i 

3i 

32 

32 

33 

33 

33 

34 

34 

35 

35 

35 

36 

36 

37 

37 

38 

25 

26 

76 

27 

27 

28 

28 

29 

^9 

29 

to 

3o 

37 

37 

32 

32 

33 

33 

33 

34 

34 

35 

35 

36 

36 

36 

37 

37 

38 

38 

39 

39 

7(7 

27 

27 

28 

28 

29 

29 

3u 

3c) 

3i 

3i 

32 

32 

32 

33 

33 

34 

34 

35 

35 

36 

36 

36 

37 

37 

38 

38 

39 

39 

40 

4v 

4i 

27 

28 

28 

29 

29 

3(. 

3u 

3i 

3i 

32 

32 

33 

33 

34 

34 

35 

35 

35 

36 

36 

37 

37 

38 

38 

39 

39 

40 

4o 

4i 

4i 

42 

42 

28 

29 

29 

3.. 

3o 

3i 

3i 

32 

32 

33 

33 

34 

34 

35 

35 

36 

36 

37 

37 

38 

38 

39 

39 

4o 

4u 

4i 

4i 

42 

42 

43 

43 

44 

29 

3o 

3i 

3i 

32 

32 

33 

33 

34 

34 

35 

35 

36 

36 

37 

37 

38 

38 

39 

39 

4o 

40 

41 

4i 

42 

42 

43 

43 

44 

44 

45 

45 

3o 

3i 

32 

32 

33 

33 

34 

34 

35 

35 

36 

36 

37 

37 

38 

38 

39 

39 

4(> 

4o 

4 1 

4i 

42 

42 

43 

43 

44 

44 

45 

45 

46 

47 

3i 

32 

33 

33 

34 

34 

35 

35 

36 

36 

37 

37 

38 

38 

39 

39 

40 

4\ 

4i 

42 

42 

43 

43 

44 

44 

45 

45 

46 

46 

47 

47 

48 

32 

33 

34 

34 

35 

35 

36 

36 

37 

37 

38 

39 

39 

4o 

4o 

4i 

4i 

4-2 

42 

43 

43 

44 

45 

45 

46 

46 

47 

47 

48 

48 

49 

5o 

33 

34 

35 

35 

3() 

36 

37 

37 

38 

39 

39 

4(1 

4u 

4i 

4i 

42 

43 

43 

44 

44 

45 

45 

46 

46 

47 

48 

48 

49 

49 

5o 

5(j 

5i 

34 

35 

36 

3(i 

37 

37 

38 

39 

l2 

4<) 

4<. 

^< 

4> 

42 

43 

43 

44 

44 

45 

46 

46 

47 

47 

48 

48 

49 

5o 

5o 

5i 

5i 

52 

53 

35 

36 

37 

3^ 

38 

38 

39 

4" 

40 

4> 

4i 

42 

43 

43 

44 

44 

45 

46 

4() 

47 

47 

48 

49 

49 

5o 

5o 

57 

57 

57 

53 

53 

54 

36 

37 

38 

38 

39 

4" 

39 

40 

A\ 

4i 

42 

43 

43 

44 

44 

45 

46 

46 

47 

47 

48 

49 

49 

5o 

5i 

5i> 

52 

52 

53 

54 

54 

55 

56 

37 

38 

39 

39 

41 

4i 

42 

42 

43 

44 

44 

45 

46 

46 

47 

48 

48 

49 

49 

5o 

5i 

5i 

52 

53 

53 

54 

54 

55 

56 

56 

57 

38 

3y 

4" 

io 

4i'A-^ 

42 

43 

44 

44 

45 

46 

46 

47 

47 

48 

49 

49 

5<> 

5i 

5i 

52 

53 

53 

54 

55 

55 

56 

57 

57 

58 

59 

39 

40 

4i 

4i 

42  43 

43 

44 

45 

45 

46 

47 

47 

4>3 

49 

49 

5o 

5i 

5i 

52 

53 

53 

54 

55 

2^ 

56 

57 

57 

58 

5y 

59 

60 

4o 

4i 

42 

42 

4^3  44 

44 

45 

46 

46 

47 

48 

49 

49 

5o 

5i 

5i 

57 

53 

53 

54 

55 

55 

56 

^7 

57 

58 

59 

59 

60 

61 

67 

4i 

42 

43 

43144  45 

46 

4^^ 

47 

48 

48 

49 

5<) 

5o 

5i 

52 

53 

53 

54 

55 

55 

56 

57 

57 

58 

59 

60 

60 

61 

62 

62 

63 

42 

43 

44 

44  45  46 

47 

4i 

48 

49 

49 

5o 

5i 

52 

52 

53 

54 

54 

55 

56 

57 

57 

58 

59 

59 

60 

61 

62 

62 

63 

64 

65 

43 

M 

45 

45 146:47 

48 

48 

49 

5o 

5i 

5[ 

52 

53 

54 

54 

55 

56 

56 

57 

58 

59 

59 

60 

61 

62 

62 

63 

64 

65 

65 

66 

44 

45 

46 

4714748 

49 

5o 

5<. 

5i 

52 

53 

53 

54 

55 

56 

56 

57 

58 

59 

59 

60 

61 

62 

62 

63 

64 

65 

65 

66 

67 

68 

45 

46 

47 

i8  4»l49 

5<i 

57 

5i 

52 

53 

54 

54 

55 

56 

57 

58 

58 

59 

60 

61 

67 

62 

63 

64 

64 

65 

66 

67 

67 

68 

69 

46 

4? 

48 

49,49150 

5i 

52 

5? 

53 

54 

55 

56 

56 

57 

58 

59 

6(. 

60 

61 

62 

63 

63 

64 

65 

66 

67 

67 

68 

69 

70 

71 

47 

48 

49 

5o.5.,5i 

52 

53 

54 

54 

55 

56 

^7 

58 

58 

59 

611 

61 

62 

62 

63 

64 

65 

66 

66 

('7 
69 

68 

69 

70 

70 

71 

72 

48 

49 

5(1 

5 1 15 1 

52 

53 

54 

55 

56 

56 

57 

58 

59 

60 

6ci 

61 

62 

63 

64 

65 

65 

66 

^7 

68 

69 

70 

71 

T2 

73 

74 

49 

5o 

5i 

52I53 

1 

53 

54 

55 

56 

57 

58 

58 

59 

60 

61 

62 

63 

63 

64 

65 

66 

67 

68 

68 

69 

7" 

7[ 

72 

73 

73 

74 

7^ 

5o 

TT 

52 

53  54 

54 

55 

56 

57 

58 

^ 

60 

60 

61 

67 

63 

(74 

65 

675 

66 

<57 

68 

^ 

7<.i 

71 

7' 

72 

73 

74 

75 

7G 

77 

5i 

52 

53 

54i55 

55156 

57 

58 

^9 

60 

61 

62 

62 

63 

64 

65 

66 

67 

68 

68 

69 

70 

71 

72 

73 

74 

75 

75 

76 

77 

78 

52 

•53 

54 

55^56 

57 

')7 

58 

59 

6.i 

61 

62 

63 

64 

64 

65 

66 

67 

68 

69 

70 

71 

72 

72 

73 

74 

7^ 

7^ 

77 

78 

79 

80 

53 

54 

55 

56'57 

58 

59 

59 

6. 

61 

62 

63 

64 

65 

66 

67 

68 

68 

69 

7<' 

71 

7' 

73 

74 

75 

76 

77 

77 

78 

79 

80I 

81 

54 

55 

56 

5758 

5?!^ 

61 

61 

62 

63 

64 

65 

66 

67 

68 

69 

70 

71 

72 

73 

73 

74 

7^ 

76 

77 

Z^ 

79 

80 

81 

8283 

55 

56 

57 

58159 

6( )  6 1 

67 

63 

63 

64 

65 

66 

67 

68 

69 

70 

71 

72 

73 

74 

75 

76 

77 

77 

78 

79 

8(1 

87 

82 

83  84 

56 

57 

58 

59 

6ii6i|62 
61  62;(33 

63 

64165 

66 

fi7 

67 

68 

69 

70 

71 

72 

73 

74 

75 

76 

77 

78 

79 

8ci 

81 

82 

83 

84 

85I86 

57 

58 

59 

6( 

64 

65  !66 

67 

68 

69 

70 

71 

72 

73 

73 

74 

75 

7G 

77 

78 

79 

8(1 

81 

82 

83 

84 

85 

86I87 

58 

59 

6( 

:()! 

6;  63  64 

65 

6667 

68 

69 

70 

71 

72 

73 

74 

75 

76 

77 

7S 

79 

80 

81 

82 

83 

84 

85 

86 

87 

8689 

60 

60 

61:6: 

63  64^65 

66 

67 '68  69 

70 

71 

72 

73 

74 

75 

76 

77 

78 

79 

80 

81 

82 

83 

84 

85 

86 

87i 

88 

89  90 

ao 


Page  234] 

TABLE 

XXX 

For  finding  the  Variation  of  the  Sun's  Ri^ht  Ascension,  of  the  Declination,  of  tlie  Equation  of  Time  or  | 

of  the  Moon's  Riglit  Ascension,  in 

any  number  of 

iiinutes  of  time,  the  Horary  Motion  being  siven  at 

the  top  of  the  page  in  seconds^ 

and  the  number  of  minutes  of 

uue  in  the  side-column  ; — 

Also,  for  finiling  the  Variation 

of  the  Moon's  Declination  in  seconds  of  time  ;  the  motion  in  one 

minute  being  given  at  the  top,  and  the  numbers  in  the 

side-column  being  taken  for  seconds. 

Horary  Motion. 

M 

II 

II 

// 

II 

n 

" 

// 

// 

/' 

// 

// 

// 

II 

II 

/; 

(/ 

II 

// 

II 

/; 

II 

// 

;/ 

// 

// 

M 

91 

92 

93 

94 

95 

96 

97 

98 

99 

100 

101 

102 

103 

104 

105 

106 

107 

108 

109 

110 

111 

112 

113 

114 

115 

I 

2 

2 

2 

2 

2 

2 

2 

2 

1 

2 

2 

2 

2 

2 

2 

2 

2 

2 

2 

2 

2 

2 

2 

2 

2 

I 

2 

3 

3 

3 

3 

3 

3 

3 

3 

3 

3 

3 

3 

3 

3 

4 

4 

4 

4 

4 

4 

4 

4 

4 

4 

4 

2 

3 

5 

5 

5 

5 

5 

5 

5 

5 

5 

5 

5 

5 

5 

5 

5 

5 

5 

5 

5 

6 

6 

6 

6 

6 

6 

3 

4 

6 

6 

6 

6 

6 

6 

6 

7 

n 

7 

7 

7 

7 

7 

7 

7 

7 

7 

7 

7 

7 

7 

8 

5 

8 

4 

5 

5  8 

8 

8 

8 

8 

8 

8 

8 

8 

8 

9 

9 

_9 

_9 

_9 

_9 

_? 

9 

_9 

_? 

9 

_9 

10 

10 

5 

6 

9^9 

9 

9 

10 

10 

10 

10 

10 

10 

10 

10 

10 

10 

II 

II 

II 

II 

II 

II 

II 

1 1 

u 

11 

12 

"6 

7 

1 1 

11 

1 1 

II 

II 

II 

II 

12 

12 

12 

12 

12 

12 

12 

12 

12 

i3 

i3 

i3 

i3 

i3 

i3 

i3 

i3 

7 

8 

'12 

12 

12 

i3 

i3 

i3 

i3 

i3 

i3 

i3 

i3 

i4 

i4 

1 4 

i4 

i4 

14 

i4 

i5 

i5 

•I  5 

i5 

i5 

i5 

i5 

8 

9 

i4 

i4 

i4 

i4 

i4 

i4 

i5 

i5 

1 5 

i5 

i5 

i5 

i5 

16 

16 

16 

16 

16 

16 

17 

17 

17 

17 

17 

17 

9 

lO 

i5 

i5 

16 

16 

16 

16 

16 

16 

17 

17 

17 

17 

17 

17 

18 

18 

18 

18 

18 

18 

_L9 

_L9 

_[9 

-19 

.1? 

10 

11 

17 

17 

17 

17 

17 

78 

78 

7s 

18 

18 

19 

19 

19 

~^ 

19 

19 

20 

20 

20 

20 

20 

21 

21 

21 

21 

1] 

12 

18 

18 

'9 

'9 

'9 

'9 

•9 

20 

20 

20 

20 

20 

21 

21 

21 

21 

21 

22 

22 

22 

22 

22 

23 

23 

23 

12 

i3 

20 

20 

20 

20 

21 

2i 

21 

21 

21 

22 

22 

22 

22 

23 

23 

23 

23 

23 

24 

24 

24 

24 

24 

25 

25 

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21 

21 

22 

22 

22 

22 

23 

23 

23 

23 

24 

24 

24 

24 

25 

25 

25 

25 

25 

26 

26 

26 

26 

27 

27 

i4 

i5 
i6 

23 

23 

23 

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24 

25 

24 

25 

24 
26 

24 
26 

25 

25 

25 

27 

25 

26 

26 

26 

26 

27 
28 

27 
29 

27 
29 

27 
29 

28 
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28 
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28 
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28 
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29 

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27 

27 

27 

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28 

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27 

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29 

29 

29 

29 

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32 

32 

32 

33 

17 

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27 

28 

28 

28 

29 

29 

29 

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3o 

3o 

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3. 

32 

32 

32 

32 

33 

33 

33 

34 

34 

34 

35 

18 

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29 

29 

29 

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3n 

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3i 

32 

32 

32 

33 

33 

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34 

34 

34 

35 

35 

35 

35 

36 

36 

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3i 

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32 

33 

33 

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34 

34 

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35 

36 

36 

36 

37 

37 

37 

38 

38 

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20 

21 

32 

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33 

33 

33 

34 

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34 

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35 

36 

36 

36 

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37 

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37 

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39 

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40 

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22 

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35 

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39 

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40 

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43 

44 

44 

23 

24 

36 

37 

37 

38 

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39 

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40 

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42 

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43 

44 

44 

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45 

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47 

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53 

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59 

60 

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61 

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32 

33 

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5i 

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52 

52 

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53 

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56 

56 

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58 

59 

59 

60 

61 

61 

62 

62 

63 

63 

33 

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52 

52 

53 

53 

54 

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58 

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59 

60 

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63 

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65 

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61 

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67 

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61 

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67 

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68 

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69 

70 

70 

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58 

58 

59 

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62 

63 

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65 

66 

67 

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68 

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69 

70 

70 

71 

72 

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60 

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61 

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67 

68 

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70 

70 

71 

72 

72 

73 

73 

74 

75 

39 

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61 

61 

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63 

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71 

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73 

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76 

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62 

63 

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75 

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77 

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78 

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64 

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65 

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67 

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78 

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80 

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66 

67 

67 

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70 

71 

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80 

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82 

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67 

68 

69 

70 

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72 

73 

73 

74 

75 

76 

76 

77 

78 

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79 

80 

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81 

82 

S3 

84 

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44 

45 

68 

69 

70 

71 

71 

72 

73 

74 

74 

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76 

77 

77 

78 

79 

80 

80 

81 

82 

83 

83 

84 

85 

86 

86 

45 

46 

70 

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71 

72 

73 

74 

74 

75 

76 

77 

77 

78 

79 

80 

81 

81 

82 

83 

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84 

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07 

"87 

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46 

47 

71  72 

73 

74 

74 

75 

76 

77 

78 

78 

79 

80 

81 

81 

82 

83 

84 

85 

85 

86 

87 

88 

89 

89 

90 

47 

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74 

75 

76 

77 

78 

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79 

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81 

82 

82 

83 

84 

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86 

87 

88 

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90 

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92 

48 

49 

74 

75 

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77 

78 

78 

79 

80 

81 

82 

82 

83 

84 

85 

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87 

87 

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89 

90 

91 

9! 

92 

93 

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49 

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76 

77 

78 

78 

79 

80 

81 

82 

83 

83 

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85 

86 

87 

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92 

93 

93 

94 

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96 

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81 

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83 

84 



85 

86 

87 

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88 

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90 

9' 

92 

93 

94 

94 

95 

96 

97 

98 

57 

52 

79 

8(1 

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81 

82 

63 

84 

85 

86 

87 

88 

83 

89 

90 

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92 

93 

94 

94 

95 

96 

97 

98 

99 

lOo 

52 

53 

8(. 

81 

82 

83 

64 

85 

86 

8- 

87 

88 

89 

90 

9' 

92 

93 

94 

95 

95 

96 

97 

98 

99 

100 

101 

102 

53 

54 

82 

83 

84 

85 

86 

86 

87 

88 

89 

90 

9' 

92 

93 

94 

95 

95 

06 

97 

98 

99 

100 

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IC? 

io3 

io4 

54 

55 

83 

84 

85 

86 

87 

88 

89 

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21 

92 

93 

_94 

94 

^^ 

96 

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_98 

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100 

lOI 

102 

i(>3 

io4 

io5 

i.j5 

55 

56 

85 

86 

87 

88 

89 

90 

9' 

9' 

92 

93 

94 

95 

96 

97 

98 

99 

100 

lOI 

102 

7^ 

io4 

7^ 

io5 

K-i') 

107 

56 

^7 

86 

87 

88 

89 

90 

91 

92 

93 

94 

95 

96 

97 

98 

99 

100 

101 

102 

io3 

io4 

io5 

io5 

106 

107 

108 

109 

57 

58 

88 

89 

9" 

91 

92 

93 

94 

95 

96 

97 

98 

99 

100 

lOI 

102 

102 

io3 

io4 

io5 

106 

107 

108 

109 

I  IIJ 

III 

58 

59 

90 

9" 

9' 

92 

93 

94 

95,9697 

98 

99 

too 

101 

102 

io3 

io4 

io5 

106 

107 

108 

109 

1 10 

1 1 1 

1  12 

ii3 

59 

60 

2L 

21 

93 

94 

£l 

96 

97  98  99 

100 

101 

I03 

io3 

io4 

io5 

106 

107 

108 

109 

IJn 

1 1 1 

1 12 

iij 

ii4 

ii5 

bo 

TABLE  XXX. 

[Page  235 

For  finding'  the  Variat 

on  of  till 

Sun's  Right  Ascension,  of  the  Declination,  of  the  Equation  of  Time  or  | 

of  the  JMoon's  Right  A 

jcensioii, 

in 

my 

number 

af  minutes  of  time,  the  Horary  Motion 

Iteing  given  at 

the  top  ol'  the  page  in  se 

conds,  and  the  number  of  minutes  of  time  in  ihe  side-column; — 

Also,  for  finding  ilie 

Variation  0 

f  ih 

e  I\Ioon' 

s  Declination  in  seconds  of  time :  the 

motion  in  one 

minute  being  given  at  the  top,  and  the 

numbers  in  the  side-column  being  taken  for  seconds. 

Ilorat-y  Motion. 

M 

/'  ■  •' 

//  1  '/ 

/; 

/ 

// 

// 

// 

II 

// 

// 

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/; 

// 

II 

// 

II 

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// 

// 

II 

II 

M 

liG 

117 

llgll!) 

120 

121 

l->2 

123 

12-1 

125 

126 

127 

128 

129 

130 

131 

132 

133 

131 

135 

13G 

j37 

138 

I 

2 

2 

2I  2 

2 

2 

2 

2 

2 

2 

2 

2 

2 

2 

2 

2 

2 

2 

2 

2 

2 

"^ 

2 

I 

4 

4 

4 

4 

4 

4 

4 

4 

4 

4 

4 

4 

4 

4 

4 

4 

4 

4 

4 

5 

5 

5 

5 

2 

3 

6 

6 

6 

6 

6 

6 

6 

6 

6 

6 

6 

6 

6 

6 

7 

7 

7 

7 

7 

7 

7 

7 

7 

3 

4 

8 

8 

8 

8 

8 

8 

8 

8 

8 

8 

8 

8 

9 

9 

9 

9 

9 

9 

9 

9 

9 

9 

9 

4 

5 
l3 

10 
12 

10 
12 

10 
12 

10 
12 

10 
12 

10 
12 

10 

12 

10 
12 

10 
12 

ID 

II 

II 

n 

II 

1 1 

1 1 
"73 

n 

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n 
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Ta 

II 

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n 

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7 

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i4 

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i4 

i4 

i4 

i4 

i4 

i4 

i5 

i5 

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i5 

i5 

i5 

i5 

i5 

16 

16 

16 

16 

16 

16 

7 

a 

i5 

16 

16 

16 

16 

16 

16 

16 

17 

17 

17 

17 

17 

17 

17 

17 

18 

18 

18 

18 

18 

18 

18 

8 

9 

n 

18 

18 

18 

18 

18 

18 

18 

19 

19 

19 

19 

19 

19 

20 

20 

20 

20 

20 

20 

20 

21 

21 

9 

10 

Jl 

20 

20 

20 

20 

20 

20 

21 

21 

21 

21 

21 

21 

22 

22 

22 

22 

22 

22 

23 

23 

23 

23 

10 

II 

21 

21 

22 

22 

22 

22 

22 

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23 

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24 

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25 

25 

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12 

23 

23 

24 

24 

24 

24 

24 

25 

25 

25 

25 

25 

26 

26 

26 

26 

26 

27 

27 

27 

27 

27 

28 

12 

i3 

25 

25 

26 

26 

26 

26 

26 

27 

27 

27 

27 

28 

28 

28 

28 

28 

29 

29 

29 

29 

29 

3o 

3o 

1 3 

i4 

27 

27 

28 

28 

28 

28 

28 

29 

29 

29 

29 

3o 

3o 

3o 

3o 

3i 

3i 

3i 

3i 

32 

32 

3-2 

32 

i4 

i5 

29 

29 

3o 

3o 

3o 

3o 

3i 

3i 

3i 

31 

32 

32 

32 

32 

33 

33 

33 

33 

34 

34 

34 

34 

35 

1 5 

i6 

3i 

3i 

3i 

li^ 

1i 

32 

~33 

33 

33 

33 

34 

34 

Ta 

34 

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'35 

35 

36 

36 

36 

37 

37 

76 

'7 

33 

33 

33 

M 

34 

M 

35 

35 

35 

35 

36 

36 

36 

37 

37 

37 

37 

38 

38 

38 

39 

39 

39 

17 

i8 

35 

35 

35 

36 

36 

36 

37 

37 

37 

35 

38 

38 

38 

39 

39 

39 

4o 

4o 

4o 

4i 

4. 

4i 

4i 

18 

'9 

37 

37 

37 

38 

38 

38 

39 

39 

39 

4o 

4o 

4o 

41 

4i 

4i 

4i 

42 

42 

42 

43 

43 

43 

44 

19 

20 

39 

^ 

39 

40 

4o 

40 

4i 

4i 

4i 

42 

42 

42 

A3 

43 

A3 

AA 

AA 

AA 

45 

45 

45 

46 

46 

20 

^^ 

4i 

4i 

4i 

42 

42 

42 

43 

43 

^3 

AA 

M 

AA 

45 

45 

46 

46 

46 

Ai 

47 

47 

48 

48 

48 

21 

■2  2 

43 

43 

43 

M 

^^ 

U 

45 

45 

45 

46 

46 

47 

47 

47 

48 

48 

48 

49 

49 

5o 

5o 

5o 

5i 

22 

23 

M 

45 

45 

46 

46 

46 

47 

47 

48 

48 

48 

49 

49 

49 

5o 

5o 

5i 

5i 

5i 

52 

52 

53 

53 

23 

^4 

46 

47 

47 

48 

48 

48 

49 

49 

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5o 

5o 

5i 

5i 

52 

52 

52 

53 

53 

54 

54 

54 

55 

55 

24 

25 

48 

49 

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5o 

5o 

5o 

51 

Si 

52 

52 

53 

53 

53 

54 

54 

55 

55 

55 

56 

56 

57 

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58 

9.5 

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5i 

5i 

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52 

52 

53 

53 

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Ta 

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55 

55 

56 

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57 

57 

58 

58 

59 

59 

59 

60 

96 

27 

52 

53 

53 

54 

54 

54 

55 

55 

56 

56 

57 

57 

58 

58 

59 

59 

59 

60 

60 

61 

6. 

62 

62 

27 

28 

54 

55 

55 

56 

56 

56 

57 

57 

58 

58 

59 

5q 

60 

60 

61 

61 

62 

62 

63 

63 

63 

64 

64 

28 

29 

56 

57 

57 

58 

58 

58 

59 

59 

60 

60 

61 

61 

62 

62 

63 

63 

64 

64 

65 

65 

66 

66 

67 

29 

3o 

58 

59 

59 

60 

60 

61 

61 

62 

62 

63 

63 

64 

64 

65 

65 

66 

66 

67 

67 

68 

68 

_69 

69 

3o 

37 

60 

60 

6i 

61 

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64 

64 

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66 

66 

67 

67 

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69 

69 

70 

70 

71 

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32 

62 

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67 

68 

68 

69 

69 

70 

70 

71 

71 

72 

73 

73 

74 

32 

33 

64 

64 

65 

65 

66 

67 

6- 

68 

68 

69 

69 

70 

70 

71 

72 

72 

73 

73 

74 

74 

75 

75 

76 

33 

34 

66 

66 

67 

67 

68 

69 

69 

70 

70 

71 

71 

72 

73 

73 

74 

74 

75 

75 

76 

77 

77 

78 

78 

34 

35 

6i 

68 

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70 

71 

71 

72 

72 

73 

74 

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Jl 

Jl 

76 

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77 

78 

78 

79 

79 

80 

81 

35 

36 

70 

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71 

71 

72 

73 

73 

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74 

75 

76 

76 

11 

11 

78 

79 

79 

80 

80 

81 

82 

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36 

37 

72 

73 

73 

73 

74 

75 

75 

76 

76 

77 

78 

78 

79 

80 

80 

81 

81 

82 

83 

83 

84 

84 

85 

37 

38 

73 

74 

73 

75 

76 

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77 

78 

79 

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80 

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81 

82 

82 

83 

84 

84 

85 

86 

86 

87 

87 

38 

39 

75 

76 

77 

77 

78 

79 

79 

80 

81 

81 

82 

83 

83 

84 

85 

85 

86 

86 

87 

88 

88 

89 

90 

39 

4o 

_77 

78 

79 

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84 

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86 

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79 

80 

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86 

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90 

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92 

92 

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42 

81 

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83 

84 

85 

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86 

87 

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89 

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90 

91 

92 

92 

93 

94 

95 

95 

96 

97 

42 

43 

83 

84 

85 

85 

86 

87 

87 

88 

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90 

90 

91 

92 

92 

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95 

95 

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97 

97 

98 

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43 

■U 

85 

86 

87 

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89 

90 

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92 

92 

93 

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96 

97 

98 

98 

99 

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92 

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101 

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100 

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io3 

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92 

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93 

94 

95 

96 

96 

97 

98 

99 

99 

100 

lOI 

102 

io3 

io3 

io4 

io5 

106 

107 

107 

108 

47 

48 

93 

94 

94 

9-) 

96 

97 

98 

98 

99 

100 

lOI 

T02 

102 

io3 

104 

io5 

106 

106 

107 

108 

109 

1 10 

1 10 

48 

49 

95 

96 

9!? 

97 

98 

99 

too 

100 

lOI 

102 

io3 

104 

[o5 

io5 

106 

107 

108 

109 

109 

1 10 

n  1 

I  [2 

ii3 

49 

JO 

_2Z 

_?^ 

_9S 

_99 

100 

lOI 

102 

io3 

io3 

1 04 

io5 

106 

l^Z 

108 

108 

109 

no 

in 

112 

ii3 

n3 

ii4 

ii5 

5o 

5i 

99 

99 

100 

101 

102 

io3 

io4 

7^ 

io5 

106 

107 

108 

109 

no 

in 

III 

112 

773 

T74 

775 

116 

116 

i'7 

57 

52 

lOI 

lOI 

102 

io3 

io4 

io5 

106 

107 

107 

108 

109 

no 

in 

112 

n3 

114 

1x4 

n5 

116 

117 

#£! 

"9 

120 

52 

53 

102 

io3 

io4 

io5 

106 

107 

108 

109 

no 

no 

III 

112 

1x3 

ii4 

n5 

116 

1x7 

117 

118 

119 

121 

122 

53 

54 

io4 

io5 

106 

107 

108 

109 

no 

in 

112 

n3 

ii3 

ii4 

ii5 

116 

117 

118 

1x9 

120 

121 

122 

122 

123 

124 

54 

55 

106 

107 

108 

109 

no 

n  I 

112 

n3 

ii4 

ii5 

116 

116 

117 

118 

112 

120 

X2I 

122 

123 

124 

125 

126 

127 

55 

56 

7o8 

109 

no 

lU 

112 

773 

774 

n5 

116 

117 

n8 

119 

119 

120 

121 

122 

7^3 

7^ 

7^ 

126 

127 

128 

129 

56 

57 

no 

1 1 1 

112 

ii3 

ii4 

n5 

116 

117 

118 

119 

120 

121 

122 

123 

124 

124 

125 

126 

127 

128 

129 

i3o 

i3i 

57 

18 

112 

T  I  3 

ii4 

ii5 

n6 

117 

118 

119 

120 

121 

122 

123 

124 

125 

126 

127 

128 

129 

i3o 

i3i 

i3. 

l32 

i33 

58 

*"^9 

ii4 

ii5 

116 

1 17 

n8 

119 

120 

121 

12a 

123 

124 

125 

126 

127 

128 

129 

i3o 

i3i 

l32 

i33 

r34 

i35 

1 36  59 

no 

116 

1 17 

118 

119 

120 

121 

122 

123 

124 

125 

126 

127 

128 

129 

i3o 

i3i 

l32 

i33  i34 

i35 

1 36 

i37 

1 38  60 

P'»g«236i                TABLE  XXX. 

For  finding  the  Variation  of  the  Sun's  Right  Ascension,  of  the  Declination,  c^f  tlie  Equation  of  Time 
or  of  the  Moon's  Rigiit  Ascension,  in  any  number  of  minutes  of  time,  the  Horary  ftlolion  being  ! 

given  at  the  top  of  the  page  in  seconds,  and  the  number  of  miimtes  of  time  in  the  side-column  ; — 

Also,  for  finding  the  Variation  of  the  Moon's  Declination  in  seconds  of  time ;  the  motion  in  one 

minute  being  given  at  the  top,  and  the  numbers  in  the  side-column  being  taken  for  seconds. 

Horarn)  Motion. 

M 

// 

II 

// 

II 

// 

;/ 

// 

// 

II 

1  // 

// 

II 

// 

'/ 

// 

;/ 

II 

// 

/' 

// 

// 

II 

M 

13!! 

140 

141 

142 

143 

144 

145 

14G 

147 

148 

140 

150 

151 

15; 

153 

154 

155 

15C 

157 

158 

15< 

160 

I 

2 

2 

2 

2 

2 

2 

2 

2 

2 

2 

2 

3 

~3 

"1 

""3 

""3 

1 

3 

~~3 

~3 

I 

2 

5 

5 

5 

5 

5 

5 

5 

5 

5 

5 

5 

5 

5 

5 

5 

r 

5 

5 

5 

5 

5 

5 

2 

3 

7 

7 

7 

7 

7 

7 

7 

7 

7 

7 

7 

8 

8 

8 

8 

6 

8 

8 

8 

8 

8 

8 

3 

4 

9 

9 

9 

9 

10 

10 

10 

10 

10 

10 

10 

10 

10 

10 

10 

IC 

IC 

10 

10 

II 

n 

1 1 

4 

5 

12 

12 

12 

12 

12 

12 

12 

12 

12 

12 

12 

i3 

i3 

i3 

i3 

i3 

i3 

i3 

i3 

i3 

i3 

i3 

5 

6 

~U 

i4 

"T4 

i4 

"74 

"74 

~75 

"75 

"75 

~i^ 

"75 

13 

l5 

"75 

~l5 

"75 

IC 

16 

16 

"76 

"76 

"]6 

6 

7 

16 

16 

16 

17 

f7 

17 

17 

17 

17 

17 

17 

18 

18 

18 

18 

i£ 

18 

18 

18 

18 

•9 

'9 

7 

8 

'9 

19 

19 

19 

19 

19 

19 

19 

20 

20 

20 

20 

20 

20 

20 

21 

21 

21 

21 

21 

2] 

21 

8 

9 

21 

21 

21 

21 

21 

22 

22 

22 

22 

22 

22 

23 

23 

23 

20 

23 

23 

23 

24 

24 

24 

24 

9 

lO 

23 

23 

24 

24 

24 

24 

j4 

^ 

25 

25 

25 

25 

25 

25 

26 

26 

26 

26 

26 

26 

27 

27 

10 

II 

25 

~^ 

26 

26 

26 

26 

27 

27 

27 

27 

27 

28 

28 

28 

28 

'Tb 

28 

29 

29 

29 

29 

29 

1 1 

12 

28 

28 

28 

28 

29 

29 

29 

29 

29 

3o 

3f 

3o 

3( 

3o 

3r 

3i 

3i 

3i 

3i 

32 

32 

32 

12 

t3 

3o 

3o 

3i 

3i 

3i 

3i 

3; 

32 

32 

32 

32 

33 

33 

33 

33 

33 

3A 

3A 

3A 

34 

34 

35 

i3 

i4 

32 

33 

33 

33 

33 

34 

34 

34 

M 

35 

35 

35 

35 

35 

36 

3G 

3b 

36 

3l 

0-, 

37 

37 

i4 

i5 

35 

35 

35 

36 

36 

36 

36 

37 

37 

37 

37 

38 

38 

38 

38 

39 

39 

39 

39 

40 

40 

4o 

i5 

Te 

37 

^7 

"38 

38 

^8 

38 

"^ 

39 

39 

39 

4o 

4r 

^ 

~A'i 

~A^ 

4i 

4i 

42 

42 

42 

Ao 

43 

76 

17 

39 

40 

40 

4o 

4i 

4i 

4i 

4i 

42 

42 

42 

A^ 

43 

43 

43 

AA 

AA 

AA 

AA 

45 

45 

45 

17 

i8 

42 

42 

42 

43 

43 

e 

M 

AA 

AA 

AA 

45 

45 

45 

46 

46 

46 

A- 

Ai 

Ai 

47 

48 

48 

18 

19 

AA 

M 

45 

45 

45 

46 

46 

46 

Ai 

Ai 

47 

48 

48 

48 

48 

49 

49 

49 

5o 

5o 

5o 

5i 

'9 

2  CI 

46 

47 

47 

47 

48 

48 

48 

49 

49 

49 

5o 

5o 

5o 

5i 

5i 

5i 

52 

52 

52 

53 

53 

53 

20 

21 

49 

49 

49 

5o 

5o 

5o 

5i 

5i 

5i 

52 

52 

53 

53 

53 

54 

54 

54 

55 

55 

"55 

"56 

56 

21 

22 

5i 

5i 

52 

52 

52 

53 

53 

54 

54 

54 

55 

55 

55 

56 

56 

56 

57 

57 

58 

58 

58 

59 

22 

23 

53 

54 

54 

54 

55 

55 

56 

56 

56 

57 

57 

58 

58 

58 

59 

59 

59 

60 

60 

6r 

61 

6. 

23 

24 

56 

56 

56 

57 

57 

58 

58 

58 

59 

59 

60 

&o 

60 

61 

6. 

62 

62 

62 

63 

63 

64 

64 

24 

25 

58 

58 

59 

59 

60 

60 

60 

61 

61 

62 

62 

63 

63 

63 

64 

64 

65 

65 

65 

66 

66 

67 

25 

^ 

60 

61 

61 

62 

l32 

62 

63 

~63 

64 

"64 

"65 

"65 

"65 

"66 

66 

67 

"67 

^68 

68 

68 

69 
72 

69 

^ 

27 

63 

63 

63 

64 

64 

65 

65 

66 

66 

67 

67 

68 

68 

68 

69 

69 

70 

70 

71 

71 

72 

11 

28 

65 

65 

66 

66 

67 

67 

68 

68 

69 

69 

70 

70 

70 

71 

71 

72 

72 

73 

73 

74 

74 

75 

28 

19 

67 

68 

68 

69 

69 

70 

70 

71 

7' 

72 

72 

73 

73 

73 

74 

74 

75 

75 

76 

76 

77 

77 

29 

3o 

70 

70 

71 

71 

72 

72 

73 

73 

74 

74 

Jl 

75 

76 

76 

77 

77 

78 

78 

79 

79 

80 

80 

3o 

37 

72 

72 

73 

73 

74 

74 

75 

75 

76 

76 

77 

78 

78 

79 

79 

80 

80 

81 

81 

82 

"81 

"83 

37 

32 

74 

75 

75 

76 

76 

77 

77 

78 

78 

79 

79 

80 

81 

8i 

82 

82 

83 

83 

84 

84 

85 

85 

32 

33 

76 

77 

78 

78 

79 

79 

80 

80 

81 

8i 

82 

83 

83 

84 

84 

85 

85 

86 

86 

87 

87 

88 

33 

34 

79 

79 

80 

80 

81 

82 

8? 

83 

83 

84 

84 

85 

86 

86 

87 

87 

88 

88 

89 

90 

90 

9' 

34 

35 

81 

82 

82 

83 

83 

84 

85 

85 

86 

86 

87 

88 

88 

_89 

89 

_90 

90 

9' 

92 

92 

k 

93 

35 

36 

83 

~84 

^ 

"85 

"86 

86 

87 

"88 

88 

"89 

"89 

~9^ 

~9^ 

91 

92 

92 

93 

94 

94 

95 

9"' 

96 

36 

37 

86 

86 

87 

88 

88 

89 

89 

90 

9' 

91 

92 

93 

93 

94 

94 

95 

96 

96 

97 

97 

98 

99 

3-^ 

38 

88 

89 

89 

90 

9' 

9' 

92 

92 

93 

94 

94 

95 

96 

96 

97 

98 

98 

99 

99 

100 

TGI 

lOI 

38 

39 

9" 

9' 

92 

92 

93 

94 

•94 

95 

96 

96 

97 

98 

98 

99 

99 

100 

lOI 

lOI 

102 

io3 

io3 

1 04 

39 

4o 

_93 

93 

94 

Jl 

95 

96 

_97 

_9Z 

j8 

__99 

_99 

100 

lOI 

101 

102 

io3 

io3 

104 

io5 

io5 

106 

107 

4u 

4i 

95 

96 

96 

97 

98 

98 

99 

100 

10(1 

lOI 

102 

77^3 

io3 

io4 

'io5 

776 

i7^ 

107 

107 

108 

109 

109 

47 

42 

97 

98 

99 

99 

100 

101 

102 

102 

io3 

io4 

io4 

io5 

106 

106 

107 

108 

109 

109 

no 

in 

I  1  1 

112 

42 

43 

100 

100 

lOI 

102 

102 

io3 

io4 

io5 

io5 

106 

107 

108 

108 

109 

1 10 

no 

in 

1 12 

ii3 

n3 

n4 

n5 

43 

U 

102 

io3 

io3 

104 

io5 

106 

106 

107 

108 

109 

109 

no 

1 1 J 

1 1 1 

112 

1 13 

iiA 

ii4 

n5 

116 

117 

117 

AA 

45 

104 

io5 

106 

107 

1?2 

108 

109 

no 

no 

III 

112 

ii3 

ii3 

iiA 

n5 

116 

116 

117 

nS 

n9 

119 

120 

45 

46 

107 

107 

ro8 

109 

110 

1 10 

1 1 1 

1 12 

f73 

773 

774 

i75 

n6 

117 

117 

778 

119 

!20 

120 

121 

122 

713 

A'^ 

47 

109 

no 

no 

III 

112 

n3 

ii4 

i^A 

n5 

116 

117 

118 

118 

119 

1 20 

121 

121 

122 

123 

124 

125 

125 

47 

48 

II I 

112 

ii3 

ii4 

ii4 

ii5 

116 

117 

n8 

118 

119 

120 

I2T 

122 

1 22 

123 

124 

125 

126 

126 

127 

128 

48 

49 

ii4 

ii4 

ii5 

116 

117 

118 

iiS 

119 

120 

121 

122 

123 

123 

124 

125 

126 

127 

:27 

128 

120 

i3o 

t3i 

49 

5o 

116 

117 

118 

118 

il? 

120 

121 

122 

123 

123 

124 

125 

126 

127 

128 

128 

129 

l3c: 

i3i 

I  32 

1 33 

i33 

5o 

57 

778 

119 

120 

121 

122 

122 

72! 

124 

125 

126 

127 

7^8 

7^8 

129 

73^ 

73T 

732 

1 33 

733 

1 34 

i35 

736 

5 1 

52 

120 

121 

;i 

,23 

124 

125 

126 

127 

127 

128 

129 

i3o 

i3i 

I  32 

1 33 

1 33 

i34 

i35 

1 35 

1 37 

1 38 

.39 

52 

53 

123 

124 

125 

126 

127 

128 

120 

i3o 

i3i 

I  32 

i33 

1 33 

1 34 

i35 

1 36 

1 37 

1 38 

139 

40 

i4o 

i4i 

53 

54 

125 

126 

127 

128 

129 

i3o 

i3i 

i3. 

I  32 

1 33 

1 34 

i35 

1 36 

i37 

1 38 

139 

i4o 

i4o 

i4i 

42 

143 

1 44 

54 

55 

127 

128 

129 

i3o 

i3i 

l32 

i33 

1 34 

i35 

1 36 

1 37 

1 38 

38 

.39 

1 40 

i4i 

142 

143 

i44_ 

45 

1 46 

i47 

55 

56 

i3o 

717 

732 

733 

<33 

734 

TT5 

i36 

737 

738 

.39 

4o 

"47 

42 

iA3 

1 44 

I45 

1 46 

1 47 

47 

1 48 

149 

56 

57 

l32 

i33 

1 34 

i35 

1 36 

.37 

i38 

.39 

1 40 

•4i 

142 

43 

i43 

AA 

145 

1 46 

1 47 

1 48 

149 

5o 

i5i 

52 

57 

58 

1 34 

i35 

1 36 

■37 

1 38 

139  i4o] 

i4i 

l42 

143 

144 

45 

46 

Ai 

48 

149 

i5o 

.5i' 

l52 

53 

1 54 

55 

58 

59 

i37 

1 38 

139 

i4o 

i4i 

142  143 

1 44 

i45i46| 

i47 

48 

48 

Aq 

i5o 

,5i 

l52 

1 53  1 54 

55 

i56 

57 

59 

60 

139 

.4u 

i4i 

14-2 

i43 

i44i45i46t47!i48| 

149 

5o  i5i| 

52 

1 53 

.54 

.551 

i56i57i58| 

1^9 

60 

So 

TABLE  XXXI.                                      [ 

Page  237 

For  finding  the  Sun's  Right  Ascension  for  any  given  number  of  hours. 

J^umher  of  hours. 

Horary 
Vivrialinn. 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

Hcirary 
Viiri:vlion. 

s 

II 

It 

n 

n 

II 

II 

II 

II 

II 

II 

// 

II 

S 

8. So 

8.5 

17.0 

25.5 

34.0 

42.5 

5i.o 

59.5 

68.0 

76.5 

85.0 

93.5 

102.0 

8.5o 

8.55 

8.6 

17. 1 

25.7 

34.2 

42.8 

5i.3 

59.9 

68.4 

77.0 

85.5 

94.1 

102.6 

8.55 

8.60 

8.6 

17.2 

25.8 

34.4 

43.0 

5i.6 

60.2 

68.8 

77.4 

86.0 

94 .61103. 2 

8.60 

8.65 

8.7 

17.3 

26.0 

34.6 

43.3 

5i.9 

60.6 

69.2 

77-9 

86.5 

93.2 

io3.8 

8.65 

8.70 

8.7 

17.4 

26.1 

34.8 

43.5 

52.2 

60.9 

69.6 

78.3 

87.0 

95.7 

104.4 

8.70 

8.75 

8.8 

17.5 

26.3 

35.0 

43.8 

52.5 

61.3 

70.0 

78.8 

87.5 

96.3 

io5.o 

8.75 

8.80 

8.8 

17.6 

26.4 

35.2 

44.0 

52.8 

61.6 

70.4 

79.2 

88.0 

96.8 

io5.6 

8.80 

8.85 

8.9 

17.7 

26.6 

35.4 

44.3 

53.1 

62.0 

70.8 

79-7 

88.5 

97.4|io6.2 

8.85 

8.00 

8.9 

17.8 

26.7 

35.6 

44.5 

53.4 

62.3 

71.2 

80.1 

89.0 

97-9 

106.8 

8.90 

8.95 

9.0 

17.9 

26.9 

35.8 

44.8 

53.7 

62.7 

71.6 

80.6 

89.5 

98.5 

107.4 

8.95 
9.00 

9.00 

9.0 

18.0 

27.0 

36.0 

45.0 

54.0 

63.0 

72.0 

81.0 

90.0 

99.0 

108.0 

9.05 

9.1 

18. 1 

27.2 

36.2 

45.3 

54.3 

63.4 

72.4 

81.5 

90.5 

99.6 

108.6 

9.05 

9.10 

9.1 

18.2 

27.3 

36.4 

45.5 

54.6 

63.7 

72.8 

81.9 

91 .0 

100. 1 

1 09 . 2 

9.10 

9.15 

9.2 

18.3 

27.5 

36.6 

45.8 

54.9 

64.1 

73.2 

82.4 

91.5 

100.7 

1 09 . 8 

9.  i5 

9.20 

9.2 

18.4 

27.6 

36.8 

46. 0 

55.2 

64.4 

73.6 

82.8 

92.0 

IO[  .2 

1 1 0 . 4 

9.20 

9.25 

9.3 

18.5 

27.8 

37.0 

46.3 

55.5 

64.8 

74.0 

83.3 

92.5 

101.8 

1 1 1 . 0 

9.25 

9.30 

9.3 

18.6 

27.9 

37.2 

46.5 

55.8 

65.1 

74.4 

83.7 

93.0 

102.3 

III. 6 

9.30 

9.35 

9-4 

18.7 

28.1 

37.4 

46.8 

56.1 

65.5 

74.8 

84.2 

93.5 

102.9 

112.2 

9.35 

9.40 

9.4 

18.8 

28.2 

37.6 

47.0 

56.4 

65.8 

75.2 

84.6 

94.0 

io3.4 

112.8 

9.40 

9-45 

9-5 

18.9 

28.4 

37.8 

47-3 

56.7 

66.2 

75.6 

85.1 

94.5 

104.0 

ii3.4 

9-45 

9.50 

9.5 

19.0 

28.5|38.o 

47.5 

57.0 

66.5 

76.0 

85.5 

95.0 

104.5 

.114.0 

9. DO 

9.55 

9.6 

19. 1 

28.7 

38.2 

47.8 

57.3 

66.9 

76.4 

86.0 

95.5 

io5.i 

114. 6 

9.55 

9.60 

9.6 

19.2 

28.8 

38.4 

48.0 

57.6 

67.2 

76.8 

86.4 

96.0 

105.6 

Il5.2 

9.60 

9.65 

9-7 

19.3 

29.0 

38.6 

48.3 

57.9 

67.6 

77.2 

86.9 

96.5 

106.2 

ii5.8 

9-65 

9.70 
9.75 

9-7 

19.4 

29.1 

38.8 

48.5 

58.2 

67.9 

77.6 

87.3 

97.0 

106.7 

1 16.4 

9.70 

9.75 

9.8 

19.5 

29.3 

39.0 

48.8 

58.5 

68.3 

78.0 

87.8 

97.5 

107.3 

1 17.0 

9.80 

t,.8 

19.6 

29.4 

39.2 

49.0 

58.8 

68.6 

78.4 

88.2 

98.0 

107.8 

1 17.6 

9.80 

9.85 

9.9 

19.7 

29.6 

39.4 

49-3 

59.1 

69.0 

78.8 

88.7 

98.5 

108.4 

118. 2 

9.85 

9.90 

9.9 

19.8 

29.7 

39.6 

49-5 

59.4 

69.3 

79.2 

89.1 

99.0 

108.9 

118.8 

9.90 

9.95 

10.00 

10. 0 

19.9 

29.9 

39.8 

49-8 

59.7 

69.7 

79.6 
80.0 

89.6 
90.0 

99.5 
100. 0 

109.5 

IIO.O 

119.4 
120.0 

9.95 

10. 0 

20.0 

3o.o 

4o.o 

5o.o 

60.0 

70.0 

10.00 

io.o5 

10. 1 

20.1 

3o.2 

4o.2 

5o.3 

60.3 

70.4 

80.4 

90.5 

100.5 

II0.6 

120.6 

10. o5 

10.10 

lO.I 

20.2 

3o.3 

40.4 

5o.5 

60.6 

70.7 

80.8 

90.9 

lOI  .0 

III. I 

121.2 

10.10 

io.i5 

10.2 

20.3 

3o.5 

40.6 

5o.8 

60.9 

71. 1 

81.2 

91.4 

loi  .5 

III. 7 

121. 8 

10. i5 

10.20 

10.2 

20.4 

3o.6 

4o.8 

5i.o 

61 .2 

71-4 

81.6 

91.8 

102.0 

112. 2 

122.4 

10.20 

10.25 

10.3 

20.5 

3o.8 

4i  .0 

5i.3 

61.5 

71.8 

82.0 

92.3 

102.5 

112. a 

123. 0 

10.25 

10. 3o 

10.3 

20.6 

3o.9 

4i  .2 

5i.5 

61.8 

72.1 

82.4 

92.7 

io3.o 

ii3.3  123.6 

io.3o 

10.35 

10.4 

20.7 

3i.i 

41.4 

5i  8 

62.1 

72.5 

82.8 

93.2 

io3.5 

113.9 

1 24 . 2 

10.35 

10.40 

10.4 

20.8 

3l.2 

4i.6 

52  0 

62.4 

72.8 

83.2 

93.6 

104.0 

114.4 

124.8 

10.40 

10.45 

10.5 

20.9 

3i.4 

4i.8 

52.3 
"52".  5 

62.7 

73.2 

83.6 

94.1 

104.5 

ii5.o 

125.4 

10.45 

io.5o 

10.5 

21 .0 

3i.5 

42.0 

03. 0 

73.5 

84.0 

94.5 

io5.o 

ii5.5 

126.0 

10. 5o 

10.55 

10.6 

21. 1 

3i.7 

42.2 

52.8 

63.3 

73  <9 

84.4 

95.0 

io5.5 

116.1 

126.6 

10.55 

10.60 

10.6 

21.2 

3i.8 

42.4 

53.0 

63.6 

74.2 

84.8 

95.4 

106.0 

1 16.6 

127.2 

10.60 

10.65 

10.7 

21.3 

32.0 

42.6 

53.3 

63.9 

74.6 

85.2 

95-9 

106.5 

117.2 

127.8 

■10.65 

10.70 
10.75 

10.7 

21.4 

32.1 

42.8 

53.5 

64.2 

74.9 

75.3 

85.6 
86.0 

96.0 

107.0 

117.7 

128.4 

10.70 

10.8 

21 .5 

32.3 

43.0 

53.8 

64.5 

96.8 

107.5 

118. 3 

129.0 

10.75 

10.80 

10.8 

21.6 

32.4 

43.2 

54.0 

64.8 

75.6 

86.4 

97.2 

108.0 

118. 8 

129.6 

10.80 

10.85 

10.9 

21.7 

32.6 

43.4 

54.3 

65.1 

76.0 

86.8 

97-7 

108.5 

119. 4 

l30.2 

10.85 

10.90 

10.9 

21.8 

32.7 

43.6 

54.5 

65.4 

76.3 

87.2 

98.1 

109.0 

119. 0 

i3o.8 

10.90 

10.95 

II  .0 

21 .9 

32.9 

43.8 

54.8 

65.7 

76.7 

87.6 

98.6 

109.5 

120.5 

i3i.4 

10.95 

11 .00 

II  .0 

22.0 

33.0 

44.0 

55.0 

66.0 

77.0 

88.0 

99.0 

IIO.O 

121 .0 

l32.0 

11.00 

11  .OD 

II. I 

22.1 

33.2 

44.2 

55.3 

66.3 

77-4 

88.4 

99.5 

no. 5 

121 .6 

132.6 

II  .o5 

II  .10 

II  .1 

22.2 

33.3 

M.A 

55.5 

66.6 

77-7 

88.8 

99.9 

III  .0 

122. 1 

i33.2 

1 1 .10 

II. i5 

11 .2 

22.3 

33.5 

44.6 

55.8 

66.9 

78.1 

89.2 

100.4 

III  .5 

122.7 

133.8 

II. i5 

II  .20 

II. 2 

22.4 

33.6 

44.8 

56.0 

67.2 

78.4 

89.6 

100.8 

112.0 

123.2 

1 34. 4 

II  .20 

11  .25 

II. 3 

22.5 

33.8 

45.0 

56.3 

67.5 

78.8 

90.0 

I0I.3 

112. 5 

123.8 

i35.o 

11.25 

1 1 .3o 

II. 3 

22.6 

33.9 

45.2 

56.5 

67.8 

79.1 

90.4 

101.7 

ii3.o 

124.3 

i35.6 

1 1. So 

11.35 

II. 4 

22.7 

34.1 

45.4 

56.8 

68.1 

79.5 

90.8 

102.2 

ii3.5 

124.9 

136.2 

11.35 

II  .40 

II. 4 

22.8 

34.2 

45.6 

57.0 

68.4 

79-8 

91 .2 

102.6 

114.0 

125.4 

i36.8 

11.40 

11.45 

II. 5 

22.9 

34.4 

45.8 

57.3 

68.7 

80.2 

91 .6 

io3.i 

114.5 

126.0 

137.4 

11.45 

J'^e*  2381                                    TABLE  XXXI. 

For  finding  the  Sun's  Right  Ascension  for  any  given  number  of  hours. 

JVumher  of  hours. 

Horary 

Variatinn. 

13 

14 

15 

16 

17 

18 

19i 

20. 

21  1  22 

23 

24 
II 

Horary 
Variation. 

s 

// 

// 

II 

II 

II 

■/ 

II 

II 

// 

// 

II 

s 

8.5o 

no. 5 

119. c 

127.5 

i36.o 

144.5 

i53.c 

161. 5 

170.0 

178. f 

187.0 

195.5 

204.0 

8.5o 

8.55 

III. 2 

119-7 

128.3 

i36.8 

145.4 

i53.(; 

162.5 

171 .0 

i79.e 

188. 1 

196.7 

205.2 

8.55 

8.60 

III. 8 

120.4 

129.0 

137.6 

146.2 

i54.8 

i63.4 

172.0 

180. c 

189.2 

IQ7.8 

206.4 

8.60 

8.65 

112.5 

121. 1 

129.8 

i38.4 

147.1 

i55.7 

164.4 

173.0 

181.' 

190.3 

199.0 

207.6 

8.65 

8.70 
8.75 

ii3.i 

121. 8 

i3o.b 

139.2 

147-9 

i56.6 

i65.3 

174.0 

182.: 

191.4 

200.1 

208.8 

8.70 

ii3.'8 

122.5 

i3i.3 

i4o.o 

148.8 

157.5 

166.3 

175.0 

i83.£ 

192.5 

201.3 

210.0 

8.75 

8.80 

114.4 

123.2 

l32.0 

i4o.8 

149.6 

i58.4 

167.2 

176.0 

184. £ 

193.6 

202.4 

211  .2 

8.80 

8.85 

lib. I 

123.9 

l32.b 

i4i.6 

i5o.5 

159.3 

168.2 

177.0 

l85.q 

194.7 

2o3.6 

212.4 

8.85 

8.90 

.15.7 

124.6 

133.5 

142.4 

i5i.3 

160.2 

169. 1 

178.0 

186. q 

iq5.8 

204.7 

213.6 

8.90 

8.95 

9...0 

116. 4 

125. J 

134.3 

143.2 

l52.2 

161 .1 

170. 1 

179.0 

188. c 

196.9 

205.9 

214.S 

8.95 

1 17.0 

126.0 

i35.o 

144.0 

i53.o 

162.0 

171 .0 

180.0 

189. c 

198.0 

207.0 

216.0 

9.00 

9.05 

117. 7 

126.7 

i3b.8 

144.8 

153.9 

162.9 

172.0 

181. 0 

190. 1 

199.1 

208.2 

217.2 

9.05 

9.10 

118.3 

127.4 

i3b.b 

145.6 

ib4.7 

i63.8 

172.9 

182.0 

191. 1 

200.2 

209.3 

218.4 

9.10 

9.15 

1 19.0 

128. 1 

137-^ 

146.4 

i55.6 

164.7 

173.9 

i83.o 

192.2 

201.3 

210.5 

219.6 

?.i5 

9.20 

119. 6 

128.8 

i38.o 

147.2 

i56.4 

i65.6 
166.5 

174.8 

184.0 

193.2 

202.4 

211. 6 

220.8 

9.20 

9.25 

120.3 

129.5 

i38.8 

i48.o 

157.3 

175.8 

i85.o 

194.3 

2o3.5 

212.8 

222.0 

9.25 

9.30 

120.9 

i3o.2 

139.5 

148.8 

i58.i 

167.4 

176.7 

186.0 

195.3 

204.6 

213.9 

223.2 

9.30 

9.35 

121. 6 

1 30.9 

140.3 

149.6 

159.0 

168.3 

177.7 

187.0 

196.4 

205.7 

2l5.I 

224.4 

9.35 

9.40 

122.2 

i3i.6 

i4i  .0 

i5o.4 

159.8 

169.2 

178.6 

188.0 

197.4 

206.8 

216.2 

225.6 

9.40 

9-45 

122.9 

132.3 

i4i.8 

i5i  .2 

160.7 

170. 1 

179.6 

189.0 

198.5 

207.9I2I7.4 

226.8 

9.45 

9.50 

123.5 

i33.o 

142.5 

I  52.0 

161.5 

171 .0 

180.5 

190.0 

199.5 

209.0^18.5 

228.0 

9.50 

9.55 

124.2 

133.7 

143.3 

i52.8 

162.4 

171.9 

181. 5 

191 .0 

200.6 

210. 1 

219.7 

229.2 

9.55 

9.60 

124.8 

134.4 

i44.o 

i53.6 

i63.2 

172.8 

182.4 

192.0 

201 .6 

211  .2 

220.8 

23o.4 

9.60 

9-65 

125.5 

i35. 1 

144.8 

154.4 

164. 1 

173.7 

i83.4 

193.0 

202.7 

212.3 

222.0 

23i.6 

9-65 

9.70 

126. 1 

135.8 

i4b.5 

i55.2 

164.9 

174.6 

184.3 

194.0 

203.7 

2i3.4 

223.1 
224.3 

232.8 

234.0 

9.70 
9.75 

9-p 

126.8 

136.5 

146.3 

i56.o 

i65.8 

175.5 

i85.3 

195.0 

204.8 

214.5 

9.80 

127.4 

137.2 

i47-o 

i56.8 

166.6 

176.4 

186.2 

196.0 

2o5.S 

2i5.6 

225.4 

235.2 

9.80 

9.85 

128. 1 

137.9 

147-8 

157.6 

167.5 

177.3 

187.2 

197.0 

206.9 

216.7 

226.6 

236.4 

9-85 

9.90 

128.7 

JJ8.6 

148. b 

i58.4 

168.3 

178.2 

188.1 

198.0 

207.9 

217.8 

227.7 

237.6 

9.90 

9.95 

129.4 

139.3 

149.3 

159.2 

169.2 

179.1 

189. 1 

199.0 

209.0 

218.9 

228.9 

238.8 

9.95 

10,00 

i3o.o 

i4o.o 

ibo.o 

160.0 

170.0 

180.0 

190.0 

200.0 

210.0 

220.0 

23o.o 

240.0 

10.00 

10. o5 

i3o.7 

140.7 

i5o.8 

160.8 

170.9 

180.9 

191. 0 

201 .0 

211 .1 

221. I 

23l.2 

241.2 

10. o5 

10.10 

i3i.3 

141.4 

i5i.5 

161. 6 

171. 7 

18J.8 

191. 9 

202.0 

212. 1 

222.2 

232.3 

242.4 

10.10 

10. i5 

l32.0 

142. 1 

lb2.3 

162.4 

172.6 

182.7 

192.9 

2o3.0 

2l3.2 

223.3 

233.5 

243.6 

10. i5 

10.20 

t32.6 

142.8 

ib3.o 

i63.2 

173.4 

i83.6 

193.8 

204.0 

214.2 

224.4 

234.6 

244.8 

10.20 

10.20 

i33.3 

143.5 

i53.8 

164.0 

174.3 

184.5 

194.8 

205.0 

2i5.3 

225.5 

235.8 

246.0 

10.25 

10. 3o 

133.9 

144.2 

ib4.5 

164.8 

175. 1 

i85.4 

195.7 

206.0 

216.3 

226.6 

236-9 

247.2 

10. 3o 

10.35 

i34.6 

144.9 

i5b.3 

165.6 

176.0 

186.3 

196.7 

207.0 

217.4 

227.7 

238.1 

248.4 

10.35 

10.40 

135.2 

145.6 

ib6.o 

166.4 

176.8 

187.2 

197.6 

208.0 

218.4 

228.8 

239.2 

249.6 

10.40 

10.45 

iJb.9 
i36.5 

146-3 
i47-o 

ib6.8 
157.5 

167.2 

177.7 

188.1 

198.6 

209.0 

219.5 

229.9 

240.4 

250.8 

10.45 

io.5o 

168.0 

178.5 

189.0 

199.5 

210.0 

220.5 

23i  .0 

241.5 

252.0 

io.5o 

10.55 

137.2 

i47-7 

i58.3 

168.8 

179.4 

l89.q 

200.5 

211  .0 

221  .6 

232.1 

242.7 

253.2 

10.55 

10.60 

137.8 

148.4 

159.0 

169.6 

180.2 

190.8 

201 .4 

212.0 

222.6 

233.2 

243.8 

254.4 

10.60 

10. 65 

i38.5 

149- 1 

ib9.8 

170.4 

181. 1 

191. 7 

202.4 

2l3.0 

223.7 

234.3 

245.0 

255.6 

10.65 

10.70 

139. 1 

149.8 

160.5 

171 .2 

181. 9 

192.6 

2o3.3 

214.0 

224.7 

235.4 

246.1 

256.8 

10.70 

10.75 

139.8 

1 5o .  5 

161. 3 

172.0 

182.8 

193.5 

204.3 

2l5.0 

225.8 

236.5 

247.3 

258.0 

10.75 

10.80 

i4o.4 

l5l.2 

162.0 

172.8 

i83.6 

194.4 

205.2 

216.0 

226.8 

237.6 

248.4 

259.2 

10.80 

10.85 

i4i  .1 

i5i  .9 

162.8- 

173.6 

184.5 

195.3 

206.2 

217.0 

227.9 

238.7 

249.6 

260.4 

10. 85 

10.90 

141.7 

ib2.6 

i63.b 

174.4 

185.3 

196.2 

207.1 

218.0 

228.9  239. 8| 

25o.7 

261 .6 

10.90 

10.95 

142.4 

i53.3 

164.3 

175.2 

186.2 

197.1 

208.1 

219.0 

23o.O 

240.9 

251.9 

262.8 

10.95 
11.00 

11 .00 

143.0 

1 54.0 

i65.o 

176.0 

1S7.0 

198.0 

209.0 

220.0 

23l  .0 

242.6 

253. 0 

264.0 

II. o5 

143.7 

154.7 

165.8 

176.8 

187.9 

198.9 

210.0 

221  .0 

232.1 

243.1 

254.2 

265.2 

II  .o5 

II. 10 

144.3 

ibb.4 

166. b 

177.6 

188.7 

199.8 

210.9 

222.0 

233.1 

244.2 

255.3 

266.4 

II  .10 

II. i5 

145.0 

i56.i 

167.3 

17B.4 

189.6 

200.7 

211  .0 

223. 0 

234.2 

245.3 

256.5 

267.6 

II. i5 

II  .20 

145.6 
146.3 

1 56.8 

168.0 

179.2 

190.4 

201 .6 

212.8 

224.0 

235.2 

246.4 

257.6 

268.8 

11.20 
11.25 

11 .25 

157.5 

168.8 

180.0 

191 .3 

202.5 

2l3.8 

225.0 

236.3 

247.5 

258.8 

270.0 

II  .3o 

146.9 

i58.2 

169.5 

180.8 

1 92. 1 

2o3.4 

214.7 

226. 0 

237.3 

248.6 

259.9 

271 .2 

II  .3o 

11.35 

147.6 

1 58. 9 

170.3 

181. 6 

193.0 

204.3 

2i5.7 

227.0 

238.4 

249.7 

261 .1 

272  .4 

11.35 

II  .4o 

148.2 

59.6  1 7 1 . 0 

182.4 

193.8 

2o5.2 

216.6 

228.0 

239.4 

25o.8 

262.2 

273.6 

II  .40 

11.45 

14S.9 

60.3J171.8  i83.2| 

194.7 

206.1 

217.6 

229.0 

240.5  25 1 .9I263.4I 

274.8 

11.45 

TABLE   XXXII. 

[Pa-e239 

Variation  of"  the  Sun's  Altitude  in 

one 

minute  from  noon. 

Lat. 

Declination  of  a  different  name 

from 

the  Latitude. 

Lat. 

0= 

1° 

9° 

3° 

40 

5° 

6° 

7° 
II 

8° 

9° 

10° 

11° 

// 

n 

II 

II 

II 
28.1 

II 

II 

i4.o 

II 

II 

II 

o° 

22.4 

18.7 

16.0 

12.4 

1 1 . 1 

10. 1 

c'' 

I 

28.1 

22.4 

18.7 

16.0 

i4.o 

12.4 

1 1 .2 

10. 1 

9.3 

I 

2 

1 

28.1 

22.4 

18.7 

16.0 

14.0 

12.5 

1 1 .3 

10.2 

9.3 

8.6 

2 

3 

28.1 

22.4 

18.7 

16.0 

i4.o 

12. D 

11 .2 

10.2 

9.3 

8.6 

8.0 

3 

•4 

28.1 

22.4 

18.7 

16.0 

i4.o 
12.5 

12.5 
II  .2 

II  .2 

10.2 

9.3 
8.6 

8.6 

8.0 

7-4 

4 

5 

22.4 

.8.7 

16.0 

i4.o 

10.2 

9.3 

8.0 

7-4 

7.0 

5 

6 

.8.7 

16.0 

i4-o 

12.5 

1 1 .2 

10.2 

9.3 

8.6 

8.0 

7.5 

7.0 

6.6 

6 

7 

16.0 

i4-o 

12.4 

II  .2 

10.2 

9.3 

8.6 

8.0 

7.5 

7.0 

6.6 

6.2 

7 

8 

i4-o 

12.4 

11.2 

10.2 

9.3 

8.6 

8.0 

7-5 

,  7.0 

6.6 

6.2 

5.9 

8 

9 

12.4 

ti  .2 

10.2 

9.3 
8.6 

8.6 

8.0 

7.5 

7.0 

6.6 

6.2 

5.9 

5.6 

9 

lO 

II .  I 

10. 1 

tl 

8.0 

7-4 

7.0 

6.6 

6.2 

5.9 

5.6 

5.3 

10 

1 1 

10. 1 

9.3 

8.0 

7-4 

7.0 

6.6 

6.2 

5.9 

5.6 

5.3 

5.1 

1 1 

12 

9.2 

8.5 

7-9 

7.4 

7.0 

6.5 

6.2 

b.9 

5.6 

5.3 

5.0 

4.8 

12 

i3 

8.5 

7-9 

7.4 

6.9 

6.5 

6.2 

5.8 

5.6 

5.3 

5.0 

4.8 

4.6 

i3 

1 4 

7-9 
7.3 

7.4 

6.9 

6.5 

6.2 

5.8 

5.5 

5.3 

5.0 

4.8 

4.6 

4.4 

i4 

i5 

6.9 

6.5 

6.1 

5.8 

5.5 

5.3 

5.0 

4.8 

4.6 

4.4 

4.1 

i5 

1 6 

6.8 

6.5 

6.1 

5.8 

5.5 

5.2 

5.0 

4.8 

4.6 

4.4 

4.2 

4.1 

16 

17 

6.4 

6.T 

5.8 

5.5 

5.2 

5.0 

4.8 

4.6 

4.4 

4.1 

4.1 

3.9 

17 

1 8 

6.0 

5.7 

5.5 

5.2 

f.o 

4.8 

4.6 

4.4 

4.2 

4.1 

3.9 

3.8 

18 

19 

5.7 

5.4 

5.2 

4.9 

4.7 

4.5 

4.4 

4.1 

4.0 

3.9 

3.8 

3.5 

'9 

20 

5.4 

5.1 

4.9 

4.7 

4.5 

4.3 

4.2 

4.0 

3.9 

3.8 

3.6 

20 

21 

5.1 

4.9 

4.7 

4.5 

4.3 

4.2 

4.0 

3.9 

3.7 

3.6 

3.5 

3.4 

21 

22 

4.9 

4.7 

4.5 

4.3 

4.1 

4.0 

3.9 

3.7 

3.6 

3.5 

3.4 

3.3 

22 

23 

4.6 

4.4 

4.3 

4.1 

4.0 

3.8 

3.7 

3.6 

3.5 

3.4 

3.3 

3.2 

23 

24 
25 

4.4 

4.2 

4.1 

3.9 

3.8 

3.7 

3.6 

3.5 

3.4 

3.3 

3.2 

3.1 

4.2 

4.1 

3.9 

3.8 

3.7 

3.5 

3.4 

3.3 

3.2 

3.1 

3.1 

3 .; 

26 

4.0 

3.9 

3.8 

3.6 

3.5 

3.4 

3.3 

3.2 

3.1 

3.0 

3.0 

2.9 

26 

27 

3.9 

3.7 

3.6 

3.5 

3.4 

3.3 

3.2 

3.1 

3.0 

2.9 

2.9 

2.8 

27 

28 

3.7 

3.6 

3.5 

3U 

3.3 

3.2 

3.1 

3.0 

2.9 

2.8 

2.8 

2.7 

28 

29 

3.5 
3.4 

3.4 

3.3 

3.2 

3.1 

3.1 

3.0 

2.9 

2.8 

2.8 

2.7 

2.6 

29 

3o 

3.3 

3.2 

3.1 

3.0 

3.0 

2.9 

2.8 

2.7 

2.7 

2.6 

2.5 

3o 

3i 

3.3 

3.2 

3.1 

3.0 

2.9 

2.9 

2.8 

2.7 

2.6 

2.6 

2.5 

2.5 

3i 

32 

3.1 

3.1 

3.0 

2.9 

2.8 

2.8 

2.7 

2.6 

2.6 

2.5 

2.5 

2.4 

32 

33 

3.0 

2.9 

2.9 

2.8 

2.7 

2.7 

2.6 

2.5 

2.5 

2.4 

2.4 

2.3 

33 

34 

2.9 

2.8 

2.8 

2.7 

2.6 

2.6 

2.5 

2.5 

2.4 

2.4 

2.3 

2.3 

34 

35 

2.8 

2.7 

2.7 

2.6 

2.5 

2.5 

2.4 

2.4 

2.3 

2.3 

2.2 

2.2 

35 

■    3G 

2.7 

2.6 

2.6 

2.5 

2.5 

2.4 

2.4 

2.3 

2.3 

2.2 

2.2 

2.1 

36 

37 

2.6 

2.5 

2.5 

2.4 

2.4 

2.3 

2.3 

2.2 

2.2 

2.2 

2.1 

2.  I 

37 

38 

2.5 

2.5 

2.4 

2.4 

2.3 

2.3 

2.2 

2.2 

2.1 

2. 1 

2.  I 

2.0 

38 

39 

2.4 

2.3 

2.4 

2.3 

2.3 

2.2 

2.2 

2.1 

2. 1 

2.1 

2.0 

2.0 

2.0 

39 

4o 

2.3 

2.2 

2.2 

2.2 

2.1 

2.1 

2.0 

2.0 

2.0 

1.9 

1.9 

40 

4i 

2.3 

2.2 

2.2 

2.  I 

2.  I 

2.T 

2.0 

2.0 

1.9 

1.9 

1.9 

1:8 

4i 

42 

2.2 

2.  I 

2.  1 

2.  I 

2.0 

2.0 

2.0 

1.9 

1.9 

1 .9 

1.8 

1.8 

42 

43 

2.  1 

2.1 

2.0 

2.0 

2.0 

1.9 

1.9 

1.9 

1.8 

1 .8 

1.8 

1-7 

43 

44 

2.0 

2.0 

2.0 

1.9 

1.9 

1.9 

1.8 

1.8 

1.8 

1-7 

1-7 

1-7 

44 
45 

45 

2.0 

1.9 

I  .9 

1.9 

1.8 

1.8 

1.8 

1  -7 

1-7 

1-7 

1-7 

1.6 

46 

1.9 

1.9 

1  .8 

1.8 

1.8 

1-7 

1-7 

1-7 

1-7 

1.6 

1.6 

I.b 

46 

47 

1.8 

1.8 

1.8 

1-7 

I  -7 

1-7 

1-7 

i.b 

1.6 

1.6 

1.6 

I.b 

47 

48 

1.8 

'•7 

'  .7 

1-7 

1-7 

1.6 

1.6 

1.6 

1.6 

1.6 

1.5 

1.5 

48 

49 

1.7 

'•7 

7 

1.6 

1.6 

1.6 

1.6 

1.5 

1.5 

1.5 

1.5 

1.5 

49 
5o 

5o 

1.6 

1.6 

6 

1.6 

1.6 

1.5 

1.5 

1.5 

1.5 

1.5 

1.4 

1.4 

52 

1.5 

1.5        .5 

1.5 

1.5 

1,4 

1.4 

1.4 

1.4 

1.4 

1.4 

1.3 

52 

54 

1.4 

1.4      i./\ 

1.4 

1.4 

I  3 

1.3 

1.3 

1.3 

1.3 

1.3 

1.3 

54 

56 

1.3 

1.3      1.3 

1.3 

1.3 

1.3 

1.2 

1 .2 

1 .2 

1 .2 

1 .2 

1 .2 

56 

58 

1 .2 

1 .2 

1 .2 

1 .2 

1 .2 

1 .2 

1 .2 

I .  I 

1 .1 

1 .1 

1 .1 

1 . 1 

58 

60 

f .  I 

I .  I 

1 .1 

1 .1 

I .  I 

I .  I 

1 . 1 

I .  I 

1 .0 

1 .0 

1 .0 

1 .0 

60 

62 

1 .0 

1 .0 

1 .0 

1 .0 

I'.O 

1 .0 

1 .0 

I.O 

1 .0 

1 .0 

1 .0 

0.9 

62 

64 

1 .0 

0.9 

0.9 

0.9 

0.9 

0.9 

c; 

0.9 

0.9 

0.9 

0.9 

0.9 

64 

66 

0.9 

0.9 

0.9 

0.9 

0.8 

0.8 

0.8 

0.8 

0.8 

0.8 

0.8 

0.8 

66 

68 

0.8 

0.8 

0.8 

0.8 

0.8 

0.8 

0.8 

0.8 

0.8 

0.7 

0.7 

0.7 

68 

70 

0.7 

0.7 

0.7 

0.7 

0.7 

0.7 

0.7 

0.7 

0.7 

0.7 

0.7 

0.7 

70 

QP 

1' 

2" 

3° 

40 

5° 

6° 

7° 

8° 

9° 

10° 

11° 

r-i=e240]  TABLE   XXXII. 

Variation  of  the  Sun's  Altitude  in  one  minute  from  noon. 


Declination  of  a 

different  name  from  the 

Latitu 

de. 

Lat. 
~0° 

12° 

13° 

14° 

15° 

16° 

17° 

18° 

19^ 

20° 

21° 

22° 

23° 

24° 

Lnt. 

// 

II 

// 

II 
7.3 

6.8 

// 

II 

II 

// 

II 

II 

II 

II 

0° 

9.2 

8.5 

7-9 

6.4 

6.0 

5.7 

5.4 

5.1 

4.9 

4.6 

4  4 

I 

8.5 

7-9 

7.4     6.9 

6.5 

6.1 

5.7 

5.4 

5.1 

4.9 

4.7 

4.4 

4  2 

1 

2 

7-9 

7.4 

6.9  :  6.5 

6.1 

5.8 

5.5 

5.2 

4.9 

4.7 

4.5 

4.6 

4  I 

2 

3 

7-4 

6.9 

6.5     6.1 

5.8 

5.5 

5.2 

4.9 

4.7 

4.5 

4.6 

4.1 

3  9 

3 

4 

7.0 

6.5 

6.2 

5.8 

5.5 

5.2 

5.2 

5.0 

5.0 

4.8 

4.7 

4.5 

4.3 

4.1 

4.0 

3  8 

4 
'  5 

5 

6.5 

6.2 

5.8 

5.5 

4.5 

4.3 

4.2 

4.0 

3.8 

3  7 

6 

6.2 

5.8 

5.5 

5.3 

5.0 

4.8 

4.6 

4.4 

4.2 

4.0 

3.9 

3.7 

3.6 

6 

7 

5.q 

5.6 

5.3 

5'.o 

4.8 

4.6 

4.4 

4.2 

4.0 

3.9 

3.7 

3.6 

3.5 

7 

8 

5.6 

5.3 

5.0 

4.8 

4.6 

4.4 

4.2 

4.0 

3.9 

3.7 

3.6 

3.5 

3.4 

8 

_9_ 

lO 

5.3 

5.0, 

4.8 

4.6 

4^4 
4.2 

4.2 
4.1 

4.1 

3.9 

3.8 

3.6 

3.5 

3.4 

3.3 

_9_ 

10 

5.0 

4.8 

4.6 

4.4 

3.9 

3.8 

3.6 

3.5 

3.4 

3.3 

3.2 

T  I 

4.8 

4.6 

4.4 

4.2 

4.1 

3.9 

3.8 

3.6 

3.5 

3.4 

6.6 

3.2 

3.1 

11 

12 

4.6 

4.4 

4.3 

4.1 

3.9 

3.8 

3.7 

3.5 

3.4 

3.3 

3.2 

3.1 

3.0 

12 

t3 

A.A 

4.3 

4.1 

3.9 

3.8 

3.7 

3.5 

3.4 

■6.6 

3.2 

3.1 

3.0 

2.9 

1 3 

i4 
i5 

4.2 

4.1 

3.9 

3.8 

3.7 

3.5 

■6.4 

3.3 

3.2 

3.1 

3.0 

2.9 

2.8 

i4 
i5 

4.1 

3.9 

3.8 

3.7 

3.5 

3.4 

3.3 

3.2 

3.1 

3.0 

2.9 

2.8 

2.8 

i6 

3.9 

3.8 

3.7 

3.5 

3.4 

3.3 

3.2 

3.1 

3.0 

2.9 

2.8 

2.8 

2.7 

16 

17 

3.8 

3.7 

3.5 

3.4 

3.3 

3.2 

3.1 

3.0 

2.9 

2.8 

2.8 

2.7 

2.6 

17 

i8 

3.7 

3.5 

3.4 

3.3 

3.2 

3.1 

3.0 

2.9 

2.9 

2.8 

2.7 

2.6 

2.5 

18 

20 

3.5 

3.4 

3.3 

3.2 

3.1 

3.0 

2.9 

2.9 

?.8 

2.7 
2.6 

2.6 

2.6 

2.5 

-9. 
20 

^.A 

3.3 

3.2 

3.1 

3.0 

2.9 

2.9 

2.8 

2.7 

2.6 

2.5 

2.4 

21 

3.3 

3.2 

3.1 

3.0 

2.9 

2.8 

2.8 

2.7 

2.6 

2.6 

2.5 

2.4 

2.4 

21 

2  2 

3.2 

3.1 

3.0 

2.9 

2.8 

2  8 

2.7 

2.6 

2.6 

2.5 

2.4 

2.4 

2.3 

22 

23 

'  3.1 

3.0 

2.9 

2.8 

2.8 

2.7 

2.6 

2.6 

2.5 

2.4 

2.4 

2.3 

2.3 

2  3 

24 
25 

3.0 

2.9 

2.8 

2.8 

2.7 

2.6 

2.5 

2.5 

2.4 

2.4 

2.3 

2.3 

2  .2 

24 

25 

2.9 

2.8 

2.7 

2.7 

2.6 

2.5 

2.5 

2.4 

2.4 

2.3 

2.3 

2.2 

2.2 

26 

2.8 

2.7 

2.7 

2.6 

2.5 

2.5 

2.4 

2.4 

2.3 

2.3 

2.2 

2.1 

2.1 

26 

27 

2.7 

2.7 

2.6 

2.5 

2.5 

2.4 

2.4 

2.3 

2.2 

2.2 

2.1 

2.  I 

2.1 

27 

28 

2.6 

2.6 

2.5 

2.5 

2.4 

2.3 

2.3 

2.2 

2.2 

•2.1 

2.1 

2.1 

.0 

28 

29 

3o 

2.6 

2.5 

2.4 

2.4 

2.3 

2.3 

2.2 

2.2 

2.  I 

i  .  I 

2.0 

2.0 

2.0 

29 

"3o 

2.5 

2.4 

2.4 

2.3 

2.3 

2.2 

2.2 

2.1 

2.1 

3.0 

2.0 

2.0 

1.9 

3i 

2.4 

2.4 

2.3 

2.3 

2.2 

2.2 

2.  I 

2.1 

2.0 

2.0 

2.0 

1.9 

1.9 

3i 

32 

2.3 

2.3 

2.2 

2.2 

2.2 

2.  I 

2.  I 

2.0 

2.0 

1.9 

1-9 

1.9 

1.8 

32 

33 

2.3 

2.2 

2.2 

2.1 

2.1 

2.1 

2.0 

2.0 

1.9 

1.9 

1-9 

1.8 

1.8 

33 

34 
35 

2.2 

2.2 

2.1 

2.1 
2.0 

2.0 

2.0 

2.0 

1.9 

1.9 

1.9 

1.8 

1.8 

1.8 

35 

2.2 

2.1 

2.1 

2.0 

2.0 

1.9 

1.9 

1.8 

1.8 

1.8 

1-7 

1-7 

36 

2.1 

2.1 

2.0 

2.0 

1.9 

1.9 

1.9 

1 .8 

1.8 

1.8 

1-7 

1-7 

1-7 

36 

37 

2.0 

2.0 

2.0 

1.9 

1.9 

1.9 

1.8 

1.8 

1.8 

1-7 

1-7 

1-7 

1.6 

37 

38 

2.0 

1.9 

1.9 

1.9 

1.8 

1.8 

1.8 

1.8 

1-7 

I  -7 

1-7 

1.6 

1.6 

38 

39 
40 

1.9 

1.9 

1.9 

1.8 

1.8 

1.8 

1-7 

1-7 

1-7 

1.6 

1.6 

1. 6 

1.6 

39 

40 

1.9 

1.8 

1.8 

1.8 

1-7 

1-7 

•■7 

1-7 

1.6 

1.6 

16 

1.6 

1.5 

4i 

1 .8 

T.8 

1.8 

>  -7 

1-7 

1-7 

1.6 

1.6 

1.6 

1.6 

1.5 

1.5 

1.5 

4i 

42 

1.8 

1-7 

1.7 

1-7 

1-7 

1.6 

1.6 

1.6 

1.6 

1.5 

1.5 

1.5 

1.5 

42 

43 

1-7 

1-7 

1-7 

1.6 

1 .3 

1.6 

1.6 

1.5 

1.5 

1.5 

1.5 

1.4 

1.4 

43 

44 
45 

1-7 

1.6 

1.6 

1.6 

1.6 

1.5 

1.5 

1.5 
1.5 

1.5 

J. 5 

1.5 

1.4 

1.4 

1.4 

44 
45 

1.6 

1.6 

1.6 

1.5 

1.5 

1.5 

1.4 

1.4 

1.4 

1.4 

1.4 

46 

1.6 

1.6 

I  .0 

1.5 

1.5 

1.5 

1.4 

1.4 

1.4 

1.4 

1.4 

1.3 

1.3 

46 

47 

1.5 

1.5 

1.5 

1.5 

1.4 

1 .4 

1.4 

1.4 

1.4 

1.3 

1.3 

1.3 

1.3 

47 

48 

1.5 

1.5 

1.4 

1.4 

1.4 

1 .4 

1.4 

1.4 

1.3 

1.3 

1.3 

1.3 

1.3 

48 

49 

5o 

1.4 

1.4 

1.4 

1.4 

1.4 

1.3 

1.3 

1.3 

1.3 

1.3 

1.3 

1 .2 

1 .2 

49 
"5o 

1.4 

1.4 

1.4 

1.3 

1.3 

1.3 

1.3 

1.3 

1.3 

1.3 

1 .2 

1 .2 

1 .2 

52 

1.3 

1.3 

1.3 

1.3 

1.3 

1.3 

1 .2 

1 .2 

1 .2 

1 .2 

1 .2 

1 . 1 

I .  I 

52 

54 

1 .2 

1 .2 

1.2 

1 .2 

1 .2 

1 .1 

1 .2 

t.i 

I .  I 

1 . 1 

1 . 1 

I .  I 

1 . 1 

54 

56 

1 .2 

I .  I 

I .  I 

1 .1 

I .  I 

1 . 1 

1 . 1 

1 .1 

1 . 1 

I .  I 

1 .0 

1 .0 

1 .0 

56 

58 

1 .1 

I .  I 

1 .1 

I .  I 

1 .0 

1 .0 

1.0 

1 .0 

1 .0 

1 .0 

1 .0 

1 .0 

1 .0 

58 
60 

1 .0 

1 .0 

I.O 

1 .0 

1 .0 

1.0 

1 .0 

0.9 

0.9 

0.9 

0.9 

0.9 

0.9 

62 

0.9 

0.9 

0.9 

0.9 

0.9 

0.9 

0.9 

0.9 

0.9 

0.9 

0.9 

0.9 

0-8 

62 

64 

0.9 

0.9 

0.9 

0.9 

0.8 

0.8 

0.8 

0.8 

O.cS 

0.8 

0.8 

0.8 

0.8 

64 

66 

0.8 

0.8 

0.8 

0.8 

0.8 

0.8 

0.8 

0.8 

0.8 

0.7 

0.7 

0.7 

66 

68 

0.7 

0.7 

0.7 

0.7 

0.7 

0.7 

0.7 

0.7 

0.7 

0.7 

68 

10_ 

0.7 

0.7 

0.7 

0.6 

0.6 

0.6 

0.6 

0.6 

_70_ 

12° 

13° 

14° 

15° 

iQP 

17° 

18° 

19° 

20° 

21° 

22° 

23° 

24° 

TABLE 

XXXII. 

[I 

age  24] 

Variation  of  the  Sun's  Altitude  in 

one 

minute  from  noon. 

Lat. 

Declination 

of  thl 

same 

name 

as  tk 

e  Latitude. 

Lat. 

0° 

1° 

2° 

3° 

40 

5° 

6° 

7° 

8° 

9" 

10° 

11° 

II 

ir 

II 

// 

II 

II 

n 

II 

II 

„ 

// 

II 
10. 1 

0° 

28.1 

22.4 

18.7 

16.0 

i4-o 

12.4 

II.I 

0° 

I 

28.0 

22.4 

18.6 

16.0 

i3.9 

12.4 

II.I 

I 

2 

28.0 

22.3 

18.6 

i5  9 

i3.9 

12.3 

2 

3 

27.9 

22.3 

18.5 

i5.8 

i3.8 

3 

4 

28.1 

27.8 

22.2 

18.5 

i5.8 
18.4 

4 

5 

22.4 

28.0 

27.7 

22. 1 

5 

6 

18.7 

22.4 

28.0 

27.6 

22.0 

6 

7 

16.0 

18. b 

22.3 

27.9 

27.4 

7 

8 

i4-o 

16.0 

18.6 

22.3 

27.8 

8 

9 

10 

12.4 

13.9 

i5.9 

18.5 

22.2 

18.5 

27.7 
22.1 

27.6 



9 

ii.i 

12.4 

i3.9 

i5.8 

10 

II 

10. 1 

II.I 

12.3 

i3.8 

i5.8 

1S.4 

22.0 

27.4 

II 

12 

9.2 

10. 1 

II.I 

12.3 

i3.8 

15.7 

18.3 

21.9 

27.3 

12 

i3 

8.5 

9.2 

10. 0 

II. 0 

12.2 

i3.7 

i5.b 

18.2 

21.7 

27.1 

i3 

i4 

7-9 

8.5 

9.2 

10. 0 

10.9 

12.1 

i3.6 

i5.5 

18.0 

21 .6 

26.9 

i4 

i5 

7.3 

7.8 

8.4 

9> 

9-9 

10.9 

12. 1 

i3.5 

i5.4 

17.9 

21.4 

26.7 

i5 

i6 

6.8 

7.3 

7.8 

8.4 

9.1 

9.8 

1U.8 

12.0 

i3.4 

i5.3 

17.8 

21.3 

16 

17 

6.4 

6.8 

7.2 

7.8 

8.3 

9.0 

9.8 

10.7 

II. 9 

i3.3 

l5.2 

17.6 

17 

i8 

6.0 

6.4 

6.8 

7  -2 

7-7 

8.3 

8.9 

9-7 

10.6 

II. 8 

l3.2 

i5.o 

18 

19 

5.7 

b.o 

6.3 

6.7 

7-2 

6.7 

7.6 

8.2 

8.9 

9.6 

10.6 

II. 7 

i3.i 

19 

20 

5.4 

5.7 

6.0 

6.3 

7.1 

7.6 

8.1 

8.8 

9.5 

10.5 

II. 6 

20 

21 

5.1 

5.4 

5.6 

5.9 

6.3 

6.6 

7.0 

7.5 

8.1 

8.7 

9.5  '10.4 

21 

22 

4.9 

5.1 

5.3 

5.6 

5.9 

6.2 

6.6 

7.0 

7.5 

8.0 

8.6 

9.4 

22 

23 

4.6 

4.8 

5.0 

5.3 

5.5 

5.8 

6.1 

6.5 

6. 9 

7.4 

7.9 

8.5 

23 

24 
25 

.4.4 

4.6 

4.8 

5.0 

5.2 

5.0 

5.5 

5.8 

6.1 

"5.7 

6.4 

6.8 

7.3 

7.8 

24 

4.2 

4.4 

4.G 

4.7 

5.2 

5.4 

6.0 

6.4 

6.8 

7.2 

25 

26 

4.0 

4.2 

4.3 

4.5 

4.7 

4.9 

5.1 

5.4 

5.7 

6.0 

6.3 

6.7 

26 

27 

3.9 

4.0 

4.1 

4.3 

4.5 

4.7 

4.9 

5.1 

5.3 

5.6 

5.9 

6.2 

27 

28 

3.7 

3.8 

4.0 

4.1 

4.i 

4.4 

4.6 

4.8 

5.0 

5.3 

5.5 

5.8 

28 

29 

3.5 

3.7 

3.8 

3.9 

4.1 

4.2 

4.0 

4.4 
4.2 

4.6 
4.3 

4.7 

5.0 

5.2 

5.5 

29 

J(l 

3.4 

3.5 

3.6 

3.7 

3.9 

4.5 

4.7 

4.9 

5.1 

3o 

3i 

3.3 

3.4 

3.5 

3.6 

3.7 

3.8 

4.0 

4.1 

4.3 

4.4 

4.6 

4.8 

3i 

3:« 

3.1 

3.2 

S.o 

■6.4 

3.5 

3.7 

3.8 

3.9 

4.1 

4.2 

4.4 

4.6 

32 

33 

3.0 

3.1 

3.2 

3.3 

3.4 

3.5 

3.6 

3.7 

3.9 

4.0 

4.2 

4.3 

33 

34 

2.9 

3.0 

3.1 

3.2 

3.2 

3.3 

3.4 

3.6 

3.7 

3.8 

3.9 

4.1 

34 

3-') 

2.8 

2.9 

3.0 

3.0 

3.1 

3.2 

3.3 

3.4 

3.5 

3.6 

3.7 

3.9 

35 

36 

2.7 

2.8 

2.8 

2.9 

3.0 

3.1 

3.2 

3.3 

3.4 

3.5 

3.6 

3.7 

36 

37 

2.6 

2.7 

2.7 

2.8 

2.9 

2.9 

3.0 

3.1 

3.2 

3.3 

3.4 

3.5 

37 

38 

2.5 

2.6 

2.6 

2.7 

2.8 

2.8 

2.9 

3.0 

3.0 

3.2 

3.2 

3.3 

38 

39 

2.4 

2.5 

2.5 

2.6 

2.7 

2.7 

2.8 

2.9 

2.9 

3.0 

3.1 

3.2 

39 
40 

40 

2.3 

2.4 

2.4 

2.5 

2.6 

2.6 

2.7 

2.7 

2.8 

2.9 

3.0 

3.0 

4i 

2.3 

2.3 

2.4 

2.4 

2.5 

2.5 

2.6 

2.6 

2.7 

2.8 

2.8 

2.9 

4i 

42 

2.2 

2.2 

2.3 

2.3 

2.4 

2.4 

2.5 

2.5 

2.6 

2.6 

2.7 

2.8 

42 

43 

2.1 

2.1 

2.2 

2.2 

2.3 

2.3 

2.4 

2.4 

2.5 

2.5 

2.6 

2-7 

43 

44 

2.0 

2.1 

2.1 

2.1 

2.2 

2.2 

2.3 

2.3 

2.4 

2.4 

2.5 

2.5 

44 

45 

2.0 

2.0 

2.0 

2.1 

2.1 

2.2 

2.2 

2.2 

2.3 

2.3 

2.4 

2.4 

45 

46 

I  .9 

1.9 

2.0 

2.0 

2.0 

2.  I 

2.1 

2.2 

2.2 

2.2 

2.3 

2.3 

46 

47 

1 .8 

1.9 

1.9 

1.9 

2.0 

2.0 

2.0 

2.1 

2.1 

2.1 

2.2 

2  .2 

47 

4» 

1.8 

1.8 

1.8 

1.9 

1.9 

1.9 

2.0 

2.0 

2.0 

2.1 

2.1 

2.1 

48 

49 
5o 

1-7 

1-7 

1.8 

1.8 

1.8 

1.8 

1.9 

1.9 

1.9 

2.0 

2.0 

2.1 

49 
5o 

1.6 

'•7 

1-7 

1-7 

1.8 

1.8 

1.8 

1.8 

1.9 

1.9 

1.9 

2.0 

52 

1.5 

1.6 

1.6 

1.6 

1.6 

1.6 

1-7 

1-7 

1-7 

1.8 

1 .8 

1.8 

52 

54 

1.4 

1.4 

1.5 

1.5 

1.5 

1.5 

1.5 

1.6 

1.6 

1.6 

1.6 

1-7 

54 

56 

1.3 

1.3 

1.4 

1.4 

1.4 

1.4 

1.4 

1.4 

1.5 

1.5 

1.5 

1.5 

56 

58 

1 .2 

1.2 

1.3 

1.3 

1.3 

1.3 

1.3 

1.3 

1.3 

1.4 

1 .4 

1.4 

58 

60 

1 .1 

I.I 

1 .2 

1.2 

1.2 

1.2 

1 .2 

1 .2 

1.2 

1.2 

1.3 

1.3 

60 

62 

1 .0 

I.O 

I.I 

1 .1 

I.I 

1 .1 

1 .1 

1 .1 

I.I 

I.I 

1 .2 

1 .2 

62 

64 

1 .0 

1 .0 

1 .0 

1.0 

1.0 

1.0 

1 .0 

1.0 

1.0 

1 .0 

1 .0 

I .  I 

64 

66 

0.9 

0.9 

0.9 

0.9 

0.9 

0.9 

0.9 

0.9 

0.9 

0.9 

0.9 

1 .0 

66 

68 

0.8 

0.8 

0.8 

0.8 

0.8 

0.8 

0.8 

0.8 

0.8 

0.8 

0.9 

0.9 

68 

70 

®-7 

0.7 

0.7 

0.7 

0.7 

0.7 

0.7 

0.7 

0.8 

0.8 

0.8 

0.8 

70 

0° 

1° 

2° 

3° 

40 

5° 

G° 

7° 

8° 

9°      10° 

11° 

■6i 

Pa;' 

u242J 

TABLE    XXXJI. 

Variation  of  the 

Sun's  Altitude 

in  one  minute  from 

noon. 

Lat. 

'0° 

Declination  of  the  same  name  as  the  Latitude. 

Lat. 
0° 

12= 

13° 

14° 

15° 

16° 

17° 

18° 

19° 

20° 

21° 

22° 

23° 

2-4° 

// 

II 

II 

II 

II 

II 

II 

II 

II 

It 

II 

II 

It 

9.2 

8.5 

7-? 

7.3 

6.8 

6.4 

6.0 

5.7 

5.4 

5.1 

4.Q 

4.0 

4.4 

I 

10. 1 

9.2 

8.5 

7.8 

7.3 

6.8 

6.4 

6.0 

5.7 

5.4 

5.1 

4.8 

4.6 

I 

2 

11. 1 

10. 0 

9.2 

8.4 

7.8 

7.2 

6.8 

6.3 

6.0 

5.6 

5.3 

5.0 

4.8 

2 

3 

12.3 

II  .0 

10. 0 

9.1 

8.4 

7.8 

7-2 

6.7 

6.3 

5.9 

5.6 

5.3 

5.0 

3 

4 
5 

i3.8 

12.2 

10.9 

9.9 

9.1 

8.3 

7-7 

7.2 

6.7 

6.3 

5.9 

5.5 

5.2 

4 

5" 

15.7 

i3.7 

12. 1 

10.9 

9.8 

9.0 

8.3 

7.6 

7-1 

6.6 

6.2 

5.8 

5.5 

6 

18.3 

i5.6 

i3.6 

12. 1 

10.8 

9.8 

8.9 

8.2 

7.6 

7.0 

6.8 

6.1 

5.8 

6 

7 

21.9 

18.2 

i5.5 

i3.5 

12.0 

10.7 

9-7 

8.9 

8.1 

7.5 

7.0 

6.5 

6.1 

7 

8 

27.3 

21.7, 

18.0 

i5.4 

i3.4 

1 1 .9 

10.6 

9.6 

8.8 

8.1 

7.5 

6.9 

6.4 

8 

_9 
10 

27.1 

21 .6 

17.9 

i5.3 

i3.3 

II. 8 

10.6 

9.5 

8.7 

8.0 

7-4 

6.8 

_9_ 
10 

26.9 

21.4 

17.8 

l5.2 

l3.2 

II. 7 

10.5 

9.5 

8.6 

7-9 

7.3 

11 

26.7 

21.3 

17.6 

i5.o 

i3.i 

II. 6 

10.4 

9-4 

8.5 

7.8 

11 

12 

26.5 

21 .1 

17.5 

14.9 

i3.o 

II. 5 

10.3 

9.3 

8.4 

12 

i3 

26.2 

20.9 

17.3 

i4.8 

12.8 

II. 3 

10. I 

9.2 

i3 

i4 
i5 

26.0 

20.7 

17. 1 

r4.6 

12.7 

II  .2 

10. 0 

i4 
i5 

25.7 

20.4 

16.9 

i4.4 

12.5 

1 1 .1 

lb 

2b. 5 

25.4 

20.2 

16.7 

i4.3 

12.4 

16 

17 

21. 1 

26.2 

25.1 

20.0 

16.5 

i4.i 

17 

18 

17.6 

20.9 

26.0 

24.8 

19.7 

16.3 

18 

_L9 

2.0 

14.9 

17.3 

20.7 

25. '7 

24.5 

19.5 

11, 
20 

i3.o 

i4.8 

17. 1 

20.4 

25.4 

24.2 

21 

II. 5 

E2.8 

i4-6 

16.9 

20.2 

25.1 

21 

22 

10.3 

II. 3 

12.7 

14.4 

16.7 

20.0 

24.8 

22 

2j 

9.3 

10. 1 

II  .2 

12.5 

i4.3 

16.5 

19.7 

24.5 

23 

24 
25 

8.4 

9.2 

10. 0 

II. I 

12.4 

14. 1 

16.3 

19.5 

24.2 

24 

25 

7-7 

8.3 

9.0 

9.9 

10.9 

12.2 

13.9 

16.1 

19.2 

23.8 

26 

7-1 

7.6 

8.2 

8.9 

0.8 

10.8 

12. 1 

i3.7 

15.9 

18.9 

23.5 

26 

27 

6.6 

7.0 

7.5 

8.1 

8.8 

9.6 

10.6 

1 1 .9 

i3.5 

i5.6 

18.6 

23.1 

27 

28 

6.2 

6.5 

7.0 

7-4 

8.0 

8.7 

9.5 

10.5 

II. 7 

i3.3 

i5.4 

18.3 

22.7 

28 

29 

3o 

5.7 

6.1 

6.4 

6.9 

7.3 

7-9 

8.6 

9-4 

10.3 

II. 5 

i3.i 

i5.i 

18.0 

29 

3o 

5.4 

5.7 

6.0 

6.4 

6.8 

7.2 

7.8 

8.4 

9.2 

10. 1 

II. 3 

12.8 

14.9 

3i 

5.1 

5.3 

5.6 

5.9 

6.3 

6.7 

7.1 

7-7 

8.3 

9.0 

10. 0 

II. I 

12.6 

3i 

32 

4.8 

5.0 

5.2 

5.5 

5.8 

6.2 

6.5 

7.0 

7.5 

8.1 

8.9 

9.8 

10.9 

32 

33 

4.5 

4.7 

4.9 

5.1 

5.4 

5.7 

6.1 

6.4 

6.9 

7-4 

8.0 

8.7 

9.6 

33 

M 
35 

•4.3 
4.0" 

A.^ 

4.6 

4.8 

5.1 

5.3 

5.6 

5.9 

6.3 

6.8 

7.3 

7.8 

8.6 

35 

4.2 

A.A 

4.5 

4.7 

5.0 

5.2 

5.5 

5.8 

6.2 

6.6 

7-1 

7.7 

36 

3.8 

4.0 

4.1 

4.3 

4.5 

4.7 

4.9 

5.1 

5.4 

5.7 

6.1 

6.5 

7.0 

36 

■37 

3.6 

3.8 

3.9 

4.0 

A.  2 

4.4 

4.6 

4.8 

5.0 

5.3 

5.6 

6.0 

6.4 

37 

38 

3.4 

3.6 

3.7 

3.8 

4.0 

4.1 

4.3 

4.5 

4.7 

4.9 

5.2 

5.5 

5.8 

38 

39 

4o 

3.3 

3.4 

3.5 

3.6 

3.8 
3.6 

3.9 

3.7 

4.0 

4.2 

4.4 

4.6 

4.8 

5.1 

5.4 

39 
40 

3.1 

3.2 

3.3 

3.4 

3.8 

4.0 

4.1 

4.3 

4.5 

4.7 

•5.0 

4i 

3.0 

3.1 

3.2 

■6.6 

3.4 

3.5 

3.6 

3.7 

3.9 

4.0 

4.2 

4.4 

4.6 

4i 

4-2 

2.9 

2.9 

3.0 

3.1 

3.2 

3.3 

3.4 

3.5 

3.7 

3.8 

4.0 

4.1 

4.3 

42 

43 

2.7 

2.8 

2.9 

3.0 

3.0 

3.1 

3.2 

3.3 

3.5 

3.6 

3.7 

3.9 

4.0 

43 

44 
45 

2.6 

2.7 

2.7 

2.8 

2.9 

3.0 

3.1 

3.2 

3.3 

3.4 
3.2" 

3.5 

3.6 

3.8 

44 
45 

2.5 

2.6 

2.6 

2.7 

2.8 

2.8 

2.9 

3.0 

3.1 

3.3 

3.4 

3.5 

4(i 

2.4 

2.4 

2.5 

2.6 

2.6 

2.7 

2.8 

2.8 

2.9 

3.0 

3.1 

3.2 

3.3 

46 

47 

2.3 

2.3 

2.4 

2.4 

2.5 

2.6 

2.6 

2.7 

2.8 

2.9 

2.9 

3.0 

3.1 

47 

48 

2.2 

2.2 

2.3 

2.3 

2.4 

2.4 

2.5 

2.6 

2.6 

2.7 

2.8 

2.9 

3.0 

48 

49 
5o 

2.  I 

2.1 

2.2 

2.2 
2.1 

2.3 

2.3 

2.4 

2.4 

2.3 

2.5 
2.4 

2.6 
2.4 

2.6 

2.7 

2.8 
2.6 

i?. 

5o 

2.0 

2.0 

2.  I 

2.2 

2.2 

2.3 

2.5 

2.6 

52 

1.8 

1.9 

1.9 

1.9 

2.0 

2.0 

2.  I 

2.1 

2.1 

2.2 

2.2 

2.3 

2.4 

52 

54 

1-7 

1-7 

1-7 

1.8 

1.8 

1.8 

1.9 

1.9 

1.9 

2.0 

2.0 

2.  I 

2. 1 

54 

56 

1.5 

1.6 

1.6 

1.6 

I  .(i 

1-7 

1-7 

1-7 

1.8 

1.8 

1.8 

1.9 

1.9 

56 

58 
60' 

1.4 

1.4 

1.5 

1.5 

1 .5 

1.5 

1.5 

1.6 

1.6 

1.6 

1.6 

1-7 

1-7 

58 
"60" 

1 .3 

1.3 

1.3 

1.3 

I  .i 

1.4 

1.4 

1.4 

r.4 

1.5 

1.5 

1.5 

1.5 

02 

1 .2 

1 .2 

1.2 

1 .3 

I  .2 

1 .2 

1.3 

1.3 

1.3 

1.3 

1.3 

1.3 

1.4 

62 

()4 

1 .1 

1 .1 

I .  I 

I .  I 

I  .  I 

I .  I 

1 . 1 

1 .2 

1.2 

1 .2 

1 .2 

1 .2 

1 .2 

64 

66 

1 .0 

1 .0 

1 .0 

1 .0 

I  .0 

1 .0 

1 .0 

1 .0 

I.O 

I.I 

1 .1 

1 .1 

1 . 1 

66 

08 

0.9 

0.9 

0.9 

0.9 

0.9 

0.9 

0.9 

0.9 

0.9 

0.9 

0.9 

1.0 

1 .0 

68 

70 

0.8 
12° 

0.8 
13° 

0.8 

0.8 
J  5° 

0.8 

0.8 

0.8 

0.8 

0.8 

0.8 

0.8 

0.8 

0.9 

70_ 
_  I 

14° 

16\ 

17° 

18° 

19° 

20° 

21° 

22° 

23° 

24° 

TABLE    XXXIII.                                      [Page  243' 

Tc 

reduce  the  numbers  of   Table  XXXII  to  other  given  intervals  of  time  | 

from  noon. 

Time  from  Noon. 

s. 

o 

0' 

]/ 

2' 

3' 

4' 

5' 

Q 

7' 

8' 

9' 

10' 

11' 

12' 

0 

0.0 

1 .0 

4.0 

9.0 

16.0 

25.0 

36. 0 

49.0 

64. 0 

81 .0 

100.0 

121. 0 

144.0 

I 

0.0 

1.0 

4.1 

9.1 

16.1 

25.2 

36.2 

49.2 

64.3 

81.3 

100.3 

121. 4 

144.4 

I 

2 

0.0 

I .  I 

4.1 

9.2 

16.3 

25.3 

■66.4. 

49.5 

64.5 

81.6 

100.7 

121 .7 

144.8 

n 

3 

0.0 

I .  I 

4.2 

9.3 

16.4 

25.5 

36.6 

49.7 

64.8 

81.9 

lOI  .0 

122. 1 

145.2 

3 

4 

0.0 

I.I 

4.3 

9-4 

16.5 

25.7 

36.8 

49.9 

65.1 

82.2 

101.3 

122.5 

145.6 

4 

5 
6 

0.0 

1.2 

4.3 

9.5 

16.7 

25.8 

37.0 

5o.2 

65.3 

82.5 

101 .7 

122.9 

i46.o 

5 
6 

0.0 

1.2 

4. A 

9.6 

16.8 

26.0 

37.2 

5o.4'65.6 

82.8 

102.0 

123.2 

146.4 

7 

0.0 

1 .2 

4.5 

9-7 

16.9 

26.2 

37.4 

5o.6'65.9 

83.1 

102.3 

123.6 

i46.8 

7 

8 

0.0 

1.3 

A.ii 

9.8 

17. 1 

26.4 

37.6 

5o.9 

66.1 

83.4 

102.7 

124.0 

l47-2 

8 

9 

0.0 

1.3 

4.6 

9.9 

17.2 

26.5 

37.8 

5i.i 

66.4 

83.7 

io3.o 

124.3 

147.6 

9 

lo 

0.0 

1.4 

4.7 

10. 0 

17.4 

26.7 

38. 0 

5i.4 

66.7 

84.0 

io3.4 

124.7 

i48.o 

10 

I  1 

12 

0.0 

1.4 

4.8 

10. 1 

17.5 

26.9 

38.2 

5i.6 

67.0 

84.3 
84.6 

io3.7 

125. I 

i48.4 

II 
12 

0.0 

1.4 

4.8 

10.2 

17.6 

27.0 

38.4 

5i.8 

67.2 

104.0 

125.4 

i48.8 

i3 

0.0 

1.5 

4.9 

10.3 

17.8 

27.2 

38.6 

02.1 

67.5 

84.9 

104.4 

125.8 

149.2 

i3 

i4 

0.1 

1.5 

5.0 

10.5 

17.9 

27-4 

38.9 

52.3 

67.8 

85.3 

104.7 

126.2 

149.7 

i4 

i5 

0.1 

1.6 

5.1 

10.6 

18. 1 

27.6 

39.1 

52.6 

68.1 

85.6 

io5.i 

126.6 

i5o.i 

i5 

i6 

0.1 

1.6 

5.1 

10.7 

18.2 

27.7 

39.3 

52.8 

68.3 

85. Q 

io5.4 

126.9 

i5o.5 

16 

17 
i8 

0. 1 

1.6 

5.2 

10.8 

18.3 
18.5 

27.9 
28.1 

39.5 
39.7 

53.0 
53.3 

68.6 
68.8 

86.2 

io5.7 

127.3 

1 50.9 

17 
■  18 

0.1 

1-7 

5.3 

10.9 

86.5 

106. 1 

127.7 

i5i.3 

19 

0.1 

1-7 

5.4 

II. 0 

18.6 

28.3 

39.9 

53.5 

69.2 

86.8 

106.4 

128. 1 

i5i.7 

19 

20 

0.1 

1.8 

5.4 

1 1 . 1 

18.8 

28.4 

4o.  I 

53.8 

69.4 

87.1 

106.8 

128.4 

l52.1 

20 

21 

0.1 

1.8 

5.5 

II. 2 

.8.9 

28.6 

40.3 

54.0 

69.7 

87.4 

107. 1 

128.8 

i52.5 

21 

22 

0.1 

1.9 

5.6 

II. 3 

19. 1 

28.8 

40.5 

54.3 

70.0 

87.7 

107.5 

129.2 

152.9 

22    > 

23 

0.1 

1.9 

5.7 

II. 4 

19.2 
19.4 

29.0 

40.7 

54.5 

70.3 
70.6 

88. 0 

107.8 

129.6 

i53.3 

23 

0.2 

2.0 

5.8 

II. 6 

29.2 

4i  -o 

54.8 

88.4 

108.2 

i3o.o 

i53.8 

24 

2  5 

0.2 

2.0 

5.8 

II. 7 

iq.5 

29.3 

4l.2 

55.0 

70.8 

88.7 

108.5 

i3o.3 

i54.2 

25 

26 

0.2 

2.1 

5.9 

11.8 

19.7 

29.5 

4i.4 

55.3 

71. 1 

89.0 

108.9 

i3o.7 

i54.6 

26 

27 

0.2 

2.1 

6.0 

II. 9 

19.8 

29-7 

4i.6 

55.5 

71.4 

89.3 

109.2 

i3i  .1 

i55.o 

27 

?8 

0.2 

2.2 

6.1 

12.0 

20.0 

f9.9 

4i.8 

55.8 

71-7 

89.6 

109.6 

i3i.5 

i55.4 

28 

29 

3.0 

0.2 

2.2 

6.2 

12. 1 

20. 1 

3o.i 

42.0 
42.2 

56.0 

72.0 

89.9 

109.9 

i3i  .9 

i55.8 

29    1 

0.2 

2.2 

6.2 

12.2 

20.2 

3o.2 

56.2 

72.2 

90.2 

no. 2 

l32.2 

i56.2 

3o  1 

3 1 

0.3 

2.3 

6.3 

12.4 

20.4 

3o.4 

42.5 

56.5 

72.5 

90.6 

no. 6 

i32.6 

i56.7 

3i    1 

3; 

0.3 

2.4 

6.4 

12.5 

20.6 

3o.6 

42.7 

56.8 

72.8 

90.9 

III.O 

i33.o 

157. 1 

32 

33 

0.3 

2.4 

6.5 

12.6 

20.7 

3o.8 

42.9 

57.0 

73.1 

91 .2 

III  .3 

i33.4 

157.5 

33 

31 

0.3 

2.5 

6.6 

12.7 

20.9 

3i  .0 

43.1 

57.3 

73.4 

91 .5 

III  .7 

i33.8 

157.9 

34- 

35 

3(3 

0.3 

2.5 

6.7 

12.8 

21  .0 

3l.2 

43.3 

57.5 

73.7 

91.8 
92.2 

112. 0 

i34.2 

i58.3 

35 
36 

0.4 

2.6 

6.8 

i3.o 

21.2 

3i.4 

43.6 

57.8 

74.0 

112. 4 

i34.6 

i58.8 

3- 

0.4 

2.6 

6.8 

i3.i 

21.3 

3i.5 

43.8 

58. 0 

74.3 

92.5 

1 12.7 

134.9 

159.2 

37 

38 

0.4 

2.7 

6.9 

l3.2 

21.5 

3i.7 

44.0 

58.3 

74.5 

92.8 

ii3.i 

i35.3 

159.6 

38 

39 

4o 

0.4 

2.7 

7.0 

i3.3 

21.6 

3i  .9 

44.2 

58.5 

74.8 

93.1 

ii3.4 

135.7 

160.0 

39 

0.4 

2.8 

7-1 

i3.4 

21.8 

32.1 

44.4 

58.8 

75.1 

93.4 

ii3.8 

i36.i 

160.4 

40 

4i 
42 

0.5 

2.8 

7.2 

i3.6 

21.9 

32.3 

44.7 

59.0 

75.4 

93.8 

114.1 

i36.5 

160.9 

4i 

42 

0.5 

2.9 

7.3 

i3.7 

22.1 

32.5 

44.9 

59.3 

75.7 

94.1 

114.5 

1 36. 9 

161.3 

43 

0.5 

2.9 

7.4 

1J.8 

22.2 

32.7 

45.1 

59.5 

76.0 

94.4 

114.8 

137.3 

161 .7 

4i 

4f 

0.5 

3.0 

7.5 

i3.9 

22.4 

32.9 

45.3 

59.8 

76.3 

94.7 

Il5.2 

137.7 

162. 1 

44 

45 

0.6 

3.1 

7.6 

i4.i 

22.6 

33.1 

45.6 

60.1 

76.6 

9D.1 

ii5.6 

i3S.i 

162.6 

45 

46 

0.6 

3.1 

7-7 

l4.2 

22.7 

33.3 

45.8 

60.3 

76.9 

95.4 

115.9 

i38.S 

i63.o 

46 

47 
48 

0.6 

3.2 

7-7 

U.i 

22.9 

33.4 

46.0 

60.6 

77.1 
77.4 

95.7 

n6.3 

i38.8 

i63.4 

47 
48 

{..6 

3.2 

7.8 

i4.4 

23.0 

33.6 

46.2 

60.8 

96.0 

116. 6 

139.2 

i63.8 

49 

0.7 

3.3 

7-9 

14.6 

23.2 

33.8 

46.5 

61. 1 

77-7 

96.4 

117. 0 

139.6 

164.3 

49 

5o 

0  7 

3.4 

8.0 

14.7 

23.4 

34.0 

46.7 

61.4 

78.0 

96.7 

117-4 

i4o.o 

164.7 

5o 

5i 

0  7 

3.4 

8.1 

14.8 

23.5 

34.2 

46. Q 

61.6 

78.3 

97.0 

117. 7 

i4o.4 

i65.i 

5i 

5t 

0.8 

3.5 

8.2 

i5.o 

23.7 

34.4 

47-2 

61 .9 

78.6 

97-4 

118. 1 

i4o.8 

i65.6 

ir2 

53 
51 

0.8 

3.5 

8.3 

ID.  I 

23.8 

34.6 

47.4 

62.1 

78.9 

97-7 
98.0 

118. 4 

l4l.2 

166.0 

63 
54 

0.8 

3.6 

8.4 

l5.2 

24.0 

34.8 

47.6 

62.4 

79.2 

118. 8 

i4i.6 

166.4 

55 

0.8 

3.7 

8.5 

i5.3 

24.2 

35.0 

47.8 

62.7 

79.5 

98.3 

119. 2 

142.0 

166.8 

65 

56 

0.9 

3.7 

8.6 

i5.5 

24.3 

35.2 

48.  T 

62.9 

79.8 

98.7 

119. 5 

142.4 

167.3 

6t) 

57 

0.9 

3.8 

8.7 

1 5. 6 

24.5 

35.4 

48.3 

63.2 

80.1 

99.0 

119. 9 

142.8 

167.7 

67 

58 

0  9 

3.P 

8.8 

ID. 7 

24.7 

35.6 

48.5 

63.5 

80.4 

99.3 

120.3 

l13.2 

168. 1 

68 

A9_ 

I.O 

3.9 

8.9 

.5.9 

■^4.8 

35.8 

48.8 

63.7 

80.7 

99-7 

120.6 

i43.6 

i68.6 

69 

0' 

1'       2' 

3'        4'       5'   1 

6' 

7' 

8' 

9' 

10 

11' 

Page  244] 

TABLES  XXXIV.,  XXXV.,  and 

XXXVI . 

TABLE   XXXIV. 

TABLE  XXXV. 

Errors  arising  from  a  deviation  ot  1'  in 

Angles 
Obs'd. 

the  parallelism  of  the  surfaces  of  the  cen- 
tral mirror. 

Angle  of  deviation. 

10' 

II 

15' 

20' 
// 

25 

II 

30' 

II 

35' 

40'  45' 
//     // 

50' 

II 

55' 
II 

60' 
II 

Obs'd. 
Anglo 

Obs.  to       Obs.  to 

Obs. 

Fifth 

D. 

0 
10 
20 

0 
0 
0 

0 

0 

I 

0 

I 
I 

2 

0 

I 
2 

~3 

0 

I 
3 

0 
2 

4 
6 

0 
2 

5 

7 

0 
3 
6 

9 

0 

4 
8 

12 

0 
5 
9 

14 

0 

5 

II 

17 

D. 

'      II 

1    II 

/      II 

/      // 

o 

10 

0 
2 

0 
I 

0 
2 

0 

0 

So 

0      I 

20 

5 

2 

4 

2 

40 

I      I 

3 

4 

6 

8 

10 

i3 

16 

19 

23 

3o 

10 

I 

6 

4 

5o 

2 

3 

5 

7 

10 

i3 

16 

30 

25 

29 

36 
4o 

4o 

16 

0 

8 

7 

60 

65 

2 
3 

4 
4 

6 

9 
10 

12 

i4 

16 
18 

20 

23 

77' 
28 

37 
34 

45 

19 

I 

9 

9 

5o 

2S   ' 

2 

II 

II 

70 

3 

5 

8 

II 

i5 

20 

25 

Si 

37 

44 

55 
6o 
65 
70 

28 

S3 
39 

46 

4 
5 

7 
10 

12 

i4 
16 
18 

i4 
17 
21 

25 

75 

~ 

3 

5 

~8 

12 

76" 

21 

27 

S3 

4i 

48 

80 
85 

2 

3 
4 
4 
4 

6 
6 

9 
10 

i3 
i4 
16 
17 

18 
20 

23 

26 

78 
So 

So 

32 

"  35" 
39 

37 
4o 

44  ~ 
48 

44 
48 

53 

58 

53 
58 
63 
69 

75 
80 

54 
I.  4 
i.i5 
1.27 

12 
16 

21 

24 
28 

32 

So 
35 

95 

2 

7 
8 

12 

23 

85 
90 

19 

23 

4i 
48 

100 

2 
2 

5 
~5~ 

8 
"9 

i3 

_i9. 
20 

2b 

33 
l6 

42 

"  46 

b2 

^7~ 

33 

59" 

75 
82 

io5 

95 

1.43 

28 

37 

56 

no 
ii5 

2 

3 

6 
6 

10 

16 

22 

2  5 

Si 

34 

40 
44 

bo 
55 

62 
68 

75 
83 

90 

100 

2.  I 

33 

44 

I.  6 

11 

yy 

io5 
no 

2.23 
2.49 

39 
46 

52 

I .   I 

1. 16 
1 .29 

120 

3 

7 

12 

19 

27 

^7 

48 

61 

76 

91 

109 

ii5 

3.23 

54 

i.i4 

1.44 

120 

4.o5 

1 

.  4 

1 .3o 

2.   3 

i3o 

2.5l 

1 40 

4.  6 

TAI 

3LE  XXXVL 

Corrections  of  the  mea 

a  refract 

on  for  various  heights  of  the  Thermometer  and  Barometer. 

Ht.  Th. 

20= 

24° 

23° 

32° 

3C 

° 

40° 

44° 

43° 

52° 

5( 

3° 

G0° 

G4° 

68° 

72° 

7G° 

Baroin- 

32.00 

31  .GG 

31.32 

30.99 

30.67 

30.3G 

30.05 

29.75 

29.4.^ 

29 

IG 

23.8t: 

^  28.G0 

28.33 

28.0 

n  27.80 

app.  alt. 

0       ' 
0      Q 

'+" 

'+" 

'+" 

'+" 

4 

Ji 

'+" 

'+" 

1 II 

/ // 

1 // 

/ II 

1 II 

1 / 

r    > II 

2   4i 

2    18 

I    55 

I    33 

I 

12 

0   5i 

0    So 

0     ID 

0    ir 

0 

29 

0  4t 

I      7 

I      25 

I   4 

32         I 

0  3o 

2    18 

I    58 

I    39 

I    20 

I 

2 

44 

26 

9 

6 

25 

4i 

58 

I    i3 

I    2 

91   44 

I       0 

I    59 

I    42 

I    26 

I      9 

53 

38 

22 

7 

22 

se 

5o 

I      3 

I    I 

7  I    So 

I   So 

I   43 

I    29 

I    i4 

I      0 

46 

33 

19 

6 

e 

19 

Si 

43 

55 

I 

5  I    18 

2      0 

I    So 

I    18 

I      5 

0    53 

4o 

29 

17 

6 

e 

16 
75 

2" 

37 

48 

5 

Si      8 

2    So 

I    20 

I      8 

57 

46 

36 

25 

i5 

5 

r 

2^ 

33 

43 

5 

I  I      0 

3     0 

I    II 

I      I 

5i 

4i 

32 

22 

i3 

4 

z; 

i3 

21 

So 

38 

4 

3      53 

4 

58 

49 

4i 

33 

26 

18 

11 

4 

/ 

10 

l- 

24 

3i 

3 

7       43 

5 

48 

4i 

35 

28 

22 

i5 

9 

3 

9 

lA 

20 

26 

3 

I       36 

6 

4i 

35 

So 

24 

18 

iS 

8 

3 

7 

12 

17 

22 

2 

1      Si 

7 

36 

3i 

26 

21 

16 

II 

7 

2 

2 

7 

II 

i5 

19 

2 

3       27 

8 

32 

27 

23 

18 

i4 

10 

6 

2 

3 

6 

IC 

i3 

17 

2 

J       24 

9 

28 

24 

20 

16 

iS 

9 

5 

2 

2 

5 

12 

i5 

I 

3       21 

10 

26 

22 

18 

i5 

11 

8 

5 

2 

2 

5 

10 

i4 

i( 

3       19 

12 

i4 

21 

18 

i5 

12 

10 

7 

4 

4 
3 

e 

9 

II 

iz 

f       16 
>      i4 

18 

16 

i3 

II 

8 

6 

3 

t 

8 

10 

i: 

16 

16 

i4 

II 

9 

7 

5 

3 

3 

5 

7 

9 

ic 

)      12 

18 

i4 

12 

10 

8 

6 

4 

3 

S 

4 

6 

c 

?      II 

21 

12 

ID 

9 

7 

5 

4 

2 

2 

4 

5 

6 

i 

)        9 

24 

10 

9 

7 

6 

5 

3 

2 

2 

3 

4 

5 

7        8 

27 

9 

8 

6 

5 

4 

3 

2 

2 

3 

4 

5 

C 

^        7 

So 

8 

7 

6 

5 

4 

3 

0 

0 

2 

3 

4 

t 

6 

35 

7 

6 

5 

4 

3 

2 

0 

0 

2 

3 

3 

I 

5 

40 

6 

5 

4 

3 

2 

2 

0 

0 

2 

2 

3 

L. 

4 

45  . 

5 

4 

3 

3 

2 

I 

0 

0 

I 

2 

2 

S 

5o 

4 

S 

S 

2 

2 

I 

0 

0 

I 

2 

2 

S 

60 

3|        2 

2 

2 

I 

I 

0 

0 

0 

I 

I 

I 

2 

70 

\        ' 

I 

I 

I 

I 

0 

0 

0 

0 

I 

I 

I 

I 

I 

80 

I         1 

I 

0 

0 

0 

0 

0 

0 

0 

o!      0 

0 

I 

I 

90 

0        0 

0 

0 

0 

0 

0 

0 

0 

0 

ol        0 

0 

c 

0 

TABLE  XXXVII. 

Longitudes  and  Latitudes  of  Stars,  for  Jan.  1830. 


[Page  245 


Names  of  STARS. 


•/  Pegasi Algcnih 

u  Andromcdffi Alpheratz 

>;  Pisciuin 

a  AlUKTIS 

a  (.x-ti Menkar 

t;  Pleiadum ilcyone 

y  Tauri 

t  Tauri 

u  Tauri Aldebaran 

1^  Orionis Rigcl 

a-  AurigiB Ctqiclla 

3  Orionis 

|-J  Tauri 

£  Orionis 

L  Orionis 

L  Tauri 

u  Orionis   Betergiiese 

1}  Geminorum 

u  Geminorum 

•/  Geminorum 

f  Geminorum 

a  Canis  Majoris Siritis 

t  Geminorum 

(5  Geminorum 

a  Geminorum Castor 

;•?  Geminorum Pollux 

a  Canis  Minoris Procyon 

a  i:  Caneri Jicuhcns 

u  flydrs Ilphard 

»j  Leonis 

a  Leonis Regulus 

i  Leonis Dcnehola 

•i  Virginis 

'/  Virginis 

y  Virginis 

a  Virginis Spic a 

a  Bootis 'Ircturus 

a  Coronae  Bor Jilphacca 

a  2  Librce Zuhcncsch 

a  Serpcntis 

•/  LibrfD 

b  Seorpii 

S  Scorpii 

rt  Seorpii 

^  Seorpii 

a  Seorpii Antares 

0  Opliiunhi 

a  Opliiuchi Ras  Alhaguc 

a  Sagiitarii 

«  Lyras Vega 

71  Sagitfarii 

y  Aquiia; 

a  Aquilae Athair 

1^  Aquiloe 

a  2  Capricorni 

[i  Capricorni 

y  Capricorni 

S  Capricorni 

a  Aquarii 

a  Pisca  Aust Fomalhaut 

a  Cygni Deneb 

a  Pesasi ]\L\re ab 


Rlag 


4.3 

2.3 
2 

3 

3 

3.4 


4.3 

2 

3.4 


3 

4.3 

3 


2.3 
2.3 
2.3 


3.4 
3 

I  .2 

3 
3 
3 
4  3 
3 
3 


Longitude. 


Ann.Var. 
aft.  1830. 


s     o      I      It 

0.  6.47.09 
o . 1 1 . 56 . 26 
0.24.26.32 

1.  5.17.07 
I .11 .56.4i 
I .27.36.57 

2.  3.25. i4 
2.  6.  4.55 
2.  7.24.45 
2.14.27.  7 


2.19.28.47 
2. 19.59. 17 
2.20. 1 1 .58 
2.21 .  5.23 
2.22.18.26 
2.22.24.32 
2.26.22.42 
3.  I.  3.54 
3.  2.55.14 
3.  6. .-(3. 35 


3.  7.33.46 
3. II. 44. 55 
3.12.36.52 
3.16.  8.43 
3.17.52.24 
3.20.52.06 
3.23.27.09 
4. II .i5.52 
4.24.54.50 
4.25.3i .40 


4.27.27.53 
5.19. i5.54 
5.24.44.16 
6.   2.27.42 

6.  7.48.04 
6.21 .28.05 
6.21 .5i .55 

7.  9.53.32 
7.12.42.48 
7. 19.41 .08 


7.22.45.27 
7.28.45.00 
8.  o. II. 44 
8.  0.33.49 
8.  0.48.49 
8.  7.23.15 

8.19.  I. 10 

8.20.  3.45 
9.10.  o.3i 
9.12.55.38 


9. i3. 52.40 
9.28.34.08 
9.29.22.38 
10,  o.o3.34 
10.   1.28.52 

10.  i.4o.i4 
10.19.24.25 
10.21.  9.29 

11.  0.58.57 
II.  1.27.58 
II.  2.59.30 
II .21 .07.05 


50.09 
49.98 
5o.  16 

50.27 
50.27 
5o.i8 

5o.2I 

5o.2o 

5o.2I 

50.24 


50.19 

50.2O 
5o.20 

5o.2o 
5o.2o 

5o.20 

50.19 
5o.2o 
5o.2o 
5o.i8 


5o.2o 
50.07 
50.19 

50.2O 

5o.23 
49.50 

50.12 

5o.i6 

50.02 

5o.23 


49.94 
5o.3o 

5o.20 

5o.2i 
5o.oo 
5o.o8 
5o.45 
5o.5i 

5o.20 

5o.32 


5o.2  2 

5o.i8 
5o.i  9 
5o.i8 

5o.20 
5o.I2 
5o.20 
5o.2I 
5o.2I 

49.89 


So.io 
5o.o3 
50.79 
5o.o5 
5o.  i5 
50.17 
5o.2i 

5o.2I 

5o.ii 
50.59 
49.42 
5o.ii 


Latitude. 


Ann.Var. 
aft.  1830. 


12.35.43  N 

25.41.   qN 

5.22.  5  N. 

9.57.40  N. 
12.35.40  S. 

4.  2.  7N. 
5.44.56  S. 
2.35.   I  S. 

5.28.41  S. 
3i.  8.39  S. 


22.52.17  N 

23.34.29  S 

5.22.3i  N 

24.3i.38  S 

25.i8.5r  S 

2.12.55  S 

16.  2.59  S 

0.54.28  S 

0.49.59  S 

6.45.36  S 


2.  3.00  N. 
39.22.26  S. 

2.  3.3i  S, 

o.  II  .5o  S. 
10.  5.  4N. 

6.40.20  N, 

15.57.43  S, 

5.  5.35  S, 

22.23.36   S, 

4.5i.2i  N 


0.27.41  N. 
12. 17.10  N. 

0.41.32  N. 

I .22.22  N. 

2.48.42  N. 

2.  2.22  S. 
3o.53.58N. 
44.20.42  N. 

0.21 .25  N. 
25. 31.27  N. 


4.24.20  N. 

5.27.49  S. 

1.57.42  S. 

5.27.  4  S. 

I.   1.52  N. 

4.32.45  S. 

1.49.  6  S. 
35.52.21  N. 

3.25.23  S. 
61.44.21  N. 


I. 27.41  N. 
31.15.39N. 
29. 18.46  N. 
26.42.28  N. 

6.56.55  N. 

4.36.28  N. 

2.32.18  S. 

2.33.52  S. 
10.40. i4  N. 
21.  6.42  S. 
59.54.55  N. 
19.24.45  N. 


4-0-I2 
4-0.16 

-)-o.25 

+0.16 

— 0.37 

+0.43 
—0.45 

—0.46 
—0.33 

-0.47 


+0., 

—0.48 

+0.48 

— 0.48 

—0.48 

—0.48 


— 0.47 
—0.47 


+0.46 
—0.45 
-^.45 
-0.44 
+0.43 
+0.26 
— 0.41 
— o.3i 
— 0.22 
+0.22 


+0.22 
+o.o3 
— 0.02 
—0.08 
— 0.1 3 
+0.17 
— 0.24 
—0.35 
— 0.37 
— 0.40 


— 0.42 
+0.44 
+0.44 
+0.45 
-0.45 
+0.42 
+0.." 
—0.48 
+0.46 
—0.45 


—0.45 
— 0.39 
+0.08 
— 0.38 
— 0.37 
— 0.37 
+0.26 

+0.25 

—0.18 

+0.21 

0.16 

+0.10 


Page  246]              TABLES 

XXXVIIL,  XXXIX.,  XL.,  AND  XLL 

TABLE  XXXVIII. 

TABLE  XXXIX. 

Reduct.  of  Lat.  and  Hor.  Par 
for  Ellipticity  _i_j 

Aberration  of  Planets  in  Longitude. 

Elong. 

LTran. 

Sat. 

Jup. 

Mars. 

Venus. 

Mercury. 

Red.   5    Hor.  Par 
Horizontal  Par. 

Elong. 

Ab. 

Elong. 

A  ph. 

Me  a. 

Peri. 

Lat. 

of  Lat. 

D 

— 

— 





D 

1 

D 





53' 

57'  '  " 

01' 

Con.  0 
i5 

25" 

24 

27" 
26 

29" 

28 

36" 
35 

S.C.  0 

m 

S.C.  0 

5 

46" 
46 

5i^" 
5i 

w 

0 

/     // 

II 

II 

i5  4i 

O 

0.   0.0 

0.0 

o.c 

0.0 

3o 

22 

24 

2b 

33 

3o  M 

ID 

M 

48 

52 

2 

0.47.9 

0.0 

0.0 

0.0 

45 

TO 

21 

23 

28 

45  19 

i5 

4i 

43 

4i 

4 

1.35.5 

O.I 

0.1 

O.J 

60 

ID 

lb 

19 

23 

Gt.El.  i4 

20 

37 

34 

6 

2.22.7 

O.I 

0.1 

0.1 

75 

10 

12 

i4 

18 

45    9 

25 

29 

8 

3;    9.2 

0.2 

0.2 

0.2 

90 

5 

6 

9 

12 

3o    0 

Gt.  El. 

18 

18 

^9 

10 

3.54.8 

0.3 
0.5 
0.6 
0.8 

I.O 
I  .2 

0.3 

0.5 
0.7 
0.9 
1 .1 
1.3 

0.4 

0.5 
0.7 
0.9 
1 .2 
1.4 

io5 

120 
i35 
i5o 
i65 

I 

5 

10 
i3 
i5 

I 

+ 

4 

8 

II 

i3 

3 

I 
5 

9 
1 1 

7 

3 

+ 
2 
3 

i5 
Inf.  C. 

3 
3^ 

25 
20 

i5 

10 

5 
Inf  C 

7 

I 

+ 
2 
5 
6 
6 

4 
4- 

4 

8 
11 

t1 

+ 
2 

1 3 
18 
^9h 

12 

i4 
i6 

i8 

20 

4.39.3 
5.22.4 

6.  3.9 

6.43.7 
7.21.5 

22 

7.57.2 
8.30.7 

1.5 

T,R 

1.6 

1-7 

Op. 180 

i5 

i3  1   II   1 

4              1 

i.y 
2.2 

2fi 

9.   1.6 
9.29.9 

9.55.4 

2-0 

0   3 

The  aberration    of    the  Sun  in  longitude  is  always   20".  1 

28 

2    3 

2  5 

2.7 
3.1 

3.8 

Tlie  apparent  place  is  given  in  the  INautical  Almanac, 

and 

3o 

2-7 

2.9 

3.2 

3.6 

by  adding  20"  the  Sun's  true  longitude  will  be  obtained. 

32 

34 

10. 18. r 
10.37.8 

3.0 

3.3 

TABLE  XL. 

TABLE  XLI. 

36 

10.54.3 

3.7 

3.Q 

4.2 

Equat.  Equino.xes  in 

Aberration  in  Long,  and  Lat.  | 

38 

II .   7.7 

4  0 

4  1 

4  6 

Longitude. 

1 

_4^ 

II. 17. 8 

4.A 

4.7 

5.0 

Arg.  long.  =  0  long.  —  :)f.  long. 
Arg.  lat.  =  Arg.  long.  —  3  signs. 

Long.  5  '3  Node. 

42 

II. 24. 7 

4.7 

S.I 

h.S 

44 

1 1. 28. 2 

5.1 

5.5 

5.9 
6.3 

6,7 

0 

46 

II. 28. 4 
II .25.1 

5  5 

5.0 
6.3 

D 

D 

48 

6.2 

+ 

+ 

+ 

+ 

+ 

+ 

5o 

II. 18. 6 

6.7 

7.2 

b 

7 

8 

b 

7 

8 

52 

II.  8.8 

6.6 

7.1 

7  6 

0 

o"o 

8"^ 

i5"5 

3o 

0 

20"o 

i7"i 

io"o 

3 

0 

54 

10.55.6 

6.0 

7  5 

8   n 

2 

o.b 

9.6 

i5.8 

28 

2 

Jo.o|i7.c 

9-^ 

2 

« 

56 

10.39.3 

7.3 

7.8 

8,4 

4 

1.2 

10. c 

lb. I 

26 

4     - 

!0.0 

ib.fc 

8.8 

26 

58 

10. 19.9 

7.6 

8.9 

8,8 

6 

1 .0 

10.5 

lb. 4 

24 

6 

9.9 

lb. 2 

8.1 

24 

6o 

9-57-4 

7-9 

8.5 

9.1 

8 

2.5 
3,T 

II  .0 
II. 5 

ib.b 
16  P 

22 

8 

9.8 

i5.fc 

t5.,1 

7.5 
6,8 

22 

62 
64 

y-/ 

9.32.0 
9.  3.8 

8.3 
8.6 

8.9 
9.2 
9.5 
9.8 
10. 1 

9.5 
9.9 
10.2 

12 

3.7 

12.0 

17.0 

18 

12 

9.6 

.4.9 

6.2 

I 

8 

66 

8.32.9 

7.59.6 

8.8 

i4 

4.3 

12.4 

17.2 

16 

i4 

9-4 

14.4 

5.5 

I 

b 

68 

9.1 

10,5 

16 

4.9 

12. Q 

17.4 

i4 

16 

9.2 

13.9 

4.8 

I 

4 

70 

7.23.8 

9-4 

10.8 

18 

5.5 
6.1 

i3.3 
i3.7 

17.5 
17.6 

12 

18     I 
20     I 

9.0 
8.8 

i3.4 

4.2 
3.5 

12 

72 
74 

6.45.9 
6.  6.0 

9.6 

9.8 

10. 0 

10.3 
10  5 

II  .0 
II    3 

12.9 

22 

6.7 

14. 1 

17-7 

8 

22     I 

8.5 

12.3 

2.8 

8 

76 

5.2.4.3 

10.7 

TI    5 

24 

7.3 

i4.5 

17.8 

6 

24     I 

8.3 

II. 8 

2.1 

b 

78 

4.41.0 

10. 1 

10.9 
II  .1 

II  .7 

26 

7.8 

i4.8 

17.9 

4 

26     I 

8.0 

II  .2 

1.4 

4 

80 

3.56.3 

10.3 

II. 8 

28 

8.4 
8.9 

l5.2 

i5.5 

17.9 

2 

28      I 
3o      1 

7-7 
7.3 

10.6 

0.7 

2 

82 

3. 10.4 
2.23.7 

10.4 
10.5 

II  .2 
11 .3 

12.0 
12. 1 

ly.y 

84 

— 

— 

— 

+ 

+ 

+ 

86 
83 

1.36.2 
0.48.2 

10.5 
10.6 

11. 3 

11. 4 

12. 1 

12.2 

5 
+ 

4 

6 

D 

- 

b 

4 

3 

D 

+ 

+ 

— 

90 

0.  0.0 

10.6 

II. 4 

12.2 

II 

10 

9 

II 

10 

9 

Table  XI-.  contains  the  etjua 

tion  of  the  equinoxes  in  lorjjitiide  to  be  applied  with  its  sign  to 
ily  bodies.     Thus  on  July  lb,  1830,  when  the  longitude  of  the  moo 

the 

inean  longilndos  of  all  the  heave 

n  s 

ascending  node  was  5s.  !2°  38'  tl 

e  equation  of  tiso  equinoxes  was  ■ —  5".  3. 

The  correction  in  Table  XLI.  ( 

orresponding  to  the  Argument  of  Longitude  being  found,  and  its  lo 

;a- 

rithm  added  to  the  loj.  secant  (Ic.s 

s  radius)  of  tiie  star's  latitude,  will  be  tlie  log.  of  the  star's  aberratiot 

in 

lowsilude,  to  be  applied  witii  its  s 

gn  to  the  mean  longitude.      The  logarithm  of  the  correction  in  Ta 

.le 

XLI.  corresponding  to  the  Arpu 

ncnt  of  Latitude  added  to  the  log.  sine  of  the  star's  latitude  will 

be 

the  aberration  of  the  star  in  latitu 

de,  to  be  apj)lied  with  its  sign  lo  the  mean  latitude. 

Example.      Re 

quired  the  Aberration  of  a  Pegasi,  July  16,  1830? 

0  long.    3s.  23°  22'. 

*  long.     U.  21.  07. 

Ar-.  loi.-.i.  02.    15.  Table  4L- 

1-10".  7  log.  1.02938.  Arg.  lat.  Is.  2°.  15  Tab.  41.— IG".  9  log.  l.tr. 

39 

*  Latitude            10°  25' 

Sec.  0.025-13. 

Sii,e  d.Si.\ 

li 

* 

Aboir.  lo 

n^^  + 

11"; 

I. 

L 

^g.  l.C 

.>W1. 

*•  Aber. 

lat. 

-5" 

.6. 

Log.  O.TWf/l  [ 

TABLE    XLII.                                                          [Page  247 

Aberration  in  Right  Ascension  and  Declination. 

PART  I. 

PART  II. 

PART  III. 

Arg.  R.  A.  =  ^  R.  A  .  —  Q  Long. 

Arg.  R.  A.  =  *  R.  A.  +  Q  Long. 

Ar.2il  T)ec.=01on4-^Dec.  /  Ad.Gsia-na 

Arg.  Drc.  =:  Arg.  R.  A.  +  3  sigug. 

Arg.  Dec.  =  Arg.  R.  A.  +  3  signs. 

Ai-.3il  Dcc.=3loii— ^Dcc.  S  if  Decl.  s. 

D 

0.     G. 

1.      7 

2.    8. 

D 

0.    G. 

1.     7. 

2.     8. 

D 

0.    G. 

1.    7. 

2.     8. 

-    + 

-     + 

-   + 

+   - 

+    - 

0"  41 

3o 

-   + 

-   + 

3". 45 

-  4- 

o 

19". 17 

I 6". 60 

9".  59 

3o 

0 

o".83 

0".72 

0 

3'-'.  98 

i".99 

3o 

I 

19  .17 

16  .43 

9  .3o 

29 

I 

0  .83 

0  .71 

0  .40 

29 

I 

3  .98 

3  .4i 

I    .93 

29 

2 

19  .16 

16  .26 

9  .00 

28 

2 

0  .83 

0   .70 

0  .39 

28 

2 

3   .98 

3   .38 

I    .87 

28 

3 

19  .i5 

16  .08 

8   .70 

27 

3 

0  .83 

0  .69 

0  .38 

27 

3 

3  .98 

3   .34 

I    .81 

27 

4 

19  .i3 

1 5   .90 

8  .40 

26 

4 

0  .82 

0  .69 

0  .36 

26 

4 

3  .97 

3   .3o 

I    .75 

26 

6 

19  .10 

0   .71 

8  .10 

25 

5 

0  .82 

0   .68 

0  .35 

25 

5 

3  .97 

3  .26 

I    .68 

25 

6 

19  .07 

i5   .5i 

7  .80 

24 

6 

0  .82 

0   .67 

0  .34 

24 

6 

3   .96 

3   .22 

I    .62 

24 

7 

19   .o3 

i5   .3i 

7  -49 

23 

7 

0  .82 

0  .66 

0    .32 

23 

7 

3  .95 

3  .18 

I    .56 

23 

« 

18   .99 

i5   .11 

7   .18 

22 

8 

0  .82 

0  .65 

0  -31 

22 

8 

3  .94 

3  .14 

I   .49 

22 

9 

18   .94 

i4  .90 

6  .87 

21 

9 

0  .82 

0  .64 

0  .3o 

21 

9 

3   .93 

3  .09 

I   .43 

21 

10 

18   .88 

i4  .69 

6  .56 

20 

10 

0  .81 

0  .63 

0  .28 

20 

10 

3  .92 

3  .o5 

I    .36 

20 

II 

18   .82 

i4  .4? 

6  .24 

19 

II 

0  .81 

0  .62 

0  .27 

19 

II 

3   .91 

3  .00 

I    .3o 

19 

12 

18   .75 

i4  .25 

5   .92 

18 

12 

0  .81 

0  .61 

0  .26 

18 

12 

3   .8q 

2   .96 

I     ..23 

18 

i3 

18   .68 

i4  .02 

5  .61 

17 

i3 

0  .81 

0  .60 

0  .24 

17 

i3 

3   .88 

2  .91 

I    .16 

IT 

i4 

18   .60 

i3  .79 

5  .28 

16 

i4 

0  .80 

0  .59 

0    .23 

16 

i4 

3   .86 

2   .86 

I    .10 

16 

i5 

18   .5? 

i3  .56 

4  .96 

i5 

i5 

0  .80 

0  .58 

0     .21 

i5 

i5 

3  .85 

2  -82 

I    .o3 

1 5 

i6 

i8  .43 

i3   .32 

4  .64 

i4 

16 

0  .79 

0  .57 

0     .20 

i4 

16 

3   .83 

2   .77 

0  .96 

i4 

17 

18  .34 

r3   .08 

4  .3i 

i3 

17 

0  .79 

0  .56 

0   .19 

i3 

17 

3  .81 

2   .72 

0  .90 

i3 

i8 

18    .23 

12   .83 

3  .99 

12 

18 

0  .79 

0  .55 

0    .17 

12 

18 

3  .79 

2   .66 

0  .83 

12 

19 

18  .i3 

12   .58 

3   .66 

II 

19 

0  .78 

0  ,54 

0    .16 

II 

19 

3   .76 

2  .61 

0  .76 

II 

20 
21 

r8   .02 

12    .32 

3   .33 

10 

20 

0  .78 

0  .53 

0  .i4 

10 

20 

3  .74 

2   .56 

0  .69 

10 

17   .90 

12   .07 

3  .00 

9 

21 

0  .77 

0    .52 

0  .i3 

9 

21 

3   .72 

2   .5i 

0  .62 

22 

17   .78 

II    .80 

2   .67 

8 

22 

0  .77 

0  .5i 

0  .12 

8 

22 

3   .69 

2  .45 

0  .55 

8 

23 

17   .65 

II    .54 

2   .34 

7 

23 

0  .76 

0  .5o 

0  .10 

7 

23 

3  .66 

2  .4o 

0  .49 

7 

24 

17    .52 

II    .27 

2   .00 

6 

24 

0  ,76 

0  ,49 

0  .09 

6 

24 

3   .64 

2   .34 

0  .42 

6 

2b 

26 

17  .38 

II    .00 

I   .67 

5 

25 

0  .75 

0  .47 

0  .07 

5 

25 

3   .61 

2   .28 

0  .35 

5 

17    .23 

10  .72 

I    .34 

4 

26 

0  .74 

0  .46 

0  .06 

4 

26 

3  .58 

2    .23 

0  .28 

4 

27 

17   .08 

10  .u 

I   .00 

3 

27 

0  .74 

0  .45 

0  .04 

3 

27 

3  .55 

2   .17 

0  .21 

3 

28 

16  .93 

10  .16 

0  .67 

2 

28 

0  .73 

0  M 

0  .o3 

2 

28 

3  .52 

2   .11 

0  .14 

2 

29 

16   .77 

9  -87 

0  .33 

I 

29 

0  .72 

0  .43 

0  .01 

I 

29 

3   .48 

2   .o5 

0   .07 

I 

3o 

16   .60 

9  -59 

0  .00 

0 

3o 

0  .72 

0  .4i 

0  .00 

0 

3o 

3  .45 

I    .99 

0  .00 

0 

-     + 

-    + 

-   + 

r» 

+    - 

4-  - 

4-  - 

D 

-   + 

-    + 

-    + 

D 

11.    5. 

10.    4. 

9.    3.     "  \ 

11.   5.1 

10.    4. 

y.   3. 

11.   5. 

10.   4. 

9.     3. 

To  J?«r/  ^/;e  Aberration  of  a  Star  in  Rioht  Ascension. — Find  the  Equations  in  Part  I.  and  II. 

corresponding  to  the  arguments  of  R.  A.   at  the  top  of  tliose  tables,  and  connect  them  ac- 

cording to  their  signs,  and  to  t!ie  log.  of  this  sum  or  difference  add  the  log.  secant  (less  ra- 

dius) of  the  star's  declination,  the  sum  will  be  tlie  log.  of  the  aberration  in  Piiglit  Ascension 

in  seconds  of  a  degree,  which  divided  by  15  will  be  reduced  to  time,  to  be  applied  to  the 

mean  R.  A. 

To  find  the  Aberration  of  a  Star  in  DecVnmlion. — Increase  the  former  arguments  of  R.  A. 

by  3  signs,  and  connect  together  the  corresponding  equations  of  Part  I.  and  II.  to  the  log.  of 

which  add  the  log.  sine  of  the  star's  declination,  the  sum  will  be  the  log.  of  arc    1st.     With 

the  arguments  at  the  top  of  Part  III.   find    in    the    Table  arcs    2d  and  3d.      Those  three 

arcs    connected  with  their  signs  will  be  the  aberration  in  declination,  to  be  applied  to  the 

mean  declination. 

Example.     Required  the  Aberration  in  R.  A.  and  Dec.  of  a  P(s-asi,  July  IG,  1830.' 

By  Tal)lc  8.  *R.  A.=22h.  5G'  13"=lls.  11°.  5'.    *Dcc.  14°  18'  N.  and  b;  N.  A.  Glonj.  Ss.  23°  22'. 

>lcR.A.  lis.  14°  5'. 

©Lon.    3.   23.22. 

Difr.        7.  20.4.S.  Part  I.4-12".M.. 

Dlff.  4- 3s=10s.20°.43'  Part  l.—U".U. 

Sum        3.     7.27.  Parlll.—  0.  11. 

4-12.  03.    log.  1.08027 

Suin+3  =6     7   27  Par',  rt.- 

-0   82 

-15.  66 

og.  1.19479 

*Dec.  14°18'                     860.0.013^7 
*Abcr.  R.  A.  +  12".  4.                      log.l.093L>4 

Arc    1st.  — 3".07. 

iiie  9.39270 

og.  0..58749 

:t:  Ah.  in  Il.A.  in  time  0"  83.  Olong-f-;f:Dec.:=ls.  7°  40'  Arc    2d+2.  43. 

©long— >};  Dcc.=3s.  9°  01'  Arc  3d--0.  62. 

^  .\berr.  in  Declination  — 0.  8?*.. 

PaB«248]                                     TABLE  XLIII. 

Nutation  in  Right  Ascension  and  Declination  to  be  applied  to  the  mean  values. 

PART  I. 

PART  TI. 

PART  III. 

Arg.  R.A.=:  >|<:  R.  A. — Lon.  D  node. 
+  6  signs  if  Dec.  is  S. 

Arg.  R.A.  =  *  R.A.  +  Lon.  D  node, 
-j-  6  signs  if  Dec.  is  S. 

Equation  Equinoxes  in 

R.A. 

Arg.  Dec.  =  Arg.  R.  A.  -|-  3  signs. 

Arg.  Dec.  =  Arg.  R.  A.  +  3  signs. 

Arg.  ^  Long.  5  node 

a- 

D 

0 

0.     C. 

1.  7. 

2.    8. 

D 

0.    G. 

1.    7. 

2.    8. 

D 

0.     C.!l.      7. 

2. 

~8. 

-    + 

-  + 

-   + 

-    + 

-    -f 

-   + 

-    + 

-    + 

— 

± 

8".  33 

7". 21 

4". 16 

3o 

0 

l".22 

i".o6 

o".6i 

3o 

0 

0"  .0 

8".  2 

1 4' 

.2 

3o 

I 

8   .33 

7  .14 

4  .04 

29 

I 

I     .22 

I   .o5 

0  .59 

29 

I 

0    .3 

8   .4 

i4 

.3 

29 

2 

8   .32 

7  .06 

3  .91 

28 

2 

1     .22 

I   .o3 

0  .57 

28 

2 

0    .6 

8   .7 

i4 

.5 

28 

3 

8  .32 

6  .99 

3   .78 

27 

3 

I     .22 

I    .02 

0  .55 

27 

3 

0    .9 

8   .9 

i4 

.6 

27 

4 

8  .3i 

6    91 

3   .65 

26 

4 

I     .22 

r   .01 

0  .53 

26 

4 

I     .1 

9  -2 

i4 

.7 

26 

5 

8   .3o 

6  .82 

3   .52 

25 

5 

I     .22 

I    .00 

0    .52 

25 

5 
6 

I     .4 

9  .4 

i4 

.8 

25 

6 

8  .28 

~6"74 

3   .39 

24 

6 

I     .21 

0  .99 

0  .5o 

24 

I     -7 

9  .6 

i5 

.0 

24 

7 

8   .27 

6   .65 

3   .25 

23 

7 

I     .21 

0  .97 

0  .48 

23 

7 

2     .0 

9  -9 

i5 

.  I 

23 

8 

8   .25 

6  .56 

3   .12 

22 

8 

I     .21 

0  .96 

0  .46 

22 

8 

2    .3 

10  .1 

i5 

.2 

22 

9 

8   .23 

6  .47 

2   .99 

21 

9 

I     .20 

0   .95 

0  .44 

21 

9 

2    .6 

10  .3 

i5 

.3 

21 

10 

8  .20 

6  .38 

2   .85 

20 

10 

I     .20 

0   .93 

0  .42 

20 

10 

2    .8 

10  .5 

i5 

.4 

20 

11 

8  .18 

6  .29 

2   .71 

II 

I     .20 

0  .92 

0  .40 

19 

II 

3    .1 

10  .7 

i5 

.5 

19 

12 

8  .i5 

6  .19 

2   .57 

18 

12 

1     .19 

0  .91 

0   .38 

18 

12 

3    .4 

II   .0 

i5 

.6 

18 

i3 

8  .12 

6  .09 

2   .44 

17 

i3 

I     .19 

0  .89 

0  .36 

17 

i3 

3    .7 

II    .2 

i5 

•  7 

17 

i4 

8  .08 

5   .99 

2   .3o 

16 

i4 

I     .18 

0  .88 

0  .34 

16 

i4 

4    .0 

II   .4 

i5 

.7I  i6i 

i5 

8   .05 

5   .89 

2   .16 

i5 

i5 

I     .18 

0  .86 

0    .32 

i5 

i5 

4    .2 

II    .6 

i5 

.8 

i5 

i6 

8   .01 

5   .79 

2   .02 

i4 

16 

I     .17 

0  .85 

0   .3o 

i4 

16 

4    .5 

II    .8 

i5 

•9 

i4 

I? 

7  -97 

5  .68 

I    .87 

i3 

17 

I     .17 

0  .83 

0  .27 

i3 

17 

4    .8 

12   .0 

16 

.0 

i3 

i8 

7   -92 

5  .57 

I    .73 

12 

18 

I     .16 

0  .82 

0    .25 

12 

18 

5    .1 

12   .2 

16 

.0 

12 

19 

7   .88 

5  .46 

I   .59 

II 

19 

I   -15 

0  .80 

0    .23 

II 

19 

5    .3 

12   .4 

16 

.1 

11 

20 

7  .83 

5   .35 

I   .45 

10 

20 

I   .i5 

0  .78 

0    -21 

10 

20 

5    .6 

12   .5 

16 

.1 

10 

21 

7   .78 

5   .24 

I    .3o 

9 

21 

I   .14 

0  .77 

0   .19 

9 

21 

5    .9 

12   .7 

16 

.2 

9 

22 

7   -72 

5  .i3 

I    .16 

8 

22 

I   .i3 

0  .75 

0  .17 

8 

22 

6    .1 

[2     .0 

16 

.2 

8 

23 

7   -67 

5  .01 

I    .02 

7 

23 

I    .12 

0  .73 

0  .i5 

7 

23 

6    .4 

i3  .1 

16 

.3 

7 

24 

7   .61 

4  .90 

0   .87 

6 

24 

I    .11 

0  .72 

0  .i3 

6 

24 

6    .7 

i3  ,3 

16 

.3 

6 

25 

7  .55 

4  .78 

0   .73 

5 

25 

I    .11 

0  .70 

0  .11 

5 

25 

6    .9 

i3  .4 

16 

.3 

3 

26 

7   -49 

4  .66 

0  .58 

~4 

26 

I    .10 

0  .68 

0  .09 

4 

'V6 

7    .2 

i3   .6 

16 

73" 

4 

27 

7     -42 

4  .54 

0   .44 

3 

27 

]:1 

0  .66 

0  .06 

3 

27 

7    .4 

i3   .7 

16 

.4 

3 

28 

7   .35 

4  .41 

0   .29 

2 

28 

0  .65 

0  .04 

2 

28 

7    -7 

i3  .9 

16 

.4 

2 

29 

7   -29 

4  .29 

0  .i5 

I 

29 

I  .07 

0  .63 

0  .02 

I 

29 

7    -9 

i4  .0 

16 

.4 

I 

3o 

7   .21 

4  .16 

0  .00 

0 

3o 

I  .06 

0  .61 

0  .00 

0 

3o 

8    .2 

i4  .2 

16 

.4 

0 

-    + 

-  + 

-  4- 

D 

-  + 

-    + 

-   + 

D 

+   - 

+   - 

+ 

— 

D 

n.  5. 

10    4. 

9.     3. 

11.    5. 

10.    4. 

9.     3. 

11.   5. 

10.   4.  9. 

3. 

To  find  the  Katatlon  of  a  Star  in  Riffht  Jisccnsion. — Find  in  Parts  I.  II.  the  Eqi 

ations 

corresponding  to  the  arguments  of  R.  A.  at  the  top  of  the  tables,  connect  them  ace 

ording 

to  the  signs,  and  to  the  log.  of  the  sum  or  difference  add  the  log  tangent  of  the  star's 

decli- 

nation,  the  sum  will  be  the  log.  of  an   arc,    to  which  apply  the  equation  of  the  equinoxes, 

Part  III.  corresponding  to  the  long,  of  the   D  's  node  (page  3,  N.  A.)  the  sum  or  difference 

will  be  the  Nutation  in  Right  Ascension   in  seconds  of  a  degree,  which  divided 

by  15 

will  be  reduced  to  seconds  of  time. 

To  find  the  Nattition  of  a  Star  in  Declination. — Increase  the  arguments  of  R.  A.  Parts  I. 

II.  by  3  signs,  and  connect  the  corresponding  equations  of  those  tables,  which  will  be  the 

nutation  of  declination.     Kote.     In  putting  the  R.  A.  of  the  star  equal  to  3  signs,  tl 

e  nu- 

tation  in  declination  will  be  the  equation  of  the  obliquity  of  the  ecliptic. 

Example.     Required  the  Nutation  of  a  Pegasi,  in  R.  A.  and  Decl.  July  16,  1830  ? 

>tR.A.Tab.8.11s.l4°  .5' 

D  Node  N.  A.  5.  12.  38 

Diff.            ().      1.  27  Part  I.+8"33 

Diff.+3s=9s  1°27'  Part  I.— 0"21 
Sum+3s=7. 20. 43PartII.+0"  07 

Sum            4.   26.43PartII.+6.97 

+15"3niog.l. 18409 

Nut.  in  Dee.      -|-0"40 
Gs.— 5  Node=Os.    17°.      Parti.— 

*Dec.  14°  18'  tang.  9.4(1(130 

-7.  !'7 

Arch    4-3".9  log.  0.59105 
Part  III.  Eq.  Arg.  5s.  12°  38'  —4.  9 

Gs.4-  D  Node=l Is.  1 3.       Part 
Eq.  Obi.  Eclif 

[I.- 

1.  17 

t.  — 

9.14 

Nutation  in  Right  Ascension.— I.  0= — C'lof  t. 

If  the  Declination  of  the  Star  was  South,  the  argument  of  Part  I.  II.  of  Right  A 

scen- 

8:on  and  Declination  must  be  increased  6  signs. 

TABLE  XLIV. 


[Page  249 


To  find  the  Augmentation  of  the  Moon's  Seniidianieter,  by  the  altitude  of  the 
Nonagesimal,  and  the  apparent  distance  of  the  Moon  therefrom. 


Arz. 
Slimi  of 
Pre.  Eq. 


l3 

i4 
i5 


PART  IV. 


d  's  Horiz.  Semi.  Diam. 


14' 


15' 


16' 


4(1" 


o.i6 

0.32 


50" 


0" 


0.480.4? 
0.640. 56 


0.80 


0.96 
1 .  12 
1.28 

1 .44j 

[  .6n 


0.70 


II 
).  12 
).24 

).36 
).48 
1.61 


70 
92 
08 
24 
2.40 
56 


0.98 
1 .12 
1 .26 
1 .41 


0.73 
0.85 
a. 97 
1 .09 


1.55 
1 .69 
[.83 
1.97 
2.11 

2.25 


10" 


o.3o 
0.4 1 
o.5i 


0.61 

0.71 


0.91 
1 .01 


1.33 
r.45 
[  .57I1 .32 
r .70  1 .42 
82I1.52 
1 .94,1 .62 


20" 


,16 


0.33 
o.4i 


0.49 

0.57 
0.65 
0.73 
0.82 


30" 


0.06 
0.12 

0.18 
0.25 
0.3 


3.90 

3.  98 
1  .06 

i.i4 


0.37 
0.43 
0.49 
0.55 
0.62 


0.68 
0.74 
0.80 
0.86 
0.92 
r.3i!o.98, 


40" 


50" 


0' 


10" 


20" 


30" 


40" 


50" 


0.04 


16 
0.21 


II 
0.02 
0.04 
0.06 
0.08 
o.  I( 


II 
0.00 
0.00 
0.00 
0.00 
0.00 


0.25 

9 

0.33 
0.37 
0.41 
0T45 
0.49 
0.54 
0.58 
0.62 


0.12 
o.i5 

17 

•9 

0.21 


0.00 
0.00 
0.00 
0.00 

).00 


// 
0.02 
0.04 
0.06 
0.08 
o.  10 

73 
o.  i5 

o.  17 

9 
0.21 


+ 

o.o4 
0.08 
i3 


4- 
II 

0.06 

o.i3 

o.  19 

0.25 

0.32 


Find  in  P.  I.  llie 
Iwo  equations  cor- 
responding to  tlie 
iirgumcnts   at    the 
top,   and    connect 
them  according  to 
their  signs.     Willi 
this  sum  or  dill'er- 
ence,  take  out  the 
corres|ionding  cor- 
rection P.  II.      In 
occLiitations,       tlie 
orrection   P.  III. 
■i  to  be  found  with 
liic  C  's  Par.  in  lat. 
at  the  top,  and  lier 
true  lat.  at  the  side, 
l>ut  in  solar  Eclip- 
cs,  this  p.  is  nolh- 
ig.  Connect  these 
iree     parts,    and 
with   the   sum   en- 
ter   the   side    col- 
lur.n  of  P.  IV.,  and 

lind  the  (J 's  Hori~. 

Semi.  Dia.  at  the  top  ;  the  corresponding  cor.  applied,  with  its  sign,  to  the  sum  of  the  three  first  parts,  will 

give  the  Autr.  of  the  d  's  S.  D. 

Thus  in  Ex.  1,  Prob.  5,  Appendi.x.     The  Alt.  Nonag.  is  2s.  7'^  59',  Dis.  Nonag.  (D.-f  P.)  20°  46',  d  S.  D. 

by  N.  A.  16'  27".  7.  Hence  Arg.  P.  I.  are  2s.  1°  59'+20°  46',  thuMs,2s.  28°  45'  and  Is.  17°  13',  to  which 

correspond  +  8"  18  +  6".01  =+  14"  19.     This  giv'^  in  P.  II.  +  x}"  21.  P.  HI.  is  0".     The  sum  of  the 

three  parts  is  +  14".  4,  with  which  and  the  d  S.  D.  16' 27".  7.    P.  IV.  is.  ncarl3'+0"8;  this  connected  with 

14".  4  gives  the  .\ug.  of  d  "s  S.  D.  15"  2.  as  in  Prob.  VI.  Appendi.x. 


0.23 
0.2  5 


0.29 
3i 

o.66|o.33 


0.00 
o  00 
o  00 
0.00 
0.00 
0.00 


0.23 
25 

0.27 

=9 
3i 

0.34 


0.25 
0.29 

0.34 
0.38 
0.42 

0T46 
o.5i 
0.55 
0.59 
0.63 
0.67 


0.38 

0.44 

0.5 

0.57 

0.63 


0.70 
0.76 
0.83 
0.89 
0.95 


4- 
,v 

0.09 

17 
0.26 
0.34 

ck4; 

o.5i 
0.60 
0.68 
0.77 
0.85 


0.94 
1 .02 
1 .  1 1 

9 
1.28 
36 


+ 


3? 

0.43 
0.53 

oT64 
0.75 
0.86 
0.96 
07 
78 

28 

39 

5< 
1 .6fi 
1. 71 


Pas'.'  250] 

TABLE  XLV. 

Equation  of  Second  Differences  to  be  appl 

led  to  the  jnean  longitude  or  latitude 

with  a  sign  contrary  to  that  of  the  mean  of  the  second  differences. 

App.  Time  after  noon 
or  midnight. 

Second  Difference. 

V 

2' 

3' 

4' 

5' 

G' 

7' 

8' 

9' 

10' 

11' 

12' 

h.m. 

h.  m. 

II 

// 

II 

II 

// 

// 

// 

II 

It 

II 

II 

II 

0.  0 

12.  0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.10 

11.50 

0.4 

0.8 

1.2 

1.6 

2.1 

2.5 

2.9 

3.3 

3.7 

4.1 

4.5 

4.9 

0.20 

11.40 

0.8 

1.6 

2.4 

3.2 

4.1 

4.9 

5.7 

6.5 

7.3 

8.1 

8.9 

9.7 

0.30 

11.30 

1.2 

2.4 

3.6 

4.8 

6.0 

7.2 

8.4 

9.6 

10.8 

12.0 

13.2 

14.4 

0.40 

11.20 

1.6 

3.1 

4.7 

6.3 

7.9 

9.4 

11.0 

12.6 

14.2 

15.7 

17.3 

18.9 

0.50 

11.10 

1.9 

3.9 

5.8 

7.8 

9.7 
11.5 

11.6 

13.6 

15.5 
18.3 

17.4 

20.6 

19.4 

22.9 

21.3 

23.3 

27.5 

1.  0 

11.  0 

2.3 

4.6 

6.9 

9.2 

13.7 

16.0 

25.2 

1.10 

10.50 

2.6 

5.3 

7.9 

10.5 

13.2 

15.8 

18.4 

21.1 

23.7 

26.3 

29.0 

31.6 

1.20 

10.40 

3.0 

5.9 

8.9 

11.9 

14.8 

17.8 

20.7 

23.7 

26.7 

29.6 

32.6 

35.6 

1.30 

10.30    ' 

3.3 

6.6 

9.8 

13.1 

16.4 

19.7 

23.0 

26.2 

29.5 

32.8 

36.1 

S9.4 

1.40 

10.20 

3.6 

7.2 

10.8 

14.4 

17.9 

21.5 

25.1 

28.7 

32.3 

36.9 

39.6 

43.1 

1.50 

10.10 

3.9 

7.8 

11.6 

15.5 

19.4 
20.S 

23.3 

27.2 
29.2 

31.1 
33.3 

34.9 
37.5 

38.8 
41.7 

42.7 
45,8 

46.6 
oO.O 

2.  0 

10.  0 

4.2 

8.3 

12.5 

16.7 

25.0 

2.10 

9..50 

4.4 

8.9 

13.3 

17.8 

22  "' 

26.6 

31.1 

35.5 

39.9 

44..4 

48.8 

63.3 

2.20 

9.10 

4.7 

9.4 

14.1 

18.8 

23.5 

28.2 

32.9 

37.6 

42.3 

47.0 

51.7 

56.4 

2.30 

[)..30 

4.9 

9.9 

14.8 

19.8 

24.7 

29.7 

34.6 

39.6 

44.5 

49.5 

54.4 

59.4 

2.40 

9.20 

5.2 

10.4 

15.6 

20.7 

25.9 

31.1 

36.3 

41.5 

46.7 

51.9 

57.0 

62.2 

2.50 

9.10 

5.4 

10.8 
11.2 

16.2 

21.6 

22.5 

27.1 

32.5 

37.9 

39.4 

43.3 
45.0 

48.7 
60.6 

- 

54.1 

59.5 

64.9 

3.  0 

9.  0 

5.6 

16.9 

28.1 

33.7 

56.2 

61.9 

67.5 

3.10 

8.50 

5.8 

11.7 

17.5 

23.3 

29.1 

35.0 

40.8 

46.6 

52.4 

58.3 

64.1 

69.9 

3.20 

8.40 

6.0 

12.0 

18.1 

24.1 

30.1 

36.1 

42.1 

48.1 

54.2 

60.2 

66.2 

72.2 

3.30 

8.30 

6.2 

12.4 

18.6 

24.8 

31.0 

37.2 

43.4 

49.6 

55.8 

62.0 

68.2 

74.4 

3.40 

8.20 

6.4 

12.7 

19.1 

25.5 

31.8 

38.2 

44.6 

60.9 

57.3 

63.7 

70.0 

76.4 

3.50 

8.10 

6.5 

13.0 

19.6 

26.1 

32.6 
33.3 

39.1 
40.0 

45.7 
46.7 

62.2 
53.3 

58.7 
60.0 

65.2 
66.7~ 

71.7 
73.3 

78.3 
80.0 

4.  0 

8.  0 

6.7 

13.3 

20.0 

26.7 

4.20 

7.40 

6.9 

1,3.8 

20.8 

27.7 

34.6 

41.5 

48.4 

55.4 

62.3 

69.2 

76.1 

83.1 

4.40 

7.20 

7.1 

14.3 

21.4 

28.5 

.35.6 

42.8 

49.9 

57.0 

64.2 

71.3 

70.4 

85.6 

5.  0 

7.  0 

7.3 

14.6 

21.9 

29  2 

36.5 

43.7 

51.0 

68.3 

65.6 

72.9 

S0.2 

87.6 

5.20 

G.40 

7.4 

14.8 

22  2 

29.6 

37.0 

44.4 

51.9 

59.3 

66.7 

74.1 

81.5 

88,9 

5.40 

6.20 

7.5 

15.0 

22.4 

29.9 

37.4 

44.9 

52.3 

59.8 

67.3 

74.8 

82  2 

89.7 

6.  0 

G.  0 

7.5 

15.0 

22.5 

30.0 

37.5 

45.0 

52.5 

60.0 

67.5 

75.0 

82.5 

90.0 

App.  Tim 
or  m 

e after  noon 
dnight. 

Sec 

ond  Difference. 

10" 

20" 

30" 

40" 

50" 

1" 
II 

2" 
II 

3" 

4" 

5" 

6" 

7" 

8" 

9" 

n..m. 

h.  m. 

II 

// 

II 

II 

" 

// 

// 

II 

II 

0.-  0 

12.  0 

0.0 

0,0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

o-.o 

0.0 

0.10 

11.50 

0.1 

0.1 

0.2 

0.3 

0.3 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.1 

0.1 

0.20 

11.40 

0.1 

0.3 

0.4 

0.5 

0.7 

0.0 

0.0 

0.0 

0.1 

0.1 

0.1 

0.1 

0.1 

0.1 

0.30 

11.30 

0.2 

0.4 

0.6 

0.8 

1.0 

0.0 

0.0 

0.1 

0.1 

0.1 

0.1 

0.1 

0.2 

0.2 

0.40 

11.20 

0.3 

0.5 

0.8 

1.0 

1.3 

0.0 

0.1 

0.1 

0.1 

0.1 

0.2 

0.2 

0.2 

0.2 

0..30 

11.10 

0.3 

0.6 

1.0 

1.3 

1.6 

0.0 
0.0 

0.1 

0.1 

0.1 
0.1 

0.1 

0.2 

0.2 
0.2 

0.2 

0.2 

0.2 

0.3 

0.3 

1.  0 

11.  0 

0.4 

0.8 

1.1 

1.5 

1.9 

0.3 

(  .3 

0.3 

1.10 

10.50 

0.4 

0.9 

1.3 

1.8 

0  2 

0.0 

0.1 

0.1 

0.2 

0.2 

0.3 

0.3 

0.4 

0.4 

1.20 

10.40 

0.5 

1.0 

1,5 

2.0 

2..5 

0.0 

0.1 

0.1 

0,2 

0.2 

0.3 

0.3 

f.4 

0.4 

1.30 

10  30 

0;5 

1.1 

1.6 

9  «> 

2.7 

0.1 

0.1 

0.2 

0.2 

03 

0.3 

0.4 

0.4 

0.5 

1.40 

10.20 

0.6 

1.2 

1.8 

2.4 

3.0 

0.1 

0.1 

0.2 

0.2 

0.3 

0  4 

0.4 

0.5 

0.5 

1.50 

10.10 

0.6 

1.3 

1.9 

2.6 

3.2 
3.5 

0.1 
0.1 

0.1 
0.1 

0.2 
0.2 

0.3 
0.3 

0.3 
0.3 

0.4 
0.4 

0.6 

0.6 

0.6 

0.6 

2.  0 

10.  0 

0.7 

1  4 

2.1 

2.8 

0.5 

0.6 

2.10 

9.50 

0.7 

15 

•5  9 

3.0 

3.7 

0.1 

0.1 

0.2 

0.3 

0.4 

0.4 

0.5 

0.6 

0.7 

2  20 

9.10 

0.8 

1,6 

2,3 

3.1 

3.9 

0.1 
0.1 

0.2 

0.2 

0.3 

0.4 

0.5 

0.5 

0.6 

0.7 

2.30 

9.30 

0.8 

1,6 

2.5 

3.3 

4.1 

0.2 

0.2 

0.3 

0.4 

0.5 

0.6 

0.7 

0.7 

2.40 

9.20 

09 

1,7 

2  6 

3.5 

4.3 

0.1 

0.2 

0.3 

0.3 

0.4 

0.5 

0.6 

0.7 

0.8 

2.50 

9.10 

0.9 

1,8 

2.7 
2^ 

3.6 
3.7 

4.5 

4.7 

0.1 
0.1 

0.2 
0.2 

0.3 

0.4 
0.4 

0.5 
0.5 

0.5 

0.6 
0.7 

0.7 
0.7 

0.8 
0.0 

3.  0 

9.  0 

0.9 

1.9 

0.6 

3.1C 

8.50 

1.0 

1.9 

2,9 

3.9 

4.9 

0.1 

0.2 

0.3 

0.4 

0.5 

0.6 

0.7 

0.8 

0.9 

3  20 

8.40 

1.0 

2.0 

3.0 

4.0 

5.0 

0.1 

0.2 

0.3 

0.4 

0.5 

0.6 

0.7 

0  8 

0.9 

3. .TO 

8..30 

1.0 

2.1 

3.1 

4.1 

5.2 

0.1 

0.2 

0.3 

0.4 

0.5 

0.6 

0.7 

0.8 

0.9 

3  40 

8.20 

1.1 

2.1 

Q  G) 

4.2 

5.3 

0.1 

0.2 

0.3 

0.4 

0.5 

0.6 

0.7 

0.8 

1.0 

3.50 

8.10 

1.1 

2.2 

3.3 
3.3 

4.3 

5.4 

5.6 

0.1 
0.1 

0.2 
0.2 

0.3 
0.3 

0.4 
0.4 

0.5 

0.7 
0.7 

0.8 
0.8 

0.9 
0.9 

1.0 
1.0 

4.  0 

8.  0 

1.1 

4.4 

0.6 

4.20 

7.40 

1.2 

2.3 

3.5 

4.6 

5.8 

0.1 

0.2 

0.3 

0.5 

0.6 

0.7 

0.8 

0.9 

1.0 

4.40 

7.20 

1.2 

2.4 

3.^ 

4.8 

5.9 

0.1 

0.2 

0.4 

0.5 

0.6 

0.7 

0.8 

1.0 

1.1 

5.  0 

7.0 

1.2 

2,4 

\G 

4.9 

6.1 

0.1 

0.2 

0.4 

0.5 

0.6 

0.7 

0.9 

1.0 

1.1 

fi.  0 

6.0 

1.2 

2.5  1 

3.7 

5.0 

6.2 

0.1 

0.2      0.4   1 

0.5 

0.6 

0.7 

0.9 

1.0 

1.1 

TABLE   XLVI.                                       [Page  251 

Table  showing  tlie 

variation  of  the   altitude  of  an  object  arising  from  a 

change  of  100  second 

3  in  the  declination.     If  the  change  move  the  body  to- 

wards  the  elevated  pole,  apply  the  correction  to  the  altitude  with  the  signs  in   1 

the  Table  ;  otherwise, 

change  the  signs. 

LATITUDE 

LATITUDE 

"iF 

Of  same  name 

-^5  declination. 

Of  different  name  from  declination. 

0- 

_a_ 

70=' 

94" 

00° 

87" 

50° 

76" 

40° 

64" 

30° 

5o" 

20° 

34" 

10° 

0° 

10° 

20° 

34" 

30° 

5o" 

40° 
64" 

50° 

76' 

G0° 

87" 

70° 

94' 

17" 

0' 

17" 

10 

95 

88 

78 

6!) 

5i 

35 

18 

0' 

18 

35 

5i 

65 

78 

88 

95 

10 

■20 

lOO 

92 

82 

68 

53 

m 

18 

0' 

18 

36 

53 

68 

82 

92 

too 

20 

30 

100 

88 

74 

57 

39 

20 

0' 

20 

39 

57 

74 

88 

too 

30 

0° 

40 

100 

84 

65 

45 

22 

0' 

23 

45 

65 

84 

100 

40 

0° 

50 

100 

78 

53 

27 

0' 

27 

53 

78 

too 

1 

50 

GO 

100 

68 

35 

0' 

35 

68 

too 

G'J 

— 

70 
~0~ 

94 

87 

77 

64 

5o 

too 
34 

5i 

0' 

5i 

100 
34 

5o 

64 

77 

87 

94 

70 

— 

17 

0 

•7 

10 

9b 

By 

77 

bb 

60 

34 

17 

— I 

18 

35 

5i 

66 

78 

88 

96 

10 

20 

99 

9' 

81 

67 

52 

35 

17 

—  I 

19 

37 

54 

69 

83 

93 

101 

20 

:J0 

1 07 

98 

87 

73 

56 

38 

18 

— 2 

22 

4i 

59 

76 

90 

102 

30 

2" 

40 

1 1 1 

9^ 

82 

63 

42 

20 

— 2 

25 

47 

68 

86 

102 

40 

90 

50 

116 

97 

74 

5o 

24 

—3 

3o 

57 

81 

io3 

50 

GO 

124 

95 

64 

3o 

D 

40 

73 

io3 

.iO 

— 

70 
0 

94 

87 

77 

64 

139 
5o 

92 
M 

43 

—8 

59 

108 
34 

5o 

64 

77 

87 

94 

70 

— 

17 

0 

17 

10 

94 

87 

77 

64 

5o 

M 

16 

—  I 

19 

36 

52 

67 

79 

89 

97 

10 

20 

9» 

90 

79 

66 

5i 

M 

16 

—3 

21 

3q 

56 

71 

84 

95 

io3 

20 

30 

io5 

gb 

8b 

70 

54 

36 

16 

—4 

24 

U 

62 

78 

93 

io4 

30 

4° 

40 

107 

94 

78 

59 

39 

17 

—6 

29 

5i 

71 

90 

106 

40 

40 

bO 

1 1 1 

92 

70 

45 

19 

—8 

35 

62 

86 

109 

50 

GO 

117 

88 

56 

23 

— 12 

47 

81 

112 

GO 

— 

70 
0 

94 

87 

77 

65 

127 
DO 

81 
34 

32 

—19 

70 

119 

34 

5o 

65 

77 

87 

94 

70 

— 

17 

— 0 

17 

10 

94 

87 

76 

64 

49 

33 

16 

— 2 

20 

37 

53 

67 

to 

90 

98 

10 

20 

97 

89 

78 

65 

5o 

■6i 

i5 

—4 

22 

40 

57 

73 

86 

96 

io4 

20 

30 

.o3 

94 

8J 

69 

52 

M 

i4 

—6 

26 

46 

64 

81 

95 

107 

30 

0" 

40 

.o5 

92 

76 

57 

36 

i4 

—9 

32 

54 

74 

93 

109 

40 

c° 

50 

107 

88 

66 

4i 

i5 

—13 

40 

66 

91 

,i3 

50 

(=0 

III 

82 

5i 

17 

—18 

53 

87 

119 

tiO 

/O 

118 

72 

22 

-29 

80 

129 

70 

0 

95 

87 

77 

65 

5()      35 

18 

—0 

18 

35 

5o 

03 

77 

87 

95 

0 

10 

94 

86 

76 

63 

49 

33 

i5 

—3 

20 

38 

54 

68 

81 

91 

09 

10 

20 

96 

88 

77 

64 

49 

32 

i4 

—5 

24 

4o 

59 

74 

87 

98 

1<:6 

20 

30 

lOI 

9^ 

81 

67 

5o 

32 

12 

—8 

28 

48 

66 

83 

97 

109 

:',() 

ti'^ 

40 

102 

89 

73 

54 

33 

II 

— 12 

35 

57 

78 

Q^ 

1 13 

40 

d° 

;j>0 

io4 

84 

62 

37 

1 1 

—  17 

d^ 

70 

95 

118 

50 

(.0 

io5 

77 

45 

II 

—24 

59 

93 

125 

i;n 

— 

/O 
0 

9^ 

88 

78 

65 

109 
5i 

62 
35 

i3 

-39 

90 

t4o 
35 

Si 

65 

7r 

88 

o5 

70 

0" 

— 

18 

— 0 

18 

10 

9i 

86 

75 

63 

48 

32 

i5 

—3 

21 

38 

55 

69 

82 

92 

too 

10 

21) 

95 

87 

76 

63 

48 

3i 

12 

—6 

25 

43 

60 

76 

89 

too 

20 

30 

100 

9T 

80 

65 

49 

3o 

10 

— 10 

3o 

5o 

69 

86 

too 

30 

10°!40 

too 

87 

70 

5i 

3t 

8 

— 15 

38 

60 

81 

100 

40 

10° 

•->0 

100 

81 

58 

33 

6 

— 21 

48 

75 

100 

50 

OO 

[00 

71 

39 

5 

— 3i 

66 

100 

GO 

-0 

too 

53 

3 

-48 

100 

70 

0 

96 

89 

78 

66 

5i 

35 

18 

— 0 

18 

35 

5i 

66 

78 

89 

96 

0 

10 

94 

86 

76 

63 

48 

32 

r4 

—4 

22 

39 

56 

70 

83 

94 

lOI 

10 

20 

94 

86 

76 

62 

47 

29 

1 1 

—8 

27 

45 

62 

78 

Q[ 

102 

20 

12" 

30 

99 

90    ;    78 

64 

47 

28 

8 

— 12 

33 

53 

71 

88 

io3 

30 

40 

108 

98  •  84 

68 

49 

28 

5 

—18 

4i 

63 

85 

io4 

40 

12° 

oO 

112 

97 

77 

54 

29 

2 

—25 

53 

80 

io5 

51) 

GO 

120 

95 

65 

33 

—  r 

—37 

72 

107 

GO 

;u 

70° 

C0° 

50° 

1 34 
40° 

91 
30° 

20° 

—6 

—58 

no 

20° 

30° 

40° 

50° 

60° 

70° 

70 

< 

-r- 

10° 

0° 

10° 

LATIT 

UDE 

LATITUDE 

Of  same  name  a 

s  dcrlination. 

Of  different  name  from  drxlination. 

' 

Page  252]                                                           TABLE     XLVI. 

Table  showing  the  variation  of  the  altitude  of  an  object  arising  from  a 

change  of  100  seconds  in  the  declination.     If  this  change  move  the  body  to- 

wards the  elevated  pole,  apply  the  correction  to  the  altitude  with  the  signs  in 

the  Table ;  otherwise,  change  the  signs. 

LATITUDE 

LATITUDE 

_Q_ 

0° 

Of  same  name  as  declination. 

Of  different  name  from  declination.     1 

0- 

0) 

3... 

70° 
97" 

60° 

80" 

50° 

79" 

40° 
66" 

30° 

52" 

20° 
35" 

10° 

0° 

10° 

20° 

30° 

40° 

50° 

C0° 

70° 

18" 

0 

18" 

35" 

52" 

66" 

79" 

89" 

97" 

10 

q4 

86 

76 

63 

48 

3i- 

i4 

—4 

23 

4o 

57 

72 

85 

95 

io3 

10 

20 

94 

86 

75 

61 

46 

27 

10 

—9 

28 

45 

64 

80 

93 

io4 

20 

30 

97 

89 

77 

62 

45 

26 

6 

—14 

35 

55 

74 

91 

106 

30 

14° 

40 

io6 

96 

82 

66 

46 

25 

2 

— 21 

44 

67 

88 

107 

4iJ 

14° 

.^0 

109 

93 

73 

5o 

25 

— 2 

— 3o 

58 

85 

no 

"0 

60 

ii5 

89 

60 

27 

—7 

—43 

79 

ii4 

i'i) 

70 

125 

82 

35 

—  16 

-69 

121 

80 

90 

98 

70 
0 

— 

0 

98 

90 

80 

67 

52 

36 

18 

— 0 

18 

36 

52 

67 

10 

94 

86 

76 

63 

48 

3i 

i3 

—5 

23 

4i 

58 

73 

86 

97 

io4 

10 

20 

94 

85 

74 

61 

45 

27 

9 

— 10 

3o 

48 

66 

82 

95 

106 

20 

30 

96 

87 

75 

61 

44 

25 

4 

—17 

37 

58 

77 

94 

109 

30 

IGo 

40 

io4 

94 

80 

63 

44 

22 

0 

-24 

48 

70 

92 

III 

40 

16*^ 

50 

106 

90 

70 

47 

21 

—6 

-34 

62 

90 

ii5 

50 

fiO 

I  [0 

84 

54 

21 

— 14 

— 5o 

86 

121 

60 

— 

70 

IT 

99 

91 

117 

73 

25 

—26 

—79 

l32 

70 
0 

— 

81 

68 

53 

36 

18 

— 0 

18 

36 

53 

68 

81 

9' 

99 

10 

95 

87 

76 

63 

48 

3i 

i3 

—6 

24 

42 

59 

74 

88 

98 

106 

10 

20 

93 

85 

74 

60 

44 

26 

8 

— 12 

3i 

5o 

68 

84 

98 

109 

20 

30 

95 

86 

74 

59 

42 

23 

2 

—19 

40 

60 

79 

97 

112 

30 

18° 

40 

102 

92 

78 

61 

4i 

20 

—3 

—27 

5i 

74 

96 

116 

40 

Ib^ 

50 

io3 

87 

66 

43 

17 

— 10 

-39 

67 

95 

121 

oO 

60 

io5 

79 

49 

i5 

— 20 

—56 

93 

128 

60 

70 

108 

64 

16 

—36 

-89 

i43 

36 

53 

68 

82 

92 

100 

70 
~0" 

— 

0 

100 

92 

82 

68 

53 

36 

18 

— 0 

18 

10 

95 

87 

76 

63 

48 

3i 

12 

—6 

25 

43 

60 

76 

89 

100 

10 

20 

93 

85 

74 

60 

4i 

2D 

6 

— 13 

33 

52 

70 

86 

[00 

20 

30 

94 

85 

73 

58 

40 

21 

0 

— 21 

42 

63 

82 

100 

30 

20° 

40 

100 

90 

76 

59 

39 

17 

—6 

— 3i 

55 

78 

100 

40 

20° 

50 

100 

83 

63 

39 

i3 

— 15 

-43 

72 

100 

oO 

60 

100 

74 

4'^ 

10 

—26 

—63 

100 

60 



70 

100 

56 

6 

-46 

— 100 

93 

lOl 

70 
0" 



0 

93 

83 

69 

54 

37 

19 

— n 

19 

37 

54 

69 

83 

10 

96 

88 

77 

63 

48 

3o 

12 

— 7 

26 

45 

62 

78 

91 

102 

10 

20 

93 

85 

73 

59 

43 

25 

5 

-i5 

35 

54 

72 

88 

io3 

20 

30 

94 

85 

72 

57 

39 

19 

— 2 

—23 

45 

66 

86 

io3 

30 

22° 

40 

98 

88 

74 

57 

36 

i4 

—9 

—34 

58 

82 

io4 

40 

22^^ 

50 

1 10 

97 

80 

60 

36 

9 

—19 

-48 

77 

106 

50 

60 

117 

95 

68 

38 

4 

—33 

—70 

107 

(;0 

— 

70 

"o" 

i3i 

92 

47 

—3 

—56 

— Ill 

VO 



q5 

84 

70 

55 

37 

19 

— 0 

19 

37 

55 

70 

84 

95 

io3 

10 

97 

88 

77 

64 

48 

3o 

II 

—8 

27 

46 

63 

79 

93 

io4 

10 

20 

93 

85 

73 

59 

42 

24 

4 

—16 

36 

56 

74 

91 

io5 

20 

30 

93 

84 

71 

56 

38 

18 

-4 

—26 

48 

69 

89 

107 

30 

24° 

40 

97 

86 

72 

54 

34 

12 

— 12 

-37 

62 

86 

109 

40 

24^ 

50 

107 

93 

77 

56 

32 

5 

—  23 

—53 

83 

III 

50 

60 

112 

91 

64 

32 

— 2 

-39 

—77 

ii5 

60 

70 

123 

83 

38 

— 13 

-67 

— 122 

96 

io5 

70 

— 

0 

96 

85 

72 

56 

38 

'9 

— 0 

19 

38 

56 

72 

85 

10 

98 

89 

78 

64 

48 

3o 

II 

—9 

28 

47 

65 

81 

95 

106 

10 

20 

95 

85 

73 

59 

4i 

23 

3 

—18 

38 

58 

77 

94 

108 

20 

30 

93 

83 

70 

54 

36 

16 

—6 

—28 

5o 

72 

92 

I II 

30 

2(;° 

40 

96 

85 

70 

52 

32 

9 

—16 

— 4i 

66 

91 

ii4 

40 

2G'-' 

50 

6!) 

io5 

92 

108 

74 
86 

53 

58 

28 

27 

I 
—8 

—28 
—46 

—58 
—84 

88 

123 

117 

50 
60 

70 

ii5 

75 

29 

—23 

-78 

—1 34 

70° 

70 

< 

0 

4) 

Q 

< 

70° 

60° 

.50° 

40° 

30°  j  20° 

10° 

0° 

10° 

20° 

30° 

40° 

50° 

00° 

LATITUDE 

LATITUDE 

Of  same  name  as  declination. 

Of  different  name  from  dec 

lination. 

TABLE  XLVII. 

[Page  253 

The  first  correction  is  always  to  be  taken  at  the  top. 

The  second  correction  is  to  be  taken  at  the  top  if  the  apparent  distance  exceed  90°. 

o 

107/ 

1°8' 

1°  9' 

1°10' 

\°\V 

1°12' 

1°13' 

1°14' 

1°15' 

1°  16' 

60 

9.8879 

9.8898 

9.8917 

9.8935 

9-8954 

9.8973 

9.8992 

9.9012 

9.9031 

9.9050 

I 

8879 

8898 

8917 

8936 

8955 

8974 

8993 

9012 

903 1 

905 1 

59 

2 

8880 

8898 

8917 

8936 

8955 

8974 

8993 

9012 

9032 

905 1 

58 

3 

8880 

8899 

8918 

893b 

8955 

8974 

8993 

9013 

9032 

905 1 

J37 

4 

5 

8880 

8899 

8918 
9.8918 

8937 
9.8987 

8956 

8975 

8994 

9013 

9032 

9052 

5b 

55 

9.8881 

9.8899 

9.8956 

9.8975 

9.8994 

9.9013 

9.9033 

9.9052 

6 

8881 

8900 

8918 

8937 

8956 

8975 

8994 

9014 

9033 

9o52 

54 

7 

8881 

8900 

8919 

8938 

8957 

8976 

8995 

9014 

9033 

9053 

53 

« 

8882 

8900 

8919 

8938 

8957 

8976 

8995 

9014 

9033 

9053 

52 

_9 

10 

8882 

8901 

8919 

8938 

8957 

8976 

8995 
9 .  8996 

9015 
9.9015 

9034 

9053 

5i 

9.8882 

9.8901 

9.8920 

9.8939 

9.8958 

9.8977 

9.9034 

9.9053 

II 

8883 

8901 

8920 

8939 

8958 

8977 

8996 

9015 

9034 

9054 

49 

12 

8883 

8902 

8920 

8939 

8958 

8977 

8996 

9015 

9035 

9054 

48 

i3 

8883 

8902 

8921 

8940 

8958 

8978 

8997 

9016 

9035 

9054 

47 

i4 
i5 

8884 

8902 

8921 

8940 

8959 

8978 

8997 

9016 

9035 

9055 

4b 
45 

9.8884 

9.8903 

9 .  892 1 

9.8940 

9.8959 

9.8978 

9.8997 

9.9016 

9.9036 

9.9055 

i6 

8884 

8903 

8922 

8940 

8959 

8978 

8998 

9017 

9o36 

9055 

44 

17 

8884 

8903 

8922 

8941 

8960 

8979 

8998 

9017 

9o36 

9o56 

4d 

i8 

8885 

8903 

8922 

8941 

8960 

8979 

8998 

9017 

9037 

9o56 

42 

19 

20 

8885 

8904 

8923 

8941 

8960 

8979 

8999 
9.8999 

9018 

9037 

9o56 

4i 
4o 

9.8885 

9 . 8904 

9.8923 

9.8942 

9.8961 

9.8980 

9 . 90 1 8 

9.9037 

9.9057 

21 

8886 

8904 

8923 

8942 

8961 

8980 

8999 

9018 

9088 

9057 

39 

22 

8886 

8905 

8924 

8942 

8961 

8980 

8999 

9019 

9o38 

905-7 

38 

23 

8886 

8905 

8924 

8943 

8962 

8981 

9000 

9019 

9088 

9o58 

37 

24 
25 

8887 

8905 

8924 

8943 

8962 

8981 

9000 

9019 

9039 

9o58 

3b 
35 

9.8887 

9.8906 

9.8924 

9.8943 

9.8962 

9.8981 

9 . 9000 

9.9020 

9.9039 

9.9058 

2b 

8887 

8906 

8925 

8944 

8963 

8982 

9001 

9020 

9039 

9059 

34 

27 

8888 

8906 

8925 

8944 

8963 

8982 

9001 

9020 

9040 

9039 

SS 

28 

8888 

8907 

8925 

8945 

8963 

8982 

9001 

9021 

9040 

9059 

32 

29 

3o 

8888 

8907 

8926 

8945 

8964 

8983 

9002 

9021 

9040 

9060 

3i 
3^ 

9.8888 

9.8907 

9.8926 

9.8945 

9.8964 

9.8983 

9.9002 

9.9021 

9.9041 

9 . 9060 

31 

8889 

8908 

8926 

8945 

8964 

8983 

9002 

9022 

9041 

9060 

29 

32 

8889 

8908 

8927 

8946 

8964 

8984 

9003 

9022 

9041 

9061 

33 

8889 

8908 

8927 

8946 

8965 

8984 

9003 

9022 

9042 

9061 

27 

34 
35 

8890 

8908 

8927 
9.8928 

8946 

8965 

8984 

9003 

9023 

9042 

9061 

2() 
I5 

9.8890 

9.8909 

9.8946 

9.8965 

9.8985 

9 . 9004 

9.9023 

9.9042 

9.9062 

3b 

8890 

8909 

8928 

8947 

8966 

8985 

9004 

9023 

9042 

9062 

24 

37 

8891 

8909 

8928 

8947 

8966 

8985 

9004 

9024 

9043 

9062 

23 

38 

8891 

8910 

8929 

8947 

8966 

8985 

9003 

9024 

9043 

9063 

22 

39 
40 

8891 

8910 

8929 

8948 

8967 
9.8967 

8986 

9005 

9024 

9043 

9063 

21 
20 

9.8892 

9.8910 

9.8929 

9.8948 

9.8986 

9.9005 

9.9024 

9.9044 

9.9063 

4i 

8892 

8911 

8929 

8948 

8967 

8986 

9006 

9025 

9044 

9064 

'9 

42 

8892 

8911 

8930 

8949 

8968 

8987 

9006 

9025 

9044 

9064 

18 

Ai 

8893 

891 1 

8930 

8949 

8968 

8987 

9006 

9025 

9045 

9064 

17 

44 
45 

8893 

8912 

8930 

8949 

8968 

8987 

9007 

9026 

9045 

9064 

lb 

i5 

9.8893 

9.8912 

9.8931 

9.8950 

9 . 8969 

9.8988 

9.9007 

9.9026 

9.9045 

9.9065 

4b 

8893 

8912 

8931 

8950 

8969 

8988 

9007 

9026 

9046 

9e>65 

14 

47 

8894 

8913 

8931 

8950 

8969 

8988 

9007 

9027 

9046 

9065 

i3 

48 

8894 

8913 

8932 

8951 

8970 

8989 

9008 

9027 

9046 

9066 

12 

49 
5o 

8894 

8913 

8932 

8951 

8970 

8989 

9008 

9027 

9047 

9066 

1 1 

10 

9.8895 

9.8913 

9.8932 

9.8951 

9.8970 

9.8989 

9 . 9008 

9.9028 

9.9047 

9 .  9<:)66 

DI 

8895 

8914 

8933 

8952 

8971 

8990 

9009 

9028 

9047 

9067 

9 

b2 

8895 

8914 

8933 

8952 

8971 

8990 

9009 

9028 

9048 

9067 

b 

53 

8896 

8914 

8933 

8952 

8971 

8990 

9009 

9029 

9048 

9067 

7 

54 
55 

8896 

8915 

8934 

8952 

8971 

8991 

9010 

9029 

9048 

90G8 

b 
"5 

9.8896 

9.8915 

9.8934 

9.8953 

9.8972 

9.8991 

9.9010 

9.9029 

9 . 9049 

9 . 9068 

Db 

8897 

8915 

8934 

8953 

8972 

8991 

9010 

9o3o 

9049 

9068 

4 

i)7 

8897 

8916 

8935 

8953 

8972 

8992 

901 1 

9o3o 

9049 

9069 

3 

58 

8897 

8916 

8935 

8954 

8973 

8992 

9011 

9o3o 

9o5o 

9069 

2 

59 

8898 

8916 

8935 

8954 

8973 

8992 

90 1 1 

903 1 

9o5o 

9069 

I 

bo 

8898 

8917 

8935 

8954 

8973 

8992 

9012 

903 1 

9o5o 

9070 

0 

8°  52' 

8°  51' 

8°  50' 

8°  49' 

8°  48' 

8°  47' 

8°  46' 

8°  45' 

8°  44' 

8°  43' 

b 

^he  second  correct 

ion  is  to  be  taken 

at  the  bottom  it'  t 

^le  apparent  dista 

ace  be  less  than  90°. 

Page  254]                      TABLE  XLVII. 

The  first  correction  is  always  to  be  lalien  at  the  top. 
Tlie  second  correction  is  to  be  taken  at  the  top  if  the  apparent  distance  exceed  90°. 

II 

0 

I 

2 

3 
4 
5 
6 

7 
8 

_9 

lO 

II 

12 

i3 
i4 
i5 
t6 

17 
i8 

19 

20 
21 

23 
24 
25 
26 
27 
28 
29 

3o 
3i 

32 

33 
34 
35 
36 

37 
38 
39 

40 
4i 

42 
43 
44 
45 
46 
47 
48 

49 
5o 
5i 

52 

53 
54 
55 
56 
57 
58 

60 

1°17' 

i°18' 

1°  19' 

1°  20' 

1°21' 

1°  22' 

1°  23' 

I°24' 

1°  25' 

1°  26' 

60 
59 
58 

57 
56 

55 
54 
53 

52 

5i 

5o 

49 
48 

47 
46 

45 

4o 
42 
4i 
40 
39 
38 
37 
36 

35 
34 
33 

32 

3i 

3o 

29 
28 

27 
26 

25 

24 

23 
22 
21 

20 
19 
18 

17 
16 

i5 
1 4 
i3 
12 
II 
10 

n 

6 
■5 
4 
3 
2 
I 
0 

// 

9.9070 
9070 
9070 
9071 
9071 

9.9089 
9090 
9090 
9090 
9091 

9.9109 
9109 
9109 
9110 
9110 

9.9128 
9129 
9129 
9129 
9i3o 

9.9148 
9149 
9149 
9149 
91 5o 

9.9168 
9168 
9169 
9169 
9169 

9.9188 
9188 
9189 
9189 
9189 

9.9208 
9209 
9209 
9209 
9210 

9.9228 
9229 
9229 
9229 
9280 

9.9249 
9249 
9249 
9250 
9250 

9.9071 
9072 
9072 
9072 
9073 

9.9091 
9091 
9091 
9092 
9092 

9.9110 
9111 
9111 
9111 
9112 

9.9130 
9i3o 
9i3i 
9i3i 
9i3i 

9.9150 
9i5o 
9i5i 
9i5i 
9i5i 

9.9170 
9170 
9170 
9171 

9171 

9.9190 
9190 
9190 
9191 
9191 

9.9210 
9210 
92 1 1 
9211 
9211 

9.9230 
9280 
9231 
9231 
9231 

9.9250 
9251 
9251 
9251 
9252 

9.90-3 

9-.; 
9074 
9074 
9074 

9.9092 
9093 
9093 
9093 
9094 

9.9094 
9094 
9095 
9095 
9095 

9.9112 
9112 
9113 
91x3 
9113 

9.9132 
9i32 
9132 
9133 
9133 

9.9152 
9i52 

0152 

9153 
9153 

9.9171 
9172 
9172 
9172 
9173 

9.9191 
9192 
9192 
9192 
9193 

9.9212 
9212 
9212 
9213 
9213 

9.9282 
9282 
9282 
9233 
9233 

9.9252 
9252 
9253 
9253 
9253 

9.9075 
9075 
9075 
9076 
9076 

9.9114 
9114 
9114 
9115 
9ii5 

9.9133 
9134 
9134 
9134 
9135 

9.9153 
9154 
9154 
9154 
9155 

9.9173 
9173 
9174 
9174 
9174 

9.9193 
9193 
9194 
9194 
9194 

9.9218 
9214 
9214 
9214 
9215 

9.9233 
9234 
9234 
9234 
9235 

9.9254 
9254 
9254 
9255 
9255 

9.9076 

9076 
9077 
9077 
9077 

9 . 9096 
9096 
9096 
9097 
9097 

9.9115 
911D 
9116 
9116 
9117 

9.9135 
0135 
9i36 
9 1 36 
9 1 36 

9.9155 
9155 
9i56 
9i56 
9i56 

9.9175 
9175 
9175 
9176 
9176 

9.9195 
9195 
9195 
9196 
9196 

9.9215 
9215 
921G 
9216 
9216 

9.9235 
9235 
9236 
9236 
9286 

9.9255 
9256 
9256 
92  56 
9257 

9.9078 
9078 
9078 
9079 
9079 

9.9097 
9098 
90  98 
9098 
9099 

9.9117 
9117 
9118 
9118 
9118 

9.9137 
9137 
9137 
9 1 38 
9i38 

9.9157 
9157 
9157 
9i58 
9i58 

9.9176 
9177 

9177 
9177 
9178 

9.9196 
9197 
9197 
9197 
9198 

9.9217 
9217 
9217 
9218 
9218 

9.9237 
9237 
9287 
9238 
9288 

9.9257 
9257 
9258 
9258 
925s 

9.9079 
9080 
90S0 
9080 
9081 

9.9099 
9099 
9100 
9100 
9100 

9.9119 
9119 
9119 
9:20 
9120 

9.9138 
9139 
9i3o 
9139 
9140 

9.9158 
9159 
9159 
9159 
91D0 

9.9178 
9178 
9179 

9179 
9179 

9.9198 
9198 
9199 
9199 
9199 

9.9218 
9219 
9219 
9219 
9220 

9.9288 
9239 
9239 
9289 
9240 

9.9259 
9259 
9259 
9260 
9260 

9 . 908 1 
9081 
90S  2 

9082 
9082 

9.9101 
9101 
9101 
9102 
9102 

9.9102 
9ig3 
9103 
9103 
9104 

9.9120 
91 2 1 
91 2 1 
9121 
9122 

9.9140 
9140 
9141 
9141 
9141 

9 . 9 1 60 
9160 
9161 
9161 
9161 

9.9180 
9180 
9180 
9181 
9181 

9.9200 
9200 
9200 
9201 
9201 

9.9220 
9220 
9221 
9221 
9221 

9.9240 
9240 
9241 
9241 
9241 

9.9260 
9261 
9261 
9261 
9262 

9.9083 

9083 
9083 
9084 
9084 

9.9122 
9122 
9123 
9123 
9123 

9.9124 
9124 
9124 
9125 
9125 

9.9142 
9142 
9142 
9143 
9143 

9.9162 
9162 
9162 
9163 
9163 

9.9181 
9182 
9182 
9182 
9183 

9.9201 
9202 
9202 
9202 
9203 

9.9222 
9222 
9222 
9223 
9223 

9.9242 
9242 
9243 
9243 
9243 

9.9244 
9244 
9244 
9245 
9245 

9.9262 
9262 
9268 
9268 
9263 

9.9264 
9264 
9265 
92G5 
9265 

9 . 9084 
9085 
9085 
9085 
9086 

9.9104 
9104 
9105 
9105 
9105 

9  9143 
9144 
9144 
9144 
9145 

9.9163 
9164 
9164 
9164 
9165 

9.9183 
9183 
9184 
9184 
9184 

9.9208 
9203 
9204 
9204 
9205 

9.9223 
9224 
9224 
9224 
9225 

9.9086 

9086 
9087 
90S  7 
9087 

9.9106 
9106 
9106 
9107 
9107 

9.9125 
9126 
9126 
9126 
9127 

9.9145 
9145 
9146 
9146 
9146 

9.9165 
9165 
9166 
9166 
9166 

9.9.85 
9185 
9185 
9186 
9186 

9.9205 
9205 
9206 
9206 
9206 

9.9225 
9225 
9226 
9226 
9226 

9.9245 
9246 
9246 
9246 

9247 

9.9266 
9266 
9266 
9267 
9267 

9.9088 
908S 
90SS 
90S9 
9.09 
9089 

9.9107 
9107 
9108 
9108 
9108 
9' 09 

9.9127 
9127 
9128 
912S 
9128 
9128 

9.9147 

9147 
9147 
9148 
9148 
9148 

9.9167 
9167 
9167 
9167 
9168 
916S 

9.9186 
9187 
9187 
9187 
9188 
918S 

9.9207 
9207 
9207 
920S 
920S 
9208 

9.9227 
9227 
9227 
9228 
9228 
9228 

9.9247 
9247 
9248 
9248 
9248 
9249 

9.9267 
9268 
9268 
9268 
9269 
9269 

8°  42' 

8°  41' 

8°  40' 

8^39' 

8°  38' 

8°  37' 

8°  30' 

8°  35' 

8°  34' 

8°  33' 

The  s/cimd  correction  is  to  be  taken  at  the  bottom  if  tlie  app-ire.il  distance  be  less  than  90°.  | 

TABLE  XLVII. 

[Page  255 

The  firs 

!  correction  is  always  to  be  taken  at  tlie  top. 

The  second  correction  is 

to  be  taken  at  the  top  if  the  apparent  distance  exceed  90°. 

// 
o 

P  27' 

1  =  28' 

1°  29' 

1°30' 
9-9331 

1°31' 

1°  32' 

9.9372 

1°  33' 

1°  34' 

P35' 

1°3G' 

60 

9.9269 

9.9289 

9.9310 

9.9351 

9.9393 

9.9414 

9.9435 

9.9456 

T 

9269 

9290 

9310 

933 1 

9352 

9372 

9393 

94i4 

9436 

9457 

59 

?- 

9270 

9290 

93 1 1 

931 

9352 

9373 

9394 

94x5 

9436 

9457 

58 

i 

9270 

9290 

93 1 1 

9332 

9352 

9373 

9394 

941 5 

9436 

9457 

37 

4 
5 

9270 
9.9271 

9291 

93 1 1 

9332 

9353 

9373 

9394 

941 5 

9437 

9458 

56 

55 

9.9291 

9.9312 

9-9332 

9.9353 

9.9374 

9.9395 

9.9416 

9.9437 

9.9458 

6 

9271 

9291 

9312 

9333 

9353 

9374 

9395 

9416 

9437 

9459 

54 

7 

9271 

9292 

9312 

9333 

9354 

9375 

9395 

9417 

9438 

9459 

53 

8 

9272 

9292 

93i3 

9333 

9354 

9375 

9396 

9417 

9438 

9459 

32 

_9 

lO 

9272 

9292 

93i3 

9334 

9354 

9375 

9396 

9417 
9.9418 

9438 

9460 

5l 

5^ 

9.9272 

9.9293 

9.9313 

9.9334 

9.9355 

9.9376 

9.9397 

9.9439 

9 . 9460 

I  r 

9273 

9293 

93i4 

9334 

9355 

9376 

9397 

9418 

9439 

9460 

49 

12 

9273 

9293 

93i4 

9335 

9355 

9376 

9397 

9418 

9439 

9461 

48 

i3 

9273 

9294 

93i4 

9335 

9356 

9377 

9398 

9419 

9440 

9461 

47 

1 4 
i5 

9274 

9294 

93i5 

9335 

9356 

9377 

939« 

9419 

9440 

9461 

46 
45 

9.9274 

9.9294 

9.9315 

9.9336 

9.9356 

9.9377 

9.9398 

9.9419 

9.9440 

9.9462 

i6 

9274 

9295 

93i5 

9336 

93^7 

937S 

9399 

9420 

9441 

9462 

AA 

'  7 

9275 

9295 

9316 

9336 

9357 

9378 

9399 

9420 

9441 

9462 

A'i 

i8 

9275 

9296 

93i6 

9337 

9358 

9378 

9399 

9420 

9342 

9463 

42 

12 

■iO 

927D 

9296 

9316 

9337 

9358 

9379 

9400 

9421 

9442 
9.9442 

9463 
9.9464 

4i 
4o 

9.9276 

9.9296 

9.9317 

9.9337 

9.9358 

9.9379 

9 . 9400 

9.9421 

71 

9276 

9297 

9317 

9338 

9359 

9379 

9400 

9421 

9443 

9464 

39 

>r>. 

9276 

9297 

9317 

933-8 

9359 

9380 

9401 

9422 

9443 

9464 

38 

pj 

9277 

9297 

9318 

9338 

9359 

9380 

9401 

9422 

9443 

9465 

37 

p4 
■i5 

9277 

9298 

9318 

9339 

9360 

9380 

9401 

9422 

9444 

9465 

36 

35 

9.9277 

9.9298 

9. 931 8 

9.9339 

9.9360 

9.9381 

9.9402 

9.9423 

9.9444 

9 . 9465 

3  b 

9278 

929S 

9319 

9340 

9360 

9381 

9402 

9423 

9444 

9466 

M 

■'7 

9278 

9299 

9319 

9340 

9361 

9381 

9402 

9424 

9445 

9466 

i6 

■'0 

9278 

9299 

9320 

9340 

9361 

9382 

94o3 

9424 

9445 

9466 

32 

!2 

9279 

9299 

9320 

9341 

9361 

9382 

94o3 

9424 

9445 

9467 

61 

3o 

9.9279 

9.9300 

9.9320 

9.9341 

9.9362 

9.9383 

9.9404 

9.9425 

9.9446 

9.9467 

u 

9279 

9300 

9321 

9341 

9362 

9383 

9404 

9425 

9446 

9467 

29 

•;■> 

92S0 

9300 

9321 

9342 

9362 

9383 

9404 

9425 

9447 

9468 

28 

) ) 

92S0 

9301 

9321 

9342 

9363 

9384 

94o5 

9426 

9447 

9468 

27 

-'4 

9280 

9301 

9322 

9342 

9363 

9384 

94o5 

9426 

9447 

9469 

26 

9.9281 

9.9301 

9.9322 

9.9343 

9.9363 

9.9384 

9.9405 

9.9426 

9.9448 

9.9469 

J(-> 

9281 

9302 

93" 

9343 

9364 

9385 

9406 

9427 

9448 

9469 

24 

9282 

9302 

9J23 

9343 

9364 

9385 

9406 

9427 

9448 

9470 

23 

J-S 

9282 

9302 

9323 

9344 

9364 

9385 

9406 

9427 

9449 

9470 

2  2 

9282 

93o3 

9323 

9344 

9365 

9386 

9407 

9428 

9449 

9470 

21 
20 

9.9283 

9.9303 

9.9324 

9.9344 

9.9365 

9.9386 

9-9407 

9.9428 

9.9449 

9.9471 

II 

9283 

93o3 

9^24 

9345 

9365 

9386 

9407 

9428 

9450 

9471 

19 

(9. 

9533 

9304 

9324 

9345 

9366 

9387 

940S 

9429 

9450 

9471 

18 

4  J 

9284 

93o4 

9325 

9345 

9366 

9387 

9408 

9429 

9450 

9472 

17 

44 
i5 

9284 

9304 

9325 

9346 

9367 

9387 

9408 

9430 

945 1 

9472 

16 

i5 

9.9284 

9.9305 

9.9325 

9.9346 

9.9367 

9. 9388 

9.9409 

9.9430 

9.9451 

9.9472 

-,(J 

9285 

93o5 

9326 

9346 

9367 

9388 

9409 

9430 

945 1 

9473 

i4 

i7 

9285 

9.0b 

9326 

9347 

9368 

9388 

9409 

943 1 

9452 

9473 

10 

!« 

92S5 

9306 

9326 

9347 

9368 

9389 

9410 

943 1 

9452 

9473 

12 

-19 
5<j 

9286 

9006 

9327 

9347 

9368 

93S9 

94  TO 

943 1 
9.9432 

9453 

9474 

II 
10 

9.9286 

9 . 9306 

9.9327 

9.9348 

9.9369 

9.9390 

9.94II 

9.9453 

9.9474 

Dl 

9286 

9307 

9327 

9348 

q369 

9390 

941  I 

9432 

9453 

9475 

9 

5? 

92S7 

93°7 

9328 

9348 

9369 

9390 

94II 

0432 

9454 

9475 

8 

jj 

9287 

930^ 

9328 

9349 

9370 

9391 

9412 

9433 

9454 

9475 

7 

j4 

9287 

9308 

9328 

9349 

9370 

9391 

9412 

9433 

9454 

9476 

b 
5 

9.9288 

9.9308 

9.9329 

9.9350 

9.9370 

9.9391 

9.9412 

9.9433 

9.9455 

9.9476 

-){J 

9288 

9309 

9329 

9350 

9371 

9392 

94i3 

9434 

9455 

9476 

4 

'J  7 

9288 

9309 

9329 

9350 

9371 

9392 

94i3 

9434 

9455 

9477 

3 

!j5 

9280 

9309 

9330 

9351 

9371 

9392 

94i3 

9434 

9456 

9477 

2 

^9 

9.89 

9310 

9330 

9351 

9372 

9393 

94i4 

9435 

9456 

9477 

I 

X) 

9289 

9310 

9331 

9351 

9372 

9393 

94i4 

9435 

9456 

9478 

0 
II 

8°  82' 

8^  31' 

8°  30' 

8°  29' 

8°  28' 

8=27' 

8°2G' 

8°  25' 

8°  24' 

8°  23' 

T 

!ie  srcnnd  correcti 

on  is  to  1 

e  taken  . 

it  the  bottom  if  th 

e  apparent  distan 

ce  be  less  than  90°. 

Page  25G] 


TABLE  XLVII. 


The  first  correction  is  always  to  be  taken  at  tlie  top. 
The  second  correction  is  to  be  taken  at  the  top  if  the  apparent  distance  exceed  90°. 


// 

1°37' 

1°38' 

1°  39' 

1°40' 

1°41' 

1°  42' 

1°  43' 

1°  44' 

1°45' 

1°4G' 

60 

0 

9-9478 

9.9499 

9.9521 

9.9542 

9.9564 

9.9586 

9.9608 

9.9630 

9.9652 

9.9675 

I 

9478 

9500 

9521 

9543 

9565 

9586 

9608 

963 1 

9653 

9b75 

S 

3 

9478 

9500 

9521 

9543 

9565 

9587 

9609 

9631 

9653 

9675 

3 

9479 

9500 

9522 

9544 

9565 

9587 

9609 

9631 

9653 

9676 

57 

4 

9479 

9501 

9522 

9544 
9.9544 

9566 

9588 

9610 

9632 

9654 

9676 

5b 
55 

5 

9 . 9480 

9.9501 

9.9523 

9.9566 

9.9588 

9.9610 

9.9632 

9.9654 

9.9677 

6 

9480 

9501 

9523 

9545 

9566 

9588 

9610 

9632 

9655 

9677 

54 

7 

9480 

9502 

9523 

9545 

9567 

9589 

9611 

9633 

9655 

9677 

53 

8 

9481 

9502 

9524 

9545 

9567 

9589 

9611 

9633 

9655 

9678 

52 

9 

9481 

95,02 

9524 

9546 
9.9546 

9567 

9589 

961 1 

9633 

9656 

9678 

5i 

lO 

9 . 948 1 

9.9503 

9.9524 

9.9568 

9.9590 

9.9612 

9.9634 

9.9656 

9.9678 

1 1 

9482 

95o3 

9525 

9546 

9568 

9590 

9612 

9634 

9656 

9679 

49 

12 

9482 

9504 

9525 

9547 

9569 

9590 

9612 

9635 

9657 

9679 

48 

i3 

9482 

95o4 

9525 

9547 

9569 

9591 

9613 

9635 

9657 

9680 

47 

i4 

9483 

9504 

9526 

9547 
9.9548 

9569 

9591 

9613 

9635 

9658 

9(JSo 

4b 
45 

i5 

9-9483 

9.9505 

9.9526 

9.9570 

9.9592 

9.9614 

9 . 9636 

9.9658 

9 .  9f)8o 

i6 

9483 

95o5 

9527 

9548 

9570 

9592 

9614 

9636 

9658 

9681 

44 

17 

9484 

95o5 

9527 

9549 

9570 

9592 

9614 

9636 

9659 

9(181 

43 

i8 

9484 

9506 

9527 

9549 

9571 

9593 

9615 

9637 

9659 

9(j8i 

42 

19 

9485 

9506 

9528 

9549 
9.9550 

9571 

9593 

9615 

9637 

9659 

9682 

41 
4o 

20 

9.9485 

9.9506 

9.9528 

9.9571 

9.9593 

9.9615 

9.9638 

9 . 9660 

9.9(;82 

21 

9485 

9507 

9528 

9550 

9572 

9594 

9616 

9638 

9660 

9683 

39 

22 

9486 

9507 

9529 

9550 

9572 

9594 

9616 

9638 

9661 

9683 

38 

23 

9486 

9507 

9529 

9551 

9573 

9594 

9617 

9639 

9^5' 

9()83 

37 

24 

9486 

9508 

9529 

955i 

9573 

9595 

9617 

9639 

9661 

9684 

jb 
35 

25 

9.9487 

9 . 9508 

9.9530 

9.9551 

9.9573 

9.9595 

9.9639 

9.9662 

9.9(^84 

26 

9487 

9509 

9530 

9552 

9574 

959b 

9618 

9640 

9662 

9684 

34 

27 

9487 

9509 

9530 

9552 

9574 

9596 

9618 

9640 

9662 

9685 

:i6 

28 

9488 

9509 

9531 

9553 

9574 

9596 

96r8 

9641 

9663 

9685 

32 

29 

3.) 

9488 

9510 

9531 

9553 

9575 

9597 

9619 

9641 

o663 

9686 

3i 

Jo 

9.9488 

9.9510 

9.9532 

9.9553 

9.9575 

9.9597 

9.9619 

9.9641 

9.9664 

9 . 96S6 

3i 

9489 

9510 

9532 

9554 

9575 

9597 

9619 

9642 

9664 

9686 

29 

32 

9489 

95i  I 

9532 

9554 

9576 

9598 

9620 

9642 

9664 

9687 

28 

33 

9490 

951 1 

9533 

9554 

9576 

9598 

9620 

9642 

9bb5 

9687 

27 

34 
35 

9490 

951 1 

9533 
9.9533 

9555 

9577 

9599 

9621 

9643 

9665 

9()87 

2b 
l5 

9 • 9490 

9.9512 

9.9555 

9.9577 

9.9599 

9 . 962 1 

9.9643 

9 . 9665 

9.9688 

36 

9491 

9512 

9534 

9555 

9577 

9599 

9621 

9643 

9666 

9(388 

24 

3- 

9491 

9512 

9534 

9556 

9578 

9600 

9622 

9644 

9666 

9(;89 

23 

38 

9491 

95i3 

9534 

9556 

9578 

9600 

9622 

9644 

9667 

9689 

22 

39 

40 

9492 

95i3 
9.9514 

9535 

9557 

9578 

9600 

9622 

9645 

9667 

9()89 

21 

20 

9.9492 

9.9535 

9.9557 

9.9579 

9.9601 

9.9623 

9.9645 

9.9667 

9 . 9(390 

4i 

9492 

95 1 4 

9536 

9557 

9579 

9601 

9623 

'^^Jl 

9668 

9690 

19 

42 

9493 

95 14 

9536 

9558 

9579 

9601 

9624 

9646 

9668 

9690 

ih 

4'i 

9493 

95i5 

9536 

9558 

9580 

9602 

9624 

9646 

9668 

9691 

l-! 

44 
45 

9493 

95i5 

9537 

9558 

9580 

9602 

9624 

9646 

9669 

9f)9i 

16 
75 

9.9494 

9.9515 

9.9537 

9.9559 

9.9581 

9 . 9603 

9.9625 

9.9647 

9 .  9669 

9.9(^92 

4b 

9494 

9516 

9537 

9559 

9581 

9603 

9625 

9647 

9669 

9(192 

14 

4? 

9495 

95i6 

9538 

9559 

9581 

9603 

9625 

9648 

9670 

9692 

i3 

48 

9495 

9516 

9538 

9560 

9582 

9604 

9626 

9648 

9670 

9693 

49 
5o 

9495 
9.9496 

9517 
9.9517 

9538 

9560 
9.9561 

95S2 

9604 

9626 

9648 

9671 

9(193 

11 

10 

9.9539 

9.9533 

9 . 9604 

9.9626 

9 . 9649 

9.9671 

9 . 9693 

5i 

9496 

9518 

9539 

9561 

9583 

9605 

9627 

9649 

9671 

9694 

9 

52 

9496 

9518 

9540 

9561 

9583 

9605 

9627 

9649 

9672 

9694 

8 

53 

9497 

9518 

9540 

9562 

9584 

9605 

9628 

9650 

9672 

9695 

7 

54 
55 

9497 

9519 

9540 

9562 

9584 

9606 

9628 

9650 

9672 

9695 

5 

9.9497 

9.9519 

9.9541 

9.9562 

9.9584 

9 . 9606 

9.9628 

9.9651 

9.9673 

9.9695 

56 

9498 

9519 

9541 

9563 

9585 

9607 

9629 

965 1 

9673 

969b 

4 

5? 

9498 

9520 

9541 

9563 

9585 

9607 

9629 

9b5i 

9674 

9(396 

J 

58 

9498 

9520 

9542 

9563 

9585 

9607 

9629 

9652 

9674 

9696 

2 

59 

9499 

9520 

9542 

9564 

9586 

9608 

9630 

9652 

9674 

9697 

I 

60 

9499 

9521 

9542 

9564 

9586 

9608 

9630 

9652 

9675 

9697 

0 

8°  22' 

8°  21' 

8°  20' 

8°  19' 

8°  18' 

8°  17' 

8°  13' 

8°  15' 

8°  14' 

8°  13' 

The  sirnnd  correct 

ion  is  to  be  taken 

at  tlie  bottom  if  t 

le  apparent  distar 

ice  be  less  than  'JO 

o_ 

TABLE  XLVII 

[Page  257 

The  first  correction  is  always  to  be 

taken  at  the  top. 

The  secojtd  correction  is  to  be  tai 

Len  at  the  top  if  t 

10  apparent  distance  e.xcecd  00°. 

// 
o 

1°  47' 

1°43 

P  49' 

1°  50' 

1°51' 

1°  52' 

1°  53' 

1°  54' 

1°  55' 

1°  5G' 

60 

9-9''97  9-9720 

9.9742 

9.9765 

9.9788 

9.9811 

9.9884 

9.9858 

9.9881 

9.9905 

I 

9098 

9720 

9743 

9766 

9788 

9812 

9885 

9858 

9881 

9905 

59 

3 

9698 

9720 

9743 

9766 

9789 

9812 

9835 

9858 

9882 

9905 

58 

3 

9698 

9721 

9744 

9766 

9789 

9812 

9885 

9859 

9882 

9906 

57 

4 
5 

9699 

9721 

9744 

9767 
9.9767 

9790 

9818 

9886 

9859 

9888 

9906 

5b 

55 

9.9699 

9.9722 

9.9744 

9.9790 

9.9818 

9.9886 

9 . 9860 

9.9888 

9.9907 

6 

9()99 

9722 

9745 

9767 

9790 

9818 

9887 

9860 

9888 

9907 

54 

9700 

9722 

9745 

97b8 

979' 

9814 

9887 

9S60 

9884 

9907 

53 

8 

9700 

9728 

974b 

9768 

9791 

9814 

9887 

9861 

9884 

9908 

52 

_9 

lO 

9701 

9723 

9746 

9769 

9792 

9815 

9888 

9861 

9885 
9.9885 

9908 
9.9908 

5i 
5^ 

9.9701 

9.9723 

9.9746 

9.9769 

9.9792 

9.9815 

9.9888 

9.9S61 

1 1 

9701 

9724 

9747 

9769 

9792 

9«.5 

9889 

9863 

9885 

9909 

t 

12 

9702 

9724 

9747 

9770 

9793 

9816 

9839 

9862 

98S6 

9909 

i3 

9702 

9720 

9747 

9770 

9798 

9816 

9889 

9S68 

9886 

9910 

47 

1 4 
I'j 

9702 

9725 

9748 

9771 

9793 

9817 

9840 

9868 

9886 

9910 

4b 
45 

9.9703 

9.9725 

9-9748 

9-977' 

9-9794 

9 . 08 1 7 

9 . 9840 

9.9868 

9.9887 

9.9910 

i6 

9703 

9726 

9748 

9771 

9794 

9817 

9841 

9864 

9887 

9911 

44 

17 

9704 

9726 

9749 

9772 

9795 

9818 

984' 

9864 

9888 

991 1 

48 

18 

9704 

9727 

9749 

9772 

9795 

98.8 

9841 

98G5 

9888 

9912 

42 

20 

9704 

9727 

9750 

9772 

9795 

9S18 

9842 

9865 

9888 

9912 

4i 
4o 

9.9705 

9.9727 

9.9750 

9-9773 

9.9796 

9.9819 

9.9842 

9.9865 

9.9889 

9.9912 

21 

970S 

9728 

9750 

9773 

9796 

9819 

9842 

9866 

9889 

99 '3 

39 

29 

9705 

9728 

975 1 

9774 

9797 

9820 

9848 

9866 

9890 

9918 

88 

23 

9706 

9728 

975 1 

9774 

9797 

9820 

9843 

9867 

9890 

9914 

37 

24 
25 

9706 

9729 

9751 

9774 

9797 

9820 

9844 

9867 

9890 
9.9S91 

9914 
9-9914 

8b 
35 

9.9707 

9.9729 

9.9752 

9-9775 

9-9798 

9.9821 

9-9844 

9.9867 

26 

9707 

9730 

97^2 

9775 

9798 

9821 

9844 

9868 

9891 

99'5 

34 

27 

9707 

9730 

9758 

9775 

9798 

9822 

9845 

98G8 

9892 

9915 

^^ 

38 

9708 

9780 

9758 

9776 

9799 

9822 

9845 

9869 

9892 

9916 

82 

29 

3o 

970S 

9781 

9753 

9776 

9799 

9822 

9846 

9869 

9892 

9916 

81 
3^ 

9.970S 

9.9781 

9.9754 

9.9777 

9.9800 

9.9828 

9.9846 

9 .  98G9 

9.9898 

9.9916 

3i 

9709 

973 1 

97^4 

9777 

9800 

9828 

9846 

9870 

9898 

9917 

29 

32 

9709 

9782 

97^5 

9777 

9800 

9828 

9847 

9870 

9894 

99' 7 

28 

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on  is  to  be  taken 

at  the  Iwttom  if  i\ 

le  ap[)are 

nt  distar 

ce  be  less  than  90°. 

33 


Page  253]                      TABLE  XLVII. 

The  first  correction  is  always  to  be  taken  at  the  top. 

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o 

1°57' 

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The  second  correction  is  to  be  taken  at  the  bottom  if  the  apparent  distance  be  less  than  90°. 

TABLE  XLVII.                     [Page  259 

The  first  correction  is  always  to  be  taken  at  the  top. 

The  second  correction  is  to  be  taken  at  the  top  if  the  apparent  distance  exceed  90®. 

o 

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5() 

0218 

0244 

0269 

0295 

032I 

o347 

0878 

0899 

0426 

0453 

o48o 

II 
10 

0.0219 

0.0244 

0.0270 

0.0293 

o.o32i 

0.0347 

0.0874 

o.o4oo 

0.0426 

0.0453 

0.0480 

bi 

0219 

0244 

0270 

0296 

o322 

o348 

0874 

o4oo 

0427 

0454 

o48o 

0 

1  :)2 

0219 

0245 

0270 

0296 

0322 

o348 

0874 

o4oi 

0427 

0454 

0481 

8 

d3 

0220 

0245 

0271 

0297 

o323 

0349 

0875 

o4oi 

0428 

c454 

o48i 

7 

55 

0220 

0246 

0271 

0297 

o323 

0349 

0875 

o4o2 

0428 

0455 

0482 

6 
5 

0.0221 

D.0246 

0.0272 

3.0297 

0.0323 

0.0349 

0.0876 

o.o4o2 

0.0429 

0.0455 

0.0482 

■jt. 

0221 

0247 

0272 

0298 

o324 

o35o 

0876 

o4o3 

0429 

o456 

o483 

4 

37 

0221 

0247 

0273 

0298 

o324 

o35o 

0877 

o4o3 

0480 

0456 

o488 

3 

58 

02  2  2 

0247 

0273 

0299 

o325 

o35i 

0877 

o4o3 

o43o 

•  0457 

0484 

2 

59 

0222   0248 1 

0273 

0299 

o325 

o35i 

0877 

o4o4 

o43o 

0457 

0484 

I 

bo 
The 

0223 

0248 

0274 

o3oo 

0326 

o352 

0878 

o4o4 

043 1 

0458 

0484 

0 
7 

7°  51' 

7°  50' 

7°  49' 

7°  48' 

7°  47' 

7°  46' 

7°  45' 

7°  44' 

7°  43' 

7°  42' 

7°  41'. 

second  correction  is  to  be  taken  at  the  bottom  if  the  apparent  distance  be  less  than  90°. 

Page  2fi0] 

TABLE  XLVII. 

The  firsi  correction  is  always  to  be  taken  at  the  top. 

The  necond  correction  is  to  be  taken  at  the  top  if  the  apparent  distance  exceed  90°. 

o 

2°  19'  2°  20' 

2=21' 

2°  22' 

2=23' 

2=24' 

2=25' 

2°2G' 

2°  27' 

2°  28' 

2=29' 

60 

0.0484  0.o5l2 

o.o53c 

0 . o566 

0 . 0594 

0.0621 

0.0649 

0.0678 

0.0706  0.0734 

0.0763 

I 

048  5 

■o5i2 

0539 

0567 

0594 

0622 

o65o 

0678 

0706 

0735 

0763 

59 

2 

o485 

05l2 

o54o 

0667 

0595 

0622 

o65o 

0678 

0707 

0735 

0764 

58 

3 

o486 

o5i3 

o54o 

o568 

0595 

0623 

o65i 

0679 

0707 

0736 

0764 

57 

4 
5 

0486 

o5i3 

o54i 

o568 

0596 

0623 

o65i 

0679 

0708 

0736 

0765 

56 
55 

0.0487 

o.o5i4 

o.o54i 

o.o568 

0.0596 

0.0624 

o.o652 

0.0680 

0.0708 

0.0737 

0.0765 

6 

0487 

o5i4 

o54i 

0569 

o5q6 

0624 

o652 

0680 

0709 

0737 

0766 

54 

7 

0488 

o5i5 

o542 

0569 

0597'  0625 

o653 

0681 

0709 

0738 

0766 

53 

8 

0488 

o5i5 

0542 

0570 

0597 

0625 

o653 

068: 

0710 

0738 

0767 

52 

9 

lO 

0489 

0S16 

o543 

0570 

059S 

0626 

o654 

0682 

0710 

0739 

0767 

5i 
5o 

0.0489 

o.o5i6 

0.0543 

0 . o57 1 

0 . 0598 

0 . 0626 

0.0654 

0.0682 

0.07 II 

0.0739 

0.0768 

1 1 

0489 

o5i7 

c/>AA 

0571 

0599 

0627 

o655 

o683 

G711 

0740 

0768 

49 

12 

0490 

o5i7 

0:..  44 

0572 

0599 

0627 

o655 

o683 

07 1 1 

0740 

0769 

48 

iJ 

0490 

o5i7 

0545 

0572 

0600 

0628 

o655 

0684 

0712 

0740 

0769 

47 

i4 
i5 

0491 

o5i8 

0545 

0573 

0600 

0628 

o656 

0684 

0712 

0741 

0770 

46 

45 

0.0491 

o.o5i8 

0 . o546 

0 . 0573 

0.0601 

0.0628 

o.o656 

0.0685 

0.0713 

0.0741 

0.0770 

i6 

0492 

o5i9 

o546 

0573 

0601 

0629 

0657 

068  5 

0713 

0742 

0771 

M 

17 

0492 

o5i9 

o546 

0574 

0602 

0629 

0657 

0686 

0714 

0742 

0771 

43 

i8 

0493 

o52o 

o547 

o574 

0602 

o63o 

o658 

0686 

0714 

0743 

0772 

42 

19 

20 

0493 

o52o 

o547 

0575 

0602 

o63o 

o658 

0686 

0716 

0743 

0772 

4i 
4o 

0.0493 

0.o52I 

0.0548 

0.0575 

o.o6o3 

o.o63i 

0.0659 

0.0687 

0.0715 

5.0744 

0.0773 

21 

0494 

052I 

o548 

0576 

o6o3 

o63i 

0659 

0687 

0716 

0744 

0773 

39 

22 

0494 

o52i 

0549 

0576 

0604 

o632 

0660 

0688 

0716 

0745 

0774 

38 

2J 

0495 

0522 

0549 

0577 

o6o4 

o632 

0660 

0688 

0717 

0745 

0774 

37 

24 
25 

0495 

o522 

o55o 

0577 

o6o5 

o633 

0661 

0689 

0717 

0746 

0774 

36 
~35 

0.0496 

o.o523 

o.o55n 

0.0578 

0.060 5 

o.o633 

0.0661 

0.0689 

0.0718 

0.0746 

0.0775 

26 

0496 

o523 

o55i 

0578 

0606 

0634 

0662 

0690 

0718 

0747 

0775 

34 

27 

0497 

o524 

o55i 

0579 

0606 

o634 

0662 

0690 

0719 

0747 

0776 

33 

28 

0497 

0624 

o552 

0579 

0607 

o634 

o663 

0691 

0719 

0748 

0776 

32 

29 

3o 

0498 

o52  5 

o552 

0579 

0607 

o635 

o663 

0691 

0720 

0748 

0777 

3 1 
3o 

0.0498 

o.o525 

o.o552 

o.o58o 

0.0608 

o.o635 

o.o663 

0.0692 

0.0720 

0.0749 

0.0777 

Ji 

0498 

0526 

o553 

o58o 

060S 

o636 

0664 

0692 

0721 

0749 

077S 

29 

32 

0499 

o526 

o553 

o58i 

0609 

o636 

0664 

0693 

0721 

0760 

0778 

28 

33 

0499 

0526 

o554 

o58i 

0609 

0637 

o665 

0693 

0721 

0750 

0779 

27 

34 

o5oo 

0527 

o554 

o582 

0609 

0637 

o665 

0694 

0722 

0751 

0779 

26 

25 

35 

o.oSoo 

0.0527 

0.0555 

o.o582 

0.0610 

o.o63S 

0.0666 

0.0694 

0.0722 

0.0751 

0.0780 

36 

o5oi 

0628 

o555 

o583 

0610 

o638 

0666 

0694 

0723 

0761 

0780' 

24 

37 

o5oi 

o528 

o556 

o583 

061 1 

0639 

0667 

0695 

0723 

0752 

0781 

23 

38 

o5o2 

0529 

o556 

o584 

0611 

0639 

0667 

0695 

0724 

0762 

0781 

2  2 

39 
4o 

o5o2 

0529 

0557 

o584 

0612 

o64o 

0668 

0696 

0724 

0753 

0782 

2  1 
20 

o.o5o2 

o.o53o 

0.0557 

o.o585 

0 . 06 1 2 

0 . 0640 

0.0668 

0.0696 

0.0725 

0.0753 

0.0782 

4i 

o5o3 

o53o 

o557 

o585 

o6i3 

0641 

0669 

0697 

0725 

0754 

0783 

19 

42 

o5o3 

o53i 

o558 

o585 

o6i3 

064 1 

0669 

0697 

0726 

0754 

0783 

18 

43 

o5o4 

o53i 

o558 

o586 

o6i4 

064 1 

0670 

0698 

0726 

0755 

0784 

17 

44 
45 

o5o4 

o53i 

0559 

o586 

0614 

0642 

0670 

0698 

0727 

0755 

0784 

16 

i5 

o.o5o5 

o.o532 

0.0559 

0.0587 

0.061 5 

0.0642 

0.0670 

0.0699 

0.0727 

0.0756 

0.0785 

46 

o5o5 

o532 

o56o 

0587 

o6i5 

0643 

0671 

0699 

0728 

0766 

0785 

i4 

47 

o5o6 

o533 

o56o 

o588 

061 5 

0643 

0671 

0700 

0728 

0767 

0786 

i3 

48 

o5o6 

o533 

o56i 

o588 

0616 

0644 

0672 

0700 

0729 

0757 

0786 

12 

49 
5» 

o5o7 

o534 

o56i 

0589 

0616 

0644 

0672 

0701 

0729 

0758 

0787 

1 1 

10 

o.o5()7 

o.o534 

0.0 50 2 

0.0589 

0 . 06 1 7 

0.0645 

0 . 0673 

0.0701 

0.0730 

0.0758 

0.0787 

5i 

o5o7 

o535 

o562 

0590 

0617 

0645 

0673 

0702 

0730 

0759 

0787 

9 

i)2 

o5o8 

o535 

o562 

0590 

0618 

0646 

0674 

0702 

0730 

0759 

0788 

8 

63 

o5o8 

o536 

o563 

0591 

0618 

0646 

0674 

0703 

0731 

0760 

0788 

7 

54 
55 

o5o9 

o536 

o563 

0591 

0619 

0647 

0675 

0703 

073  r 

0760 

0789 

6 
5 

o.o5()9 

o.o536 

0.0 564 

0.0591 

0 . 06 1 9 

0 . 0647 

0.0675 

0.0703 

0.0732 

0.0761 

0.0789 

5b 

o5io 

o537 

o564 

0692 

0620 

0648 

0676 

0704 

0732 

0761 

0790 

4 

^7 

o5io 

o537 

o565 

0592 

0620 

o648 

0676 

0704 

0733 

0762 

0790 

3 

58 

o5ii 

o538 

o565 

0693 

0621 

0648 

0677 

0705 

0733 

0762 

0791 

2 

59 

o5ii 

o538 

o566 

0593 

0621 

0649 

0677 

0705 

0734 

0769 

0791 

I 

bo 

05l2 

0539 

o56fi 

0594 

0621 

0649 

0678 

0706 

0734 

0763 

0792 

0 
// 

7°  40' 

7°  39' 

7°  38' 

7=37' 

7°  30' 

7°  35' 

7=31' 

7°  33' 

7°  32' 

7^31' 

7°3(y 

}  secmid 

correct 

on  is  to 

be  take 

n  at  th 

3  hottom 

if  the  f 

ipparen 

t  distanc 

e  be  Ic. 

\«  t/ia-n  i 

0°. 

T.ABLE  XLVII. 

[Page  2Cl 

H\ie  first  correction  is  always  to  be  taken  at  the  top. 

The  second  correction  is  to  be  taken  at  tlie  toj)  if  the 

apparent  distance  exceed  90°. 

II 

0 

2°3t/ 

2^31' 

2°  32' 

2^33' 

2=34' 

2°  35' 

2°3G' 

2°  37' 

2°  38' 

2°  39' 

2°  40' 

60 

0.079a 

0.0821 

o.oS5o 

0.0880 

0 .  091  )9 

0.0939 

0 . 0969 

0 . 0999 

0.1 o3o 

0. I06I 

0. 1091 

I 

0792 

0821 

08  5 1 

0880 

0910 

0940 

0970 

1000 

io3o 

1061 

1092 

59 

2 

0793 

0822 

o85i 

0881 

0910 

09.40 

0970 

1000 

io3i 

1062 

1093 

5fi 

3 

0793 

0822 

08  52 

0881 

091 1 

0941 

0971 

lOOI 

io3i 

1 062 

1093 

57 

4 
5 

0794 

0823 

o852 

0882 

091 1 

0941 

0971 

lOOI 

1032 

io63 

1094 

56 
55 

0.0794 

0.0823 

o.oS53 

0.0882 

0.0912 

0.0942 

0.0972 

0.1002 

0.1032 

0.1 o63 

0. 109.5 

6 

0795 

0824 

o853 

088  3 

0912 

0942 

0972 

1002 

io33 

1064 

1095 

54 

7 

0795 

0824 

o854 

o883 

0913 

0943 

0973 

ioo3 

io33 

1064 

1095 

53 

8 

0796 

0825 

oS54 

o883 

0913 

0943 

0973 

ioo3 

io34 

io65 

1 096 

52 

_9. 

10 

0796 

0825 

08  5  5 

0S84 

0914 
0.0914 

0944 

0974 

ioo4 

1034 

io65 

1 09() 

5i 
'So 

0.0797 

0.0826 

0.0855 

0.0884 

0.0944 

0.0974 

0.1004 

o.io35 

0 . 1 066 

0.1097 

11 

0797 

0826 

o855 

088  5 

0915 

0945 

0975 

ioo5 

io35 

1066 

1097 

49 

12 

0798 

0827 

o856 

088  5 

0915 

0945 

0975 

ioo5 

io36 

1067 

1098 

48 

i3 

0798 

0827 

o856 

0886 

0916 

0946 

0976 

1006 

io36 

1067 

1098 

47 

i4 
i5 

0799 

0828 

0857 

0886 

0916 

0946 

0976 

1006 

io37 

1068 

1099 

46 
45 

0  0799 

0.0S28 

0.0S57 

0.0887 

0.0917 

0.0947 

0.0977 

0.1007 

0.1037 

0.1068 

0.1099 

i6 

0800 

0829 

oS5S 

0887 

0917 

0947 

0977 

1007 

io38 

1069 

1100 

A4 

17 

0800 

0829 

oS58 

0S88 

0918 

0948 

0978 

1008 

1039 

1069 

1 100 

43 

18 

0801 

o83o 

0859 

08S8 

0918 

0948 

0978 

1008 

1039 

1070 

IlOl 

42 

19 

20 

0801 

o83o 

0859 

0889 

0919 

0949 

0979 

1009 

I  o4(j 

1070 

1101 

4i 
4o 

0.0801 

o.o83i 

0.0860 

0.0889 

0.0919 

0.0949 

0.0979 

0 . I 009 

0 . 1 o4o 

0.1071 

0.1102 

21 

0802 

o83i 

0860 

0S90 

0920 

0950 

0980 

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io4i 

1071 

1102 

39 

22 

0802 

o832 

086 1 

0890 

0920 

0950 

0980 

1011 

1 04 1 

1072 

iio3 

38 

23 

o8o3 

o832 

0861 

0891 

0921 

0951 

0981 

1011 

1042 

1072 

iio3 

37 

24 

25 

o8o3 

o833 

0862 

0891 

0921 

0951 

0981 

I0I2 

1042 

1073 

iio4 

36 

0 . 0804 

0.0833 

0.0862 

0.0892 

0.0922 

0.0952 

0.0982 

0.1012 

0.1043 

0.1073 

0.1104 

26 

0804 

o834 

o863 

0892 

0922 

0952 

0982 

ioi3 

1043 

1074 

HOD 

34 

27 

o8o5 

o834 

o863 

0893 

0923 

0953 

0983 

10x3 

1044 

1074 

iio5 

33 

28 

o8o5 

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0864 

0893 

0923 

0953 

0983 

ioi4 

1044 

1075 

1106 

32 

29 

3o 

0806 

o835 

0864 
o.o865 

0894 
0 . 0S94 

0924 

0954 

0984 

ioi4 

1045 

1075 

1106 

3i 
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0.0806 

0.0835 

0.0924 

0.0954 

0.0984 

o.ioi5 

0.1045 

0.1076 

0.1107 

3[ 

0807 

o836 

086  5 

0S95 

0925 

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09S5 

ioi5 

io46 

1076 

1108 

29 

32 

0807 

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0S66 

0895 

0925 

0955 

0985 

1016 

io46 

1077 

1108 

28 

33 

0808 

o837 

0866 

0896 

0926 

0956 

0986 

1016 

1047 

1078 

1 109 

27 

34 
35 

0808 

0837 

0867 

0896 

0926 

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0986 

1017 

1047 

1078 

1109 

56 

25 

0.0809 

0.08 38 

o.o8t)7 

0.0897 

0.0927 

0.0957 

0.0987 

0.1017 

0.1048 

0.1079 

0 . 1 1 1 0 

36 

0S09 

o838 

0868 

0897 

0927 

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0987 

1018 

1048 

1079 

IIIO 

24 

37 

0810 

0839 

0868 

0898 

0928 

0958 

0988 

1018 

1049 

1080 

nil 

23 

38 

0810 

0839 

0869 

0898 

0928 

0958 

0988 

1019 

1049 

1080 

nil 

22 

39 
4o 

0811 

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0S69 

0899 

0929 

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1019 

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21 
20 

0.081 1 

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0.0870 

0.0899 

0.0929 

0.0959 

0 .  0989 

0.1020 

0 . 1 o5o 

0 . 1 08 1 

0.1112 

4i 

0812 

084 1 

0870 

0901) 

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1082 

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19 

42 

0812 

084 1 

0871 

0900 

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io5i 

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18 

43 

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1114 

17 

45 

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0872 

0901 

0931 

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1 114 

16 
75" 

0.0814 

0.0843 

0.0872 

0.0902 

0.0932 

0 . 0962 

0.0992 

0. 1022 

o.io53 

0.1084 

o.iii5 

46 

0814 

0843 

0873 

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0932 

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1 023 

io53 

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oSi5 

0844 

0873 

0903 

0933 

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io54 

io85 

1116 

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48 

o8i5 

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0874 

0903 

0933 

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1024 

io54 

io85 

1116 

12 

49 
5o 

0816 

0845 

0874 

0904 

0934 

0964 

0994 

1024 

io55 

1086 

1117 

11 
10 

o.c«8i6 

0.0845 

0.0875 

0.0904 

0.0934 

0.0064 

0.0994 

0.  I025 

0. to55 

0.1086 

o.ii  17 

5i 

(j8i6 

o846 

0875 

0905 

0935 

0965 

0993 

1025 

io56 

1087 

1118 

9 

52 

0817 

0846 

0876 

0905 

0935 

0965 

0995 

1026 

io5u 

1087 

1118 

b 

53 

0817 

0847 

0876 

'0906 

C936 

0966 

0996 

1026 

1067 

1088 

1119 

n 

54 
55 

0818 

0847 

0877 

0906 

0936 

0966 

0996 

T027 

1057 

1088 

1119 

6 

0.0818 

0.0848 

0.0877 

0.0907 

0.0937 

0.0967 

0.0997 

0.1027 

o.io58 

0. 1089 

0.1120 

56 

0819 

0848 

0878 

0907 

0937 

0967 

0997 

1028 

io58 

1089 

1 1 20 

4 

57 

0819 

0849 

0878 

0908 

0938 

0968 

0998 

1028 

1059 

1090 

1121 

3 

58 

0820 

0849 

0879 

0908 

0938 

0968 

0998 

1029 

1060 

1090 

1122 

2 

59 

0820 

o85o 

0879 

0909 

0939 

0969 

0999 

1029 

ic6o 

1091 

1122 

I 

60 

0821 

o85o 

0880 

0909 

0939 

0969 

0999 

io3o 

ic6i 

1 09 1 

II23 

0 

II 

7°  29'  7°  28' 

7°  27' 

7°2G' 

7°  25' 

7°  24' 

7°  23' 

7°  22' 

7°21' 

7°  20' 

7°  19' 

The 

;  second  correction  is  to  be  taken  at  the  bottom  if  the 

ipparen 

t  distan 

ze  be  le 

'^s  than  90°. 

P'^esea]               TABLE  XLVII. 

The  first  correction  is  always  to  be  taken  at  the  top. 
The  second  correction  is  to  be  taken  at  the  top  if  the  apparent  distance  exceed  90°. 

1/ 

o 

I 

2 

3 
4 
5 
6 

7 
8 

9 

10 

1 1 

12 

i3 
i4 
i5 
i6 

17 
i8 

19 
20 
21 
22 

23 

24 

25 

26 
27 
28 
29 

3o 
3i 

32 

33 
34 

00  4j/ 

2°  42' 

2°43- 

2°  44' 

2°  45'  2°  46' 

2°  47' 

2°  48' 

2°  49' 

2°  50' 

2°  51' 

60  • 
59 
58  . 

57  . 
56 

55 
54 
53 

52 

5i 

5o 

49 
48 

47 
46 

45 
44 
43 

42 
4i 
4o 

39 
38 

37 
36 

35 
34 
33 

32 

3i 

1^ 
29 
20 

27 
26 

25 

24 

23 
22 
21 

20 
19 
18 

17 
16 

i5 
i4 
i3 
12 
11 
10 

7 
6 

5 
4 
3 
2 
I 
0 

0.1123 
1123 

1124 
1124 
ri25 

O.I154 
ii54 
ii55 
ii56 
ii56 

0.1186 
1 186 
11S7 
1187 
1188 

0.11S8 
1189 
1189 
1 190 
1 1 90 

0. 1217 
1218 
1218 
1219 
1219 

0.1249 

125o 
125o 
125l 
1252 

0.1282 
1282 
1283 
1283 
1284 

o.i3i4 
i3i5 
i3i5 
i3i6 
i3i6 

0.1347 

1 348 
i348 

1 349 
1 349 

0.  i38o 
i38i 
i38i 
i382 
i382 

o.i4i3 
i4i4 
i4i4 
i4i5 
i4i6 

0.1447 
1447 
i448 
1449 
1449 

0.  1125 

1 126 
1126 
1127 
1127 

0.1157 
ii57 
ii58 
ii58 
1159 

0. 1220 
1221 
1221 
1222 
1222 

0.1252 

1253 
1253 
1254 
1254 

0.1284 
1285 
1285 
1286 
1287 

o.i3i7 
i3i7 
i3i8 
i3i9 
i3i9 

o.i35o 
i35o 
i35i 
i35i 
i352 

o.i383 

1 383 
i384 

1 384 
i385 

o.i4i6 
1417 
1417 
i4i8 
i4i8 

o.i45o 
i45o 
i45i 
i45i 
1452 

0.1128 
II 28 
1129 
1129 
ii3o 

0.1 159 
1160 
1 1 60 
1161 
1161 

0.1191 
1191 
1190 
1192 
1193 

0. 1223 
1223 
1224 
1224 
1225 

0.1255 
1255 
1256 
1256 
1257 

0.1287 
1288 
1288 
1289 
1289 

0. l320 
l320 
l321 
l321 
l322 

o.i352 
i353 
1 354 
i354 
i355 

0.1386 
i386 
1387 
1387 
1 388 

0.1419 
1419 
1420 
1421 
1421 

0.1452 
1453 
1454 
1454 
1455 

o.ii3o 
ii3i 
ii3i 

Il32 

ir32 

0.1162 
1 162 
ii63 
ii63 
ii64 

0.1193 
1 194 
1195 
1195 
1196 

0.1225 
1226 
1226 
1227 
1227 

0.1257 
1258 
1259 
1259 
1260 

0.1290 
1290 
1291 
1291 
1292 

0.l322 

i323 
i323 
i324 
i325 

0.1355 
i356 
i356 
i357 
i357 

o.i388 
1389 
1389 
1390 
1391 

0.1422 
1422 
i423 
x423 
1424 

0.1455 
i456 
i456 
1457 
i458 

o.ii33 
1134 
ii34 
ii35 
ii35 

0.1164 
ii65 
ii65 
1166 

•  1 167 

0.1196 
1197 
1197 
1198 
1 198 

0.1228 
1229 
1229 
123o 
123o 

0.1260 
1261 
1261 
1262 
1262 

0.1292 
1293 
1294 
1294 
1295 

o.i325 
i326 
i326 
i327 
i327 

0.-I358 
1359 
1359 
i36o 
i36o 

0.1391 
1392 
1392 
1393 
1393 

0.1424 
i425 
1426 
1426 
1427 

0.1458 
1459 
1459 
1 460 
i46o 

o.ii36 
ii36 
ii37 
1137 
ii38 

0.1167 
1168 
1168 
1 169 
1169 

0.1199 
1199 
1200 
1200 
1201 

0.1 23 1 
I  23  I 
1232 
1232 

1233 

0.1263 
1263 
1264 
1264 
1265 

0.1295 
1296 
1296 
1297 
1297 

0.1328 
i328 
i329 
1329 

i33o 

o.i36i 
i36i 
i362 

1 362 

1 363 

0.1394 
1394 
1395 
1396 
1396 

0.1427 
1428 
1428 
1429 
1429 

0.1461 
i46i 
1462 
1 463 
1 463 

o.ii38 
1139 
1139 
I  i4o 
ii4o 

0.1170 
1170 
1171 
1171 
1172 

0. 1201 
1202 

1202 
I203 
I204 

0.1233 
1234 
1234 
1235 
1235 

0 . 1 266 
1266 
1267 
1267 
1268 

0.1298 
1298 
1299 
i3oo 
i3oo 

o.i33i 
i33i 
i332 
i332 
i333 

o.i363 

1 364 

1 365 
1 365 
i366 

0.1897 
1397 
1398 
1398 
1399 

o.i43o 
i43i 
i43i 
i432 
i432 

o.i464 

1 464 

1 465 

1 465 

1 466 

35 
36 

37 
38 
39 

4o 
4i 
42 
43 
44 
45 
46 
47 
48 

49 
5o 
5i 

52 

53 
54 
55 
56 

57 
58 
59 
60 

o.it4i 
ii4i 

Il42 

II42 
II43 

0.1172 
1173 
1173 
1174 
1 174 

0 . I io4 

I  205 
1205 

1206 
1206 

0. 1236 
1237 
1237 
1238 
1238 

0.1268 
1269 
1269 
1270 
1270 

o.i3oi 
i3oi 

l3o2 
l302 

i3o3 

O.I333 
1 334 
i334 
i335 
i335 

0.1 366 
i367 

1 367 

1 368 
1 368 

0.1399 
i4oo 
i4oi 
i4oi 

l402 

0.1433 
1433 
1434 
1435 
1435 

0.1467 
1467 
1 468 
1 468 
1469 

0.1143 
ii44 
1145 
ii45 
ii46 

0.1175 
1175 
1176 
1177 
1177 

0.1207 
1207 

1208 
I20S 

1209 

0.1239 
1239 
1240 
1240 
1241 

0.1 27 1 
1271 
1272 
1273 
1273 

o.r3o3 
i3o4 
i3o4 
i3o5 
i3o6 

o.i336 
i337 
i337 
1 338 
1 338 

0.1369 
1370 
1370 
1371 
1371 

0. l402 

i4o3 
i4o3 
i4o4 
i4o4 

0.1436 
i436 
1437 
1437 
i438 

0.1469 
1470 
1470 
1471 
1472 

0.1146 
ii47 
1147 
ii48 
ii48 

0.1178 
1178 
1179 

1179 
1 180 

0.1209 

I2IO 
I2IO 
1211 
121  I 

0.1241 
1242 
1242 
1243 
1243 

0.1274 
1274 
1275 
1275 
1276 

0 . 1 3o6 
i3o7 
i3o7 
i3o8 
i3o8 

0.1339 
1339 
1 340 
1 340 
i34i 

0.1372 
1372 
1373 
1373 
1374 

o.i4o5 
i4o6 
i4o6 
1407 
1407 

0.1438 
1439 
i44o 
i44o 
i44i 

0.1472 
1473 
1473 
1 474 
i474 

0.1149 
1149 
ii5o 
ii5o 
ii5i 

0.1 1  So 
1181 
1181 
II 82 
1182 

0.1212 
I2l3 
I2l3 
12l4 
12l4 

0 . 1 244 
1245 
1245 
1246 
1246 

0.1276 
1277 

1277 
1278 
1278 

0.1 309 
1 309 
i3io 
i3io 
i3ii 

0.1342 
i342 
1 343 

1 343 

1 344 

0.1374 
1375 
1376 
1376 

1 377 

o.i4o8 
i4o8 
1409 
1409 
i4io 

o.i44i 
1442 
1442 
1443 
1443 

0.1475 
1476 
1476 

i477 
i477 

0.1 i5i 

Il52 
Il52 

ii53 
ii53 

ii54 

o.ii83 
ii83 
1184 
ii84 
ii85 
1186 

0.12l5 
I2l5 

I2I6 
I2I6 

I2I7 
1217 

0.1247 
1247 
1248 
1248 
1249 
1249 

0.1279 
1280 
1280 
1281 
1281 
1282 

o.i3ii 

l3l2 

i3i3 
i3i3 
i3i4 
i3i4 

0.1344 
1 345 

1 345 

1 346 

1 346 

1 347 

0.1377 
1378 
1378 
i379 
1 379 
i38o 

o.i4ii 
i4ii 

l4l2 
l4l2 

i4i3 
i4i3 

0.1444 
1445 
i445 

1 446 
i446 

1 447 

0.1478 
1478 
1 479 
1479 
i48o 
i48i 

7=18' 

7°  17' 

7°1G' 

7°  IS'- 

7°  14' 1 7°  13' 

7°  12' 

7°  11' 

7°  10' 

7^  9' 

70  g/ 

// 

The  second  correction  is  to  be  taken  at  the  bottom  if  the  apparent  distance  be  less  than  90°. 

TABLE  XLVII.                    [Page  263 

The  first  correction  is  always  to  be  taken  at  the  top. 
The  second  correction  is  to  be  taken  at  the  to])  if  the  apparent  distance  e.Yceed  90°. 

// 

o 

I 

2 

3 

4 
5 
6 

7 
8 

lO 

II 

12 

i3 

i4 
i5 
i6 

17 
i8 

19 

20 
21 
22 

23 

24 

25 

26 
27 
28 
29 

3o 
3i 

32 

33 
34 
35 
36 

37 
38 
39 

40 
4i 
42 
43 
44 
45 
46 
47 
48 
49 
5o 
5i 

52 

53 
54 
55 

56 

57 
58 

?9 
60 

2=52' 

2=53' 

2°  54' 

2°  55' 

2°  56' 

2°  57' 

2°  58' 

2°  59' 

3°0' 

3°  1' 

3°  2' 

60 
59 
58 

57 
56 

55 
54 
53 

52 

5i 

5o 

49 
48 

47 
46 

45 
44 
43 
42 
4i 
4o 
39 
38 

37 
36 

35 
34 
33 

32 

3i 

29 

28 
27 
26 

25 

24 

23 
23 
21 

20 

19 
18 

17 
16 

75" 
14 
i3 
12 
II 
10 

7 
6 

5 
4 
3 
2 

1 
0 

o.i48i 
i48i 
1482 
1482 
1 483 

o.i5i5 
i5i5 
i5i6 
i5i6 

i5i7 

0 . 1 549 
i55o 
i55o 
i55i 
i55i 

o.i584 

1 584 

1 585 
i585 

1 586 

0. 1619 
1619 
1620 
1620 
1621 

0.1654 
i654 
i655 
i655 
i656 

0.1689 
1690 
1690 
1691 
1692 

0.1725 
1725 
1726 
1727 
1727 

0.1761 
1762 
1762 
1763 
1763 

0.1797 
1798 
1798 
1799 
1800 

0.1834 
i835 
i835 
1 836 
i836 

0.1837 
i838 
i838 
1839 
1839 

o.i483 
1 484 
i485 
i485 
i486 

o.i5i8 
i5i8 
i5i9 
i5i9 
I  Sao 

o.i552 
i552 

1 553 

1 554 
1 554 

0.15S7 
i587 
1 588 
1 588 
1589 

0.1621 
1622 
1623 
1623 
1624 

0 . 1 657 
i657 
1 658 
1 658 
1659 

0.1692 
1693 
1693 
1694 
1694 

0.1728 
1728 
1729 
1730 
1730 

0.1764 
1765 
1765 
1766 
1766 

0. 1800 
1801 
1802 
1802 
i8o3 

0.1486 
1487 
1487 
i488 
14S9 

0.l520 
l521 
I  522 
l522 

i523 

0.1555 
i555 
1 556 

1 556 

1 557 

0.1589 
1590 
1591 
1591 
1592 

0.1624 
1625 
1626 
1626 
1627 

0.1660 
1660 
1661 
1661 
1662 

0.1695 
1696 
1696 
1697 
1697 

0.1731 
1731 
1732 
1733 
1733 

0.1767 
1768 
1768 
1769 
1769 

o.i8o3 
1804 
i8o5 
i8o5 
1806 

0 . 1 840 
i84i 
]84i 
1842 
1843 

0.1489 
1490 
1490 
1491 
1491 

0.  i523 
1 524 
i524 
i525 
i526 

o.i558 
i558 
1559 
1559 
i56o 

0.1592 
1593 
1593 
1594 
1595 

0.1627 
1628 
1628 
1629 
i63o 

0.1663 
i663 
16&4 
1664 
1 665 

0.1698 
1699 
1699 
1700 
1700 

0.1734 
1734 
1735 
1736 
1736 

0.1770 
1771 
1771 
1772 
1772 

0.1806 
1807 
1808 
1808 
1809 

0.1843 
1844 
1 844 
1845 
1 846 

0.1492 
1493 
1493 
1494 
1494 

0.1526 
1527 
i527 
1528 
i528 

o.i56i 
i56i 
1 562 

1 562 

1 563 

0.1595 
1596 
1596 

1597 
1598 

0.1598 
1599 
1599 
1600 
1600 

0.1 63o 
i63i 
i63i 
i632 
i633 

0.1665 
1666 
1667 
1667 
1668 

0.1701 
1702 
1702 
1703 
1703 

0.1737 
1737 
1738 
1739 
1739 

0.1773 
1774 
1774 
1775 
1775 

0.1809 
1810 
1811 
1811 
1812 

0.1846 
1847 
1847 
1848 
1849 

0.1495 
1495 
1496 
1496 
1497 

0.1529 
i53o 
i53o 
i53i 
i53i 

O.I563 

1 564 
i565 

1 565 

1 566 

0.1633 
i634 
1 634 
i635 
i635 

0.1668 
1669 
1670 
1670 
1671 

0.1704 
1705 
1705 
1706 
1706 

0.1740 
1740 
1741 
1742 
1742 

0.1776 
1777 
1777 
1778 
1778 

0.1812 
i8i3 
i8i4 
i8i4 
i8i5 

0.1849 
i85o 
i85o 
165: 
i852 

0.1498 
1498 
1499 
1499 
i5oo 

o.i532 
i532 
i533 
1 534 
i534 

0 . 1 568 
1567 
1 587 

1 568 

1 569 

0.1601 
1602 
1602 
i6o3 
i6o3 

0.1636 
1637 
i637 
i638 
1 638 

0.1671 
1672 
1673 
1673 

1674 

0.1674 
1675 
1676 
1676 
1677 

0.1707 
1708 
1708 
1709 
1709 

0.1743 
1743 
1744 
1745 
1745 

0.1779 
1780 
1780 
1781 
1781 

0.1816 
1816 
1817 
1817 
1818 

o.iS52 
i853 
1 854 
i854 
i855 

o.iSoo 
i5oi 

l5o2 
l502 

i5o3 

O.I535 
i535 
1 536 
1 536 
i537 

0 . 1 569 
1570 
1570 
1571 
i57i 

0 . 1 6o4 
i6o5 
i6o5 
1606 
1606 

0.1639 
1640 
1640 
i64i 
i64i 

0.1710 
1711 
1711 
1712 
1712 

0.1746 
1746 
1747 
1748 
1748 

0.1782 
1783 
1783 
1784 
1785 

0.1819 
1819 
1820 
1820 
1821 

0.1855 
1 856 
1857 
1857 
i858 

0 . I 5o3 
i5o4 
i5o4 
i5o5 
i5o6 

0.1538 

1 538 

1 539 
1539 

1 540 

0.1572 
1573 
1573 

1 574 
1 574 

0.1607 
1607 
1608 
1609 
1609 

0.1642 
1643 
1643 
1 644 
1644 

0.1.677 
1678 
1678 
1679 
1680 

0.1713 
1714 
1714 
1715 
I7i5 

0.1749 
1749 
1750 
1751 
1751 

0.1785 
1786 
1786 
1787 
1788 

0.1822 
1822 
1823 
1823 
1824 

0.1859 
1859 
i860 
i860 
1861 

0 . I 5o6 
i5o7 
1 507 
i5o8 
i5o8 

0 . 1 540 
i54i 
1 542 

1 542 

1 543 

0.1575 
1576 
1576 
1 577 
1 577 

0.1610 
1610 
1611 
1612 
1612 

0.1645 
1645 
i646 
1647 
1647 

0.1680 
1681 
1681 
1682 
1 683 

0. 1716 
1717 
1717 
1718 
1718 

0.1752 
1752 
1753 
1754 
1754 

0.1788 
1789 
1789 
1790 
1791 

0.1825 
1825 
1826 
1827 
1827 

0. 1862 
1862 
1 863 
1 863 
1864 

0.1 509 
1 5 1 0 
i5io 
i5ii 
i5ii 

0.1543 
1 544 

1 544 

1 545 

1 546 

0.1578 
1578 
1 579 
i58o 
i58o 

o.i6i3 
i6i3 
i6i4 
i6i4 
i6i5 

0.1648 
1 648 
1649 
i65o 
i65o 

0.1683 
1684 

1 684 

1 685 
1686 

0.1719 
1719 
1720 
1721 

1721 

0.1755 
1755 
1756 

,757 

1757 

0.1791 
1792 

1792 
1793 
1794 

0.1828 
1828 
1829 
i83o 
i83o 

0.1 865 
1 865 
1866 
1867 
1867 

0.l5l2 

l5l2 

i5i3 
i5i4 
i5i4 
i5i5 

0.1 546 
1 547 

1 547 

1 548 

1 548 

1 549 

o.i58i 
i58i 
1 582 

1 582 

1 583 

1 584 

0.1616 
1616 
16.7 
1617 
1618 
1619 

o.i65i 
i65i 
i652 

3652 

1 653 
i654 

0.1686 
1687 
1687 
1688 
16S9 
1689 

0.1722 
1722 
1723 
1724 
1724 
1725 

0.1758 
1759 
1759 
1760 
1760 
1761 

0.1794 
1795 
1795 
1796 
1797 
1797 

o.i83i 
i83i 
i832 
i833 
1 833 
i834 

0.1868 
1868 
1S69 
1870 
1870 
1871 

7°  7' 

7°  & 

7°   5' 

70  4/ 

7°  3' 

7°  2' 

7°  1' 

7°  0' 

6^59' 

6°  58' 1 6°  57', 

The  second  correction  is  to  be  taken  at  the  bottom  if  the  apparent  distance  be  less  than  90^. 

Page  264] 

TABLE  XLVII. 

The  first  correction  is  always  to  be  taken  at  the  top. 

The  second  correction  is  to  be  Uken  at  the  top  if  the  apparent  distance  exceed  90°. 

o 

3°  3' 

30  4/ 

3°  5' 

3°  & 

3°  7' 

3°  8' 

3°  9' 

3°  10' 

3°  11' 

3°  12' 

3°  13' 

60 

0.1871 

0 . 1 9f  )8 

0 . 1 946 

0 . 1 984 

0.2022 

0.2061 

0 .  2099 

0.2139 

0.2178 

0.2218 

0.2259 

I 

1871 

1909 

1946 

i9«4 

2023 

2061 

2100 

2139 

2179 

2219 

2260 

59 

2 

1872 

1909 

1947 

1985 

2023 

2062 

2101 

2i4o 

2180 

2220 

2260 

58 

J 

1873 

1910 

1948 

1986 

2024 

2062 

2101 

2l4l 

2180 

2220 

2261 

57 

4 
5 

1873 

1911 

1948 

1986 

2025 

2o63 

2102 

2l4l 

2181 

2221 

2262 

56 

55 

0.1874 

0. 191 1 

0 . 1 949 

0.1987 

0.2025 

0.2064 

o.2io3 

0.2142 

0.2182 

0.2222 

0.2262 

b 

1875 

1912 

1950 

1987 

2026 

2064 

2io3 

2143 

2182 

2223 

2263 

54 

7 

1875 

1913 

1950 

1988 

2026 

2o65 

2104 

2143 

2i83 

2223 

2264 

53 

8 

1876 

1913 

1951 

1989 

2027 

2066 

2io5 

2144 

2184 

2224 

2264 

52 

9 

10 

1876 

i9l4 

1951 

1989 

2028 

2066 

2io5 

2145 

2184 

2225 

2265 

5i 
5o 

0.1877 

0.1914 

0.1952 

0.1990 

0.2028 

0.2067 

0 . 2  I  06 

0.2145 

0.2185 

0  2220 

0. 2266 

II 

1878 

1915 

1953 

1991 

2029 

2068 

2107 

2i46 

2186 

2226 

2266 

49 

12 

1878 

1916 

1953 

1991 

2o3o 

2068 

2107 

2147 

2186 

2227 

2267 

48 

iJ 

i«79 

1916 

1954 

1992 

2o3o 

2069 

2108 

2147 

2187 

2227 

2268 

47 

14 
i5 

1880 

1917 

1955 

1993 

2o3l 

2070 

2109 

2148 

2188 
0.2188 

2228 

2268 

46 

45 

0.1880 

0.1918 

0.1955 

0. 1993 

O.2o32 

0.2070 

0 . 2  I  09 

0.2149 

0.2229 

0.2269 

lb 

1881 

1918 

1956 

1994 

2032 

2071 

2110 

2149 

2189 

2229 

2270 

44 

17 

1881 

1919 

1956 

1994 

2033 

1072 

2III 

2l5o 

2190 

2  23o 

2270 

46 

i8 

1882 

1919 

1957 

1995 

2o33 

•072 

2II1 

2l5l 

2190 

223l 

2271 

42 

19 
20 

i883 

1920 

1958 

1996 

2o34 

'.073 

2112 

2l5l 

2191 

223l 

2272 

4i 
4o 

0.1883 

0.1921 

0.1958 

0 . 1 996 

o.2o35 

0.2073 

0.21l3 

0.21 52 

0.2192 

0  2232 

0.2272 

21 

1884 

1921 

1959 

1997 

2o35 

2074 

2Il3 

2i53 

2192 

2  233 

2273 

39 

22 

1 884 

1922 

i960 

1998 

2o36 

2075 

21l4 

2i53 

2193 

2233 

2274 

38 

2j 

i885 

1923 

1 960 

•1998 

2037 

2075 

21l5 

2i54 

2194 

2234 

2274 

37 

24 
2,5 

1886 

1923 

1961 

1999 

2o37 

2076 

2Il5 

2i55 

2194 

2235 

2275 

36 
35 

0.1886 

0. 1924 

0. 1962 

0.2000 

o.2o38 

0.2077 

0.21 16 

o.2i55 

0.2195 

0.2235 

0.2276 

2b 

1887 

1924 

1962 

2000 

2039 

2077 

2116 

2i56 

2196 

2236 

2277 

34 

27 

1S88 

1925 

1963 

2001 

2039 

2078 

2117 

•  2i57 

2196 

2237 

2277 

66 

28 

1 888 

1926 

1963 

2001 

204o 

2079 

2118 

2 1 57 

2197 

2237 

2278 

32 

29 

3o 

1889 

1926 

1964 

2002 

204l 

2079 

2118 

2i58 

2198 

2238 

2279 

3i 
3o 

0. 1889 

0.1927 

0.1965 

0 . 20o3 

0.204 1 

0.2080 

0.2119 

0.2159 

0.2198 

0.2239 

0.2279 

Ji 

1890 

1928 

1965 

20o3 

2042 

2081 

2120 

2159 

2199 

2239 

2280 

29 

Ja 

1891 

1928 

1966 

20o4 

2042 

2081 

2X20 

2160 

2200 

2240 

2281 

28 

J:i 

1891 

1929 

1967 

20o5 

2043 

2082 

2121 

2161 

2200 

224l 

2281 

27 

34 
35 

1892 

1929 

1967 

2oo5 

2044 

2o83 

2122 

2161 

2201 

2241 

2282 

26 

25 

0.1893 

0 . 1 930 

0 . 1 968 

0.2006 

0 . 2o44 

o.2o83 

0.2122 

0.2162 

0.2202 

0.2242 

0.2283 

Jb 

1893 

1931 

1968 

2007 

2045 

2084 

2123 

2i63 

2202 

2243 

2283 

24 

37 

1894 

1 93 1 

1969 

2007 

2046 

2o85 

2124 

2i63 

2203 

2243 

2284 

23 

38 

1894 

1932 

1970 

2008 

2o46 

2o85 

2124 

2164 

2  204 

2244 

2285 

22 

J9 

4o 

1895 

1933 

1970 

2009 

2047 

2086 

2125 

21 65 

2204 

2245 

2285 

21 
20 

0. 1896 

0.1933 

0.1971 

0.2009 

0.2048 

0 .  2086 

0.2126 

o.2i65 

0.2205 

0.2245 

0.2286 

41 

1896 

1934 

1972 

2010 

2048 

2087 

2126 

2166 

2206 

2246 

2287 

19 

42 

1897 

1934 

1972 

2010 

2049 

2088 

2127 

2167 

2206 

2247 

2287 

18 

43 

1898 

1935 

1973 

201  I 

2o5o 

2088 

2128 

2167 

2207 

2247 

2288 

17 

44 
"45 

1898 

1936 

1974 

2012 

2o5o 

2089 

2128 

2168 

220S 

2248 

2289 

16 

i5 

0.1899 

0.1936 

0.1974 

0.201 2 

0.2o5l 

0.2090 

0.2129 

0.2169 

0.2208 

0.2249 

0.2289 

4b 

1899 

1937 

1975 

20l3 

2o52 

2090 

2l3o 

2169 

2209 

2249 

2290 

i4 

47 

1900 

1938 

1975 

20 1 4 

2052 

2091 

2i3o 

2170 

2210 

225o 

2291 

i3 

48 

1 90 1 

1938 

1976 

2014 

2o53 

2092 

2l3l 

2170 

2210 

225l 

2291 

12 

49 
5o 

1901 

1939 

1977 

20 1 5 

2o53 

2092 

2l32 

2171 

22II 

225  I 

2292 

II 

10 

0. 1902 

0.1939 

0.1977 

0.2016 

o.2o54 

0.2093 

0.2l32 

0.2172 

0.22  12 

0.2252 

0.2293 

5i 

1903 

1940 

1978 

2016 

2o55 

2094 

2i33 

2172 

2212 

2253 

2294 

9 

52 

1903 

194 1 

1979 

2017 

2o55 

2094 

2i34 

2173 

22l3 

2253 

2294 

8 

53 

1904 

1941 

'979 

2017 

2o56 

2095 

2i34 

2174 

22l4 

2  2  54 

2295 

7 

54 
55 

1904 

1942 

1980 

2018 

2o57 

2096 

2i35 

2174 

22l4 

2255 

2296 

0 

~5' 

0.1905 

0.1943 

0 . 1 98 1 

0.2019 

0.2057 

0 .  2096 

0. 2 1 36  0.2 1 75 1 

0.22l5 

0.2256 

0.2296 

bb 

1906 

1943 

1981 

2019 

2o58 

2097 

2i36 

2176 

2216 

2256 

2297 

4 

f)7 

1906 

1944 

1982 

2020 

2059 

2098 

2 1 37 

2176 

2216 

2257 

2298 

3 

58 

1907 

1944 

1982 

2021 

2059 

2098 

2i37 

2177 

2217 

2258 

2298 

2 

59 

1908 

1945 

1983 

2021 

2060 

2099 

2i38 

2178 

2218 

2258 

2299 

I 

60 

1 
The 

1908 

1946 

1984 

2022 

2061 

2099 

2139 

2178 

2218 

2259 

23oo 

0 
If 

(P  5(;' 

6°  5-^' 

G°  54' 

6°  53' 

6°  52' 

G°  51' 

G°  50' 

6°  49' 

6°  48' 

6°  47' 

6°4G' 

second 

correcti 

on  is  to  be  taken  at  the  Iiottom  if  the  apparent  distance  be  less  than  90°. 

TABLE  XLVIL               [fas^  205 

Tlie  ^rsi  correction  is  always  to  be  taken  at  tlie  top. 

The  second  correction  is  to  be  taken  at  the  top  if  the  apparent  distance  exceed  90°. 

// 
o 

3°  14' 

3^5' 

3°  16' 

3°  17' 

3°  18' 

3°  19' 

3°  20' 

3°  21' 

3®  22' 

3°  23' 

3°  24' 

60 

0.23u() 

0.2341 

0.2382*0.2424 

0.2467 

0.25lO 

0.2553 

0.2596 

0.2640 

o.2()85 

0.2730 

I 

23oO 

2342 

2383 

2425 

2467 

25lO 

2553 

2597 

2641 

2686 

2781 

5q 

2 

23oi 

2342 

2384 

2426 

2468 

25 11 

2554 

2598 

2642 

2687 

2732 

58 

3 

200:^ 

2343 

2384 

2426 

2469 

2612 

2555 

2599 

2643 

2687 

2732 

57 

4 
5 

23o2 

2344 

2385 

2427 

2470 

25l2 

2556 

2599 

2643 

2688 

2733 

56 
55 

o.23o3 

0.2344 

0.2386 

0.2428 

0.2470 

o.25i3 

0.2556 

0 . 2600 

0 . 2644 

0.2689 

0.2734 

6 

23o4 

2345 

2387 

2429 

2471 

25i4 

2557 

2601 

2645 

2689 

2735 

54 

7 

23o4 

2346 

2387 

2429 

2472 

25i5 

2558 

2601 

2646 

2690 

2735 

58 

8 

23o5   2346 

2388 

243o 

2472 

25i5 

2559 

2602 

2646 

2691 

2736 

52 

9 

10 

23061  2347 

2389 

243 1 

2473 

25  16 

2559 

2603 

2647 

2692 

2737 

5i 
5o 

0.2307 

0.234s 

0.2389 

0.2431 

0.2474 

0.2517 

0.2560 

0.2604 

0.2648 

0 . 2692 

0.2788 

II 

2307 

2348 

2390 

2432 

2475 

25i7 

256i 

2604 

2649 

2693 

2788 

49 

12 

23oS 

2349 

2391 

2433 

2475 

25i8 

256i 

2  60  5 

2649 

2694 

2789 

48 

IJ 

2309 

23  5o 

2391 

2433 

2476 

2519 

2562 

2606 

265o 

2695 

274(J 

47 

i4 
i5 

2309 

235o 

2392 

2434 

2477 

2520 

2563 

2607 

265 1 

2695 

2741 

46 
45 

0.23lO 

o.235i 

0.2393 

0.2435 

0.2477 

0.2520 

0.2564 

0 . 2607 

0.2652 

0 . 269C 

0.2741 

i6 

23ll 

2352 

2394 

2436 

2478 

2521 

2564 

2608 

2652 

2697 

2742 

A^ 

17 

23ll 

2353 

2394 

2436 

2479 

2522 

2565 

2609 

2653 

2698 

2743 

43 

18 

23l2 

2353 

2395 

2437 

2480 

2522 

2566 

2610 

2654 

2698 

2744 

42 

19 
20 

23i3 

2354 

2396 

2438 

2480 

2523 

2566 

2610 

2655 

2699 

2744 

41 
40 

o.23i3 

0.2355 

0.2396 

0.2438 

0.2481 

0.2524 

0.2567 

0.2611 

0.2655 

0.2700 

0.2745 

21 

23 1 4 

2355 

2397 

2439 

2482 

2525 

2568 

2612 

2656 

2701 

2746 

3q 

22 

23i5 

2356 

239S 

2440 

2482 

2525 

2569 

2612 

2657 

2701 

2747 

38 

23 

23i5  2357 

2398 

2441 

2483 

2526 

2569 

2613 

2657 

2702 

2747 

37 

25 

23i6 

235- 

2399 

2441 

2484 

2527 

2570 

2614 

2658 

2703 

2748 

36 
35 

o.23i7 

0.2358 

0 . 2400 

0.2442 

0.2485 

0.2527 

0.2571 

0.2615 

0.2659 

0.2704 

0.2749 

26 

23i7 

2359 

2401 

2443 

2485 

2528 

2572 

.  261 5 

2660 

2704 

2750 

34 

27 

23[8 

2359 

2401 

2443 

2486 

2529 

2572 

2616 

2660 

2705 

2750 

Zi 

28 

23i9 

236o 

2402 

-^-iU 

2487 

253o 

2573 

2617 

2661 

2706 

2751 

32 

29 

3o 

2320 

236i 

24o3 

244;" 

2487 

253o 

2574 

2618 

2662 

2707 

2752 

3i 
■3^7 

0.2320 

0.2362 

o.24u3 

0.2445 

0.2488 

0.253 1 

0.2574 

0.2618 

0 . 2663 

0.2707 

0.2753 

3i 

2321 

2362 

24o4 

2446 

2489 

2532 

2575 

2619 

2663 

2708 

2753 

29 

3a 

2322 

2363 

24o5 

2447 

24S9 

2533 

2576 

2620 

2664 

2709 

2754 

28 

33 

2322 

23()4 

24o5 

2448 

2490 

2533 

2577 

2621 

2665 

2710 

2755 

27 

34 
35 

2323 

2364 

2406 

2448 

2491 

25?4 

2577 

2621 

2666 

2710 

2756 

26 

25 

0.2324 

0.2365 

0.2407 

0.2449 

0 .  2492 

0.2535 

0.2578 

0.2622 

0.2666 

0.271 1 

0.2756 

36 

2394 

2366 

2408 

245o 

2492 

2535 

2579 

2623 

2667 

2712 

2757 

24 

37 

2325 

2366 

2408 

245o 

2493 

2536 

258o 

2624 

2668 

2713 

2758 

23 

38 

2326 

2367 

2409 

245 1 

2494 

2537 

2  58o 

2624 

2669 

2713 

2750 

22 

39 
4o 

2326 

2368 

2410 

2452 

2494 

2538 

258i 

2625 

2669 

2714 

2760 
0.2760 

21 
20 

0.2327 

0.2368 

0.2410 

0.2453 

0.2495 

0.2538 

0.2582 

0.2626 

0.2670 

0.2715 

4i 

2328 

2369 

24 1 1 

2453 

2496 

2539 

2583 

2626 

2671 

2716 

2761 

19 

42 

232S 

2370 

2412 

2454 

2497 

2540 

2583 

2627 

2672 

2716 

2762 

18 

43 

2329 

2371 

2412 

2455 

2497 

2540 

2584 

2628 

2672 

2717 

2768 

17 

44 
45 

233o 

2371 

24i3 

2455 

2498 

254 1 

2585 

2629 

2673 

2718 

2768 

16 

i5 

o.233( 

0.2372 

0.2414 

0.2456 

0 .  2499 

0.2542 

0.2585 

0.2629 

0.2674 

0.2719 

0.2764 

4b 

233 1 

2373 

24i5 

2457 

2499 

2543 

2586 

263o 

2675 

2719 

2765 

i4 

47 

2332 

2373 

24i5 

2458 

2  5  00 

2543 

2587 

263 1 

2675 

2790 

2766 

i3 

48 

2333 

2374 

2416 

2458 

25oi 

2544 

2588 

2682 

2676 

2721 

2766 

12 

49 
5o 

9333 

2375 

2417 

2459 

25o2 

2545 

2588 

2632 

2677 

2722 

2767 

11 
10 

0.2334 

0.2375 

0.2417 

0 . 2460 

0.2502 

0.2545 

^0.2589 

0.2633 

0.2678 

0.2'722 

0.2768 

5i 

2335 

2376 

2418 

2460 

2  5o3 

2546 

2590 

2634 

2678 

2723 

2769 

9 

52 

2335 

2377 

2419 

2461 

25o4 

2547 

2591 

2635 

2679 

2724 

2769 

8 

53 

2336 

2378 

2419 

2462 

2  5o4 

2548 

2591 

2635 

2680 

2725 

2770 

7 

54 
55 

2337 

2378 

2420 

2462 

25o5 
o.25o6 

2548 

2592 

2636 

2681 

2725 

2771 

6 

5 

0.2337 

0.2379 

0.2421 

0.2463 

0.2549 

0.2593 

0.263-' 

0.2681 

0.2726 

0.2772 

5b 

2338 

238o 

2422 

2464 

25o7 

255o 

2593 

2638 

2682 

2727 

2772 

4 

^7 

2339 

238o 

2422 

2465 

2507 

255i 

2594 

2638 

2683 

2728 

2778 

3 

58 

233y 

238i 

2423 

2465 

25o8 

255i 

2595 

2639 

2684 

2729 

2774 

2 

59 

2340 

2382 

2424 

2466 

2509 

2552 

2596 

2640 

2684 

2729 

2775 

I 

bo 

2341 

2382 

2424 

2467 

25io 

2553 

2596 

2640 

2685 

273(< 

2775 

0 

6°  45' 

6°  44' 

6°  43' 

6°  42' 

6°  41' 

G°40' 

6°  39' 

6°  38' 

6°  37' 

6°3G'  6°  35' 

Th€ 

sccund  correct 

on  is  to  be  taken  at  the  bottom  if  the  apparent  distance  be  less  than  90°. 

34 


Page  266]                      TABLE  XLVII. 

The_^r5J  correction  is  always  to  be  taken  at  the  to-p. 

The  second  correction  is  to  be  taken  at  the  top  if  the  apparent  distance  exceed  90°. 

// 
o 

3°  25' 

3°  26' 

3°  27' 

3°  28' 

3°  29' 

3°  30' 

3°  31' 

3°  32' 

3°  33' 

3°  34' 

3°  35' 

60 

0.2775 

0.2821 

0.286S 

0.2915 

0.2962 

o.3oio 

o.3o59 

0.8108 

o.3i58 

0.3208 

0.8259 

I 

2776 

2822 

2S69 

2916 

2963 

3oii 

3o6o 

3109 

3x58 

8209 

8259 

59 

2 

2777 

2823 

2869 

2916 

2964 

3oi2 

3  060 

3iio 

8x59 

3209 

8260 

58 

3 

2778 

2824 

2870 

2917 

2965 

3oi3 

3o6i 

3iio 

3i6o 

3210 

326X 

57 

4 
5 

.  2779 

2825 

2871 

2918 

2965 

3oi4 

8062 

3iii 

3i6x 

8211 

8262 
0.3263 

56 
55 

0.2779 
2780 

0.2825 

0.2872 

0.2919 

0 . 2966 

o.3oi4 

o.3o63 

0.3ll2 

0.8162 

0.8212 

6 

2826 

2873 

2920 

2967 

3oi5 

3o64 

3ii3 

3i63 

32i3 

8264 

54 

7 

2781 

2827 

2873 

2920 

2968 

3oi6 

3o65 

3ii4 

3x63 

8214 

3265 

53 

8 

2782 

2828 

2874 

2921 

2969 

3oi7 

3o65 

3ii4 

3i64 

8214 

8265 

52 

9 

10 

2782 

2828 

2875 

2922 

296^ 

^  3oi8 

3o66 

8ii5 

3i65 

32x5 

3266 

5i 
5o 

0.2783 

0.2829 

0.2876 

0.2923 

0.2970 

o.3oi8 

0.3067 

o.3ii6 

o.3i66 

0.8216 

0.8267 

II 

2784 

283o 

2876 

2924 

2971 

3019 

3o68 

8117 

8167 

8217 

8268 

49 

12 

2785 

283 1 

2877 

2924 

2972 

3020 

8069 

3ii8 

3x68 

3218 

8269 

4^ 

13 

2785 

283 1 

2878 

2925 

2973 

302I 

3069 

3119 

3x68 

8219 

8270 

47 

i4 
1 5 

2786 

2832 

2879 

2926 

2973 

3022 

3070 

8119 

8x69 

8220 

8270 

46 

45 

0.2787 

0.2833 

0.2880 

0.2927 

0.2974 

0.3o2  2 

0.8071 

0.3l20 

0.8x70 

0.8220 

0.327X 

Tfi 

2788 

2834 

2880 

2927 

2975 

3o23 

3072 

3l2I 

8171 

8221 

8272 

4A 

17 

2788 

2835 

2881 

2928 

2976 

3o24 

3073 

3l22 

8x72 

8222 

8278 

43 

t8 

2789 

2835 

2882 

2929 

2977 

3o25 

8078 

3i23 

8x78 

8228 

3274 

42 

19 

20 

2790 

2836 

2883 

2930 

2977 

3026 

8074 

3i24 

8x78 

3224 

8275 

4i 
4o 

0.2791 

0.2837 

0.2883 

0.2931 

0.2978 

0.8026 

0.8075 

0.3 1 24 

0.8x74 

0.8225 

0.8276 

21 

2792 

2838 

2884 

2931 

2979 

8027 

3076 

3i25 

8x7b 

8225 

8276 

39 

22 

2792 
2793 

2838 

2885 

2932 

2980 

3028 

3077 

8126 

8x76 

8226 

8277 

88 

23 

2839 

2886 

2933 

2981 

8029 

3078 

3i27 

8177 

3227 

8278 

il 

24 

25 

2794 

2840 

2887 

2934 

2981 

3o3o 

3078 

8128 

8x78 

8228 

8279 

3b 
85 

0.2795 

0.2841 

0.2887 

0.2935 

0.2982 

0 . 3o3o 

o.3o79 

0.3129 

0.8x78 

0.3229 

0.3280 

26 

2795 

2842 

2888 

2935 

2983 

3o3i 

3o8o 

8129 

8179 

8280 

8281 

84 

27 

2796 

2842 

2889 

2936 

2984 

3o32 

3o8i 

3i3o 

3x8o 

823x 

3282 

66 

28 

2797 

2843 

2890 

2937 

2985 

3o33 

3082 

3i3i 

8x81 

8281 ,  82S2 

32 

29 

3o 

2798 

2844 

2891 
0.2891 

2938 

2985 

3o34 

3082 

8i32 

8x82 
0.3x83 

3232J  3283 

3i 
3o 

0 .  2798 

0.2845 

0.2939 

0.2986 

o.3o34 

o.3o83 

o.3i33 

0.8233 

0.8284 

3 1 

2799 

2845 

2892 

2939 

2987 

3o35 

3o84 

3i33 

3x83 

8284 

3285 

29 

32 

2S00 

2846 

2893 

2940 

2988 

3o36 

3o85 

3x34 

3x84 

3235 

3286 

28 

33 

2801 

2847 

2894 

2941 

2989 

3o37 

3o86 

3x35 

3x85 

3236 

8287 

27 

34 
3-^ 

2801 

2848 

2894 

2942 

2989 

3o38 

8087 

3x36 

3x86 
0.8x87 

8236 

3288 

2b 

25 

0.2802 

0.2848 

0.2895 

0.2942 

0.2990 

0.8089 

0.8087 

0.3 1 37 

0.8287 

0.8288 

3fi 

2803 

2849 

2896 

2943 

2991 

3089 

3o88 

3x38 

8x88 

8238 

8289 

24 

37 

2804 

285o 

2897 

2944 

2992 

3o4o 

8089 

3x38 

3x88 

8289 

3290 

28 

38 

2805 

285i 

2898 

2945 

2993 

3o4) 

8090 

3x39 

3x89 

8240 

8291 

22 

39 
40 

2805 

2852 

2898 

2946 

2993 

3o42 

8091 

3i4o 

8x90 

8241 

8292 

21 
20 

0.2806 

0.2852 

0.2899 

0.2946 

0.2994 

o.3o43 

0.8091 

o.3x4i 

0.8x91 

0.8242 

0.8298 

4t 

2807 

2853 

2900 

2947 

2995 

3o43 

8092 

3x42 

8x92 

3242 

3294 

19 

42 

2808 

2854 

2901 

2948 

2996 

3o44 

8098 

3x43 

8x93 

8243 

8294 

18 

43 

2808 

2855 

2901 

2949 

2997 

3o45 

3094 

8143 

8x93 

8244 

829b 

17 

2809 

2855 

2902 

2950 

2997 

3o46 

8095 
0.8096 

^iM 

8194 

8245 

3296 

16 

i5 

0.2810 

0.2856 

0.2908 

0.2950 

0.2998 

o.3o47 

c.3x45 

0.8x95 

0.8246 

0.8297 

40 

2811 

2857 

2904 

2951 

2999 

3o47 

8096 

3x46 

8x96 

3247 

8298 

14 

47 

2811 

2858 

2905 

2952 

3ooo 

3o48 

8097 

3x47 

8x97 

8247 

3299 

i3 

48 

2812 

2859 

2905 

2953 

3ooi 

3o49 

8098 

3x48 

8x98 

3248 

33oo 

12 

49 
5o 

2813 

2859 

2906 

2954 

3ooi 

3o5o 

8099 

3x48 

8x98 
0 . 3  X  99 

8249 
o.325o 

3  3  00 
o.33ox 

II 
10 

0.2814 

0.2860 

0.2907 

0.2954 

o.3oo2 

0  3o5i 

0.3  IOC 

0.3 1 49 

'ii 

28i5 

2861 

2908 

2955 

3oo3 

3o5? 

3ioi 

3x5o 

8200 

825i 

33o2 

9 

'io 

2815 

2862 

2909 

2956 

3oo4 

3o52 

3ioi 

8x5i 

3201 

3252 

33o3 

8 

^■^ 

2816 

2862 

2909 

2957 

3oo5 

3o53 

3l02 

3i52 

8202 

3253 

33o4 

7 

54 

2817 

2863 

2910 

2958 

3oo5 

3o54 

3io3 

3x53 
o.3i53 

32o3 

8253;  33o5 

6 

5 

G.2818I0.2864 

0.2911 

0.2958 

o.3oo6 

o.3o55 

0 . 3 1 04 

0.8204 

0.3254 

o.33o6 

'ifi 

2818 

2865 

2912 

2959 

3007 

3o56 

8io5 

3x54 

32o4 

3255 

33u6 

4 

^^7 

2819 

2866 

2912 

2960 

3oo8 

3o56 

3io5 

3x55 

32o5 

3256 

3307 

3 

58 

2820 

2866 

2913 

2961 

3009 

3o57 

3 1 06 

3x56 

8206 

8257 

33o8 

2 

5o 

2821 

2867 

2914 

2962 

3009 

3o58 

8107 

3x57 

8207 

3258 

3309 

I 

^ 

2821 

2868 

2915 

2962 

3oio 

3o59 

3io8 

3x58 

8208 

8259 

33x0 

0 

6°  34' 

G°33' 

6°  32' 

6°  31' 

6°  30' 

G°29' 

6°  28' 

C°27' 

(3°2G' 

6°  25' 

G°24' 

Th 

e  second  correction  is  to  be  taken  at  the  bottom  if  the  apparent  distance  be  less  than  90°. 

—■'— 

TABLE  XL VII.               t^^s^^oy 
The  first  correction  is  always  to  be  taken  at  the  top. 

The  second  correction  is  to  be  taken  at  the  top  if  the  apparent  distance  exceed  90°. 

o 

3=36' 

3°  37' 

3°  38' 

3°  39' 

3°  40' 

3°  41' 

3°  42' 

3=43' 

3°  44' 

0.3745 

3°  45'  3°  46' 

60 

o.33io 

0.3362 

o.34i5 

0.3468 

0.3522 

0.3576 

0.3632 

0.3688 

o.38o2 

0 . 386o 

I 

33ii 

3363 

34 1 5 

3469 

3523 

3577 

3633 

3689 

3746 

38o3 

386 1 

59 

2 

33i2 

3364 

3416 

3470 

3524 

3578 

3634 

3690 

3746 

38o4 

3862 

58 

3 

33i3 

3365 

3417 

3471 

3525 

357Q 

3635 

3691 

3747 

3So5 

3863 

i)7 

4 
5 

33i3 

3365 

3418 

3471 

3525 

358o 

3635 
0.3636 

3692 
0.3693 

3748 

38o6 

3864 

56 
"5"5" 

o.33i4 

0.3366 

0.3419 

0.3472 

0.3526 

0.3581 

0.8749 

0.8807 

0.3865 

6 

33i5 

3367 

3420 

3473 

3527 

3582 

3637 

3693 

3750 

38o8 

3866 

54 

7 

33 16 

3368 

3421 

3474 

3528 

3583 

3638 

3694 

3751 

3809 

0867 

53 

8 

33i7 

3369 

3422 

3475 

3529 

3584 

3639 

3695 

3752 

38 10 

3868 

52 

9 

10 

33i8 

3370 

3423 

3476 

353o 

3585 

364o 

3696 

3753 

38ii 

8869 

01 

"5^ 

0.3319 

0.3371 

0.3423 

0.3477 

o.353i 

0.3586 

0.364 1 

0.3697 

0.3754 

0.3S12 

0.3870 

II 

3319 

3372 

3424 

3478 

3532 

3587 

3642 

3698 

3755 

38i3 

3871 

49 

12 

3320 

3372 

3425 

3479 

3533 

3587 

3643 

3699 

3756 

38i4 

3872 

48 

i3 

3321 

3373 

3426 

3480 

3534 

3588 

3644 

3700 

3757 

38i5 

8878 

47 

i4 
i5 

3322 

3374 

3427 

3480 
0.3481 

3535 

3589 

3645 

3701 

3758 
0.8759 

38 16 

3874 

46 
45 

0.3323 

0.3375 

0.3428 

0.3535 

0.3590 

0.3646 

0.3702 

0.8817 

0.3875 

i6 

3324 

3376 

3429 

3482 

3536 

3591 

3647 

3703 

8760 

38i8 

8876 

44 

17 

3325 

3377 

343o 

3483 

3537 

3592 

3648 

3704 

3761 

3819 

3877 

4i 

i8 

3325 

3378 

343 1 

3484 

3538 

3593 

3649 

3705 

3762 

8820 

3878 

42 

19 

20 

3326 

3379 

343 1 

3485 

3539 

3594 

3649 

3706 

8768 
0.8764 

3820 

3879 
0 .  388o 

4i 
4o 

0.3327 

0.3379 

0.3432 

0.3486 

0.3540 

0.3595 

o.365o 

0.8707 

0.3821 

21 

3328 

338o 

3433 

3487 

3541 

3596 

365 1 

3708 

3765 

8822 

388 1 

39 

22 

3329 

338i 

3434 

3488 

3542 

3597 

3652 

3709 

3766 

8828 

3882 

38 

23 

333o 

3382 

3435 

3488 

3543 

3598 

3653 

3709 

3767 

3824 

3883 

87 

24 
25 

333i 

3383 

3436 

3489 

3544 

3598 

3654 

8710 

8768 

3825 

3834 

36 
"35" 

0.3332 

0.3384 

0.3437 

0 .  3490 

0.3545 

0.3599 

0.3655 

0.871 1 

0.8768 

0.8826 

0.38S5 

26 

3332 

3385 

3438 

3491 

3545 

36oo 

3656 

8712 

8769 

8827 

3886 

34 

27 

3333 

3386 

3438 

3492 

3546 

36oi 

3657 

3713 

8770 

8828 

38S7 

6i 

28 

3334 

3386 

3439 

3493 

3547 

36o2 

3658 

3714 

8771 

3829 

3888 

62 

29 

3o 

3335 

3387 

3440 

3494 

3548 

36o3 

3659 

3715 

8772 

383o 

3889 

3i 
3o 

0.3336 

0.3388 

0.3441 

0.3495 

0.3549 

0 . 36o4 

o.366o 

0.3716 

0.8778 

0.3831 

0.3890 

3i 

3337 

3389 

3442 

3496 

355o 

36o5 

366 1 

3717 

3774 

3832 

8891 

29 

32 

3338 

3390 

3443 

3497 

355i 

36o6 

3662 

3718 

3775 

3833 

3892 

28 

33 

3338 

3391 

3444 

3497 

3552 

3807 

3663 

3719 

3776 

3834 

3S93 

27 

34 
35 

3339 

3392 

3445 

3498 

3553 

36o8 

3663 

8720 

3777 

3835 

8894 

26 

25 

0.3340 

0.3393 

0.3446 

0.3499 

0.3554 

0.3609 

0 . 3664 

0.3721 

0.8778 

0.3S36 

0 . 3898 

36 

3341 

3393 

3446 

35oo 

3555 

36io 

3665 

8722 

3779 

3837 

3896 

24 

37 

3342 

3394 

3447 

35oi 

3555 

36io 

3666 

3723 

8780 

3838 

8897 

23 

38 

3343 

3395 

3448 

35o2 

3556 

36ii 

3667 

3724 

3781 

3839 

8898 

22 

39 
40 

3344 

3396 

3449 

35o3 

3557 

36i2 

3668 

3725 

8782 

3840 

8899 

21 
20 

0.3345 

o.33q7 

0.3450 

o.35o4 

0.3558 

o.36i3 

0 . 3669 

0.8726 

0.3788 

0.3841 

0.8900 

4i 

3345 

3398 

345i 

35o5 

3559 

36i4 

3670 

3727 

3784 

3842 

8901 

19 

42 

3346 

3399 

3452 

35o6 

356o 

36i5 

3671 

3727 

3785 

3843 

8902 

18 

Ai 

3347 

3400 

3453 

35o6 

356i 

36i6 

3672 

3728 

8786 

3844 

8903 

17 

44 
'45 

3348 

3400 

3454 

3507 

3562 
0.3563 

36 17 

3673 

3729 

3787 
0.3788 

3845 

8904 
0.3905 

16 
l5 

0.3349 

0.3401 

0.3454 

o.35o8 

o.36i8 

0.36740.3730 

0.3846 

46 

33  5o 

3402 

3455 

3509 

3564 

3619 

3675 

;?73i 

3789 

3847 

89(16 

l4 

47 

335i 

34o3 

3456 

35io 

3565 

3620 

36-6 

J732 

8790 

3848 

8907 

i3 

48 

335i 

3404 

3457 

35ii 

3565 

3621 

36-7 

3733 

8791 

3849 

8908 

12 

49 
5o 

3352 

34o5 

3458 

35i2 

3566 

3622 

3677 

3734 

8792 

385<. 

8909 

II 
10 

0.3353 

0.3406 

0.3459 

o.35i3 

0.3567 

0.3623 

0.3678,0.3735 

0.8792 

o.385i 

0.3910 

5i 

3354 

3407 

3460 

35i4 

3568 

3623 

3679!  3736 

3793 

3852 

391 1 

9 

52 

3355 

3408 

3461 

35i5 

3569 

3624 

368o 

3737 

3794 

3853 

8912 

8 

53 

3356 

3408 

3462 

35 1 5 

3570 

3625 

368 1 

8788 

3795 

3854 

8913 

7 

54 
55 

3357 

3409 

3463 

35i6 
0.3517 

3571 

3626 

3682 

3739 

3796 

3855 

3914 

6 

5 

0.3358 

0.3410 

0.3463 

0.3572 

0.3627 

0.3683  0.3740 

0.8797 

0.3856 

0.8915 

56 

3358 

3411 

3464 

35i8 

3573 

3628 

3684 

3741 

8798 

3856 

8916 

4 

57 

3359 

3412 

3465 

3519 

3574 

3629 

3685 

3742 

8799 

3857 

8917 

3 

58 

336o 

34i3 

3466 

3520 

3575 

363o 

3686 

37.43 

38oo 

3858 

3918 

2 

59 

336i 

34i4 

3467 

3521 

3576 

363 1 

3687 

3744 

38oi 

3859 

3919 

I 

60 

336-i 

341  5 

3468 

3522 

3576 

3632 

3688 

3745 

38o2 

386c 

3919 

0 

6°  23' 

G°22' 

6°  21' 

6=20' 

G°19' 

6°  18' 

6°  17' 

6°  16' 

6°  15' 

6°  14' 

6°i:y 

II 

Th 

e  second  correct 

ion  is  to  be  taken  at  the  lottom  if  the  apparent  distance  be  less  than  <!0°. 

Page  268]                      TABLE  XLVII. 

The  Jirst  correction  is  always  to  be  taken  at  the  top. 
The  second  correction  is  to  be  taken  at  the  top  if  the  apparent  distance  exceed  90°. 

o 

I 

2 

3 

4 
5 
6 

7 
8 

9 

10 

II 

12 

i3 
i4 
i5 
i6 

I? 
i8 

'9 

20 
21 
22 
23 
24 
25 
26 
27 
28 
29 

3o 
3i 

32 

33 
34 
35 
36 

37 
38 
39 

4o 
4i 
42 
43 
44 
45 
46 
47 
48 

49 
5o 
5i 

52 

53 
54 
55 
56 

57 
58 

3°  47' 

3°  48' 

3°  49' 

3°  50' 

3°  51' 

3°  52' 

3°  53' 

3°  54' 

3°  55' 

3°5G' 

3°  57' 

60 

59 

58 

57 
56 

"55" 
54 
53 

52 

5i 

49 

48 

47 
46 

45 
44 
43 
42 
41 
40 
39 
38 

37 
36 

35 
34 
33 

32 

3i 

3o 
29 
28 
27 
26 

25 

24 

23 
22 
21 

20 

19 

18 

17 
16 

l5 
i4 
i3 
12 
1 1 
10 

9 

8 

7 
6 

5 
4 
3 

■2 

I 
0 
II 

0.3919 

3920 
3921 
3922 
3923 

3.3979 
39S0 
3981 
3982 
3983 

o.4o4t' 
4o4i 
4042 
4043 
4o44 

0.4lO2 

4io3 
4io4 
4io5 
4io6 

0.4107 
4 1 08 
4109 
4i  10 
4i  1 1 

0.4164 
4i65 
4166 
4167 
4i68 

3.4228 
4229 
423o 
423i 

4232 

0.4292 
4293 
4294 
4295 
4296 

3.4357 
4358 
4359 
436 1 
4362 

0.4424 
4425 
4426 
4427 
4428 

3.4491 < 
4492 
4493 
4494 
4495 

3.4559 
456() 
4562 
4563 
4564 

0.3924 
3925 
3926 
3927 
3928 

0.3984 
39S5 
3986 
3987 
3988 

0.3989 
3990 
3991 

3992 
3993 

0.4045 
4o46 
4o47 
4o48 
4o49 

0.4169 
4171 
4172 
4173 
4174 

0.4233 
4234 
4235 
4236 
4237 

0.4297 
4298 
43oo 
43oi 
43o2 

0.4363 
4364 
4365 
4366 
4367 

0.4368 
4369 
4370 
4372 
4373 

0.4429 
4430 
443 1 
4433 
4434 

0.4435 
4436 
4437 
4438 
4439 

0.4497 
4498 
4499 
45oo 
45oi 

3.4565 
4566 
4567 
4569 
4570 

0.4571 
4572 
4573 
4574 
4575 

0.3929 
3930 
3931 
3932 
3933 

o.4o5o 
4o5i 
4o52 
4o53 
4o54 

0.4112 
4ii3 
4ii4 
4ii5 
4ii6 

0.4175 
4176 
4177 
4178 
4179 

0.4238 
4239 
4240 
4i4\ 
4243 

o.43o3 
43o4 
43o5 
43o6 
4307 

0.4502 
45o3 
45o5 
45o6 
4507 

0.3934 
3935 
3936 
3937 
3938 

0.3995 
3996 
3997 
3998 
3999 

o.4o55 
4o56 
4o58 
4o59 
4o6o 

0.4117 
4ii8 
4119 
4 1 20 

4l2I 

0.4180 
4i8i 
4182 
4i83 
4i84 

0.4244 
4245 
4246 

4247 
4248 

0.4308 
4309 
43 10 
43ii 
43i3 

0.4374 
4375 
4376 
4377 
4378 

0.4440 
4441 
4443 
4444 
4445 

0.4508 
4509 
45 10 
45ii 
45i2 

0.4577 
4578 

4579 
458.) 
458i 

0.3939 
3940 
3941 
3942 
3943 

0.4000 
4ooi 
4002 
4oo3 
4oo4 

0.4061 
4062 
4o63 
4064 
4o65 

0.4122 
4124 
4i25 
4126 
4127 

o.4i85 
4186 
4187 
4i88 
4189 

0.4249 
425o 
425i 
4252 
4253 

o.43i4 
43i5 
43i6 
43i7 
43i8 

0.4379 
438o 
438 1 
4383 
4384 

0.4446 
4447 
4448 
4449 
44  5o 

o.45i4 
45i5 
45i6 
4517 
45i8 

0.4582 
4584 
4585 
4586 
4587 

0.3944 
3945 
3946 
3947 
3948 

o.4oo5 
4oo6 
4007 
4008 
4009 

0 . 4o66 
4067 
4068 
4069 
4070 

0.4128 
4129 
4i3o 
4i3i 
4i32 

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4556 
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4618 
4619 
4621 
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4625 
4626 
4628 
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3975 
3976 
3977 
3978 
3979 

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4o38 
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4223 
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4227 
4228 

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4288 
4289 
4290 
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4292 

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4353 
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4356 
4357 

0,4418 

4419 
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4422 
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G°12' 

()°  11' 

G°10' 

6°  9' 

6°  8' 

G°  7' 

6°  G' 

6°  5' 

6°  4' 

6°  3' 

G°  2' 

The  second  correction  is  to  be  taken  at  the  bottom  if  the  apparent  distance  be  less  than 

t)0°. 

TABLE  XLVII.               [Page  209 

The  first  correction  is  always  to  be  taken  at  the  tov. 

Tlie  second  correction  is  to  be  taken  at  the  top  if  the  apparent  distunce  exceed  90°. 

II 
o 

3=^58' 

3°  59' 

40  0' 

40  y 

40  2/ 

40  3/ 

40  4A 

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0  J339 

40  7/ 
0.53 10 

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5218 

5399 

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8 

53 

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54 
55 

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0.4693 

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0.4765 

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b 
5 

0.4838 

0.4912 

0.498S 

o.5o64 

o.5i43 

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0.5386 

0.5470 

56 

4695 

4766 

4839 

4913 

4989 

5o66 

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5223 

53o5 

5387 

5471 

4 

57 

4696 

4768 

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49i5 

4990 

5067 

5i45 

5225 

53o6 

5389 

5473 

3 

58 

4697 

4769 

4842 

4916 

4991 

5o68 

5i46 

5226 

53o7 

5390 

5474 

2 

59 

4698 

4770 

4843 

4917 

4993 

5070 

5i48 

5227 

5309 

5391 

5476 

I 

60 

4699 

477  > 

4844 

4918 

4994 

5071 

5i49 

5329 

53io 

5393 

5477 

0 

6^  1' 

6°  0' 

5°  59' 

5=58' 

5°  57' 

5=  ^0 

5°  55'  j  5°  54' 

5°  53' 

5°  52' 

5°  51' 

II 

Th 

e  srcoml  correction  is  to  be  taken  at  the  hottmn  if  the  apparent  distance  be  less  than  90". 

Page  270]                 TABLE  XLVII. 

The  Jir St  correction  is  always  to  be  taken  at  the  top. 

The  second  correction  is  to  be  taken  at  the  top  if  the  apparent  distance  exceed  90°. 

n 

0 

40  Q, 

0.5477 

4°  10' 

0.5563 

4°  11' 

4°  12' 

4°  13' 

4°  14' 

4°  15' 

4°  16' 

4°  17' 

4°  18' 

4°  19' 

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57x3 

58o4 

5897 

5992 

6089 

6x88 

6289 

6893 

65oo 

x8 

43 

5538 

5626 

57x5 

58o6 

5898 

5993 

6090 

6190 

6291 

6895 

65oi 

17 

44 
45 

5540 

5627 

57x6 

5807 

5900 

5995 

6092 

6191 

6293 

6897 

65o3 

x6 
i5 

0.5541 

0.5629 

0.5718 

0 . 5809 

0.5902 

0.5997 

0 . 6094 

0.6193 

0.6294 

0.6898 

o.65o5 

46 

5543 

563o 

5719 

58x0 

5903 

5998 

6095 

6x95 

6296 

64oo 

65o7 

14 

47 

5544 

5632 

5721 

58x2 

5905 

6000 

6097 

6x96 

6298 

64o2 

65o9 

x3 

48 

5546 

5633 

5722 

58x3 

5906 

600  X 

600Q 

6198 

63oo 

64o4 

65io 

12 

49 
5o 

5547 

5635 

5724 

58x5 

5908 

6oo3 

6100 

6200 

63ox 
o.63o3 

64o6 

65x2 

XX 

10 

0.5549 

0 . 5636 

0.5725 

o.58i6 

0 . 5909 

o.6oo5 

c.6xo2 

0.620X 

0.6407 

o.65x4 

5 1 

555o 

5637 

5727 

58x8 

591 X 

6006 

6io3 

6208 

63o5 

6409 

65x6 

9 

52 

555i 

5639 

5728 

58x9 

59x3 

6008 

6io5 

6205 

63o6 

64ix 

65x8 

8 

53 

5553 

564o 

5730 

5821 

5914 

6009 

6x07 

6206 

63o8 

64x3 

65x9 

7 

54 
55 

5554 
0.5556 

5642 

573 1 

5823 

5916 

60XX 

6x08 

6208 

63x0 
0.63x2 

64x4 

6521 

b 
5 

0.5643 

0.5733 

0.5824 

0.59x7 

o.6oi3 

0.61x0 

0.6210 

0.6416 

0.6523 

56 

5557 

5645 

5734 

5826 

59x9 

6ox4 

6112 

62x1 

63i3 

64i8 

6525 

4 

57 

5559^  5646 

5736 

5327 

5920 

6016 

6ii3 

62x3 

63i5 

6420 

6527 

3 

58 

55601  5648 

5737 

5829 

5922 

6017 

61x5 

62x5 

63x7 

642  X 

6529 

2 

59 

5562  5649 

5739 

583o 

5924 

60x9 

6x17 

6216 

63 1 9 

6423 

653o 

I 

60 

5563   565 r 

574c 

5832 

5925 

602  X 

6118 

6218 

6320 

642  5 

6532 

0 
II 

5°  50' 1 5°  49' 

5°  48' 

5°  47' 

5=  46' 

5°  45' 

5°  44' 

5°  43' 

5°  42' 

5°  41' 

5^40' 

Th 

e  second 

correc 

ion  is  t 

0  be  tak 

sn  at  th 

e  bottom 

if  the 

apparen 

t  distan 

ce  be  Ic 

5S  than 

30°. 

TABLE  XLVII. 

[Pas.  ^71 

The  first 

correction  is  always  to  be  taken  at  the  top. 

The  second  correction  ig 

to  be  talien  at  the  top  if  the  apparent  dista 

nee  exceed  90°. 

o 

4°  20' 

4°  21' 

4°  22' 

4^23' 

4°  24' 

4°  25' 

4°  20' 

4°  27' 

4°  28' 

4°  2[y 

^^9 

60 

0.6532 

0.6642 

0.6755 

0.6871 

0.6990 

0.7112 

0.7238 

0.7368 

0.7501 

I 

6534 

6644 

6757 

6873 

6992 

7114 

7240 

7370 

75o3 

7641 

59 

2 

6536 

6646 

6759 

6875 

6994 

7116 

7242 

7372 

7506 

7G44 

58 

3 

6538 

6648 

6761 

6877 

6996 

7118 

7244 

7374 

7508 

7646 

57 

4 
5 

6539 

665o 

6763 

6879 

6998 

7120 

7246 

7376 

75x0 

7648 

56 
55 

0.6541 

0.665I 

0.6764 

0.6881 

0 . 7000 

0.7122 

0.7249 

0.7379 

o.75i3 

0.7651 

6 

6543 

6653 

6766 

688  2 

7002 

7124 

725x 

738  X 

75x5 

7653 

54 

7 

6545 

6655 

6768 

6884 

7004 

7127 

7253 

7383 

75x7 

7655 

53 

8 

6547 

6657 

6770 

6886 

7006 

7129 

7255 

7385 

7519 

7658 

52 

_9 

10 

6548 

6659 

6772 

6888 

7008 

7i3i 

7257 

7387 

7522 

7660 

5x 

5^ 

0.6550 

0.6661 

0.6774 

0.6890 

0.7010 

0.7x33 

0.7259 

0.7390 

0.7524 

0.7663 

II 

6552 

6663 

6776 

6892 

7012 

7x35 

726X 

7392 

7526 

7665 

49 

12 

6554 

6664 

677S 

6894 

7014 

7137 

7264 

7394 

7528 

7667 

48 

i3 

6556 

6666 

6780 

6896 

7016 

7139 

7266 

7396 

753i 

7670 

47 

i4 
i5 

6558 

6668 

6782 

6898 

7018 

7141 

7268 

7398 

7533 

7672 

46 
45 

0.6559 

0.6670 

0.6784 

0 . 6900 

0.7020 

0.7143 

0.7270 

0.740X 

0.7535 

0.7674 

i6 

656 1 

6672 

6785 

6902 

7022 

7145 

7272 

74o3 

7538 

7677 

AA 

17 

6563 

6674 

6787 

6904 

7024 

7147 

7274 

74o5 

7540 

7679 

43 

i8 

6565 

6676 

6789 

6906 

7026 

7149 

7276 

7407 

7542 

7681 

42 

!9 

20 

6567 

6677 

6791 

6908 

7028 

7x52 

7279 

7409 

7544 

7G84 

4i 
4o 

0.6568 

0.6679 

0.6793 

0.6910 

o.7o3o 

0 . 7 1 54 

0.7281 

0.7412 

0.7547 

0.7686 

21 

6570 

6681 

6795 

6912 

7032 

7x56 

7283 

74x4 

7549 

7688 

39 

22 

6572 

6683 

6797 

6914 

7034 

7x58 

7285 

7416 

755x 

7691 

38 

23 

6574 

6685 

6799 

6916 

7o36 

7x60 

7287 

74i8 

7554 

7693 

37 

24 
25 

6576 

6687 

6801 

6918 

7o38 

7162 

7289 

7421 

7556 

7696 

36 
35 

0.6578 

0.6689 

o.68o3 

0.6920 

0 . 7040 

0.7x64 

0.729X 

0.7423 

0.7558 

0 . 7698 

26 

6579 

6691 

68o5 

6922 

7042 

7x66 

7294 

7425 

7560 

7700 

'M 

27 

658 1 

6692 

6807 

6924 

7044 

7168 

7296 

7427 

7563 

7703 

'66 

2b 

6583 

6694 

6809 

6926 

7046 

7x70 

7298 

7429 

7565 

7705 

32 

29 

3o 

6585 

6696 
"0T6698" 

6810 

6928 

7048 

7172 

73oo 

7432 

7567 

7707 

3i 

3o 

0.6587 

0.6812 

0.6930 

0.7050 

0.7175 

0.7302 

0.7434 

0.7570 

0.7710 

3i 

6589 

6700 

68x4 

6932 

7o52 

7177 

73o4 

7436 

7572 

77x2 

29 

32 

6590 

6702 

68 1 6 

6934 

7o55 

7179 

7307 

7438 

7574 

77x4 

28 

33 

6592 

6704 

6818 

6936 

7057 

7x8x 

7309 

7441 

7577 

7717 

27 

35 

6594 

6706 

6820 

6938 

7059 

7x83 

73  XX 

7443 

7579 

7719 

26 

25 

0.6596 

0.6708 

0.6822 

0 . 6940 

0.7061 

0.7x85 

o.73i3 

0.7445 

0.7581 

0.7722 

36 

6598 

6709 

6824 

6942 

7063 

7x87 

73x5 

7447 

7583 

7724 

24 

3- 

6600 

6711 

6826 

6944 

7065 

7x89 

73x7 

745o 

7586 

7726 

23 

3S 

6601 

6713 

6828 

6946 

7067 

7191 

7320 

7452 

7588 

7729 

22 

09 
40 

66o3 

6715 

683o 

6948 

7069 

7x93 

7322 

7454 

7590 

773 1 

2X 
20 

o.66o5 

0.6717 

0.6832 

0.6950 

0.7071 

0.7x96 

0.7324 

0.7456 

0.7593 

0.7734 

4i 

6607 

6719 

6834 

6952 

7073 

7198 

7326 

7458 

7595 

7736 

19 

42 

6609 

6721 

6836 

6954 

7075 

7200 

7328 

7461 

7597 

7738 

x8 

43 

661  r 

6723 

6838 

6956 

7077 

7202 

7330 

7463 

7600 

774 1 

17 

45 

6612 

6725 

684o 

6953 

7079 

7204 

7333 

7465 

7602 

7743 

x6 
75 

0.6614 

0.6726 

0.6841 

0.6960 

0.7081 

0.7206 

0.7335 

0.7467 

0 . 7604 

0.7745 

46 

6616 

6728 

6843 

6962 

7083 

7208 

7337 

7470 

7607 

7748 

i4 

47 

6618 

6730 

6845 

6964 

7085 

7210 

7339 

7472 

7609 

775o 

i3 

48 

6620 

6732 

6847 

6966 

7087 

7212 

7341 

7474 

761 1 

7753 

X2 

49 

5o 

6622 

6734 

6849 

6968 

7089 

72x5 

7344 

7476 

76r3 

775:)  _ 

IX 
10 

0.6624 

0.6736 

0.685I 

0.6970 

0.7091 

0.72x7 

0.7346 

0.7479 

0 . 76 1 6 

0.7757 

5i 

6625 

6738 

6853 

6972 

7093 

7219 

7348 

748  X 

76x8 

7760 

9 

52 

6627 

6740 

6855 

6974 

7096 

722X 

7350 

7483 

7620 

7762 

8 

53 

6629 

6742 

6857 

6976 

7098 

7223 

7352 

7485 

7623 

7765 

7 

54 
55 

663 1 

6743 

6859 

6978 

7100 

7225 

7354 

7488 

7625 

7767 

6 
5 

0.6633 

0.6745 

0.6861 

0.6980 

0.7102 

0.7227 

0.7357 

0 . 7490 

0.7627 

0.7769 

56 

6635 

6747 

6863 

6982 

7104 

7229 

7359 

7492 

7630 

7772 

4 

i)7 

6637 

6749 

6865 

6984 

7106 

7232 

736i 

7494 

7632 

7774 

d 

58 

6638 

6751 

6867 

6986 

7108 

7234 

7363 

7497. 

7634 

7777 

2 

59 

6640 

6753 

6869 

6988 

7110 

7236 

7365 

7499 

7637 

7779 

I 

60 

6642 

6755 

6871 

6990 

7112 

7238 
5=34' 

7368 

7D0X 

7639 

7782 

0 

5°  39' 

5°  38' 

5°  37' 

S^'  36' 

5°  35' 

5°  33' 

5°  32' 

5=  31' 

5=  30' 

T 

he  seconi 

I  correct) 

on  is  to 

38  taken 

at  the  bottom  if  tl 

le  apparexit  distai 

ice  be  la 

s  than  90°. 

Page  a7-2] 

TABLE  XLVII. 

The  first  correction  is  always  to  be  taken  at  the  top. 

The  second  correction  is 

to  be  taken  at  tlie  top  if  the  apparent  distance  exceed  90°. 

// 
o 

4°  30' 

4°  31' 

4=  32' 

4°  33' 

4°  34' 

4°  35' 

4°  36' 
0.8751 

4°  37' 
0.8935 

4°  38' 

4°  39' 

60 

0.7782 

0.7929 

0.8081 

0.8239 

o.84o3 

0.8573 

0.9128 

0.9381 

I 

7784 

7931 

8084 

8242 

84o6 

8576 

8754 

8939 

9182 

9334 

5q 

2 

7786 

7934 

8086 

8244 

8409 

8579 

8757 

8942 

9135 

9887 

58 

3 

77S9 

7936 

8089 

8247 

84ii 

8582 

8760 

8945 

9i38 

9341 

57 

4 
5 

7791 

7939 
0.7941 

8091 
0 . 8094 

8250 

84i4 

8585 

8763 

8948 

9142 

9344 

56 
55 

0.7794 

0.8253 

0.8417 

0.8588 

0.8766 

0.8951 

0.9145 

0.9348 

b 

7796 

7944 

8097 

8255 

8420 

8591 

8769 

8954 

9148 

985 1 

54 

7 

-^798 

7946 

8099 

8258 

8423 

8594 

8772 

8958 

9152 

9355 

53 

« 

7S01 

7949 

8102 

8261 

8425 

8597 

8775 

8961 

9155 

9358 

52 

_9 

10 

7803 

795 1 

8io4 

8263 

8428 

8599 

8778 

8964 

9i58 

9862 

5i 
5o 

0 . 7806 

o.79?4 

0.8107 

0.8266 

o.843i 

0.8602 

0.8781 

0.8967 

0.9162 

0.9865 

1 1 

7808 

7956 

8110 

8269 

8434 

86o5 

8784 

8970 

9165 

9869 

4q 

12 

7811 

7959 

8112 

8271 

8437 

8608 

8787 

8973 

9168 

9872 

48 

]J 

7813 

7961 

8ii5 

8274 

8439 

861 1 

8790 

8977 

9171 

9376 

47 

i4 
i5 

7815 

7964 

8117 

8277 

8442 

86i4 

8793 

8980 

9175 

9379 

46 
45 

0.7818 

0 . 7966 

0.8120 

0.8279 

0.8445 

0.8617 

0.8796 

0.8983 

0.9178 

0.9088 

lO 

7820 

7969 

8123 

8282 

8448 

8620 

8799 

8986 

9181 

9386 

44 

17 

7823 

7971 

8125 

8285 

845 1 

8623 

8802 

8989 

9185 

9890 

4i 

i8 

7S25 

7974 

8128 

8288 

8453 

8626 

88o5 

.  8992 

9188 

9898 

42 

■9 

2U 

7S2S 

7976 

8i3i 

8290 

8456 

8629 

8808 

8996 

9191 
0.9195 

9897 

4i 

40 

0.7830 

0.7979 

o.8i33 

0.8293 

0.8459 

0.8632 

0.8811 

0 . 8999 

0 . 9400 

21 

7832 

7981 

8i36 

8296 

8462 

8635 

88i4 

9002 

9198 

9404 

39 

22 

7835 

7984 

8i38 

8298 

8465 

8637 

8817 

9005 

9201 

9407 

38 

23 

7837 

7987 

8i4i 

83oi 

8467 

864o 

8821 

9008 

9205 

941 1 

^7 

24 
25 

7840 
0.7842 

7989 

8144 

83o4 

8470 

8643 

8824 

9012 

9208 

94  i  4 

36 
35 

0.7992 

0.8146 

o.83o7 

0.8473 

0.8646 

0.8827 

0.9015 

0.9212 

0 .  94 1 8 

26 

7845 

7994 

8149 

83o9 

8476 

8649 

883o 

9018 

9275 

9421 

84 

27 

7847 

7997 

8i52 

83i2 

8479 

8652 

8833 

9021 

9218 

9425 

66 

.a8 

78  5o 

7999 

8i54 

83i5 

8482 

8655 

8836 

9024 

9222 

9428 

32 

29 

3o 

7S52 

8002 

8i57 

83i8 

8484 

8658 

8839 

9028 

9225 

9432 

81 
3o 

0.7855 

0.8004 

0.8159 

0.8320 

0.8487 

0.8661 

0.8842 

0.9031 

0.9228 

0.9435 

3i 

7857 

8007 

8162 

8323 

8490 

8664 

8845 

9034 

9282 

9439 

29 

J  2 

7859 

8009 

8i65 

8326 

8493 

8667 

8848 

9037 

9235 

9442 

28 

33 

7862 

8012 

8167 

8328 

8496 

8670 

885i 

9041 

9238 

9446 

27 

34 
35 

7864 

8014 

8170 
0.8173 

833i 

8499 

8673 

8854 

9044 

9242 
0.9245 

9449 
0.9453 

2b 

l5 

0.7867 

0.8017 

0.8334 

o.85o2 

0.8676 

0.8857 

0.9047 

36 

7869 

8020 

8175 

8337 

85o4 

8679 

8861 

90  5o 

9249 

9456 

24 

37 

7S72 

8022 

8178 

8339 

85o7 

8682 

8864 

9053 

9252 

9460 

23 

38 

7874 

8025 

8181 

8342 

85io 

8685 

8867 

9057 

9255 

9464 

22 

39 

4o 

7877 

8027 

8i83 

8345 
0.8348 

85i3 

8688 

8870 

9060 

9259 

9467 

21 
20 

0.7879 

o.8o3o 

0.8186 

o.85i6 

0.S691 

0.8873 

0 . 9063 

0.9262 

0.947' 

4i 

7882 

8o32 

8188 

835o 

85i9 

8694 

8876 

9066 

9266 

9474 

'9 

42 

7S84 

8o35 

8191 

8353 

8522 

8697 

8879 

9070 

9269 

9478 

lb 

43 

7887 

8087 

8194 

8356 

8524 

8700 

8882 

9073 

9272 

9481 

17 

44 
45 

7889 

8o4o 

8196 

8359 

8527 
0.8530 

8703 
0.8706 

8885 

9076 

9276 

9485 

lb 
i5 

0.7891 

0.8043 

0.8199 

0.836I 

0  8888 

0.9079 

0.9279 

0.948S 

46 

7894 

8045 

8202 

8364 

8533 

8709 

8892 

9083 

9283 

9492 

14 

47 

7896 

8o48 

S204 

8367 

8536 

8712 

8895 

9086 

9286 

9496 

tJ 

■18 

7899 

8o5o 

8207 

3370 

8539 

8715 

8898 

9089 

9289 

9499 

12 

49 
5o 

7901 
0.7904 

8o53 

8210 

8372 

8542 

8718 

8901 

9092 

9293 

95o3 

11 
10 

o.8o55 

0.8212 

0.8375 

0.8544 

0.8721 

0.8904 

0.9096 

0.9296 

0.9506 

Dl 

7906 

8o58 

8215 

8378 

8547 

8724 

8907 

9099 

9800 

9510 

9 

5  2 

7909 

8061 

8218 

838i 

855o 

8727 

8910 

9102 

93o3 

95i4 

8 

53 

791 1 

8o63 

8220 

8384 

8553 

8730 

8913 

9106 

9806 

9517 

/ 

54 
55 

7914 

8066 
0.8068 

8223 

8386 

8556 
0.8559 

8733 
0.8736 

8917 

9109 

9810 

9521 

b 

~5 

0.7916 

0.8226 

0.8389 

0.8920 

0.9112 

0 . 93 1 3 

0.9524 

b6 

7919 

8071 

8228 

8392 

8562 

8739 

8923 

9115 

9817 

9528 

A 

i)7 

7921 

8073 

823i 

8395 

8565 

8742 

8926 

9119 

9820 

9"^ 

3 

58 

7924 

8076 

8234 

8397 

8568 

8745 

8929 

9122 

9824 

9585 

2 

59 

7926 

8079 

•  8236 

8400 

8570 

8748 

8932 

9125 

9827 

9539 

I 

()o 

7929 

8()8i 

8239 

84o3 

8573 

8751 

8935 

9128 

9381 

9542 

0 

5=  2ct' 

.5°  28' 

5°  27' 

5°  26' 

5°  25' 

5°  24' 

5°  23' 

5=  22' 

5°  21' 

5°  20 

The  sccom 

I  correct! 

on  is  to  1 

)e  taken 

at  the  bottom   if  tl 

e  appare 

nt  distan 

ce  be  Ics 

s  than  90 

3_ 

TABLE 

XLVII. 

[I'wgti  -273 

The  first 

correction  is  always  to  be  taken  at  the  toj). 

Tha  St 

cond  correction  is 

to  be  taken  at  tlie  top  if  the  apparent  distance  exceed  90°. 

II 

0 

40  40' 

4°  41' 

40  42/ 

40  43/ 

40  44/ 

4°  45' 

4°  46' 

40  47, 

40  48' 

40  49/ 

0.9542 

0.9765 

I . 0000 

1.0248 

I .05l2 

1.0792 

I . 1091 

I  .i4i3 

I .1761 

I  .2139 

60 

I 

9546 

9769 

ooo4 

0252 

o5i6 

0797 

1097 

1419 

1767 

2145 

59 

2 

9550 

9773 

0008 

0257 

o52i 

c8oi 

1102 

1424 

1773 

2l52 

58 

3 

9553 

9777 

0012 

0261 

o525 

0806 

1107 

i43o 

1779 

2159 

57 

4 
5 

9557 

9780 

0016 

0265 

o53o 

0811 

1112 

i436 

1785 

2i65 

56 
55 

0.9561 

0.9784 

I .0020 

I .0270 

1.0534 

I. 0816 

1.1117 

i.i44i 

1.1791 

1 .2172 

6 

9^64 

9788 

0024 

0274 

0539 

0821 

II23 

1447 

1797 

2178 

54 

7 

9568 

9792 

0028 

0278 

0543 

0826 

1 1 28 

1452 

i8o3 

2i85 

53 

8 

9571 

9796 

oo32 

0282 

o548 

o83i 

ii33 

1458 

1809 

2192 

52 

lO 

9575 

9800 

oo36 

0287 

o552 

o835 

ii38 

1 464 

1816 

2198 

5i 
5o 

0.9579 

0 .  981)3 

1 . oo4o 

I .0291 

1  .o557 

I .0840 

1.1143 

1.1469 

1.1822 

1 .2  2o5 

u 

9582 

9S07 

0044 

O29D 

o562 

0845 

1149 

1475 

1828 

2212 

49 

12 

9586 

981 1 

0049 

o3oo 

o566 

o85o 

ii54 

i48i 

i834 

2218 

48 

i3 

9590 

9S15 

oo53 

o3o4 

0571 

0855 

1159 

i486 

i84o 

2225 

47 

i4 
i5 

95y3 

9819 

00  5  7 

o3o8 

0575 

0860 

ii64 

1492 

1 846 

2232 

46 
45 

0.9597 

0.9823 

I .0061 

i.o3i3 

I .o58o 

1.0865 

1 .1170 

1.1498 

1.1852 

I .2239 

i6 

9(5f)i 

9827 

oo65 

o3i7 

o585 

0870 

1175 

i5o3 

1859 

2245 

44 

17 

9()o4 

9830 

0069 

o32i 

0589 

0875 

1 180 

1 509 

1 865 

2252 

43 

lb 

9f)o8 

9S34 

0073 

0326 

0594 

0880 

1 186 

i5i5 

1871 

2259 

42 

11 

20 

9012 

9838 

0077 

o33o 

0598 

0884 

1191 

l520 

1877 

2266 
I .2272 

4i 
40 

0.9615 

0.9842 

I .0081 

1.0334 

I .o6o3 

1.0889 

1 . 1 1 96 

1.1526 

1.1883 

21 

9619 

9846 

oo85 

0339 

0608 

0894 

1201 

i532 

1889 

2279 

39 

22 

9623 

9850 

0089 

0343 

0612 

0899 

1207 

i538 

1896 

2286 

38 

23 

9626 

98  54 

0093 

o347 

0617 

0904 

1212 

i543 

1902 

2293 

^-i7 

24 
25 

963c. 

9S58 

0098 

o352 

0621 

0909 
I .0914 

1217 

1 549 

1908 

23oo 

I .2307 

36 
35 

0.9634 

0 . 986 1 

1 .0102 

I .o356 

I .0626 

I .1223 

1.1555 

1. 1914 

26 

9638 

9865 

0106 

o36o 

o63r 

0919 

1228 

i56i 

1921 

23 1 3 

M 

^7 

9641 

9869 

01 10 

o365 

o635 

0924 

1233 

i566 

'927 

2320 

6i 

28 

9(345 

98/3 

oii4 

0369 

0640 

0929 

1239 

1572 

1933 

2327 

32 

29 

3o 

9649 

9877 

0118 

o374 

0645 

0934 

1244 

1578 

1939 

2334 

3i 
3^ 

0.9652 

0.9881 

1 .0122 

1.0378 

1 .0649 

1 .0939 

I.124Q 

1 . 1 584 

1 . 1 946 

I .2341 

3i 

91)56 

9885 

0126 

o382 

o654 

0944 

1255 

1589 

1902 

2  348 

29 

32 

9660 

9889 

oi3i 

o387 

0659 

0949 

1260 

159! 

1958 

2355 

28 

33 

9664 

9S93 

oi35 

0391 

o663 

0954 

1266 

1601 

1965 

2362 

27 

34 
35 

9667 

9897 
0.9901 

0139 

0395 

0668 

0959 

1271 

1607 

1971 

2368 
1.2875 

2b 

0 . 967 1 

I. 0143 

I  .o4oo 

I .0673 

1 . 0964 

1 .1276 

i.i6i3 

1.1977 

36 

9675 

990D 

oi47 

o4o4 

0678 

0969 

1282 

1619 

19S4 

2382 

24 

37 

9678 

9908 

oi5i 

0409 

0682 

0974 

1287 

1624 

1990 

2,389 

23 

38 

9()82 

99' 2 

oi56 

o4i3 

0687 

0979 

1292 

i63o 

1996 

2396 

22 

39 
40 

9686 

99.6 

0160 

o4i8 

0692 

0984 

1298 

1 636 

3'X)3 

24o3 
1. 2410 

21 
20 

0 . 9(590 

0.9920 

I .0164 

I .0422 

1 . 0696 

I .0989 

I .i3o3 

i.:642 

I . 2009 

4i 

9693 

9924 

0168 

0426 

0701 

0994 

1 309 

1648 

2016 

2417 

19 

42 

9697 

9928 

0172 

043 1 

0706 

0999 

i3i4 

1 654 

2022 

2424 

18 

43 

9701 

9932 

0176 

0435 

071 1 

1004 

l320 

1660 

2028 

243 1 

17 

45 

9705 

9936 

oi8r 

o44o 

0715 

1009 

1 325 

i665 

2o35 

2438 

lb 
i5 

0 .  97C)8 

0.9940 

1.0185 

I .0444 

1 .0720 

1 . 1 0 1 5 

i.i33i 

I . 1671 

1 .  204 1 

I .2445 

40 

9712 

9944 

0189 

0449 

0725 

1020 

1 336 

1677 

2048 

2453 

14 

47 

9716 

9948 

0193 

0453 

0730 

1025 

i342 

i683 

2o54 

2460 

i3 

48 

9720 

9952 

0197 

0458 

0734 

io3o 

1347 

16S9 

2061 

2467 

12 

49 
5o 

9723 

9956 

0202 

0462 

0739 

io35 

i352 

1695 

2067 

2474 

II 
10 

0.9727 

0.9960 

I .0206 

I . 0467 

1.0744 

I .1040 

i.i358 

I .1701 

1 .2073 

I. 2481 

5 1 

9731 

9964 

0210 

0471 

0749 

1045 

1 363 

1707 

2080 

2488 

9 

52 

9735 

9968 

02l4 

0475 

0753 

io5o 

1369 

1713 

20S6 

2495 

8 

53 

9739 

9972 

0219 

0480 

0758 

io55 

1 374 

1719 

2093 

2502 

V 

54 
55 

9742 

9976 

0223 

o484 

0763 

1061 

i38o 

1725 

2099 

25lO 

,b 
5 

0.9-/46 

0 . 9980 

I .0227 

1 .0489 

1 .0768 

1 . 1 066 

i.i3S6 

1 .1731 

1 . 2 1 06 

1 .2517 

56 

9750 

9984 

023l 

0493 

0773 

1071 

1391 

1737 

2Il3 

2524 

4 

57 

9754 

9988 

0235 

0498 

0777 

1076 

1397 

1743 

21  19 

253i 

3 

58 

9758 

9992 

0240 

o5o2 

0782 

1081 

l402 

1749 

2126 

2538 

2 

59 

9761 

9996 

0244 

o5o7 

0787 

10S6 

1 408 

1755 

2l32 

2545 

I 

60 

9765 

1 .  0000 

0248 

o5i2 

0792 

1 09 1 

i4i3 

1761 

2139 

2553 

0 

5°  10' 

5°  18' 

5°  17' 

Tp  m 

5°  15' 

5=  14' 

5°  13' 

5°  12' 

5°  11' 

5°  10' 

T 

he  sicoiu 

/  correct 

on  is  to 

be  taken 

at  the  ho 

tlom  if  t^ 

le  appar« 

;nt  distar 

1C3  be  le. 

-5  llian  90 

o_ 

•x> 


p 

ige  274] 

TABLE 

XLVII. 

The  fust  correction  is  always  to  be  taken  at  the  top. 

The  second  correction  is  to  be  taken  at  tlie  top  if  the  apparent  distance  exceed  90''.    { 

o 

4°  50' 

4°  51' 

4°  52' 

4°  53' 

4°  54' 

4°  55' 

1.5563 

4°  5G' 

4°  57' 

4°  58' 

4°  59' 

60 

1.2553 

I .3oio 

1.3522 

I .4102 

1.4771 

1.6532 

1.7782 

I .9542 

2.2553 

I 

256o 

3o[8 

353i 

4lI2 

4783 

5578 

655o 

7806 

9570 

2626 

5q 

2 

2567 

3026 

3540 

4l22 

4795 

5592 

6568 

783o 

9615 

2700 

58 

3 

2574 

3o34 

3549 

4i33 

4808 

5607 

6587 

7S55 

9652 

2775 

57 

4 
5 

2582 

3o43 

3558 

4r43 

4820 

5621 
1.5636 

66o5 

7879 

9690 

2852 

56 
55 

1.2589 

1 .3o5i 

1.3567 

i.4i54 

1.4832 

I .6624 

1.7904 

1.9727 

2.2931 

6 

2596 

3o59 

3576 

4 1 64 

4844 

565i 

6642 

7929 

9765 

3oio 

54 

7 

2604 

3067 

3586 

4175 

4856 

5666 

6661 

7954 

9803 

3091 

53 

8 

261 1 

3075 

3595 

4i85 

4869 

568o 

6679 

7979 

9842 

3i74 

52 

_9 

10 

2618 

3oS3 

36o4 

4196 

4881 

5695 

6698 

8004 

9881 

3259 

5i 
5^ 

1.2626 

I .8091 

I .36i3 

1 .4206 

I .4894 

I .5710 

1-6717 

1 .8000 

I .9920 

2.3345 

II 

2633 

3 1 00 

3623 

4217 

4906 

5725 

6736 

8o55 

9960 

3432 

49 

12 

2640 

3io8 

3632 

4228 

4918 

5740 

6755 

8081 

2.0000 

3522 

48 

i3 

2648 

3ii6 

364 1 

4238 

4931 

5755 

6774 

8107 

oo4o 

36i3 

47 

i4 
i5 

2655 

3i24 

365o 

4249 

4943 

5771 

6793 

8i33 

0081 

3707 

46 
45 

I . 2663 

i.3i33 

i.366o 

I .4260 

I .4956 

1.5786 

I. 6812 

I. 8159 

2.0122 

2.3802 

i6 

2670 

3i4i 

3669 

4270 

4969 

58oi 

6832 

8186 

0164 

3900 

44 

17 

2678 

3i49 

3678 

4281 

4981 

58i6 

685 1 

8212 

0206 

4000 

43 

i8 

2685 

3i58 

3688 

4292 

4994 

5832 

6871 

8239 

0248 

4l02 

42 

12 

20 

2692 
I .2700 

3i66 
1.3174 

3697 

43o3 

5007 

5847 

6890 

8266 

0291 

4206 

4i 
4o 

1.3707 

i.43i4 

I .5019 

1.5863 

I .6910 

1 .8293 

2.0334 

2.43i4 

21 

2707 

3i83 

3716 

4325 

5o32 

5S78 

6930 

8320 

0378 

4424 

39 

22 

2715 

3191 

3726 

4335 

5o45 

5894 

6950 

8348 

0422 

4536 

38 

23 

2722 

3199 

3735 

4346 

5o58 

5909 

6970 

8375 

0467 

4652 

37 

24 
25 

2730 

3208 

3745 

4357 
I .4368 

5071 

5925 
1. 5941 

6990 

84o3 

o5i2 

4771 

ib 
35 

1.2738 

I .3216 

1.3754 

I . 5o84 

1 .7010 

I.843I 

2.0557 

2.4894 

26 

2745 

3225 

.  3764 

4379 

5097 

5957 

7o3o 

8459 

o6o3 

5019 

M 

27 

2753 

3233 

3773 

4390 

5iio 

5973 

70  5o 

8487 

0649 

5i49 

33 

28 

2760 

3242 

3783 

44oi 

5i23 

5989 

7071 

85i6 

0696 

5283 

32 

29 

3o 

2768 

325o 
I .3259 

3792 

4412 
1.4424 

5i36 

6oo5 

7091 

8544 

0744 

5421 

3i 
3o 

1.2775 

1.3802 

i.5i49 

1 .6021 

1 .7112 

1.8573 

2.0792 

2.5563 

3i 

2783 

3267 

38i2 

4435 

5162 

6087 

7i33 

8602 

0840 

5710 

29 

32 

2791 

3276 

3821 

4446 

5175 

6o53 

7i54 

8632 

0889 

5863 

28 

33 

279S 

3284 

383i 

4457 

5189 

6069 

7175 

8661 

0939 

6021 

27 

34 
35 

2806 

3293 
I .33oi 

384i 

4468 

5202 

6o85 

7196 

8691 

0989 

6i85 

2b 
25 

1. 2814 

1.3851 

1 .4480 

I.52I5 

I .6102 

J. 7217 

I .8721 

2. io4o 

2.6355 

36 

2821 

33io 

386o 

4491 

6229 

6118 

7238 

8751 

1091 

6532 

24 

37 

2829 

3319 

3870 

45o2 

5242 

6i35 

7259 

8781 

ii43 

6717 

23 

38 

2837 

3327 

388o 

45i4 

5256 

6i5i 

7281 

881 1 

1 1 96 

6910 

22 

39 
4o 

2845 

3336 

3890 

4525 
1.4536 

5269 
1.5283 

6168 
i.6i85 

73o2 

8842 

1249 

7112 

21 
20 

1.2852 

1.3345 

I .3900 

1.7324 

1.8873 

2 . I 3o3 

2.7324 

4i 

2860 

3353 

3910 

4548 

5296 

6201 

7346 

8904 

1 358 

7547 

19 

42 

2868 

3362 

3919 

4559 

53io 

6218 

7368 

8935 

i4i3 

7782 

18 

43 

2876 

3371 

3929 

4571 

5324 

6235 

7390 

8967 

1469 

8o3o 

17 

44 
45 

2883 
i.289r 

3379 
1.3388 

3939 

4582 
1.4594 

5337 

6252 
I .6269 

7412 

8999 

i526 

8293 

lb 
75 

1.3949 

i.535i 

1.7434 

I .9031 

2.1584 

2.8573 

46 

2899 

3397 

3959 

4606 

5365 

6286 

7456 

9063 

1642 

8873 

i4 

47 

2907 

3406 

3969 

4617 

5379 

63o3 

7479 

9096 

1701 

9195 

i3 

48 

2915 

34i5 

3979 

4629 

5393 

6320 

75oi 

9128 

1761 

9542 

12 

49 

5o 

2923 

3423 
1.3432 

3989 

464o 

5407 

6338 

7524 

9162 

1822 

9920 
3.0334 

II 
10 

I .2981 

I . 4ooo 

I .4652 

1. 5421 

1 .6355 

1.7547 

1. 9195 

2.1883 

5i 

2939 

3441 

4oio 

4664 

5435 

6372 

7570 

9228 

1946 

0792 

9 

52 

2946 

345o 

4020 

4676 

5449 

6390 

7593 

9262 

2009 

i3o3 

0 

53 

2954 

3459 

4o3o 

4688 

5463 

6407 

7616 

9296 

2073 

i883 

7 

54 
55 

2962 

3468 

4o4o 

4699 

5477 

6425 

7639 

9331 

2139 

2553 

b 
"5 

1.2970 

1.3477 

I .4o5o 

1. 471 1 

1. 5491 

I .6443 

1.7663 

1.9365 

2.2205 

3.3345 

56 

2978 

3486 

4o6i 

4723 

55o6 

646o 

7686 

9400 

2272 

43i4 

4 

57 

2986 

3495 

4071 

4735 

5520 

6478 

7710 

9435 

234i 

5563 

6 

58 

2994 

35o4 

4o8i 

4747 

5534 

6496 

7734 

9471 

2410 

7324 

2 

59 

3o02 

35i3 

4091 

4759 

5549 

65i4 

7757 

9506 

2481 

4.0334 

I 

60 

3oio 

3522 

4l02 

4771 

5563 

6532 

7782 

9542 

2553 

0 

5°  9' 

5°  8' 

5°  7' 

5°  6' 

5°  5' 

5°  4' 

5°  3' 

5°  2' 

5°  1' 

5°  0' 

T 

he  scconc 

I  correct 

on  is  to 

je  taken 

at  the  bo 

ttom  if  tl 

le  appare 

nt  distar 

ice  be  les 

s  than  90°. 

TABLE  XLVIIl 

. 

[Page  275 

Third  Correction.  Apparent  Distance  20°. 

A  pp. 

apparent  Altitude  of  the  Sun,  Star  or 

Planet. 

I>'s 
App. 

III. 

6° 

«" 

10" 

12" 

16° 

20° 

24° 

28° 

32° 

36° 

42° 

50° 

58" 

66° 

74" 

82" 

Alt. 

o 

/  II 

/  // 

/  ;; 

/  ^r 

/  n 

/  ;/ 

1  II 

1  II 

/  // 

/  . 

// 

1  II 

/  11 

II 

1   II 

0 

6 

I  38 

I  46 

,2  7 

2  34 

3  43 

4  5i 

5  59 

6 

7 

I  46 

r  4o 

I  53 

2  12 

3  I 

3  57 

4  5o 

7 

8 

I  55 

I  36 

I  44 

I  56 

2  35 

3  17 

4  0 

4  42 

« 

9 

2  8 

I  4o 

I  39 

I  45 

2  12 

2  47 

3  23 

3  58 

9 

lO 

2  23 

I  46 

I  36 

I  39 

I  56 

2  24 

2  53 

3  23 

10 

II 

2  38 

I  54 

I  38 

I  37 

I  46 

2  8 

2  32 

2  56 

3  16 

II 

12 

2  53 

2  3 

I  4i 

I  35 

I  4i 

I  56 

2  16 

2  35 

2  52 

12 

i3 

3  q 

2  r3 

I  46 

r  37 

I  37 

I  48 

2  4 

2  19 

2  32 

i3 

i4 

3  25 

2  23 

t  52 

I  3q 

I  34 

I  42 

I  54 

2  5 

2  16 

14 

lb 

3  4i 

2  34 

I  58 

I  42 

I  33 

I  38 

I  45 

I  54 

2  3 

2  i3 

i5 

1(3 

3  58 

2  45 

2  4 

I  46 

I  32 

I  34 

I  38 

I  46 

I  53 

I  59 

16 

I? 

4  i5 

2  56 

2  10 

I  5o 

I  33 

I  32 

I  34 

I  39 

I  44 

I  46 

17 

iS 

4  32 

3  7 

2  17 

I  54 

I  34 

I  3o 

I  3i 

I  34 

I  37 

I  40 

18 

19 

4  49 

3  18 

2  24 

I  58 

r  35 

I  20 
I  28 

I  29 

I  32 

I  33 

I  34 

19 

20 

5  5 

3  28 

2  3r 

2  2 

I  37 

t  28 

I  3o 

I  3o 

I   29 

I  26 

20 

21 

5  21 

3  39 

2  38 

2  6 

I  39 

I  29 

r  27 

I  27 

I  27 

I  25 

I  21 

21 

22 

5  36 

3  49 

2  46 

2  II 

I  4o 

I  29 

I  25 

I  25 

I  24 

I  22 

I  18 

22 

2J 

5  5i 

3  59 

2  53 

2  16 

I  42 

I  29 

I  25 

I  24 

I  22 

r  20 

I  i5 

23 

24 

6  5 

4  9 

3  0 

2  22 

I  43 

I  3o 

I  24 

I  23 

I  21 

I  18 

I  12 

24 

25 

6  19 

4  18 

3  7 

2  26 

I  45 

I  3o 

I  24 

I  21 

I  19 

I  16 

I  9 

25 

26 

6  32 

4  27 

3  i4 

2  3i 

I  47 

I  3i 

I  25 

I  21 

I  17 

r  i4 

I  7 

26 

27 

6  454  35 

3  20 

2  35 

I  49 

I  32 

I  25 

I  21 

I  17 

I  i3 

I  6 

27 

28 

4  42 

3  26 

2  38 

I  5o 

I  33 

I  25 

I  21 

I  17 

I  i3 

I  4 

5o 

28 

29 

4  49 

3  32 

2  4i 

I  52 

I  33 

I  25 

I  21 

I  17 

I  i4 

I     5 

5o 

29 

3o 
3i 

3  37 

2  45 

I  54 
I  56 

I  34 
I  34 

r  25 

I  25 

r  21 

I  20 

I  18 

I  17 

I  i5 
I  i5 

I  7 
I  7 

5o 
5r 

3o 
3i 

3  42 

2  49 

32 

2  52 

I  58 

I  34 

I  24 

I  19 

I  17 

I  i4 

I  7 

5i 

32 

33 

2  55 

X  59 

I  a3 

I  24 

I  19 

I  16 

I  i3 

I  8 

52 

33 

34 

I  5q 

I  33 

I  23 

I  18 

I  i5 

I  i3 

I  8 

53 

34 

35 

I  59 

I  32 

I  22 

I  17 

I  i4 

I  12 

I  8 

53 

35 

36 

I  59 

I  3i 

I  20 

I  i5 

I  i3 

I  II 

I  7 

54 

36 

■66 

J7 

I  59 

I  So 

I  19 

I  i4 

I  12 

I  10 

I  6 

54 

37 

37 

38 

I  29 

I  18 

I  i3 

I  II 

I  9 

I  6 

55 

38 

38 

39 

I  28 

I  17 

I  II 

I  10 

I  5 

55 

39 

39 

4o 

I  27 

I  i5 

I  10 

I  9 

I  8 

I  4 

55 

39 

40 

4i 

1  26 

I  i3 

I  9 

I  8 

I   7 

I  3 

55 

39 

4i 

42 

I  II 

I  7 

I  7 

I  6 

I  2 

55 

4o 

42 

43 

I  10 

I  5 

I  5 

I  5 

I  2 

55 

40 

4i 

44 

I  9 

I  3 

I  4 

I  4 

I  I 

55 

40 

29 

44 

46 

I  7 

I  0 

I  I 

I  2 

I  0 

54 

4i 

3o 

46 

48 

56 

56 

59 

58 

53 

43 

3i 

48 

5o 

52 

52 

55 

55 

5i 

43 

33 

5o 

52 

48 

5o 

5i 

49 

43 

35 

24 

52 

54 

44 

45 

47 

47 

43 

36 

25 

54 

56 

4o 

44 

45 

42 

35 

27 

56 

58 

35 

4o 

43 

40 

M 

27 

58 

60 

36 

4i 

39 

33 

26 

21 

60 

62 

33 

38 

38 

32 

26 

22 

62 

64 

3o 

35 

37 

32 

27 

22 

64 

66 

32 

36 

3i 

27 

23 

66 

68 

29 

M 

3o 

26 

23 

68 

70 

27 

32 

29 

26 

22 

70 

72 

25 

29 

28 

25 

21 

72 

74 

27 

27 

24 

21 

7-i 

76 

25 

26 

24 

20 

76 

78 

23 

25 

23 

20 

78 

8n 

21 

24 

22 

20 

80 

8'; 

23 

21 

82 

84 

22 

21 

84 

86 

21 

20 

tb 

G° 

8°  10° 1 

12° 

1G° 

20° 

24° 

28° 

32° 

36° 

42° 

50° 

58° 

66° 

74° 

82° 

Pase  276]               TABLE  XLVJ  XL  . 

Third  Correction.  ApparenL  Distance  24°. 

5's 
App. 

Jlpparent  Altitude  of  the  Sun,  Star  or  Planet. 

])'s 
App. 

Alt. 

G" 

7°  ,   ' 

y" 

y^ 

lU^ 

IP 

1^2^ 

140 

IG^ 

18" 

yu" 

22° 

24° 

26° 

28° 

30° 

Ait. 

0 

(  / 

/  /^ 

1  II 

/  ?/ 

/  // 

1   II 

1   II 

/  II 

/  // 

/  // 

/  // 

/  // 

1  II 

/  II 

/  // 

/  II 

0 

6 

I  28 

I  3i 

I  35 

I  42 

I  52 

2    3 

2  16 

2  46 

3  16 

}  47  4  19I 

i   5o 

5  20 

5  5o 

6  206  5o| 

6 

-7 

I  35 

I  27 

I  3o 

I  34 

I  39 

I  46 

I  54 

2  i5 

2  38 

3  3 

3  29 

i   55 

4  20 

4  46 

5  10 

5  34 

7 

8 

I  45 

I  32 

I  26 

I  28 

I  00 

I  3i) 

I  4i 

I  58 

2  17 

2  37 

2  58 

i   18 

3  39 

4  I 

4  204  3q| 

8 

9 

I  56 

I  39 

I  3o 

I  25 

I  26 

I  29 

I  34 

I  44 

I  59 

2  i5 

2  3i 

2  48 

3  6 

3  24 

3  40 

3  56 

9 

10 

2  8 

I  48 

I  36 

I  29 

I  25 

I  26 

I  28 

1-35 

I  45 

I  57 

2   i3 

2  27 

2  43 

2  58 

3  12 

3  26 

10 

1 1 

2  21 

I  58 

I   43 

I  34 

I  28 

I  24 

I  26 

I  3o 

I  36 

I  46 

I  58 

2  1 1 

2  24 

2  37 

2  4q 

3  0 

II 

12 

2  36 

2  9 

I  52 

I  4i 

I  33 

I  27 

I  24 

I  26 

I  3o 

1  07 

I  47 

I  58 

2  9 

2  20 

2  29 

2  38 

12 

r3 

2  5i 

2  20 

2   I 

I  48 

I  38 

I  3i 

I  27 

I  24 

I  27 

I  32 

I  4o 

I  48 

I  57 

2  6 

2  i4 

2  22 

i3 

i4 

3  6 

2  3i 

2'  10 

I  55 

I  43 

I  35 

I  3o 

I  23 

I  25 

I  28 

I  33 

I  4o 

I  48 

I  55 

2  2 

2  10 

i4 

i5 

3  21 

2  42 

2  20 

2  2 

I  5o 

I  3q 

I  33 

I  24 

I  23 

I  25 

I  24 

I  34 

I  40 

I  46 

I  52 

I  59 

i5 

i6 

3  36 

2  54 

2  3o 

2  9 

I  56 

I  44 

I  36 

I  26 

I  22 

I  23 

I  25 

I  29 

I  33 

I  38 

I  44 

I  5o 

16 

17 

3  5i 

J  6 

2  40 

2  17 

2  2 

I  49 

I  39 

I  28 

I  23 

I  21 

I  23 

I  26 

I  29 

I  34 

I  39 

I  43 

17 

j8 

4  6 

3  18 

2  49 

2  25 

2  8 

I  54 

I  43 

I  3i 

I  24 

I  20 

I  21 

I  23 

I  26 

I  3o 

I  3A 

I  37 

18 

19 

4  21 

3  3o 

2  59 

2  33 

2  i4 

I  59 

I  47 

I  33 

I  25 

I  21 

I  20 

I  22 

I  24 

I  27 

I  3o 

I  32 

19 

20 

4  3b 

J  42 

^  9 

2  4i 

2  21 

2  5 

I  52 

I  36 

I  27 

;  a 

I  19 

I  20 

I  22 

I  24 

I  26 

I  28 

20 

21 

4  5u 

3  54 

3  19 

2  5o 

2  28 

2  11 

I  56 

I  39 

I  29 

I  23 

I  20 

I  19 

I  20 

I  21 

I  23 

I  25 

21 

22 

5  4 

4  6 

3  28 

2  58 

2  3b 

2  17 

2  I 

I  42 

I  3i 

I  24 

I  20 

I  18 

I  19 

I  IQ 

I  20 

I  22 

22 

23 

5  19 

4  18 

3  38 

3  6 

2  43 

2  23 

2  6 

I  46 

I  33 

I  25 

I  21 

I  18 

I  18 

I  18 

I  18 

I  19 

23 

24 

5  33 

4  29 

3  48 

3  i4 

2  5i 

2  29 

2  12 

I  5o 

I  36 

I  27 

I  22 

I  19 

I  17 

I  17 

I  17 

I  17 

24 

25 

5  47 

4  4i 

3  57 

3  22 

2  58 

2  35 

2  17 

I  53 

I  38 

I  28 

I  23 

I  20 

I  18 

I  16 

I  16 

I  16 

25 

26 

6  I 

4  52 

4  6 

3  3o 

3  4 

2  4i 

2  22 

I  57 

I  4i 

I  3o 

I  24 

I  20 

1  18 

I  16 

I  i5 

I  i5 

26 

27 

6  i4 

5  4 

4  lb 

3  38 

3  10 

2  47 

2  27 

2  0 

I  43 

I  32 

I  25 

I  21 

I  18 

I  i5 

I  i4 

I  i3 

27 

28 

6  27 

5  i5 

4  23 

3  45 

3  16 

2  53 

2  32 

2  4 

I  46 

I  34 

I  27 

I  21 

I  18 

I  i5 

I  i3 

I  12 

28 

29 

6  38 

5  26 

4  32 

3  53 

3  22 

2  58 

2  38 

2  8 

I  49 

I  36 

I  28 

I  22 

I  18 

I  i5 

1  i3 

I  1 1 

29 

So 

6  5o 

5  36 

4  41 

4  0 

3  28 

3  3 

2  44 

2  12 

I  52 

I  38 

I  29 

I  23 

I  19 

r  i5 

I  i3 

I  1 1 

3o 

3i 

7  0 

5  45 

4  5o 

4  7 

3  34 

3  8 

2  4q 

2  16 

I  55 

I  40 

1  3o 

I  24 

I  IQ 

I  i5 

I  i3 

I  1 1 

3i 

32 

5  53 

4  58 

4  i4 

3  40 

3  i3 

2  54 

2  19 

I  57 

I  41 

I  3i 

I  24* 

I  19 

I  i5 

I  i3 

I  II 

32 

33 

5  5 

4  20 

3  46 

3  18 

2  58 

2  22 

I  59 

I  42 

I  3i 

I  24 

I  19 

I  i5 

I  i3 

I  II 

33 

34 

4  35 

3  5i 

3  22 

3  I 

2  24 

2  1 

I  43 

I  32 

I  25 

I  20 

I  i5 

I  i3 

I  II 

.^4 

35 

3  56 

3  26 

3  32  26 

2  2 

I  45 

I  33 

I  25 

I  20 

I  i5 

t  i3 

I  II 

35 

36 

3  3o 

3  52  28 

2  4 

I  46 

I  34 

I  25 

I  20 

I  i5 

I  12 

I  10 

36 

37 

3  7  2  3o 

2  6 

I  47 

I  35 

I  25 

I  20 

I  i5 

I  12 

I  10 

37 

38 

2  32 

2  7 

1  48 

I  35 

I  25 

I  20 

I  i5 

I  12 

I  10 

38 

39 

2  34 

2  8 

I  49 

I  35 

I  25 

I  19 

I  i5 

I  12 

I  10 

39 

4o 

2  9 

I  5o 

I  35 

I  2b 

I  19 

I  i5 

I  II 

I  9 

4o 

4i 

2  ID 

I  5o 

I  35 

I  25 

I  19 

I  i5 

I  II 

I  8 

4i 

42 

I  5i 

I  36 

I  2b 

I  19 

I  i4 

I  10 

I  7 

42 

43 

I  52 

I  36 

I  2b 

I  18 

I  i3 

I  9 

I  6 

43 

44 

I  36 

I  2b 

I  18 

I  i3 

I  8 

I  5 

44 

46 

I  36 

I  2b 

I  17 

I  12 

I  7 

I  3 

46 

48 

I  25 

I  17 

I  10 

I  5 

I  I 

48 

5o 

I  17 

I  8 

I  4 

59 

5o 

52 

I  7 

I  3 

58 

52 

54 

I  2 

57 

54 

56 

56 

56 

58 

58 

60 

60 

62 

62 

64 

64 

66 

66 

68 

68 

70 

70 

72 

72 

74 

74 

76 

78 









76 

78 

80 

80 

82 

82 

84 

84 

86 

G° 

70 

S° 

9= 

10° 

11° 

12° 

14° 

16° 

18° 

20° 

22° 

86 

24° 

26° 

28° 

30° 

TABLE 

XLVIII 

[Page  277 

Third  Correction 

Apparent  Distance  24°. 

D's 
A,.p. 

All. 

Apparent  Altitude  of  the  Sun,  Sta 

r  or  Planet. 

D's 
App. 
Alt. 

32° 

34° 

36° 

38° 

42° 

46° 

50° 

54° 

58° 

62° 

66° 

70° 

74° 

78° 

82° 

86° 

o 

/ 

.  ., 

/  II 

/  // 

/  II 

/ 

/  // 

1  II 

1 

/  II 

1   II 

/  II 

/  n 

/  // 

/  / 

0 

6 

6 

7 

7 

8 

4  58 

8 

9 

4  12 

9 

10 

3  39 

3  5i 

10 

II 

3  II 

3  21 

3  3o 

II 

12 

2  48 

2  56 

3  5 

3  12 

12 

i3 

2  3o 

2  37 

2  44 

2  49 

i3 

i4 

■2   16 

2  22 

2   27 

2  32 

i4 

i5 

2  4 

2  9 

2  i4 

2  18 

i5 

itj 

I  54 

1  59 

2  3 

2  6 

2  II 

16 

17 

I  46 

r  5o 

I  53 

I  56 

2  0 

17 

18 

I  40 

I  43 

I  45 

I  47 

I  5r 

18 

19 

I  35 

I  37 

I  39 

I  4i 

I  43 

19 

20 

I  3o 

I  32 

I  33 

I  34 

I  36 

I  38 

20 

21 

I  26 

I  27 

I  28 

I  29 

I  3o 

I  3i 

21 

22 

I  22 

I  23 

I  24 

I  24 

I  25 

I  25 

22 

23 

I  20 

I  20 

I  21 

I  21 

I  21 

I  21 

23 

24 

I  18 

I  18 

I  19 

I  19 

I  18 

I  17 

I  i5 

24 

2I) 

I  16 

I  16 

I  17 

I  17 

I  16 

I  i4 

I  II 

25 

26 

I  i4 

I  i4 

I  i4 

I  i4 

I  i3 

I  II 

I  8 

26 

27 

I  i3 

I  i3 

I  12 

I  12 

I  II 

I  9 

I  6 

27 

28 

I  12 

I  12 

I  11 

I  10 

I  9 

I  7 

I  4 

I  I 

28 

29 

I  II 

I  II 

I  10 

I  9 

I  8 

I  5 

I  2 

D9 

29 

3o 

I  II 

I  10 

I  9 

I  8 

I  7 

I  4 

I  0 

67 

3o 

3i 

I  10 

I  9 

I  8 

I  8 

I  6 

I  2 

58 

55 

3i 

32 

I  9 

I  9 

I  8 

I  7 

I  5 

I  I 

57 

54 

5i 

32 

66 

I  9 

I  8 

I  7 

I  6 

I  4 

I  I 

57 

53 

5o 

■66 

M 

I  9 

I  7 

I  6 

I  5 

I  3 

I  0 

57 

53 

49 

34 

3t) 

I  9 

I  7 

I  6 

I  5 

I  2 

I  0 

56 

52 

48 

35 

36 

I  8 

I  7 

I  6 

I  4 

I  2 

I  0 

56 

5i 

47 

44 

36 

^7 

I  8 

I  6 

I  5 

I  3 

I  I 

58 

55 

5i 

46 

43 

37 

38 

I  8 

I  6 

I  5 

I  3 

I  0 

57 

54 

5o 

46 

43 

38 

39 

I  8 

I  6 

I  4 

I  2 

59 

56 

52 

48 

45 

42 

39 

4o 

I  7 

I  5 

I  4 

I  2 

59 

55 

5i 

47 

44 

4i 

39 

40 

4i 

1  6 

I  4 

I  3 

I  I 

58 

54 

5o 

47 

44 

4i 

38 

4i 

42 

I  5 

I  4 

I  3 

I  I 

57 

54 

5o 

47 

44 

4i 

38 

42 

43 

I  4 

I  3 

I  2 

I  0 

56 

53 

5o 

47 

43 

40 

37 

34 

43 

44 

I  3 

I  2 

I  I 

59 

56 

53 

5o 

47 

43 

40 

37 

34 

44 

46 

I  I 

I  0 

59 

58 

55 

52 

49 

46 

43 

40 

37 

34 

32 

46 

48 

59 

59 

58 

57 

54 

5i 

4q 

46 

43 

4o 

37 

34 

32 

48 

bo 

57 

57 

56 

55 

53 

5o 

48 

45 

43 

40 

37 

34 

32 

?.o 

5o 

52 

55 

54 

53 

52 

5i 

49 

47 

45 

43 

40 

37 

34 

32 

3o 

52 

54 

54 

52 

5i 

5o 

49 

47 

46 

44 

42 

39 

37 

34 

32 

29 

27 

54 

56 

53 

5i 

49 

48 

47 

45 

44 

43 

4i 

38 

36 

34 

3i 

29 

27 

56 

58 

52 

49 

47 

46 

45 

44 

43 

42 

4o 

37 

35 

33 

3i 

29 

27 

26 

58 

60 

47 

45 

44 

43 

42 

4i 

40 

38 

36 

34 

32 

3o 

28 

27 

26 

60 

62 

43 

43 

4i 

40 

39 

38 

37 

35 

33 

3i 

29 

28 

27 

26 

62 

64 

42 

39 

38 

38 

37 

36 

34 

32 

3o 

29 

28 

27 

26 

64 

66 

38 

37 

37 

36 

35 

33 

3i 

29 

28 

27 

26 

25 

66 

68 

37 

35 

35 

34 

34 

33 

3i 

29 

28 

27 

26 

25 

68 

70 

34 

34 

33 

33 

32 

3o 

28 

27 

26 

25 

25 

70 

72 

33 

33 

32 

32 

3i 

29 

28 

26 

25 

24 

25 

72 

74 

32 

3i 

3i 

3o 

29 

28 

26 

25 

■^4 

74 

76 

3i 

3o 

3o 

29 

28 

27 

25 

24 

24 

76 

78 

29 

29 

29 

28 

27 

25 

24 

78 

80 

28 

28 

28 

27 

26 

25 

24 

80 

82 

27 

27 

26 

25 

24 

82 

84 

26 

26 

25 

25 

24 

84 

86 

26 

25 

25 

86 

32= 

34° 

36° 

38° 

42° 

46° 

50° 

54° 

58° 

62= 

m° 

70° 

74° 

78° 

82° 

86° 

Page  278]                      TABLE  XLVIIL 

Third  Correction.  Apparent  Distance  28°. 

App. 

Apparent  Mtitude  of  the  Sui 

,  Star  or  Planet. 

App. 

Alt. 
o 

6° 

70 

/  II 

10° 

/  // 

ir 

1  II 

12^ 

14^ 

Iti" 
/  // 

18° 
1  II 

20^ 

1    1 

22° 

/  II 

24" 

26° 
/  // 

28° 
/  // 

ciU° 

1   II 

Alt. 
0 

/  II 

6 

I  20 

I  23 

I  27 

I  33 

I  4o 

I  49 

2  00 

2   28 

2  56 

3  24 

i   53 

i   21 

4  48 
3  58 

5  i5 

5  42 

5   q 

6 

7 

I  25 

I  20 

I  23 

I  27 

I  32 

I  38 

I  45 

2  5 

2  26 

2  49 

i   i3 

3  36 

i   20 

4  43 

5  6 

7 

8 

I  32 

I  24 

I  20 

I  22 

I  25 

I  29 

I  35 

I  5o 

2  7 

2  26 

2  46 

3  4 

3  23 

3  42 

4  I 

4  20 

8 

9 

I  4i 

I  29 

I  23 

I  20 

I  22 

I  24 

I  28 

I  39 

I  52 

2  7 

2  32 

2  37 

2  53 

3  9 

3  25 

3  41 

9 

lO 

i  53 

I  37 

I  28 

I  23 

I  20 

I  21 

I  23 

I  3o 

I  39 

I  52 

2   5 

2  18 

2  3i 

2  44 

2  58 

3  II 

10 

II 

2  6 

I  46 

I  34 

I  27 

I  23 

I  20 

I  21 

I  24 

I  3i 

I  4i 

I  52 

2  4 

2  i5 

2  26 

2  37 

2  48 

II 

12 

2  19 

I  56 

I  4i 

r  32 

I  26 

I  22 

I  19 

I  21 

I  26 

I  33 

I  42 

I  52 

2  I 

2  ID 

2  20 

2  3o 

12 

i3 

2  32 

1    6 

I  49 

I  38 

I  3o 

I  25 

I  21 

I  20 

I  23 

I  28 

I  34 

I  42 

I  49 

I  57 

2  6 

2  i5 

i3 

i4 

2  46 

2  17 

I  58 

I  44 

I  M 

I  28 

I  23 

I  19 

I  21 

I  24 

I  28 

I  34 

I  4o 

I  47 

I  55 

2  3 

i4 

lb 

3  00 

2  28 

2,  7 

I  5i 

I  39 

I  32 

I  25 

I  20 

I  19 

I  21 

I  24 

1   28 

I  33 

1 39 

I  45 

I  52 

i5 

i6 

3  14 

2  39 

2  j6 

I  58 

I  45 

I  36 

I  28 

I  21 

I  18 

I  19 

I  21 

I  24 

I  28 

I  33 

I  38 

I  44 

16 

17 

3  28 

2  5i 

2  25 

2  5 

I  5i 

1  4i 

I  32 

I  23 

I  19 

I  18 

I  19 

I  21 

I  24 

I  28 

I  33 

I  38 

17 

18 

3  4i 

3  2 

2  35 

2  i3 

I  58 

I  46 

I  36 

I  25 

I  20 

I  17 

I  18 

I  19 

I  21 

I  24 

I  28 

I  33 

18 

19 

3  55 

3  i3 

2  45 

2  21 

2  5 

I  52 

I  4i 

I  27 

I  21 

I  18 

I  16 

I  17 

I  18 

I  21 

I  24 

I  28 

19 

20 

4  9 

3  24 

2  55 

2  29 

2  II 

1 57 

I  46 

I  3o 

I  23 

I  18 

I  16 

I  i5 

I  16 

I  18 

I  21 

I  24 

20 

21 

4  23 

3  35 

3  4 

2  37 

2  17 

2   3 

I  5i 

I  33 

I  25 

I  19 

I  16 

I  i4 

I  i5 

I  16 

I  18 

I  20 

21 

22 

4  36 

3  46 

3  i3 

2  45 

2  24 

2   9 

I  56 

I  36 

I  27 

I  20 

I  16 

I  i3 

I  i4 

I  i5 

I  16 

I  17 

22 

2J 

4  49 

3  57 

3  22 

2  53 

2  3i 

2  i4 

2  I 

I  4o 

I  29 

I  22 

I  17 

I  i3 

I  i3 

I  i3 

I  i4 

I  i5 

23 

24 

5  2 

4  8 

3  3i 

3  0 

2  37 

2  20 

2  6 

I  43 

I  3i 

I  24 

I  18 

I  i4 

t  12 

I  12 

I  12 

I  i3 

24 

2b 

5  16 

4  19 

3  40 

3  8 

2  43 

2  26 

2  II 

I  47 

I  34 

I  26 

I  19 

I  i5 

I  i3 

I  II 

I  II 

I  12 

25 

26 

5  29 

4  3o 

3  49 

3  i5 

2  5o 

2  32 

2  16 

I  5i 

I  36 

I  28 

I  20 

I  i5 

1  i3 

I  II 

I  II 

I  II 

26 

27 

542 

4  4i 

3  58 

3  23 

2  57 

2  38 

2  21 

I  55 

I  39 

I  3o 

I  21 

I  16 

I  i3 

I  11 

I  10 

I  10 

27 

28 

5  55 

4  52 

4  7 

3  3o 

3  4 

2  44 

2  26 

I  59 

I  42 

I  32 

I  22 

I  17 

I  14 

I  II 

I  10 

I  10 

28 

29 

6  7 

5  3 

4  16 

3  38 

3  II 

2  5o 

2  3i 

2  3 

I  45 

I  34 

I  24 

I  18 

I  i4 

I  12 

I  10 

I  10 

29 

3o 

6  19 

5  i3 

4  25 

3  45 

3  18 

2  55 

2  36 

2  7 

I  47 

I  36 

I  26 

I  19 

I  i5 

I  12 

I  10 

I  9 

3o 

3i 

6  3i 

5  23 

4  34 

3  52 

3  25 

3  I 

2  4i 

2  10 

I  5o 

I  38 

I  27 

[  20 

I  i5 

I  12 

I  10 

I  9 

3i 

32 

6  42 

5  32 

4  43 

3  59 

3  3i 

3  7 

2  46 

2  i3 

I  53 

I  40 

I  29 

I  21 

I  16 

I  12 

I  10 

I  9 

32 

33 

6  53 

5  4i 

4  5i 

4  6 

3  37 

3  12 

2  5i 

2  17 

I  56 

I  42 

I  3i 

I  22 

I  16 

I  12 

I  10 

I   9 

33 

■M 

7  4 

5  5o 

4  58 

4  i3 

3  43 

3  17 

2  55 

2  20 

I  58 

I  44 

I  32 

I  23 

I  17 

I  12 

I  10 

34 

35 

7  i5 

559 

5  5 

4  20 

3  48 

3  21 

2  59 

2  23 

2  00 

I  46 

I  33 

I  23 

I  17 

I  i3 

I  10 

I   8 

35 

36 

6  8 

5  II 

4  26 

3  53 

3  25 

3  3 

2  26 

2  3 

I  47 

I  34 

I  24 

I  18 

I  i3 

I  10 

I  8 

36 

37 

5  17 

4  32 

3  58 

3  29 

3  7 

2  29 

2  5 

I  49 

I  35 

I  25 

I  18 

I  i3 

I  10 

I  8 

37 

38 

4  38 

4  2 

3  33 

3  10 

2  32 

2  7 

I  5i 

I  36 

I  26 

I  IQ 

I  i4 

I  10 

I  8 

38 

39 

4  6 

3  37 

3  12 

2  34 

2  9 

I  52 

I  37 

I  27 

I  19 

I  i4 

I  II 

I  8 

39 

40 

3  4i 

3  i5 

2  37 

2  II 

I  53 

I  38 

I  27 

I  20 

I  i5 

I  II 

I  8 

40 

41 

3  17 

2  40 

2  i3 

I  54 

I  39 

I  28 

I  20 

I  i5 

I  II 

I  8 

4i 

42 

2  42 

2  i5 

I  55 

I  40 

I  29 

I  21 

I  i5 

I  ID 

I  7 

42 

43 

2  44 

2  17 

I  56 

I  4o 

I  29 

I  21 

I  i5 

I  10 

I  7 

43 

44 

2  18 

I  57 

I  4i 

I  3o 

I  21 

I  i5 

I  10 

I  7 

44 

46 

48 

2  19 

1  59 

2  0 

I  42 
I  43 

I  3o 
I  3i 

I  22 
I  22 

I  i5 
I  i5 

I  10 
I  10 

I  7 
I  6 

46 
48 

5o 

I  44 

I  32 

I  23 

I  i5 

I  ID 

I  6 

5o 

52 

I  33 

I  24 

I  i5 

I   9 

I  5 

52 

54 

I  25 

I  i5 

I   9 

I  5 

54 

56 
58 

I  i5 

I  9 
I   9 

I  4 
I  3 

56 

58 

60 

I  3 

60 

62 

62 

64 

64 

66 
68 

66 
68 

70 

70 

72 

72 

74 

74 

76 

76 

78 

78 

bo 

80 

82 

82 

84 

84 

86 

86 

6° 

7° 

8° 

9° 

10^ 

|ll° 

12° 

ll4° 

16° 

18° 

20° 

22° 

24° 

26° 

28° 

30° 

TABLE 

XLVIU 

[Page  279 

Third  Correction.  Appar 

ent  Distance  28°. 

D's 
A  pp. 

Apparent  Altitude  of  the  Sun,  Star  or  Planet. 

App. 

All. 

;j-2^ 

34" 

3G^ 

38'^ 

42" 

4G° 

50° 

54° 

58° 

(32° 

66° 

70-' 

74° 

78^ 

82° 

86^ 

Alt. 

o 

/  /,' 

1  II 

1  II 

/  // 

/  II 

/  // 

1  II 

/  // 

/  // 

II 

/  II 

/  // 

/  / 

;  /I 

1  1. 

/  // 

0 

b 

6  37 

1    4 

f 

7 

5  28 

5  49 

6  8 

7 

8 

4  40 

4  57 

3  II 

8 

9 

3  58 

4  i3 

4  26 

4  38 

9 

lO 

3  25 

3  38 

3  5o 

4  2 

10 

II 

3  0 

3  12 

3  23 

3  33 

II 

12 

2  4o 

2  5n 

2  59 

3  7 

3  22 

12 

i3 

2  24 

2  33 

2  4i 

2  48 

3  0 

1 3 

i4 

2  II 

2  18 

2  25 

2  3i 

2  42 

i4 

i5 

I  59 

2  6 

2  12 

2  17 

2  27 

i5 

i6 

I  5o 

1  56 

2  I 

2  6 

2  i4 

2  21 

16 

17 

I  43 

I  48 

I  52 

I  56 

2  3 

2  9 

17 

i8 

I  37 

I  4i 

I  4f) 

I  48 

I  54 

I  59 

18 

19 

I  3i 

I  35 

I  38 

I  4i 

I  46 

I  5o 

19 

20 

I  26 

I  29 

I  32 

I  M 

I  38 

I  42 

I  45 

20 

21 

I  22 

I  25 

I  27 

I  29 

I  32 

I  36 

I  38 

21 

22 

I  19 

I  21 

I  23 

I  25 

I  28 

I  3o 

I  32 

22 

23 

I  17 

I  18 

I  20 

I  22 

I  24 

I  26 

I  27 

23 

24 

I  i5 

I  16 

I  17 

I  18 

I  20 

I  22 

I  23 

I  24 

24 

25 

I  i3 

I  i4 

I  i4 

I  i5 

I  16 

I  18 

I  19 

I  19 

25 

26 

I  11 

I  12 

I  12 

I  i3 

I  i3 

I  i4 

I  i5 

I  i5 

26 

27 

I  10 

I  II 

r  II 

I  II 

I  11 

I  II 

I  12 

I  12 

27 

28 

I  10 

I  10 

I  10 

I  10 

I  10 

I  9 

I  9 

I  9 

I  9 

28 

29 

I  10 

I  10 

I  10 

I  9 

I  9 

I     8 

I  7 

I  6 

I  b 

29 

3o 

I  9 

I  9 

I  9 

I  8 

I  8 

I  7 

I  6 

I  4 

I  3 

3o 

3i 

I  8 

I  8 

I  7 

I  7 

I  6 

I  5 

I  4 

I  2 

I  I 

3i 

32 

I  8 

I  7 

I  6 

I  6 

I  5 

I  4 

I  3 

I  I 

I  0 

59 

32 

33 

I  7 

I  6 

I  5 

I  5 

I  4 

I  3 

I  2 

I  0 

58 

56 

33 

34 

I  7 

I  5 

I  4 

I  4 

I  3 

I  2 

I  I 

59 

57 

54 

■ 

34 

35 

I  7 

I  5 

I  4 

I  3 

r  2 

I  I 

I  0 

58 

55 

53 

35 

36 

I  6 

I  5 

I  4 

I  3 

I  I 

I  0 

58 

56 

54 

52 

5i 

36 

37 

I  6 

I  4 

I  3 

I  2 

I  0 

59 

57 

55 

53 

5i 

5o 

37 

38 

I  6 

I  4 

I  3 

I  I 

59 

58 

56 

54 

52 

5o 

49 

38 

39 

I  6 

I  4 

I  2 

I  0 

59 

57 

55 

53 

5i 

49 

47 

39 

4o 

I  b 

I  4 

I  2 

I  0 

58 

57 

55 

52 

5o 

48 

46 

44 

4o 

4i 

I  6 

I  4 

I  2 

I  0 

58 

56 

54 

5i 

49 

47 

45 

AZ 

4i 

42 

I  6 

I  4 

I  2 

59 

57 

55 

53 

5o 

48 

46 

44 

42 

42 

43 

I  6 

I  3 

I  I 

59 

57 

55 

53 

5o 

48 

46 

44 

42 

4i 

43 

/d 

I  6 

I  J 

I  I 

59 

56 

54 

52 

5o 

47 

45 

43 

4i 

40 

44 

46 

I  4 

I  2 

I  0 

58 

55 

53 

5i 

49 

47 

44 

42 

4o 

39 

46 

48 

I  3 

I  I 

59 

57 

54 

52 

5o 

48 

46 

43 

4i 

39 

38 

37 

48 

5o 

I  3 

I  I 

58 

56 

53 

5i 

49 

47 

45 

42 

40 

38 

37 

36 

5o 

52 

I  2 

I  0 

57 

55 

52 

5o 

48 

46 

44 

42 

4o 

38 

36 

35 

34 

52 

54 

I  2 

59 

56 

54 

5i 

t 

47 

45 

43 

4i 

3q 

37 

35 

34 

33 

54 

56 

I  I 

58 

55 

53 

5o 

46 

44 

42 

40 

38 

36 

35 

M 

33 

32 

56 

58 

I  0 

57 

54 

52 

49 

47 

45 

43 

4i 

3g 

37 

36 

35 

34 

32 

3. 

58 

60 

58 

55 

53 

5i 

48 

46 

44 

42 

40 

3fe 

37 

36 

35 

34 

32 

3i 

60 

62 

56 

i)4 

52 

5o 

47 

45 

43 

4i 

3q 

38 

37 

36 

35  34 

32 

3i 

62 

64 

62 

5o 

49 

46 

A4 

42 

40 

38 

37 

36 

35 

34  33 

32 

3o 

64 

66 

48 

48 

45 

43 

4i 

39 

38 

37 

36 

35 

34  33 

3i 

29 

66 

68 

46 

43 

4i 

4o 

38 

37 

36 

35 

34 

33 

32 

3o 

28 

68 

70 

42 

4o 

39 
38 

38 

37 

36 

35 

34 

6i 

3i 

28 

70 

72 

4i 

39 

37 

36 

35 

34 

33 

32 

3o 

72 

74 

39 

37 

36 

35 

34 

33 

32 

3o   28 

74 

76 

38 

36 

35 

M 

M 

33 

3i 

29 

27 

76 

78 

3(. 

M 

34 

33 

32 

3o 

28 

78 

80 

35 

M 

33 

32 

3i 

3o 

28 

80 

82 

33 

32 

3i 

3o 

29 

82 

84 

32 

32 

3i 

3o 

2Q 

84 

86 

.3i 

3o 

29 

86 

32° 

34° 

3G°  38°  1 

42° 

46° 

50° 

54° 

58° 

62° 

66° 

70°  74° 

78° 

82° 

86° 

P^?e230]               TABLE  XLVIII 

Third  Correction 

Apparent  Distance  32°. 

D's 
App. 

Apparent  Altitude  of  the  Sun,  Stai-  or  Planet. 

D's 
App. 

Alt. 

6^ 

.;u 

8^ 

V' 

lU" 

11^ 

12" 

14" 

IG" 

18° 

20" 

22" 

2-1° 

2G° 

28° 

30° 

Ait. 

o 

1   II 

/  II 

1   II 

1  II 

/  // 

/  // 

/  // 

/  // 

/  II 

/  // 

/  II 

/  /; 

/  II 

/  // 

/  // 

/  II 

0 

6 

I  18 

I  21 

I  25 

I  3o 

I  37 

I  47 

I  59 

2  23 

1   48 

3  i3 

3  39 

4  5 

4  3o 

4  55 

5  20 

5  45 

6 

7 

I  23 

I  18 

I  21 

I  24 

I  28 

I  'd^ 

I  42 

2  0 

2  18 

2  37 

2  58 

3  20 

3  4-2 

4  4 

4  25 

4  46 

7 

b 

I  3(. 

I  22 

1  18 

I  20 

I  22 

I  25 

I  29 

I  42 

I  57 

2  i4 

2  32 

2  5o 

3    8 

3  26 

3  44 

4  2 

8 

9 

I  38 

I  27 

I  20 

I  18 

I  19 

I  21 

I  23 

I  3i 

I  44 

I  58 

2  12 

2  26 

2  4i 

2  56 

3  II 

3  26 

9 

10 

I  47 

I  6^ 

I  23 

I  20 

I  18 

I  19 

I  20 

I  25 

I  34 

I  45 

I  57 

2  9 

2  21 

2  34 

2  4G 

2  59 

10 

II 

I  57 

I  4i 

I  28 

I  23 

I  19 

I  17 

1  18 

I  21 

I  27 

I  36 

I  46 

I  56 

2  6 

2  17 

2  28 

2  39 

II 

12 

2  9 

I  5o 

1  M 

I  27 

I  22 

I  19 

I  17 

I  19 

I  23 

I  29 

I  37 

I  46 

I  55 

2  4 

2  i3 

2  23 

12 

iJ 

2  21 

I  59 

I   4i 

1  32 

I  26 

I  21 

I  18 

I  17 

I  20 

I  24 

I  3o 

I  37 

I  45 

I  53 

2  I 

2  9 

i3 

i4 

2  34 

2  8 

l   5o 

I  38 

I  3o 

I  24 

I  20 

I  16 

I  18 

I  21 

I  25 

I  3o 

I  36 

I  43 

I  5i 

I  58 

i4 

if) 

2  47 

2  18 

I  59 

I  45 

1  35 

I  28 

I  22 

I  17 

I  16 

I  18 

I  21 

I  25 

I  3n 

I   35 

I  42 

I  49 

i5 

i6 

2  59 

2  28 

2  7 

I  52 

I  4i 

I  32 

I  25 

I  19 

I  i5 

I  16 

I  18 

I  21 

I  25 

I  29 

I  35 

I  4i 

16 

17 

3  12 

2  38 

2  16 

I  59 

I  47 

I  36 

I  28 

I  21 

I  16 

I  i5 

I  16 

I  18 

I  21 

I  25 

I  3o 

I  35 

17 

i8 

3  25 

2  48 

2  25 

2   7 

I  52 

I  4i 

I  32 

I  23 

I  17 

I  i4 

I  i5 

I  17 

I  19 

I  22 

I  25 

I  29 

18 

19 

3  38 

2  58 

2  34 

2  i4 

I  58 

I  46 

I  36 

I  25 

I  18 

I  i5 

I  i4 

I  i5 

I  17 

I  19 

I  22 

I  25 

19 

20 

3  5u 

3  9 

2  4i 

2  21 

2  4 

I  5i 

I  4o 

I  27 

I  20 

I  16 

I  i3 

I  i4 

I  i5 

I  17 

I  19 

I  21 

20 

21 

4  3 

3  ,9 

2  52 

2  28 

2  10 

I  56 

I  45 

I  3o 

I  22 

I  17 

I  14 

I  i3 

I  14 

I  i5 

I  16 

I  iS 

21 

22 

4  i5 

i  3o 

3  0 

2  35 

2  17 

2  2 

I  5o 

I  33 

I  24 

I  18 

I  i4 

I  II 

I  12 

I  i3 

I  i4 

I  16 

22 

23 

4  28 

3  4o 

3  9 

2  42 

2  24 

2  7 

I  55 

I  36 

I  26 

I  IQ 

I  i5 

I  12 

I  II 

I  12 

I  i3 

I  i4 

23 

24 

4  4o 

3  5i 

i   17 

2  5o 

2  3o 

2  i3 

I  59 

I  39 

I  28 

I  21 

I  16 

I  12 

I  ID 

I  II 

I  1 1 

I  12 

24 

2b 

4  52 

4  I 

3  26 

2  57 

2  36 

2  iS 

2  4 

I  42 

I  3o 

I  22 

I  17 

I  i3 

1  II 

I  10 

I  10 

I  10 

25 

26 

5  4 

4  12 

3  34 

3  5 

2  43 

2  24 

2  8 

I  46 

I  33 

I  24 

I  18 

I  i3 

I  II 

I  9 

I  Q 

I   Q 

26 

27 

5  16 

4  22 

3  43 

3  12 

2  5o 

2  3o 

2  i3 

I  5o 

I  34 

I  26 

I  19 

I  i4 

I  II 

I   9 

I     8 

I  8 

27 

28 

5  28 

4  33 

3  52 

3  20 

2  57 

2  35 

2  17 

I  53 

I  37 

I  27 

I  20 

I  i5 

I  II 

I   9 

I  7 

I  8 

28 

29 

5  4i 

4  44 

4  I 

3  28 

3  3 

2  4i 

2  21 

I  57 

I  4o 

I  29 

I  21 

I  16 

I  12 

I  ID 

I  8 

I   7 

29 

3o 

5  53 

4  54 

4  10 

3  35 

3  9 

2  46 

2  26 

2  0 

I  43 

1  3i 

I  23 

I  17 

I  l3 

1  10 

I  8 

1  6 

3o 

3i 

6  5 

5  4 

4  19 

3  42 

3  i5 

2  52 

2  3i 

2  4 

I  46 

I  33 

I  24 

I  18 

I  i3 

I  10 

I  8 

I  6 

3i 

32 

6  ,7 

5  i4 

4  27 

3  49 

3  21 

2  57 

2  36 

2  8 

I  4q 

I  36 

I  26 

I  19 

I  i4 

I  II 

I  9 

I   7 

32 

33 

6  29 

5  23 

4  35 

3  56 

3  27 

3  2 

2  4i 

2  12 

I  52 

I  38 

I  27 

I  20 

I  i5 

I  II 

I  9 

I   7 

33 

34 

6  4o 

5  32 

4  43 

4  3 

3  32 

3  7 

2  46 

2  i5 

I  55 

I  40 

I  29 

I  21 

I  16 

I  12 

I  9 

I   7 

34 

35 

6  5o 

5  4o 

4  5o 

4  9 

3  38 

3  12 

2  5o 

2  19 

I  58 

I  4-3 

I  3i 

I  22 

I  17 

I  l3 

I  9 

I   7 

35 

36 

6  59 

5  48 

4  57 

4  i5 

3  43 

3  16 

2  54 

2  22 

2  I 

I  45 

I  32 

I  23 

I  18 

I  i3 

I  10 

I   7 

36 

37 

7  7 

5  56 

5  4 

4  21 

3  49 

3  21 

2  59 

2  25 

2  4 

I  47 

I  34 

I  24 

I  19 

I  i4 

I  10 

I   7 

37 

38 

7  13 

6  3 

5  10 

4  29 

3  54 

3  25 

3  3 

2  28 

2  6 

I  49 

I  35 

I  25 

I  19 

I  i4 

I  10 

1   7 

38 

39 

7  22 

6  10 

5  16 

4  33 

3  59 

3  3o 

3  7 

2  3i 

2  8 

I  5. 

I  36 

I  26 

I  20 

I  i5 

I  10 

I  7 

39 

4o 

6  17 

5  21 

4  38 

4  4 

3  34 

3  II 

2  3^ 

2  10 

I  52 

I  38 

I  27 

I  20 

I  i5 

I  11 

I  8 

40 

4i 

5  26 

4  43 

4  8 

3  38 

3  i5 

2  36 

2  i3 

I  54 

I  39 

I  28 

I  21 

I  16 

I  12 

I  8 

4i 

42 

4  47 

4  12 

3  42 

3  18 

2  39 

2  16 

I  56 

I  4i 

I  29 

I  22 

I  16 

I  12 

I  8 

42 

43 

4  16 

3  46 

3  21 

2  42 

2  18 

I  58 

I  42 

I  3o 

I  22 

I  16 

I  12 

I  8 

43 

44 

3  5o 

3  24 

2  45 

2  20 

2  0 

I  4i 

I  3i 

I  23 

I  17 

I  12 

I  8 

44 

46 

3  27 

2  5o 

2  23 

2  2 

I   45 

I  3, 

I  24 

I  17 

I  12 

I  8 

46 

48 

2  54 

2  26 

2  4 

Li  47 

1  34 

I  25 

I  18 

I  12 

I  8 

48 

bo 

2  29 

2  6 

I  49 

I   36 

I  26 

I  19 

I  i3 

I  8 

5o 

b2 

2  8 

I  5i 

I  38 

I  28 

I  19 

I  i3 

I  S 

52 

64 

I  53 

I  39 

I  29 

I  20 

I  i4 

I  8 

54 

b6 

I  4o 

I  3o 

I  21 

I  14 

I  8 

56 

58 

I  3o 

I  21 

I  i4 

I  8 

58 

60 

I  21 

I  i4 

I  8 

60 

62 

I  i4 

I  8 

62 

64 

I  8 

64 

66 

66 

68 

68 

70 

70 

72 

72 

74 

74 

76 
78 



76 

78 

80 

80 

82 

82 

84 

84 

86 

86 

G° 

7° 

8° 

9° 

10° 

IP 

12° 

140 

1G° 

18° 

20° 

22° 

24° 

2G° 

28° 

30° 

TABLE 

XLVIII 

[Page  281 

Third  Correction. 

Apparent  Distance  32°. 

D's 
App. 
Alt. 

Apparent  JlltiUide  of  t 

he  Su 

I,  Sta 

r  or 

Plane 

D's 
App. 

Alt. 

32° 

34° 

36° 

38° 

42° 

46° 

50° 

54° 

58° 

62° 

66° 

70° 

743 

78^ 

82=' 

86° 

o 

(  // 

1  II 

1  II 

/  // 

/  // 

/  /( 

/  " 

/  II 

1   1 

/  /( 

/  // 

/  u 

/  // 

/  ;/ 

,  / 

/  II 

0 

6 

6  10 

6  33 

6  55 

7  i5 

6 

7 

5  7 

5  26 

5  44 

6  2 

7 

8 

4  20 

4  37 

4  52 

5  7 

5  35 

8 

9 

3  4i 

3  56 

4  10 

4  24 

4  5o 

9 

10 

3  12 

3  25 

3  38 

3  5o 

4  12 

10 

11 

2  5i 

3  2 

3  i3 

3  23 

3  42 

. 

II 

12 

2  33 

2  43 

2  5i 

3  00 

3  17 

3  33 

12 

i3 

2  18 

2  26 

2  34 

2  42 

2  56 

3  9 

i3 

i4 

2  5 

2  12 

2  19 

2  27 

2  39 

2  5o 

14 

i5 

I  55 

2  2 

2  8 

2  i4 

2  23 

2  35 

i5 

i6 

1  47 

I  53 

I  58 

2  3 

2  l3 

2  22 

2  3o 

16 

17 

I  40 

I  45 

I  5o 

I  54 

2   2 

2  II 

2  18 

17 

18 

I  34 

I  38 

I  42 

I  46 

I  53 

2  0 

2  7 

18 

ip 

I  29 

I  33 

I  36 

I  39 

I  45 

I  5i 

I  57 

19 

20 

I  25 

I  28 

I  3i 

I  33 

I  38 

I  43 

I  49 

I  54 

20 

21 

I  21 

I  24 

I  26 

I  28 

I  32 

I  37 

I  42 

I  46 

21 

22 

I  18 

I  20 

I  22 

I  24 

I  27 

I  3i 

I  35 

1  39 

22 

23 

I  i5 

I  17 

I  19 

I  20 

I  23 

I  27 

I  3o 

I  34 

23 

24 

I  i3 

I  i4 

I  16 

I  17 

I  20 

I  23 

I  26 

I  29 

I  32 

24 

25 

I  II 

I  12 

I  i3 

I  i5 

I  17 

I  19 

>  21 

I  24 

I  26 

25 

26 

I  9 

I  10 

I  II 

I  12 

I  i4 

I  16 

I  17 

I  19 

I  21 

26 

27 

I  8 

I   Q 

I  9 

I  10 

I  12 

I  i3 

I  i4 

I  16 

I  17 

27 

28 

r  8 

I  8 

I  8 

I  9 

I  10 

I  II 

1  12 

I  i3 

I  14 

I  i5 

28 

29 

I   7 

I   7 

I  7 

I  7 

I  8 

I  9 

I  9 

I  10 

I  II 

I  II 

29 

3o 

I  6 

I  6 

I  6 

I  6 

I  6 

I  7 

I  7 

I  7 

I  8 

I  8 

3o 

3i 

I  6 

I  6 

I  6 

I  5 

I  5 

I  5 

I  5 

I  5 

I  5 

I  5 

3i 

32 

I  6 

I  5 

I  5 

I  4 

I  4 

I  4 

I  4 

I  4 

I  3 

I  3 

I  3 

32 

33 

I  5 

I  4 

I  4 

I  3 

I  3 

I  2 

I  2 

[  2 

I  I 

I  I 

I  I 

33 

34 

I  5 

I  4 

I  3 

I  2 

I  2 

I  I 

I  0 

I  0 

59 

59 

59 

34 

35 

I  5 

I  3 

t  3 

I  2 

I  I 

I  0 

59 

58 

57 

57 

57 

35 

36 

I  5 

I  3 

I  2 

I  I 

I  I 

I  0 

58 

57 

56 

56 

55 

54 

36 

37 

I  5 

I  3 

I  I 

I  0 

I  0 

59 

57 

56 

55 

55 

54 

53 

37 

38 

I  5 

I  3 

I  I 

I  0 

59 

58 

56 

55 

54 

54 

53 

52 

38 

39 

I  5 

I  3 

I  I 

59 

58 

57 

56 

54 

53 

52 

5i 

5o 

39 

40 

I  5 

I  2 

I  0 

59 

58 

56 

55 

53 

52 

5i 

5o 

49 

48 

40 

4i 

I  5 

I  2 

I  0 

59 

58 

56 

54 

52 

5i 

5o 

49 

48 

47 

4i 

42 

I  5 

I  2 

I  0 

59 

57 

55 

53 

5i 

5o 

49 

48 

47 

47 

42 

43 

I  5 

I  2 

I  0 

58 

56 

54 

52 

5i 

49 

48 

47 

47 

4b 

45 

43 

44 

I  5 

I  2 

I  0 

58 

55 

53 

5i 

5o 

49 

48 

47 

46 

45 

44 

44 

46 

r  5 

I  2 

I  0 

58 

55 

52 

5i 

5o 

48 

47 

46 

45 

44 

43 

4b 

48 

I  5 

I  2 

59 

57 

55 

52 

5o 

49 

47 

46 

45 

44 

43 

42 

4i 

48 

5o 

I  5 

I  2 

59 

57 

54 

5i 

49 

48 

47 

46 

44 

4i 

42 

4i 

40 

5o 

52 

I  4 

I  I 

58 

56 

53 

5i 

49 

47 

46 

45 

43 

42 

4i 

4o 

39 

38 

52 

54 

I  4 

I  I 

58 

56 

53 

5o 

48 

46 

45 

44 

42 

4i 

4o 

39 

38 

37 

54 

56 

I  4 

I  I 

58 

56 

52 

49 

47 

45 

44 

41 

4i 

4o 

39 

38 

37 

36 

5b 

58 

I  4 

I  I 

58 

56 

52 

49 

47 

45 

43 

4i 

4o 

39 

38 

37 

36 

35 

58 

60 

I  4 

I  0 

57 

55 

5i 

48 

46 

44 

42 

40 

39 

38 

37 

3b 

35 

3b 

bo 

62 

I  3 

59 

56 

54 

5i 

48 

45 

43 

4i 

39 

38 

37 

36 

35 

34 

34 

62 

64 

I  3 

59 

56 

54 

5o 

47 

45 

43 

4i 

38 

38 

37 

36 

35 

34 

33 

64 

66 

I  3 

59 

56 

54 

5o 

47 

44 

42 

40 

38 

37 

36 

35 

M 

33 

bo 

68 

59 

55 

53 

48 

46 

44 

42 

4o 

38 

37 

36 

35 

34 

33 

68 

70 

55 

52 

48 

45 

43 

4i 

39 

37 

36 

35 

M 

33 

70 

72 

52 

47 

44 

42 

40 

38 

37 

36 

35 

33 

32 

72 

l4 

47 

44 

42 

4o 

38 

36 

35 

M 

32 

74 

76 

47 

43 

4i 

39 

38 

36 

35 

34 

32 

7b 

78 

4i 

4i 

39 

37 

35 

34 

33 

78 

80 

43 

4i 

39 

37 

35 

M 

33 

80 

82 

4o 

38 

36 

34 

33 

82 

84 

39 

38 

36 

34 

33 

84 

86 

37 

35 

34 

86 

32° 

34° 

36° 

38° 

42° 

46° 

50° 

54° 

58° 

62° 

m° 

70° 

74° 

78° 

82^ 

86° 

36 


rage  262]                 TABLE  XLVIII 

Third  Correction.  Apparent  Distance  3G°. 

5's 
A  pp. 
All. 

Apparent  Altitude  of  the  Sun,  Star  or  Planet. 

3's 
App. 

Alt. 

6" 

7° 

8° 

9° 

10° 

11° 

12° 

14° 

1G° 

18° 

20° 

22° 

24° 

2G° 

28° 

30° 

o 

1   II 

1  II 

/  » 

/  /; 

/  // 

1     II 

/  // 

1  II 

/  II 

1  II 

/  /; 

1  II 

/  // 

1  II 

/  /I 

/  // 

0 

b 

I  17 

I  19 

I  22 

1  27 

I  33 

I  42 

I  52 

2  i3 

2  34 

2  56 

3  19 

3  43 

4  7 

4  3i 

4  55 

5  18 

6 

7 

I  20 

I  17 

1  19 

1  22 

1  26 

1  3i 

1  37 

1  52 

2  10 

2  28 

2  48 

3  8 

3  27 

3  46 

4  6 

4  25 

7 

8 

I  2b 

I  20 

I  17 

1  19 

1,  21 

I  23 

1  27 

I  39 

1  53 

2  8 

2  24 

2  4o 

2  57 

3  i4 

3  3o 

3  46 

8 

9 

I  32 

I  24 

1  19 

I  17 

I  18 

1  19 

I  21 

1  29 

I  4o 

I  52 

2  5 

2  19 

2  33 

2  47 

3  2 

3  16 

9 

lO 

I  42 

I  3o 

1  23 

1  19 

1  lb 

1  17 

1  18 

1  23 

1  3i 

1  40 

1  5i 

2  2 

2  14 

2  27 

2  4o 

2  52 

10 

11 

I  52 

I  37 

1  28 

1  22 

1  18 

1  16 

I  17 

1  19 

I  35 

1  33 

I  42 

1  5i 

2  1 

2  12 

2  23 

2  33 

11 

12 

2   3 

I  4b 

1  34 

I  26 

1  20 

1  17 

I  lb 

1  17 

I  21 

I  27 

I  34 

1  4i 

1  5o 

1  59 

2  8 

2  17 

12 

iJ 

2  i4 

I  ti6 

I  4o 

1  3o 

1  23 

I  19 

1  lb 

1  i5 

I  18 

1  23 

I  28 

1  34 

I  4i 

I  4q 

I  57 

2  5 

i3 

14 

2  25 

2  I 

I  47 

I  6'j 

1  26 

I  21 

1  18 

I  i4 

I  16 

I  19 

I  24 

I  29 

1  35 

I  4i 

I  4q 

1  55 

i4 

lb 

2  3b 

2  10 

i.b4 

1   4i 

1  3o 

1  25 

I  21 

1  16 

1  i5 

1  17 

I  21 

1  25 

I  3o 

I  35 

1  4i 

I  46 

i5 

i6 

2  48 

2  20 

2  2 

I  47 

I  35 

I  29 

1  24 

1  18 

I  i3 

I  i5 

1  18 

1  21 

I  25 

1  2q 

I  3A 

T  3g 

ifi 

17 

3  0 

2  3o 

2  ID 

I  53 

I  4o 

1  33 

I  28 

I  20 

1  i5 

I  i4 

1  16 

1  18 

I  21 

I  24 

I   28 

I  3,"^ 

17 

i8 

3  12 

2  4o 

2  18 

2  0 

1  46 

1   38 

I  32 

1  22 

1  16 

I  i3 

1  i5 

1  16 

T  18 

I  20 

1  23 

1  27 

18 

^9 

3  24 

2  49 

2  27 

2  7 

I  5i 

1  43 

I  36 

1  25 

1  18 

I  i5 

I  i4 

1  i5 

1  t6 

I  18 

I  20 

[  23 

19 

20 

21 

20 
21 

3  3b 
3  46 

2  b9 

3  9 

2  3b 
2  43 

2  14 
2  21 

1  57 

2  3 

I  48 
1  53 

1  4o 

I  AA 

1  28 

1  3i 

I  21 
1  23 

I  16 

I  17 

I  12 
1  i3 

1  i3 
1  12 

I  i4 

1  16 

I  18 

1  20 

I  i3 

I  i4 

1  16 

I  18 

22 

3  b7 

i   18 

2  5l 

2  28 

2  9 

1  58 

I   48 

1  34 

1  25 

1  18 

I  i4 

1  11 

I  12 

1  i3 

I  i4 

1  i5 

22 

2J 

4  9 

;i  28 

2  59 

2  35 

2  16 

2  3 

1  52 

I  36 

1  26 

I  19 

I  14 

1  11 

I  10 

1  11 

1  12 

I  i3 

2  3 

24 

4  20 

i   37 

i    7 

2  42 

2  22 

2  8 

I  5b 

I  39 

1  28 

1  20 

1  i5 

I  11 

I  9 

I  0 

1  10 

1  1 1 

24 

26 

4  32 

i  47 

3  lb 

2  49 

2  28 

2  i3 

2  0 

1  42 

1  3o 

1  22 

I  i5 

I  11 

I  9 

I  8 

1  8 

I  9 

25 

26 

4  43 

3  56 

3  23 

2  56 

2  34 

2  18 

2  4 

1  45 

I  32 

I  23 

I  16 

1  12 

I  9 

I  7 

I  7 

26 

27 

4  bb 

i    6 

3  3i 

3  3 

2  4o 

2  23 

2  9 

1  48 

I  35 

1  25 

I  17 

1  12 

'f  9 

1  7 

1  6 

1  6 

27 

28 

b  6 

4  lb 

3  39 

3  10 

2  46 

2  28 

2  i3 

1  52 

I  38 

1  27 

I  iq 

1  i3 

I  9 

I  7 

I  6 

I  6 

28 

29 

b  17 

4  2b 

3  47 

3  17 

2  52 

2  3A 

2  18 

I  56 

I  4o 

I  2q 

I  20 

1  i3 

I  9 

1  7 

I  6 

I  5 

29 

do 

b  28 

4  M 

3  b4 

3  24 

2  58 

2  39 

2  23 

2  0 

I  A3 

I  3i 

I  21 

I  i4 

I  10 

1  8 

I  6 

1  5 

3o 

3i 

5  39 

4  43 

4  2 

3  3i 

3  4 

2  44 

2  28 

2  4 

I  46 

I  33 

1  23 

1  16 

1  11 

1  8 

I  6 

I  5 

3i 

32 

b  49 

4  b2 

4  10 

3  37 

3  10 

2  49 

2  33 

2  7 

I  49 

I  35 

1  25 

1  17 

1  12 

I  9 

I  7 

I  5 

32 

J3 

b  b9 

b  0 

4  18 

3  AA 

3  lb 

2  54 

2  37 

2  10 

1  5i 

1  37 

1  27 

1  19 

1  i4 

I  10 

I  7 

1  5 

33 

34 

6  9 

b  8 

4  25  3  5o 

3  22 

2  59 

2  4i 

2  i3 

I  53 

.  39 

1  29 

1  21 

I  i5 

I  11 

1  8 

1  6 

34 

3b 

6  19 

b  lb 

4  32 

3  56 

3  28 

3  4 

2  46 

2  16 

I  56 

I  4i 

I  3o 

1  22 

I  16 

1  11 

1  8 

1  6 

35 

36 

6  28 

5  24 

4  38 

4  2 

3  33 

3  9 

2  5o 

2  19 

I  59 

I  43 

1  32 

I  23 

I  17 

1  12 

I   9 

1  6 

3b 

^7 

6  38 

b  32 

4  4b 

4  8 

3  39 

3  14 

2  54 

2  22 

2  I 

I  45 

1  33 

I  24 

I  18 

I  i3 

I  9 

I  6 

37 

38 

6  47 

3  4o 

4  b2 

4  i4 

3  AA 

3  18 

2  5t; 

2  25 

2  4 

1  47 

1  35 

I  26 

I  19 

I  i4 

I  10 

I  6 

38 

39 

6  b7 

b  48 

4  b9 

4  20 

3  49 

3  23 

3  2 

2  28 

2  6 

I  49 

1  36 

1  27 

1  20 

1  i4 

1  10 

I  7 

39 

40 

7  b 

b  bb 

b  5 

4  25 

3  54 

3  27 

3  6 

2  3i 

2  8 

1  5. 

I  38 

1  28 

1  21 

1  i5 

1  11 

I  7 

4o 

4i 

7  16 

5  4 

5  12 

4  3i 

3  59 

3  3i 

3  10 

2  33 

2  1 1 

I  53 

1  40 

1  3o 

I  22 

I  i5 

1  11 

1  8 

4i 

■  42 

7  25 

3  12 

b  18 

4  36 

4  3 

3  35 

3  i3 

2  36 

2  i4 

I  55 

1  42 

1  3i 

I  22 

1  16 

I  11 

1  8 

42 

Ai 

7  33 

J  1.9 

b  24 

4  4i 

4  8 

3  39 

3  17 

2  39 

2  16 

1  57 

1  43 

1  32 

I  23 

I  16 

1  11 

I  8 

43 

AA 

b  2b 

b  3o 

4  46 

4  12 

3  43 

3  20 

2  42 

2  18 

I  5q 

I  45 

I  33 

I  24 

I  17 

I  12 

I  9 

AA 

46 

b  4i 

4  55 

4  20 

3  5o 

3  26 

2  47 

2  22 

a  2 

I  47 

I  35 

1  25 

1  18 

I  i3 

I  9 

46 

48 

4  27 

3  57 

3  32 

2  52 

2  26 

2  5 

I  49 

I  37 

1  27 

1  20 

I  i4 

1  10 

48 

bo 

3  38 

2  57 

2  3o 

2  8 

1  5i 

I  39 

I  29 

1  21 

I  i5 

I  10 

5o 

•  b2 

3  1 

2  33 

2  II 

I  53 

I  4i 

1  3i 

1  22 

1  lb 

I  II 

52 

b4 

2  36 

2  i3 

I  55 

I  A3 

1  32 

1  23 

1  lb 

I  11 

54 

bb 

2  i5 

I  57 
I  59 

I  AA 
I   45 
I  46 

1  33 

I  34 
1  35 

I  24 

I  25 

I  26 

I  17 
I  18 
I  18 

1  11 
1  12 
1  12 

56 
58 
60 

TaUc  P.  Effect  of  Sun's  Par. 

A'l  i  the  Nuiiihers  above  the  lines 

I  3b 

1  26 

I  18 

1  12 

62 

tu  Third  Corrt-clinii  ;  sublnict 
thri  oMiers. 

I  26 

1  19 
I  19 

I  12 

T  l3 

64 
66 

D'g 

Srin'fi  Appiireiit  Altitude. 

Arp. 
AH. 

1  i3 

68 
70 
72 

5  1 

0  20  31 

•10 

50 

0  70 

HO 

90 

5 

0 

3  5 

7 

" 

10 

1 

T  2  4 

5 

74. 

20 

4 

i  I  1 

3 

4 

76 

30 

6 

3  1 

0 

•2 

3 

78 

40 

9  i 

S  3 

•2 

0 

1  '2 

80 

SO 

7  5 

4 

2 

1  0 

1 

82 

60 

7 

5 

4 

3  2 

1 

0 

84 

70 

6 

3 

4  3 

2 

86 

80 

6 

4 

11° 

12° 

14° 

1G° 

18° 

20° 

22° 

24° 

2G° 

28° 

30-- 

TABLE  XLVllI 

[Page  283 

Third  Correction 

Apparent  Distance  36°. 

D's 
Alt. 

Apparent  Altitude  of  the  Sun.  Star  or 

Planet. 

li's 
App. 

Alt. 

32° 

34° 

36^ 

38° 

42° 

46° 

50° 

54° 

58° 

62° 

66° 

70° 

74° 

78° 

82° 

86° 

o 

(  // 

/  // 

1   II 

1  II 

1  II 

/  // 

/  // 

/  II 

/  // 

/  /' 

/  // 

/  // 

/  // 

/  II 

/  // 

/  // 

0 

6 

5  40 

6  I 

6  22 

6  43 

7  24 

6 

7 

4  43 

5  I 

5  195  36 

6  II 

7 

8 

4  I 

4  16 

4  3i  4  46 

5  16 

5  45 

8 

9 

3  20 

3  42 

3  55  4  8 

4  33 

4  58 

9 

10 

3  4 

3  16 

3  273  38 

359 

4  20 

10 

II 

2  43 

2  54 

3  43  i3 

3  32 

3  5c. 

II 

12 

2  27 

2  36 

2  45  2  53 

3  10 

3  25 

3  40 

12 

i3 

2  i3 

2  21 

2  29  2  37 

2  5i 

3  4 

3  16 

i3 

i4 

2  2 

2  9 

2  16  2  23 

2  36 

2  47 

2  57 

i4 

i5 
i6 

r  53 

I  45 

I  59 
1  5o 

2  5 

I  56 

2  II 
2  I 

2  23 

2  12 

2  33 
2  21 

2  42 
2  29 

2  36 



i5 

16 

17 

I  38 

I  42 

I  47 

I  53 

2  2 

2  10 

2  17 

2  24 

'7 

i8 

I  32 

I  36 

I  4o 

I  45 

I  53 

2  I 

2  7 

2  i3 

18 

19 

I  27 

I  3o 

I  34 

I  38 

I  45 

I  52 

I  58 

2  3 

19 

20 

I  23 

I  26 

I  29 

I  33 

I  38 

I  4A 

I  49 

I  54 

I  58 

20 

21 

I  20 

I  22 

I  25 

I  28 

I  33 

I  38 

I  43 

I  47 

I  5i 

21 

22 

I  17 

I  18 

I  20 

I  23 

I  28 

I  33 

I  37 

I  4i 

I  45 

22 

23 

I  i4 

I  i5 

I  17 

I  19 

1.24 

I  28 

I  32 

I  36 

I  39 

23 

24 

I  II 

I  12 

I  i4 

I  16 

I  20 

I  23 

I  27 

I  3i 

I  'M 

I  37 

24 

25 

26 

I  9 

I  10 

I  II 

I  i3 

I  16 
713 

I  19 
I  16 

I  22 

I  18 

I  26 
I  21 

I  29 

I  24 

I  3i 
I  26 

25 

26 

I  8 

I  8 

I  9 

I  II 

27 

I  7 

I  7 

I  8 

I  9 

I  II 

I  i3 

I  i5 

I  17 

I  20 

I  22 

27 

28 

I  6 

I  6 

I  7 

I  8 

I  9 

I  II 

I  12 

I  i4 

I  16 

I  18 

I  20 

28 

29 

I  6 

I  6 

I  6 

I  7 

I  8 

I  9 

I  10 

I  II 

I  i3 

I  i4 

I  16 

29 

So 

I  5 

I  5 

I  5 

I  6 

I  7 

I  7 

I  8 

I  9 

I  10 

I  II 

I  i3 

3o 

3i 

I  5 

I  5 

I  5 

I  5 

I  6 

I  6 

I  6 

I  7 

I  8 

I  9 

I  ID 

3i 

32 

I  4 

I  4 

I  5 

I  5 

I  5 

I  5 

I  5 

I  5 

I  6 

I  7 

I  8 

I  9 

32 

33 

I  4 

I  4 

I  4 

I  4 

I  4 

I  4 

I  4 

I  4 

I  4 

I  5 

I  5 

I  6 

36 

34 

I  4 

I  3 

I  3 

I  3 

I  3 

I  3 

I  3 

I  3 

I  3 

I  3 

I  3 

I  3 

34 

35 

I  4 

I  3 

I  3 

I  3 

I  2 

I  I 

I  I 

I  I 

I  I 

I  I 

I  I 

I  I 

35 

36 

I  4 

I  3 

I  2 

I  2 

I  I 

I  0 

I  0 

I  0 

I  0 

I  0 

I  0 

I  0 

I   0 

36 

37 

I  4 

I  3 

I  2 

I  1 

59 

59 

59 

59 

59 

59 

59 

59 

58 

37 

38 

I  4 

I  3 

I  0 

58 

58 

58 

58 

58 

68 

58 

58 

57 

38 

39 

I  5, 

I  3 

I  0 

58 

58 

58 

58 

57 

57 

57 

56 

56 

39 

40 

I  5 

I  3 

I  0 

58 

57 

57 

57 

57 

56 

56 

55 

54 

53 

40 

4i 

I  6 

I  3 

59 

57 

56 

56 

56 

56 

55 

54 

53 

52 

52 

4i 

42 

I  6 

I  3 

59 

57 

56 

55 

55 

55 

54 

53 

52 

5i 

5i 

42 

43 

I  6 

I  3 

59 

56 

55 

54 

54 

54 

53 

52 

5i 

5o 

5o 

49 

43 

44 

I  6 

I  3 

59 

56 

54 

53 

53 

53 

52 

5i 

5o 

49 

49 

48 

44 

46 

I  6 

I  3 

59 

56 

54 

53 

52 

5i 

5o 

49 

48 

48 

47 

47 

46 

48 

I  7 

I  3 

59 

56 

54 

52 

5i 

49 

48 

47 

46 

46 

45 

45 

45 

48 

5o 

I  7 

I  3 

59 

56 

53 

5i 

5o 

48 

47 

46 

45 

45 

44 

44 

44 

5o 

52 

I  7 

I  3 

59 

55 

52 

5o 

49 

48 

47 

46 

45 

44 

43 

42 

42 

52 

54 

I  7 

I  3 

59 

55 

52 

5o 

48 

47 

46 

45 

44 

43 

42 

4i 

4i 

54 

56 

I  7 

I  3 

X  0 

58 

55 

52 

49 

48 

47 

46 

45 

44 

43 

42 

41 

40 

5b 

58 

I  7 

I  3 

I  0 

58 

55 

52 

49 

47 

46 

45 

44 

43 

42 

4i 

40 

39 

58 

60 

I  7 

I  3 

I  0 

58 

55 

5i 

48 

46 

45 

44 

43 

42 

4i 

40 

39 

38 

bo 

62 

I  7 

I  3 

I  0 

58 

54 

5i 

48 

46 

44 

43 

42 

4i 

4o 

39 

38 

62 

64 

I  7 

I  3 

r  0 

58 

54 

5i 

48 

46 

44 

43 

42 

4o 

39 

38 

37 

64 

66 

I  8 

I  3 

r  0 

57 

54 

5o 

47 

45 

43 

42 

4i 

39 

38 

37 

bb 

68 

I  8 

I  3 

I  0 

57 

54 

5o 

47 

45 

43 

42 

40 

39 

38 

37 

68 

70 

I  8 

I  3 

I  0 

57 

53 

5g 

47 

44 

42 

4i 

40 

39 

38 

70 

72 

I  3 

I  0 

57 

53 

5o 

46 

43 

4i 

4o 

39 

38 

72 

74 

I  0 

57 

52 

49 

46 

43 

4i 

4o 

39 

38 

74 

76 

57 

52 

48 

45 

43 

4i 

39 

38 

37 

7b 

78 

5i 

48 

45 

42 

40 

39 

37 

78 

80 

5i 

47 

M 

42 

40 

39 

37 

80 

82 

47 

44 

4i 

40 

38 

82 

84 

47 

44 

4i 

39 

38 

84 

86 

44 

4i 

39 

86 

32° 

34° 

36° 

38° 

42° 

46° 

50° 

54° 

58° 

62° 

6G° 

70° 

74° 

78° 

82°j86° 

P«s''234]               TABLE  XLVIII 

Third  Correction.  Apparent  Distance  40°. 

J)'s 
App. 

All. 

0 

Apparent  Altitude  of  the  Sun,  Star  or 

Planet. 

A^'l?.- 
0 

1    II 

7° 

8° 

9° 
/  // 

10° 

11° 

12° 

14° 

16° 

/  // 

18° 
/  // 

20° 
'  II 

22° 

1    II 

24° 

26° 

28° 
/  // 

30° 

/  II 

/  » 

/  // 

/  // 

/  // 

6 

T  l6 

I  i8 

I  21 

I  25 

I  3i 

I  39 

I  47 

2  5 

2  26 

2  48 

3  10 

3  32 

3  54 

4  16 

4  38 

A  59 

6 

7 

I  iq 

I  i6 

I  18 

I  21 

I  24 

I  28 

I  M 

I  48 

2  4 

2  22 

2  40 

2  58 

3  16 

3  3A 

3  52 

4  10 

7 

8 

I  24 

I  IQ 

I  16 

I  18 

I  20 

I  22 

I  26 

I  36 

I  5o 

a  4 

2  18 

2  33 

2  48 

3  4 

3  20 

3  36 

8 

9 

I  3i 

I  23 

I  19 

I  16 

I  18 

I  19 

I  21 

I  27 

I  38 

I  49 

2  I 

2  i3 

2  25 

2  38 

2  52 

3  5 

9 

10 

I  4o 

I  29 

I  23 

I  19 

I  ifa 

I  17 

I  18 

I  21 

I  29 

I  38 

I  48 

I  58 

2  9 

2  20 

2  32 

2  A4 

10 

II 

I  5o 

I  36 

I  28 

I  22 

I  18 

I  l5 

I  16 

I  18 

I  23 

I  3i 

I  39 

I  48 

I  57 

2  7 

2  17 

2  27 

II 

12 

2   I 

I  M 

I  34 

I  26 

I  20 

I  17 

I  i5 

I  17 

I  20 

1  26 

I  33 

I  40 

I  48 

I  57 

2  5 

2  i3 

12 

i3 

2  I  I 

I  52 

I  4o 

I  3o 

I  23 

I  19 

T  16 

I  16 

I  18 

I  22 

I  28 

I  34 

I  4i 

I  48 

I  55 

2  2 

i3 

i4 

2  21 

2  0 

I  46 

I  M 

I  26 

I  2] 

I  17 

I  i5 

I  17 

I  19 

I  23 

X  28 

I  34 

1  40 

I  46 

I  53 

i4 

i5 

2  3l 

2  8 

I  '5? 

I  39 

I  3o 

I  23 

I  19 

I  16 

I  i5 

I  17 

I  20 

I  23 

I  27 

I  32 

I  38 

I  44 

i5 

i6 

2  4i 

2  16 

I  58 

I  A^ 

I  34 

I  26 

I  21 

I  17 

I  i4 

I  i5 

I  17 

I  19 

I  22 

I  26 

I  3i 

I  37 

16 

17 

2  52 

2  24 

2  4 

I  49 

I  38 

I  3o 

I  24 

I  19 

I  i5 

I  14 

I  i5 

I  17 

I  19 

I  22 

I  2fa 

I  3i 

17 

i8 

3  3 

2  32 

2  1 1 

I  54 

I  43 

1  iA 

I  28 

I  21 

I  16 

I  i3 

I  i4 

I  i5 

I  17 

I  19 

I  22 

I  26 

18 

19 

3  i4 

2  4i 

2  18 

2  0 

I  48 

I  39 

I  32 

I  23 

I  17 

I  i4 

I  i3 

I  i4 

I  i5 

I  17 

1  19 

I  22 

19 

20 

3  25 

2  5o 

2  25 

2  6 

I  53 

I  Ai 

I  36 

I  25 

I  19 

I  i5 

I  12 

I  12 

I  i3 

I  i5 

I  16 

I  19 

20 

21 

3  36 

2  59 

2  32 

2  12 

I  58 

I  47 

I  39 

I  27 

I  20 

I  16 

I  i3 

I  11 

I  12 

I  i3 

I  i4 

I  16 

21 

22 

3  47 

3  8 

2  4o 

2  18 

2  4 

I  52 

I  43 

I  3o 

I  22 

I  17 

I  i3 

I  II 

I  II 

I  12 

I  i3 

I  i4 

22 

23 

3  58 

3  17 

2  48 

2  25 

2  10 

I  57 

I  47 

I  33 

I  24 

I  18 

I  i4 

I  12 

I  10 

I  10 

I  II 

I  12 

23 

24 

4  9 

3  26 

2  56 

2  32 

2  i5 

2  2 

I  5i 

I  37 

I  26 

I  19 

I  i5 

I  12 

I  9 

I  9 

I  9 

I  10 

24 

25 

4  2(1 

3  35 

3  4 

2  39 

2  21 

2  7 

I  56 

-I  4o 

I  28 

I  21 

I  16 

I  i3 

I  10 

I  8 

I  8 

I  9 

25 

26 

4  3o 

3  U 

3  12 

2  45 

2  27 

2  12 

2  0 

I  43 

I  3o 

I  22 

I  17 

I  i3 

I  10 

I  8 

I  8 

r-i  9 

26 

27 

4  4i 

3  53 

3  20 

2  52 

2  33 

2  17 

2  4 

I  47 

I  33 

I  24 

I  18 

I  i4 

I  II 

I  8 

I  7 

I  8 

27 

28 

4  5i 

4  2 

3  28 

2  59 

2  39 

2  23 

2  8 

I  5o 

I  35 

I  25 

I  19 

I  i4 

I  II 

I  8 

I  7 

I  7 

28 

29 

5  I 

4  11 

3  36 

3  6 

2  45 

2  28 

2  12 

I  53 

I  38 

I  27 

I  20 

I  i5 

I  12 

1  9 

I  7 

I  7 

29 

3o 

5  12 

4  20 

3  M 

3  i3 

2  5o 

2  33 

2  17 

I  56 

I  4o 

I  29 

I  21 

I  i5 

I  12 

I  9 

'  7 

I  6 

3o 

3i 

5  23 

4  29 

3  52 

3  20 

2  56 

2  38 

2  21 

2  0 

I  43 

I  3o 

I  22 

I  16 

I  12 

I  9 

I  7 

I  6 

3i 

32 

5  33 

4  38 

3  59 

3  27 

3  I 

2  43 

2  26 

2  3 

I  45 

I  32 

I  23 

I  17 

I  i3 

I  ID 

I  7 

I  6 

32 

33 

5  43 

4  46 

4  6 

3  33 

3  7 

2  48 

2  3o 

2  6 

I  47 

I  34 

I  24 

I  18 

I  i4 

I  ID 

I  8 

I  6 

33 

34 

5  52 

4  54 

4  i3 

3  39 

3  i3 

2  53 

2  34 

2  9 

I  49 

I  36 

I  26 

I  19 

I  i5 

I  II 

I  8 

I  6 

34 

35 

6  I 

5  2 

4  20 

3  45 

3  19 

2  58 

2  38 

2  12 

I  5i 

I  38 

I  27 

I  20 

I  i5 

I  II 

I  8 

I  b 

35 

36 

6  lo 

5  10 

4  26 

3  5i 

3  24 

3  2 

2  42 

2  i5 

I  54 

I  4o 

I  29 

I  22 

I  16 

I  12 

I  8 

I  6 

36 

37 

6  i8 

5  17 

4  32 

3  57 

3  29 

3  7 

2  46 

2  18 

I  57 

I  42 

I  3i 

I  23 

I  17 

I  12 

I  9 

1  7 

37 

38 

6  26 

5  24 

4  38 

4  3 

3  33 

3  11 

2  5o 

2  21 

2  0 

I  44 

I  33 

I  25 

I  18 

I  l3 

I  9 

I  7 

38 

39 

6  34 

5  3i 

4  44 

4  8 

3  38 

3  i5 

2  54 

2  24 

2  2 

I  46 

I  35 

I  26 

I  19 

I  i4 

I  10 

I  7 

39 

4o 

6  42 

5  38 

4  5o 

4  i3 

3  42 

3  .9 

2  58 

2  27 

2  5 

I  48 

I  37 

I  28 

I  20 

I  14 

I  10 

I  7 

40 

4i 

6  5o 

5  45 

4  56 

4  19 

3  47 

3  24 

3  2 

2  3o 

2  8 

I  5i 

I  39 

I  29 

I  21 

I  i5 

I  II 

I  8 

4i 

42 

6  58 

5  52 

5  2 

4  24 

3  5i 

3  28 

3  6 

2  33 

2  10 

I  53 

I  4i 

I  3o 

I  22 

I  16 

I  II 

I  8 

42 

43 

7  7 

5  59 

5  8 

4  29 

3  56 

3  323  10 

2  36 

2  i3 

I  55 

I  43 

I  32 

I  23 

I  17 

I  12 

I  9 

43 

U 

7  i6 

6  6 

5  i4 

4  34 

4  0 

3  36 

3  i3 

2  39 

2  i5 

I  57 

I  AA 

I  33 

I  24 

I  18 

I  i3 

I  9 

AA 

46 

7  33 

6  21 

5  26 

4  44 

4  9 

3  A^ 

3  20 

2  AA 

2  19 

2  I 

I  47 

I  35 

I  27 

I  20 

I  i4 

I  10 

46 

48 

7  5o 

6  35 

5  38 

4  54 

4  18 

3  5i 

3  27 

2  49 

2  23 

2  5 

I  5o 

I  37 

I  29 

I  22 

I  i5 

I  1 1 

48 

5o 

5  5o 

5  3 

4  27 

3  58 

3  33 

2  54 

2  27;2   8 

I  52 

I  39 

I  3i 

I  23 

I  17 

I  12 

5o 

52 

4  36 

4  5 

3  39 

2  59 

2  3l 

2  II 

I  54 

I  42 

I  32 

I  24 

I  18 

I  i3 

52 

54 

3  45 

3  4 

2  35 

2  i4 

I  56 

I  A4 

I  34 

I  26 

I  19 

I  14 

54 

56 

3  9 

2  39 
2  43 

2  17 
2  19 

1  58 

2  0 

I  46 
I  48 

I  36 
I  37 

I  28 
I  29 

I  20 
I  21 

I  14 
I  i5 

56 
58 

Table  v.     Effect  of  Sun's  Far. 

2  21 

2  2 

I  49 

I  38 

I  3o 

I  22 

I  i5 

60 

Aiitl  the  Numbers  above  the  lines 

2  4 

I  5o 

I  39 

I  3o 

I  22 

I  16 

62 

the  others. 

I  5i 

I  40 
I  4o 

I  3i 
I  3i 

I  23 

I  24 

I  16 

I  17 

64 
66 

D's 
App. 
Ak. 

Si.n's  Appaa-eiU  Altitude. 

5 

10  -iO  3 

U  40 

30 

50  70 

80 

90 

I  3i 

I  24 

I  17 

68 

t: 

r.  ., 

.. 

'■ 

.' 

I  24 

I  17 

70 

5 

0 

1  2 

6 

I  17 

72 

10 

1 

0   1 

4 

6 

74 

20 

■1 

3  I 

2 

3 

4 

76 

30 

6 

5  3  5 

0 

1 

2  3 

78 

40 

8 

7  5  4 

2 

80 

50 

9  7  3 

4 

2 

1  0 

0 

0 

, 

82 

60 

9  7 

5 

4 

3  2 

2 

84 
86 

70 
80 

8 

6 

7 

5 
6 

4  3 

4 

90 

G 

11° 

12° 

14° 

1G° 

18° 

20° 

22° 

24° 

26° 

28° 

-30° 

TABLE  XLVIII 

[Page  ith 

Third  Correction.  Apparent  Distance  40°. 

])'s 
App. 
Alt. 

Apparent  Altitude  of  the  Sun,  Sta 

r  or  Planet. 

D's 
App, 
Alt. 

32° 

34° 

3G= 

38° 

42° 

46° 

50° 

54° 

58° 

62° 

6G° 

70° 

74° 

82° 

86° 

o 

(  // 

1  II 

1   II 

// 

/  II 

1  II 

/  // 

/  // 

/  // 

1  II 

1   It 

1  II 

/   / 

/  // 

/  / 

/  // 

0 

6 

5  19 

5  39 

5  59 

3  19 

6  57 

7  33 

b 

7 

4  27 

4  44 

5  I 

b  18 

5  5i 

6  20 

7 

8 

3  5i 

4  6 

4  20 

4  34 

5  I 

5  26 

5  5o 

8 

9 

3  20 

3  34 

3  46 

J  58 

4  22 

4  44 

5  5 

9 

10 

2  56 

i    8 

3  19 

J  Jo 

3  5o 

4    9 

4  27 

10 

II 

2  37 

2  47 

2  57 

3  6 

3  25 

3  42 

3  58 

11 

12 

2  22 

2  3o 

2  39 

2  48 

3  5 

3  20 

3  33 

3  46 

12 

i3 

2  10 

2  17 

2  25 

2  32 

2  47 

3  I 

3  i3 

3  25 

i3 

i4 

2  0 

2  6 

2  12 

2  18 

2  32 

2  44  2  55 

3  4 

14 

i5 

I  5o 

I  56 

2  I 

2  7 

2  19 

2  3o  2  4o 

2  48 

i5 

i6 

I  42 

I  47 

I  52 

I  58 

2   8 

2  18  2  27 

2  35 

2  42 

16 

I? 

I  36 

I  4o 

I  45 

I  5o 

I  59 

2  8:2  16 

2  23 

2  3o 

17 

i8 

I  3i 

I  34 

r  38 

1  43 

I  5i 

I  59  2  6 

2  12 

2  19 

18 

^9 

I  26 

I  29 

I  33 

I  36 

I  44 

I  5i)i  58 

2  3 

2  9 

19 

20 

I  22 

I  24 

I  27 

I  3o 

I  37 

I  44,1  5o 

I  5b 

2  0 

2  5 

20 

21 

I  18 

I  20 

I  23 

I  26 

I  32 

I  38|i  44 

I  49 

I  53 

I  57 

21 

22 

I  i5 

I  17 

I  19 

I  22 

I  28 

I  33  I  38 

I  4'i 

I  47 

I  5o 

22 

23 

I  i3 

I  14 

I  16 

I  19 

I  24 

I  291  33 

I  38 

I  42 

I  45 

23 

24 

r  II 

I  12 

I  i4 

I  16 

I  21 

I  25Ji  29 

I  33 

I  37 

I  4o 

I  43 

24 

25 

I  10 

I  II 

I  12 

I  i4 

I  18 

I  21  I  25 

I  29 

I  32 

I  35 

I  37 

25 

26 

I  9 

I  10 

I  II 

I  12 

I  i5 

I  18 

I  21 

I  25 

I  28 

I  3o 

I  32 

26 

27 

I  8 

I  9 

I  9 

I  10 

I  i3 

I  i5 

I  18 

I  21 

I  24 

I  26 

I  27 

27 

28 

I  7 

I  8 

I  8 

I  9 

I  11 

I  i3 

I  16 

I  18 

I  20 

I  22 

I  23 

I  0.4 

28 

29 

I  7 

I  7 

I  7 

I  8 

I  9 

I  II 

I  i3 

I  i5 

I  16 

I  18 

I  19 

I  20 

29 

3o 
3i 

I  6 
I  6 

I  6 
I  6 

I  6 
I  6 

I  7 
I  7 

I  8 

I  9 

I  1 1 

I  12 

I  i3 
I  II 

I  i5 
I  i3 

I  16 

I  i4 

I  17 
I  i5 

3o 
3i 

I  7 

I  8 

I  9 

I   10 

32 

I  6 

I  6 

I  6 

I  6 

I  6 

I  6 

I  7 

I  8 

I  9 

I  10 

I  II 

I  12 

I  i3 

32 

33 

I  5 

I  5 

I  5 

I  5 

I  5 

I  5 

I  6 

I  6 

I   7 

I  8 

I  9 

I  10 

I  10 

ii 

34 

I  5 

I  4 

I  4 

I  4 

X  4 

I  4 

I  5 

I  5 

I  6 

I  7 

I  7 

I  8 

I  8 

34 

35 

I  5 

I  4 

I  4 

I  4 

I  4 

I  4 

I  4 

I  4 

I  4 

I  5 

I  5 

I  6 

I  6 

35 

36 

I  5 

I  A, 

I  3 

I  3 

I  3 

I  3 

I  3 

I  3 

I  3 

I  4 

I  4 

I  4 

I  4 

I  4 

36 

37 

I  5 

I  4 

I  3 

I  2 

I  2 

I  2 

I  2 

I  I 

I  I 

I  2 

I  2 

I  2 

I  2 

I  2 

il 

38 

I  5 

I  4 

I  2 

I  I 

I  I 

I  I 

I  I 

I  0 

I  0 

I  0 

I  0 

I  I 

I  I 

I  I 

38 

3q 

I  5 

I  4 

I  2 

I  I 

I  0 

I  0 

I  0 

59 

59 

59 

59 

59 

59 

^9 

39 

4o 

I  5 

I  4 

I  2 

I  I 

I  0 

59 

59 

58 

58 

67 

S7 

57 

!57 

57 

^7 

40 

4i 

I  6 

I  4 

I  2 

I  I 

5q 

58 

58 

57 

57 

56 

56 

56 

56 

56 

56 

4i 

42 

I  6, 
I  6 

I  4 

I  2 

I  0 

58 

57 

57 

56 

56 

55 

55 

55 

55 

55 

55 

42 

43 

I  4 

I  2 

I  0 

58 

57 

56 

55 

55 

54 

54 

54 

54 

54 

54 

54 

4i 

44 

I  6 

I  4 

I  2 

I  0 

58 

56 

55 

54 

54 

53 

53 

53 

53 

53 

53 

53 

44 

46 

I  7 

I  4 

I  2 

I  0 

58 

«56 

54 

53 

53 

52 

52 

5i 

5i 

5i 

5i 

5i 

46 

48 

I  8 

i  5 

I  2 

I  0 

58 

55 

53 

52 

52 

5i 

5i 

5o 

49 

49 

49 

49 

48 

5o 

I  8 

I  '^ 

I  2 

I  0 

57 

54 

52 

5i 

5i 

5o 

49 

48 

48 

48 

48 

48 

5o 

52 

I  9 

I  5 

I  2 

I  0 

57 

54 

52 

5o 

So 

49 

48 

47 

47 

46 

46 

46 

52 

54 

I  9 

I  5 

I  2 

I  0 

57 

54 

5i 

49 

49 

48 

47 

46 

46 

45 

45 

45 

54 

56 

I  10 

I  6 

I  3 

I  0 

56 

53 

5i 

49 

48 

47 

46 

45 

45 

44 

44 

44 

56 

58 

t  10 

I  6 

I  3 

I  0 

56 

53 

5o 

48 

47 

46 

45 

45 

44 

43 

43 

58 

60 

r  10 

I  7 

I  4 

56 

52 

5o 

48 

47 

45 

44 

44 

Ai 

42 

42 

60 

62 

I  II 

I  7 

I  4 

56 

52 

5o 

48 

46 

45 

44 

43 

42 

42 

62 

64 

I  II 

I  7 

I  4 

56 

52 

49 

47 

45 

44 

43 

42 

4i 

4i 

64 

66 

I  12 

I  7 

I  4 

56 

52 

49 

47 

45 

43 

42 

42 

4i 

66 

68 

I  12 

I  8 

I  4 

56 

52 

49 

47 

45 

43 

42 

42 

4i 

68 

70 

I  12 

I  8 

I  4 

55 

5i 

48 

46 

44 

43 

42 

42 

70 

72 

I  i3 

I  8 

I  4 

55 

5i 

48 

46 

44 

43 

42 

4i 

72 

74 

I  i3 

I  8 

I  4 

55 

5i 

48 

46 

44 

43 

42 

74 

76 

I  8 

I  4 

55 

5i 

48 

46 

44 

42 

4i 

7b 

78 

I  4 

55 

5i 

48 

46 

43 

42 

78 

80 

55 

5i 

48 

46 

43 

4i 

80 

82 

55 

5i 

48 

46 

43 

82 

84 

55 

5i 

48 

46 

43 

84 

86 

5i 

48 

45 

8b 

32° 

34° 

36° 

38° 

42° 

46° 

50° 

54° 

58° 

62° 

GG° 

70° 

740 

78° 

82° 

86° 

1 

^^^^^^^^                                    TABLE  XLVIIL 

Third  Correction.  Apparent  Distance  44° 

D's 
App. 
Alt. 

Apparent  Altitude  of  the  Sun,  Star  or  Planet. 

D's 
App. 

Alt. 

6° 

7° 

8° 

9° 

10° 

11° 

12° 

14° 

16° 

18° 

20° 

22° 

24° 

26° 

28° 

30° 

o 

/  // 

1  II 

/  » 

/  II 

/  // 

1     II 

/  // 

/  // 

/  // 

/  // 

/  II 

/  II 

/  // 

/  // 

/  /; 

/  // 

0 

b 

X  It 

)  I  18 

[  21 

I  25 

I  3i 

I  37 

I  45 

2  3 

2  23 

2  44 

3  5 

3  25 

3  45 

4  5 

4  25 

4  44 

6 

7 

I  2C 

)  I  16 

I  18 

I  20 

I  24 

1  28 

I  33 

I  46 

2  1 

2  17 

2  34 

2  5i 

3  8 

3  25 

3  42 

3  59 

7 

8 

I  2i 

I  19 

I  lb 

I  17 

I  19 

I  22 

I  25 

I  35 

I  47 

2  0 

2  i4 

2  29 

2  43 

2  58 

3  12 

3  27 

8 

9 

I  3i 

I  23 

I  lb 

I  i5 

I  lb 

I  18 

I  21 

I  27 

I  36 

I  47 

I  5a 

2  12 

2  24 

2  36 

2  48 

3  0 

9 
10 

lO 

I  35 

^I  28 

I  21 

I  17 

I  i5 

I  lb 

I  18 

I  22 

I  29 

I  38 

I  48 

I  58 

2  8 

2  18 

2  29 

2  39 

II 

I  4t 

I  34 

I  25 

I  20 

I  17 

I  i5 

I  16 

I  19 

I  24 

I  3i 

I  39 

I  47 

I  56 

2  5 

2  i4 

2  24 

II 

12 

I   5fc 

I  4i 

I  3o 

I  23 

I  19 

I  lb 

I  i5 

I  17 

I  20 

I  25 

I  32 

I  38 

I  46 

I  54 

2  2 

2  II 

12 

i3 

2  t 

I  48 

I  35 

I  27 

I  22 

I  18 

I  16 

I  i5 

I  17 

I  21 

I  26 

I  32 

I  38 

I  45 

I  52 

I  5o 

t3 

i4 

2  It 

I  5b 

i'4i 

I  3i 

I  25 

I  20 

I  17 

I  i4 

I  i5 

I  18 

I  22 

I  27 

I  32 

I  38 

I  44 

I  49 

t4 

i5 

2  2fc 

2  4 

I  47 

I  6b 

I  29 

I  23 

I  19 

I  i5 

I  14 

I  16 

I  19 

I  23 

I  27 

I  32 

I  37 

I  42 

i5 

i6 

2   ,..,2  12 

I  53 

I  4i 

I  33 

I  26 

I  21 

I  17 

I  i4 

I  i5 

I  17 

I  20 

I  23 

I  27 

I  32 

T  36 

ifi 

17 

2  48 

2  20 

2  0 

I  47 

I  37 

I  3o 

I  24 

I  19 

I  i5 

I  i5 

I  16 

I  18 

I  20 

)  23 

I  26 

I  3o 

17 

i8 

2  5b 

2  28 

2  8 

I  53 

I  42 

I  34 

I  27 

I  20 

I  16 

I  t4 

I  i5 

I  16 

I  18 

I  20 

I  22 

T  25 

18 

19 

3  8 

2  37 

2  i5 

I  59 

I  47 

I  38 

I  3o 

I  22 

I  17 

I  i4 

I  i4 

I  i5 

I  16 

I  17 

I  19 

I  22 

19 

20 

3  lb 

2  45 

2  22 

3  5 

I  52 

I  42 

I  34 

I  25 

I  19 

r  i5 

I  i3 

I  i4 

I  i4 

I  i5 

I  17 

I  19 

20 

21 

3  29 

2  54 

2  3o 

2  12 

I  57 

I  46 

I  37 

I  27 

I  21 

I  17 

I  i4 

I  12 

1  i3 

I  i4 

I  i5 

I  17 

21 

22 

3  39 

3  2 

2  37 

2  18 

2  3 

I  5i 

I  4i 

I  3o 

I  23 

I  18 

I  i4 

I  II 

I  12 

I  i3 

I  i4 

1  16 

22 

23 

3  49 

3  II 

2  45 

2  24 

2  8 

I  55 

I  45 

I  33 

I  25 

I  19 

I  i5 

I  12 

I  II 

I  12 

I  i3 

I  i4 

23 

24 

4  0 

3  19 

2  52 

2  3i 

2  14 

2  0 

I  49 

I  36 

I  27 

I  20 

I  16 

I  12 

I  10 

I  10 

I  11 

I  i3 

?4 

25 

4  10 

3  28 

2  59 

2  37 

2  20 

2  5 

I  53 

I  39 

I  29 

I  21 

I  17 

I  i3 

I  10 

I  9 

I  10 

I  II 

25 

26 

4  20 

3  36 

3  6 

2  43 

2  25 

2  10 

I  57 

I  42 

I  3i 

I  22 

I  17 

I  i3 

I  10 

I  8 

I   9 

I  0 

26 

27 

4  3o 

3  45 

3  i3 

2  49 

2  3i 

2  i5 

2  I 

I  45 

I  32 

I  23 

I  18 

1  i4 

I  II 

I  9 

I  8 

I  8 

27 

28 

4  39 

3  53 

3  20 

2  55 

2  36 

2  20 

2  5 

I  47 

I  34 

I  25 

I  19 

I  i5 

I  12 

I  9 

I  7 

I   7 

28 

29 

4  48 

4  I 

3  27 

3  I 

2  4i 

2  24 

2  9 

I  49 

I  36 

I  27 

I  20 

I  i5 

I  12 

1  9 

I  7 

I  6 

29 

So 

4  57 

4  9 

3  34 

3  7 

2  4b 

2  29 

2  i4 

I  52 

I  38 

I  28 

I  21 

I  16 

I  i3 

I  9 

I  7 

I  6 

3o 

3i 

5  7 

4  17 

3  4i 

3  i3 

2  5i 

2  34 

2  19 

I  55 

I  40 

I  3o 

I  22 

I  17 

I  i3 

I  10 

I  8 

I  6 

3 1 

32 

5  16 

4  25 

3  48 

3  19 

2  56 

2  38 

2  23 

I  58 

I  42 

I  3i 

I  23 

I  18 

I  i4 

I  10 

I  8 

I  6 

32 

33 

5  25 

4  33 

3  54 

3  25 

3  I 

2  43 

2  27 

2  I 

I  44 

I  33 

I  24 

I  IQ 

I  i5 

I  II 

I  9 

I   7 

33 

34 

D  34 

4  40 

4  I 

3  3o 

3  b 

2  47 

2  3i 

2  4 

I  47 

I  35 

I  26 

I  20 

I  i5 

I  II 

I  9 

I  7 

34 

35 

5  43 

4  48 

4  8 

3  m 

3  II 

2  52 

2  35 

2  7 

I  5o 

I  37 

I  27 

I  21 

I  16 

I  12 

:  9 

I  7 

35 

36 

5  5i 

4  55 

4  i4 

3  42 

3  i5 

2  56 

2  39 

2  II 

I  53 

I  3q 

I  28 

I  22 

I  17 

I  i3 

I  10 

I   7 

36 

37 

6  0 

5  3 

4  21 

3  47 

3  20 

3  0 

2  43 

2  i5 

I  56 

I  4i 

I  3o 

I  23 

I  17 

I  i3 

I  10 

I  8 

37 

38 

6  9 

5  10 

4  27 

3  52 

3  24 

3  4 

2  47 

2  18 

I  58 

I  43 

I  32 

I  24 

I  18 

I  14 

I  II 

I  8 

3S 

39 

6  18 

5  18 

4  33 

3  58 

3  29 

3  8 

2  5i 

2  21 

2     I 

I  45 

I  33 

I  25 

I  19 

I  i4 

I  II 

I  8 

39 

4o 

6  27 

5  25 

4  39 

4  3 

3  33 

3  12 

2  54 

2  24 

2  3 

I  46 

I  35 

I  26 

I  20 

I  i5 

I  II 

I  8 

40 

4i 

6  36 

5  32 

4  45 

4  8 

3  38 

3  16 

2  58 

2  27 

2  6 

I  48 

I  37 

I  27 

I  21 

I  16 

I  12 

I  9 

4t 

42 

6  45 

5  3q 

4  5i 

4  i3 

3  42 

3  20 

3  I 

2  3o 

2  8 

I  5o 

I  39 

I  20 

I  22 

I  16 

I  12 

I  9 

42 

43 

6  53 

5  46 

4  57 

4  18 

3  47 

3  24 

3  4 

2  33 

2  10 

I  52 

I  4o 

I  3o 

I  23 

I  17 

I  i3 

I  9 

43 

44 

7  0 

5  53 

J  5 

4  23 

3  5i 

3  28 

3  7 

2  35 

2  12 

I  54 

I  42 

I  32 

I  24 

I  18 

I  i3 

I  10 

44 

46 

7  14 

6  6 

5  i4 

4  33 

4  0 

3  35 

3  i4 

2  4o 

2  17 

I  58 

I  4§ 

I  35 

I  26 

I  20 

I  i4 

I  10 

46 

48 

7  27 

6  i8 

5  25 

4  43 

4  9 

3  43 

3  21 

2  45 

2  21 

2  2 

I   48 

I  37 

I  28 

I  21 

I  i5 

I  II 

48 

bo 

7  4o 

6  29 

5  35 

4  52 

4  18 

3  5o 

3  27 

2  5o 

2  25 

2.  6 

I  52 

I  4o 

I  3i 

I  23 

r  16 

I  II 

5o 

52 

7  52 

6  40 

5  45 

5  I 

4  26 

3  57 

3  33 

2  55 

2  29 

2  10 

I  56 

I  43 

I  33 

I  25 

I  18 

I  12 

52 

54 

3  55 

5  9 

4  34 

4  4 

3  39 

3  0 

2  33 

2  i4 

I  59 

I  46 

I  35 

I  26 

I  19 

I  i3 

54 

56 

4  42 

4  10 

3  45 
3  5o 

3  5 

3  ID 

3  i4 

2  37 
2  4i 
2  44 

2  17 

2  20 
2  22 

2  2 

2  4 
2  5 

I  49 
I  5i 

I  52 

I  37 
I  39 
I  4o 

I  27 

1 29 

I  3o 

I  20 
I  21 

I  23 

I  i4 

I  ID 
I  16 

56 
58 
60 

1 

Table  P.  Effect  of  Sun's  Pa 

^. 

Add  the  Numbers  above  tlie  lines 

2  47 

2  24 

2  b 

I  53 

I  42 

I  3i 

I  23 

I  17 

62 

lo  Tl}ird  Correction  ;  subtract 
tlie  others. 

2  26 

2  7 
9  8 

I  54 
I  55 

I  43 

T  44' 

I  32 

I  33 

I  24 
T  95 

I  18 

64 
66 

1  19 

Act 
All 

''   Sun's  Apparent  Altitude. 

I  56 

I  45 
I  45 

I  34 
I  35 

I  26 

68 

5  1 

20 

30 

0  50  6 

0  70  80 

90 

u      t 

,, 

,, 

,,  ,, 

70 

r 

0  1 

9 

4 

5  6 

I  3b 

I  29 

I  22 

72 

in 

T  0 

I 

? 

4  5 

I  3o 

I  22 

74 

20 

3  3 

T 

"1 

2  3 

1 

I  23 

7b 

30 

5  5 

3 

2 

0  1 

J  2 

78 

40 

7  7 

5 

4 

2  1 

3  0  1 

80 

so 

9  8 

7 

5 

4  3 

i    I  1 

82 

60 

8 

7 

=!^ 

3  3  2 

84 

70 

8(1 

1 

8 

6  S 

3 

86 

11° 

12° 

14° 

16° 

18° 

20° 

22° 

24° 

26° 

28° 

30° 

TABLE  XLVIII 

[Page  2E7 

Third  Correction.  Apparent  Distance  44°. 

D's 

Apixircnt  Altitude  of  the  Sun,  Star  or  Planet. 

Add. 
Alt. 

Api.. 
Alt. 

32= 

34° 

36° 

38° 

42° 

46°  50° 

54° 

58° 

62° 

66° 

70° 

74° 

78° 

82° 

86° 

o 

1   II 

1  II 

/  II 

/  // 

1    n 

1  II   1  II 

/  // 

1  II 

/  II 

/  // 

/  II 

/  II 

J  II 

/  // 

/  // 

0 

6 

5  3 

5  22 

5  4i 

5  59 

6  36 

7  107  4o 

6 

7 

4  i5 

4  3i 

4  ^7 

5  2 

5  '66 

6  1,6  29 

7 

8 

3  40 

3  53 

4  6 

4  20 

4  46 

5  ii5  35 

5  58 

8 

9 

3  12 

3  24 

3  35 

3  47 

4  10 

4  3i'4  5i 

5  10 

9 

lO 

2  5o 

3  ol 

3  10 

3  20 

3  39 

3  58;4  17 

4  34 

10 

II 

2  33 

2  42 

2  52 

3  0 

3  17 

3  33  3  48 

4  3 

II 

12 

2  J9 

2  27 

2  36 

2  44 

2  59 

3  i3  3'26 

339 

3  5i 

12 

i3 

2  6 

2  i3 

2  21 

1   29 

2  43 

2  563  9 

3  20 

3  29 

i3 

i4 

I  55 

2   2- 

2   Q 

2  16 

2  29 

2  4i 2  53 

3  2 

3  10 

i4 

i5 

I  47 

I  53 

I  59 

2  5 

2  17 

2  28,2  38 

2  47 

2  54 

i5 

i6 

I  4c 

I  45 

I  5o 

I  56 

2  7 

2  17*2  26 

2  34 

2  4i 

2  47 

16 

I? 

I  34 

I  38 

I  43 

I  48 

I  58 

2  7215 

2  22 

2  29 

2  3b 

17 

i8 

I  29 

I  33 

I  37 

1  42 

I  5i 

I  59 

2  6 

2  12 

2  18 

2  24 

18 

19 

I  25 

I  28 

I  32 

I  36 

r  44 

I  52 

I  59 

2  4 

2  9 

2  i4 

19 

20 

I  22 

I  25 

I  28 

I  3i 

1   38 

I  46  I  52 

I  57 

2  1 

2  6 

2  II 

20 

21 

I  IQ 

I  22 

I  25 

I  27 

I  33 

I  4o'i  46 

I  5i 

I  55 

I  59 

2  2 

21 

22 

I  17 

I  19 

I  22 

I  24 

I  29 

I  35!i  4o 

I  45 

I  49 

I  53 

I  55 

22 

23 

I  i5 

I  17 

I  IQ 

I  21 

1  25 

I  3o:i  35 

I  4o 

I  44 

I  47 

I  49 

23 

24 

I  i4 

I  i5 

I  16 

I  18 

I  22 

I  26' I  3o 

r  35 

I  39 

I  42 

I  44 

I  46 

24 

25 

I  12 

I  i3 

I  i4 

i  16 

1  19 

I  22 

I  26 

I  3o 

I  34 

I  3- 

I  39 

I  4o 

25 

26 

I  10 

I  II 

I  12 

I  i4 

I  16 

I  IQ 

I  22 

I  26 

I  3o 

I  32 

I  34 

I  35 

26 

27 

I   9 

I  10 

I  II 

I  12 

I  i4 

I  16 

I  19 

1.23 

I  26 

I  28 

I  3o 

I  3i 

27 

28 

I  8 

I  9 

I  10 

I  II 

I  12 

I  i4 

I  17 

I  20 

I  22 

I  24 

I  26 

I  27 

I  28 

28 

29 

I  7 

I  8 

I  8 

I  9 

I  10 

I  12 

I  i5 

I  17 

I  19 

I  21 

I  22 

I  23 

I  25 

29 

3o 

I  6 

I  7 

I  7 

I  8 

I  9 

I  10 

I  12 

I  i4 

I  16 

I  18 

I  19 

I  20 

I  22 

3o 

3i 

I  6 

I  6 

I  6 

I  7 

I  8 

I  8 

I  10 

I  12 

I  14 

I  i5 

1  17 

I  18 

I  19 

3i 

32 

I  5 

I  6 

I  6 

I  6 

I   7 

I  7 

I  8 

I  10 

I  12 

I  i3 

I  i4 

I  i5 

I  16 

I  17 

32 

33 

I  5 

1     5 

I  5 

I  5 

I  6 

I  6 

I  7 

I  8 

I  9 

I  10 

I  II 

I  12 

I  i3 

I  i4 

33 

34 

I  5 

I  4 

I  4 

I  4 

I  5 

I  5 

I  6 

I  6 

I  7 

I  8 

I  9 

I  10 

I  II 

I  12 

M 

35 

I  5 

I  4 

I  4 

I  4 

I  4 

I  4 

I  5 

I  5 

I  5 

I  6 

I  7 

I  8 

I  9 

I  10 

35 

36 

I  5 

I  4 

I  3 

I  3 

I  3 

I  3 

I  4 

I  4 

I  4 

I  5 

I  5 

I  6 

I  7 

I  8- 

I  9 

36 

37 

I  6 

I  4 

I  3 

I  2 

I  2 

I  2 

I  3 

I  3 

I  3 

I  4 

I  4 

I  5 

I  5 

I  6 

I  7 

37 

38 

I  6 

I  4 

I  2 

I  I 

I  I 

I  2 

I  2 

I  2 

I  3 

I  3 

I  4 

I  4 

I  4 

I  5 

38 

39 

I  6 

I  4 

I  2 

I  0 

I  0 

I  I 

I  I 

I  I 

I  2 

I  2 

I  3 

I  3 

I  3 

I  4 

39 

4o 

I  6 

I  4 

I  2 

I  0 

I  0 

I  I 

I  I 

I  1 

I  I 

I  I 

I  I 

I  I 

I  I 

I  2 

I  3 

40 

4i 

I  5 

I  3 

I  0 

I  0 

I  0 

I  0 

I  0 

I  0 

I  0 

I  0 

I  0 

I  0 

I  0 

I   I 

4r 

42 

I  7 

I  5 

I  3 

59 

59 

59 

59 

59 

59 

59 

59 

59 

59 

59 

59 

42 

43 

I  7 

I  5 

I  3 

59 

b9 

58 

58 

58 

58 

58 

58 

58 

58 

58 

58 

43 

44 

I  7 

I  5 

I  3 

59 

58 

57 

57 

57 

57 

57 

57 

57 

57 

57 

57 

44 

46 

I  7 

I  5 

I  3 

59 

57 

56 

56 

56 

56 

55 

55 

55 

55 

55 

55 

46 

48 

I  8 

I  6 

t  4 

I  2 

59 

57 

55 

55 

55 

54 

54 

54 

53 

53 

53 

53 

48 

5o 

I  8 

I  6 

I  4 

I  2 

59 

57 

55 

54 

54 

53 

53 

53 

52 

52 

52 

52 

5o 

52 

I  9 

I  6, 

I  4 

I  2 

59 

56 

54 

53 

53 

52 

52 

5i 

5i 

5i 

5o 

5i 

52 

54 

I  10 

I  7 

I  4 

I  2 

59 

56 

54 

53 

52 

5i 

5i 

5o 

5o 

49 

49 

54 

56 

I  10 

I  7 

I  5 

I  2 

59 

56 

54 

52 

5i 

5o 

5o 

49 

49 

48 

4- 

56 

58 

I  II 

I  8 

I  5 

I  3 

59 

56 

53 

5i 

5o 

49 

49 

48 

48 

47 

58 

60 

I  II 

I  8 

I  5 

I  3 

5.9 

56 

53 

5i 

5o 

49 

48 

47 

47 

46 

60 

62 

I  12 

I  9 

I  6 

I  3 

59 

56 

53 

5i 

49 

48 

47 

47 

46 

62 

64 

I  i3 

I  9 

I  6 

I  3 

59 

56 

53 

5i 

49 

48 

47 

46 

45 

64 

66 

I  i4 

I  10 

I  7 

I  4 

59 

56 

53 

5i 

49 

48 

47 

46 

66 

68 

I  i5 

I  II 

I  7 

I  4 

59 

56 

53 

5i 

49 

47 

46 

45 

68 

70 

I  16 

I  II 

^  7 

I  4 

59 

55 

53 

5i 

49 

47 

46 

70 

72 

I  16 

I  12 

I  8 

I  4 

59 

55 

52 

5o 

48 

46 

45 

72 

74 

I  16 

I  12 

r  8 

I  4 

59 

55 

52 

5o 

48 

46 

74 

76 

r  17 

I  12 

I  8 

I  5 

59 

55 

52 

49 

47 

46 

76 

78 

I  17 

I  12 

I  8 

I  5 

59 

55 

52 

49 

47 

78 

80 

I  12 

I  8 

I  5 

59 

55 

52 

49 

47 

80 

82 

I  8 

I  5 

59 

55 

52 

49 

82 

84 

I  5 

59 

55 

52 

49 

84 

86 

59 

55 

52 

86 

32° 

34° 

36° 

38° 

49° 

46° 

50° 

54° 

58° 

62° 

GGP 

70° 

74° 

78° 

82° 

86° 

' : — — — - 

P^'gesssj               TABLE  XLVIII. 

Third  Correction.  Apparent  Distance  48°. 

5's 
App. 
Alt. 

.Apparent  Altitude  of  the  Sun,  Star  or  Planet. 

\pp. 

Alt. 

6^ 

7° 

8° 

9° 

10° 

11° 

12° 

14° 

1G° 

18° 

20° 

22° 

24° 

26° 

28° 

30° 

o 

1    II 

>  // 

/  » 

f  // 

/  // 

1  II 

/  // 

/  // 

/  II 

/  // 

/  // 

/  // 

1  II 

/  II 

/  // 

1  II 

0 

6 

r  t6 

I  17 

t  19 

I  23 

I  29 

I  36 

t  43 

2  I 

2   20 

I  39: 

58: 

16  3  35  3  54|4  i3|4  82 1 

6 

7 

I  19 

T  ?4 

I  16 

I  17 

I  19 

I  23 

I  28 

[  33 

I  46 

2  0 

2  16  2  32]; 

47  3  2: 

18  3  34: 

5o 

7 

8 

I  19 

I  16 

I  17 

I  19 

I  22 

I  26 

I  35 

I  47 

[  59 ; 

I   12  : 

25 

I   39  = 

53  3  73  21 

8 

9 

10 

T  3n 

I  23 

I  18 

I  16 

I  17 

I  19 

[  21 

I  28 

I  37 

I  47 

58: 

-  9 

2  20  5 

32  2  44  2  55 

9 

I  37 

I  27 

I  21 

I  18 

I  16 

I  17 

I  18 

I  23 

I  3o 

I  38 

47 

56 

2  b: 

16 

I   2b: 

36 

10 

II 

I  45 

[  33 

I  25 

I  21 

I  18 

I  16 

I  17 

I  20 

I  25 

I  02 

[  39 

47 

I  55  2  4\ 

2  i3  2  22 

II 

12 

T  53 

r  3o 

I  3o 

I  24 

I  21 

I  18 

I  16 

I  19 

I  22 

I  27 

[  33 

4o 

I  47 

54 

2  2210 

12 

1 3 

2  2 

I  46 

I  36 

I  28 

I  24 

I  20 

I  18 

I  17 

I  19 

I  23 

I  28 

34 

I  4o 

4b 

[  53  2  0 

i3 

i4 

2  II 

I  54 

1 .42 

I  33 

I  27 

I  23 

I  20 

I  16 

I  17 

I  20 

I  24 

29 

I  34 

39 

I  45 

5i 

i4 

i5 

2  20 

2  I 

I  48 

I  37 

I  3o 

I  26 

I  22 

I  17 

I  16 

I  18 

I  21 

[  24 

I  29 

[  33 

I  38 

[  43 

i5 

i6 

2  3u 

2   Q 

I  54 

I  42 

I  34 

I  29 

I  24 

I  18 

I  16 

I  17 

I  18 

[  20 

I  24 

t  28 

I  32 

I  37 

16 

I? 

9  40 

2  17 

2  0 

I  47 

I  38 

I  32 

I  27 

I  20 

I  17 

I  16 

I  17 

I  18 

I  21 

[  25 

I  28 

I  32 

17 

iR 

2  5o 

2  25 

2  7 

I  52 

I  42 

I  3b 

I  3o 

I  22 

I  18 

I  i5 

I  lb 

I  17 

I  19 

I  22 

I  25 

I  28 

18 

19 

3  n 

2  32 

2  i4 

I  58 

I  46 

I  39 

I  33 

I  24 

I  19 

I  16 

I  i5 

I  lb 

I  17 

I  19 

I  22 

I  24 

19 

20 

21 

3  9 

3  18 

2  4o 
2  48 

2  20 
2  26 

2  3 
2  9 

I  5i 
I  56 

I  43 
I  47 

I  36 
I  4o 

I  27 
I  3o 

I  21 

I  23 

I  17 
I  18 

I  14 
I  i5 

I  i5 
I  i4 

I  lb 

I  17 

I  1^ 

1  21 

20 
21 

I  i5 

I  16 

I  17 

I  19 

22 

3  on 

9  56 

2  33 

2  i5 

2  2 

I  52 

I  43 

I  32 

I  24 

I  19 

I  lb 

I  i3 

I  14 

I  i5 

I  lb 

I  18 

22 

23 

3  37 

3  3 

2  4o 

2  21 

2  7 

I  56 

I  46 

I  35 

I  26 

I  20 

I  16 

I  i4 

I  i3 

I  i4 

I  i5 

I  lb 

23 

0.4 

3  46  3  II 

2  47 

2  26 

2  12 

2  0 

I  5o 

I  37 

I  27 

I  21 

I  17 

I  i4 

I  12 

I  i3 

I  13 

I  i4 

24 

25 

3  56 

3  19 

2  54 

2  32 

2  17 

2  5 

I  54 

I  4o 

I  29 

I  22 

I  18 

I  i5 

I  i3 

I  12 

I  12 

I  i3 

25 

"^fT 

4  'i 

3  27 

3  I 

2  38 

2  22 

2  9 

I  58 

I  42 

I  3i 

I  24 

I  19 

I  16 

I  i3 

I  II 

I  II 

I  12 

26 

27 

4  i5 

3  34 

3  8 

2  M 

2  27 

2  i4 

2  2 

I  44 

I  6^ 

I  25 

I  20 

I  lb 

I  i3 

I  II 

I  10 

I  II 

27 

28 

4  ?4 

3  42 

3  i5 

2  5o 

2  32 

2  18 

2  b 

I  47 

I  35 

I  27 

I  21 

I  17 

I  14 

I  12 

I  10 

I  10 

28 

3o 

4  33 

3  5( 

3  21 

2  56 

2  37 

2  23 

2  10 

I  5o 

I  37 

I  28 

I  22 

I  18 

I  i5 

I  12 

I  10 

I  9 

29 

4  42 

3  5b 

3  28 

3  2 

2  42 

2  27 

2  i3 

I  53 

I  4o 

I  3o 

I  23 

I  19 

I  i5 

I  12 

1  9 

3o 

3i 

4  5i 

A    6 

3  35 

3  8 

2  47 

2  3i 

2  17 

I  57 

I  42 

I  32 

I  25 

I  20 

I  lb 

I  i3 

I  II 

I  9 

3i 

32 

5  0 

i  1 3 

3  4?- 

3  i4 

2  52 

2  35 

2  20 

2  0 

1  44 

I  33 

I  2b 

I  21 

I  lb 

I  i3 

I  II 

I  9 

32 

33 

5  9 

5  t8 

4  21 

3  49 

3  20 

2  57 

2  39 

2  23 

2  3 

I  46 

I  35 

I  27 

I  22 

I  17 

I  i4 

I  12 

I  10 

33 

34 

4  2& 

3  55i3  25 

3  2 

2  44 

2  27 

a     6 

I  49 

I  37 

I  28 

I  23 

I  18 

I  14 

I  12 

I  10 

34 

35 

5  27 

4  36  4  I 

3  3i 

3  7 

2  48 

2  3i 

2  9 

I  52 

I  39 

I  3o 

I  24 

I  19 

I  i5 

I  12 

I  10 

35 

36 

5  35 

4  43  4  8 

3  37 

3  12 

2  52 

2  35 

2  12 

I  54 

I  4i 

I  3i 

I  25 

I  19 

I  i5 

1  12 

I  10 

36 

37 

5  U 

4  5c 

4  14 

3  42 

3  17 

2  57 

2  39 

2  :6 

I  56 

I  4'S 

I  33 

I  26 

I  20 

I  lb 

I  i3 

I  II 

37 

38 

5  52 

4  5- 

4  20 

3  47 

3  22 

3  I 

2  Ai 

2  19 

I  59 

I   4b 

I  34 

I  27 

I  21 

I  17 

I  14 

I  11 

38 

39 

40 

6  0 

5  4 

4  26 

3  53 

3  26 

3  5 

2  47 

2  22 

2  2 

I  47 

I  35 

I  28 

I  22 

I  17 

I  14 

I  II 

39 

6  8 

5  II 

4  32 

3  58 

3  3o 

3  10 

2  5i 

2  25 

2  5 

I  49 

I  37 

I  29 

I  23 

I  18 

I  i5 

I  12 

40 

4i 

6  16 

-TIT 

4  38 

4  3 

3  35 

3  14 

2  55 

2  28 

2  7 

I  5i 

I  39 

I  3i 

I  24 

I  19 

I  16 

I  i3 

4i 

4? 

6  24 

5  2^ 

4  M 

4  8 

3  4o 

3  18 

2  58 

2  3i 

2  10 

I  53 

I  4i 

I  33 

I  2b 

I  20 

I  lb 

I  1 3 

42 

43 

6  32 

5  3t 

4  5o 

4  i3 

3  44 

3  22 

3  2 

2  6i 

2  12 

I  55 

I  43 

I  34 

I  27 

I  21 

I  17 

I  i4 

43 

44 

6  39 

5  3- 

4  55 

4  18 

3  48 

3  26 

3  5 

2  36 

2  i4 

I  57 

I  45 

I  36 

I  28 

I  22 

I  18 

I  i5 

44 

46 

6  53 

5  4r 

,5  5 

4  28 

3  56 

3  34 

3  12 

2  4i 

2  18 

2  1 

I  4^ 

I  38 

I  3o 

I  23 

I  18 

I  i5 

46 

48 

7  7 

6 

5  15 

4  37 

4  4 

3  4i 

3  18 

2  46 

2  22 

2  5 

I  5i 

I  4o 

I  3i 

I  24 

I  19 

I  16 

48 

5o 

7  21 

6  I,' 

5  25 

4  46 

4  12 

3  47 

3  24 

2  5i 

2  26 

2  8 

I  53 

I  42 

I  33 

I  25 

I  20 

I  lb 

5o 

59 

7  34 

6  3. 

i5  34 

4  54 

4  20 

3  53 

3  3o 

2  56 

2  3o 

2  12 

I  5b 

I  44 

I  35 

I  27 

t  22 

I  17 

52 

54 

7  47 

6  3- 

5  5  43 

5  I 

4  27 

3  59 

3  36 

3  I 

2  34 

2  i5 

I  59 

I  4b 

I  37 

I  29 

I  23 

'I  19 

54 

56 

8  0 

6  4( 

55  5i 

5  8 

4  34 

4  5 

3  4i 

3  6 

2  38 

2  18 

2  2 

I  49 

I  39 

I  3i 

I  25 

I  20 

56 

58 

5  59 

5~r5 

4  4o 
4  46 

4  II 
4  16 

3  46 
3  5o 
3  54 

3  10 
3  i3 
3  i5 

2  42 

2  45 
2  47 

2  21 
2  23 
2  25 

2  4 
2  6 
2  8 

I  5i 
I  53 
I  54 

I  41 
I  42 
I  43 

I  33 

I  33 
I  34 

I  26 

1  21 

58 
60 
62 

TabJc  P.  Effect  of  Sun's  Par. 

I  27 

I  28 

I  22 

AJ.I  tl>e  Nuiiibers  ;>ljo»e  ilie  lines 
to  Third  Correcliun  j  subtract 

3  17 

2  49 

2  27 

2  10 

I  56 

I  45 

I  35 

I  28 

1  22 

64 

the  0  liiTs. 

2  5l 

2  29 

2  3l 

2  II 
2  12 
2  i3 

I  57 
I  58 
I  59 

I  4b 

I  47 
I  48 

I  3b 

I  37 
I  38 

I  29 

I  3o 

I  3o 

I  23 

I  24 
I  24 

66 

68 

70 

Ayn. 
Alt. 

Sun's  Apipareiit  Allilutle. 

5 

lU  . 

U  3L'  -10 

50 

60  r 

0  80 

90 

" 

,  ,. 

" 

2  0 

I  49 

I  39 

I  3i 

I  24 

72 

5 

U 

3  4 

5 

I  5o 

I  4o 

I  32 

1  25 

74 

10 

1 

1 

y  3 

4 

3 

I  4i 

I  33 

I  25 

76 

20 

:h 

H 

0  1 

V 

3 

30 

fi 

.1 

^i   T 

0 

1 

I  33 

1  2b 

78 

40 

7 

6  . 

4  3 

i 

T 

]  0 

0 

I  2b 

80 

50 

9  8 

S  4 

3 

2 

2  1 

82 

60 

9 

6  .5 

4 

3 

84 

70 

9  7  6 

5 

4 

8b 

SO 
90 

1 

8  7 

7 

6 

10° 

11° 

12° 

U° 

16° 

18° 

20° 

22° 

24° 

26° 

28° 

30° 

TABLE  XLVill 

[I'ajjf  -Jti) 

Third  Correction.  Apparent  Distance  48°. 

D's 
A  pp. 
All. 

Jlpjmrent  Altkuilc  of  tlie  Sun,  Uti 

r  or 

Planet. 

D's 
AliC- 

.32° 

:m° 

3G° 

38° 

42° 

46° 

.^0° 

54° 

58° 

(i2° 

6(i° 

70° 

74° 

78° 

82° 

86" 

o 

1  II 

/  II 

/  // 

1  II 

/  II 

'  II 

1   II 

/  '/ 

/  // 

/  /; 

/  II 

/  II 

/  II 

/  // 

/  II 

/  // 

0 

6 

4  5i 

5  10 

5  28 

5  46 

6   18 

6  49,7  J9 

7  47 

6 

7 

4  6 

4  21 

4  36 

4  5i 

5  19 

5  45!6  II 

6  35 

7 

8 

3  34 

3  48 

4  I 

4  14 

4  38 

5  I 

5  22 

5  42 

6  I 

8 

9 

3  7 

3  19 

3  3o 

3  4i 

4  3 

4  24 

4  43 

5  0 

5  17 

9 

lO 

2  47 

2  57 

3  7 

3  17 

3  36 

3  54 

4  II 

4  26 

4  40 

10 

II 

2  3i 

2  4" 

2  49 

2  57 

3  i4 

3  3o 

3  44 

3  57 

4  w 

II 

12 

2  17 

2  25 

2  -63 

2  4o 

2  55 

3  9 

3  22 

3  34 

3  45 

3  55 

12 

i3 

2  6 

2  i3 

2  20 

2  27 

2  40 

2  52 

3  4 

3  i5 

3  25 

3  32 

i3 

i4 

I  57 

2  4 

2  10 

2  16 

2  27 

2  38 

2  49 

2  59 

3  8 

3  i5 

i4 

i5 

I  49 

I  55 

2  I 

2  b 

2  16 

2  26 

2  35 

2  44 

2  53 

3  0 

i5 

i6 

I  42 

I  47 

I  52 

I  57 

2  7 

2  i5 

2  23 

2  32 

2  4() 

2  46 

2  52 

16 

17 

I  36 

1  4i 

I  45 

I  5o 

I  59 

2  6 

2  i4 

2  22 

2  29 

2  34 

2  4o 

17 

i8 

I  3i 

I  35 

I  39 

1  43 

I  5i 

I  59 

2  6 

2  i3 

2  19 

2  24 

2  29 

18 

iQ 

I  27 

I  3i 

I  34 

I  38 

I  45 

I  52 

I  58 

2  4 

2  10 

2  i5 

2  19 

19 

20 

I  24 

I  27 

I  3o 

I  33 

I  39 

I  45 

I  5i 

I  57 

2  2 

2  7 

2  II 

2  i5 

20 

21 

I  22 

I  24 

I  27 

I  29 

I  34 

I  40 

I  45 

I  5i 

I  56 

2  0 

2  4 

2  7 

21 

22 

I  20 

I  22 

I  24 

I  26 

I  3o 

I  35 

I  40 

1  45 

1  5o 

.  54 

I  57 

I  59 

22 

23 

I  18 

I  19 

I  21 

I  23 

I  27 

I  3i 

I  36 

I  40 

I  45 

I  49 

I  5i 

I  53 

23 

24 

I  16 

I  17 

I  19 

I  21 

I  25 

I  28 

I    32 

I  36 

I  40 

I  44 

I  46 

I  48 

I  5o 

24 

25 
26 

I  i4 
I  12 

I  i5 
I  i3 

I  16 

I  i4 

1  18 
I  16 

I  22 
I  19 

I  25 
I  23 

I  29 
I  26 

I  32 

I  29 

I  36 

I  32 

I  39 

I  34 

I  4i 
I  36 

I  43 
1  38 

I  45 

25 

I  4o 

26 

27 

I  11 

I  12 

I  i3 

I  i4 

I  17 

I  20 

I  23 

I  26 

I  28 

I  3o 

I  32 

I  34 

I  36 

27 

28 

I  10 

I  II 

I  12 

I  1 3 

I  i5 

I  18 

I  21) 

I  23 

I  25 

I  27 

I  28 

I  3c) 

I  32 

I  34 

28 

29 

I  9 

I  10 

I  II 

I  12 

I  i4 

I  16 

I  18 

1  20 

I  22 

I  24 

I  25 

I  27 

I  28 

1  3o 

29 

3o 

I  9 

I  10 

I  10 

I  11 

I  12 

I  i4 

I  16 

I  18 

I  19 

I  21 

I  22 

I  24 

I  25 

I  26 

3o 

3i 

I  9 

I  9 

I  9 

I  10 

I  II 

I  12 

I  i4 

I  16 

I  17 

I  19 

I  20 

I  21 

I  22 

I  23 

3i 

32 

I  8 

I  8 

I  8 

I  9 

I  10 

I  II 

I  i3 

I  i4 

I  i5 

i   17 

I  18 

I  19 

I  19 

I  20 

I  21 

32 

33 

I  8 

I  7 

I  7 

I  8 

I  9 

I  10 

I  II 

I  12 

I  i3 

I  i5 

I  16 

I  17 

I  17 

I  17 

I  18 

33 

34 

r  8 

I  6 

I  6 

I  7 

I  8 

I  9 

I  10 

I  ]  I 

I  12 

I  i3 

I  i4 

I  i4 

I  l5 

I  i5 

I  16 

34 

35 

I  8 

I  6 

I  5 

I  6 

I  7 

I  8 

I  9 

I  9 

I  10 

I  II 

I  12 

I  12 

I  i3 

I  i3 

I  14 

35 

36 

I  8 

I  6 

I  5 

I  5 

I  5 

I  6 

I  7 

I  7 

I  8 

I  9 

I  10 

I  10 

I  II 

I  II 

I  ^2 

I  i3 

36 

37 

I  9 

I  7 

I  5 

I  4 

I  4 

I  5 

I  (J 

I  6 

I  7 

I  7 

I  8 

I  8 

I  9 

1  9 

I  10 

I  II 

37 

33 

I  9 

I  7 

I  5 

I  3 

I  3 

I  4 

I     5 

I  5 

I  6 

I  6 

I  7 

I   7 

I  8 

I  8 

I  8 

I   9 

38 

39 

I  9 

I  7 

I  5 

I  3 

I  3 

I  3 

I  4 

I  4 

I  5 

I  5 

I  6 

I  6 

I  6 

I  7 

I  7 

I  7 

39 

4o 

I  9 

I  7 

I  5 

I  3 

I  2 

I  2 

I  3 

I  3 

I  4 

I  4 

I  5 

I  5 

I  5 

I  6 

I  6 

I  6 

4o 

4 1 

I  10 

I  8 

I  5 

I  3 

I  I 

I  2 

I  2 

I  3 

I  3 

I  4 

I  4 

I  4 

I  5 

I  5 

I  5 

4i 

42 

I  10 

I  8 

I  5 

I  3 

I  I 

I  I 

I  2 

I  2 

I  2 

I  3 

I  3 

I  3 

I  4 

I  4 

I  4 

42 

43 

I  II 

I  8 

I  6 

I  4 

I  0 

I  0 

I  I 

I  I 

I  I 

I  2 

I  2 

I  2 

I  3 

I  3 

I  3 

43 

44 

I  12 

I  9 

I  6 

I  4 

I  0 

I  0 

I  0 

I  0 

I  0 

I  I 

I  I 

I  I 

I  I 

I  I 

I  I 

44 

4) 

I  12 

I  9 

I  6 

I  4 

59 

59 

59 

59 

59 

59 

59 

59 

59 

59 

59 

46 

48 

I  i3 

I  10 

f  7 

I  4 

59 

58 

58 

58 

58 

57 

57 

57 

57 

57 

57 

48 

5o 

I  i3 

I  10 

I  7 

I  5 

59 

57 

57 

57 

57 

56 

56 

56 

56 

56 

5o 

52 

I  i4 

I  II 

I  8 

I  5 

59 

57 

56" 

56 

56 

55 

55 

54 

54 

54 

52 

54 

I  i5 

I  II 

I  8 

I  6 

I  2 

59 

57 

56 

55 

55 

54 

54 

53 

53 

54 

56 

I  i5 

I  II 

I  8 

I  6 

I  2 

59 

57 

55 

54 

54 

53 

53 

52 

52 

56 

58 

I  16 

I  12 

I  9 

I  6 

I  2 

59 

57 

55 

54 

53 

52 

52 

5i 

58 

6<> 

I  16 

I  12 

I  9 

I  6 

I  2 

59 

57 

55 

53 

52 

52 

5i 

5o 

60 

62 

I  17 

I  i3 

I  10 

I  7 

I  2 

59 

57 

55 

53 

52 

5i 

5i 

62 

64 

I  17 

1  1 3 

1  10 

I  7 

I  2 

59 

57 

55 

53 

52 

5i 

5o 

64 

66 

I  18 

I  i4 

I  10 

I  7 

I  3 

59 

57 

54 

52 

5i 

5o 

66 

68 

I  18 

I  i4 

I  10 

I  7 

I    J 

59 

56 

54 

52 

5i 

5o 

68 

70 

I  19 

I  i5 

I  II 

I  8 

I  3 

59 

56 

54 

52 

5i 

70 

72 

I  19 

I  i5 

I  II 

I  8 

I  3 

59 

56 

54 

52 

5o 

72 

74 

I  20 

I  i5 

I  1 1 

I  8 

I  3 

59 

56 

53 

5i 

74 

76 
78 

I  20 
I  21 

I  16 

I  ifi 

I  12 
I  12 

I  8 
•  9 

I  3 

59 

56 

53 

5i 

76 
78 

I  4 

59 

56 

53 

80 

I  21 

I  16 

I  12 

I  9 

I  4 

59 

56 

53 

80 

83 

I  21 

I  16 

I  12 

I  9 

I  4 

59 

56 

82 

84 

I  16 

I  12 

I  9 

I  4 

59 

56 

84 

86 

I  12 

I  9 

I  4 

59 

86 

300 

.34° 

3G° 

38° 

42° 

46° 

50° 

54° 

58° 

62° 

60" 

70° 

74° 

78° 

82° 

86° 

37 


1 

I'age290]                      TABLE  XLVIIl 

1 

Third  Correction 

Apparent  Distance 

52°. 

1 

App. 

Jl 

pparcnt  Altitude  of  the  Sun,  Sta 

r  or 

Planet. 

D  's 
App. 

Alt. 

d" 

70 

8^ 

y^ 

10° 

11" 

12" 

14" 

IG" 

lb" 

20" 

22° 

24" 

2(i" 

28" 

yo° 

Alt. 

o 

1   II 

1     •! 

/  \i 

/  // 

1   II 

/  II 

/  ;/ 

1  II 

/  II 

/  .'/ 

/  // 

/  // 

/  // 

1  II 

/  // 

1  It 

0 

6 

I   18 

I    19 

t  21 

I  24 

I  3o 

I  37 

I  44 

2  0 

2  17 

2  34 

2  5i 

3  10 

3  28 

3  4- 

4  6 

4  24 

6 

7 

I  21 

I  18 

I  19 

I  21 

I  24 

I  29 

I  34 

I  46 

2  0 

2  i4 

2  28 

2  42 

2  57 

3   12 

3  27 

3  43 

7 

8 

I  25 

I  21 

I  18 

I  19 

I  21 

I  24 

I  27 

I  36 

I  47 

I  58 

2  II 

2  23 

2  36 

2  5o 

3  3 

3  16 

8 

9 

I  3o 

I  24 

I  20 

I  18 

I  19 

I  21 

I  23 

t  29 

I  3- 

I  47 

I  57 

2  8 

2  19 

2  3i 

2  42 

2  53 

9 

lO 

I  37 

I  28 

I  23 

I  20 

I  18 

I  19 

I  21 

I  2D 

I  3o 

I  38 

I  46 

I  56 

2  6 

2  16 

2  26 

3  36 

Id 

1 1 

I  45 

I  34 

I  28 

I  23 

I  20 

I  18 

I  19 

I  22 

I  26 

I  32 

I  39 

I  47 

I  56 

2  4 

2  i3 

2  22 

11 

12 

r  54 

I  4i 

I  33 

I  27 

I.  22 

I  20 

I  18 

I  20 

I  23 

I  27 

I  33 

I  40 

I  47 

I  54 

2  2 

2  10 

12 

i3 

2  2 

I  48 

I  38 

I  3. 

I  25 

I  22 

I  19 

I  19 

I  21 

I  24 

I  29 

I  35 

I  4i 

I  47 

I  54 

2  I 

i3 

i4 

2  II 

I  55 

I  44 

I  35 

I  28 

I  24 

I  21 

I  18 

I  19 

I  23 

I  26 

I  3o 

I  35 

I  4i 

I  47 

I  52 

i4 

i5 

2  19 

2  2 

I  5o 

I  39 

I  32 

I  27 

I  23 

I  19 

I  18 

I  20 

I  23 

I  26 

r  3o 

I  35 

I  40 

I  44 

i5 

i6 

2  28 

2  q 

I  55 

I  44 

I  35 

I  3o 

I  25 

I  20 

I  17 

I  18 

I  20 

I  23 

I  26 

I  3o 

I  34 

I  38 

16 

I? 

2  37 

2  16 

2  0 

I  48 

I  39 

I  33 

I  27 

I  21 

I  18 

I  17 

I  18 

I  20 

I  23 

I  26 

I  3o 

I  33 

17 

i8 

2  46 

2  23 

2  6 

I  53 

I  43 

I  36 

I  3o 

I  23 

I  19 

I  16 

I  17 

I  18 

I  20 

I  23 

I  26 

I  29 

18 

'9 

2  56 

2  3o 

2  12 

I  59 

I  48 

I  4o 

I  33 

I  25 

I  20 

I  17 

I  16 

I  17 

I  18 

I  20 

I  23 

I  26 

J9 

20 

3  5 

2  37 

2  18 

2  4 

I  52 

I  44 

I  37 

I  27 

I  22 

I  18 

I  i5 

I  16 

I  17 

I  18 

I  20 

I  23 

20 

21 

3  i4 

2  44 

2  24 

2  9 

I  57 

1   48 

I  4o 

I  29 

I  23 

I  19 

I  16 

I  16 

I  16 

I  17 

I  18 

I  20 

21 

22 

3  23 

2  52 

2  3i 

2  i5 

2  I 

I  52 

I  44 

I  32 

I  2b 

I  20 

I  16 

I  i5 

I  i5 

I  lb 

I  17 

I  18 

22 

23 

3  32 

2  59 

2  38 

2  20 

2  6 

I  56 

I  47 

1 34 

I  26 

I  21 

I  17 

I  i5 

I  i4 

I  i5 

I  16 

I  17 

23 

24 

3  4i 

3  7 

2  44 

2  26 

2  II 

2  0 

I  5i 

1 37 

I  28 

I  23 

I  18 

I  i5 

I  i4 

I  i4 

I  i5 

I  16 

24 

2.5 

3  5o 

3  i4 

2  5i 

2  3i 

2  16 

2  4 

I  54 

I  4o 

I  3o 

I  23 

I  19 

I  lb 

I  14 

I  i3 

I  i4 

I  i5 

25 

26 

3  5q 

3  22 

2  58 

2  37 

2  21 

2  8 

I  58 

I  42 

I  32 

I  25 

I  20 

I  16 

I  i4 

I  i3 

I  i3 

1  i4 

26 

27 

4  8 

3  3o 

3  5 

2  42 

2  26 

2  12 

2  2 

I  45 

I  33 

I  26 

I  21 

I  17 

I  i5 

I  14 

I  i3 

I  i3 

27 

28 

4  17 

3  38 

3  12 

2  48 

2  3i 

2  16 

2  6 

I  48 

I  35 

I  28 

I  22 

I  18 

I  i5 

I  14 

I  i3 

I  i3 

28 

2Q 

4  26 

3  45 

3  19 

2  53 

2  36 

2  21 

2  10 

I  5i 

I  37 

I  29 

I  23 

I  19 

I  lb 

1  14 

I  i3 

I  12 

29 

3o 

4  34 

3  53 

3  25 

2  59 

2  4i 

2  25 

2  i3 

I  54 

I  39 

I  3i 

I  24 

I  19 

I  lb 

I  i4 

I  i3 

I  J2 

3o 

3 1 

4  43 

4  0 

3  32 

3  5 

2  45 

2  29 

2  17 

I  57 

I  4i 

I  32 

I  25 

1  20 

I  17 

I  i5 

I  i3 

I  12 

3i 

32 

4  52 

4  8 

3  38 

3  10 

2  5o 

2  34 

2  20 

I  59 

I  43 

I  34 

I  27 

I  21 

I  17 

I  i5 

I  i3 

I  12 

32 

33 

5  0 

4  i5 

3  44 

3  16 

2  55 

2  38 

2  24 

2  2 

I  45 

I   36 

I  29 

I  23 

I  18 

I  i5 

I  i3 

I  12 

33 

34 

5  9 

4  22 

3  5o 

3  21 

2  59 

2  42 

2  27 

2  5 

I  48 

I  38 

I  3o 

I  24 

I  19 

I  16 

I  i4 

I  12 

34 

35 

5  17 

4  29 

3  56 

3  27 

3  4 

2  46 

2  3i 

2  7 

I  5i 

I  40 

I  32 

I  25 

I  20 

I  17 

I  i4 

I  12 

35 

36 

5  26 

4  36 

4  2 

3  32 

3  9 

2  5o 

2  34 

2  10 

I  53 

I  42 

I  33 

I  26 

I  21 

I  17 

I  i4 

I  12 

36 

37 

5  34 

4  42 

4  8 

3  37 

3  i4 

2  54 

2  38 

2  i3 

I  56 

I  44 

I  34 

I  27 

I  22 

I  18 

1  i5 

I  l3 

37 

38 

5  42 

4  49 

4  i3 

3  42 

3  18 

2  58 

2  42 

2  16 

I  58 

I  46 

I  36 

I  28 

I  22 

I  18 

I  i5 

I  l3 

38 

3q 

5  5o 

4  56 

4  19 

3  47 

3  23 

3  2 

2  46 

2  19 

2  I 

I  48 

I  38 

I  3o 

I  23 

I  18 

I  i5 

I  i3 

39 

4o 

5  58 

5  3 

4  24 

3  52 

3  27 

3  6 

2  49 

2  22 

2  3 

I  5o 

I  39 

I  3i 

I  25 

I  19 

I  lb 

I  14 

4o 

4i 

6  6 

5  9 

4  3o 

3  57 

3  32 

3  10 

2  53 

2  25 

2  6 

I  52 

I  4i 

I  32 

I  26 

I  20 

I  16 

I  14 

4i 

42 

6  i4 

5  i5 

4  35 

4  2 

3  36 

3  i4 

2  56 

2  28 

2  8 

I  54 

I  42 

I  34 

I  27 

I  21 

I  17 

I  i5 

42 

43 

6  21 

5  21 

4  4i 

4  7 

3  40 

3  18 

3  0 

2  3i 

2  1 1 

I  56 

I  44 

I  35 

I  28 

I  22 

I  18 

I  i5 

43 

44 

6  28 

5  27 

4  46 

4  12 

3  44 

3  22 

3  3 

2  34 

2  i3 

I  58 

I  45 

I  37 

I  29 

I  23 

I  19 

I  16 

44 

46 

6  42 

5  39 

4  56 

4  21 

3  52 

3  29 

3  10 

2  39 

2  18 

2  I 

I  48 

I  39 

I  3i 

I  24 

I  20 

I  17 

46 

48 

6  55 

5  5i 

5  6 

4  3o 

3  5q 

3  36 

3  16 

2  44 

2  22 

2  5 

I  5i 

I  4i 

I  33 

I  26 

I  21 

I  18 

48 

5o 

7  8 

6  2 

5  16 

4  38 

4  7 

3  43 

3  23 

2  49 

2  26 

2  8 

I  54 

I  43 

I  35 

I  27 

I  22 

I  19 

5o 

52 

7  21 

6  i3 

5  25 

4  46 

4  i5 

3  5o 

3  29 

2  54 

2  3o 

2  II 

I  57 

I  45 

I  3b 

I  29 

I  24 

I  20 

52 

54 

7  33 

6  23 

5  34 

4  53 

4  22 

3  56 

3  35 

2  59 

2  34 

2  i4 

2  0 

I  48 

I  38 

I  3i 

I  25 

I  21 

54 

56 

7  44 

6  33 

5  43 

4  59 

4  29 

4  2 

3  4o 

3  4 

2  38 

2  17 

2  2 

I  5o 

I  4o 

I  32 

I  20 

I  22 

56 

58 

7  £3 

6  42 

5  5o 

5  6 

4  35 

4  7 

3  45 

3  8 

2  42 

2  20 

2  5 

I  53 

I  42 

I  33 

I  27 

I  23 

58 

6o 

8  2 

6  4q 

5  56 

5  12 

4  4o 

4  12 

3  5o 

3  12 

2  46 

2  23 

2  7 

I  55 

I  44 

I  35 

I  29 

I  24 

60 

62 

6  2 

5  17 

4  45 

4  16 

3  54 

3  i5 

2  49 

2  26 

2  9 

I  57 

I   46 

I  36 

I  3o 

1  25 

62 

64 

4  5o 

4  20 

3  58 

3  18 

2  5i 

2  28 

2  II 

I  59 

I  48 

I  37 

I  3i 

t  26 

64 

66 

4  I 

3  20 

2  53 

2  3o 

2  i3 

2  0 

I  49 

I  39 

I  32 

I  26 

66 

68 

3  22 

2  54 

2  32 

2  i5 

2  I 

I  5o 

I  4o 

I  33 

I  27 

68 

70 

2  55 

2  33 

2  16 

2  2 

I  5i 

I  4i 

I  34 

I  28 

70 

72 

2  34 

2  17 

2  3 

I  52 

I  42 

I  34 

I  28 

72 

74 

2  18 

2  4 

I  53 

I  43 

I  35 

I  29 

74 

76 

2  5 

I  54 

I  44 

I  36 

I  29 

76 

78 

I  55 

I  44 

I  36 

I  3o 

78 

80 

I  45 

I  37 

I  3o 

80 

89 

I  38 

I  3o 

82 

84 

I  3i 

84 

86 

10° 

11° 

12° 

14° 

1G° 

18° 

20° 

22° 

24° 

26° 

28° 

30° 

86 

6° 

T 

8° 

9° 

TABLE  XLVIII.              [PasoSQi 

Third  Correction.  Apparent  Distance  52°. 

D's 
A  pp. 

Apparent  Jlltitadc  of  the  Sun,  Star  or  Planet. 

D's 
App. 

Alt. 

32° 

34^^ 

36^^ 

38^ 

42° 

4G^ 

5U° 

54^ 

5b° 

62" 

m° 

70° 

74° 

78° 

82° 

86° 

Ah. 

0 

/  II 

1  II 

1  II 

/  // 

/  II 

/  // 

/  II 

/  // 

/  // 

/  /; 

1  II 

/  // 

/  // 

/  // 

/  // 

1  II 

0 

6 

4  43 

5  I 

5  18 

5  34 

6  6 

6  36 

7    4 

7  29 

7  53 

6 

7 

3  59 

4  14 

4  29 

4  43 

5  9 

5  M 

5  58  6  20 

6  42 

7 

8 

3  3o 

3  4^ 

3  55 

4  8 

4  3o 

4  52 

5  i3 

5  32 

5  5o 

6  6 

8 

9 

3  4 

3  i5 

3  26 

3  37 

3  58 

4  17 

4  36 

4  5i 

5  5 

5  18 

9 

lO 

2  45 

2  54 

3  4 

3  i4 

3  32 
3  II 

3  48 
3  26 

4  4 
3  4o 

4  20 
3  54 

4  33 

4  45 

10 
II 

II 

2  3o 

2  38 

2  4i 

2  55 

4  6 

4  16 

12 

2  17 

2  25 

1   32 

2  4o 

2  54 

3  7 

3  20 

3  32 

3  43 

3  52 

4  I 

12 

i3 

2  7 

2  i3 

2  20 

2  26 

2  39 

2  5i 

3  3 

3  i4 

3  24 

3  32 

3  38 

i3 

i4 

I  58 

2  3 

2  9 

2  14 

2  26 

2  37 

2  48 

2  58 

3  7 

3  i4 

3  20 

i4 

i5 

I  49 

I  54 

I  59 

2  4 

2  i5 

2  26 

2  35 

2  44 

2  52 

2  59 

3  5 

i5 

]6 

I  42 

I  47 

I  5i 

I  56 

2  7 

2  16 

2  24 

2  32 

2  4o 

2  46 

2  52 

2  57 

16 

17 

I  37 

I  4i 

I  45 

I  5o 

2  0 

2  8 

2  i5 

2  22 

2  29 

2  35 

2  4o 

2  44 

17 

i8 

I  32 

I  36 

I  40 

I  45 

I  53 

2  0 

2  7 

2  i3 

2  19 

2  25 

2  3o 

2  33 

18 

19 

I  29 

I  32 

I  36 

I  4o 

I  47 

I  53 

2  0 

2  6 

2  II 

2  16 

2  21 

2  24 

19 

20 

I  26 

I  29 

I  32 

I  35 

I  4i 

I  47 

I  53 

I  59 

2  4 

2  9 

2  i3 

2  16 

2  19 

20 

21 

I  23 

I  26 

I  28 

I  3i 

I  37 

I  42 

I  47 

I  53 

I  58 

2  2 

2  6 

2  9 

2  II 

21 

22 

1  21 

I  23 

I  25 

I  28 

I  ZZ 

I  37 

I  42 

I  47 

I  52 

I  56 

I  59 

2  2 

2  4 

22 

23 

I  19 

I  21 

I  23 

I  25 

I  29 

I  33 

I  38 

I  42 

I  47 

I  5i 

I  54 

I  56 

I  58 

23 

24 

I  17 

I  19 

I  21 

I  23 

I  26 

I  3o 

I  34 

I  38 

I  42 

I  46 

I  49 

I  5i 

I  53 

I  55 

24 

25 

I  16 

I  17 

I  19 

I  20 

I  23 

I  27 

I  3o 

I  34 

I  37 

I  4i 

I  44 

I  46 

I  48 

I  49 

25 

26 

I  i5 

I  16 

I  17 

I  18 

I  21 

I  24 

I  27 

I  3o 

I  33 

I  36 

I  39 

1  4i 

I  43 

I  44 

26 

27 

I  i4 

I  i5 

I  16 

I  17 

I  19 

I  22 

I  24 

I  27 

I  3o 

I  32 

I  35 

I  37 

I  39 

I  4o 

27 

28 

I  i3 

I  i4 

I  i5 

I  16 

I  17 

I  20 

I  22 

I  24 

I  27 

I  29 

I  3i 

I  33 

I  35 

I  36 

I  37 

28 

29 

I  12 

I  i3 

I  14 

I  i5 

I  16 

I  18 

I  20 

I  22 

I  24 

I  2b 

I  28 

I  3o 

I  3i 

I  32 

I  33 

29 

3o 
3i 

I  12 
I  II 

I  12 
r  II 

I  i3 
I  12 

I  i3 
1  12 

1  14 
I  i3 

I  16 
I  i5 

I  18 
I  16 

I  20 
I  18 

I  22 

I  24 

I  25 

I  27 

I  28 

I  25 

I  29 
I  26 

I  3o 

I  27 

3o 
3i 

I  20 

I  22 

I  23 

I  24 

32 

I  II 

I  II 

I  II 

I  II 

I  12 

I  i4 

I  13 

I  16 

I  18 

I  20 

I  21 

I  22 

I  23 

I  23 

I  24 

I  25 

32 

33 

I  II 

I  10 

I  10 

I  10 

I  II 

I  i3 

I  i4 

I  i5 

I  17 

I  18 

I  19 

I  20 

I  21 

I  21 

I  22 

I  22 

33 

M 

I  II 

I  10 

I  10 

I  10 

I  II 

I  12 

I  i3 

I  i4 

I  ]6 

I  17 

I  17 

I  18 

I  19 

I  19 

I  20 

I  20 

34 

3b 

I  11 

I  10 

I  10 

I  10 

I  10 

I  II 

I  12 

I  i3 

I  14 

I  i5 

I  i5 

I  16 

I  17 

I  17 

I  18 

I  18 

35 

36 

I  II 

I  10 

I  9 

I  9 

I  9 

I  10 

I  II 

I  11 

I  12 

I  i3 

I  i3 

I  14 

I  i5 

t  i5 

I  16 

I  16 

36 

-il 

I  II 

I  10 

I  9 

I  9 

I  9 

I  10 

I  10 

I  II 

I  II 

I  12 

I  12 

I  i3 

I  i3 

I  i4 

1  i4 

37 

3d 

I  II 

I  10 

I  9 

I  8 

I  8 

I  9 

I  9 

I  9 

I  10 

I  10 

I  II 

III 

I  1 1 

I  II 

I  12 

I  12 

38 

39 

I  1 1 

I  10 

I  9 

I  8 

I  8 

I  8 

I  8 

I  8 

I  9 

I  9 

I  10 

I  10 

I  10 

I  10 

I  10 

I  10 

39 

4o 

I  12 

I  10 

I  9 

I  8 

I  7 

I  7 

I  7 

I  7 

I  8 

I  8 

I  9 

I  9 

I  9 

I  9 

I  9 

I  9 

4o 

4i 

I  12 

I  II 

I  9 

I  8 

I  7 

I  7 

:  7 

I  7 

I  7 

I  7 

I  8 

I  8 

I  8 

I  8 

I  8 

I  8 

4i 

42 

I  i3 

I  II 

I  9 

I  8 

I  6 

I  6 

I  6 

I  6 

I  6 

I  6 

I  7 

I  7 

I  7 

I  7 

I  7 

t  7 

42 

43 

X  i3 

I  II 

I  9 

I  8 

I  5 

I  6 

I  6 

I  6 

I  6 

I  6 

I  6 

I  6 

I  6 

I  6 

I  6 

I  6 

43 

44 

I  i4 

I  II 

I  9 

E   8 

I  6 

I  5 

I  5 

I  5 

I  5 

I  5 

I  5 

I  5 

I  5 

I  5 

I  5 

I  5 

44 

46 

I  i4 

I  12 

I  10 

I  9 

I  6 

I  4 

I  4 

I  4 

I  4 

I  4 

I  4 

I  3 

I  3 

I  3 

I  3 

46 

48 

I  i5 

I  i3 

I  II 

I  9 

I  6 

I  4 

I  3 

I  3 

I  3 

I  2 

I  2 

I  I 

I  I 

I  I 

I  I 

48 

5o 

I  lb 

I  i4 

I  II 

I  9 

I  6 

I  4 

I  2 

I  2 

I  2 

I  I 

I  1 

t  0 

I  0 

I  0 

5o 

52 

I  17 

I  i5 

I  12 

I  9 

I  6 

I  4 

I  2 

I  I 

I  I 

I  0 

I  0 

59 

58 

58 

52 

54 

I  18 

I  i5 

I  12 

I  9 

I  6 

I  4 

I  2 

I  I 

I  0 

59 

59 

58 

57 

54 

:jo 

I  18 

I  i5 

I  12 

I  10 

I  6 

I  4 

I  2 

I  0 

59 

58 

58 

57 

56 

56 

58 
Ho 

I  19 

!  16 
I  16 

I  i3 
I  t3 

I  10 

I  6 

I  4 
T  A 

I  2 

I  0 

59 

58 

58 
5-7 

57 
56 

56 
55 

1 

62 
04 

I  21 
I  22 

I  17 
I  18 

I  i3 

I  i4 

I  10 
I  II 

I  7 
I  7 

I    4 
I  4 

I  I 
I  I 

59 

58 
57 

56 
56 

55 
54 

Table  P.  Effect  of  Sun  3  Par. 
Add  the  Numbers  fibcve  (he  lines 
to  Third  Correction  ;  sutjtract 

(),S 

I  22 

I  18 

I  i4 

I  II 

I  7 

I  7 

I  7 

T  4 

I  J 

59 
58 

57 
56 

55 

the  Cillers. 

68 

I  22 

r  18 

I  i^ 

I  II 

I  ^ 

54 

App. 
Alt. 

Sun's  Apparent  Altitude.   i 

70 

I  23 

I  18 

I  14 

I  u 

I  3 

I  0 

58 

56 

5  10  2e 

30'40L 

0  60 

™ 

30  90  1 

72 

I  23 

I  19 

I  i5 

I  II 

I  7 

I  3 

I  0 

57 

55 

"  "  ' 

" 

" 

" 

74 

I  24 

I  19 

I  i5 

I  II 

I  7 

I  3 

I  0 

57 

5 

0  1  2 

3 

4 

4 

-6 

I  24 

I  19 

I  t5 

I  12 

I  7 

I  3 

I  0 

56 

10 

T  1  1 

•2 

3 

i    4 

20 

3  3  1 

0 

1 

i    '2 

3 

78 

I  24 

I  19 

I  i5 

I  12 

I  7 

I  3, 

I  0 

30 

5  4  3 

"2 

T 

0  n 

I 

80 

I  24 

I  19 

I  i5 

I  12 

I  7 

I  3 

I  0 

40 

7  6  5 

4 

2  T 

T 

0 

0 

82 

1  25 

I  20 

I  16 

I  12 

I  7 

I  3 

50 

8  8  6 

5 

4 

3  3 

2 

2 

84 

1  25 

I  20 

I  lb 

I  12 

I  7 

I  3 

60 

9  7 

6 

5 

i  i 

3 

80 

.  25 

I  21 

I  16 

I  12 

I  7 

70 

8 

7 

6 

5  5 

32°  .34°  1 

36° 

38° 

42° 

46° 

50°  54° 

58° 

62° 

66° 

70° 

80 

90 

8 

7 

7 

S 

PageOiK]               TABLE  XLVm 

Third  Correction 

Apparent  Distance  5G°. 

App. 

Apparent  Altiiude  of  the  Sun,  Star  or 

Planet. 

5  's 
App. 

Alt. 

6" 

r 

b" 

y^ 

10" 

11" 

12" 

14" 

16" 

18" 

20" 

22° 

24" 

20" 

28" 

yo° 

Alt. 

o 

II 

1  II 

?  II 

/  // 

/  // 

1  II 

/  // 

/  // 

/  // 

1   II 

1  II 

/  // 

/  // 

/  // 

/  // 

1  II 

0 

6 

I  20 

I  22 

I  25 

I  29 

I  35 

I  4i 

I  48 

2  2 

2  18 

2  35 

2  52 

3  10 

3  27 

3  45 

4  3 

A  20 

6 

7 

I  23 

I  20 

I  22 

I  24 

I  27 

I  32 

I  37 

I  48 

2  I 

2  i5 

2  29 

2  43 

2  58 

3  12 

3  27 

3  42 

7 

8 

r  28 

1  23 

I  20 

I  21 

I  23 

I  26 

I  29 

I  38 

I  48 

2  0 

2  12 

2  23 

2  35 

2  48 

3  I 

3  i4 

8 

9 

I  34 

I  27 

I  22 

I  20 

I  21 

I  23 

I  25 

I  3i 

I  39 

I  48 

I  58 

2  8 

2  18 

2  29 

2  40 

2  5o 

9 

10 

I  4<> 

I  3i 

I  25 

I  22 

I  20 

I  21 

I  22 

I  26 

I  32 

I  39 

I  48 

I  56 

2  5 

2  i5 

2  24 

2  33 

10 

11 

I  47 

I  36 

I  29 

I  25 

I  22 

I  20 

I  21 

I  23 

I  27 

I  33 

I  4o 

I  47 

I  55 

2  4 

2  12 

2  20 

II 

12 

I  54 

I  42 

I  33 

I  28 

I  24 

I  21 

I  20 

I  21 

I  24 

I  28 

I  34 

I  40 

I  47 

I  55 

2  2 

2  9 

12 

i3 

2  2 

I  48 

I  38 

I  3i 

I  26 

I  23 

I  21 

I  20 

I  22 

I  25 

I  3o 

I  35 

I  4i 

I  47 

I  54 

2  0 

i3 

i4 

2  10 

I  54 

I  43 

I  35 

I  29 

I  25 

I  22 

I  19 

I  20 

I  23 

I  27 

I  3i 

I  36 

I  4i 

I  47 

I  52 

i4 

i5 

2  18 

2  I 

I  48 

I  39 

I  33 

I  28 

I  24 

I  21 

I  19 

I  21 

I  24 

I  27 

I  32 

I  36 

I  4i 

I  46 

i5 

i6 

2  27 

2  8 

I  53 

I  43 

I  36 

I  3i 

I  26 

I  22 

I  19 

I  19 

I  21 

I  24 

I  28 

I  32 

I  36 

I  40 

16 

17 

2  35 

2'l5 

I  59 

I  47 

I  4o 

I  u 

I  29 

I  23 

I  20 

I  I'd 

I  19 

I  22 

I  25 

I  28 

I  32 

I  35 

17 

18 

2  44 

2  22 

2  4 

I  52 

I  43 

I  37 

I  3i 

I  25 

I  20 

I  17 

I  18 

I  20 

I  22 

I  25 

I  28 

I  3i 

18 

19 

2  53 

2  29 

2  10 

I  57 

I  47 

I  4o 

I  34 

I  26 

I  21 

I  18 

I  17 

I  19 

I  20 

I  23 

I  25 

I  28 

19 

20 

3  2 

2  36 

2  16 

2  2 

I  5i 

I  AA 

I  37 

I  28 

I  22 

I  19 

I  17 

I  18 

I  19 

I  21 

I  2j 

I  25 

20 

21 

3  II 

2  M 

2  22 

2  8 

I  55 

I  47 

I  4o 

I  3o 

I  24 

I  20 

I  18 

I  17 

I  18 

I  19 

I  21 

I  23 

21 

22 

3  20 

1   5i 

2  29 

2  i3 

2  0 

I  5i 

I  43 

I  32 

I  25 

I  21 

I  18 

I  16 

I  17 

I  18 

I  19 

I  21 

22 

23 

3  2Q 

2  58 

2  35 

2  18 

2  5 

I  55 

I  46 

I  35 

I  27 

I  22 

I  19 

i   17 

I  16 

I  17 

I  18 

I  19 

23 

24 

3  38 

3  5 

2  42 

2  23 

2  9 

I  59 

I  5o 

I  37 

I  29 

I  24 

I  20 

I  17 

I  16 

I  16 

I  17 

I  18 

24 

25 

3  47 

3  i3 

2  49 

2  29 

2  i4 

2  3 

I  53 

I  39 

I  3i 

I  25 

I  21 

I  18 

I  16 

I  16 

I  16 

I  17 

25 

26 

3  55 

3  20 

2  55 

2  M 

2  19 

2  7 

I  57 

I  42 

I  33 

I  27 

I  22 

I  19 

I  17 

I  16 

I  16 

I  16 

26 

27 

4  4 

3  27 

3  I 

1   39 

2  24 

2  12 

2  I 

I  45 

I  35 

I  28 

I  23 

I  19 

I  17 

I  16 

I  16 

I  16 

27 

28 

4  12 

3  34 

3  8 

2  45 

2  29 

2  16 

2  5 

I  48 

I  37 

I  3o 

I  24 

I  20 

I  18 

I  16 

I  i5 

I  16 

28 

29 

4  21 

3  4i 

3  i4 

2  5o 

2  33 

2  20 

2  8 

I  5i 

I  39 

I  3i 

I  25 

I  21 

I  18 

I  16 

I  i5 

I  i5 

29 

3o 

4  29 

3  48 

3  20 

2  55 

2  38 

2  24 

2  12 

I  54 

I  4i 

I  33 

I  26 

I  21 

I  18 

I  16 

I  i5 

I  i5 

3o 

3 1 

4  38 

3  55 

3  26 

3  0 

2  A'^ 

2  28 

2  16 

I  57 

I  44 

I  34 

I  28 

I  22 

I  18 

I  16 

I  16 

I  i5 

3i 

32 

4  46 

4  2 

3  32 

3  6 

2  48 

2  32 

2  19 

2  0 

I  46 

I  36 

I  29 

I  23 

I  19 

I  17 

I  16 

I  i5 

32 

33 

4  54 

4  9 

3  39 

3  II 

2  53 

2  36 

2  23 

2  3 

I  49 

I  38 

I  3i 

I  25 

I  20 

I  17 

I  16 

I  i5 

33 

34 

5  2 

4  16 

345 

3  16 

2  57 

2  4o 

2  26 

2  6 

1  5. 

I  40 

1  32 

I  26 

I  21 

I  18 

I  16 

I  i5 

34 

35 

5  10 

4  23 

3  5i 

3  22 

3  2 

2  44 

2  3o 

2  9 

I  53 

I  42 

I  34 

I  27 

I  22 

I  18 

I  16 

I  i5 

35 

36 

5  18 

4  3o 

3  57 

3  27 

3  6 

2  48 

2  33 

2  12 

I  55 

I  AA 

I  35 

I  28 

I  23 

I  19 

I  17 

I  16 

36 

37 

5  26 

4  37 

4  3 

3  32 

3  10 

2  52 

2  37 

2  i5 

I  58 

I  46 

I  37 

I  29 

I  24 

I  20 

I  18 

I  16 

37 

38 

5  33 

4  43 

4  8 

3  37 

3  i4 

2  56 

2  41 

2  17 

2  0 

I  48 

I  38 

I  3o 

I  25 

I  21 

I  18 

I  16 

38 

39 

5  4i 

4  5o 

4  i4 

3  42 

3  19 

3  0 

2  45 

2  20 

2  2 

I  5o 

I  39 

I  3i 

I  25 

I  21 

I  18 

I  16 

39 

4o 

5  48 

4  56 

4  19 

•^  47 

3  23 

3  4 

2  48 

2  23 

2  4 

I  5i 

I  40 

I  32 

I  26 

I  22 

I  19 

I  16 

4o 

4i 

5  55 

5  2 

4  25 

3  52 

3  28 

3  8 

2  5i 

2  25 

2  6 

I  53 

I  42 

I  33 

I  27 

I  23 

I  20 

I  17 

4i 

42 

6  2 

5  8 

.4'  3o 

3  57 

3  32 

3  II 

2  54 

2  28 

2  9 

I  55 

I  43 

I  M 

I  28 

I  24 

I  20 

I  17 

42 

43 

6   Q 

5  i4 

4  35 

4  2 

3  36 

3  i5 

2  58 

2  3l 

2  12 

I  57 

I  AA 

I  35 

I  29 

I  25 

I  21 

I  17 

43 

44 

6  16 

5  20 

4  40 

4  7 

3  40 

3  19 

3  I 

2  34 

2  i4 

I  59 

I  46 

I  37 

I  3i 

I  26 

I  22 

I  18 

A^ 

46 

6  29 

5  32 

4  5o 

4  16 

3  48 

3  26 

3  8 

2  4o 

2  18 

2  2 

I  49 

I  40 

I  33 

I  28 

I  23 

I  19 

46 

48 

6  42 

5  43 

4  59 

4  24 

3  56 

3  33 

3  i4 

2  45 

2  22 

2  6 

I  52 

I  43 

I  36 

I  3o 

I  25 

I  20 

48 

5o 

6  54 

5  54 

5  8 

4  32 

4  3 

3  40 

3  19 

2  5o 

2  26 

2  9 

I  55 

I  45 

I  38 

I  32 

I  26 

I  21 

5o 

52 

7  6 

6  4 

5  17 

4  39 

4  10 

3  46 

3  24 

2  55 

2  3o 

2  12 

I  58 

I  48 

I  4o 

I  33 

I  27 

I  22 

52 

54 

7  18 

6  t4 

5  25 

4  46  4  16 

3  52 

3  29 

2  59 

2  34 

2  i5 

2  0 

I  5o 

I  42 

I  35 

I  29 

I  24 

54 

56 

7  29 

6  24 

5  33 

4  53  4  22 

3  57 

3  34 

3  3 

2  37 

2  19 

2  3 

I  52 

I  43 

I  36 

I  3o 

I  25 

56 

58 

7  40 

6  33 

5  41 

5  0  4  28 

4  2 

3  39 

3  7 

2  41 

2  22 

2  6 

I  54 

I  45 

I  37 

I  3i 

I  26 

58 

60 

7  5o 

6  4i 

5  48 

5  7  4  34 

4  7 

3  43 

3  II 

2  44 

2  25 

2  8 

I  56 

I  47 

I  39 

I  32 

I  27 

60 

62 

7  58 

6  48 

5  55 

5  i3  4  40 

4  12 

3  48 

3  i5 

2  47 

2  28 

2  II 

I  58 

I  48 

I  4o 

I  33 

I  28 

62 

64 

8  6 

6  55 

6  1 

5  19  4  45 

4  17 

3  52 

3  18 

2  5o 

2  3o 

2  i3 

2  0 

I  5o 

I  4i 

I  34 

I  29 

64 

66 

6  7 

5  24 

4  5o 

4  21 

3  56 

3  20 

2  53 

2  32 

2  i5 

2  2 

I  5i 

I  42 

I  .-5 

I  29 

66 

(38 

4  55 

4  25 

4  0 

3  22 

2  55 

2  34 

2  17 

2  4 

I  52 

I  43 

I  36 

I  3o 

68 

70 

4  4 

3  24 

2  57 

2  36 

2  18 

2  5 

I  53 

I  AA 

I  37 

I  3i 

70 

72 

3  26 

2  59 

2  37 

2  19 

2  6 

I  54 

I  45 

I  38 

I  32 

72 

•74 

3  I 

2  38 

2  20 

2  7 

I  55 

I  46 

I  39 

I  32 

74 

76 

2  39 

2  21 

2  8 

I  56 

I  47 

I  39 

I  '66 

76 

78 

2  22 

2  8 

I  57 

I  48 

I  40 

I  33 

78 

80 

2  9 

I  58 

I  48 

I  4o 

r  34 

80 

82 

I  58 

I  48 

I  40 

I  34 

82 

84 

I  49 

I  4i 

I  34 

84 

86 

I  4i 

I  M 

86 

G" 

7= 

8° 

9° 

10° 

11° 

12° 

14° 

1G° 

18° 

20° 

22° 

24° 

26° 

28° 

30° 

\ 

TABLE  XLVm.                                     ^^''=0293 

Third  Correction.     Apparent  Distance  56°. 

Ap;.. 

Jipparcnt  Altitude  of  the  Sun,  Star  or  Planet. 

D's 
App. 

All. 

:32° 

\iA^ 

3G° 

cid^ 

42" 

4t)" 

5U" 

54" 

58" 

02" 

m° 

70" 

74° 

78" 

82" 

86" 

Ah. 

c 

/    n 

1  II 

/   II 

1  II 

/  /' 

/  II 

/   // 

1  II 

1  II 

/  // 

1  II 

1  II 

/  // 

1  II 

/  // 

/     '/ 

0 

C^ 

4  37 

4  54 

5    TO 

5  26 

5  56 

6  25 

6  5i 

1  i5 

7  37 

7  58 

6 

7 

3  57 

4  n 

4  25 

4  38 

5    3 

5  29 

5  52 

b  12 

6  3i 

6  48 

7 

ft 

3  26 

3  38 

3  5i 

4    3 

4  26 

4  47 

5    5 

5  23 

5  40 

5  55 

6    8 

8 

9 

lO 

3     I 

3  12 

3  23 

3  33 

3  53 

4  12 

4  3o 

4  46 

5    0 

5  i3 

5  25 

9 

•2  43 

2  53 

3     2 

3  II 

3  28 

3  45 

4     I 

4  i5 

4  27 

4  39 

4  5o 

10 

II 

2  29 

2  37 

2  45 

2  53 

3     9 

3  24 

3  38 

3  So 

4    I 

4  12 

4  21 

1 1 

12 

2  16 

2   23 

2  3o 

2  38 

2    52 

3     6 

3  18 

3  28 

3  38 

3  47 

3  56 

4    4 

12 

1 3 

2     6 

2   12 

2  18 

2    25 

2  37 

2    5n 

3     I 

3  10 

3   19 

3  28 

3  36 

3  42 

1 3 

i4 

I  57 

2     3 

2     8 

2  i4 

2    2D 

2    3(i 

2  47 

2  56 

3     4 

3  12 

3  19 

3  24 

i4 

i5 

I  5o 

I  55 

I  59 

2     5 

2    l5 

2    25 

2  35 

2  A4 

2  5i 

2  5b 

3    4 

3  10 

i5 

i6 

I  44 

I  48 

I  53 

I  58 

2       7 

2     16 

2    25 

2  33 

2  39 

2  45 

2  5i 

2  57 

3     2 

16 

'7 

I  39 

I  43 

I  48 

I     52 

2      0 

2       8 

2  16 

2  24 

2  3o 

2  35 

2  4o 

2  45 

2  49 

17 

i8 

I  35 

I  39 

I  4-i 

I  47 

I  54 

2       I 

2     8 

2  i5 

2  21 

2  26 

2  3i 

2  35 

2  38 

18 

19 

I  3i 

I  35 

I  38 

I  42 

I  48 

I  65 

2     I 

2    7 

2  i3 

2  18 

2   23 

2  27 

2  3u 

'9 

20 
21 

I  28 

I     25 

I  3i 

I   27 

I  34 
1  3o 

I   37 
I  33 

I  43 
I  38 

.  49 
I  44 

I  55 
I  49 

2    0 

I  54 

2     6 
I  59 

2  10 
2     3 

2  i5 
2     7 

2  19 
2  11 

2  22 

2  i4 

2  24 
2  iC 

20 
21 

22 

I   22 

I  24 

I  27 

I  3o 

I  34 

I  39 

I  44 

I  48 

I    52 

I  56 

2     0 

2    4 

2    6 

2    8 

22 

23 

I   20 

I  22 

I  24 

I   27 

I  3i 

I  3d 

I  40 

I  44 

I   47 

I  5i 

I  54 

I  57 

2    0 

2     2 

23 

24 

I   19 

I  20 

I  22 

I     25 

I  28 

I    32 

I  36 

I  4o 

I  A6 

I  46 

I  49 

I    52 

I  54 

I  5C 

I  58 

24 

25 

I    18 

I   19 

I  21 

1    23 

I  26 

I   29 

I  33 

I  36 

I  39 

I  42 

I  44 

I  47 

I  49 

I  5i 

I  53 

25 

26 

I   17 

I   18 

I   19 

I     21 

I  24 

I   27 

I  3o|i  33 

I  35 

I  38 

I  40 

I  42 

I  44 

I  4( 

I  48 

26 

27 

I   16 

I  17 

I   18 

I     19 

I  22 

I     25 

I  27 

I  3o 

I    32 

I  35 

I  37 

I  39 

I  40 

I  42 

1  44 

27 

28 

I   16 

I   lb 

I   17 

I     18 

I  20 

I    23 

I    25 

I   27 

I  29 

I    32 

I  M 

I  36 

I  37 

I    3q 

I  40 

I  4i 

28 

29 

I   i5 

I   i5 

I   16 

I     17 

I   19 

I     21 

I     23 

I     25 

I  27 

I  29 

I  3i 

I  33 

I  34 

I  35 

I  36 

I  37 

29 

3o 

I   i5 

I   lb 

I   16 

I   lb 

I   17 

I     19 

I  21 

I     23 

I     25 

I  27 

I  29 

I  3o 

I  3i 

I    32 

I  33 

I  34 

3o 

3i 

I  i4 

I   i4 

I   i5 

I   i5 

I   16 

I     18 

I   19 

I     21 

I    23 

I     25 

I  27 

I   28 

I  29 

I   2c; 

I  3o 

I  3i 

3i 

32 

I  14 

I   14 

I  i4 

I   i4 

I   i5 

I     17 

I    18 

I     19 

I     21 

I    23 

I    25 

I  26 

I  27 

I   2- 

I  27 

I  28 

32 

33 

I  14 

I   i3 

I   i3 

I   i3 

I  i4 

I   lb 

I   17 

I     18 

I     20 

I    21 

I    23 

I  24 

I    25 

I    25 

I    25 

I  26 

33 

34 

I  i4 

I   i3 

I   i3 

I  i3 

I  i4 

I   i5 

I   16 

I     17 

I     19 

I    20 

I  21 

I  22 

I    23 

I    23 

I    23 

I  24 

34 

35 

I  14 

I  i3 

I   i3 

I   i3 

I  i3 

I   14 

I   i5 

I     16 

I    17 

I     18 

I   19 

I  20 

I    21 

I    21 

I  21 

I  22 

35 

36 

I  i4 

I   i3 

1   12 

I   12 

I  12 

I   i3 

I   14 

I   i5 

I    16 

I     16 

I   17 

I   18 

I     19 

I    ic; 

I   19 

I  20 

36 

37 

I  14 

I    i3 

I   12 

I   12 

I  12 

I   12 

I   i3 

I   i4 

I   i5 

I     l5 

I   16 

I   16 

I     17 

I    17 

I   17 

I   18 

37 

38 

I  14 

I   i3 

I  12 

I   II 

I  II 

I   12 

I   i3 

I   i3 

I  i4 

I   i4 

I   i5 

I   i5 

I     16 

I    iC 

I   16 

I   17 

38 

39 

I  i4 

I   i3 

I   12 

I   II 

I  1 1 

I   II 

I   12 

I   12 

I   i3 

I   i3 

I   i3 

I   i3 

I  i4 

I   i4 

I   i5 

I   i5 

39 

4o 

I  14 

I   i3 

I   12 

I   II 

I  10 

I    10 

I   II 

I   II 

I    12 

I   12 

I   12 

I  12 

I  12 

I   i3 

I   i3 

I   i3 

4o 

4i 

I   i5 

I   14 

I   12 

I   II 

I  10 

I     ID 

I   10 

I   10 

I    II 

I   n 

I   II 

I   II 

I  II 

I   12 

I   12 

4i 

42 

I   lb 

I  i4 

I   12 

I   II 

I     9 

I     9 

I     9J1     9 

I   10 

I   10 

I   10 

I   10 

I  10 

I   II 

I   II 

42 

4i 

I   i5 

1  i4 

I   12 

I   II 

I     9 

I      Q 

I     9|i     9 

I     9 

I     9 

I     9 

I     9 

I     9 

I     IC 

I   10 

43 

44 

I   lb 

I  14 

I  12 

I  II 

I     9 

I     8 

I     81     8 

I     8 

I     8 

I     8 

I     8 

I     8 

I      c; 

'     9 

44 

46 

I   17 

I   i5 

I   i3 

I     12 

I     9 

I     7 

I     71     7 

I     b 

1     b 

I     6 

I     7 

I     7 

I     7 

46 

48 

1   17 

1   i5 

I   i3 

I     12 

I     9 

I     7 

I     61     6 

I     5 

I     5 

I     5 

I     5 

I     5 

I     ( 

48 

5o 

I   18 

I   16 

I   14 

I     12 

I     9 

I     b 

I     5i     5 

I    4 

I    4 

I    4 

I     4 

I     4 

5o 

52 

I   19 

I   17 

I   i5 

I     l3 

I     9 

I     b 

r    4i     4 

I     3 

I     3 

I     3 

I     3 

I     3 

52 

54 

I  20 

I   17 

I   i5 

I   i3 

I     9 

I     b 

I    4i     3 

I     3 

I     2 

I     2 

I     2 

54 

56 

58 
60 

1     21 
I    22 
I    23 
I    24 

I   18 
I   19 
I   19 

I   16 
I   16 
I   16 

I   i4 
I  i4 
I  i4 

I     K) 
I     10 
1     10 

I     b 

I     6 
I     6 
I     6 

I    4i     2 
I    4|i     2 
I    4.     2 

I     2 

I     I 

I     I 

I     I 

56 

I     I 
I     I 

I     0 

I     0 

I     0 
I     0 

Talile  P.     Effect  of  Sun's  Par 

62 

I  20 

1   17 

I   i4 

I     10 

I    4i     2 

I     I 

I     0 

Adii  tlie  Numbers  above  the  liiu-s 

64 
66 

I    24 
I    25 

I  20 
I  21 

I   17 
I   18 

I  i4 
I  i5 

I     7 

I      7 

I    4i     2 
I    4 1     2 

I     0 
I     0 

I     0 

10  Tliird  Correcliori  ;    subtract 
the  others. 

68 
70 

I    25 
I    26 

1  21 
I  22 

t   19 

I   i5 
I   16 

..  .  1 

5)'^ 

Sun's  Apparent  Ahiluile. 

I     7 
I     7 

I     4 
I    4 

I     2 

App. 
All. 

5 

0  20  3 

D  JO 

50  6 

0  70 

80 

- 

90 

72 
74 

76 

I    27 
I    27 
I    28 

I    23 
I     23 
I    23 

I   19 
I   19 
I   19 

I   16 
I   16 
I   16 

I     7 
I     7 
I      7 

I    4 
I    4 
I    4 

I     2 

5 
10 
20 

0 

T 
3 

0    1    2 

i  0  _i 
3  2   1 

3 

2 
0 

4 
3 

78 

I     28 

I     23 

I  20 

I   17 

1     7 

30 

5 

i  3  2 

I 

I     ( 

0 

1 

80 

I    29 

I  24 

I  20 

I   17 

I     7 

40 

s 

5    5    4 

3 

2    • 

1 

1 

0 

82 

I     29 

I  24 

I  20 

I   17 

50 

3 

7    6    5 

4 

4 

3 

84 

I    29 

I  24 

I  20 

I   17 

60 

9 

3    7    6 

5 

5    4 

86 

!_i9 

1  24 

I  20 

r   17 

70 
80 

9    8    7 
H    S 

6 
7 

6 

32° 

y4" 

30" 

US'" 

42° 

4(j" 

50" 

54° 

58° 

62° 

06° 

90 

8 

P^^.s^^J               TABLE  XLVIII. 

Third  Correction.  Apparent  Distance  60° 

D's 
App. 

Alt. 

o 

6 

7 
8 

9 

10 

II 

12 

i3 
i4 
i5 

i6 

17 
i8 

19 
20 

21 
22 

23 

24 

25 

26 
27 

28 

29 

So 
3i 

32 

33 
34 
35 

36 

37 
38 

39 

40 

4i 
42 
43 

46 

48 
5o 

52 

54 
56 

58 
60 
62 
64 
66 

68 
70 
72 
74 
1^ 
78 
80 
82 
84 
86 

Apparent  Mtitudc  of  the  Sun,  Star  or  Planet. 

D's 
4pp. 
Alt.  . 

0 

6 

7 
8 

9 
10 

11 
12 
i3 
i4 
i5 

16 

17 
18 

19 

20 

21 
22 

23 

24 

25 

26 

27 
28 

=9 
3o 

3i 

32 

33 
34 
35 

36 

37 
38 
39 
40 

4i 
42 
43 
4^ 
46 

48 
5o 

52 

54 
56 

58 
60 
62 

66 

68 
70 
72 
74 
76 

78 
80 
82  ( 

84  1 

^1 

6° 
/  // 
I  22 

I  24 
I  28 
I  33 
I  4o 

I  47 

1  55 

2  3 
2  10 
2  18 

2  26 
2  34 
2  42 
2  5o 

2  59 

3  7 
3  i5 
3  24 
3  32 
3  4} 

3  49 

3  58 

4  6 
4  i5 
4  23 
4  3i 
4  39 
4  47 

4  55 

5  3 

5  10 
5  18 
5  25 
5  32 
5  39 

5  46 

5  53 

6  0 

6  7 
6  21 

6  34 
6  47 

6  59 

7  II 
7  22 

7  3i 
7  4o 
7  48 

7  56 

8  3 

8  10 
6° 

7° 
t  II 

I  23 

I  22 

I  24 
I  28 
I  33 
I  38 
I  43 
I  49 

1  55 

2  I 

2  7 
2  i3 
2  20 
2  27 
2  34 
2  4i 
2  48 

2  55 

3  2 
3  9 

3  16 
3  23 
3  3o 
3  37 
3  44 
3  5i 

3  58 

4  5 
4  12 
4  18 

4  24 
4  3i 
4  38 
4  45 
4  5i 

4  57 

5  3 

5  i5 

5  26 

5  37 
5  48 

5  58 

6  8 
6  17 
6  25 
6  32 
6  39 
6  46 
6  53 

6  59 

7° 

8° 
1  II 

I     25 
I  23 

I  22 
I  24 
I  27 

I  3i 
I  36 
I  4o 
I  45 
I  5o 

1  55 

2  0 
2  5 
2  II 

2  17 

2  23 
2  29 
2  35 
2  4i 
2  47 
2  53 

2  59 

3  5 
3  II 
3  17 
3  23 
3  29 
3  34 
3  40 
3  46 
3  52 

3  58 

4  4 
4  10 
4  i5 
4  21 
4  26 
4  3i 
4  36 
4  46 

4  55 

5  4 
5  i3 
5  22 
5  3o 

5  37 
5  45 
5  52 

5  58 

6  2 
6  6 
6  10 

8° 

9° 

f  II 
I   28 

I  25 
I  23 
I  22 

I  24 
I  27 

I  3o 

I  34 
I  38 
I  42 

I  46 

1  5o 
t  54 
.  59 

2  4 
2  9 
2  i4 
2  19 
2  24 
2  29 

2  34 
2  39 
2  44 
2  49 
2  54 

2  59 

3  4 
3  9 
3  14 
3  19 

3  24 
3  29 
3  34 
3  39 
3  44 

3  49 
3   53 

3  58 

4  3 
4  12 

4  20 
4  28 
4  36 
4  44 
4   5i 

4  58 

5  4 
5  10 
5  i5 
5  20 

^^ 

5  27 

9° 

10° 

/  // 
I  33 

I  28 

I  25 

I  24 

I  23 

I  24 

I  26 
I  29 

I  32 

I  36 
I  39 
I  43 
I  46 
I  5o 
I  54 

1  58 

2  2 

2  7 
2  10 
2  i5 
2  20 
2  25 

2  29 

2  33 
2  38 
2  42 
2  47 
2  52 

2  56 

3  0 

3  4 
3  8 
3  12 

3  17 
3  21 

3  26 
3  3o 
3  35 
3  39 
3  47 

3  54 

4  I 
4  8 
4  i5 
4  21 
4  27 

4  32 

4  38 
4  43 
4  47 
4  5i 
4  54 
4  57 

10° 

11' 

1  II 
I  4o 
I  33 
I  28 

I  25 

I  24 
713 
I  24 
I  26 
I  28 
I  3i 

I  34 
I  37 
I  4o 
I  43 
I  46 
7"5^ 
I  53 

1  57 

2  I 
2  4 
2  8 
2  12 
2  16 
2  20 
2  24 

2  28 

2  33 

2  36 
2  4o 
2  44 

2  48 
2  52 
2  55 

2  59 

3  3 

I,] 

3   i5 
3  19 
3  26 

3  32 
3  37 
3  43 
3  49 

3  55 

4  I 
4  6 
4  II 
4  i5 
4  19 
4  23 
4  26 
4  29 

11° 

12° 

1  II 
I  4i 
1   37 
I  3i 
I  27 

I  25 

I  24 

I  23 

I  24 

I  25 

I  27 

1 29 

I  3i 
I  34 
I  36 
.  39 

I  42 
I  45 
I  48 

I  52 

I  55 

1  59 

2  3 

2  7 
2  II 
2  i4 
2  18 
2  21 

2  25 

2  28 

2  32 

2  35 
2  39 
2  42 
2  46 
2  49 

2  52 

2  55 

2  58 

3  I 
3  7 
3  i3 
3  19 
3  25 
3  3o 
3  35 

3  4o 
3  45 
3  5o 
3  55 

3  59 
14  2 

4  4 
4  6 
4  8 

12° 

14° 

/  II 
2     I 

I  47 
I  39 
I  33 
I  29 

I  26 

I  25 

I  24 

I  23 

I  24 

I  25 

I  26 
I  27 

1 29 

I  3i 
I  33 
I  35 
I  37 
I  4o 
I  42 

I  45 
I  48 
I  5i 
I  53 
I  56 

1  59 

2  2 
2  5 
2  8 
2  II 

2  i4 

2  17 
2  20 
2  22 

2  25 

2  27 
2  3o 

2  32 

2  35 
2  4o 

2  45 
2  5o 
2  55 

2  59 

3  4 
3  8 
3  12 
3  16 
3  19 
3  22 

3  25 
3  27 
3  28 
3  29 
3  3o 

14° 

1(3° 

1  II 

2  16 
I  59 
I  48 
I  4o 
I  34 
I  3o 
I  28 
I  26 

I  25 
I  23 
I  22 
I  22 
I  23 
I  24 
I  25 

I  26 

I  28 

I  3o 
I  3i 
I  33 

I  35 
I  38 
I  40 
I  42 
I  44 
1   46 
I  48 
I  5i 
I  53 
I  55 

I  57 

1  59 

2  2 
2  4 
2  6 

2  8 
2  10 
2  i3 
2  i5 
2  19 

2  23 

2  27 
2  3i 
2  35 
2  38 

2  4i 
2  44 
2  48 
2  5i 
2  54 
2  56 

2  58 

3  c. 
3  2 
3  3 

3  4 
1G° 

18° 
/  II 

1  33 

2  i3 
I  59 
I  49 
I  4i 
I  36 

I  32 

I  29 

I  27 

I  25 

r  23 
I  22 
I  21 
I  22 
I  22 

I  23 

I  24 

I  25 

I  26 
I  27 

1 29 

I  3i 

I  32 

I  34 
I  35 

I  37 
I  38 
I  40 
I  4i 
I  43 

I  45 
I  47 

\f, 

I  53 

I  55 
I  56 

1  58 

2  0 
2  4 
2  8 
2  II 
2  i4 
2  18 
2  21 

2  24 
2  27 
2  29 
2  3i 
2  33 
2  35 
2  36 
2  38 
2  39 
2  4i 
2  42 
2  43 

18° 

20° 

/  II 

1  5u 

2  27 
2  11 
I  5b 
I  49 
I  42 
I  37 
I  33 
I  3o 
I  27 

I  25 
I  23 

I  22 
I  21 
I  20 
I  21 
I  21 
I  22 

I  23 

I  24 

I  25 

I  26 
I  27 
I  28 
I  29 

I  3o 
I  3i 
I  33 
I  34 
I  35 

I  37 
I  38 
I  4o 
I  42 
I  43 

I  45 
I  46 
I  48 
I  49 

I  52 

I  56 

1  59 

2  2 
2  4 
2  7 
2  10 
2  12 
2  i4 
2  16 
2  18 
2  19 
2  20 
2  21 
2  22 

2  23 

2  24 

2  25 
2  26 

20° 

22° 

/  II 
3  8 
2  41 

2  23 

2  8 
I  57 

I  49 
I  43 
I  38 
I  34 
I  3o 

I  27 

I  25 
I  23 

I  22 
I  21 
I  20 
I  20 
I  20 
I  21 
I  22 

I  22 

I  23 
I  23 

I  24 
I  24 

I  25 

I  26 
I  27 
I  28 
I  29 

I  3i 

I  32 

I  33 
I  35 
I  36 

I  37 
I  38 
I  40 
I  4i 
I  43 
I  46 
I  48 
1  5i 
I  53 
I  56 

1  58 

2  0 
2  2 
2  4 
2  5 

2  6 
2  7 
2  8 
2  9 
2  10 

2  II 
2  12 
2  12 

2  12 

22° 

24° 

/  // 
3  25. 
2  55. 
2  35 
2  18 
2  6 
I  57 
I  49 
I  43 
I  38 
I  34 
I  3u 
I  28 

I  25 
I  23 
I  22 

I  21 
I  20 
I  20 
I  20 
I  20 

I  20 
I  21 
I  21 
I  21 
I  21 
I  22 
I  22 
I  23 
I  24 
I  25 

I  26 

I  27 
I  28 

I  29 
I  3o 

I  3i 

I  32 

I  34 
I  35 
I  37 
I  39 
I  4i 
I  43 
I  45 
I  47 

I  49 

I  5() 

I  52 

I  53 
I  55 
I  56 
I  57 
I  58 
I  59 

1  59 

2  0 
2  I 
2  I 
2  2 

2  2 

24° 

2li° 

1  II 
3  4i 

3  9. 

2  48 
2  29 
2  i5 
2  5 
I  56 
I  49 
I  43 
I  38 

I  34 
I  3i 

I  28 

I  26 
I  24 
I  22 
I  21 
I  20 
I  20 
I  19 

I  19 
I  19 
I  19 
I  19 
I  19 
I  20 
I  20 
I  20 
I  21 
I  22 
I  22 

I  23 

I  24 

I  25 

I  26 

I  27 
I  28 

1 29 

I  3o 
I  3i 

I  33 
1  35 
I  36 
I  38 
I  40 

I  4i 
I  42 
I  44 
I  45 
1  46 

I  47 

I  48 
I  49 
I  5o 
I  5o 

I  5i 
I  5i 

I  52 
I  52 
I  52 

26° 

28° 
/  ri\ 
5  58. 
3  23 
3  0. 
239 
2  25 
>  i3 
2  3 
I  55 
I  49 
I  43 

I  38 
I  34 
I  3i 
I  28 
I  26 

I  24 
I  22 
I  21 
I  20 
I  19 

I  19 
I  19 
I  18 
I  18 
r  18 

I  18 
I  19 
I  19 
I  19 
I  20 

I  20 
I  21 

I  21 
I  22 
I  22 

I  23 
I  24 
I  25 
I  26 

I  27 

I  28 

^^9 
I  3i 
I  33 
I  34 
I  35 
I  36 
I  37 
I  38 
I  39 

I  40 
I  4i 
I  4i 
I  42 
I  42 
I  43 
I  43 
I  44 
I  44 
I  44 

28° 

30° 

f  i5 
}  37 
5  12 
2  5o 
2  34 
2  21 
2  II 
2  2 
I  54 
I  48 

I  43 
I  38 
I  34 
I  3i 
I  28 

I  25 
I  23 

I  22 
I  21 
I  20 

I  19 
I  19 
I  18 
I  18 
1  18 
I  18 
I  18 
I  18 
I  18 
I  18 

I  18 
I  19 
I  19 
I  20 
I  20 

I  20 
I  21 
I  22 
I  22 

I  23 

I  24 

I  25 

1  27 
I  28 
I  29 

I  3o 
I  3i 

I  32 

I  33 

1  34 

1  34 
I  35 
I  35 
I  36 
I  36 

I  37 
I  37 
I  38 
I  38 
1  38 

30° 

TABLE  XLVllI.               [Page  295 

Third  Correction.  Apparent  Distance  60°. 

Al)[). 

All. 
o 

Apparent  Altilude       thn  Sun,  Star  or  Plmirf. 

5's 
App 
All. 

3:2°  34° 
1  II   1  II 

3G° 
1  II 

38° 

42° 

/  // 

46° 

1   II 

50° 
/  II 

64° 
/  II 

68°  62° 

66° 

70" 

/4° 

78° 

82° 

86° 
/  t 

;  ir    1   /' 

1  II 

1 

II 

0 

6 

4  32  4  48 

5  3 

5  19 

5  49 

6  17 

6  44 

7  '' 

7  28I7  47 

8  3 

6 

7 

3  5i 

4  5 

4  19 

4  32 

4  58 

5  22 

5  44 

6  4 

6  22 

6  38 

6  53 

7 

8 

3  23 

•J  35 

3  47 

3  59 

4  22 

4  42 

5  I 

5  19 

5  35 

5  5o 

6  2 

6  i3 

8 

9 

3  0 

3  10 

3  20 

3  3o 

3  49 

4  8 

4  25 

4  41 

4  55 

5  8 

5  19 

5  3o 

9 

lO 

2  43 

2  5i 

3  0 

3  9 

3  26 

3  42 

3  58 

4  12 

4  24|4  35 

4  45 

4  54 

10 

1 1 

2  29 

2  37 

2  44 

2  52 

3  7 

3  21 

3  35 

3  48 

3  59 

4  9 

4  18 

4  26 

II 

12 

2  18 

2  25 

2  32 

2  39 

2  52 

3  5 

3  17 

3  29 

3  3g 

3  48 

3  56 

4  3 

4  8 

12 

i3 

2  8 

2  i5 

2  21 

2  28 

2  39 

2  5i 

3  2 

3  12 

3  21 

3  3o 

3  38 

3  44 

3  48 

i3 

i4 

2  0 

2  6 

2  12 

2  18 

2  28 

2  38 

2  48 

2  57 

3  6 

3  14 

3  21 

3  26 

3  29 

i4 

i5 

I  53 

I  58 

2  3 

2   8 

2  18 

2  27 

2  36 

2  45 

2  53 

3  0 

3  6 

3  11 

3  i5 

i5 

i6 

I  47 

1  5i 

I  55 

2   0 

2  9 

2  18 

2  26 

2  34 

2  4i 

2  48 

2  53 

2  58 

3  2 

3  6 

16 

17 

I  42 

I  45 

I  49 

I  53 

2  1 

2  9 

2  17 

2  24 

2  3i 

2  37 

2  42 

2  46 

2  5c 

2  53 

17 

i8 

I  37 

I  4o 

I  44 

I  47 

I  54 

2  1 

2  9 

2  16 

2  22 

2  27 

2  32 

2  36 

2  4o 

2  42 

18 

'9 

I  33 

1  36 

I  39 

I  42 

t  48 

I  55 

2  2 

2  9 

2  i5 

2  19 

2  24 

2  28 

2  3i 

2  33 

19 

20 
21 

I  3o 
I  27 

I  32 

I  29 

I  35 

I  32 

I  38 
I  35 

I  44 
I  40 

I  5o 
I  46 

I  56 
I  5i 

2  2 

I  56 

2  8 
2  I 

2  12 
2  6 

2  16 
2  10 

2  20 
2  i3 

2  23 

2  i5 

2  25 

2  27 
2  19 

20 
21 

2  17 

22 

I  25 

I  27 

I  29 

I  32 

I  37 

I  42 

I  47 

I  5i 

I  56 

2  0 

2   4 

2  6 

2  8 

2  10 

2  12 

22 

23 

I  23 

I  25 

I  27 

I  3(. 

I  34 

I  38 

I  43 

I  47 

I  5i 

I  55 

I  59 

2  I 

2  3 

2  4 

2   6 

2  3 

24 

I  22 

I  23 

I  25 

I  27 

I  3i 

I  35 

I  40 

I  44 

I  47 

I  5i 

I  54 

I  56 

I  58 

I  59 

2   I 

2  3 

24 

25 

I  21 

I  22 

1  23 

I  25 

I  29 

I  32 

I  36 

I  40 

I  43(1  47 

I  49 

I  5i 

1  53 

I  54 

I  56 

I  57 

25 

26 

I  20 

I  21 

I  22 

1  23 

I  26 

1  29 

I  33 

I  37 

I  40 

I  43 

I  45 

I  47 

I  4q 

I  5o 

I  5i 

I  52 

26 

27 

I  19 

I  20 

I  21 

I  22 

I  24 

I  27 

I  3o 

I  34 

I  37 

I  40 

I  42 

I  43 

I  45 

I  46 

I  47 

I  48 

27 

28 

I  19 

I  19 

I  20 

I  21 

I  23 

I  25 

I  28 

I  01 

I  34 

I  37 

I  89 

I  4o 

I  4i 

I  42 

I  43 

I  44 

28 

29 

I  18 

I  18 

I  19 

I  20 

I  22 

I  23 

I  26 

I  29 

I  3i 

I  34 

I  36 

I  37 

I  38 

I  39 

I  4o 

I  4i 

29 

Jo 

I  18 

I  18 

I  18 

I  19 

1  20 

I  22 

I  24 

I  27 

I  29 

I  3i 

I  33 

I  34 

I  35 

I  36 

I  37 

1  38 

3o 

3i 

I  18 

I  18 

I  18 

I  18 

I  19 

I  20 

I  22 

I  25 

I  27 

I  29 

I  3o 

I  3i 

I  32 

I  33 

I  34 

I  35 

3i 

■3  2 

I  17 

I  17 

I  17 

I  17 

I  18 

I  19 

I  21 

I  23 

I  25 

I  27 

I  28 

I  29 

I  3( 

I  3i 

I  3i 

I  32 

32 

33 

I  17 

I  16 

I  16 

I  16 

I  17 

I  18 

I  19 

I  21 

I  23 

I  25 

I  26 

I  27 

I  28 

I  29 

I  29 

I  3c 

33 

34 

I  17 

I  16 

I  16 

I  16 

I  16 

I  17 

I  18 

I  20 

I  22 

I  23 

I  24 

I  25 

I  26 

I  27 

I  27 

I  28 

34 

35 

I  17 

I  16 

I  16 

I  16 

I  16 

I  16 

I  17 

I  18 

I  20 

I  21 

r  22 

I  23 

I  24 

I  25 

I  25 

I  26 

35 

36 

I  17 

I  16 

1  i5 

1  16 

I  16 

I  16 

I  16 

I  17 

I  18 

1  19 

I  20 

I  21 

I  22 

I  23 

I  23 

I  24 

36 

37 

I  17 

I  16 

I  ID 

I  i5 

I  i5 

I  i5 

I  i5 

I  16 

I  17 

I  18 

I  19 

I  20 

I  21 

I  21 

I  22 

37 

38 

I  17 

I  16 

I  i5 

I  i4 

I  14 

I  14 

I  i4 

1  i5 

I  16 

I  17 

I  18 

I  19 

I  20 

I  20 

I  21 

38 

39 

I  18 

I  16 

I  i5 

I  i4 

I  i3 

I  13 

I  i3 

I  14 

I  i5 

I  16 

I  17 

I  17 

I  18 

I  18 

I  19 

3q 

4<) 

I  18 

I  16 

I  i5 

I  14 

I  i3 

I  i3 

I  i3 

I  i4 

I  i4 

I  i5 

I  16 

I  16 

I  17 

I  17 

I  17 

40 

4i 

I  iS 

I  16 

I  i5 

I  i4 

I  12 

I  12 

I  12 

I  i3 

I  i3 

I  14 

I  i5 

I  i5 

I  16 

I  16 

4i 

42 

I  18 

I  16 

I  i5 

I  i4 

I  12 

I  12 

I  12 

I  12 

I  12 

I  i3 

I  i4 

I  i4 

I  i5 

I  i5 

42 

43 

I  19 

I  17 

I  16 

I  i4 

I  12 

I  II 

1  II 

I  II 

I  II 

I  12 

I  i3 

I  i3 

I  i4 

I  i4 

43 

44 

I  19 

I  17 

I  16 

I  14 

I  12 

I  II 

I  II 

I  II 

I  II 

I  1 1 

I  12 

I  12 

I  i3 

I  i3 

44 

46 

I  30 

I  18 

I  16 

I  i4 

I  12 

I  If 

I  ID 

I  10 

I  10 

1  10 

I  II 

I  1 1 

I  II 

46 

48 

1  21 

1  19 

I  17 

I  i5 

I  12 

I  10 

I  9 

I  9 

1  9 

I  9 

I  10 

I  10 

I  10 

48 

5o 

I  22 

I  19 

I  17 

I  i5 

I  12 

I  10 

I  9 

I  8 

I  8 

I  8 

I  8 

I  8 

5o 

52 

I  23 

I  20 

I  17 

I  i5 

I  12 

I  10 

I  8 

I  8 

I  8 

I  7 

I  7 

<■     7 

52 

54 

I  24 

I  21 

I  18 

I  16 

I  i3 

I  10 

I  8 

I  7 

I  7 

I  6 

I  6 

"14 

56 

58 
60 
62 

I  25 

I  26 

I  27 
I  28 

I  22 

I  23 

I  24 
I  24 

I  19 

I  20 
I  21 
1  21 

I  16 
I  17 
I  18 
I  18 

I  i3 
1  i3 

I  14 
1  14 

I  10 

I  10 
I  10 
I  10 

I  8 

I  8 
I  8 
I  8 

<■     7 

I  7 
I  7 
I  6 

t  7 

I  6 

I  6 

56 

I  6 
I  6 
I  5 

I  5 
I  5 

'I'uble  P.  Effect  of  Sun's  Par. 
AM  the  Nn.nlwrs  al.ove  Iho  lines 

64 

I  29 

I  25 

I  21 

I  18 

I  14 

I  10 

I  8 

I  6 

I  5 

to  TliirtI  Corictlioii  ;  suLilnicl 

66 

68 
70 

I  29 
,  2_9 

I  3u 

1  25 

I  21 
1  22 
I  22 

I  18 
I  19 
I  19 

1  14 
I  i5 
I  i5 

I  8 

I  6 

the  othf-rs. 

I  25 

I  26 

I  II 
I  II 

I  8 
I  8 

I  6 

D's   S 

im's  Apparent  Al'.itude. 

Anp. 
Alt.   5 

lU  2ul30 

-10  50  fi 

0  70 

so 

so 

72 

I  3() 

I  26 

I  23 

I  20 

I  t5 

I  1 1 

T   8 

74 
7<^) 

I  3i 
T  3r 

I  27 

I  27 

I  23 
I  93 

I  20 
:  20 

I  i5 
r  t5 

I  1 1 
I  1 1 

5   0 
10   1 

0  1  -i 
\    0  1 

3  3 
2  2 

!  3 

~7tr 

I  i5 

20   3 

30   5 

3  2  1 

4  3  3 

2  7  ' 

7 

0 

I  32 

I  28 

I  24 

I  20 

80 

I  3} 

I  28 

I  24 

I  21 

I  i5 

40   6 

6  5  4 

3  3  ' 

2 

2 

82 

I  33 

I  28 

I  24 

I  21 

50   7 

7  6  5 

5  4  . 

3 

84 

1  33 

I  28 

I  2.4 

I  21 

60   3 

8  7  6 

6  5  4 

86 

I  33 

I  28 

I  24 

70 

9  8  7 

6  6 

32^ 

34" 

36°  38° 

42° 

46° 

50°  1 

54° 

58° 

62° 

m'' 

BO 

1  '*■ 

1 

—  ... 

Page  293]                       TABLE  XLVIII 

Third  Correction.  Appai-ent  Distance  64*^. 

D's 

Apparent  Altitude  of  the  Sun,  Star  or  Planet. 

5's 
App. 

Alt. 

A  pp. 

Alt. 

6=- 

7° 

8° 

9° 

1U° 

11° 

12° 

14° 

16° 

18° 

20° 

22° 

24° 

20° 

28° 

ao° 

o 

/  // 

1  II. 

?  // 

/  II 

/  // 

/  II 

/  // 

'  // 

/  II 

/  II 

/  // 

/  II 

/  // 

/  // 

/  // 

/  // 

0 

6 

I  26 

I  27 

I  29 

I  32 

I  36 

I  42 

I  49 

2  3 

2  19 

2  35 

2  5i 

3  8 

3  24 

3  4o 

3  56 

4  12 

6 

7 

T  28 

I  26 

I  27 

I  29 

I  32 

I  35 

I  4<J 

I  5i 

2  3 

2  i5 

2  28 

2  4? 

2  56 

3  9 

3  22 

3  36 

7 

8 

T  32 

I  28 

I  26 

I  27 

I  29 

I  3i 

I  34 

I  4i 

I  5i 

2'    2 

2  i3 

2  24 

2  36 

2  48 

3  0 

3  II 

8 

9 

r  37 

I  3i 

I  28 

I  26 

1  27 

I  28 

I  3o 

I  36 

I  43 

I  52 

2  1 

2  10 

2  20 

2  3i 

2  4i 

2  5i 

9 

10 

1 43 

I  35 

I  3o 

I  27 

I  26 

I  27 

I  28 

I  32 

I  37 

I  44 

I  5i 

I  59 

2  8 

2  17 

2  26 

2  35 

10 

II 

I  5o 

I  4o 

I  36 

I  29 

I  27 

I  26 

I  27 

I  29 

I  33 

I  38 

I  44 

I  5i 

I  59 

2  7 

2  14 

2  22 

II 

12 

I  57 

I  45 

I  37 

I  32 

I  29 

I  27 

I  26 

I  28 

I  3u 

I  34 

I  38 

I  44 

I  5i 

I  58 

2  5 

2  12 

12 

i3 

2  4 

I  5o 

I  4i 

I  35 

I  3i 

I  29 

I  27 

I  27 

I  28 

I  3i 

I  34 

I  38 

I  44 

I  5o 

I  57 

2  3 

i3 

i4 

2  12 

I  56 

1  46 

I  39 

I  34 

I  3i 

I  29 

I  26 

I  27 

I  29 

I   3i 

I  34 

I  39 

I  44 

I  5o 

1  55 

i4 

i5 

2  20 

2  2 

I  5i 

I  43 

I  37 

I  33 

I  3u 

I  27 

I  26 

I  27 

I  29 

I  3i 

I  35 

I  40 

I  44 

1  49 

ID 

i6 

2  27 

2  8 

I  56 

I  47 

I  41 

I  36 

I  32 

1  28 

I  25 

I  26 

I  27 

1  29 

I  32 

I  36 

I  40 

I  44 

16 

I? 

2  35 

2  i4 

2  I 

I  5i 

I  45 

I  89 

I  34 

I  29 

I  26 

I  25 

I  26 

I  28 

I  3o 

I  33 

I  3b 

I  4o 

17 

i8 

2  42 

2  21 

2  6 

I  56 

I  48 

I  42 

I  37 

I  3i 

I  27 

I  25 

I  25 

I  26 

I  28 

I  3o 

I  33 

I  36 

18 

19 

2  5i 

2  27 

2  12 

2  0 

I  52 

I  45 

I  39 

I  32 

I  28 

I  25 

I  25 

I  25 

I  27 

I  28 

I  3o 

I  33 

19 

20 

2  59 

2  34 

2  17 

2  5 

I  56 

I  49 

I  42 

I  34 

I  29 

I  26 

I  24 

I  24 

I  25 

I   26 

I  28 

1  3o 

20 

21 

3  7 

2  4i 

2  23 

2  10 

2  0 

I  52 

I  45 

I  36 

I  3o 

I  26 

I  24 

i  23 

I  2A 

I  25 

1  26 

I  28 

21 

22 

3  1 5 

2  48 

2  29 

2  i5 

2  4 

I  55 

I  48 

I  38 

I  3i 

I  27 

I  25 

I  23 

I  23 

I  24 

I  25 

I  26 

22 

23 

3  23 

2  55 

2  35 

2   20 

2  8 

I  59 

I  5i 

I  4o 

I  33 

I  28 

I  25 

I  23 

I  23 

I  24 

I  24 

I  25 

23 

24 

3  3i 

3  2 

2  4i 

2  25 

2  12 

2  2 

I  54 

[  42 

I  34 

I  29 

I  26 

I  24 

I  23 

I  23 

I  24 

I  25 

24 

25 

3  39 

3  8 

2  47 

2  3o 

2  17 

2  6 

I  57 

I  44 

I  36 

I  3o 

I  26 

I  24 

1  23 

I  23 

I  23 

1  24 

25 

26 

3  47 

3  i5 

2  53 

2  35 

2  21 

2  10 

2  0 

I  47 

I  38 

I  32 

I  27 

I  25 

I  23 

I  23 

X  23 

I  23 

26 

27 

3  56 

3  22 

2  59 

2  4o 

2  26 

2  14 

2  4 

I  bo 

I  4o 

I  33 

I  28 

I  25 

I  23 

I  20 

I  22 

I  23 

27 

28 

4  4 

3  29 

3  5 

2  45 

2  3o 

2  18 

2  7 

I  53 

I  42 

I   35 

I  29 

I  2b 

I  24 

I  23 

I  22 

I  22 

28 

29 

4  12 

3  36 

3  11 

2  5o 

2  35 

2  22 

2  II 

I  55 

I  44 

I  36 

I  3o 

I  27 

I  25 

I  23 

1  22 

I  22 

29 

So 

4  20 

3  42 

3  17 

2  55 

2  39 

2  26 

2  i5 

I   58 

I  46 

I  38 

I  32 

I  28 

I  25 

I  24 

I  23 

I  22 

3o 

3 1 

4  28,3  49 

3  23 

3  0 

2  43 

2  3o 

2  18 

2  0 

I  48 

I  40 

I  33 

1  29 

I  26 

I  24 

I  23 

I  22 

3i 

3? 

4  36 

3  55 

3  28 

3  5 

2  48 

2  34 

2  22 

2  3 

I  5o 

I  4i 

I  34 

I  3o 

I  2b 

I  24 

I  23 

I  22 

32 

33 

4  44 

4  2 

3  34 

3  10 

2  52 

2  38 

2  26 

2  6 

I  53 

I  43 

I  36 

I  3o 

I  27 

I  24 

I  23 

I  22 

33 

34 

4  52 

4  8 

3  39 

3  i5 

2  56 

2  4i 

2  29 

2  8 

I  55 

I  44 

I  37 

I  3i 

I  28 

I  25 

I  23 

I  22 

34 

35 

5  0 

4  i5 

3  45 

3  20 

3  I 

2  45 

2  33 

2  II 

I  57 

I  46 

I  38 

I  32 

I  28 

I  25 

I  23 

I  22 

35 

36 

5  7 

4  21 

3  5i 

3  25 

3  5 

2  49 

2  36 

2  i4 

I  59 

I  47 

I  39 

I  33 

129 

I  26 

I  24 

I  23 

36 

37 

5  i4 

4  28 

3  57 

4  2 

3  3o 

3  9 

2  53 

2  4o 

2  17 

2  2 

I  49 

I  4. 

I  34 

I  3u 

I  27 

I  25 

I  23 

37 

38 

5  2. 

4  34 

3  35 

3  i4 

2  57 

2  43 

2  20 

2  4 

I  52 

I  43 

I  3b 

I  3i 

I  27 

I  25 

I  23 

38 

39 

5  28 

4  41 

4  7 

3  39 

3  18 

3  I 

2  46 

2  23 

2  6 

I  54 

I  45 

I  37 

I  32 

I  28 

I  25 

I  23 

39 

40 

5  35 

4  47 

4  12 

3  44 

3  22 

3  4 

2  49 

2  26 

2  9 

I  56 

I  46 

I  38 

I  33 

I  29 

I  26 

I  24 

40 

4r 

5  42 

4  53 

4  17 

3  49 

3  26 

3  8 

2  52 

2  29 

2  1 1 

I  58 

£  48 

I  40 

1  34 

I  29 

I  26 

I  24 

4i 

42 

5  49 

4  5q 

4  22 

3  53 

3  3o 

3  II 

2  55 

2  3i 

2  i3 

2  0 

I  49 

I  41 

I  35 

I  3o 

I  27 

I  24 

42 

43 

5  56 

5  5 

4  27 

3  58 

3  34 

3  i5 

2  59 

2  34 

2  i5 

2  2 

I  5i 

I  42 

I  3b 

I  3i 

I  28 

I  25 

43 

44 

6  2 

5  II 

4  32 

4  3 

3  38 

3  19 

3  2 

2  36 

2  17 

2  3 

I  52 

I  44 

I  38 

I  32 

I  29 

I  2b 

44 

46 

6  i5 

5  21 

4  42 

4  II 

3  45 

3  26 

3  8 

2  4i 

2  22 

2  6 

I  55 

I  47 

I  40 

1 34 

I  3o 

I  27 

46 

48 

(i  28 

5  32 

4  52 

4  19 

3  53 

3  32 

3  i4 

2  ^5 

2  26 

2  10 

I  58 

I  49 

I  42 

I  36 

I  32 

I  28 

48 

5o 

6  4o 

5  42 

5  I 

4  27 

4  0 

3  38,3  20 

2  5o 

2  29 

2  i4 

2  I 

I  5i 

1  44 

I  37 

I  33 

I  29 

5o 

52 

6  52 

5  52 

5  10 

4  35 

4  7 

3  44 

3  25 

2  55 

2  33 

2  17 

2  4 

I  54 

I  46 

1  39 

I  34 

I  3o 

52 

54 

7  3 

6  I 

5  18 

4  42 

4  i4 

3  5o 

3  3o 

2  59 

2  37 

2  20 

2  7 

I   56 

I  48 

I  41 

I  35 

I  3i 

54 

56 

7  i4 

6  10 

5  26 

4  49 

4  20 

3  55 

3  35 

3  3 

2  4i 

2  23 

2  9 

I  58 

I  49 

I  43 

I  37 

I  32 

56 

58 

7  24 

6  18 

5  34 

4  56 

4  25 

4  0 

3  3v 

3  7 

2  44 

2  26 

2  II 

2  0 

I  52 

1   45 

I  38 

I  33 

58 

60 

7  32 

6  26 

5  4i 

5  2 

4  3o 

4  5 

3  44 

3  II 

2  47 

2  29 

2  i4 

2  2 

I  54 

I  47 

I  4o 

I  35 

60 

62 

7  4o 

6  33 

5  47 

5  7 

4  35 

4  10 

3  49 

3  i5 

2  5o 

2  3i 

2  16 

2  4 

I  55 

I  48 

I  4i 

I  36 

62 

64 

7  48 

6  40 

5  53 

5  12 

4  40 

4  i5 

3  53 

3  .9 

2  52 

2  34 

2  19 

2  6 

I  56 

I  49 

I  42 

I  37 

64 

66 

7  55 

6  47 

5  59 

5  17 

4  45 

4  19 

3  57 

3  22 

2  54 

2  36 

2  21 

2  8 

I  57 

I  5o 

1  43 

1  38 

66 

68 

8  I 

6  53 

6  4 

5  22 

4  49 

4  23 

4  1 

3  24 

2  56 

2  38 

2  22 

2  9 

I  59 

I  5i 

I  44 

I  38 

68 

70 

8  7 

6  59 

6  8 

5  26  4  53 

4  26 

4  4 

3  26 

2  58 

2  40 

2  23 

2  10 

2  0 

I  52 

I  45 

I  39 

70 

7'- 

8  12 

7  4 

6  II 

5  30*4  56 

4  29 

4  6 

3  28 

3  0 

2  4i 

2  24 

2  II 

2  I 

I  53 

I  46 

I  39 

72 

74 

6  i4 

5  3314  59 

4  3i 

4  8 

3  3o 

3  2 

2  4.' 

2  25 

2  12 

2  2 

I  54 

I  47 

I  40 

74 

76 

5  I 

4  33 

4  9 

3  32 

3  4 

2  43 

2  26 

2  i3 

2  3 

I  54 

I  47 

I  4o 

76 

-78 

4  10 

3  33 

3  6 

2  44 

2  27 

2  i4 

2  3 

I  54 

I  47 

I  4i 

78 

80 

3  34 

3  7 

2  45 

1    28 

2  i5 

2  4 

I  55 

I  47 

I  4i 

80 

8? 

3  8 

2  46 

2  2y 

2  16 

2  4 

I  55 

I  48 

I  42 

82 

84 

2  47 

2  29 

2  16 

2  5 

1  56 

I  49 

I  42 

84 

86 

2  29 

2  16 

2  6 

I  5b 

I  49 

I  42 

8b 

iP 

7° 

8° 

9° 

10° 

11° 

12° 

14° 

10° 

18° 

20^ 

22° 

24° 

26° 

28° 

30° 

TABLE  XL VIII.               t^'-s*^'-^^ 

Third  Correction.  Apparent  Distance  64°. 

])'s 
Add. 

Jjppare 

nt  JUtitudc  of  the  Sun,  Star  or  Planet. 

D's 
App 

Alt. 

32° 

34^ 

36° 

3d" 

42" 

46° 

50" 

54" 

58" 

62" 

(JG" 

70" 

74° 

78" 

82" 

m^ 

All. 

o 

1  II 

1  II 

1  II 

/  // 

/  // 

/  II 

/  II 

/  // 

/  // 

/  // 

1  II 

/  II 

/  // 

/  // 

/  // 

1  II 

0 

6 

4  29 

4  45 

5  0 

5  i5 

5  43 

6  10 

6  36 

6  59 

7  20 

7  39 

7  54 

8  7 

6 

7 

3  49 

4  2 

4  i5 

4  28 

4  53 

5  16 

5  37 

5  57 

6  i5 

6  32 

6  46 

6  59 

7 

8 

3  22 

3  34 

3  45 

3  56 

4  18 

4  38 

4  57 

5  i5 

5  3i 

5  46 

5  58 

6  7 

6  16 

8 

9 

3  0 

3  10 

3  20 

3  3o 

3  49 

4  7 

4  23 

4  38 

4  52 

5  5 

5  i6 

5  26 

5  34 

9 

lO 

2  43 

2  52 

3  I 

3  10 

3  27 

3  42 

3  56 

4  9 

4  21 

4  32 

4  42 

4  5i 

4  59 

10 

II 

2  3o 

2  37 

2  45 

2  54 

3  9 

3  22 

3  35 

3  47 

3  57 

4  7 

4  16  4  24 

4  3i 

II 

12 

2  19 

2  25 

2  33 

2  4o 

2  53 

3  5 

3  17 

3  27 

3  37 

3  47 

3  56 

4  3 

4  8 

4  i3 

12 

i3 

2  ^ 

2  l5 

2  22 

2  28 

2  40 

2  5i 

3  1 

3  II 

3  20 

3  29 

3  37 

3  43 

3  47 

3  5i 

i3 

i4 

2     I 

2   7 

2  i3 

2  18 

2  29 

2  39 

2  48 

2  57 

3  6 

3  i4 

3  20 

3   25 

3  29 

3  33 

i4 

i5 

I  54 

2   0 

2  5 

2  10 

2  19 

2  29 

2  37 

2  45 

2  53 

3  0 

3  5 

3  10 

3  .4 

3  18 

i5 

i6 

I  48 

I  53 

I  58 

2  3 

2  II 

2  20 

2  28 

2  35 

2  42 

2  48 

2  53 

2  57 

3  I 

3  5 

3  8 

16 

17 

I  43 

I  47 

I  52 

i56 

2  4 

2  12 

2  2o]2  26 

2  32 

2  38 

2  43 

2  47 

2  5i 

2  54 

2  56 

17 

i8 

I  39 

I  43 

I  47 

I  5o 

I  58 

2  5 

2  12 

2  18 

2  24 

2  3o 

2  35 

2  39 

2  42 

2  44 

2  46 

18 

19 

I  36 

I  39 

I  42 

I  46 

I  52 

I  59 

2   5 

2  II 

2  17 

2  22 

2  27 

2  3i 

2  34 

2  36 

2  38 

19 

20 

I  33 

I  36 

I  38 

I  42 

I  48 

I  54 

I  59 

2  5 

2  II 

2  i5 

2  20 

2  23 

2  26 

2  28 

2  3o 

2  32 

20 

21 

I  3o 

I  33 

I  35 

138 

I  44 

I  49 

1  54 

2  0 

2  5 

2  9 

2  i3 

2  16 

2  18 

2  20 

2  22 

2  23 

21 

22 

I  28 

I  3o 

I  33 

I  35 

I  40 

I  45 

I  5o 

I  55 

I  59 

2  0 

2  6 

2  0 

2  II 

2  i3 

2  i5 

2  16 

22 

23 

I  27 

I  28 

I  3o 

I  32 

I  37 

I  4i 

I  46 

I  5i 

I  54 

I  58 

2  I 

2  3 

2  5 

2  7 

2  Q 

2  ID 

23 

24 

I  s6 

I  27 

I  26 

I  3o 

I  34 

I  38 

I  42 

I  47 

I  5o 

I  54 

I  57 

I  59 

2  0 

2  2 

2  4 

2   5 

24 

25 

I  25 

I  26 

I  27 

I  28 

I  32 

I  35 

I  39 

I  43 

I  47 

I  5o 

I  53 

I  55 

I  56 

I  58 

I  59 

2   0 

25 

26 

I  24 

I  25 

I  26 

I  27 

I  3o 

I  33 

I  36 

I  4o 

I  44 

I  47 

I  49 

I  5i 

I  52 

1  54 

1  55 

I  56 

26 

27 

I  23 

I  24 

I  25 

I  26 

I  28 

I  3i 

I  34 

I  37 

I  4i 

I  44 

I  46 

I  47 

I  4q 

I  5o 

I  5i 

I  52 

27 

28 

I  23 

I  23 

I  24 

I  25 

I  26 

I  29 

I  32 

I  35 

I  38 

I  4i 

I  43 

I  44 

I  45 

I  46 

1  47 

I  48 

28 

29 

I  22 

I  22 

I  23 

I  24 

I  25 

I  27 

I  3o  I  32 

I  35 

I  38 

I  4o 

I  4i 

I  42 

1  43 

I  44 

I  45 

29 

3o 

1  22 

I  22 

r  23 

I  23 

I  24 

I  26 

I  28  I  3o 

I  33 

I  35 

I  37 

I  38 

I  39 

I  4o 

I  4" 

I  42 

3o 

3i 

I  22 

I  22 

I  22 

I  22 

I  23 

I  24 

I  261  28 

I  3i 

I  33 

I  34 

I  35 

1  36 

1  3- 

1  38 

1  4o 

3i 

32 

I  21 

I  21 

I  21 

I  21 

I  22 

I  23 

I  25  I  27 

I  29 

I  3i 

I  32 

I  33 

I  34 

I  35 

I  36 

I  38 

32 

33 

I  21 

I  21 

I  21 

I  21 

I  21 

I  22 

I  24 

I  26 

I  27 

I  29 

I  3o 

I  3i 

I  32 

I  33 

I  34 

33 

M 

I  21 

I  20 

I  20 

I  20 

I  20 

I  21 

I  23 

I  25 

I  26 

I  27 

I  28 

I  29 

I  3o 

I  3i 

I  32 

34 

35 

I  21 

I  20 

I  20 

I  20 

I  20 

I  21 

I  22 

I  23 

I  24 

[  25 

I  26 

I  27 

I  28 

1  29 

1  3o 

35 

36 

I  21 

I  20 

I  19 

I  19 

I  19 

I  20 

I  21 

I  22 

I  23 

I  24 

I  25 

I  26 

I  26 

1  2-1 

I  28 

36 

37 

I  21 

I  20 

I  19 

I  19 

I  19 

I  19 

I  20 

I  21 

I  22 

I  23 

I  24 

I  25 

I  25 

I  j6 

37 

38 

I  21 

I  20 

I  19 

I  18 

I  18 

I  18 

I  19 

I   20 

I  21 

I  22 

I  23 

I  24 

I  24 

I  25 

38 

39 

I  21 

I  20 

I  19 

I  18 

I  18 

I  18 

I  18 

I  19 

I  20 

I  21 

I  21 

I  22 

t  22 

I  23 

39 

4o 

I  22 

I  20 

I  19 

I  18 

I  17 

I  17 

I  18 

r  18 

I  19 

I  20 

I  20 

I  21 

I  21 

I  22 

40 

4i 

I  22 

I  20 

I  19 

I  18 

I  17 

I  17 

I  17 

I  17 

I  18 

I  19 

I  19 

1  20 

I  20 

4i 

42 

I  22 

I  20 

I  19 

I  18 

I  16 

I  16 

I  16 

I  17 

I  17 

I  18 

I  18 

I  19 

I  19 

42 

43 

I  23 

I  21 

I  19 

I  18 

I  16 

r  16 

I  16 

I  16 

I  16 

I  17 

I  17 

I  18 

I  18 

43 

44 

I  23 

I  21 

I  19 

I  18 

I  16 

I  16 

I  16 

I  16 

I  16 

I  16 

I  16 

I  17 

I  17 

44 

46 

I  24 

I  22 

I  20 

I  18 

I  16 

I  i5 

I  i5 

I  i5 

I  i5 

I  i5 

I  i5 

I  16 

46 

48 

I  25 

I  22 

I  20 

I  19 

I  16 

I  i5 

I  i5 

I  i4 

I  i4 

I  i4 

I  i4 

I  i4 

48 

5o 

I  26 

I  23 

I  21 

I  19 

I  16 

I  i5 

I  i4 

I  i3 

I  i3 

I  i3 

I  i3 

5o 

52 

I  27 

1 24 

I  22 

I  20 

I  17 

I  i5 

I  i3 

I  12 

I  12 

I  12 

I  12 

5? 

54 

I  28 

I  25 

I  22 

I  20 

I  17 

I  i5 

I  i3 

I  12 

I  II 

I  II 

54 

56 

58 
60 
62 

I  29 

1 29 

I  3o 
I  3i 

I  26 
I  26 

I  27 

,  28 

I  23 
I  23 

I  a4 

I  25 

I  21 
I  21 
I  22 
I  22 

I  17 
I  18 
I  18 
I  18 

I  i5 
I  i5 
I  i5 
I  i5 

I  i3 
I  i3 
I  i3 
I  i3 

I  12 

I  II 
I  II 
r  II 

I  II 

I  1 1 

56 

I  10 
I  10 

TaUe  P.  Effect  of  Sun's  Par. 

AM  Ihn  Numl)ers  above  llip  lines 

64 

I  32 

1  2b 

r  2,5 

I  22 

I  18 

r  t5 

I  i'^ 

I  1 1 

to  Tliird  Correclion  ;  subtract 

66 

r  33 

r  29 

I  26 

I  23 

I  18 

I  ifi 

I  t3 

tlie  others. 

68 

70 

I  33 

.  29 
I  3o 

I  26 

3   

I  23i I  19 

I  24  I  19 

I  16 
I  16 

I  i3 

D'.s 

Smi'a  Appivrcnt  Altiluile. 

I  34 

I  27 

All. 

5 

020 

30 

40 

50  t 

0  70l SO 

90 

72 

I  34 

I  3o 

I  27 

I  241 1  19 

I  16 

74 

76 

I  35 
I  35 

I  3i 
I  3i 

I  28 
I  28 

I  24 i I  19 
I  25, I  20 

5 
10 

20 

1 
3 

1  0 
3  2 

0 

r 

1 
I 

2 

0 

I    2 
0  I 

1 

78 

I   30 

I  32 

I  28 

I  25i 

30 

5 

4  3 

3 

2 

2 

1  1 

1 

0 

80 

I  36 

1  32 

I  2S]I  25 

40 

S 

6  5 

4 

4 

3 

3  2 

2 

82 

I  37 

I  32 

I  28 

50 

r 

7  6 

5 

5 

4 

4 

84 

I  37 

I  32 

60 

s 

8  7 

6 

6 

5 

86 

1^37 

70 
80 

9 

9  8  7 
8  8 

7 
7 

6 

32° 

34" 

3b'" 

38" 

42" 

46" 

50" 

54° 

58° 

62° 

m'' 

90 

8 

38 


Pase  29ej                       TABLE 

XLVIII 

_..,  _....  ^ 

Third  Correction.  Apparent  Distance  68°. 

1 

App. 

Apparent  Altitude  of  the  Su}i,  Star  or 

Planet. 

D's 
Ajjp. 

Alt. 

6° 

70 

8^ 

9" 

lU'^ 

IP 

12" 

14" 

16" 

18° 

2U° 

22° 

24" 

26" 

28" 

;jO° 

Alt. 

o 

1    II 

/  // 

?  II 

/  // 

/  // 

/  II 

/  // 

/  // 

/  // 

/  ;/ 

/  /( 

/  II 

/  // 

/  // 

;  ;/ 

/  // 

0 

6 

I  2Q 

I  3i 

I  34 

I  37 

I  4i 

I   46 

I  52 

2  6 

2  21 

2  36 

2  52 

3  8 

3  24 

3  39 

3  54 

4  10 

6 

7 

I  32 

I  20 

I  3i 

I  33 

I  36 

I  39 

I  43 

I  54 

2  5 

2  17 

2  3o 

2  43 

2  56 

3  9 

3  22 

3  36 

7 

8 

I  36 

I  3i 

r  29 

I  3o 

I  32 

I  34 

I  37 

I  45 

I  54 

2  4 

2  14 

2  25 

2  37 

2  48 

2  59) 

3  II 

8 

9 

I  4i 

I  34 

I  3, 

I  29 

I  3c) 

I  3i 

I  33 

I  38 

I  46 

I  54 

2  3 

2  12 

2  22 

2  32 

2  42 

2  52 

9 

lO 

I  46 

I  38 

I  33 

I  3o 

I  29 

I  3(j 

I  3i 

I  M 

I  40 

I  47 

I  54 

2  2 

2  10 

2  19 

2  28 

2  36 

10 

II 

I  52 

I  43 

I  36 

I  32 

I  3o 

I  29 

I  3o 

I  32 

I  36 

I  4i 

I  47 

I  54 

2  1 

2  9 

2  16 

2  24 

It 

12 

I  59 

I  48 

I  4o 

I  35 

I  32 

I  3o 

I  29 

I  3o 

I  33 

I  37 

I  42 

I  48 

I  54 

2  0 

2  7 

2  i4 

12 

i3 

2  6 

I  53 

I  44 

I  38 

I  34 

I  32 

I  3o 

I  29 

I  3i 

I  34 

I  38 

I  43 

I  48 

I  53 

I  59 

2  5 

i3 

i4 

2  14 

I  59 

I  49 

I  42 

I  37 

I  34 

I  3i 

I  29 

I  3o 

I  32 

I  35 

I  39 

I  44 

I  48 

I  53 

I  58 

i4 

i5 

2  21 

2  5 

.  54 

I  46 

I  4o 

I  36 

I  33 

I  3o 

I  3o 

I  3i 

I  33 

I  36 

I  40 

I  44 

I  48 

I  53 

i5 

i6 

2  2b 

2  1 1 

I  59 

I  5o 

I  44 

I  39 

I  35 

I  3i 

I  29 

I  3o 

I  32 

I  M 

I  37 

I  4o 

I  44 

1  48 

16 

17 

2  36 

2  17 

2  4 

I  54 

I  47 

I  42 

I  38 

I  32 

I  29 

I  2() 

I  3o 

I  32 

I  34 

I  37 

I  4o 

I  44 

17 

i8 

2  4i 

2  24 

2  10 

I  59 

I  5i 

I  45 

I  4o 

I  M 

I  3o 

I  28 

I  29 

I  3o 

I  32 

I  35 

I  37 

I  40 

18 

19 

2  52 

2  3o 

2  i5 

2  4 

I  55 

I  48 

I  4i 

I  35 

I  3i 

I  28 

I  28 

I  29 

I  3i 

I  Si 

I  35 

I  37 

'9 

20 

3  0 

2  36 

2  21 

2  8 

I  59 

I  52 

I  46 

I  37 

I  32 

I  29 

I  28 

I  29 

I  3o 

I  3i 

I  33 

I  35 

20 

•21 

3  8 

2  43 

2  26 

2  i3 

2  3 

I  55 

I  48 

I  39 

I  33 

I  3o 

I  28 

I  28 

I  29 

I  3o 

I  3i 

I  33 

21 

22 

3  i5 

2  49 

2  32 

2  17 

2  7 

I  58 

I  5i 

I  4i 

I  35 

I  3i 

I  29 

I  27 

I  28 

I  29 

I  3o 

I  3i 

22 

23 

3  23 

2  56 

2  37 

2  22 

2  11 

2  2 

I  54 

I  4i 

I  37 

I  32 

1  29 

I  27 

I  27 

I  28 

I  29 

I  3o 

23 

24 

3  3i 

3  3 

2  43 

2  27 

2  i5 

2  5 

I  57 

I  46 

I  39 

I  34 

I  3o 

I  28 

I  27 

I  28 

I  28 

I  29 

24 

25 

3  39 

3  9 

2  48 

2  32 

2  19 

2  9 

2  0 

I  48 

I  4i 

I  35 

I  3i 

I  29 

I  27 

I  27 

I  27 

I  28 

25 

26 

3  47 

3  16 

2  54 

2  37 

2  23 

2  12 

2  4 

I  5i 

I  43 

I  36 

I  32 

I  3o 

I  28 

I  27 

I  27 

I  27 

26 

27 

3  55 

3  23 

3  0 

2  42 

2  27 

2  16 

2  7 

I  54 

I  44 

I  37 

I  33 

I  3o 

I  28 

I  27 

I  26 

I  27 

27 

28 

4  2 

3  29 

3  5 

2  47 

2  3i 

2  19 

2  10 

I  56 

I  46 

I  39 

I  34 

I  3i 

I  29 

I  27 

I  26 

I  26 

28 

2P 

4  10 

3  36 

3  11 

2  52 

2  35 

2  23 

2  14 

I  59 

I  48 

I  4i 

I  35 

I  32 

I  29 

I  27 

I  26 

I  26 

29 

3o 

4  17 

3  42 

3  16 

2  57 

2  4o 

2  27 

2  17 

2  I 

I  5o 

I  42 

I  36 

I  32 

I  29 

I  27 

I  26 

I  26 

3o 

3i 

4  25 

3  4q 

3  22 

3  2 

2  44 

2  3i 

2  20 

2  3 

I  52 

I  43 

I  37 

I  33 

I  3o 

I  28 

I  27 

I  26 

3i 

32 

4  32 

3  55 

3  27 

3  7 

2  49 

2  34 

2  23 

2  6 

I  54 

I  45 

I  38 

I  ii 

I  3o 

I  28 

I  27 

I  26 

32 

33 

4  4o 

4  2 

3  33 

3  12 

2  53 

2  38 

2  26 

2  9 

I  56 

I  47 

1  39 

I  34 

I  3i 

I  29 

I  27 

I  26 

33 

34 

4  48 

4  8 

3  39 

3  16 

2  57 

2  42 

2  3o 

2  12 

I  58 

I  48 

I  4i 

I  35 

I  32 

I  3o 

I  28 

I  26 

34 

35 

4  55 

4  i5 

3  45 

3  21 

3  2 

2  46 

2  34 

2  i5 

2  0 

I  5o 

I  43 

I  37 

I  66 

I  60 

I  28 

I  26 

35 

36 

5  2 

4  21 

3  5o 

3  26 

3  6 

2  5o 

2  3? 

2  17 

2  3 

I  52 

I  44 

I  38 

I  34 

I  3i 

I  28 

1  26 

36 

37 

5  10 

4  27 

3  56 

3  3o 

3  to 

2  53 

2  4i 

2  20 

2  5 

I  54 

I  46 

I  39 

I  35 

I  3i 

I  28 

I  26 

37 

38 

5  11 

4  33 

4  I 

3  35 

3  i4 

2  57 

2  44 

2  22 

2  7 

I  56 

I  48 

I  4i 

I  36 

I  32 

I  29 

I  27 

38 

3g 

5  24 

4  39 

4  6 

3  4o 

3  18 

3  I 

2  47 

2  25 

2  9 

I  58 

I  5o 

I  43 

I  37 

I  'di 

I  3q 

I  27 

39 

4o 

5  3i 

4  45 

4  II 

3  45 

3  22 

3  5 

2  5o 

2  27 

2  II 

2  0 

I  5i 

I  44 

I  3S 

I  M 

1  3i 

.  28 

40 

4i 

5  58 

4  5i 

4  16 

3  49 

3  26 

3  9 

2  53 

2  3o 

2  i4 

2  2 

I  53 

I  45 

I  39 

I  35 

I  3i 

I  28 

4i 

42 

5  44 

4  57 

4  21 

3  53 

3  3o 

3  12 

2  56 

2  32 

2  16 

2  4 

I  54 

I  46 

I  4o 

I  36 

I  32 

I  29 

42 

43 

5  5(1 

5  2 

4  26 

3  58 

3  34 

3  16 

2  59 

2  34 

2  19 

2  6 

I  56 

I  48 

I  4i 

I  37 

I  ii 

I  3o 

43 

44 

5  57 

5  8 

4  3i 

4  2 

3  38 

3  19 

3  3 

2  37 

2  21 

2  8 

I  57 

I  49 

I  43 

I  38 

I  34 

I  3i 

44 

46 

6  10 

5  19 

4  4i 

4  10 

3  46 

3  26 

3  9 

2  42 

2  25 

2  II 

I  59 

I  5i 

I  45 

I  4o 

I  35 

I  3i 

46 

48 

6  22 

5  29 

4  5o 

4  18 

3  53 

3  32 

3  i5 

2  47 

2  29 

2  i4 

2  2 

I  54 

I  47 

I  4i 

I  36 

I  32 

48 

5o 

6  34 

5  3q 

4  59 

4  26 

3  5q 

3  38  3  21 

2  52 

2  33 

2  18 

2  5 

I  56 

I  49 

I  43 

I  38 

I  33 

5o 

52 

6  45 

5  48 

^  7 

4  33 

4  6 

3  44  3  26 

2  56 

2  36 

2  21 

2  8 

I  58 

I  5i 

I  45 

.  39 

I  35 

52 

54 

6  56 

5  57 

5  i4 

4  4o  4  12 

3  5o3  3i 

3  0 

2  39 

2  24 

2  II 

2  0 

I  52 

I  46 

I  40 

I  36 

54 

56 

7  6 

6  6 

5  21 

4  46j4  18 

3  55 

3  36 

3  4 

2  42 

2  27 

2  14 

2  2 

I  54 

I  47 

I  4i 

I  37 

56 

'ItH' 

7  i5 

6  i4 

5  28 

4  52  4  24 

4  0 

3  4i 

3  8 

2  45 

2  29 

2  16 

2  4 

I  56 

I  49 

I  43 

I  38 

58 

60 

7  24 

6  22 

5  35 

4  58  4  29 

4  5 

3  45 

3  12 

2  48 

2  32 

2  18 

2  6 

I  58 

I  5i 

I  45 

I  39 

60 

Cn 

7  33 

6  29 

5  42 

5  3  4  34 

4  10 

3  49 

3  i5 

2  5i 

2  34 

2  20 

2  8 

I  59 

I  52 

I  46 

I  4o 

62 

64 

7  4i 

6  35 

5  48 

5  8 

4  39 

4  14 

3  53 

3  18 

2  54 

2  36 

2  22 

2  10 

2  I 

I  53 

I  47 

I  4i 

64 

66 

7  48 

6  41 

5  53 

5  i3 

4  4i 

4  18 

3  5- 

3  21 

2  56 

2  38 

2  24 

2  12 

2  2 

I  54 

I  48 

I  42 

66 

58 

7  55 

6  47 

5  58 

5  17 

4  4i 

4  22 

4  0 

3  24 

2  59 

2  4o 

2  26 

2  i4 

2  3 

I  55 

I  49 

I  43 

68 

70 

8  I 

6  52 

6  3 

5  21 

4   5i 

4  25 

4  3 

3  27 

3  1 

2  42 

2  27 

2  i5 

2  4 

I  56 

I  5o 

I  44 

70 

72 

8  7 

6  57 

6  8 

5  25  4  55 

4  28 

4  6 

3  3o 

3  3 

2  44 

2  28 

2  i5 

2  5 

I  57 

I  5i 

I   45 

72 

74 

8  12 

7  I 

6  12 

5  29  4  58 

4  3o 

4  8 

3  32 

3  5 

2  45 

2  29 

2  16 

2  5 

I  57 

I  5i 

I  45 

74 

76 

8  17 

7  5 

6  t5 

5  32  5  I 

4  32 

4  10 

3  34 

3  7 

2  46 

2  3o 

2  17 

2  6 

I  58 

I  5i 

I  45 

76 

7« 

6  18 

5  35'5  3 

4  34 

4  12 

3  35 

3  9 

2  47 

2  3i 

2  18 

2  7 

I  58 

I  52 

I  46 

78 

80 

5  5 

4  36 

4  i3 

3  36 

3  10 

2  48 

2  32 

2  18 

2  7 

I  59 

I  52 

I  46 

80 

82 

4  i4 

3  37 

3  II 

2  49 

2  32 

2  19 

2  8 

.1  59 

I  52 

I  46 

82 

84 

3  38 

3  n 

2  5o 

2  33 

2  20 

2  9 

2  0 

I  53 

I  46 

84 

86 

3  12 

2  5o 

2  33 

2  20 

2  9 

2  0 

I  53 

86 

G° 

7° 

8° 

9° 

10° 

11° 

12° 

14° 

16° 

Ife-^ 

20° 

22° 

24° 

26° 

28° 

30° 

—  1 
TABLE  XLVIII.               [iv.ge  201. 

Third  Correction.  Apparent  Distance  68°. 

D's 
App. 

Jlpparent  Altitude  of  the  Sun,  Star  or  Plunct. 

App. 

Alt. 

32° 

;j4" 

yG'^ 

yb^ 

42° 

4G° 

50° 

54° 

58° 

G2° 

6G° 

70° 

74° 

78° 

82° 

8G° 

Alt. 

o 

1   II 

1  II 

/  II 

1  II 

/  // 

/  // 

/  II 

/  II 

/  // 

/  // 

/  // 

/  // 

/  II 

/  // 

/  // 

0 

6 

4  25 

4  40 

4  55 

5  II 

5  4o 

6  5 

6  29 

6  5i 

7  11 

7  29 

7  45 

8  0 

S  i4 

6 

7 

3  49 

4  I 

4  i4 

4  27 

4  52 

5  i5 

5  35 

5  53 

6  ic 

6  25 

6  38 

6  5o 

7  1 

7 

8 

3  22 

3  33 

3  44 

3  55 

4  17 

4  37 

4  55 

5  II 

5  25 

5  38 

5  5i 

6  2 

6  10 

6  iS 

8 

9 

3  I 

3  II 

3  21 

3  3o 

3  48 

4  6 

4  22 

4  36 

4  49 

5  0 

5  10 

5  19 

5  28 

5  36 

9 

10 

2  44 

2  53 

3  2 

3  10 

3  25 

3  4i 

3  55 

4  9 

4  21 

4  3i 

4  4o 

4  49 

4  57 

5  3 

10 

II 

2  3i 

2  39 

•2  47 

2  54 

3  8 

3  22 

3  35 

3  47 

3  5b 

4  8 

4  16 

4  23 

4  29 

4  34 

1 1 

12 

2  20 

2  27 

2  34 

2  4i 

2  53 

3  6 

3  18 

3  29 

3  39 

3  48 

3  55 

4  2 

4  7 

4  II 

4  i5 

12 

i3 

2  II 

2  17 

2  23 

2  29 

2  4i 

2  52 

3  2 

3  12 

3  22 

3  3o 

3  37 

3  43 

3  48 

3  52 

3  56 

i3 

i4 

2  3 

2  9 

2  14 

2  19 

2  3o 

2  4o 

2  49 

2  58 

3  6 

3  i4 

3  20 

3  26 

3  3i 

3  35 

3  38 

14 

i5 

I  57 

2  2 

2  6 

2  II 

2  21 

2  3o 

2  38 

2  46 

2  54 

3  I 

3  7 

3  12 

3  16 

3  20 

3  23 

i5 

i6 

I  52 

I  56 

2  0 

2  4 

2  i3 

2  21 

2  29 

2  37 

2  4/ 

2  5o 

2  55 

3  0 

3  4 

3  8 

3  10 

3  12 

16 

I? 

I  47 

I  5i 

I  55 

I  58 

2  6 

2  i4 

2  21 

2  29 

2  35 

2  4o 

2  45 

2  49 

2  53 

2  57 

2  59 

3  0 

17 

i8 

I  43 

I  47 

I  5o 

I  54 

2  I 

2  8 

2  i4 

2  21 

2  27 

2  32 

2  36 

2  4o 

2  44 

2  47 

2  49 

2  5o 

iS 

19 

I  40 

I  4^ 

r  46 

I  5o 

I  56 

2  2 

2  8 

2  i5 

2  20 

2  2D 

2  29 

2  32 

2  36 

2  39 

2  4i 

2  4? 

'9 

20 

I  37 

I  4o 

I  4'6 

I  46 

t  D2 

I  57 

2  3 

2  9 

2  i4 

2  18 

2  22 

2  25 

2  28 

2  3i 

2  33|2  34 

20 

21 

I  35 

I   37 

I  4<j 

I  43 

I  48 

I  53 

I  58 

2  3 

2  8 

2  12 

2  16 

2  19 

2  21 

2  23 

2  25 

2  26 

21 

22 

I  6i 

I  33 

I  37 

I  4o 

I  44 

I  49 

I  54 

I  58 

2  2 

2   6 

2  10 

2  l3 

2  i5 

2  17 

2  19 

2  20 

22 

23 

I  3i 

I  :i-^ 

I  35 

I  37 

I  4i 

I  46 

I  5o 

I  54 

I  57 

2   I 

2  5 

2   8 

2  10 

2  12 

2  14 

2  i5 

23 

24 

I  3o 

I  3i 

I  33 

I  35 

I  39 

I  43 

I  47 

I  5o 

I  53 

I  57 

2  0 

2   3 

2  5 

2  7 

2   9 

2  10 

24 

25 

I  29 

I  3o 

I  3. 

I  33 

I  37 

I  4o 

I  44 

I  47 

I  5o 

I  53 

I  56 

I  59 

2  I 

2  2 

2   4 

2  5 

25 

26 

I  28 

I  29 

I  3o 

1  32 

I  35 

I  38 

I  4i 

I  44 

I  47 

I  5o 

I  53 

I  55 

I  57 

I  58 

I  59 

2  0 

26 

27 

I  27 

I  28 

I  29 

I  3o 

I  33 

I  36 

I  38 

I  4i 

I  44 

I  47 

I  5o 

I  52 

I  53 

I  54 

I  55 

I  56 

27 

28 

I  27 

I  27 

I  28 

I  29 

I  3i 

I  M 

I  36 

I  39 

I  4i 

I  44 

I  47 

1  49 

I  5o 

I  5i 

I  52 

I  52 

28 

29 

I  26 

I  26 

I  27 

1  28 

I  29 

I  32 

I  34 

I  37 

I  39 

I  4i 

I  44 

I  46 

I  47 

I  48 

I  49 

29 

Jo 

I  26 

I  26 

I  26 

I  27 

I  28 

I  3o 

I  32 

I  35 

I  37 

I  39 

I  4> 

I  43 

I  44 

I  45 

I  46 

3o 

3i 

I  25 

I  25 

I  26 

I  26 

I  27 

I  29 

I  3i 

I  33 

I  35 

I  37 

I  39 

I  4o 

I  4i 

I  42 

I  43 

3i 

32 

I  25 

I  25 

I  25 

I  25 

I  26 

I  28 

I  29 

I  3i 

I  33 

I  35 

I  37 

I  38 

I  39 

I  40 

t  4i 

32 

33 

I  25 

I  24 

I  25 

I  25 

I  26 

I  27 

I  28 

I  3o 

I  3i 

I  33 

I  35 

I  36 

T  37 

T  38 

33 

M 

I  25 

I  24 

I  24 

I  24 

I  25 

I  26 

I  27 

I  29 

I  3o 

I  3i 

I  33 

I  34 

I  35 

I  36 

34 

36 

I  25 

I  24 

I  24 

I  24 

I  24 

I  25 

I  26 

I  28 

I  29 

I  3o 

I  3i 

I  32 

I  33 

I  34 

35 

36 

I  25 

1  24 

I  23 

I  23 

I  23 

I  24 

I  25 

I  27 

I  28 

I  29 

I  3o 

I  3o 

1  3i 

I  3? 

36 

J7 

I  2D 

I  24 

I  23 

I  23 

I  23 

I  23 

I  24 

I  26 

I  27 

I  28 

I  29 

I  29 

I  3o 

37 

38 

I  25 

I  24 

I  23 

I  22 

I  22 

I  23 

I  24 

I  25 

I  26 

I  27 

I  28 

I  28 

I  29 

38 

39 

I  23 

I  24 

I  23 

I     22 

1  22 

I  23 

I  23 

I  24 

I  25 

I  26 

I  27 

I  27 

I  27 

39 

4o 

I  26 

I  25 

I  24 

I  23 

I  22 

I  22 

I  23 

I  23 

I  24 

I  25 

I  26 

I  26 

I  26 

40 

4i 

I  26 

I  25 

I  24 

I  23 

I  21 

I  21 

1  22 

I  22 

I  23 

I  24 

I  25 

I  25 

4i 

42 

I  27 

I  25 

I  24 

I  23 

I  21 

I  21 

I  21 

I  22 

I  23 

I  23 

I  24 

I  24 

42 

43 

I  27 

I  251 

I  24 

I  23 

I  21 

I     21 

I  21 

I  21 

I  22 

I  22 

I  23 

I  23 

43 

44 

I  28 

I  26 

I  24 

I  23 

I  21 

I  20 

I  20 

I  20 

I  21 

I  21 

I  22 

I  22 

44 

46 

I  28 

I  26! 

I  25 

I  24 

I  21 

I  19 

I  19 

I  19 

I  20 

I  20 

I  20 

46 

48 

I  29 

I  27J 

I  25 

1  24 

I  22 

I  19 

I  18 

I  18 

I  19 

I  19 

I  19 

48 

bo 

I  3<) 

I  28 

I  26 

I  25 

I  22 

I  20 

I  18 

I  18 

I  18 

I  18 

5o 

52 

I  3i 

I  29 

I  27 

I  25 

I  22 

!  20 

I  18 

I  17 

I  17 

I  17 

52 

54 

I  3a 

I  29 

I  27 

I  20 

I  23 

I  20 

I  18 

I  17 

I  16 

54 

56 
58 

I  33 
I  34 

I  3o 
I  3i 

I  28 

I  29 

I  26 

I  27 

I  23 
1  23 

I  20 
I  20 

I  18 
I  18 

I  16 
I  16 

I  i5 

56 

1 

60 
62 

I  35 
I  36 

1  32 

I  33 

I  29 

I  3o 

I  27 
I  28 

I  23 
I  23 

I  2(.) 
I  20 

I  18 
I  18 

I  16 

Table  P.  Effect  of  Sun's  Par. 
Add  tlip  Niiinltprs  al.ove  the  lines 

64 

I  37 

I  63 

I  3f. 

I  28 

I  24 

I  10 

I  17 

10  Tliird  Ciirreciinn  ;  siiLlracI 

66 

I  38 

1  34 

I  3i 

r  28 

1  24 

I  20 

the  others. 

68 

I  38 

I  34 
I  35 

I  3i 

I  28 

I  24 
I  24 

D's 

Sun's  Apparent  Allilmle. 

70 

I  39 

I  32 

I  29 

App- 
Alt. 

5 

u  -io 

30 

•10 

5U6 

a  70 

?0 

ao 

72 

I  39 

I  35 

I  32 

1  29 

I  24 

74 

I  40 

1  36 

I  32 

I  29 

5 

1 

u  u 

' 

-  ~ 

76 

I  40 

I  36 

I  3? 

I  29 

10 

'•i 

0 

- 

20 

H 

1  'J 

1 

' 

0  1 

0 

u 

0 

78 

1  4i 

I  36 

I  32 

30 

4 

4 

3 

2 

2  2 

1 

1 

80 

I  4i 

I  36 

40 

6 

;  s 

4 

4 

3  3 

3 

82 

I  4i 

SO 

7 

r   6 

6 

5 

5  4 

84 

GO 

8 

i    7 

7 

6 

6 

86 



70 
SO 

9 

i   8 

1  s 

7 

8 

7 

w 

;{'<!" 

'.n^ 

'3b^ 

38" 

42° 

4()° 

50° 

54° 

58° 

62° 

GG° 

90 

8 

F^^m                                    TABLE  XLVIII. 

Third  Correction.  Apparent  Distance  72°. 

D's 

Apparent  Altitude  of  tile  Sun,  Star  or  Planet. 

D  's 
Vpp. 

All. 

0 

App. 
Alt. 

o 

6° 

7° 

8° 

9^ 

10° 

/  // 

11° 

/  II 

12° 

/  // 

14° 

/  // 

16° 

/  // 

18° 

20° 

Si2° 
1   II 

24° 

/  II 

26° 
/  // 

28° 

30° 

/  ti 

/  // 

It 

1  II 

6 

T  33 

I  35 

I  37 

40 

I  44 

1   5o 

I  56 

2  9 

2  23 

I   38  3 

53  3  9I 

3  243  4ol 

3  56  z 

'  12 

6 

7 

I  35 

£  331 1  34 

[  36 

I   39 

I  4i 

I  47 

I  56 

2   8 

I   21  ' 

I  34: 

i  4i 

3  0. 

]   12. 

i   25: 

i  38 

7 

8 

I  39 

I  35  I  33 

[  34 

I  3b 

I  38 

I  4i 

I  48 

I  58 

I     8: 

I   19: 

I  3o 

2  4t 

I    52 

3  33  i4| 

8 

9 

I  44 

I  38 

I  35 

I  33 

I  34 

I  35 

I  37 

I  42 

I  5o 

I  58 

i  7 

I   17 

2  26 

2  35 

2  44'. 

I   54 

9 

10 

I  5o 

I  42 

I  37 

I  34 

I  33 

I  M 

I  35 

I  38 

I  44 

I  5o 

58 

2  6 

2  i4 

2  22 

2  3o 

2  39 

10 

II 

I  56 

I  46 

I  40 

I  36 

I  34 

I  33  I  34 

I  36 

I  40 

I  45 

[  5i 

58 

2  5 

2  12 

2   20 

2  27 

1 1 

12 

2  2 

I  5i 

I  44 

I  39 

I  30 

I  34' I  33 

I  35 

I  37 

I  4i 

.  46 

[  52 

I  58 

2  4 

2  II  2  17] 

1  2 

i3 

2  9 

I  56 

I  48 

I  42 

I  39 

I  36 

I  34 

I  34 

I  35 

I  38 

[  42 

V   47 

I  D2 

I  58 

2  4  2  (^1 

i3 

i4 

2  iti 

2  2 

I  53 

I  46 

I  42 

I  39 

1   36 

I  33 

I  34 

I  36 

.  39 

[  43 

I  47 

I  52 

I  57 

2   2 

i4 

i5 

2  23 

2  8 

I  58 

I  5o 

I  45 

I  4i 

I  33  I  341 

I  36 

I  34 

I  36 

I   39 

I  43 

I  47 

I  5i 

1  56 

i5 

i6 

2  3o 

2  i4 

2  3 

I  54 

I  48 

I  43 

I  40 

I  35 

I  33 

I  33 

I  34 

I  36 

1 39 

I  43 

1  47 

I  52 

16 

17 

2  37 

2  20 

2  8 

I  58 

I  5i 

I  46 

I  42 

I  36 

I  34 

I  33 

I  34 

I  35 

1 37 

1  4«' 

I  44 

I  48 

17 

t8 

2  45 

2  27 

2  i3 

2  2 

I  54 

I  48 

I  44 

I  37 

I  34 

I  33 

I  33 

I  34 

I  36 

I  38 

I  4i 

I  44 

18 

19 

2  53 

2  33 

2  18 

2  7 

I  58 

I  5i 

I  46 

I  39 

I  35 

I  33 

I  33 

I  34 

I  35 

I  37 

I  39 

I  4i 

19 

20 

3  I 

2  4o 

2  24 

2  II 

2  2 

I  54 

I  49 

I  4i 

I  36 

I  34 

I  33 

I  33 

I  34 

I  35 

I  37 

I  39 

20 

21 

3  9 

3  17 

2  46 

2  29 

2  16 

2  6 

I  58 

I  52 

I  43 

I  37 

I  34 

133 

I  33 

I  33 

I  34 

I  35 

I  37 

21 

22 

2  53 

2  35 

2  20 

2  10 

2  2 

I  55 

I  45 

I  39 

I  35 

I  33 

I  32 

I  33 

I  33 

I  3A 

I  35 

22 

23 

3  25 

2  59 

2  4o 

2   25 

2  i4 

2  5 

I  58 

I  47 

I  4o 

I  36 

I  34 

I  32 

I  32 

I  33 

I  33 

I  34 

23 

24 

3  33 

3  6 

2  46 

2  3o 

2  18 

2  8 

2  I 

I  5o 

I  42 

I  37 

I  34 

I  32 

I  32 

I  32 

I  33 

I  34 

24 

25 

3  4i 

3  12 

2  5i 

2  35 

2  23 

2  12 

2  4 

I  52 

I  44 

I  38 

I  35 

I  33 

I  32 

I  32 

I  32 

I  33 

25 

26 

3  48 

3  18 

2  57 

2  40 

2  27 

2  16 

2  8 

I  55 

1   46 

I  40 

I  36 

1   33 

I  32 

I  3i 

I  3i 

I  32 

26 

27 

3  56 

3  25 

3  2 

2  45 

2  3i 

2  20 

2  12 

I  57 

I  48 

I  4i 

1  37 

I  34 

I  32 

I  3i 

I  3i 

1  3i 

27 

28 

4  3 

3  3i 

3  7 

2  49 

2  35 

2  24 

2  i5 

2  0 

I  5o 

I  43 

I  38 

I  34 

I  32 

I  3i 

I  3c^ 

I  3i 

28 

29 

4  II 

3  37 

3  i3 

2  54 

2  39 

2  27 

2  18 

2  2 

I  52 

I  45 

I  39 

I  35 

I  33 

I  32 

I  3i 

I  3o 

29 

3o 

4  18 

3  44 

3  19 

2  59 

2  43 

2  3i 

2  21 

2  5 

I  54 

I  46 

I  4o 

I  36 

I  34 

I  32 

I  3i 

I  3o 

3o 

3t 

4  26 

3  5o 

3  24 

3  4 

2  47 

2  34 

2  24 

2  8 

I  56 

I  48 

I  4i 

1  37 

I  34 

I  32 

I  3i 

I  3o 

3i 

32 

4  33 

3  56 

3  29 

3  9 

2  5i 

2  38 

2  27 

2  II 

I  58 

I  5o 

I  43 

I  38 

I  35 

I  33 

I  32 

I  3i 

32 

33 

4  40 

4  2 

3  35 

3  14 

2  56 

2  42 

2  3o 

2  i4 

2  0 

I  5i 

I  44 

I  39 

I  35 

I  33 

I  32 

I  3i 

33 

34 

4  47 

4  9 
4  i5 

3  4i 

3  18 

3  0 

2  45 

2  33 

2  16 

2  2 

I  53 

I  46 

I  4o 

I  36 

I  34 

I  32 

I  3i 

34 

35 

4  54 

3  46 

3  23 

3  4 

2  49 

2  37 

2  18 

2  4 

I  54 

I  47 

I  4i 

I  37 

I  34 

1  32 

I  3i 

35 

3(S 

5  r 

4  21 

3  5i 

3  27 

3  8 

2  53 

2  4o 

2  20 

2  7 

I  56 

I  48 

I  42 

1  38 

I  35 

I  33 

I  32 

36 

37 

5  0 

4  27 

3  56 

3  32 

3  12 

2  57 

2  43 

2  23 

2  9 

I  58 

I  5o 

I  44 

I  39 

I  36 

I  33 

1  32 

37 

38 

5  16 

4  33 

4  I 

3  37 

3  16 

3  0 

2  47 

2  26 

2  II 

2  0 

I  52 

I  45 

I  4o 

I  37 

I  34 

I  32 

38 

39 

5  23 

4  3q 

4  6 

3  4i 

3  20 

3  4 

2  5o 

2  28 

2  i3 

2  2 

I  53 

I  46 

I  4i 

I  38 

I  34 

I  32 

39 

40 

5  3n 

4  45 

4  II 

3  46 

3  24 

3  7 

2  54 

2  3o 

2  i5 

2  4 

I  54 

I  48 

I  43 

,  39 

I  35 

I  33 

4o 

4i 

5  37 

4  5i 

4  16 

3  5o 

3  28 

3  II 

2  57 

2  32 

2  18 

2  6 

I  56 

I  49 

I  44 

I  4o 

I  36 

I  33 

4i 

42 

5  44 

4  57 

4  21 

3  54 

3  32 

3  i5 

3  0 

2  35 

2  20 

2  8 

I  58 

I  5o 

I  45 

I  4i 

I  37 

I  34 

42 

43 

5  5i 

5  2 

4  26 

3  59 

3  36 

3  18 

3  3 

2  37 

2  22 

2  10 

I  59 

I  5i 

I  46 

I  42 

I  3b 

I  34 

43 

44 

5  57 

5  7 

4  3o 

4  3 

3  4o 

3  22 

3  6 

2  4i 

2  24 

2  12 

2  I 

I  53 

I  47 

I  43 

I  39 

I  35 

44 

46 
48 

6  95  17 
6  21  5  27 

4  39 
4  48 

4  II 
4  19 

3  47 
3  54 

3  29 

3  12 

2  45 
2  5o 

2  28 

2  i5 

2  4 

I  55 

I  49 
I  5i 

I  44 
1  45 

I  4" 
1  4i 

I  36 
I  38 

46 
48 

3  35'3  18 

2  32 

2  18 

2  7 

I  58 

5o 

6  32  5  37 

4  57 

4  26 

4  I 

3  4i 

3  23 

2  55 

2  35 

2  21 

2  10 

2  0 

I  53 

I  47 

I  43 

i  39 

5o 

52 

6  43  5  46 

5  6 

4  33 

4  7 

3  46 

3  2h 

2  59 

2  39 

2  24 

2  12 

2  2 

I  55 

I  49 

I  44 

I  40 

52 

54 

6  54^5  55 

5  i4 

4  4o 

4  i3 

3  52 

3  33 

3  3 

2  43 

2  27 

2  i5 

2  5 

I  57 

I  5o 

I  45 

I  4i 

54 

56 

7  46  4 

5  22 

4  47 

4  19 

3  57 

3  3b 

3  7 

2  47 

2  3i 

2  18 

2  7 

I  59 

I  52 

I  46 

I  42 

56 

58 

7  i3J6  12 

5  29 

4  53 

4  25 

4  2 

3  4313  II 

2  5( 

2  34 

2  21 

2  9 

2  0 

I  53 

I  47 

I  43 

58 

60 

7  22 

6  2C 

5  35 

4  58 

4  3o 

4  7 

3  47i3  i5 

2  53 

2  37 

2  23 

2  11 

2  2 

I  54 

I  49 

I  44 

60 

62 

7  3i 

6  27 

5  4i 

5  3 

4  35 

4  II 

3  5i 

3  19 

2  56 

2  39 

2  25 

2  i3 

2  4 

I  56 

I  5c 

I  45 

62 

64 

7  3c 

6  33 

5  47 

5  8 

4  4f 

4  i5 

3  55 

3  22 

2  59 

2  4i 

2  27 

2  i5 

2  5 

I  57 

I  5i 

I  46 

64 

66 

7  4t 

6  39 

5  53 

5  i3 

4  44 

4  19 

359 

3  25 

3  I 

2  43 

2  29 

2  16 

2  6 

I  5b 

I  52 

I  47 

66 

68 

7  52 

6  45 

5  58 

5  18 

4  48 

4  23 

4  2 

3  28 

3  3 

2  45 

2  3o 

2  18 

2  7 

I  5q 

1  52 

I  47 

68 

70 

7  5£ 

6  5r 

6  3 

5  22 

4  52 

4  2t 

4  4 

3  3o 

3  5 

2  47 

2  3l 

2  19 

2  8 

2   0 

I  53 

I  48 

70 

72 

8  ^ 

6  5f 

6  7 

5  26 

4  55 

4  29 

4  ' 

3  32 

3  7 

2  48 

2  33 

2  20 

2  9 

2   I 

I  54 

I  48 

72 

74 

8  q 

)7  c 
7  ^ 

6  10 

5  3o 

4  5fc 

4  3i 

4  c. 

3  34 

3  9 

2  49 

2  34 

2  21 

2  ic 

2   2 

I  55 

I  49 

74 

76 

8  il 

6  i4 

5  33 

5  I 

4  3i 

4   II 

3  35 

3  II 

2  5c 

2  35 

2  22 

2  II 

2   2 

I  bt 

I  49 

76 

78 

8  !( 

>7  ' 

6  17 

5  36 

5  : 

4  35|4  i: 

3  37 

3  12 

2  5i 

2  36 

2  23 

2  12 

2   3 

I  5f 

I  5o 

78 

80 

8  ic 

57  K 

)6   15 

5  38 

5  e 

4  3- 

4  14 -i  38 

3  i3 

2  52 

2  37 

2  24 

2  i3 

2  4 

I  5- 

I  5i 

80 

82 

6  21 

5  40 

5  - 

74  3c 

;4  i6  3  :-9 

3  i32  53 

2  38 

2  24 

2  ij 

2  4 

i  5- 

82 

84 

5  c 

?4  4 

4  173  4(> 

3  i4  2  54 

2  38 

2  24 

2  IZ 

2   5 

84 

86 

4  t8I3  4i 

3  i5  2  54 

2  38 

2  24 

2  1/ 

86 

G° 

1  7° 

8° 

9° 

10° 

11°  12°  14° 

K)°  18° 

20° 

22° 

24° 

26° 

28° 

30° 

1 

TABLE  XLVIII.               f'''^se3oi 

Third  Correction.  Apparent  Distance  72°. 

D's 

Aun. 

Jijjparent  MlUudc  of  the  Sun,  Star  or  Planet. 

I 
A 

's 

aYi. 

32^ 

;J4" 

30^ 

36" 

42° 

4'o° 

50" 

54" 

5a° 

02" 

m°   70° 

740 

78" 

82-' 

b'o-^ 

All. 

0 

1     1 

/  // 

1  II 

/  // 

/  /' 

1  II 

1   II 

II 

1  I' 

/  II 

1  I'   1  II 

/  // 

/  // 

/  // 

/  // 

0 

6 

4  27 

4  4\ 

4  56 

5  II 

5  38 

6  3 

6  27 

6  48 

7  8 

7  27 

7  427  55 

8  6 

8  16 

6 

7 

3  5i 

4  3 

4  16 

4  28 

4  5. 

5  12 

5  32'5  5 1 

6  8 

6  23 

6  36  6  48 

6  58 

7  7 

7 

8 

3  25 

3  3(i 

3  47 

3  58 

4  18 

4  36 

4  54|5  II 

5  26 

5  39 

5  5i 

6  I 

6  9 

6  16 

6  22 

8 

9 

3  4 

3  i4 

3  24 

3  33 

3  5i 

4  8 

4  23 

4  37 

4  5o 

5  I 

5  II 

5  20 

5  28 

5  35 

5  4i 

9 

lO 

2  48 

2  57 

3  6 

3  x4 

3  29 

3  44 

3  58 

4  10 

4  22 

4  33 

4  42 

4  5(> 

4  57 

5  3 

3  7 

KJ 

II 

2  35 

2  43 

2  5i 

2  58 

3  II 

3  25 

3  37 

3  48 

3  59 

4  9 

4  17 

4  24 

4  3o 

4  35  4  39I 

I  I 

12 

2  24 

2  3i 

2  38 

2  45 

2  57 

3  9 

3  20 

3  3i 

3  4. 

3  49 

3  57 

4  3 

4  8 

4  12 

4  i6'4  ?o 

12 

i3 

2  ID 

2  21 

2  27 

2  -6-^ 

2  45 

2  56 

3  6 

3  16 

3  24 

3  32 

3  39 

3  45 

3  49 

3  53 

3  56  3  59 

i3 

i4 

2   7 

2  i3 

2  iS 

2  24 

2  34 

2  44 

2  54 

3  2 

3  10 

3  18 

3  24j3  29 

3  33 

3  36 

3  393  4 1 

1 4 

i5 

*  I 

2  6 

2  1 1 

2  16 

2  25 

2  34 

2  4^ 

2  5i 

2  58 

3  5 

3  n 

3  16 

3  20 

3  23 

3  25  3  27 

i5 

i6 

I  5(j 

2  I 

2  5 

2  9 

2  18 

2  26 

2  33 

2  4i 

2  48 

2  54 

2  59 

3  4 

3  8 

3  II 

3  i3 

3  i5 

16 

17 

I  52 

I  56 

I  59 

2  3 

2  1 1 

2  19 

2  25 

2  32 

2  39 

2  45 

2  5() 

2  54 

2  57 

3  0 

3  2 

3  4 

17 

i8 

I  48 

1  5i 

I  54 

I  58 

2  6 

2  i3 

2  19 

2  25 

2  3i 

2  37 

2  42 

2  46 

2  48 

2  5o 

2  52 

2  54 

18 

19 

1  44 

1  47 

I  5o 

I  54 

2  1 

2  7 

2  i3 

2  19 

2  25 

2  3c) 

2  35 

2  38 

2  4" 

2  4^ 

2  44 

2  45 

19 

20 

I  4i 

I  44 

I  47 

I  5o 

I  56 

2  2 

2  7 

2  l3 

2  19 

2  23 

2  28 

2  3i 

2  33 

2  35 

2  3b 

2  37 

20 

21 

1  39 

I  41 

1  44 

I  46 

I  52 

I  57 

2  2 

2   8 

2  i3 

2  17 

2  21 

2  24 

2  26 

2  28 

2  29 

2  ,'0 

21 

22 

.  37 

.  39 

1  4i 

I  43 

I  48 

I  53 

I  58 

2   3 

2  7 

2  II 

2  i5 

2  18 

2  20 

2  22 

2  23 

2  0.4 

22 

23 

I  36 

I  37 

I   39 

I  4i 

I  45 

I  5o 

I  54 

I  59 

2  2 

2  6 

2  10 

2  i3 

2  i5 

2  16 

2  17 

2  18 

23 

24 

I  35 

I  3d 

I  37 

I  39 

I  43 

I  47 

I   5i 

I  55 

I  58 

2  2 

2  5 

2  8 

2  10 

2  II 

2  12 

2  i3 

24 

25 

I  M 

r  35 

I  36 

I  38. 

I  4i 

I  44 

I  48 

I  5i 

I  54 

I  58 

2  I 

2  3 

2  5 

2  6 

2  8 

25 

26 

I  33 

1  M 

I  35 

I  36 

I  39 

I  42 

I  45 

I  48 

1  5r 

I  54 

1  57 

I  59 

2  I 

2  2 

2  4 

26 

27 

I  32 

I  33 

I   34 

I  35 

I  37 

I  4o 

I  43 

I  45 

I  48 

I  5i 

I  54 

I  56 

I  57 

I  58 

2  0 

27 

28 

I  32 

I  32 

I  33 

I  34 

I  35 

I  38 

I  4 1 

I  43 

I  46 

I  48 

I  5i 

I  53 

I  54 

I  55 

I  56 

28 

29 

I  3i 

I  32 

I  32 

I  33 

I  34 

I  36 

.  39 

I  4i 

I  44 

I  46 

r  48 

I  5o 

I  52 

I  53 

29 

3o 

I  3i 

I  3i 

I  32 

I  32 

I  33 

I  35 

I  37 

I  39 

I  42 

I  44 

1  46 

I  47 

I  49 

I  5o 

3o 

3i 

I  3o 

I  3i 

I  3i 

I  3i 

I  32 

I  34 

I  36 

I  38 

I  4<> 

I  42 

I  44 

I  45 

1  46 

I  47 

3i 

32 

I  29 

I  3o 

I  3o 

I  3o 

I  3i 

I  33 

I  35 

I  36 

I  38 

I  40 

I   42 

I  43 

I  44 

I  45 

32 

33 

I  29 

I  29 

I  29 

I  3o 

I  3i 

I  32 

I  33 

I  34 

I  36 

I  38 

I  40 

I  4i 

I  42 

33 

34 

I  3o 

I  29 

I  29 

I  29 

I  3o 

I  3i 

I  32 

I   33 

I  34 

I  36 

1  38 

I  09 

I  4o 

34 

35 

I  3o 

I  29 

I  29 

I  29 

I  3o 

I  3o 

I  3i 

I  3s 

I  33 

I  35 

I  36 

.  37 

1  38 

35 

36 

I  3i 

I  29 

1  28 

I  28 

I  29 

I  3o 

I  3i 

I  32 

I  33 

I  34 

I  35 

I  36 

I  36 

36 

37 

I  3i 

I  3.,. 

I  28 

I  28 

I  29 

I  29 

I  3o 

I  3i 

I  32 

I  33 

I  34 

I  35 

37 

38 

I  3i 

1  3() 

I  28 

I  27 

I  28 

I  29 

I  3o 

I  3i 

I  32 

I  33 

I  33 

I  34 

38 

39 

I  3i 

I  3o 

I  29 

I  28 

I  28 

I  28 

I  29 

I  3o 

I  3i 

I  32 

I  32 

I  32 

39 

4o 

I  3i 

I  3() 

t  29 

I  28 

I  27 

I  28 

I  28 

I  29 

I  3o 

I  3o 

r  3o 

I  3o 

4o 

4i 

I  3i 

1  3o 

I  29 

I  28 

I  27 

I  27 

I  27 

I  28 

I  28 

1  29 

I  29 

4i 

42 

1  ii 

I  3i 

I  29 

I  28 

I  26 

I  26 

I  26  I  27 

I  27 

I  28 

I  28 

42 

43 

1  32 

I  3. 

1  29 

.  28 

I  26 

I  26 

1  26  I  26 

I  26 

I  27 

1  27 

43 

44 

t  33 

1  3i 

I  3o 

I  28 

I  26 

I  26 

r  25 

I  25 

I  25 

I  26 

[  26 

44 

46 

I  34 

I  32 

I  3o 

I  29 

I  27 

I    25 

I  25 

I  25 

I  25 

I  25 

46 

48 

I  35 

I  32 

I  3o 

I  29 

1  27 

I  25 

I  24 

I  24 

I  24 

I  24 

48 

5o 

I  3() 

I  33 

I  3i 

I  3o 

I  27 

I  25 

t.24 

I  23 

I  23 

5o 

52 

I  37 

I  34 

I  3i 

I  3o 

I  27 

I  25 

I  23 

I  22 

I  23 

52 

54 

I   37 

I  M 

I  32 

I  3i 

I  28 

I  25 

I    23 

I  22 

54 

56 
58 
60 

I  38 

.  39 
1  39 

I  35 
1  36 
I  36 

I  33 

I  34 
I  34 

I  3i 

I  32 

I  3a 

I  28 
I  28 
I  28 

I  25 
I  25 
I  25 

r  23 

I  23 
I  23 

I  22 

— 

56 

■ruUe  p.  F.fecl  of  Sun's  Par. 

62 

I  4(1 

I  in 

I  3b 

I  32 

I  28 

I  25 

Ail.l  Die  Numbers  nbove  tlie  lines 

fM 

I  4i 

I  38 

I  36 

t  33 

t  ?K 

1  25 

to  Third  Correction  ;  siiblnicl 

66 

68 

70 

1  A) 

1  38 

■  39 
1  39 

1  36 

I  36 
I  36 

I  33 

I  M 
I  34 

I  28 

tlie  others. 

I  4J 
I  43 

App- 
Alt. 

Suns  Arr^irc'it  Allilmle. 

I  29 

I 

U  20 

30 

iO 

.50  S 

0  70 

^0 

90 

72 

I  44 

1  4<' 

I  36 

I  34 

74 
76 

I  44 
I  45 

I  40 

I  4c. 

I  36 

5 
10 
20 

■2 
3 

r  7 

3  2 

0 
2 

0 

1 

T  ' 

I 

1 
0 

0 

78 

1  4:j 

30 

4 

4  4 

3 

3 

■i 

!  2 

2 

80 

40 

6 

6  5 

5 

4 

4 

3 

82 

50 

7 

7  6 

6 

5 

5  . 

84 

60 

8 

8  7 

7 

6 

6 

86 

70 

9 

8  3 

7 

7 



80 

9  8 

8 

yy^ 

34" 

30° 1 38° 

42° 

4G° 

50° 

54° 

58° 

62° 

(J()" 

90 

9 

1 

P^"«302j                TABLE  XLVIII. 

Third  Correction.  Apparent  Distance  76°. 

i 
I 

B's 

Apparent  Mtitude  of  the  Sun,  Star  or  Planet. 

D'i 
A  pp. 

A  pp. 

All. 

6° 

7" 

8" 

y^ 

lU" 

11^ 

12*^ 

14" 

16^ 

18" 

20" 

22° 

24^ 

2G" 

2b" 

yo° 

All. 

o 

/  // 

/  // 

1  II 

1  II 

1   II 

1   II 

?  II 

/  // 

1  II 

/  ;/ 

/  /; 

/  (/ 

1  II 

/  // 

1  II 

'  // 

0 

6 

I  37 

I  39 

I  4i 

I  44 

I  48 

I  54 

2  0 

2  i3 

2  27 

2  42 

2  57 

3  i3 

3  28 

3  4'i 

3  58 

4  i3 

6 

7 

I  4o 

I  37 

I  38 

I  4o 

I  4'6 

I  47 

I  5i 

2  I 

2  12 

2  24 

2  37 

2  5o 

3  3 

3  i5 

3  28 

3  4o 

7 

8 

I  44 

I  4o 

I  37 

I  38 

I  4o 

I  42 

I  45 

I  52 

2  2 

2  12 

2  22 

2  33 

2  4^ 

2  54 

3  5 

3  16 

8 

9 

I  4q 

1  43 

I  39 

I  37 

I  38 

I  39 

I  4i 

I  46 

I  54 

2  2 

2  II 

2  20 

2  3o 

2  89 

2  48 

2  58 

9 

10 

II 

1  54 

2  0 

I  46 
I  5o 

I  4i 
I  44 

I  39 
I  4i 

I  3- 

I  6^ 

I  39 

I  42 
I  4o 

I  48 
I  44 

I  55 
I  49 

2  2 

I  55 

2  10 
2  2 

2  18 
2  9 

2  26 
2  16 

2  34 

2  23 

2  43 
2  3i 

10 
II 

I  39 

I  37 

I  38 

12 

2  6 

I  55 

I  48 

I  44 

I  4i 

I  38 

I  37 

I  38 

I  4i 

I  45 

I  5o 

I  56 

2  2 

2  8 

2  i5 

2  21 

]2 

i3 

2  12 

2  0 

I  52 

I  47 

I  43 

I  4o 

I  38 

I  37 

I  39 

I  42 

I  46 

I  5i 

I  56 

2  2 

2  8 

2  i3 

i3 

i4 

2  19 

2  6 

I  56 

I  5o 

I  45 

I  42 

I  4o 

I  37 

I  38 

I  4o 

I  43 

I  47 

I  52 

I  57 

2  2 

2  7 

i4 

i5 

2  26 

2  12 

2   i 

I  54 

I  48 

I  44 

I  42 

I  38 

I  37 

I  39 

I  4i 

I  45 

I  49 

I  53 

I  57 

2  1 

i5 

i6 

2  J-, 

i  18 

2   6 

I  58 

I  5i 

I  4i 

I  4A 

I  39 

I  37 

I  38 

I  40 

I  43 

I  46 

I  49 

I  53 

I  56 

10 

I? 

2  41 

2  24 

2  II 

2  2 

I  54 

I  49 

I  46 

I  40 

I  38 

I  37 

I  39 

I  41 

I  4i 

I  4b 

I  49 

I  52 

n 

i8 

2  49 

2  3o 

2  17 

2  6 

I  58 

I  52 

I  48 

I  42 

I  39 

I  36 

I  38 

I  39 

I  4i 

I  4i 

I  4b 

I  49 

iS 

19 

2  5- 

2  36 

2  22 

2  10 

2  2 

I  5b 

I  5o 

I  43 

I  40 

I  37 

I  37 

I  38 

I  39 

I  4i 

I  43 

I  4ii 

19 

20 

3  5 

2  43 

2  27 

2  i5 

2  6 

I  58 

I  52 

I  45 

I  4i 

I  38 

I  36 

I  37 

I  35 

.  39 

I  4. 

I  4'd 

20 

21 

3  12 

2  49 

2  33 

2  20 

2  10 

2  2 

I  55 

I  47 

I  42 

I  39 

I  37 

I  36 

I  37 

I  38 

I  39 

I  4i 

21 

22 

3  20 

2  56 

2  38 

2  24 

2  14 

2  6 

I  6b 

I  49 

I  44 

I  40 

I  38 

I  ib 

I  36 

I  3- 

I  38 

I  39 

22 

23 

3  28 

3  3 

2  44 

2  29 

2  18 

2  9 

2  I 

I  5i 

I  45 

I  41 

I  38 

I  36 

I  35 

I  36 

I  37 

I  38 

23 

24 

3  36 

3  9 

2  49 

2  34 

2  22 

2  12 

2  4 

I  54 

I  47 

I  42 

I  39 

I  37 

I  35 

I  36 

I  36 

I  37 

24 

25 

3  44 

3  i5 

2  54 

2  39 

2  26 

2  16 

2  7 

I  56 

I  49 

I  44 

I  4o 

I  07 

I  ib 

I  36 

I  36 

I  37 

25 

26 

3  5 1 

3  21 

3  0 

2  44 

2  3o 

2  20 

2  II 

I  59 

I  5i 

I  45 

I  4i 

I  38 

I  36 

I  35 

I  35 

I  36 

26 

27 

3  59 

3  28 

3  5 

2  49 

2  34 

2  23 

2  14 

2  2 

I  53 

I  47 

I  42 

I  39 

I  37 

I  36 

I  35 

I  35 

27 

28 

4  6 

3  34 

3  10 

2  54 

2  38 

2  27 

2  17 

2  4 

I  54 

I  48 

I  Ai 

I  39 

I  37 

I  36 

I  35 

I  35 

28  ' 

29 

4  t3 

3  4o 

3  i5 

2  58 

2  42 

2  3i 

2  21 

2  7 

I  56 

I  49 

I  44 

I  40 

I  38 

I  3b 

I  35 

I  34 

29 

3o 

4  20 

3  46 

3  21 

3  3 

2  47 

2  34 

2  24 

2  9 

I  58 

I  5i 

I  45 

I  41 

I  39 

I  37 

I  35 

I  34 

3o 

3t 

4  27 

3  52 

F^ 

3  7 

2  5i 

2  38 

2  28 

2  12 

2  0 

I  52 

I  46 

I  42 

I  39 

I  3- 

I  35 

I  34 

3i 

32 

4  34 

3  58 

3  3i 

3  12 

2  55 

2  42 

2  3i 

2  i4 

2  2 

I  54 

I  48 

I  Ai 

I  4o 

I  38 

I  36 

I  35 

32 

33 

4  4i 

4  413  37 

3  16 

2  59 

2  45 

2  34 

2  17 

2  4 

I  55 

I  49 

I  44 

I  4i 

I  38 

I  36 

I  35 

33 

34 

4  48 

4  10,3  42 

3  20 

3  3 

2  49 

2  37 

2  19 

2  6 

I  57 

I  5o 

I  45 

I  42 

I  39 

I  37 

I  35 

34 

35 

4  55 

4  16 

3  47 

3  25 

3  7 

2  52 

2  4i 

2  22 

2  8 

I  59 

I  52 

I  46 

I  42 

I  39 

I  37 

I  35 

35 

36 

5  2 

4  22 

3  53 

3  29 

3  II 

2  56 

2  44 

2  24 

2  II 

2  I 

I  53 

I  47 

I  43 

I  40 

1  38 

I  36 

36 

37 

5  9 

4  27 

3  58 

3  34 

3  i5 

3  0 

2  47 

2  27 

2  i3 

2  3 

I  55 

I  48 

I  44 

I  4i 

I  38 

I  36 

37 

38 

5  16 

4  33 

4  3 

3  38 

3  19 

3  3 

2  5o 

2  29 

2  i5 

2  4 

I  56 

I  49 

I  45 

I  42 

I  39 

I  37 

38 

39 

5  23 

4  38 

4  8 

3  43 

3  23 

3  7 

2  53 

2  3i 

2  17 

2  6 

I  58 

I  5i 

I  46 

I  42 

I  39 

I  37 

39 

40 

5  3o 

4  44 

4  i3 

347 

3  27 

3  10 

2  56 

2  34 

2  19 

2  8 

I  59 

I  52 

I  47 

I  Ai 

I  4o 

I  38 

40 

4i 

5  37 

4   5o 

4  18 

3  5i 

3  3i 

3  i4 

2  59 

2  36 

2  22 

2  10 

2  0 

I  53 

I  48 

I  44 

I  4i 

I  38 

4i 

42 

5  43 

4  55 

4  23 

3  55 

3  34 

3  17 

3  2 

2  39 

2  24 

2  12 

2  I 

I  54 

I  49 

I  45 

I  42 

I  39 

42 

43 

5  49 

5   T 

4  28 

3  59 

3  38 

3  20 

3  5 

2  4i 

2  26 

2  i4 

2  3 

I  56 

I  5o 

I  4b 

I  4i 

I  40 

43 

AA 

5  55 

5  6 

4  33 

4  3 

3  4i 

3  24 

3  8 

2  44 

2  28 

2  i5 

2  4 

I  57 

I  5i 

I  47 

I  4i 

I  4o 

44 

46 

6  7 

5  16 

4  42 

4  II 

3  49 

3  3i 

3  i4 

2  49 

2  32 

2  18 

2  7 

I  59 

I  53 

1  48 

I  44 

1  4i 

46 

48 

6  19 

5  26 

4  5i 

4  19 

3  56 

3  37 

3  20 

2  54 

2  35 

2  21 

2  10 

2  2 

I  55 

I  5o 

1   46 

I  43 

48 

5o 

6  3o 

5  36 

4  59 

4  27 

4  3 

3  43 

3  25 

2  58 

2  39 

2  25 

2  i3 

2  4 

I  57 

I  5i 

I  47 

I  44 

5o 

52 

6  4i 

5  46 

5  7 

4  34 

4  10 

3  49 

3  3o 

3  3 

2  43 

2  28 

2  16 

2  6 

I  59 

I  53 

I  49 

I  45 

52 

54 

6  5i 

5  55 

5  i5 

4  4i 

4  17 

3  55 

3  35 

3  7 

2  47 

2  3i 

2  19 

2  9 

2  1 

I  55 

t  5o 

I  46 

54 

56 

7  1 

6  4 

5  22 

4  48 

4  23 

4  0 

3  4o 

3  II 

2  5o 

2  34 

2  22 

2  12 

2  3 

I  5b 

I  5i 

I  47 

56 

58 

7  II 

6  12 

5  29 

4  54 

4  28 

4  5 

3  45 

3  i5 

2  53 

2  37 

2  25 

2  i4 

2  5 

I  57 

I  52 

I  48 

58 

60 

7  20 

6  20 

5  36 

5  o!4  33 

4  9 

3  49 

3  19 

2  56 

2  4o 

2  27 

2  16 

2  6 

I  59 

I  53 

I  49 

60 

62 

7  28 

6  27 

5  42 

5  5  4  37 

4  i4 

3  53 

3  22 

2  59 

2  43 

2  29 

2  18 

2  8 

2  0 

I  54 

I  5o 

62 

64 

7  36 

6  34 

5  48 

5  10  4  4i 

4  18 

3  57 

3  25 

3  2 

2  45 

2  3i 

2  20 

2  10 

2  2 

I  56 

1  5i 

64 

66 

7  43 

6  4o 

5  54 

5  i5  4  45 

4  22 

4  1 

3  28 

3  5 

2  47 

2  33 

2  21 

2  II 

2  3 

I  57 

I  52 

66 

68 

7  49 

6  45 

5  59 

5  19  4  49 

4  26 

4  5 

3  3i 

3  8 

2  49 

2  35 

2  23 

2  i3 

2  4 

I  58 

I  53 

68 

70 

7  5d 

6  5o 

6  3 

5  2314  53 

4  29 

4  8 

3  M 

3  10 

2  5i 

2  36 

2  24 

2  i4 

2  5 

I  58 

I  53 

70 

72 

8  I 

6  54 

6  7 

5  27 

4  57 

4  32 

4  II 

3  37 

3  19 

2  52 

2  37 

2  25 

2  i5 

2  6 

I  5v 

I  54 

72 

74 

8  6 

6  58 

6  10 

5  3o 

5  0 

4  34 

4  i3 

3  39 

3  i3 

2  53 

2  38 

2  26 

2  16 

2  7 

2  0 

I  54 

74 

76 

8  II 

7  2 

6  i3 

5  33 

5  3 

4  36 

4  i5 

3  4. 

3  i4 

2  54 

2  39 

2  26 

2  16 

2  7 

2  I 

I  55 

76 

78 

8  i5 

7  6 

6  16 

5  36 

5  5 

4  38 

4  17 

3  4- 

3  i5 

2  55 

2  40 

2  27 

2  17 

2  8 

2  I 

78 

80 

8  18 

7  9 

6  19 

5  38 

5  7 

4  4o 

4  19 

3  43 

3  16 

2  56 

2  4o 

2  28 

2  18 

2  9 

80 

8?. 

8  20 

7  11 

6  21 

5  40 

5  9 

4  42 

4  20 

3  44 

3  17 

2  67 

2  4i 

2  28 

2  18 

82 

84 

S  22 

7  i3 

6  23 

5  42 

D  IC 

4  43 

4  21  3  45 

3  18 

2  58 

2  41 

2  28 

84 

86 

6  20 

5  44 

■:!  II 

4  44 

4  22  3  45 

3  18 

2  58 

2  42 

86 

6= 

7== 

8° 

9° 

1C° 

ir 

12°  14° 

1C° 

18° 

20°. 

22° 

24' 

26° 

28° 

30° 

TABLE  XLVIII.               iv.^so^o'i 

Third  Correction.  Apparent  Distance  7G°. 

App 
Alt. 

o 

'Apparent  JUtilude  of  the  Su7i,  Star  or  Planet. 

B's 
App. 
Alt. 

32° 

1    II 

34° 

/  II 

3G° 

1  II 

38° 
/  // 

42° 

/  // 

4G° 
/  // 

50° 
/  // 

54° 
1  II 

58° 
/ 

62° 
/  // 

06° 

1  II 

70° 

74° 

78° 

82° 

86° 

/  // 

1  II 

/  /; 

/  // 

0 

6 

4   28 

-\  42 

4  57 

5  11 

5  37 

5  2 

6  26 

5  47 

7  6 

7  24 

7  4o 

7  54 

8  5 

8  10 

8  2C 

6 

7 

3  53 

4  5 

4  17 

4  29 

4  52 

5  i3 

5  33 

5  52 

6  9 

6  24 

6  37 

6  48 

6  b7 

7  b 

7  12 

7 

8 

3  27 

3  38 

3  49 

3  59 

4  19 

4  38 

4  56 

5  12 

5  26 

5  39 

5  5i 

6  I 

6  9 

6  16 

6  21 

6  26 

8 

9 

3  8 

3  17 

3  26 

3  35 

3  52 

4  8 

4  24 

4  38 

4  5i 

5  3 

5  i3 

5  22 

5  29 

b  3o 

5  4fJ 

b  44 

9 

10 

2  52 

3  0 

3  8 

3  16 

3  3i 

3  46 

4  1 

4  i4 

4  25 

4  35 

4  44 

4  b2 

4  59 

b  4 

b  8 

5  1 1 

10 

II 

2  39 

2  46 

2  53 

3  0 

3  14 

3  27 

3  4" 

3  5i 

4  2 

4  12 

4  20 

4  27 

4  33 

4  38 

4  42 

4  45 

11 

12 

2  28 

2  34 

2  4i 

2  47 

3  0 

3  12 

3  23 

3  34 

3  43 

3  52 

4  o4  6 

4  11 

4  lb 

4  19 

4  22 

12 

i3 

2  19 

2  25 

2  3o 

2  36 

2  48 

2  59 

3  9 

3  19 

3  28 

3  36 

3  433  48 

3  53 

3  57 

4  c 

4  2 

i3 

i4 

2  12 

2  17 

2  22 

2  27 

2  38 

2  48 

2  58 

3  6 

3  i4 

3  22 

3  28 

3  33 

3  37 

3  4i 

3  43 

3  4b 

i4 

i5 

2  5 

2  10 

2  i5 

2  19 

2  29 

2  38 

247 

2  55 

3  3 

3  9 

3  i5 

3  20 

3  24 

3  27 

3  29 

3  3i 

i5 

i6 

2  0 

2  4 

2  9 

2  i3 

2  21 

2  29 

2  37 

2  45 

2  52 

2  58 

3  4 

3  8 

3  12 

3  i5 

3  17 

3  19 

16 

17 

I  56 

I  59 

2  3 

2  7 

2  i4 

2  22 

2  29 

2  36 

2  43 

2  49 

2  54 

2  58 

3  2 

3  4 

3  e 

3  8 

17 

18 

I  52 

I  55 

I  58 

2  2 

2  9 

2  16 

2  23 

2  3o 

2  36 

2  42 

2  46 

2  5o 

2  53 

2  55 

2  5- 

2  b8 

18 

19 

I  49 

I  5i 

I  54 

I  58 

2  4 

2  II 

2  17 

2  24 

2  3o 

2  35 

2  39 

2  42 

2  4b 

2  47 

2  4<; 

2  5() 

19 

20 

I  46 

I  48 

I  5i 

I  54 

2  0 

2  6 

2  12 

2  18 

2  24 

2  28 

2  32 

2  3b 

2  37 

2  39 

2  4i 

2  42 

20 

21 

I  4^ 

I  45 

I  48 

I  5i 

I  56 

2  2 

2  7 

2  i3 

2  18 

2  22 

2  26 

2  29 

2  3i 

2  33 

2  3/ 

21 

22 

I  41 

I  43 

I  46 

I  48 

I  53 

I  58 

2  3 

2  8 

2  i3 

2  17 

2  20 

2  23 

2  25 

2  27 

2  2!: 

22 

23 

I  4o 

I  42 

I  44 

I  46 

I  5o 

I  55 

.  5g 

2  3 

2  8 

2  12 

2  lb 

2  17 

2  19 

2  21 

2  2.. 

23 

24 

I  39 

I  4o 

I  42 

I  44 

I  48 

I  52 

I  56 

I  5q 

2  4 

2  7 

2  10 

2  12 

2  14 

2  16 

2  it 

24 

25 

I  38 

.39 

I  40 

I  42 

I  46 

I  49 

I  53 

I  56 

2  0 

2  3 

2  6 

2  8 

2  10 

2  12 

25 

26 

I  37 

I  38 

I  39 

I  4i 

I  44 

I  47 

I  5c> 

I  53 

I  56 

I  5q 

2  2 

2  4 

2  6 

2  8 

26 

27 

I  36 

I  37 

I  38 

I  40 

I  42 

I  45 

I  48 

I  5o 

I  53 

I  56 

I  59 

2  I 

2  3 

2  b 

27 

28 

I  36 

I  37 

I  38 

I  39 

I  4i 

I  43 

I  46 

I  48 

I  5o 

I  53 

I  56 

I  58 

2  0 

2  2 

28 

29 

I  35 

I  36 

I  37 

I  38 

I  4o 

I  42 

I  44 

1  46 

I  48 

I  5o 

I  53 

I  bb 

I  57 

29 

3o 

I  35 

I  35 

I  36 

I  37 

I  38 

I  40 

I  42 

I  44 

I  46 

I  48 

I  5o 

I  52 

I  54 

3o 

3i 

I  34 

I  M 

I  35 

I  36 

I  37 

I  3q 

I  40 

I  42 

I  44 

I  4& 

I  48 

I  5o 

I  52 

3i 

32 

I  M 

I   34 

I  34 

I  35 

I  36 

I  38 

I  39 

I  4i 

I  43 

I  44 

I  46 

I  48 

I  bo 

32 

33 

I  34 

I  33 

I  34 

I  35 

I  35 

I  37 

I  38 

I  40 

I  42 

I  43 

I  45 

I  46 

33 

34 

I  34 

I  33 

I  33 

I  34 

I  35 

I  36 

I  37 

I  3q 

I  4i 

I  42 

I  44 

I  4b 

34 

35 

I  34 

I  33 

I  33 

I  33 

I  34 

I  35 

I  36 

I  38 

I  39 

I  40 

I   42 

I  43 

35 

36 

I  35 

I  34 

1   33 

I  33 

I  33 

I  34 

I  35 

I  37 

I  38 

I  39 

I  4o 

I  4i 

36 

37 

I  35 

I  34 

I  33 

I  32 

I  33 

I  33 

I  34 

I  36 

I  37 

I  38 

I  39 

37 

38 

I  35 

I  34 

I  33 

I  32 

I  32 

I  33 

I  34 

1  35 

I  36 

I  37 

I  38 

38 

39 

I  36 

I  34 

I  33 

I  32 

I  3a 

I  33 

I  33 

I  34 

I  35 

I  36 

I  36 

39 

40 

I  36 

I  35 

I  34 

I  33 

I  32 

I  32 

I  33 

I  34 

I  34 

I  35 

I  35 

40 

4i 

I  37 

I  35 

I  34 

1  33 

I  32 

I  32 

1  32 

I  33 

I  33 

I  34 

4i 

42 

I  37 

I  35 

I  34 

I  33 

I  3i 

I  3i 

r  32 

I  32 

I  33 

I  33 

42 

43 

I  37 

I  35 

I  34 

I  33 

I  3i 

I  3o 

I  3i 

I  3i 

I  32 

I  32 

43 

44 

I  38 

I  36 

I  34 

I  33 

I  3i 

I  3o 

I  3o 

I  3i 

I  3i 

I  3i 

44 

46 

,  39 

I  37 

I  35 

I  34 

I  3i 

I  29 

I  29 

I  3o 

I  3o 

46 

48 

I  4o 

I  38 

I  36 

I  34 

I  3i 

I  29 

I  29 

I  29 

I  29 

48 

So 

52 

I  4i 
I  42 
I  43 

I  38 
I  4o 

.  37 
I  37 
I  38 

I  35 
I  35 

T  36 

I  32 
I  32 

I  33 

I  3o 
I  3o 

I  3c, 

I  29 
I  29 

I  29 
I  28 

5o 

i 

1  2y 
I  29 

56 

I  44 

I  4i 

r  38 

T  36 

I  33 

T  3o 

Table  P.  Effect  of  Sun's  Par. 

Add  Ihe  Numbers  above  llie  line? 

58 
60 
62 

I  4b 
I  46 
T  46 

I  42 
I  43 
I  ^3 

139 
1  4o 

t  4n 

I  37 
I  37 
I  37 
I  38 

1  33 
I  33 
I  33 

I  3o 
I  3o 

to  Third  Correction  ;  subtr.ict 
tlie  others. 

D's 

Alt. 

Sun's  Apparent  Altitmle. 

64 

I  47 

I  44 

I  4i 

I  33 

5 

10 

20  3 

9  -10  50 

so 

ro  s 

90 

66 

I  48 

I  44 

I  4i 

I  38 

68 

I  49 

.  45 

I  4i 

I  38 

5 

1  r 

0  0 

0 

n  1 

70 

I  49 

I  4b 

I  4i 

15 

2 

1  0 

0 

0  u 

0 

72 

I  49 

I  45 

20 

25 

3  : 

2  2 

9  2 

74 

I  bo 

SO 
35 

4  i 

A     "i 

3  3 

4  4 

3 
3 

3 

3 

76 

40 
45 

5  5 

4  4 

5  5 

4 

78 

50 

6  1 

6  5 

80 

55 
60 

8 

8 

7  ( 
7  - 

6  6 

7 

82 

65 

8 

8 

a  - 

7 

84 

70 

8 
9 

8 
9 

8  t 

8  f 

86 

SO 

9 

9 

8 

32= 

34° 

36° 

38° 

42° 

46° 

50° 

54° 

58° 

62° 

66° 

90 

9 

8 

P=^g«304]               TABLE  XLVIII. 

Third  Correction.  Apparent  Distance  80°. 

D's 
App. 
Alt. 

o 
6 

7 
8 

9 

10 

II 

12 

i3 
i4 
i5 

i6 

I? 
i8 

19 
20 

21 
22 

23 

24 

25 

26 
27 
28 

^9 

3o 

3i 

32 

33 
34 
35 

36 

37 
38 
39 
40 

4i 
42 
43 
44 
46 

48 
5o 

52 

54 
56 

58 
Go 
62 
64 
66 

68 

•,'0 

72 
74 
76 

"78 
80 
82 
84 
86 

Apparent  Mtitude  of  the  Sun,  Star  or  Planet.                                   | 

])'s 
App. 

Alt. 

0 
6 

7 
8 

9 

10 

II 
12 
i3 
i4 
i5 

16 

17 
18 

19 

•20 

2] 
22 
23 

24 
25 

26 

27 
28 

^9 
3o 

3i 

32 

33 
34 
35 

36 

37 
38 

39 

40 

4i 
42 
43 

44 
46 

48 
5o 

52 

54 
56 

58 
60 
62 

64 
66 

68 
70 
72 
74 
76 

78 
80 
82 
84 
86 

0° 

/  // 
1  4i 

I  44 
I  48 

I  52 

1  57 

2  3 
2  9 
2  16 
2  23 
2  3o 

2  37 
2  45 

2  53 

3  0 
3  8 
3  16 
3  23 
3  3i 
3  38 
3  46 

3  53 

4  I 
4  8 
4  i5 
4  22 

4  29 
4  36 
4  43 
4  5o 

4  57 

5  4 
5  II 
5  18 
5  25 
5  3i 

5  38 
5  44 
5  5i 

5  57 

^^ 

6  20 
6  3i 

6  4i 

6  5i 

7  I 

7  I' 

7  20 
7  28 
7  36 
7  43 

7  49 

7  55 

8  c 
8  5 
8  9 
8  i3 
8  16 
8  19 
8  22 
8  24 

r 

1 43 
1 41 
1 43 

I  46 
I  5o 

I  54 

1  59 

2  4 
2  10 
2  16 
2  22 
2  28 
2  34 
2  4i 
2  47 

2  54 

3  0 
3  6 
3  12 
3  18 

3  24 
3  3i 
3  37 
3  43 
3  49 

3  55 

4  I 
4  7 
4  12 
4  18 

4  24 
4  29 
4  35 
4  4i 

4  47 

4  52 

4  57 

5  3 
5  8 
5  18 

5  28 
5  38 
5  47 

5  56 

6  5 

6  i4 
6  22 
6  29 
6  35 
6  41 
6  46 
6  5i 
6  55 

6  59 

7  3 
7  6 
7  9 
7  12 
7  i4 
7  i£ 

7° 

(  // 
I  46 
I  43 
I  41 
I  43 
I  46 

I  49 

I  53 

1  56 
2.  0 

2  5 

2  10 
2  i5 
2  21 
2  26 
2  3i 

2  37 
2  43 
2  47 
2  53 

2  58 

3  4 
3  10 
3  i5 
3  20 
3  25 
3  3o 
3  35 
3  4o 
3  45 
3  5o 

3  55 

4  0 
4  5 
4  10 
4  i5 
4  20 
4  25 
4  3o 
4  35 
4  44 

4  53 

5  I 
5  9 
5  17 
5  24 

5  3i 
5  38 
5  44 

5  5o 
^  55 

6  0 
6  5 

6  9 
6  i3 
6  16 

6  19 
6  21 
6  23 

6  25 

6  27 

8° 

9° 

1  II 
I   5o 
I  45 
I  42 
I  41 
I  4^ 

I  45 
I  48 
I  5i 
I  54 

1  58 

2  2 
2  6 
2  11 
2  i5 
2  20 

2  14 

1  29 

2  33 

2  38 
2  42 

2  47 

2  52 

2  56 

3  1 
3  5 
3  10 
3  i4 
3  19 
3  23 
3  28 
3  32 
3  37 
3  42 
3  46 
3  5o 

3  54 

3  58 

4  2 
4  6 
4  i4 
4  22 
4  3o 
4  37 
4  44 
4  5o 

4  56 

5  2 

5  7 
5  12 
5  17 
5  21 
5  25 
5  29 
5  32 
5  35 

5  37 
5  4o 
5  42 
5  44 
5  46 

9° 

10° 
1  II 

I  54 
I  48 
I  44 
I  42 
I  4i 
I  /^i 
I  45 
I  48 
I  5o 
I  53 

I  56 

1  59 

2  3 

2  7 
2  10 

2  i4 
2  18 
2  22 

2  25 

2  29 

2  33 
2  37 
2  41 
2  46 
2  5o 

2  54 

2  58 

3  2 
3  6 
3  10 

IT4 
3  19 
3  23 
3  27 
3  3i 

3  35 
3  38 
3  42 
3  46 

3  53 

4  0 
4  6 
4  12 
4  18 
4  24 
4  3o 
4  35 
4  4o 
4  44 
4  49 
4  53 

4  57 

5  0 
5  3 
5  6 

5  8 
5  10 
5  12 
5  i3 
5  i4 

10° 

11° 

1   II 
159 
I  5i 

I  46 
I  44^ 
I   42 

I  4i 
I  43 
I  45 
I  47 
I  49 

I  52 

I  54 

1  57 

2  0 
2  3 

2  6 
2  9 
2  i3 
2  16 
2  19 

2  23 

2  26 
2  3o 
2  34 
2  38 

2  4i 
2  45 
2  49 

2  52 

2  56 

3  0 
3  3 

3  7 
3  II 
3  i4 
3  18 
3  21 
3  25 
3  28 
3  35 

3  4i 
3  47 
3  53 

3  58 

4  3 
4  8 
4  i3 
4  18 
4  22 
4  26 
4  3o 
4  33 
4  36 
4  38 
4  4o 
4  42 
4  44 
4  45 
4  46 
4  47 

11° 

12° 

/  // 

2  4 
I  55 
I  49 
I  46 
I  44 
I  42 
I  4i 
I  42 
I  44 
I  46 

I  48 
I  5o 

I  52 

I  54 

iJl 

1  59 

2  2 
2  5 
2  8 
2  12 

2  i5 
2  19 
2  22 
2  26 
2  29 

2  32 

2  35 
2  38 
2  4i 

2  44 

2  47 
2  5o 
2  54 

2  58 

3  I 

3  4 
3  7 
3  10 
3  i3 
3  19 

3  25 
3  3o 
3  35 
3  39 
3  44 

3  49 
3  54 

3  58 

4  2 
4  6 

4  9 

4  12 

4  i4 
4  16 
4  18 

4  19 
4  21 

4  22 
4  23 
4  24 

12° 

14° 

1  II 

2  1- 
2  5 
I  56 
I  5i 
I  47 
I  45 
I  43 
I  42 
I  4i 
I  42 

I  43 
I  45 
I  4i 
I  48 
I  5o 

I  52 

I  54 
I  57 

1  59 

2  I 

2  3 
2  6 
2  8 
2  II 
2  i4 
2  17 
2  19 
2  22 
2  24 
2  27 

2  29 

2  32 

2  34 
2  36 
2  38 
2  4i 
2  44 
2  46 
2  48 

2  53 

r58 

3  3 

3  7 
3  II 
3  i5 

3  19 
3  23 
3  27 
3  3i 
3  34 
3  37 
3  39 
3  4i 
3  43 
3  44 
3  45 
3  46 
3  4- 
3  48 
3  49 

14° 

16° 

'  II 

2  32 

2  17 
2  6 
I  58 
I  53 

I  49 
I  46 
I  44 
I  43 
I  42 

I  4i 
I  42 
I  4^ 
I  44 
I  46 

I  47 

;^? 

I  52 

I  54 
I  55 
I  57 

1  59 

2  I 
2  3 
2  5 

2  7 
2  9 
2  11 

2  i4 
2  16 
2  18 
2  20 
2  22 
2  24 
2  26 
2  28 
2  3o 
2  32 

2  36 
2  39 
2  43 
2  47 
2  5i 
2  54 

2  57 

3  0 
3  3 
3  6 
3  9 
3  12 
3  i4 
3  16 
3  18 
3  19 

3  20 
3  21 
3  22 
3  23 
3  24 

16° 

18° 
/  II 

2  47 
2  29 
2  16 
2  6 
I  59 

I  54 
I  5o 
I  47 
I  45 
I  44 
I   43 
I  42 
I  4i 
I  42 
I  43 

I  44 
I  45 
I  47 
I  48 
I  49 
I  5o 
I  5i 
I  53 
I  55 
I  56 
I  58 

1  59 

2  I 
2  2 
2  4 
2  6 
2  8 
2  9 
2  10 
2  12 

2  i4 
2  16 
2  17 
2  19 

2  23 

2  26 
2  29 

2  32 

2  35 
2  38 

2  4i 
2  44 
2  47 
2  49 
2  5i 

2  53 

2  55 
2  57 
2  58 

2  59 

3  0 
3  I 
3  2 
3  3 

1 

Il8° 

20° 

1  II 
3  2 

2  4i 
2  26 
2  i5 
2  6 

I  59 
I  54 
I  5i 
I  48 
I  46 

I  45 
I  43 
I  42 
I  4i 
I  41 
I  42 
I  42 
I  43 
I  44 
I  45 

I  46 
I  47 
I  48 

\t 

I  52 

I  53 
I  54 
I  56 
I  57 

1  58 

2  0 
2  I 
2  2 
2  4 
2  5 
2  7 
2  8 
2  10 
2  i3 

2  i5 
2  18 
2  21 
2  24 
2  26 

2  29 
2  3i 
2  33 
2  35 
2  37 
2  39 
2  4i 
2  42 
2  43 
2  44 
2  45 
2  45 
2  46 

20° 

22° 
/  // 
3  17 
2  54 
2  37 

2  25 

2  i4 

2  6 
2  0 
I  56 

I  52 

1  49 

I  47 
I  45 
I  44 
1   43 
I  42 

I  41 
I  40 
I  4i 
I  4i 
I  42 

I  43 
I  43 
I  44 
I  45 
I  46 

I  47 
I  48 

149 
I  5o 
I  5i 

I  52 

I  53 
I  54 
I  55 
I  57 
I  58 

1  59 

2  1 
2  2 
2  4 

2  7 
2  9 
2  12 

2  i4 
2  17 

2  19 
2  21 
2  22 

2  24 
2  26 
2  27 
2  29 
2  3o 
2  3i 

2  32 
2  32 

2  33 
22° 

24° 

/  // 
3  32 
3  6 
2  48 
2  34 
2  22 

2  i3 
2  6 
2  I 

I  57 
I  53 

I  5o 
I  48 
I  46 
I  44 
I  43 

I  42 
I  4i 
I  40 
1  4o 
I  4o 

I  4i 
I  4i 
I  42 
I  43 
I  44 

I  44 
1   45 
I  46 
I  47 
I  47 
I  48 
I  49 
149 
I  5o 
I  5i 

I  52 

X  53 

I  55 
I  56 

1  58 

2  0 
2  2 
2  4 
2  6 
2  8 

2  10 
2  12 
2  i3 
2  i5 
2  16 
2  17 
2  19 
2  20 
2  21 
2  22 
2  22 

24° 

26° 
/  // 
3  47 
3  19 
2  59 
2  46 
2  3o 
2  20 
2  12 
2  6 
2  2 
I  58 

I  54 
I  5i 
I  48 
I  46 
I  45 
I  43 
I  42 
I   41 
I  4i 
I  40 

I  40 
I  40 
I  41 
I  4i 
I  42 

I  42 
I  43 
I  44 
I  44 
I  44 
I  45 
I  46 
I  46 
I  47 
I  47 
I  48 

149 
I  5o 
I  5i 
I  53 
I  55 
I  56 
I  58 

1  59 

2  I 

2  3 
2  5 
2  6 

2  7 
2  8 

2  9 
2  10 
2  II 
2  12 
2  i3 

26° 

28° 
/  II 
4    2 
3  3x 
3  10 
2  52 
2  38 
2  27 
2  19 
2  12 
2  7 
2  2 

I  58 
I  54 
I  5i 
I  49 
I  47 
I  45 
I  43 
I  42 
I  42 
I  4i 

1  4i 
I  40 
I  40 
I  40 
I  40 

I  4o 
I  4i 
I  42 
I  42 
I  42 

I  4'i 
I  44 
I  44 
I  45 
I  45 
I  46 
I  40 
I  47 
I  48 
I  49 
I  5i 
I  57 
I  54 
I  55 
I  57 
I  58 

1  59 

2  0 
2  I 
2  2 

2  3 
2  4 
2  5 
2  5 

28° 

oO° 
/  /; 
4  16 
3  44 
3  so 
3  I 
2  45 

^4 
2  25 
2  18 
2  12 

2  7 
2  2 
I  58 
I  54 
I  5i 
I  49 

I  47 
I  45 
I  44 
I  43 
I  42 

I  42 
I  4i 
I  4o 
I  39 
I  39 

139 
I  39 
I  40 
I  4o 

I  40 

I  4i 
I  42 
I  42 
I  43 
143 

I  44 

I  44 
I  45 
I  45 
I  46 

I  48 
I  49 
I  5o 

I  52 

I  53 

I  54 
I  55 
I  56 
I  56 
I  57 
I  58 

1  59 

2  0 



30° 

TABLE  XL VI II.                                      [r.ge305 

Third  Correction.     Apparent  Di.stance  80°. 

])'s 
A  pp. 

Apparent  Altitude  of  tlie  Sun,  Star  or  Planet. 

App. 

All. 

32° 

34^ 

30" 

38" 

42° 

46° 

50° 

54° 

58° 

(>2° 

Gij'^ 

70° 

74°   7 

8°   82° 

80° 

Alt. 

o 

1  II 

/  // 

/  // 

/   II 

/  /' 

1  II 

/    // 

/   // 

/   // 

/  // 

1  II 

/   II 

/   //    / 

//    /  // 

/   II 

0 

G 

4  3o 

4  44 

4  58 

5  12 

5  39 

6    4 

6  28 

n  49 

7    8 

7  26 

7  4i 

7  54 

8     58 

i38  K 

?8  24 

6 

1 

3  56 

4    8 

4  19 

4  3o 

4  52 

5  i4 

5  35 

5  54 

6  II 

6  26 

6  39 

6  5c, 

6  597 

67  i: 

7  lb 

7 

8 

3  3i 

3  4i 

3  52 

4    2 

4  23 

4  42 

4  59 

5  i5 

5  29 

5  42 

5  54 

6    4 

6126 

186  2; 

b  27 

8 

9 

3  II 

3  21 

3  3o 

3  39 

3  56 

4  12 

4  28 

4  42 

4  54 

5     5 

5  i5 

5  24 

5  32  5 

38  5  4: 

5  46 

9 

lU 

2  54 

3    3 

3  12 

3  20 

3  35 

3  5o 

4    4 

4  16 

4  28 

4  39 

4  48 

4  56 

5     25 

75  11 

D   i4 

10 

II 

2  42 

2  49 

2  57 

3    5 

3  19 

3  32 

3  44 

3  56 

4    7 

4  16 

4  24 

4  3i 

4  36  4 

4i4  4J 

4  47 

II 

12 

2    32 

2  38 

2  45 

2   52 

3     5 

3  17 

3  28 

3  38 

3  48 

3  57 

4    5 

4  II 

4  i54 

194  2: 

4  25 

12 

i3 

2  24 

2  3o 

2  36 

2  42 

2  53 

3    4 

3  i4 

3  23 

3  32 

3  40 

3  47 

3  53 

3  574 

i4    ^ 

4    6 

i3 

i4 

2  18 

2    23 

2  28 

2  33 

2  43 

2  53 

3     2 

3  II 

3  19 

3  26 

3  32 

3  38 

3  42  3 

46  3  4f 

3  49 

i4 

i5 

2  12 

2  16 

2  21 

2    25 

2  34 

2  43 

2   52 

3    0 

3     7 

3  i3 

3  .9 

3  25 

3  29  3 

32  3  3^ 

'3  3b 

i5 

If.) 

2     6 

2  10 

2  i4 

2  18 

2  26 

2  34 

2  42 

2  5o 

2  56 

3     2 

3     8 

3  i3 

3  173 

20  3  2: 

3  24 

16 

17 

2     I 

2    4 

2     8 

2  12 

2  20 

2  27 

2  34 

2  4i 

2  47 

2  53 

2  58 

3    3 

3    63 

9  3  I 

17 

18 

I  57 

2    0 

2     3 

2     7 

2  i4 

2  21 

2  28 

2  34 

2  4'J 

2  46 

2  5o 

2  54 

2  573 

o3     : 

18 

19 

I  54 

I  56 

I  59 

2     2 

2    9 

2  16 

2  22 

2  28 

2  34 

2  39 

2  43 

2  47 

2  5o  2 

52  2    5; 

19 

20 

I  5i 

I  53 

I  5b 

I  58 

2     5 

2  II 

2  17 

2  22 

2  28 

2  33 

2  37 

2  4o 

2  432 

45  2  4t 

) 

20 

21 

I  49 

I  5i 

I  53 

I  55 

2     I 

2     7 

2  12 

2  17 

2  22 

2  27 

2  3i 

2  34 

2  37  2 

38 

21 

22 

I   47 

I  49 

I  5i 

I  53 

I  58 

2     3 

2     8 

2  i3 

2  17 

2  21 

2   25 

2  28 

2  3i  2 

32 

22 

23 

I  46 

I  47 

I  49 

I  5i 

I  55 

2     0 

2    4 

2     9 

2  i3 

2  17 

2  20 

2    23 

2  26  2 

27 

23 

24 

I  45 

I  46 

I  47 

I  49 

I  53 

I  57 

2     I 

2     5 

2     0 

2  i3 

2  16 

2  19 

2  21  2 

22 

24 

25 

I  44 

I  45 

I  46 

I  48 

I  5i 

I  54 

I  58 

2     1 

2     5 

2     9 

2   12 

2  i4 

2  16 

25 

26 

I  43 

1  44 

I  45 

I  46 

I  49 

I     52 

I  55 

I   58 

2     2 

2     5 

2     8 

2  10 

2  12 

26 

27 

I  42 

I  43 

I  44 

I  45 

I   47 

I  5o 

I  53 

I  56 

I  59 

2     2 

2     5 

2    7 

2     8 

27 

28 

I  4i 

I  42 

I  43 

I  44 

I  46 

I  48 

I  5i 

I  54 

I  57 

I  59 

2     2 

2    4 

2     5 

28 

29 

I  4o 

I  4i 

I  4i 

I  42 

1  44 

I  46 

I   49 

I    52 

I  55 

I  57 

I  59 

2    I 

29 

3o 

I  39 

I  40 

I  40 

I  4i 

I  43 

I  45 

1  48 

I  5i 

I  53 

I  55 

I  57 

I  59 

3o 

3i 

I  39 

I  40 

I  40 

I  4i 

I  42 

I  44 

I  46 

I   49 

I  5i 

I  53 

I  55 

I  57 

3i 

32 

I  39 

1  39 

I  39 

I  40 

I  4i 

I  4^ 

I  45 

X  47 

I   49 

I  5i 

I  53 

I  55 

32 

33 

I  39 

I  39 

I  39 

I  40 

i4i 

I  42 

I  44 

I  46 

I  48 

I  49 

I  5i 

36 

34 

I  39 

I  39 

I  39 

I  4o 

I  4i 

I  42 

I  43 

I  45 

I  47 

I  48 

I  49 

M 

35 

t  39 

I  39 

I  39 

I  39 

I  4o 

I  4. 

I  42 

I  44 

I  45 

I  46 

I  47 

35 

36 

I  4o 

I  39 

I  39 

I  39 

I  4o 

I  4i 

I  42 

I  43 

I  44 

I  45 

I  46 

36 

37 

I  4i 

I  4o 

,  39 

I   38 

I  39 

I  4o 

I  4. 

I  42 

I  43 

I  44 

37 

38 

I  4i 

I  40 

I  39 

I   38 

I  39 

I  4u 

I  4i 

I  42 

I  42 

I  4'^ 

38 

39 

I  4i 

I  40 

I  39 

I   38 

I  39 

I  39 

I  40 

I  4i 

I  4i 

1  42 

39 

40 

I  4i 

I  40 

I  39 

i   38 

I  38 

I   38 

I  39 

I  40 

I  4<-> 

I  4i 

4o 

4i 

I  42 

I  4i 

I  4o 

I   39 

I  38 

I   38 

I  38 

I  39 

I  39 

4i 

42 

I  42 

I  4i 

I  4o 

I  39 

I  37 

I  37 

I  37 

I  38 

I  38 

42 

43 

I  43 

I  4i 

I  4o 

I  39 

I  37 

I  37 

I  37 

I  37 

I  38 

4S 

A^ 

I  43 

I  42 

I  40 

I   39 

I  37 

I  37 

I  36 

I  37 

I  37 

44 

46 

I  44 

I  42 

I  4i 

I  4o 

I  38 

I   J7 

I  36 

I  36 

4b 

48 

I  45 

I  43 

I  4i 

I  4o 

I  38 

I   37 

I  36 

I  36 

48 

5o 

52 

I  46 
I  47 

I  44 
I  45 

I  42 
r  43 

I  4i 
I  4i 

I  38 
I  38 

I  36 
I  36 

I  36 
I  35 

5o 

1 

54 

I  48 

1  46 

I  44 

r  42 

I  38 

I  36 

5G 

I  49 

I   47 

I  44 

I  42 

I  38 

I  36 

raWe  P.     Effect  of  Sun's  Par. 

Adil  Ihe  Numbers  above  Che  lines 

58 

I  5o 

I    47 

I  45 

I  42 

I  38 

to  Third  Correction  ;    sublract 

60 

69 

I  5i 

I     52 

I  48 
I  49 

I  45 
I  46 

I  43 
I  43 

I  38 

the  olhers. 

D'3 

Sun's  Apparent  Altitude. 

64 

66 

68 

I    52 

I  53 
I  54 

I  49 
I  49 
I  5o 

I  46 
I  46 

I  43 

Alt. 
10 

5    10' 

0  30  40  50 

60  70 

80 

1 
0 

90 

1     I 
1     1 

10    0    1 

1     1 

I     I     1     0 

n  0 

70 

I   5-5 

15 
20 

2    2 
S    3 

2  2    11 

3  2    2    2 

1  1 

2  2 

1 
2 

0 

72 

25 

4    4 

3    3    3    3 

3  2 

nA 

30 

1    4 

14    4    3 

3    3 

35 

5    i> 

5    4    4    4 

4 

70 





40 
43 

6    6 
fi    6 

5  5    5    5 

6  6    5    5 

4 

78 

50 

7    7 

6    6    6    6 

80 

55 

7    V 

7    7    6 

60 

8    8 

7    7    7 

89 

1 

65 

S    8 

i    8 

84 

70 
75 

8  8 

9  9 

8    8 
8 

86 

80 

9    9 

8 

32^ 

34° 

36° 

38° 

42° 

4(;° 

50° 

54° 

58° 

62° 

00° 

__ 

'.\9 


iX-esou]                TABLE  XLVIII 

. 

1 

Third  Correction.  Apparent  Distance  84°. 

1 

! 
1 

App. 

Apparent  Mtitude  of  the  Sun,  Stt 

?•  or  Planet. 

D's 
Add. 

All. 

6° 

70 

8" 

9^ 

10^ 

11^ 

12" 

14" 

l(i° 

18" 

20° 

22" 

24" 

2<j" 

28" 

yo° 

Alt. 

o 

/  II 

/  II 

1  II 

/  // 

/  // 

1  II 

/  // 

/  II 

/  // 

/  ;; 

II 

/ 

1   II 

<  /( 

1   II 

1  »■ 

0 

6 

r  47 

I  49 

I  5i 

I  54 

I  5o 

2  4 

2  10 

2  22 

2  36 

2  5o 

3  5 

3  20 

3   35 

3  5o 

4  5 

4  20 

6 

7 

1  5o 

I  47 

I  48 

I  5o 

I  53 

I  56 

2  0 

2  10 

2  21 

2  33 

2  45 

2  57 

3  10 

3  23 

3  35 

3  48 

7 

8 

I  53 

I  49 

I  47 

I  48 

I   5o 

I  52 

I  55 

2  2 

2  11 

2  21 

2  3i 

2  42 

2  53 

3  3 

3  i4 

3  25 

8 

9 

I  57 

I  52 

I  49 

I  47 

I  48 

1  5o 

I  52 

I  57 

2  4 

2  12 

2  21 

2  3o 

2  39 

2  48 

2  58 

3  7 

9 

lO. 

2  2 

I   55 

I  5i 

I  49 

I  47 

I  48 

I  5o 

I  53 

I  59 

2  5 

2  12 

2  20 

2  27 

2  35 

2  44 

2  52 

10 

11 

2  8 

I  59 

I  54 

I  5i 

I  49 

I  47 

I  48 

1  5i 

I  55 

t  59 

2  5 

2  12 

2  18 

2  26 

2  33 

2  41 

II 

12 

2  i4 

2  4 

I  57 

I  53 

I  5i 

1  48 

I  47 

I  49 

I  52 

1  55 

I  59 

2  5 

2  II 

2  18 

2  25 

2  3i 

12 

i3 

2  20 

2  q 

2  I 

I  56 

I  53 

I  5o 

I  48 

I  48 

1  5o 

I  52 

I  55 

2  0 

2  6 

2  II 

2  17 

2  23 

i3 

i4 

2  27 

2  i4 

2  5 

I  59 

I  55 

I  52 

I  5o 

1 47 

I  48 

I  bo 

I  53 

I  57 

2  2 

2  6 

2  II 

2  16 

i4 

i5 

2  34 

2  20 

2  10 

2  3 

I  58 

I  54 

I  5i 

I  48 

I  47 

I  49 

I  5i 

I  54 

I  58 

2  2 

2  7 

2  I  I 

i5 

i6 

2  42 

2  26 

2  i5 

2  7 

2  1 

I  56 

I  53 

I  49 

I  47 

I  48 

I  5o 

I  52 

I  55 

1  59 

2  3 

2   7 

16 

17 

2  49 

2  32 

2  20 

2  II 

2  4 

r  59 

I  55 

I  5o 

I  48 

I  47 

I  48 

I  5o 

I  53 

I  56 

2  0 

2   3 

17 

i8 

2  57 

2  38 

2  25 

2  16 

2   8|2   2 

I  57 

I  52 

I  49 

I  46 

I  47 

1  49 

I  5i 

I  54 

1  57 

2   0 

18 

iQ 

3  4 

2  44 

2  3i 

2  20 

2  12 

2  5 

I  59 

I  53 

I  5o 

1  47 

1  46 

I  48 

1  49 

I  52 

1  54 

I  57 

19 

20 

3  12 

2  5o 

2  36 

2  25 

2  i5 

2  8 

2  2 

I  55 

I  5i 

I  48 

I  46 

I  47 

I  48 

I  5o 

I  52 

I  55 

20 

21 

3  20 

2  57 

2  42 

2  29 

2  19 

2  II 

2  5 

I  57 

I  52 

1  49 

I  47 

I  46 

I  47 

1  48 

I  5o 

I  52 

21 

22 

3  27 

3  3 

2  47 

2  34 

2  23 

2  i4 

2  8 

I  59 

I  54 

I  5o 

I  47 

I  46 

I  46 

I  47 

I  49 

1  5o 

22 

23 

3  35 

3  9 

2  52 

2  38 

2  27 

2  18 

2  II 

2  I 

I  56 

I  52 

1  48 

I  46 

I   46 

1.47 

I  48 

I  4q 

23 

24 

3  42 

3  i5 

2  57 

2  42 

2  3o 

2  21 

2  i4 

2  3 

1  57 

I  53 

I  49 

I  46 

I  46 

I  46 

I  47 

I  48 

24 

25 

3  49 

3  21 

3  3 

2  47 

2  34 

2  25 

2  17 

2  6 

I  59 

I  54 

I  5o 

I  47 

I  46 

I  46 

I  46 

I  47 

25 

26 

3  56 

3  27 

3  8 

2  52 

2  38 

2  28 

2  20 

2  8 

2  0 

I  55 

I  5i 

1  48 

I  47 

I  46 

I  46 

I  46 

26 

27 

4  4 

3  34 

3  i3 

2  56 

2  42 

2  32 

2  24 

2  11 

2  2 

I  56 

r  52 

I  49 

I  47 

I  46 

I  45 

I  46 

27 

28 

4  11 

3  4o 

3  18 

3  I 

2  46 

2  35 

2  27 

2  i3 

2  4 

1  58 

I  53 

I  49 

I  47 

I  46 

I  45 

I  45 

28 

29 

4  19 

3  47 

3  24 

3  5 

2  5l 

2  39 

2  3o 

2  16 

2  6 

I  59 

I  54 

X  5o 

1   48 

I  46 

I  45 

I  45 

29 

3o 

4  26 

3  53 

3  29 

3  10 

2  55 

2  43 

2  33 

2  18 

2  8 

2  I 

I  55 

I  5i 

I  49 

I  47 

I  46 

I  45 

3o 

3i 

4  33 

3  59 

3  35 

3  i4 

2  59 

2  46 

2  36 

2  21 

2  10 

2  3 

I  57 

I  52 

I  49 

I  47 

I  46 

I  45 

3i 

32 

4  4o 

4  5 

3  4o 

3  19 

3  3 

2  5o 

2  39 

2  24 

2  12 

2  4 

I  58 

1  53 

I  5o 

1  48 

I  46 

1  45 

32 

33 

4  47 

4  II 

3  45 

3  24 

3  7 

2  54 

2  42 

2  27 

2  i4 

2  5 

I  59 

1  54 

I  5o 

1  48 

I  46 

I  45 

33 

34 

4  54 

4  16 

3  5o 

3  28 

3  II 

2  57 

2  45 

2  29 

2  16 

2  7 

2  0 

I  55 

I  5i 

I  48 

I  47 

I  46 

34 

35 

5  1 

4  22 

3  55 

3  33 

3  i5 

3  I 

2  49 

2  32 

2  19 

2  9 

2  2 

I  56 

I  52 

I  49 

I  47 

I  46 

35 

36 

5  8 

4  28 

4  0 

3  37 

3  19 

3  5 

2  52 

2  34 

2  21 

2  10 

2  3 

I  58 

I  53 

I  49 

I  47 

I  46 

36 

37 

5  i5 

4  34 

4  5 

3  42 

3  23 

3  8 

2  56 

2  37 

2  23 

2  12 

2  4 

I  59 

I  54 

I  5o 

I  48 

I  47 

37 

38 

5  21 

4  40 

4  10 

3  46 

3  27 

3  12 

2  59 

2  39 

2  25 

2  i4 

2  6 

2  0 

I  55 

I  5i 

I  49 

I  47 

38 

39 

5  28 

4  45 

4  i5 

3  5i 

3  3i 

3  i5 

3  2 

2  42 

2  27 

2  16 

2  7 

2  I 

I  56 

I  52 

I  49 

I  47 

39 

4o 

5  34 

4  5i 

4  20 

3  55 

3  35 

3  19 

3  5 

2  44 

2  29 

2  18 

2  9 

2  3 

1  57 

I  52 

I  49 

I  47 

40 

4i 

5  4i 

4  56 

4  25 

3  59 

3  39 

3  23 

3  8 

2  47 

2  3l 

2  20 

2  II 

2  4 

I  58 

I  53 

I  5o 

I  48 

4i 

42 

5  47 

5  I 

4  3o 

4  3 

3  43 

3  26 

3  II 

2  49 

2  33 

2  21 

2  12 

2  5 

1  59 

I  54 

I  5i 

149 

42 

Ai 

5  53 

5  7 

4  35 

4  7 

3  47 

3  3o 

3  i4 

2  52 

2  35 

2  23 

2  i3 

2  7 

2  0 

I  55 

I  52 

I  5o 

43 

U 

6  0 

5  12 

4  4o 

4  11 

3  5o 

3  34 

3  17 

2  54 

2  37 

2  25 

2  i5 

2  8 

2  I 

1  56 

I  53 

1  5i 

44 

46 

6  12 

5  22 

4  49 

4  19 

357 

3  4o 

3  23 

2  59 

2  4i 

2  29 

2  18 

2  10 

2  3 

I  58 

I  55 

1  52 

46 

48 

6  24 

5  32 

4  58 

4  27 

4  4 

3  46 

3  29 

3  4 

2  45 

2  32 

2  21 

2  12 

2  5 

2  0 

I  56 

I  53 

48 

5o 

6  35 

5  42 

5  6 

4  35 

4  II 

3  52  3  35 

3  9 

2  4g 

2  35 

2  24 

2  i5 

2  8 

2  2 

I  58 

1  55 

5o 

52 

6  45 

5  5i 

5  i4 

4  42 

4  17 

3  58 

3  4o 

3  i3 

2  53 

2  38 

2  27 

2  18 

2  10 

2  4 

2  0 

I  57 

52 

54 

6  55 

6  0 

5  22 

4  49 

4  23 

4  4 

3  45 

3  17 

2  57 

2  4i 

2  3o 

2  20 

2  12 

2  6 

2  2 

I  58 

54 

56 

7  5 

6  9 

5  29 

4  55 

4  29 

4  9 

3  5o 

3  21 

3  I 

2  44 

2  32 

2  22 

2  i4 

2  8 

2  3 

I  59 

56 

58 

7  i4 

6  17 

5  36 

5  I 

4  34 

4  i4 

3  55 

3  25 

3  4 

2  47 

2  35 

2  24 

2  16 

2  9 

2  4 

2  0 

58 

60 

7  22 

6  25 

5  42 

5  6 

4  39 

4  19 

3  59 

3  29 

3  7 

2  5o 

2  37 

2  26 

2  17 

2  ID 

2  5 

2  I 

60 

62 

7  3o 

5  32 

5  48 

5  II 

4  44 

4  23 

4  3 

3  33 

3  10 

2  53 

2  39 

2  28 

2  19 

2  12 

2  7 

2  2 

62 

64 

7  38 

6  39 

5  54 

5  16 

4  49 

4  27 

4  7 

3  36 

3  i3 

2  56 

2  4i 

2  29 

2  20 

2  l3 

2  8 

2  3 

64 

66 

7  45 

6  45 

6  d 

5  21 

4  54 

4  3i 

4  II 

3  39 

3  16 

2  58 

2  43 

2  3i 

2  22 

2  l5 

2  9 

2  3 

66 

68 

7  5i 

6  5o 

6  5 

5  25 

4  58 

4  35 

4  i5 

3  4i 

3  19 

3  0 

2  45 

2  33 

2  24 

2  16 

2  10 

2  4 

68 

70 

7  57 

6  54 

6  9 

5  29 

5  2 

4  39 

4  18 

3  44 

3  21 

3  2 

2  46 

2  34 

2  25 

2-  17 

2  10 

70 

72 

8  2 

6  58 

6  i3 

5  3d 

5  6 

4  42 

4  21 

3  46 

3  23 

3  3 

2  47 

2  35 

2  26 

2  18 

72 

74 

8  6 

7  2 

6  17 

5  36 

5  94  44 

4  23 

3  48 

3  24 

3  4 

2  48 

2  36 

2  27 

74 

76 

8  10 

7  5 

6  20 

5  39 

5  II 

4  46 

4  25 

3  5o 

3  25 

3  5 

2  49 

2  37 

76 

78 

8  14 

7  8 

6  23 

5  42 

5  i3 

4  48 

4  26 

3  5i 

3  26 

3  6 

2  5o 

78 

80 

8  18 

7  II 

6  26 

5  45 

5  j5 

4  5o 

4  27 

3  52 

3  27 

3  7 

80 

82 

8  21 

7  i4 

6  28 

5  47 

5  17 

4  5i 

4  28 

3  53 

3  28 

82 

84 

8  24 

7  17 

6  3o 

5  49 

5  18 

4  52 

4  29 

3.54 

84 

86 

8  26 

7  19 

6  3i 

5  5o 

5  19 

4  53 

4  3o 

86 

6° 

7°  1  8°  1  9° 

10° 

11° 

12° 1 14° 

IG° 

18° 

20° 

22° 

24° 

26° 

28° 

30° 

TABLE  XLVIU.                                      ^i^^'o^oy 

5's 
App. 

Third  Concction.     Apparent  Distance  84°. 

.Ijiparcnt  Altitude  of  the  Sun,  Star  or  riunei. 

D's 

A  nr. 

Ail. 

32° 

34^ 

30° 

3ti° 

42° 

4G° 

50" 

54° 

5S° 

62° 

GG' 

70^ 

74° 

78° 

82° 

8()° 

All. 

o 

1    II 

/  // 

/  II 

1  II 

/  // 

/  II 

/   // 

1  II 

/   // 

/  '/ 

1  II 

/   /; 

/     II 

/  II 

/  // 

/     // 

0 

6 

4  34 

4  48 

5     2 

5  i5 

5  4. 

6     C 

6  29 

6  5i 

7  10 

7  27 

7  42 

7  55 

8     6 

8  i4 

8  21  f 

i    27 

6 

7 

4    0 

4  12 

4  24 

4  'i6 

4  58 

5  19 

5  39 

5  58 

6  i5 

6  3o 

6  42 

6  53 

7     2 

7     9 

7  i5- 

7    19 

7 

8 

3  36 

3  47 

3  57 

4    1 

4  27 

4  46 

5    4 

5  20 

5  34 

5  46 

5  58 

6    9 

6   .7 

6  23 

6  28 1 

i  3i 

8 

9 

3  16 

3  25 

3  34 

3  43 

4    0 

4  17 

4  33 

4  47 

4  59 

5  10 

5  20 

5  29 

5  36 

5  42 

5  47' 

)  5o 

9 

lo 

3     1 

3    9 

3  17 

3  25 

3  4i 

3  55 

4    9 

4  21 

4  33 

4  43 

4  53 

5     I 

5     7 

b  i3 

5  17^ 

)  19 

10 

II 

2  48 

2  55 

3     3 

3   10 

3  24 

3  37 

3  5o 

4    2 

4  12 

4  21 

4  3o 

4  37 

4  43 

4  48 

4  5i. 

i  54 

II 

12 

2  38 

2  44 

2  5i 

2  58 

3  10 

3  22 

3  34 

3  45 

3  54 

4    2 

4  10 

4  16 

4  22 

4  26 

4  28- 

i  3o 

12 

i3 

2  29 

2  35 

2  4i 

2  47 

2  58 

3     9 

3  20 

3  29 

3  38 

3  45 

3  52 

3  58 

4    3 

4    7 

4     9 

i3 

i4 

2  22 

2  27 

2  33 

2  38 

2  48 

2  58 

3     8 

3  16 

3  24 

3  3i 

3  37 

3  43 

3  47 

3  5i 

3  53 

■  4 

i5 

2  16 

2  21 

2  26 

2  3o 

2  39 

2  48 

2  57 

3     5 

3  12 

3  19 

3  25 

3  3o 

3  34 

3  37 

3  4o 

i5 

i6 

2  II 

2  i5 

2  20 

2  ■i4 

2  32 

2  4o 

2  48 

2  56 

3     2 

3     9 

3  i5 

3  19 

3  23 

3  26 

3  29 

16 

I? 

2     7 

2  10 

2  i4 

2  18 

2  26 

2  34 

2  4i 

2  48 

2  54 

3     0 

3     5 

3     9 

3  i3 

3  i5 

>7 

i8 

2     3 

2     6 

2  10 

2  i3 

2  21 

2  28 

2  34 

2  4o 

2  46 

2   52 

2  57 

3     I 

3    4 

3    6 

18 

19 

2     0 

2     3 

2    6 

2     0 

2  16 

2    23 

2  29 

2  34 

2  4o 

2  45 

2  49 

2  53 

2  56 

2  58 

19 

20 

I  57 

2     0 

2     2 

2     5 

2  12 

2    18 

2  24 

2  29 

2  34 

2  38 

2  42 

2  45 

2  48 

2  5o 

20 

21 

I  54 

1.57 

I  59 

2     2 

2    8 

2    l3 

2  19 

2  24 

2  29 

2  33 

2  36 

2  39 

2  4i 

21 

22 

I    52 

I  54 

I  56 

I  5q 

2    4 

2      9 

2  i4 

2  10 

2  24 

2  28 

2  3i 

2  34 

2  36 

22 

23 

I  5o 

I    52 

I  54 

I  56 

2     I 

2       5 

2    50 

2  lb 

2  19 

2    23 

2  26 

2  29 

2    32 

2'3 

24 

I  49 

I  5o 

I    52 

I  54 

I  58 

2       2 

2      7 

2  II 

2  i5 

2  19 

2  22 

2    25 

2  28 

24 

25 

I  48 

I  49 

I   bo 

I    52 

I  56 

2       0 

2      4 

2    8 

2  12 

2  i5 

2  18 

2    21 

2b 

26 

I  47 

I  48 

1 49 

I  5i 

I  54 

I  58 

2       2 

2     5 

2     9 

2  12 

2  i5 

2     17 

26 

27 

I   47 

I  48 

1 49 

I  5o 

I  53 

I  56 

2       0 

2     3 

2     6 

2     9 

2  12 

2  i4 

27 

28 

I  46 

r  47 

I  48 

I  4q 

I  5i 

I  54 

I  58 

2     I 

2     3 

2     6 

2     P 

2    II 

28 

20 

I  46 

I   47 

I  47 

I  48 

I  5o 

I  53 

I  56 

I  59 

2     I 

2    4 

2    6 

29 

3o 

I  45 

I  46 

I  46 

I  47 

I  49 

I    52 

I  55 

I  57 

2     0 

2     2 

2     3 

3o 

3i 

I  45 

I  45 

I  46 

I   47 

I  49 

I  5i 

I  54 

I  56 

I  58 

2    0 

2     I 

3i 

32 

I  45 

I  45 

I  45 

I  46 

I  48 

I  5o 

I     52 

I  54 

I  56 

I  58 

I  59 

32 

33 

I  45 

I  45 

I  45 

I  46 

I   47 

I  49 

I  5i 

I  53 

I  54 

I  56 

33 

34 

I  45 

I  44 

I  44 

I  45 

I  46 

I  48 

I  5o 

I    52 

I  53 

I  54 

34 

35 

I  45 

I  44 

I  44 

I  45 

I  46 

I  47 

r  49 

I  5o 

I  5i 

I    52 

3b 

36 

I  46 

I  45 

1  44 

I  44 

I  45 

I  46 

I  48 

I  49 

I  5o 

I  5o 

36 

^7 

I  46 

I  45 

I  44 

I  44 

I  45 

I  45 

I   47 

I  48 

I  49 

37 

38 

I  46 

I  45 

I  44 

I  44 

I  44 

I  45 

I  46 

I   47 

I  48 

38 

^9 

I  46 

I  45 

I  44 

I  44 

I  44 

I  44 

I  45 

I  46 

I  47 

39 

4o 
4i 

I  46 
I  47 

I  45 
I  46 

I  4b 
I  45 

I  45 
I  45 

1  44 
I  44 

I  44 
I  44 

I  45 
I  44 

I  45 
I  44 

I  46 



40 
4i 

42 

I  48 

I  47 

I  46 

I  45 

I  43 

I  43 

I  44 

I  44 

42 

43 

>  49 

I  48 

I  46 

I  45 

I  43 

I  43 

I  44 

I  44 

43 

44 

I  49 

I  48 

I  47 

I  45 

I  43 

I  43 

I  43 

44 

46 

.  5o 

I  49 

I   47 

I  45 

I  43 

I  43 

I  43 

46 

48 

I  5i 

I  5o 

I  48 

I  46 

I  44 

I  43 

I  42 

48 

5o 

52 

I  53 
I  54 

I  5i 
I  5i 

149 
I  49 

I  47 
I   47 

I  44 
I  44 

1  43 

I  42 

bo 

1 

54 

I  55 

I    52 

I  49 

I  47 

I  44 

56 

I  56 

I  53 

I  5o 

I  48 

I  44 

Table  P.     Eject  of  Sun's  Par. 

58 
60 
62 

I  56 

I  57 
I  58 

I  53 

I  5o 
I  5i 

r   5i 

I  48 
I  48 

Tu  be  subtracted  from  the  third 
Correction. 

1  54 
I  54 
I  55 

t)'s 

Arp. 

Alt. 

Suii'b  Apparent  Altitude. 

64 

I  59 

5 

0  20  3 

)  JO 

50  6t 

70  SO 

90 

66 
68 

I  59 



3 

10 

1 

1    1    0 

1  1  1 

0 
1 

0  0 

1  1 

0  0 

1  I 

0 

:'. 

<!    2    U 

2 

70 

20 

3 

3    3    3 

2 

2    2 

2 

72 

2.5 

4 
4 

4    3    3 
4    4    4 

3 
4 

3  3 

4  4 

3 
4 

74 

35 

5 

5    5    5 

5 

4    4 

76 

40 

6 

6 

5    5 

6,S    6|6 

■  H 

« 

78 

50 

7 

7 

/    7    6 
7    7    7 

|6 
7 

80 

60 

8 

8    7    7 

82 

65 
70 

8 
8 

8    8    8 
8    8 

84 

75 

9 

9    8 

86 

80 
90 

9 
9 

1 

32° 

34° 

36° 

38° 

42° 

46° 

50° 

54° 

58° 

62° 

66"" 

' 

_ 

1 

^^^303]                                     TABLE  XLVIIl. 

Third  Correction.  Apparent  Distance  SS°. 

D's 
A  pp. 
All. 

o 

6 

7 
8 

9 

10 

II 

12 

i3 
i4 
i5 

i6 

17 
i8 

19 
20 

21 
22 

23 

24 

25 

26 

27 
28 

=9 
So 

3i 

32 

33 
34 
35 

36 

3- 
38 
39 
40 

4i 
42 
43 
44 
46 
48 
5o 

52 

54 
56 

58 
60 
62 
64 
66 

68 
70 
72 
74 
76 
78 
80 
82 
84 
86 

Apparent  Altitude  of  the  Sun,  Star  or  Planet. 

])'s 
App. 
Alt. 

0 
6 

7 
8 

9 

10 

II 
12 
i3 
i4 
i5 

16 

17 
18 

'9 
20 

21 
22 
23 
24 
25 

26 

27 
28 

3o 
3i 

32 

34 
35 

36 

37 
38 
39 

4o 

4i 
42 
43 
44 
46 

48 
5o 

52 

54 
56 

58 
60 
62 
64 
66 

68 
70 
72 
74 
76 

78 
80 
82 
84 
86 

1   II 
I  53 
I  55 

1  58 

2  2 
2  7 

2  i3 
2  19 
2  26 
2  33 
2  4o 

2  47 

2  54 

3  2 
3  10 
3  17 
3  25 
3  32 
3  4o 
3  47 

3  55 

4  2 
4  10 
4  17 
4  24 
4  3i 

4  39 
4  40 

4  53 

5  0 
5  7 
5  i3 
5  20 
5  27 
5  34 
5  4o 

5  47 

5  53 

6  0 
6  6 
6  18 
6  29 
6  4o 

6  5i 

7  1 
7  10 

7  19 
7  28 
7  36 
7  44 
7  5i 

7  58 

8  4 
8  10 
8  i5 
8  19 
8"^ 
8  25 
8  28 
8  3c 
8  32 

G° 

7° 
1  II 
I  54 
I  53 
I  55 

1  58 

2  I 

2  5 
2  10 
2  i5 
2  21 
2  26 

2  32 

2  37 
2  43 
2  49 

2  55 

3  2 
3  8 
3  i5 
3  21 
3  27 
3  33 
3  39 
3  45 
3  5i 

3  57 

4  3 
4  9 
4  i5 
4  21 
4  27 
4  33 
4  39 
4  45 
4  5. 

4  56 

5  2 
5  7 
5  i3 
519 
5  29 

5  39 
5  48 

5  57 

6  6 
6  i5 
6  23 
6  3i 
6  38 
6  45 
6  5i 

6  56 

7  I 
7  5 
7  9 
7  i3 

7  16 

7  19 
7  22 

7  24 
7  26 

7° 

8° 
f  II 
I  56 
I  54 
I  53 
I  55 

1  57 

2  0 
2  4 
2  8 
2  12 
2  16 
2  20 
2  25 
2  3o 
2  35 
2  4i 
2  46 
2  52 

2  57 

3  2 
3  8 

3  i3 
3  18 
3  23 
3  28 
3  34 
3  4o 
3  45 
3  5i 

3  56 

4  I 
4  6 
4  II 
4  16 
4  21 
4  26 
4Ti' 
4  36 
4  4i 
4  46 

4  55 

5  12 
5  20 

5  28 
5  35 

M2 
5  48 

5  54 

6  0 
6  5 
6  10 
6  i5 
6  19 
6  23 
6  26 
6  29 
6  3i 
6  33 
6  35 
6  37 

8° 

9° 

1  II 
I  59 
I  56 
I  54 
I  53 
I  55 

1  57 

2  0 
2  3 
2  6 
2  9 
2  i3 
2  17 
2  21 

2  25 

2  29 

2  34 
2  39 
2  43 
2  48 

2  52 

2  57 

3  2 
3  6 
3  II 
3  i5 
3  20 
3  25 
3  29 
3  34 
3  38 

3  43 
3  48 
3  52 

3  57 

4  I 
4  5 
4  9 
4  i4 
4  18 
4  26 

4  34 
4  4i 
4  48 

4  55 

5  I 

5  7 
5  12 
5  17 
5  22 
5  27 

5  32 
5  36 
5  40 
5  43 
5  46 

5  49 
5  52 
5  54 
5  56 

9° 

10° 

1  II 

2  4 

'59 
I  5o 
I  54 
I  53 

I  55 

1  57 
.  59 

2  I 
2  4 

2  7 
2  10 

1  16 

2  20 

2  24 
2  28 
2  32 

2  36 
2  40 

2  44 
2  48 
2  52 

2  56 

3  0 

3  4 
3  8 

3  12 

3  17 
3  21 

3  25 
3  29 
3  33 
3  37 
3  4i 
3  45 
3  49 
3  53 

3  57 

4  4 
4  II 
4  17 
4  23 
4  29 
4  35 

4  40 
4  45 
4  5o 

4  55 

5  0 

5  4 
5  8 
5  I. 

5  i4 

5  17 

519 
5  21 
5  23 
5  25 

10° 

11° 

/  (/ 

2  ID 
2   3 
I  59 

I  56 
I  54 
I  53 
I  54 
I  56 

1  58 

2  0 
2  2 
2  5 

2  7 
2  10 
2  i3 

2  17 
2  20 
2  24 
2  27 
2  3i 

2  35 
2  38 
2  42 
2  46 
2  49 
2  53 

2  56 

3  0 
3  4 
3  7 
3  II 
3  i5 
3  18 
3  22 
3  25 

3  29 
3  32 
3  36 
3  39 
3  46 
3  52 

3  58 

4  2 
4  8 
4  i4 

4  19 
4  24 
4  29 
4  33 
4  37 

4  4i 
4  44 
4  47 
4  49 
4  5i 

4  53 
4  55 

4  57 

11° 

12° 

/  // 
2  i6 

2  7 
2  2 
I  58 
I  56 

I  54 
I  53 
I  54 
I  56 
I  57 

1  59 

2  I 
2  3 
2  5 
2  8 
2  II 
2  i4 
2  17 
2  20 

2  23 

2  27 
2  3o 
2  33 
2  37 
2  4o 

^43 
2  46 
2  5o 
2  53 

2  56 

259 

3  2 
3  5 
3  8 
3  11 

3  i4 
3  17 
3  20 
3  23 
3  29 

3  35 
3  4i 
3  47 
3  52 

3  57 

4  2 
4  6 
4  10 
4  i4 
4  18 

4  21 
4  23 
4  25 
4  27 
4  29 

4  3i 
4  33 
4  35 

12° 

14° 

1  II 

2  28 
2  16 
2  8 
2  3 
2  0 

I  57 
I  55 
I  54 
I  53 
I  54 
I  55 
I  56 
I  58 

1  59 

2  I 

2  3 
2  5 

2  7 
2  9 
2  II 

2  i4 
2  17 
2  19 
2  22 
2  24 
2  27 
2  29 
2  3i 
2  34 
2  37 
2  4o 
2  43 
2  46 
2  49 
2  5i 

2  54 
2  56 

2  59 

3  I 
3  6 
3  II 
3  i5 
3  19 
3  23 
3  27 
3  3i 
3  35 
3  38 
3  42 
3  45 
3  48 
3  5o 
3  52 
3  54 
3  56 

3  57 
3  58 

14° 

16° 

/  // 
2  42 
2  27 

2  17 
2  10 
2  5 
2  I 
I  58 
I  56 
I  55 
I  54 
I  53 
I  53 
I  54 
I  55 
I  56 
I  58 

1  59 

2  I 
2  2 
2  4 
2  6 
2  8 
2  10 
2  12 
2  i4 
2  16 
2  18 
2  20 
2  22 
2  24 

2  26 
2  28 
2  3i 
2  33 
2  35 

2  38 
2  4o 
2  42 
2  44 
2  48 
2  52 
2  56 

2  59 

3  3 
3  7 
3  10 
3  i3 
3  16 
3  19 
3  22 

3  25 
3  27 
3  29 
3  3o 
3  3i 
3  32 

1G° 

18° 
/  i\ 

1  56 

2  39 
2  27 
2  18 
2  II 

2  6 
2  2 
I  59 

I  57 
I  55 

I  54 
I  53 

I  52 

I  53 
I  54 
I  55 
I  5& 
I  57 

1  58 

2  0 
2  I 
2  2 
2  4 
2  5 
2  6 
2  8 
2  9 
2  II 
2  i3 
2  i5 
2  17 
2  19 
2  21 
2  22 
2  24 

2  26 
2  28 
2  3o 

2  32 

2  35 
2  39 
2  42 
2  45 
2  48 
2  5i 

2  54 
2  57 

2  59 

3  2 
3  4 
3  6 
3  8 
3  9 

3  ID 

3  II 

18° 

20° 

1  II 
3  II 

2  5i 
2  37 
2  26 
2  18 
2  12 

2  7 
2  3 
2  0 
I  58 
I  56 
I  55 
I  54 
I  53 

I  52 

I  53 
I  53 
I  54 
I  55 
I  56 

I  57 
I  58 

1  59 

2  0 
2  I 
2  2 
2  3 
2  5 

2  7 
2  8 

2  10 
2  II 

2  i3 

2  14 
2  16 

2  17 
2  19 
2  20 
2  22 

2  25 

2  28 
2  3i 
2  34 
2  36 
2  39 

2  42 
2  44 
2  46 
2  48 
2  5o 

2  5i 
2  53 
2  53 

2  54 

20° 

22° 
/  // 
3  26 
3  4 
2  48 
2  35 
2  25 

2  18 
2  12 

2  7 
2  3 
2  0 

I  58 
I  57 
I  56 
I  54 
I  53 

I  52 
I  52 
I  52 

I  53 
I  53 

I  54 
I  55 
I  55 
I  56 
I  57 
I  58 

1  59 

2  0 
2  I 
2  2 

2  4 
2  5 
2  6 
2  7 
2  9 
2  10 
2  II 
2  12 
2  i3 
2  16 

2  18 
2  21 
2  24 
2  27 
2  29 

2  3i 
2  33 
2  35 
2  37 
2  39 

2  4o 
2  4i 
2  42 

22° 

24° 
/  // 
3  4i 
3  16 
2  59 
2  45 
2  34 

2  25 
2  18 
2  12 
2  7 
2  4 
2  I 
I  59 
I  58 
I  56 
I  54 
I  53 
I  53 

I  52 
I  52 
I  52 

I  53 
I  53 
I  53 
I  53 
I  54 
1  55 
I  56 
I  56 
I  57 
I  58 

1  59 

2  0 
2  I 
2  2 
2  3 

2  4 
2  5 
2  6 

2  7 
2  9 

2  II 
2  i3 
2  16 
2  18 
2  20 
2  22 
2  24 
2  26 
2  28 
2  3o 

2  3i 
2  32 

24° 

26° 
/  // 
3  56 
3  28 

2  54 
2  43 

2  32 

2  24 
2  18 
2  12 
2  8 
2  5 
2  2 
2  0 
I  58 
I  56 
I  55 
I  54 
I  53 

I  52 
I  52 
I  52 
I  52 
I  52 
I  52 

I  53 
I  53 
I  54 
I  54 
I  55 
I  56 

I  56 
I  57 
I  58 
I  58 

1  59 

2  0 
2  I 
2  2 
2  3 
2  5 

2  7 
2  9 
2  10 
2  12 
2  i4 
2  16 
2  17 
2  19 
2  20 
2  22 

2  23 

2G° 

28° 
/  i[ 
4   II 
3  4o 
3  19 
3  3 
2  5o 

2  39 
2  3o 

2  23 

2  17 
2\,3 

2   9 
2   5 
2   2 
2   0 

I  58 

I  57 
I  55 
I  54 
I  53 
I  53 

I  52 
I  52 
I  52 
I  52 
I  52 

I  52 

I  53 
I  53 
I  54 
I  54 

I  54 
I  55 
I  56 
I  56 

I  57 

I  57 
I  58 

1  59 

2  0 
2  2 

2  3 
2  5 
2  6 
2  8 
2  9 
2  II 
2  12 
2  i3 
2  i4 
2  i5 

28° 

30° 
/  // 
4  25 
3  52 
3  3o 
3  12 
2  58 

2  47 
2  37 
2  29 
2  22 
2  17 

2  i3 

2  9 
2  5 
2  3 
2  I 

159 
I  57 
I  55 
I  54 
I  54 
I  53 

I  52 
I  52 
I  52 
I  52 

I  52 
I  52 

I  53 
I  53 
I  53 

I  53 
I  54 
I  54 
I  55 
I  55 

I  55 
I  56 
I  57 
I  58 

1  59 

2  0 
2  2 
2  3 
2  4 
2  5 

2  6 
2  7 
2  8 
2  9 

30° 

TABLE  XLVIII.                                      \y^z-^^^ 

Third  Correction.     Apparent  Distance  88°. 

D's 
A  pp. 
All. 

Apparent  Altitude  of  the  Sun,  Star  or  Planet. 

D's 
App. 
All. 

3-3° 

34° 

36° 

38° 

42° 

46° 

50° 

54° 

58° 

62° 

66° 

70° 

74° 

78° 

82° 

86" 

o 

1  II 

1  II 

/  // 

1  II 

/   // 

1  II 

/  II 

/   // 

/  // 

1  II 

/  // 

/  // 

/  II 

/  // 

/  // 

1  II 

0 

6 

4  40 

4  54 

5    8 

5  22 

5  48 

6  i3 

6  36 

6  57 

7  16 

734 

7  49 

8     2 

8  i3 

3  21 

8  27 

8  32 

6 

7 

4    4 

4  16 

4  28 

4  4o 

5     3 

5  25 

5  45 

6    4 

6  21 

6  36 

6  49 

7     0 

7     9 

7   16 

7  22 

7  27 

7 

8 

3  41 

3  52 

4    3 

4  i3 

4  33 

4  52 

5  10 

5  26 

5  4o 

5  53 

6    6 

6  i5 

6  23 

5  29 

6  34 

6  37 

8 

9 

3  22 

3  3i 

3  4i 

3  5o 

4    8 

4  24 

4  39 

4  53 

5     5 

5  16 

5  26 

5  35 

5  43 

5  49 

5  54 

9 

10 

3    6 

3  i4 

3  22 

3  3o 

3  46 

4    I 

4  i5 

4  27 

4  38 

4  49 

4  58 

5    7 

5  i4 

5  19 

5  23 

10 

II 

2  54 

3    2 

3    9 

3  16 

3  3o 

3  43 

3  56 

4    7 

4  17 

4  27 

4  36 

4  43 

4  49 

4  53 

4  57 

II 

12 

2  44 

2  5i 

2  58 

3     4 

3  16 

3  28 

3  4o 

3  5o 

4    0 

4    8 

4  16 

4  23 

4  28 

4  32 

4  3b 

12 

i3 

2  35 

2  4i 

2   47 

2  53 

3    4 

3  i5 

3  26 

3  35 

3  44 

3  52 

3  59 

4    5 

4  10 

4  i3 

i3 

i4 

2  27 

2  33 

2  38 

2  44 

2  54 

3    4 

3  i4 

3  22 

3  3o 

3  37 

3  44 

3  5o 

3  54 

3  57 

i4 

i5 

2  22 

2  27 

2   32 

2  36 

2  46 

2  55 

3    4 

3  II 

3  18 

3  25 

3  3i 

3  37 

3  4i 

3  44 

lb 

i6 

2  17 

2  21 

2    26 

2  3o 

2  39 

2  47 

2  55 

3     2 

3    9 

3  i5 

3  21 

3  26 

3  3o 

3  33 

16 

17 

2  12 

2  16 

2    21 

2   25 

2  33 

2  4o 

2  47 

2  54 

3    0 

3     6 

3  12 

3  16 

3  19 

17 

i8 

2     8 

2  12 

2    16 

2  20 

2  27 

2  34 

2  4i 

2  47 

2  53 

2  58 

3    3 

3     7 

3  10 

18 

19 

2     5 

2    8 

2    12 

2  16 

2  22 

2  29 

2  35 

2,4l 

2  47 

2   52 

2  56 

2  59 

3     2 

19 

20 

2     3 

2    6 

2       9 

2    12 

2  18 

2  24 

2  3o 

2  35 

2  4i 

2  46 

2  49 

2   52 

2  54 

20 

21 

2     1 

2     3 

2      6 

2       8 

2  14 

2  19 

2    25 

2  3o 

2  35 

2  40 

2  43 

2  46 

21 

22 

I    5q 

2     I 

2       3 

2       5 

a  10 

2  i5 

2    20 

2    25 

2  3o 

2  35 

2  38 

2  4i 

22 

23 

I    57 

I  5q 

2       I 

2       3 

2     7 

2  12 

2    16 

2  21 

2  26 

2  3o 

2  33 

2  36 

23 

24 

I  56 

I  57 

I    59 

2       I 

2    5 

2    9 

2    l3 

2  17 

2  22 

2  26 

2  29 

2  3i 

24 

25 

1  55 

I  56 

I    57 

1 59 

2     3 

2    6 

2    10 

2  i4 

2  18 

2  22 

2    25 

25 

26 

1  54 

I  55 

I  56 

1  58 

2     I 

2    4 

2       8 

2  12 

2  i5 

2   18 

2  21 

26 

27 

I  53 

I  54 

I  55 

1 57 

2     0 

2     3 

2      6 

2  10 

2  i3 

2  i5 

2  17 

27 

28 

I  53 

I  54 

r  55 

I  56 

I  58 

2     I 

2      4 

2     8 

2  II 

2  i3 

2  i4 

28 

29 

1    52 

I  53 

I  54 

I  55 

I  57 

2    0 

2       3 

2    6 

2     8 

2  10 

29 

3o 

1    52 

r  53 

I  53 

I  54 

I  56 

I  59 

2       2 

2    4 

2    6 

2     8 

3o 

3i 

I    52 

I     52 

I    52 

I  53 

I  55 

I  58 

2      0 

2     2 

2    4 

2     5 

3i 

32 

I  5i 

1    52 

I    52 

I  53 

I  55 

I  57 

I    59 

2     I 

2     2 

2     3 

32 

33 

i     52 

I  5i 

f  5i 

I     52 

I  54 

I  56 

I  58 

I  59 

2    0 

di 

34 

I     52 

I  5i 

I  5i 

I    52 

I  53 

I  55 

I  57 

I  58 

I  59 

M 

35 

I     52 

I  5i 

I  5i 

I  5i 

I    52 

I  54 

I  56 

I  57 

I  57 

35 

36 

I  53 

I    52 

T    5l 

I  5i 

I     52 

I  53 

I  55 

I  56 

I  56 

36 

37 

I  53 

I     52 

I  5i 

I  5i 

I  5i 

I     52 

I  54 

I  55 

37 

38 

I  53 

I    52 

I  5i 

I  5o 

I  5i 

I    52 

I  53 

I  54 

38 

3q 

I  54 

I    52 

I  5i 

I  5o 

I  5i 

I    52 

I    52 

I  53 

39 

40 

I  54 

I  53 

I     52 

I  5i 

I  5o 

I  5i 

I    52 

I    52 

40 

4i 

I  54 

I  53 

T    52 

I  5i 

I  5o 

I  5. 

I  5i 

4i 

42 

I  5k( 

I  53 

I    52 

I  5i 

I  5o 

I  5i 

I  5i 

42 

43 

I  55 

I  54 

I  53 

I     52 

I  5i 

I  5i 

I  5i 

43 

Ai 

I  56 

.  54 

T  53 

I     52 

I  5i 

I  5o 

I  5o 

44 

46 

I  57 

I  55 

I  53 

I     52 

I  5i 

I  5o 

4b 

48 

.  58 

I  56 

1  54 

I  53 

I  5i 

I  5o 

48 

5o 

52 

54 

1  59 

2  0 

I  57 
I  58 
I  58 

I  55 
I  55 
I  56 

I  53 
I  53 

I  54 

I  5i 

I     52 

5o 

1 

56 

2     2 

I  59 

I  56 

I  54 

Table  P.     i^^fct  of  Sari's  Par. 
To  be  subtracietl  from  the  tliinl 

58 
60 
62 

2     3 
2     3 

2    4 

I  59 
,  59 

I  56 

Correction. 

])'3 

A  pp. 
Alt. 

Sun's  Apparent  Altituile. 

64 

5 

0  20  a 

3  40 

.30 

30  70 

sn 

90 

66 

5 

1 

I  1  1 

1 

1 

1    0 

0 

63 

10 
15 

2 

2 

1  1 

2  2  s 

1 
2 

1 
2 

2    2 

2 

U 

70 

20 

3 

i  3  I 

3 

3 

3    3 

72 

as 

4 
4 

1    4    4 
4    4    4 

4 
4 

4 
4 

4    4 
4 

74 

35 

5 

>    S    5 

5 

5 

b 

76 

40 

6 

J    B    t 

(i 

45 

K 

3    K    1 

h 

78 

50 

55 

7 

7 

/    7    7 
7    7    ■ 

7 

Ho 

GO 

8 

i    8    S 

82 

65 
70 

8 
8 

B    8 
8    8 

84 

75 

9 

9 

86 

32° 

34° 

36° 

38° 

42° 

46° 

50° 

54° 

58° 

80 
90 

9 

62° 

6(j° 



1 
I'^-^^ioj               TABLE  XLVm.                    ' 

Third  Correction.  Apparent  Distance  92°. 

App. 
Alt. 

o 
6 

7 
8 

9 

lO 

11 

12 

i3 

i4 
i5 

i6 

17 
i8 

19 

20 

21 
22 

23 

24 

25 

26 

27 
28 

=9 
3o 

3i 

32 

33 
34 
35 

36 

37 
38 
39 
40 

4i 
42 
43 

45 
46 
47 
48 
5o 

52 

54 
56 
58 
60 
62 

64 
66 
68 
70 
72 

74 
76 
78 
80 
82 

jipparetit  Altitude  of  the  Sun,  Star  or  Planet. 

0 
6 

7 
8 

9 

10 

II 
12 
i3 

i4  ' 
i5  ; 

16 
17 
iS 

19 

20 

21 
22 
23 
24 

25  : 

26 

27 
28 

3o 

"3r: 

32 

33 
34 
35 

36 

4i 

42 
43 

45 
46 

47 
48 
5o 

52 

"5^ 
56 
58 
60 
62 

64 
66 

68 
70 

72 

74 
76 
78 
80 
82 

6° 

/  H 

1  59 

2  I 
2   4 
2   8 
2  l3 
2  19 
2  25 
2  32 
2  39 

2  46 

2  53 

3  0 
3  8 
3  16 
3  23 

3  3i 
3  38 
3  46 

3  53 

4  I 

4  9 
4  17 
4  24 
4  3i 
4  38 

4  46 

4  53 

5  0 
5  7 
5  i4 

5  21 
5  28 
5  34 
5  4i 
5  47 

5  54 

6  0 
6  7 
6  i3 
6  19 

6  25 
6  3i 
6  37 
6  47 

6  57 

7  7 
7  17 
7  27 
7  36 
7  45 

7  53 

8  I 
8  8 
8  i4 
8  20 

8  25 
8  29 
8  32 
8  34 
8  36 

6° 

7° 

1    H 
2   I 

1  69 

2  I 

2  4 

2   7 
2  11 
2  16 
2  21 

2  27 
2  32 

2  38 
2  44 
2  5o 

2  56 

3  2 

3  9 

3  i5 
3  22 
3  28 
3  34 
3  4o 
3  46 
3  52 

3  58 

4  4 
4  10 
4  16 
4  22 
4  28 
4  34 

4  4o 
4  46 
4  52 

4  58 

5  3 

5  i4 
5  20 
5  25 
5  3i 
5  36 
5  4i 
5  46 

5  56 

6  5 

6  i4 
6  23 
6  3i 
6  39 
6  46 
6  53 

6  59 

7  5 
7  ic 
7  i5 

7  19 
7  23 
7  26 
7  28 
7  3c 

7° 

8° 

1     H 
2   3 
2   I 

1  59 

2  I 
2   3 

2~6 
2  10 

2  I4 
2  18 
2  22 

2  27 
2  32 
2  37 
2  42 

2  48 

2  54 

2  59 

3  4 
3  9 
3  i5 
3  20 
3  26 
3  3i 
3  36 
3  4i 

3  47 
3  52 

3  58 

4  3 
4  8 
4  i3 
4  18 

4  23 

4  28 
4  33 
4  38 
4  43 
4  48 
4  53 

4  58 

5  2 

5  7 
5  II 
5,9 

5  27 

5  35 
5  43 
5  49 

5  56 

6  2 

6  8 
6  i3 
6  18 
6  23 
6  28 

6  3i 

6  34 
6  37 
6  39 
6  4i 

8° 

9° 

/  // 
2  6 
2  3 
2  0 

1  59 

2  I 

2  3 
2  6 
2  9 
2  12 
2  i5 
2  19 
2  23 
2  27 
2  3i 
2  36 

2  4i 
2  45 
2  5o 
2  54 

2  59 

3  3 
3  8 
3  i3 
3  18 
3  22 
3  27 
3  32 
3  37 
3  4i 
3  46 

3  5o 

3  55 

4  0 
4  4 
4  8 
4  12 
4  16 
4  21 
4  25 
4  29 
4  33 
4  37 
4  4i 
4  48 

4  55 

5  2 
5  9 
5  i5 
5  21 
5  26 
5  3i 
5  36 
5  4i 
5  45 
5  49 
5  53 
5  56 

5  58 

6  0 

9° 

10° 

/  // 
2  10 
2  5 
2  2 
2  0 

1  59 

2  I 
2  3 
2  5 
2  7 
2  10 

2  i3 
2  16 
2  19 
2  22 
2  26 
2  3o 
2  34 
2  38 
2  42 
2  46 
2  5o 
2  55 

2  59 

3  3 
3  7 
3  12 
3  16 
3  20 
3  24 
3  28 

3  32 
3  36 
3  4o 
3  M 
3  48 
3  52 
3  55 

3  59 

4  3 
4  7 
4  10 
4  i3 
4  17 
4  24 
4  3o 

4  36 
4  42 
4  47 
4  53 

4  58 

5  3 
5  8 

5  T2 

5  16 
5  19 
5  22 
5  25 
5  27 
5  25 

10° 

11° 

/  ti 
2  i5 
2  9 
2  4 
2  2 
2  0 

1  59 

2  I 
2  2 

2  4 
2  6 

2  8 
2  11 
2  i4 
2  16 
2  19 

2  23 

2  26 
2  3o 
2  34 
2  37 

2  4i 
2  45 
2  48 
2  52 

2  56 

3  0 
3  4 
3  8 
3  II 
3  i5 
3  18 
3  22 
3  25 
3  29 
3  32 

3  35 
3  39 
3  42 
3  46 
3  49 
3  52 
3  55 

3  59 

4  5 
4  II 
4  16 
4  21 
4  26 
4  3i 
4  36 

4  4i 
4  45 
4  49 
4  52 
4  55 

4  57 

4  59 

5  I 

IP 

12° 

/  // 
2  21 
2  i3 

2  7 
2  4 
2  2 

2  0 

1  59 

2  0 
2  2 
2  3 
2  5 

2  7 
2  9 
2  II 

2  i4 

2  17 
2  20 
2  23 
2  27 
2  3o 

2  33 
2  36 
2  39 
2  43 
2  46 
2  5o 
2  53 

2  57 

3  0 
3  3 
3  6 

3  12 
3  i5 
3  18 
3  21 
3  24 
3  27 
3  3o 
3  33 

3  36 
3  39 
3  42 
3  48 
3  53 

3  58 

4  3 
4  8 
4  i3 
4  17 
4  21 

4  25 

4  28 
4  3i 
4  33 
4  35 
4  37 
4  39 

12° 

14° 

/  // 
2  34 
2  22 
2  i4 
2  9 
2  6 

2  3 
2  I 
2  0 

1  59 

2  0 
2  I 
2  3 
2  4 
2  6 
2  8 

2  10 
2  12 
2  i4 
2  16 
2  19 

2  22 
2  24 
2  27 
2  29 

2  32 

2  35 
2  37 
2  4o 
2  42 
2  45 

2  47 
2  5o 
2  53 
2  55 

2  58 

3  0 
3  2 
3  5 
3  8 
3  II 
3  i3 
3  16 
3  18 
3  22 
3  26 
3  3o 
3  34 
3  38 
3  42 
3  46 
3  5o 
3  53 
3  56 

3  58 

4  0 

4  I 
4  2 

14° 

16° 

/  // 
2  48 
2  33 

2  23 
2  16 
2  II 

2  7 
2  4 
2  2 
2  I 
2  0 

1  59 

2  0 
2  I 
2  2 
2  « 

2  5 
2  6 
2  8 
2  9 
2  II 

2  i3 
2  i5 
2  17 
2  19 
2  21 

2  23 
2  25 
2  27 
2  29 
2  3l 

2  33 
2  36 
2  38 
2  4o 
2  42 

2  45 
2  47 
2  49 
2  5i 
2  53 
2  55 
2  57 

2  59 

3  4 
3  8 

3  II 

3  i4 
3  17 
3  20 
3  23 

3  26 
3  29 
3  32 
3  34 
3  35 

3  36 
16° 

18° 
/  // 
3  3 
2  45 
2  33 
2  24 
2  17 

2  12 
2  8 
2  5 
2  3 
2  I 

2  0 
159 

1  59 

2  0 
2  0 
2  I 
2  2 
2  3 
2  4 
2  5 

2  7 
2  9 
2  II 
2  12 
2  i3 
2  i5 
2  16 
2  18 
2  20 
2  22 

2  24 

2  25 

2  27 
2  29 
2  3i 
2  33 
2  34 
2  36 
2  38 
2  40 
2  42 
2  44 
2  46 
2  49 
2  53 

2  56 

2  59 

3  2 
3  5 
3  8 
3  10 
3  12 
3  i4 
3  i5 
3  16 

18° 

20° 

/  // 
3  18 
2  58 
2  A^ 
2  33 
2  24 
2  18 
2  i3 
2  9 
2  6 
2  4 
2  2 
2  I 
2  0 
2  0 
I  59 

1  59 

2  0 
2  0 
2  I 
2  2 

2  4 
2  5 
2  6 

2  7 
2  8 

2  9 
2  II 
2  12 
2  i4 
2  i5 

2  17 
2  18 
2  20 
2  21 
2  22 

2  24 

2  25 

2  27 
2  28 
2  3o 

2  3i 
2  33 
2  35 
2  37 
2  4i 
2  44 
2  47 
2  49 

2  52 

2  54 
2  56 
2  57 
2  58 
2  59 

20° 

22° 

/  // 
3  33 
3  II 
2  55 
2  42 

2  32 

2  24 
2  18 
2  i4 
2  10 
2  7 
2  4 
2  3 
2  2 
2  I 
2  0 

I  59 
I  59 

1  59 

2  0 
2  0 
2  I 
2  2 
2  2 
2  3 
2  4 
2  5 
2  7 
2  8 
2  9 
2  10 

2  II 
2  12 

2  i4 
2  i5 
2  16 
2  17 
2  18 
2  20 
2  21 
2  22 

2  24 

2  25 

2  27 
2  29 

2  32 

2  34 
2  37 
2  39 
2  4i 
2  43 

2  44 
2-45 
2  46 

22° 

24° 
/  (/ 
3  48 
3  24 
3  5 
2  5i 
2  4o 

2  3i 
2  24 
2  19 
2  i4 
2  10 

2  7 
2  5 
2  4 
2  2 
2  I 

2  0 
2  0 
I  59 

I  59 
I  59 

1  59 

2  0 
2  0 
2  I 
2  I 

2  2 
2  3 
2  4 
2  5 
2  6 

2  7 
2  8 
2  9 
2  10 
2  II 

2  12 
2  i3 
2  i4 
2  i5 
2  16 
2  18 
2  19 
2  20 
2  22 
2  24 

2  26 
2  29 
2  3i 

2  32 

2  34 
2  35 
2  37 

24° 

26° 
/  // 
4  3 
3  36 
3  16 
3  I 
2  48 
2  39 
2  3i 
2  24 
2  19 
2  i5 
2  II 
2  8 
2  6 
2  4 
2  3 

2  2 
2  I 

2  0 
159 
I  59 

I  59 

1  59 
r  59 

2  0 
2  0 

2  0 
2  I 
2  I 
2  2 
2  3 

2  4 
2  5 
2  6 
2  7 
2  7 
2  8 
2  9 
2  10 
2  II 
2  12 

2  i3 
2  i4 
2  i5 
2  16 
2  18 

2  20 
2  22 
2  24 
2  25 
2  26 
2  27 

26° 

28° 
/  // 
4  18 
3  48 
3  26 
3  10 
2  57 

2  46 
2  37 
2  3o 
2  24 
2  19 

2  i5 
2  12 
2  9 

2  7 
2  5 

2  4 
2  2 
2  I 
2  0 
2  0 

I  59 
I  59 
I  59 
I  59 

1  59 

2  0 
2  0 
2  I 
2  I 

2  2 
2  3 
2  4 
2  4 
2  5 
2  5 
2  6 
2  7 
2  8 
2  8 
2  9 
2  10 
2  II 
2  12 
2  i3 

2  i5 
2  16 
2  18 
2  20 
2  21 

28° 

30° 
/  /' 
4  33 
4  0 
3  37 
3  19 
3  5 
2  54 
2  44 
2  36 
2  29 
2  24 
2  19 
2  i5 
2  12 
2  10 
2  8 
2  6 
2  4 
2  2 
2  I 
2  0 

2  0 
2  0 
I  59 
I  59 
I  59 

159 
I  59 

1  09 

2  0 
2  0 

2  I 
2  I 
2  2 
2  2 
2  3 
2  3 
2  4 
2  4 
2  5 
2  5 

2  6 
2  7 
2  8 
2  9 
2  10 

2  II 
2  12 
2  i3 
2  i4 

30° 

TABLE  XLVIII.                                     [Page  311 

Third  Correction.     Apparent  Distance  92°. 

1 

D's 

A  Dp. 

Alt. 

Apparent  Altitude  <if  the  Sun,  Star  or  Planet. 

D's 
App. 
All. 

32° 

34° 

36° 

38° 

40° 

42° 

4G° 

50° 

54° 

58° 

62° 

(j6° 

70° 

74° 

78° 

82° 

0 

1  II 

1  II 

/  II 

1  II 

/  // 

/  II 

1   II 

/  // 

/   II 

^  // 

/  // 

/   // 

/     // 

/  // 

;    II 

/  // 

0 

6 

4  47 

5     2 

5  16 

5  3o 

5  44 

5  57 

6  21 

6  44 

7     5 

7  24 

7  42 

7   59 

8  12 

8  22 

8  3o 

8  36 

6 

7 

4  12 

4  24 

4  36 

4  48 

5    0 

5  II 

5  33 

5  53 

6  12 

6  29 

6  44 

6  58 

7   10 

7    18 

7    25 

7  3o 

7 

8 

3  48 

3  59 

4  10 

4  20 

4  3o 

4  4o 

4  59 

5  17 

5  33 

5  48 

6     I 

6  i3 

6  23 

6  3] 

6  37 

6  4i 

8 

9 

3  29 

3  38 

3  48 

3  57 

4    6 

4  i4 

4  3o 

4  45 

5    0 

5  i3 

5  25 

5  35 

5  44 

5  52 

5  58 

9 

10 

3  i3 

3  21 

3  3(. 

3  38 

3  45 

3  52 

4    8 

4  22 

4  34 

4  46 

4  57 

5     7 

5  i5 

5    23 

5  27 

10 

II 

3     1 

3     8 

3  16 

3  22 

3  29 

3  36 

3  5o 

4    3 

4  i4 

4  24 

4  34 

4  43 

4  5i 

4  57 

5     2 

II 

12 

2  5i 

2  57 

3    4 

3  10 

3  17 

3  23 

3  35 

3  47 

3  57 

4    6 

4  i5 

4  23 

4  3o 

4  36 

4  4i 

12 

i3 

2  42 

247 

2  53 

2  59 

3    6 

3  12 

3  23 

3  33 

3  42 

3  5i 

4    0 

4    7 

4  i3 

4  17 

i3 

i4 

2  34 

2  39 

2  44 

2  5o 

2  56 

3     2 

3  12 

3  21 

3  3o 

3  38 

3  46 

3  52 

3  57 

4     I 

i4 

i5 

2  28 

2  33 

2  38 

2  43 

2  48 

2  53 

3     2 

3  II 

3  19 

3  26 

3  33 

3  39 

3  44 

3  49 

i5 

16 

2   23 

2  28 

2    32 

2  37 

2  42 

2  46 

2  54 

3     2 

3     9 

3  16 

3  22 

3  28 

3  33 

3  38 

16 

17 

2  19 

2   23 

2  27 

2   32 

2  36 

2  4o 

2  47 

2  54 

3     I 

3     7 

3  i3 

3   19 

3  24 

•7 

18 

2  16 

2  19 

2   23 

2  27 

2  3i 

2  34 

2  4i 

2  48 

2  54 

3     0 

3    6 

3  II 

3  i5 

18 

19 

2  i3 

2  16 

2  19 

2    23 

2  26 

2  29 

2  36 

2  42 

2  48 

2  54 

2  59 

3    4 

3     7 

19 

20 

2   10 

2   13 

2  16 

2    19 

2  22 

2  25 

2  3i 

^  37 

2  43 

2  48 

2  53 

2  57 

3     0 

20 

21 

2     8 

2   10 

2  i3 

2    16 

2  18 

2  21 

2  26 

2   32 

2  38 

2  43 

2  47 

2  5i 

21 

22 

2     6 

2     8 

2  10 

2    l3 

2  i5 

2  17 

2  22 

2  28 

2  33 

2  38 

2  42 

2  45 

22 

23 

2    4 

2    6 

2    8 

2    10 

2  12 

2  i4 

2  19 

2  24 

2  29 

2  34 

2  38 

2  4i 

23 

24 

2     2 

2    4 

2    6 

2       8 

2  10 

2  12 

2  16 

2  21 

2  26 

2  3o 

2  34 

2  38 

24 

25 

2     I 

2     3 

2    4 

2       6 

2     8 

2  10 

2  14 

2  18 

2  22 

2  26 

2  3o 

25 

26 

2     I 

2     2 

2    3 

2       5 

2     6 

2    8 

2  12 

2  i5 

2  19 

2    23 

2  26 

26 

27 

2    0 

2     I 

2     2 

2       4 

2     5 

2    7 

2  10 

2  i3 

2  16 

2  20 

2    23 

27 

28 

I  59 

2    0 

2     I 

2       3 

2    4 

2     6 

2     8 

2  11 

2  14 

2  17 

2  20 

38 

29 

I  59 

I  59 

2    0 

2       2 

2    3 

2     5 

2     7 

2  10 

2  12 

2  i5 

29 

3o 

I  59 

I  59 

2    0 

2        I 

2     2 

2    4 

2    6 

2     9 

2  II 

2  i3 

3o 

3i 

I  59 

.  59 

I  59 

2       0 

2     I 

2     3 

2     5 

2     7 

2    9 

2  11 

3i 

32 

I  b9 

.   59 

I  59 

2       0 

2     I 

2     2 

2    4 

2     6 

2     7 

2     c^ 

32 

33 

I   59 

I   59 

I  59 

I    59 

2     0 

2     I 

2     3 

2     5 

2     6 

33 

U 

I   59 

I  58 

I  59 

I    59 

2    0 

2     I 

2     2 

2    4 

2     5 

34 

35 

I   D9 

I  58 

I  59 

I    59 

2    0 

2    0 

2     I 

2     3 

2    4 

35 

36 

2     0 

I  59 

I  59 

I    59 

2     0 

2    0 

2     I 

2     2 

2     3 

36 

37 

2     0 

I  59 

I  59 

I  58 

I  59 

I  59 

2     0 

2     I 

37 

38 

2     0 

I  59 

I  59 

I  58 

I  59 

I  59 

2    0 

2     1 

38 

39 

2     I 

2     0 

I  59 

I  5b 

I  58 

I  59 

I  59 

2    0 

39 

Ao 

2     1 

2     0 

I  69 

I  58 

I  58 

I  58 

I  59 

2     0 

40 

4i 

2     I 

2     0 

I  59 

I  59 

I  58 

I   58 

I  58 

4i 

42 

2     2 

2     0 

I  59 

I  59 

I  58 

I  58 

I  58 

42 

43 

2     2 

2     1 

2    0 

I  59 

I  58 

I  58 

I  58 

43 

/^A 

2     3 

2     1 

2    0 

I  59 

I  58 

I  58 

I   57 

44 

45 

2     3 

2     2 

2     I 

2     0 

.  59 

I  58 

45 

46 

2    4 

2     2 

2     I 

2     0 

I  59 

I   58 

46 

47 

2    4 

2     2 

2     1 

2     0 

I  59 

I   58 

47 

48 

2    5 

2     3 

2     2 

2     I 

I  59 

I  59 

1 

5o 

52 

2    6 

2    7 

2    4 
2    5 

2     I 
2     I 

1 

2     3 

r«6ie  r.     Efcct  of  Sun's  Par. 

54 

2     8 

2    5 

2    3 

To  be  subtracted  from  the  Ihird 

')ri 

2     9 
2   10 

9     5 

58 

D's 

Sun's  Apparent  Altitude. 

fio 

Ah. 

5 

U'20  3 

40 

50  6 

0  70 

SO 

90 

62 

6,4 



5 
10 

2 

1  1     ! 

2  2    2 

1 

I 

1  1 

2  2 

1 
2 

0 

66 

13 

20 

2 

2  2    2 

3  3    3 

3 
3 

3 
3 

i    b 
1   3 

68 

25 
30 

4 

4 

4    4    4 
4    5    5 

4 

4 

5 

5 

70 

35 

5 

i    5    5 

5 

5 

40 

6 

5    6    6 

6 

6 

45 

H 

7 

74 

50 

7 
7 

7    7    7 

7    7    8 

7 

-7h 

60 

8 

S    8   a 

78 

65 
70 

9 

8    8 
8    8 

80 

75 

9 

9 

82 

80 
90 

33° 

34° 

3(]° 

38° 

40° 

42° 

46° 

50° 

54° 

58° 

62° 

P'^^^i-^]                TABLE  XLVIII. 

Third  Correction.  Apparent  Distance  96°. 

D's 
App. 
Alt. 

o 
6 

7 
8 

9 

10 

II 

12 

i3 

i4 
i5 

i6 

17 
]8 

19 

20 

21 
22 

23 

24 

25 

26 
27 
28 
29 

3o 
3i 

32 

33 
34 
35 

36 

37 
38 

39 

4o 

4i 
42 
43 
44 
45 

46 
47 
48 
49 
5(j 

5i 

52 

54 
56 
58 

60 
62 
64 
66 
68 

70 

72 
74 

76 
78 

Apparent  Altitude  of  the  Sun,  Sta7-  or  Planet. 

D's 
App. 
Alt. 

0 
6 

7 
8 

9 

ID 

II 
12 
l3 

i4 
i5 

16 

17 
18 

19 
20 

21 
22 

23 

24 

25 

26 
27 
28 

=9 

3o 

3i 

32 

33 
34 
35 

36 

37 
38 
39 
4o 

4i 
42 
43 
44 
45 

46 
47 
48 
49 
5o 

IT 

52 

54 
56 
58 

60 
62 
64 
66 
68 

70 
72 
74 
76 
78 

6° 

2  6 
2  9 
2  12 
2  16 

2  20 
2  26 

2  32 

2  39 
2  46 

2  53 

3  I 
3  8 
3  i5 
3  23 
3  3o 
3  38 
3  46 

3  54 

4  I 
4  9 
4  16 
4  24 
4  3i 
4  39 
4  46 

4  53 

5  0 
5  7 
5  i4 
5  21 

5  28 
5  35 

5  42 
5  49 

5  55 

6  2 
6  8 
6  i4 
6  20 
6  26 
6  32 
6  38 
6  44 
6   5o 

6  55 

7  0 
7  5 
7  i5 
7  25 

7  35 
7  45 

7  54 

8  2 
8  10 
8  17 

8  24 
8  29 
8  33 
8  37 
8  4o 

7° 

2  8 
2  6 
2  8 
2  II 

2  i4 

2  18 
2  23 
2  28 
2  33 
239 

2  45 
2  5i 

2  57 

3  3 
3  9 

3  16 
3  22 
3  28 
3  34 
3  4i 

3  47 

3  53 

4  0 
4  6 
4  12 
4  18 
4  24 
4  3o 
4  36 
4  42 
4  48 

4  54 

5  0 
5  6 
5  12 

5  18 
5  23 
5  29 
5  34 
5  39 

5  44 
5  49 
5  54 

5  59 

6  4 
6  9 
6  i4 
6  23 
6  3i 
6  39 

6  46 
5  53 

7  0 
7  7 
7  i3 

7  18 
7  23 
7  27 
7  3. 

7  34 

7° 

8° 
/ 
2  10 
2  8 
2  6 
2  8 
2  10 

2  i3 

2  17 

2  21 

2  25 
2  29 

2  34 
2  39 

2  44 
2  49 

2  54 

3  0 
3  5 
3  II 
3  16 
3  22 

3  27 
3  33 
3  38 
3  44 
3  49 

3  55 

4  0 
4  5 
4  II 
4  16 

4  21 
4  26 
4  3i 
4  36 
4  4i 
4  46 
4  5i 

4  56 

5  I 
5  6 
5  10 
5  i5 
5  19 
5  23 
5  27 
5  3i 
5  35 
5  43 
5  5i 

5  58 

6  4 
6  10 
6  16 
6  21 
6  26 
6  3i 
6  36 
6  4o 
6  43 
6  46 

8° 

9° 

/  // 
2  i3 
2  10 
2  7 
2  6 
2  8 
2  10 
2  i3 
2  16 
2  19 
2  22 

2  26 
2  3o 
2  34 
2  38 
2  43 
2  48 
2  52 

2  57 

3  1 
3  6 

3  11 
3  i5 
3  20 
3  25 
3  29 

3  3i 
3  39 
3  44 
3  49 
3  54 

3  59 

4  3 
4  8 
4  12 
4  16 
4  20 
4  24 
4  29 
4  33 
4  37 

4  4i 
4  45 
4  49 
4  53 

4  57 

5  I 
5  4 
5  II 
5  18 
5  24 
5  3o 
5  35 
5  40 
5  45 
5  5o 

5  54 

5  58 

6  I 
6  4 

9° 

10° 

/  // 

2  17 
2  12 
2  9 
2  7 
2  6 

2  7 
2  9 
2  12 

2  i4 
2  17 
2  20 

2  23 

2  26 
2  3o 
2  34 
2  37 

2  4i 
2  45 
2  49 
2  53 

2  57 

3  I 
3  6 
3  10 
3  i4 
3  18 
3  23 
3  27 
3  32 
3  36 
3  40 
3  44 
3  48 
3  52 

3  56 

4~ 

4  4 
4  8 
4  II 
4  i4 
4  18 
4  22 
4  25 
4  29 
4  32 

4  36 
439 
4  45 
4  5i 

4  56 

5  1 
5  6 
5  II 
5  16 
5  21 

5  25 
5  28 
5  3i 
5  34 

10° 

11° 

1  II 

2  22 
2  16 
2  12 
2  9 
2  7 
2  6 

2  7 
2  9 
2  II 

2  i4 

2  16 
2  19 
2  21 

2  24 
2  27 

2  3o 
2  33 
2  37 
2  4i 
2  45 
2  48 

2  52 

2  55 

2  59 

3  3 

3  7 
3  II 
3  i5 
3  19 
3  23 

3  26 
3  29 
3  33 
3  36 
3  40 

3  44 
347 
3  5i 
3  54 

3  57 

4  0 
4  3 
4  7 
4  10 
4  i3 
4  16 
4  19 
4  25 
4  3o 
4  35 
4  39 
4  44 
4  48 
4  52 

4  56 

5  0 
5  3 
5  6 

11° 

12° 

/  // 
2  28 
2  20 
2  i5 
2  12 
2  9 

2  7 
2  6 

2  7 
2  9 
2  1 1 

2  i3 
2  i5 
2  17 
2  19 
2  22 

2  25 

2  28 
2  3i 

2  35 
2  38 

2  41 
2  44 
2  47 
2  5o 
2  53 

2  67 

3  I 
3  4 
3  7 
3  II 

3  i4 
3  17 
3  20 
3  23 
3  26 
3  3o 
3  33 
3  36 
3  39 
3  42 

3  45 
3  48 
3  5i 
3  54 

3  57 

4  0 
4  3 
4  8 
4  i3 
4  17 
4  21 

4  25 

4  29 

4  33 
4  36 
4  39 
4  4i 
4  43 

12° 

14° 

'  // 
2  4i 
2  29 
2  22 
2  17 
2  i3 
2  10 
2  8 
2  7 
2  6 
2  7 
2  8 
2  9 
2  II 
2  i3 
2  i5 

2  17 
2  19 
2  21 

2  23 
2  26 
2  29 
2  3i 
2  34 
2  36 
2  38 

2  4i 
2  44 
2  46 
2  48 
2  5i 

2  54 
2  57 

2  59 

3  2 
3  5 

3  7 
3  10 
3  i3 
3  16 
3  ,9 
3  21 
3  24 
3  26 
3  28 
3  3o 
3  32 
3  34 
3  38 
3  42 
3  46 
3  5o 
3  54 

3  58 

4  I 
4  3 

4  5 
4  7 

14° 

1G° 

1  II 

2  55 
2  4o 
2  3i 
2  24 
2  18 

2  i4 
2  II 
2  9 

2  8 
2  7 
2  5 

2  7 
2  8 
2  9 
2  II 

2  12 

2  i4 
2  i5 

2  17 
2  19 

2  21 

2  23 

2  24 
2  26 
2  28 

2  3o 

2  32 

2  34 
2  36 
2  38 

2  4o 
2  43 
2  45 
2  47 
2  5o 

2  52 

2  54 
2  57 

2  59 

3  I 
3  3 
3  5 
3  7 
3  9 
3  II 
3  i3 
3  i5 
3  19 
3  22 
3  26 

18° 
/  II 
3  10 
2  52 
2  4o 
2  3i 
2  24 
2  19 
2  i5 
2  12 
2  10 
2  9 

2  8 

2  7 
2  6 

2  7 
2  8 

2  9 

2  10 
2  II 
2  12 

2  14 

2  16 
2  17 
2  18 

2  2C 
2  21 

2  23 
2  25 
2  26 
2  28 

2  3o 

2  32 

2  33 
2  35 
2  37 
2  39 

2  41 

2  43 
2  45 
2  47 

2  48 
2  5o 

2  52 

2  53 
2  55 
2  56 

2  58 

3  0 
3  2 
3  6 
3  9 

20° 

/  /I 
3  26 
3  5 
2  5i 
2  4o 

2  32 

2  25 
2  20 
2  16 
2  l3 
2  II 

2  10 
2   9 
2   8 
2   7 
2   6 

2   7 
2   8 
2   8 
2   9 
2  II 

2  12 
2  l3 

2  i4 

2  l5 
2  16 

2  18 
2  19 
2  20 
2  21 
2  23 
2  25 
2  26 
2  27 
2  29 

2  3o 

2  32 
2  34 

2  35 

2  37 

2  38 
2  39 
2  4i 
2  42 
2  44 
2  45 

2  47 
2  48 
2  5i 

2  54 
2  57 

22° 
/  It 
3  4i 
3  18 
3  2 
2  49 
2  40 

2  32 
2  26 
2  21 
2  18 
2  i5 

2  i3 
2  II 
2  10 
2  8 

2  7 
2  7 
2  6 
2  6 
2  7 
2  8 

2  9 
2  10 
2  II 
2  12 
2  i3 

2  i4 
2  i5 
2  16 
2  17 
2  18 
2  20 
2  21 
2  22 

2  23 

2  24 

2  25 
2  27 
2  28 
2  29 

2  3o 

2  3l 

2  33 
2  34 
2  36 
2  37 
2  38 
2  39 
2  42 
2  45 
2  47 

24° 
/  // 
3  56 
3  3i 
3  i3 
2  59 
2  48 
2  89 

2  32 

2  26 
2  22 
2  19 

2  16 
2  i4 
2  12 
2  10 
2  9 

2  8 
2  7 
2  6 
2  6 
2  6 

2  7 
2  8 
2  9 

2  ID 
2  10 
2  II 
2  12 
2  l3 

2  i4 

2  l5 

2  l6 
2  17 
2  18 
2  19 
2  20 
2  21 
2  22 
2  23 
2  24 
2  25 

2  26 
2  27 
2  28 
2  29 
2  3l 
2  32 

2  33 
2  35 

2  37 
2  39 

2  4i 

2  42 

26° 
/  // 
4  II 

3  43 
3  24 
3  8 
2  56 

2  46 
2  38 
2  32 

2  27 

2  23 

2  20 

2  17 
2  l5 
2  l3 
2  II 

2   9 
2   8 

2   7 
2   7 
2   7 
2   6 
2   6 

2   7 
2   8 
2   8 
2   9 
2   9 
2  10 
2  II 
2  12 
2  l3 

2  i4 

2  l5 
2  16 
2  16 
2  17 
2  18 
2  19 
2  20 
2  21 
2  22 
2  23 
2  24 
2  25 
2  26 
2  27 
2  28 
2  29 

2  3o 

2  3l 
2  32 

28° 

1  II 
4  26 
3  56 
3  35 
3  18 
3  4 

2  53 
2  45 
2  38 

2  32 

2  28 

2  24 
2  21 
2  18 
2  i5 
2  i3 
2  11 
2  10 
2  9 
2  8 
2  8 

2  7 
2  6 
2  6 
2  7 
2  7 
2  8 
2  8 
2  8 
2  9 
2  10 
2  II 
2  II 
2  12 
2  i3 
2  i3 

2  i4 
2  i5 
2  16 

2  17 

2  17 

2  18 
2  19 
2  20 
2  21 
2  21 

2  22 

2  23 

2  24 

2  25 
2  25 

30° 

/  // 
4  4i 
4  8 
3  45 
3  27 
3  12 

3  I 

2  52 

2  44 
2  37 

2  32 

2  28 
2  24 

2  21 

2  18 
2  l5 

2  l3 
2  12 
2  II 
2  10 
2   9 

2   8 
2   7 
2   7 
2   6 
2   6 

2   7 
2   7 
2   7 
2   8 
2   8 
2   9 
2   9 
2  ID 
2  II 
2  II 

2  12 
2  l3 
2  l3 
2  l4 
2  l4 
2  l5 
2  16 
2  16 
2  17 
2  17 

2  18 
2  18 
2  19 
2  19 

3  29 
3  32 
3  35 
3  38 
3  4o 
3  42 

1G° 

3  12 
3  i5 
3  17 
3  19 
3  21 

18° 

2  59 

3  I 
3  3 
3  4 

20° 

2  49 
2  5o 
2  5i 

22° 

24° 

26° 

28° 

30° 

TABLE  XLVni.                                     '^^se3i3 

Third  Correction.     Apparent  Distance  96°. 

Apparent  Jlltitudc  of  the  Sun,  Star  or  Planet. 

D's 
A  pp. 
All. 

32° 

34° 

36° 

38° 

40° 

42° 

44° 

46° 

50° 

54° 

58° 

62° 

66° 

70°  '.  74°  :  73° 

0 

(    '/ 

/  II 

/  II 

/  // 

/   /' 

/  // 

/    /; 

/   // 

/  /' 

/   // 

/  // 

/  /' 

1   n 

/Hi      1.  '.      1 

0 

6 

4  56 

5  10 

5  24 

5  38 

5  5i 

3    4 

S  17 

6  29 

6  52 

7  i4 

7  34 

7   52 

8     9 

8  23  8 

33  8  4o 

b 

7 

4  20 

4  32 

4  44' 

i  56 

5    8 

5  20 

5  3i 

5  42 

6     2 

5  20 

5  37 

b    52 

7     6 

7   187 

287  35 

7 

8 

3  56 

4    7 

4  18- 

i  29 

4  39 

4  49 

4  59 

5     8 

5  26 

5  42 

5  bb 

b    9 

6  21 

6  3i  6 

406  47 

8 

9 

3  37 

3  46 

3  55. 

i    4 

4  i3 

4  23 

4  32 

4  4o 

4  56 

5  10 

5  23 

5  34 

b  44 

5  53  6 

I 

9 

lO 

3  22 

3  3o 

3  37 

i  45 

3  53 

4    2 

4  10 

4  17 

4  3i 

4  43 

4  5b 

b     6 

b  16 

5  245 

3i 

10 

II 

3     9 

3  17 

3  24 

3  3i 

3  38 

3  45 

3  52 

3  5q 

4  11 

4  22 

4  33 

4  43 

4  52 

5    o5 

7 

II 

12 

2  59 

3    6 

3  12 

3  19 

3  25 

3  32 

3  38 

3  45 

3  55 

4    5 

4  lb 

4  24 

4    32 

4  394 

46 

12 

i3 

2  5o 

2  56 

3     2 

3     8 

3  i4 

3  20 

3  26 

3  32 

3  42 

3  5i 

4    0 

4    8 

4  lb 

4    2t 

i3 

i4 

2  4?- 

2  48 

2  53 

2  58 

3     A 

3     9 

3  i5 

3  20 

3  3o 

3  39 

3  48 

3  5b 

4    I 

4    6 

i4 

i5 

2  36 

2  41 

2  46 

2  5o 

2  55 

3    0 

3     5 

3  10 

3  19 

3  28 

3  36 

3  43 

3  49 

3  54 

i5 

i6 

2   32 

2  36 

2  4fi 

2  44 

2  48 

2  53 

2  57 

3     2 

3  10 

3  18 

3  25 

3  32 

3  38 

3  44 

16 

17 

2  28 

2  3i 

2  35 

2  39 

2  43 

2  47 

2  5i 

2  55 

3     3 

3  10 

3  16 

3  22 

3    2b 

17 

i8 

2  24 

2  27 

2  3i 

2  35 

2  38 

2  42 

2  45 

2  49 

2  56 

3     2 

3    8 

3  i4 

3  ,9 

18 

19 

2  21 

2  24 

2  27 

2  3i 

2  34 

2  37 

2  4o 

2  44 

2  5o 

2  56 

3     2 

J     7 

3  11 

19 

20 

2  18 

2  21 

2  24 

2  27 

2  3o 

2  33 

2  36 

2  39 

2  45 

2  5i 

2  56 

3     I 

3    4 

20 

21 

2   16 

2  19 

2  21 

2  24 

2  26 

2  20 

2   32 

2  35 

2  4i 

2  46 

2  5i 

2  55 

21 

22 

2  i4 

2  17 

2  -19 

2  21 

2  23  2  26 

2  28 

2  3i 

2  37 

2  42 

2  46 

2  5o 

22 

23 

2  i3 

■2  i5 

2   17 

2  19 

2  21  2   23 

2   25 

2  28 

2  33 

2  38 

2  42 

2  4b 

23 

24 

2  II 

2  i3 

2   i5 

2  17 

2    192    21 

2    23 

2   25 

2  3o 

2  35 

2  38 

2  4i 

24 

25 

2  10 

2  1 1 

2  i3 

2  i5 

2    172    19 

2    21 

2    23 

2  27 

2  3i 

2  3b 

25 

26 

2     9 

2  10 

2  12 

2  i3 

2    l5 

2  17 

2    19 

2    21 

2    25 

2  28 

2  3i 

26 

27 

2     8 

2     9 

2  II 

2  12 

2    l4 

2  16 

2    18 

2    20 

2   23 

2  2b 

2  27 

27 

28 

2     8 

2     9 

2  10 

2  II 

2     l3 

2  i5 

2     17 

2    18 

2  21 

2   23 

2  24 

28 

29 

2     7 

2     8 

2     9 

2  10 

2    12 

2  i3 

2    l5 

2    17 

2  19 

2    21 

29 

3o 
3i 

2     7 
2     6 

2     8 

2     7 

2    9 

2     8 

2  10 
2     9 

2    II 

2     10 

2  12 
2  II 

2    l4 
2     12 

2    l5 
2    l4 

2  17 
2  16 

2     19 





3o 
3i 

2    17 

32 

2    6 

2     7 

2     7 

2     8 

2       9 

2  10 

2    II 

2    12 

2  i4 

2     16 

32 

33 

2    6 

2    6 

2     7 

2     7 

2      8 

2    9 

2    10 

2    II 

2  i3' 

33 

34 

2     7 

2    6 

2     7 

2     7 

2       8 

2     9 

2    10 

2    II 

2  12 

34 

35 

2     7 

2    6 

2     6 

2     7 

2       7 

2     8 

2      9 

2    ID 

2  II 

35 

36 

2     8 

2     7 

2    6 

2     6 

2       7 

2     8 

2       9 

2      9 

2  10 

36 

37 

2     8 

2     7 

2    6 

2    6 

2      7 

2     7 

2      8 

2      8 

i7 

38 

2928 

2    7 

2     6 

2      6 

2     7 

2      8 

2      8 

38 

39 

2    92    8 

2     7 

2     6 

2      6 

2     7 

2      7 

2      8 

39 

4o 

2  10  2    8 

2     7 

2     6 

2      6 

2    6 

2       7 

2       7 

4o 

4i 

2102     9 

2     8 

2     7 

2      7 

2    6 

2       6 

4i 

42 

2  II 

2     9 

2     8 

2     7 

2       7 

2    6 

2      6 

42 

43 

2  11 

2   10 

2     8 

2     7 

2      7 

2    6 

43 

44 

2  12 

2   10 

2     8 

2     7 

2      7 

2     6 

44 

45 

2  12 

2   10 

2     9 

2     8 

2      7 

45 

46 

2  i3 

2   II 

2     9 

2     8 

2       7 

46 

47 
48 

2   i3 
2  i3 
2  i4 

2   11 
2   II 

2     9 
2     9 

2     8 
2     8 

47 

49 

2    12 

2  10 

5o 

2  t4 

2    12 

2  10 

5 

^a4;e  P.     Effect  of  Suji's  Par. 

To  be  subtracled  from  the  Third 

5i 

52 

54 
56 
58 

2  i4 
2  i5 
2  i5 

2   12 
2   12 

Correclion. 

D's 

Sun's  Apparent  Altitude. 

Alt.     5 

lU  20  30  -1 

0  50  60 

70  b 

3  90 

5       1 

1     1    1 

1     I 

2    2 

" 

60 

10       2 
15       2 

2    2    2 
2    3    3 

J    3    3 

a 

62 

20       3 

3    3    4 

4    4 

64 

25       4 

4  4    4 

5  5    5 

5    5 

;     1 

66 

35      5 

5    5    6 

6 

;         J 

68 

40       6 
45       6 

50       7 

6  6    6 

7  7    7 
7    7    7 

r 

1 

1         I 
1         , 

70 

65     |7 

8    8    8 

\         1 

72 

60       8 

8    8 

j 

74 

65       8 
70       9 

8    8 
9 

76 

75       9 

78 

¥2° 

34= 

36° 

38° 

40° 

42° 

44° 

46° 

50° 

54° 

(58° 

90 

I 

I 

_— 

40 


f'^^-^^Hj               TABLE  XLVIIL 

Third  Correction.  Apparent  Distance  100°. 

D's 
App. 

Alt. 

o 
6 

7 
8 

9 

lO 

II 

12 

i3 

i4 
i5 

i6 

I? 
i8 

19 
20 

21 
22 

23 

24 

25 

26 

27 
28 

=9 
So 

3i 

32 

33 
34 
35 

36 

37 
38 
39 

4o 

4i 
42 
43 
44 
45 

46 

47 
48 
49 

Do 
52 

53 

54 
55 

56 
58 
60 
62 
64 
66 
68 
70 
72 
74 

Jipparent  Mtitude  of  the  Sun,  Sta?-  or  Planet. 

5's 
App. 

Ait. 

0 
6 

7 
8 

9 

ID 
II 

i3 
i4 
i5 

16 

17 
18 

19 

20 

21 
22 

23 

24 

25 

26 

27 
28 

29 

3o 
3i 

32 

33 
34 
35 

36 

37 
38 
39 
4o 

4i 
42 
43 
44 
45 

46 
47 
48 

49 
5o 

52 

53 

54 
55 

56 
58 
60 
62 
64 

66 
68 
70 

72 
74 

/  ti 
2   i3 
2  16 
2  19 

2  23 

2  28 
2  33 
2  4o 
2  47 

2  54 

3  I 

3  8 
3  i5 
3  23 
3  3o 
3  38 

3  45 

3  53 

4  I 

4  9 
4  16 

4  24 
4  3i 
4  39 
4  46 

4  54 

5  I 
5  8 
5  16 
5  23 
5  3o 

5  37 
5  44 
5  5i 

5  58 

6  4 
6  u 
6  18 
6  24 
6  3o 
6  36 
6  42 
6  48 

6  54 

7  0 
7  5 

7  II 
7  16 
7  21 
7  26 
7  3i 
7  36 
7  46 

7  56 

8  5 
8  i3 

8  21 
8  28 
8  35 
8  4(- 
8  4/ 

G° 

7° 
/  // 
2  i5 
2  i3 
2  15 
2  18 
2  22 

2  26 
2  3o 
2  35 

2  4o 
2  46 

2  52 

2  58 

3  4 
3  II 

3  17 

3  24 
3  3o 
3  36 
3  42 
3  49 

3  55 

4  2 
4  8 
4  14 
4  20 

4  26 
4  33 
4  39 
4  45 
4  5i 

4  57 

5  3 
5  9 
5  i5 
5  21 

5  27 
5  33 
5  38 
5  44 
5  49 
5  54 

5  59 

6  4 
6  9 
6  i4 
6  19 
6  24 
6  29 
6  34 
6  39 

6  43 
6  5i 

6  58 

7  5 
7  12 

7  .? 

7  2D 

7  3o 
7  35 
7  4o 

7° 

8° 
/  II 
2  18 
2  i5 
2  i3 
2  i5 
2  18 
2  21 
2  24 
2  28 

2  32 

2-36 

2  4i 
2  46 
2  5i 

2  56 

3  2 

3  8 
3  i3 

3  19 

3  24 
3  3o 

3~3"5 
3  4i 
3  46 
3  52 

3  57 

4  3 
4  8 
4  i4 
4  19 
4  24 

4  29 
4  35 
4  4o 
4  45 
4  5o 

4  55 

5  0 
5  5 
5  9 
5  i4 
5  18 
5  23 
5  27 
5  32 
5  36 

5  4i 
5  45 
5  49 
5  53 

5  57 

6  0 
6  7 
6  i4 
6  20 
6  26 
6  32 
6  38 
6  43 
6  47 
6  5i 

8° 

9° 

1  II 

2  21 
2  17 
2  i4 
2  i3 
2  i5 
2  17 
2  20 

2  23 

2  26 
2  29 

2  33 
2  37 
2  4i 
2  45 
2  5o 

2  54 

2  59 

3  4 
3  9 
3  i4 

3  19 

3  24 
3  28 
3  33 
3  38 

3  42 
3  47 
3  52 

3  57 

4  2 

4  7 
4  12 
4  16 
4  21 
4  25 
4  29 
4  33 
4  38 
4  42 
4  46 
4  5o 
4  54 

4  58 

5  2 
5  6 
5  10 
5  i4 
5  17 
5  21 
5  24 
5  27 
5  33 
5  39 
5  45 
5  5i 

5  57 

6  2 
6  7 
6  II 

9° 

10° 
/  // 

2  25 

2  20 
2  16 
2  14 
2  i3 

2  i5 
2  17 
2  19 
2  21 
2  24 
2  27 
2  3o 
2  33 
2  37 
2  4i 
^45 
2  49 
2  53 

2  58 

3  2 

3  6 
3  II 
3  i5 
3  19 
3  23 

3  27 
3  3i 
3  36 
3  4o 
3  44 
3  48 
3  52 

3  56 

4  0 
4  4 
4  8 
4  12 
4  16 
4  20 
4  24 
4  27 
4  3i 
4  34 
4  38 
4  41 
4  45 
4  48 
4  52 
4  55 

4  58 

5  I 

5  7 
5  12 
5  17 
5  22 

5  27 
5  32 
5  36 
5  40 

10° 

11° 

'  // 
2  3i 
2  24 
2  19 
2  16 
2  i4 
2  i3 
2  i4 
2  16 
2  iS 
2  21 

2  23 
2  25 
2  28 
2  3l 
2  34 

2  38 
2  4i 

2  45 
2  49 

2  53 

2  56 

3  0 
3  4 
3  7 
3  II 

3  i5 
3  18 
3  22 
3  26 
3  3o 

3  34 
3  38 
3  4i 
3  45 
3  48 
3  52 

3  55 
359 

4  2 
4  6 

4  9 
4  12 
4  i5 
4  18 
4  21 

4  24 
4  27 
4  3o 
4  33 
4  36 
4  39 
4  44 
4  49 
4  54 

4  59 

5  3 
5  7 
5  II 

11° 

12° 

/  // 
2  37 
2  29 

2  23 

2  19 
2  16 

2  i4 
2  i3 
2  14 
2  16 
2  18 
2  20 
2  22 
2  24 
2  26 
2  29 

2  32 

2  35 
2  38 
2  42 
2  45 
2  48 
2  5i 
2  54 

2  58 

3  I 

3  5 
3  8 
3  II 
3  i5 
3  18 
3  22 
3  25 
3  28 
3  3i 
3  34 
3  38 
3  4i 
3  44 
3  47 
3  5o 

3  53 
3  56 

3  59 

4  2 
4  5 
4  8 
4  II 
4  i4 
4  16 
4  19 
4  22 
4  26 
4  3i 
4  36 
4  4o 
4  43 
4  45 
4  48 

12° 

14° 

/  // 
2  49 
2  38 
2  3i 
2  25 

2  21 

2  18 
2  16 
2  i4 
2  i3 
2  14 

2  16 
2  17 
2  19 
2  20 
2  22 

2  24 
2  26 
2  28 
2  3i 
2  33 

2  36 
2  38 
2  4o 
2  43 
2  45 
2  48 
2  5i 
2  54 
2  56 

2  59 

3  2 
3  5 
3  8 
3  II 
3  i4 

3  17 
3  19 
3  22 
3  24 
3  27 

3  29 
3  32 
3  34 
3  37 
3  39 

3  42 
3  44 
3  46 
3  48 
3  5o 

3  52 

3  56 

4  0 
4  4 
4  7 
4  10 
4  i3 

14° 

16° 

/  // 
3  3 
2  49 
2  39 

2  32 

2  26 

2  22 
2  19 
2  17 
2  i5 
2  14 
2  i3 
2  i4 
2  i5 
2  16 
2  18 
2  19 
2  21 
2  23 

2  24 
2  26 

2  28 
2  3o 

2  32 

2  34 
2  36 

2  38 
2  4o 
2  42 
2  44 
2  46 

2  49 
2  5i 

2  54 
2  56 

2  58 

3  I 
3  3 
3  6 
3  8 
3  10 

3  12 
3  i4 
3  16 
3  18 
3  20 

3  22 
3  24 
3  26 
3  28 
3  3o 

3  32 
3  36 
3  39 
3  42 
3  45 

3  47 
16° 

18° 

1  11 
3  18 
3  1 

2  49 
2  40 
2  33 

2  27 

2  23 

2  20 
2  18 
2  16 

2  i5 
2  i4 
2  i3 
2  i4 
2  i5 

2  16 
2  18 
2  19 
2  20 
2  21 

2  23 
2  24. 
2  25 
2  26 
2  28 

2  3o 

2  32 
2  33 

2  35 

2  37 

239 

2  4i 

2  43 
2  45 
2  47 

2  49 
2  5l 

2  53 
2  55 
2  57 

2  5o 

3  0 
3  2 
3  4 
3  5 

3  7 
3  9 
3  II 
3  12 
3  14 
3  16 

3  '9 
3  22 

3  24 
3  26 

|18° 

20° 

/  // 
3  33 
3  i3 
2  59 
2  48 
2  4o 
2  33 
2  28 
2  24 
2  21 
2  19 

2  17 
2  16 
2  i5 
2  14 
2  i3 

2  i4 
2  i5 
2  16 
2  17 
2  18 
2  19 
2  20 
2  21 
2  22 
2  24 
2  25 
2  27 
2  28 
2  29 
2  3i 

2  32 

2  34 
2  36 
2  37 
2  38 

2  40 
2  4i 
2  43 
2  45 
2  47 
2  48 
2  5o 
2  5i 
2  53 
2  54 
2  55 
2  57 

2  59 

3  0 
3  I 
3  2 
3  5 
3  8 
3  10 

|20° 

22° 
/  ;/ 
3  48 
3  25 
3  10 
2  58 
2  48 
2  4o 
2  34 
2  29 

2  25 
2  22 

2  20 
2  18 
2  17 
2  16 
2  l5 

2  l4 
2  i3 
2  l3 

2  i4 

2  l5 

2  16 
2  17 
2  18 
2  19 
2  21 

2  22 
2  23 
2  24 
2  25 
2  26 

2  28 
2  29 

2  3o 

2  3l 
2  32 

2  34 
2  35 
2  36 
2  38 
2  39 

2  4i 
2  42 
2  43 
2  45 
2  46 
2  47 
2  49 
2  5o 
2  5i 

2  52 

2  53 
2  55 
2  57 

22° 

24° 

1  II 
4    4 
3  38 
3  21 

3  7 

2  56 

2  47 
2  40 
2  34 
2  3o 
2  26 

2  23 

2  21 
2  19 
2  17 
2  16 

2  i5 
2  i4 
2  i3 
2  i3 
2  i4 

2  i4 
2  i5 
2  16 
2  17 
2  18 
2  19 
2  20 
2  21 
2  22 

2  23 

2  24 

2  25 
2  26 
2  27 
2  28 
2  29 

2  3o 

2  3l 
2  32 

2  33 

2  35 
2  36 
2  37 
2  38 
2  89 

2  4o 
2  42 
2  43 
2  44 
2  45 
2  46 
2  47 

24° 

26° 
/  // 
4  19 
3  5o 
3  32 
3  16 
3  4 
2  54 
2  46 
2  4o 
2  35 
2  3o 

2  26 
2  24 
2  22 
2  20 
2  18 
2  17 
2  16 
2  i5 
2  i4 
2  i4 
2  i4 
2  14 
2  i5 
2  i5 
2  16 
2  17 
2  17 
2  18 
2  19 
2  20 

2  20 
2  21 
2  22 

2  23 

2  24 
2  25 
2  26 
2  27 
2  28 
2  29 

2  3o 
2  3i 
2  82 
2  33 
2  34 
2  35 
2  36 
2  37 
2  37 
2  38 

2  39 
26° 

28° 
/  // 
4  34 
4  3 
3  43 
3  26 
3  i3 

3~2 

2  53 
2  45 
2  89 
2  34 
2  3o 
2  27 
2  25 
2  22 
2  20 
2  19 
2  18 
2  17 
2  16 
2  i5 
2  i5 
2  14 
2  14 
2  14 
2  i5 
2  i5 
2  16 
2  16 
2  17 
2  18 

2  18 
2  19 
2  20 
2  21 
2  22 
2  22 
2  23 

2  24 
2  25 
2  26 
2  27 
2  28 
2  28 
2  29 
2  3o 

2  3i 
2  3i 

2  32 
2  32 

28° 

30° 
/  /( 

4  49 
4  16 
3  54 
3  35 
3  21 

3  10 
3  0 

2  5i 
2  44 
2  39 

2  35 
2  3i 
2  28 

2  25 
2  23 

2  21 

2  20 
2  19 
2  18 

2  17 

2  16 
2  l5 
2  l5 
2  l4 

2  i4 

2  14 
2  l5 
2  l5 
2  16 
2  16 
2  17 
2  18 
2  18 
2  19 
2  20 
2  20 
2  21 
2  22 
2  22 
2  23 

2  24 
2  25 
2  25 
2  26 
2  26 

2  27 
2  27 

30° 

TABLE  XLVIII.                                     f^=^=''''' 
Third  Correction.     Apparent  Distance  100°. 

J)'s 

o 

6 

7 
8 

9 

lO 

II 

12 

i3 

i4 
i5 

i6 

17 
i8 

19 
20 

21 
22 

23 

24 

25 

26 
27 
28 

=9 
3o 

3i 

32 

33 
34 
35 

36 

37 
38 
39 
4o 

4i 
42 
43 
44 
45 

■46 
47 
48 
49 
5o 

"57~ 

52 

53 
54 
55 

56 
58 
60 
62 
64 
66 
68 
70 
7? 
74 

Apparent  AltiUule  of  the  Sun,  Star  or  Planet. 

D's 
iSpp. 
Alt. 
o~ 

6 

7 
8 

9 

10 

n 
12 
i3 
i4 
i5 

16 

17 
18 

19 
20 

21 

22 

23 

24 

25 

26 

27 

28 

^9 
3o 

3i 

32 

33 
34 
35 

36 

37 
38 

39 
40 

4i 
42 

43 
44 
45 

46 
47 

32° 

(   II 
5    4 
4  29 
4    5 
3  45 
3  3() 

3  18 

3    7 
2  58 
2  5o 
.  44 
2  i'g 
2  35 
2  3i 
2  28 
2    25 

2    23 
2  22 
2  21 
2  20 
2  19 

2  18 
2  17 
2   16 
2  i5 
2   i5 
2  i4 
2  i4 
2  i5 
2  i5 
2  i5 

2  16 
2  17 
2  17 
2  18 
2  18 

2  19 
2  19 
2  20 
2  20 
2  21 

2  21 
2  22 
2  22 
2  2: 

2   23 

34° 

^  // 
5  19 

4  4i 
4  16 
3  55 
3  39 

3  26 
3  i4 
3    4 
2  56 
2  49 

2  44 
2  39 
2  35 
2  3i 
2  28 

2  26 
2  24 
2    23 

2  22 
2  20 
2  19 
2  18 
2  17 
2  16 
2  16 
2  i5 
2  i5 
2  i5 
2  i5 
2  i5 

2  i5 
2  16 
2  16 

2  17 
2  17 

2  18 
2  18 
2  18 
2  19 
2  19 
2  19 
2  19 
2  19 

34= 

36° 

/  II 
5  34 
4  54 
4  27 
4    5 
3  47 
3  33 
3  21 
3  10 
3     I 
2  54 
2  48 
2  43 
2  38 
2  34 
2  3i 

2  29 

2    2T 
2    25 
2    23 
2    21 

2    20 
2    19 
2    18 
2     17 
2    17 

2     16 
2     16 
2     l5 
2     l5 
2     l5 

2     l5 
2     l5 
2     l5 
2    16 
2    16 

2     17 
2    17 
2     17 
2    17 
2    17 
2     17 

3G° 

38° 

1  II 
5  48 
5    6 
4  38 
4  i5 
3  55 

3  4o 
3  27 
3  16 
3     7 

2  59 

2    52 

2  47 
2  42 
2  38 
2  35 

2   32 

2  29 
2  27 

2    25 
2    23 
2    21 
2    20 
2    19 
2     18 
2     18 
2    17 
2     17 
2    16 
2    16 
2l5 
2     l5 
2    l5 
2    l5 
2    16 
2    16 

2     16 
2    16 
2     16 
2    16 

38° 

40° 

1  II 
6    2 
5  18 
4  48 
4  24 
4    3 

3  47 
3  34 
3  22 
3  12 
3    4 

2  57 
2  5i 
2  46 
2  42 
2  38 

2  35 

2   32 

2  29 

2  27 
2  25 

2    23 
2    22 
2    21 
2    20 
2    19 

2    18 
2    18 
2     17 
2    17 
2     16 

2    16 
2     16 
2     16 
2     16 
2    16 

2     16 
2    16 

.40° 

42° 

»  // 
5  i5 
5  3o 
4  58 
4  32 
4  II 
3  54 
3  40 
3  28 
3  iS 
3    9 
3     2 
2  56 
2  5o 
2  45 
2  4i 
2  38 
2  35 

2   32 

2  29 

2  27 

2    25 
2    2^ 
2    2^ 
2    22 
2    21 

2    20 
2     19 
2     lb 
2     iS 

2    17 

2    17 
2    16 
2    16 
2    16 
2    16 

42° 

44° 

1  II 
6  28 
5  4i 
5    8 
4  4i 
4  19 
4     I 
3  47 
3  34 
3  23 
3  i4 

3     7 
3    0 

2  54 
2  49 
2  44 
2  4o 
2  37 
2  34 

2   32 
2  3o 

2    28 
2    26 
2    24 
2    23 
2    22 

2    21 
2    20 
2    19 
2    19 
2    18 
2    17 
2     17 
2     17 

44° 

40° 
/  II 
6  4i 
5  52 
5  17 
4  49 
4  26 

4    8 
3  53 
3  4o 
3  29 
3  19 
3  u 
3    4 
2  58 

2   52 

2  47 
2  43 
2  4o 
2  37 
2  35 

2   32 

2  3o 
2  28 
2  26 
2    25 
2    24 
2    22 
2    21 
2    20 
2    19 
2    18 

ns 

4G° 

48°. 

1  II 
6  53 
6    3 
5  26 
4  57 
4  33 

4  i5 
3  59 
3  46 
3  34 
3  24 
3  i5 
3    8 
3     I 

2  55 
2  5o 

2  46 
2  43 
2  40 
2  37 
2  34 

2   32 
2  3o 
2  28 
2  26 
2   25 
2    23 
2  22 

2    2J 

2    20 

50° 

/  // 
7    4 

3  i3 
5  35 
5    4 

4  4o 
4  21 
4    4 
3  5i 
3  39 
3  28 

3,9 
3  12 
3     5 
2  59 
2  54 

2  49 
2  45 
2  42 
2  39 
2  36 

2  34 

2   32 

2  3o 
2  28 
2  26 
2  24 

2    23 

54° 

II 

7   25 

3  32 

5  52 
5  19 

4  54 
4  33 
4  i5 
4    0 
3  48 
3  37 

3  27 
3.9 
3  12 
3     5 
3    0 

2  55 
2  5o 
2  46 
2  43 
2  4o 
2  37 
2  34 
2  3i 

58° 
f  II 
7  46 

3  5o 
5     7 
5  33 
5     6 

4  44 
4  25 

4    9 
3  56 
3  45 

3  35 
3  26 
3  18 
3  II 
3     5 
3    0 
2  55 
2  5o 
2  46 

62° 

/  /' 
8    5 
7     6 
6  20 
5  45 
5  16 

4  54 
4  34 
4  17 
4    4 
3  52 

3  42 
3  33 
3  24 
3  16 
3  10 

66° 

/  II 

3  20 

D    32 

5  56 
5  26 
5    3" 

4  43 
4  25 
4  ic 
3  59 

3  49 

70° 

/   /; 
8  33 
7  3o 
6  43 
6    7 
5  36 

5  12 
4  52 

74° 

1  '/ 
8  44 
7  40 
6  52 

7 

''able  P.     Efcct  of  Sun's  Par. 

To  be  sublracied  from  Ihc  Third 

Correclioii. 

i 

D's 
ipp. 

All. 

5 
10 
15 
20 
25 
30 
35 
40 
45 
50 
55 
60 
65 
70 
75 

Sun's  Apparent  Allitwiie. 

5 

1 
2 
2 
3 
4 
5 
5 
6 
7 
7 
S 
8 
8 
9 
9 

10  20J3 

1  I  2 

2  2    5 

3  3    : 

3  4    4 

4  4    5 

5  5    5 

5  6    6 

6  6    7 

7  7    - 

7  7    i 

a  8 

8  8 
9 

9 

D  40 

2 
3 
3 
4 
5 
6 
6 
7 

50 

2 
3 
4 
4 
5 
6 
6 

60  70  7 

2  2 

3  3 

4  4 
4 

5 

5  50 

48° 

50° 

54° 

P'^-^^iGj               TABLE  XLVIII. 

Third  Correction.  Apparent  Distance  104°. 

])'s 
App. 

Alt. 

o 

6 

7 
8 

9 

lO 

II 

12 

i3 
i4 
i5 

i6 

17 
i8 

19 
20 

21 
22 

23 

24 

25 

26 

27 
28 

3o 
3i 

32 

33 
34 
35 

36 

37 
38 
39 
4o 

4i 

42 
43 
44 
45 

46 
47 
48 

5o 
5i 

52 

53 
54 
55 

56 

57 
58 

60 

67 

64 

66 

68 

70 

1 

1  ... 

Jljipareiit  Altitude  of  the  Sun,  Stai-  or  Planet. 

»  's 
App. 

Alt. 

0 

6 

7 
8 

9 
10 

II 
12 
i3 
i4 
i5 

16 

17 
18 

19 

20 

21 
22 

23 

24 

25 

~^ 

27 
28 

29 

3o 
3i 

32 

33 

34 
35 

36 

37 
38 

39 
40 

4i 
42 
43 
44 
45 

46 
47 
48 
49 
5o 

52 

53 

54 
55 

56 

57 
58 

60 
62 
64 
66 
68 
70 

6= 

/  // 
2  20 

2  23 

2  26 
2  3o 
2  36 

2  42 
2  48 

2  55 

3  2 
3  9 

3  16 
3  23 
3  3i 
3  38 
3  46 

3  54 

4  2 
4  10 
4  18 
4  26 
4  33 
4  4i 

4  49 
457 

5  4 

5  12 

519 
5  27 

5  34 
5  42 

5  49 

5  56 

6  3 
6  10 
6  16 
6  23 
6  3o 
6  37 
6  43 
6  5o 

6  56 

7  2 
7  8 
7  i4 
7  20 

7  26 
7  32 
7  37 
7  42 
7  47 
7  5p 

7  57 

8  2 
8  6 
8  10 

8  19 
8  27 
8  35 
8  43 
8  49 

6° 

7° 
/  II 
2  22 
2  20 
2  22 

2  25 

2  29 

2  34 
239 
2  44 
2  49 

2  54 

3  0 
3  6 
3  i3 
3  19 
3  25 
3  32 
3  38 
3  45 
3  5i 

3  58 

4  4 
4  II 
4  18 
4  24 
4  3o 

4 '3^ 
4  44 
/,  5i 

4  58 

5  4 
5  iJ 
5  16 
5  22 
5  28 
5  33 
5  39 
5  44 
5  5o 

5  55 

6  I 

6  6 
6  12 
6  17 
6  23 
6  28 

6  33 
6  38 
6  43 
6  48 
6  53 

6  57 

7  I 
7  5 
7  9 

7  19 
7  26 
7  33 
7  39 
7  45 

7° 

8° 
/  u 

2  25 
2  22 
2  20 
2  22 
2  25 

2  28 
2  32 
2  36 
2  4o 

2*45 
2  5o 

2  55 

3  0 
3  5 
3  II 

3  16 
3  22 
3  27 
3  33 
3  39 

3  44 
3  5o 

3  56 

4  I 
4  7 
4  i3 
4  19 
4  25 
4  3o 
4  36 

4  4i 
4  46 
4  5i 

4  56 

5  I 
5  6 
5  II 
5  16 
5  21 
5  26 
5  3i 
5  36 
5  4o 
5  45 
5  49 
5  53 

5  57 

6  2 
6  6 
6  10 

6  i4 
6  18 
6  22 
6  26 
6  3o 

6  36 
6  42 
6  47 
6  52 
6  57 

8° 

9° 

/  II 

1  29 

2  25 
2  22 
2  21 
2  22 

2  24 

2  27 
2  3o 
2  33 
2  37 

2  41 
2  45 
2  49 
2  53 

2  58 

3  3 
3  8 
3  i3 
3  18 
3  22 

3  27 
3  32 
3  37 
3  42 
3  47 
3  52 

3  57 

4  2 
4  7 
4  12 

4  17 
4  21 
4  26 
4  3i 
4  36 

4  4o 
4  44 
4  49 
4  53 

4  58 

5  2 
5  6 
5  10 
5  i4 
5  18 

5  22 
5  26 
5  29 
5  33 
5  36 

5  4o 
5  44 
5  47 
5  5o 
5  53 

5  59 

6  4 
G  9 
6  i4 

9° 

10° 

/  // 
2  33 
2  28 
2  24 
2  22 
2  21 

2  22 
2  24 
2  26 
2  29 

2  32 

2  35 
2  38 
2  4i 
2  45 
2  49 
2  53 

2  57 

3  2 
3  6 
3  10 

3  i5 
3  19 
3  23 
3  28 
3  32 

3  36 
3  4i 
3  46 
3  5o 
3  55 

3  59 

4  3 
4  7 
4  II 
4  i5 

4  19 
4  23 
4  27 
4  3i 
4  35 
4  39 
4  43 
4  46 
4  5o 
4  53 

4  57 

5  0 
5  4 
5  7 
5  10 

5  i3 
5  16 
519 
5  22 
5  25 
5  3o 
5  35 
5  4o 
5  45 

10° 

11° 

/  // 
2  39 

2  32 

2  27 
2  24 
2  22 
2  21 
2  22 
2  24 
2  26 
2  28 

2  3i 
2  33 

2  36 
2  39 
2  43 

2  46 
2  5o 
2  54 

2  57 

3  I 

3  5 
3  9 
3  12 
3  16 
3  20 

3  24 
3  28 
3  32 
3  36 
3  4o 

3  44 
3  47 
3  5i 
3  55 

3  59 

4  3 
4  6 
4  10 
4  i3 
4  17 
4  20 
4  24 
4  27 
4  3o 
4  33 
4  36 
4  39 
4  42 
4  45 
4  48 
4  5i 
4  54 

4  57 

5  0 
5  2 

5  6 
5  10 
5  i4 

11° 

12° 

/  II 
2  45 
2  36 
2  3o 
2  26 
2  24 
2  22 
2  21 
2  22 
2  24 
2  26 

2~^8 

2  3o 
2  33 
2  35 
2  38 

2  4i 
2  44 
2  47 
2  5o 
2  54 

2  57 

3  0 
3  3 

3  7 
3  10 

3  i4 
3  17 
3  21 
3  24 
3  27 
3  3i 
3  35 
3  38 
3  4i 
3  45 

3  49 
3  53 
3  56 

3  59 

4  2 
4  5 
4  8 
4  II 
4  i4 
4  17 
4  20 
4  23 
4  26 
4  29 
4  32 

4  34 
4  37 
4  39 
4  4i 
4  43 

4  47 
4  5i 
4  54 

12° 

14° 

/  II 
2  58 
2  46 
2  38 

2  32 

2  28 
2  25 

2  23 
2  22 
2  22 
2  23 

2  24 
2  26 
2  27 
2  29 
2  3l 

2  33 
2  35 

2  38 
2  4o 

2  42 

2  44 

2  47 
2  49 
2  52 

2  55 

2  58 

3  0 
3  3 
3  5 
3  8 
3  II 
3  i4 
3  17 
3  20 
3  23 

3  26 
3  29 
3  32 
3  35 
3  38 

3  40 
3  43 
3  45 
3  47 
3  5o 

3  52 
3  54 
3  56 

3  58 

4  0 
4  3 
4  5 
4  7 
4  9 
4  II 

4  i5 
4  19 

14° 

16° 

/  // 
3  i3 
2  57 
2  47 

2  39 
2  34 
2  3o 
2  27 
2  25 
2  23 
2  32 
2  -,2 
2  .■jS 
2  24 
2  25 
2  27 

2  28 

2  3o 

2  32 
2  33 

2  35 
2  36 
2  38 

2  4o 
2  42 

2  44 
2  46 

2  49 
2  5l 

2  54 

2  56 

2  58 

3  I 
3  4 
3  6 
3  9 

3  II 
3  i3 
3  i5 
3  18 
3  20 
3  22 
3  24 
3  26 
3  28 
3  3o 
3  32 
3  34 
3  36 
3  38 
3  4o 
3  42 
3  44 
3  46 
3  48 
3  5o 

3  52 
16° 

18° 
/  /; 
3  28 
3  10 
2  57 
2  48 
2  4i 
2  35 
2  3i 

2  28_, 

2  26 

2  25 

2  24 
2  23 
2  22 
2  23 
2  24 
2  25 
2  26 
2  28 
2  29 

2  3o 

2  3l 
2  32 

2  34 

2  35 
2  37 

2  39 
2  4i 
2  43 
2  45 
2  47 

2  49 
2  5i 
2  53 
2  55 
2  57 

2  59 

3  1 
3  3 
3  5 
3  6 

3  8 
3  10 
3  12 
3  i4 
3  16 

3  18 
3  20 
3  22 
3  23 
3  25 

3  26 
3  28 
3  29 
3  3i 
3  32 

18° 

20° 

/  // 
3  43 
3  23 
3  8 
2  57 
2  48 

2  41 
2  36 

2  32 
2  3o 
2  28 
2  26 
2  25 
2  24 
2  23 
2  22 

2  23 
2  24 
2  25 
2  26 
2  27 

2  28 
2  29 

2  3o 

2  3l 

2  33 

2  34 
2  36 
2  37 
2  39 
2  4o 

2  42 
2  43 
2  45 
2  47 
2  49 
2  5i 

2  52 

2  54 
2  55 
2  57 

2  58 

3  0 
3  I 
3  3 
3  4 
3  6 
3  8 
3  9 
3  10 
3  12 
3  i3 
3  i4 
3  i5 

20° 

22° 

/  // 
3  59 
3  36 
3  20 

3  7 
2  56 

2  48 
2  42 
2  38 
2  34 
2  3i 

2  29 
2  27 
2  26 
2  24 
2  23 

2  22 
2  22 

2  23 
2  24 
2  24 

2  25 
2  26 
2  27 
2  28 

2  3o 

2  3l 
2  32 

2  33 
2  34 
2  35 

2  37 

2  38 

2  39 

2  4l 
2  42 

2  44 
2  45 

2  47 

2  48 
2  5o 

2  5l 
2  52 

2  53 
2  55 
2  56 
2  58 

2  59 

3  0 
3  I 
3  I 
3  2 

22° 

24° 

/  // 
4  i5 
3  48 
3  3i 
3  17 
3  5 

2  55 

2  48 
2  43 
2  39 
2  35 

2  32 
2  3o 
2  28 
2  26 
2  25 

2  24 
2  23 
2  22 
2  22 
2  23 

2  24 
2  24 
2  25 
2  26 
2  27 

2  28 
2  29 

2  3o 

2  3l 
2  32 

2  33 
2  34 
2  35 
2  36 
2_37_ 
2  39 
2  4o 
2  4i 
2  42 
2  44 
2  45 
2  46 
2  48 
2  49 
2  5o 

2  5i 

2  52 
2  52 

2  53 
24° 

26° 
/  // 
4  3o 
4  I 
3  42 
3  26 
3  i4 

3  4 
2  56 
2  49 
2  43 
2  39 

735 
2  33 
2  3] 
2  29 
2  27 

2  26 
2  25 

2  24 

2  23 
2  23 
2  23 
2  23 
2  24 
2  25 
2  25 

2  26 
2  27 
2  28 
2  29 

2  3o 
2  3o 

2  3l 
2  32 

2  33 

2  34 

2  35 
2  36 

2  37 

2  38 

2  39 
2  40 

2  4i 

2  42 
2  43 

2  44 

2  45 
2  45 

26° 

28° 
/  // 
4  46 
4  i4 
3  53 
3  36 
3  23 

3  II 
3  2 

2  55 
2  48 
2  43 

2  39 
2  36 
2  34 
2  3i 
2  29 
2  28 
2  27 
2  26 

2  25 
2  24 

2  24 
2  24 
2  24 
2  24 
2  24 
2  25 
2  26 
2  26 
2  27 
2  28 
2  28 
2  29 

2  3o 

2  3l 
2  32 
2  32 

2  33 

2  34 

2  35 
2  35 

2  36 

2  37 

2  38 
2  38 
2  39 

28° 

30° 
/  /' 
5  I 
4  27 
4  4 
3  46 
3  3i 

319 

\    ? 

2  54 
2  48 

2  44 
2  4o 
2  37 
2  34 

2  32 
2  3o 

2  29 
2  28 
2  27 
2  26 
2  26 
2  25 
2  25 
2  25 
2  24 
2  24 
2  25 
2  25 
2  26 
2  27 

2  27 
2  27 
2  28 
2  29 

2  3o 
2  3o 

2  3l 
2  3l 
2  32 
2  32 

2  33 
2  34 
2  34 

30° 

TABLE  XLVIII. 
Third  Correction.     Apparent  Distance  104<^. 

._ 

[Page  317 

o 

6 

7 
8 

9 

10 

II 

12 

i3 
i4 
i5 

i6 

'7 
i6 

'9 

20 

21 
22 
23 

24 
25 

26 

27 
28 

L' 

3i 

32 

33 
34 
35 

36 

37 
38 
39 
4o 

4i 

42 

43 
44 
45 

46 
47 
48 
49 
5o 

5i 

52 

53 
54 
55 

56 

57 
58 

6o 
62 
64 
66 
68 
70 

Apparent  Mtitude  of  the  Sun,  Star  or  Planet. 

D's 
App. 

Alt. 

0 

6 

7 
8 

9 

10 

II 
12 
i3 
i4 
i5 

16 

17 
iS 

19 
20 

21 

23 

24 

25 

27 
28 

3o 
3i 

32 

33 

34 
35 

36 
37 
38 
39 
4o 

4i 
42 
43 
44 
45 

46 
47 

32° 

1  1 
5  16 
4  42 
4  16 
3  55 
3  4o 

3  27 
3  16 
3     7 

2  59 
2  53 

2  48 
2  44 
2  4i 
2  38 
2  35 
2  33 
2  3i 
2  3o 
2  29 
2  28 
2  27 
2  26 
2  26 
2    25 
2    25 

2    24 
2    24 
2    24 
2    25 
2    26 
2    26 
2    26 
2    27 
2    27 
2    28 

2    28 
2    29 
2    29 

2  3o 
2  3o 

2    3l 

34° 

f  // 
5  3i 
4  56 
4  28 
4    5 
3  49 
3  35 
3  23 
3  i3 
3     5 
2  58 

2  53 
2  49 
2  45 
2  4i 
2  38 

2  36 
2  34 

2   32 

2  3i 
2  29 

2  28 
2  27 
2  27 
2  26 
2  26 

2    25 
2    25 
2    24 
2    2Zi 
2    25 
2    25 
2    25 
2    26 
2    26 
2    27 

2    27 
2    27 
2    27 
2    28 

36° 

1  II 
5  45 
5    8 
4  39 
4  i5 
3  58 

3  43 
3  3o 
3  20 
3  II 
3    4 

2  58 
2  53 
2  49 
2  45 
2  42 
2  39 
2  36 
2  34 
2  33 
2  3[ 

2  3o 
2  29 

2  28 
2  27 

2  27 

2  26 
2  26 
2    25 
2    25 
2    25 

2    25 
2    25 
2    26 
2    26 
2    26 

2    26 
2    26 

38° 
/  II 
6    0 
5  20 
4  49 
4  25 
4    7 
3  5i 
3  37 
3  26 
3  17 
3    9 
3     3 
2  58 
2  53 
2  49 
2  45 

2  42 
2  39 
2  37 
2  35 
2  33 

2   32 

2  3i 
2  3o 
2  29 
2  28 

2  27 
2  27 
2  26 
2  26 
2  26 
2  26 
2  26 
2  26 
2  26 
2  26 

40° 
/  // 
6  i4 
5  3i 
4  59 
4  34 
4  i5 

3  58 
3  44 
3  33 
3  23 
3  i5 

3    8 
3     2 
2  57 
2  53 
2  49 
2  45 
2  42 
2  39 
2  37 
2  35 

2  34 

2   32 

2  3. 
2  3o 
2  29 

2  28 
2  28 
2  27 
2  27 
2  26 
2  26 
2  26 
2  26 

42° 

/  // 
6  28 
5  42 
5    9 
4  43 
4  23 

4    5 
3  5i 
3  39 
3  29 
3  20 
3  12 
3    6 
3     1 
2  56 

2   52 

2  48 
2  45 
2  42 
2  4o 
2  38 

2  36 
2  34 
2  33 

2   32 

2  3i 
2  3o 
2  29 
2  28 
2  27 
2  27 
2  27 

44° 

f  II 
6  4i 
5  53 
5  ,9 
4  52 
4  3i 

4  12 
3  58 
3  45 
3  34 
3  25 

3  17 
3  10 
3    4 
2  59 
2  55 
2  5i 
2  47 

2  44 
2  42 
2  40 
2  38 
2  36 
2  35 
2  33 
2   32 

2  3i 
2  3o 
2  29 

2    2.8 

46° 

/    /; 
6  54 
6    4 
5  28 
5    0 
4  38 

4  19 
4    4 
3  5i 
3  39 
3  29 

3  21 
3  i4 
3    8 
3     3 

2  58 

2  54 
2  5o 
2  47 
2  44 
2  42 

2  4o 
2  38 
2  36 
2  34 
2  33 

2   32 

2  3i 

48° 
/  II 

1  6 
6  i5 
5  38 
5    8 
4  45 
4  26 
4  10 
3  56 
3  44 
3  M 
3  25 
3  17 
3  II 
3    6 
3     I 

2  57 
2  53 
2  5o 

2  47 
2  44 

1  42 

2  4o 
2  38 
2  36 
a  34 

50° 
/  // 
7  18 
6  26 
5  47 
5  16 
4  52 

4   32 

4  17 
4    2 
3  49 
3  38 

3  29 
3  21 
3  i5 

\    9 
3    4 

3    0 

2  56 
2  53 
2  5o 
2  47 

2  44 
2  4i 
2  39 

52° 

1  II 

1  29 
6  37 
5  57 
5  24 
4  59 
4  38 

4   23 

4    7 
3  54 
3  43 

3  33 
3  25 
3  18 
3  12 
3    7 
3    3 

2  59 
2  55 

2   52 

2  49 
2  46 

54° 

;  // 
7  4o 
6  47 
6    6 
5  32 
5    6 

4  44 
4  28 
4  12 
3  59 
3  47 
3  37 
3  29 
3  22 
3  16 
3  10 

3     5 
3     I 

2  57 
2  54 

58° 

/  // 
8     0 
7    4 
6  21 
5  46 
5  18 

4  55 
4  38 

4    29 

4    8 
3  56 

3  45 
3  36 
3  29 
3  22 
3  16 

62° 
/  // 
8  19 

7   19 
6  34 
5  58 
5  3o 
5     6 
4  47 
4  3o 
4  i5 
4    3 
3  52 

66' 

1 
8  35 
7  33 
6  46 
6    9 
5  4" 
5  16 
4  56 

70° 
;   // 
8  49 
7  46 
6  57 

48= 

"50°" 

TahU  P.     Effect  of  Sun's  Par. 

To  be  subtrncled  from  the  Third 

Correction. 

D'3 
App. 
Alt. 

5 
10 
15 

20 
25 
30 
3.5 

40 
45 
50 
S5 
60 
65 

ro 

Sun'a  Apparent  Altitude. 

5 

1 
2 
2 
3 
4 
5 
6 
6 
7 
7 
8 
8 
9 
9 

10 

2 
3 
4 

4 

5 

6 
6 
7 
8 
8 
8 
9 

20  3 

2  2 

21  3 

3  3 

4  4 

5  5 

5  6 

6  6 

7  7 
7    8 
8 

8 

J  40 

2 
3 
4 

5 

1 
7 
7 

50  ( 

2 
3 
4 
5 
6 
6 

0  70 

3  3 

4  4 

4    4 
5 

dO 

so 

40° 

42° 

44° 

46° 

32° 

34° 

30° 

38° 

52° 

1 

^"^'^'^^                                   TABLE  XLVIII. 

Third  Correction.  Apparent  Distance  108° 

5's 
A  pp. 
Alt. 

o 
6 
7 
8 

9 

10 

II 

12 

i3 

a 

i5 
i6 

17 
i8 

19 
20 

21 
22 

23 

24 

25 

26 

27 
28 

3o 
3i 

32 

33 
34 
35 

36 

37 
38 
39 
4o 

4i 
42 
43 
44 
45 

46 
47 
48 

5o 
5i 

52 

53 
54 
55 

56" 

57 
58 

60 

61 
62 
63 
64 
66 

Apparent  Mtitude  of  the  Sun,  Star  or  Planet. 

A  pp. 

Alt. 

0 

6 

7 
8 

9 

10 

II 
12 
i3 
i4 
i5 

16 

17 
18 

19 
20 

21 
22 

23 

24 

25 

26 

27 
28 

3o 
3i 

32 

33 

34 
35 

36 

37 
38 
39 
4o 

4i 
42 
43 
44 
45 

46 
47 
48 
49 
5o 

IT 

52 

53 
54 
55 

56 
57 
58 

60 

61 
62 
63 
64 
66 

6° 
/  // 
2  3o 
2  33 
2  36 
2  40 
2  46 

2  52 

2  59 

3  6 
3  i3 
3  20 

3  35 
3  43 

3  5o 

3  58 

4  6 
4  i4 
4  22 
4  3o 
4  38 
4  46 

4  54 

5  2 
5  10 
5  18 

5  26 
5  33 
5  4i 
5  48 

5  56 

6  3 
6  10 
6  17 
6  24 
6  3i 

6  38 
6  45 
6  52 

6  59 

7  6 
7  12 
7  18 
7  24 
7  3o 
7  36 

7  42 
7  47 
7  53 

7  58 

8  4 

^  9 
8  14 
8  19 

8  24 
8  28 

8  33 
8  37 
8  4i 
8  45 
8  53 

6° 

70 
/  // 

2  32 

2  3o 

2  32 

2  35 
2  39 

2  44 
2  49 
2  54 

2  59 

3  5 

J  II 
3  17 
3  24 
3  3i 

3  37 

3  44 
3  5i 

3  58 

4  4 
4  II 
4  18 
4  25 
4  3i 
4  37 
4  44 
4  5i 

4  58 

5  5 

5  IT 

5  18 

5  24 
5  3o 
5  36 
5  42 
5  48 

5  54 

5  59 

6  5 
6  II 
6  17 
6  22 
6  27 
6  32 
6  37 
6  42 

6  47 
6  52 

6  57 

7  2 
7  7 
7  II 
7  16 
7  20 
7  25 
7  39 
7  33 
7  37 
7  4o 
7  43 
7  46 

70 

8° 
r 
2  35 

2  32 

2  3o 

2  32 

2  35 
2  38 
2  42 
2  46 

2'5l 

2  56 

3  I 
3  6 
3  II 
3  17 
3  22 

3  28 
3  34 
3  4o 
3  46 
3  5i 

3  57 

4  3 
4  9 
4  i5 
4  21 

4  27 
4  33 
4  38 
4  43 
4  49 

4  55 

5  0 
5  5 
5  10 
5  i5 
5  20 
5  25 
5  3o 
5  36 
5  4i 
5  46 
5  5i 

5  56 

6  I 
6  5 
6  10 
6  i4 
6  18 
6  22 
6  26 

6  3o 
6  34 
6  38 
6  42 
6  45 

8  48 
6  5i 
6  54 

6  57 

7  0 

8° 

9° 

// 
2  39 
2  35 

2  32 

2  3i 

2  33 
2  35 
2  38 
2  4i 
2  44 
2  48 

2  52 

2  56 

3  0 
3  5 
3  10 

3  i4 
3  19 
3  24 
3  29 
3  34 
3  39 
3  44 
3  49 
3  54 

3  59 

4  4 
4  9 
4  i4 
4  19 
4  24 
4  29 
4  34 
4  39 
4  44 
4  49 
4  54 

4  58 

5  3 

5  7 
5  12 

5  16 
5  20 
5  24 
5  2.8 
5  32 

5  36 
5  40 
5  43 
5  47 
5  5i 

5  54 

5  58 

6  I 
6  4 
6  8 

6  II 

6  i4 
6  17 
6  20 

9*^ 

10° 

/  // 
2  44 
2  39 
2  35 
2  33 
2  3i 
2  33 
2  35 
2  37 
2  4o 
2  43 

2  46 
2  49 
2  53 

2  57 

3  I 

3  4 
3  8 
3  12 

3  17 
3  22 

3  26 
3  3i 
3  35 
3  40 
3  44 
3  48 
3  52 

3  57 

4  I 
4  5 
4  10 
4  i4 
4  19 
4  24 
4  28 
4  33 
4  37 
4  4i 
4  45 
4  49 

4  53 
457 

5  0 
5  4 
5  7 
5  u 
5  i4 
5  18 
5  21 
5  24 

5  27 
5  3o 
5  33 
5  36 
5  39 

5  42 
5  45 
5  48 

11° 

/  11 
2  5o 
2  43 
2  38 
2  35 
2  33 

2  32 

2  33 
2  35 
2  37 
2  39 

2  42 
2  45 
2  48 
2  5i 
2  54 

2  57 

3  0 
3  4 
3  8 
3  12 

3  16 
3  20 
3  24 
3  28 
3  32 

3  36 
3  4o 
3  44 
3  47 
3  5i 

3  55 

3  59 

4  3 
4  7 
4  II 
4  i5 
4  18 
4  22 
4  26 
4  3o 

4  34 
4  37 
4  4i 
4  44 
4  47 
4  5o 
4  53 
4  56 

4  59 

5  2 

5  5 
5  8 
5  II 
5  i4 
5  16 

5  19 
5  21 
5  23 

12° 

/  // 
2  56 
2  48 
2  42 
2  38 
2  35 
2  33 

2  32 

2  33 
2  35 
2  37 

2  39 
2  42 
2  44 
2  46 
2  49 

2  52 
2  55 

2  58 

3  I 
3  4 
3  8 
3  II 
3  i5 
3  18 
3  22 

3  25 
3  28 
3  32 
3  36 
3  39 
3  42 
3  46 
3  5o 
3  54 

3  57 

4  I 
4  5 

4  9 
4  12 
4  16 

4  19 
4  22 
4  25 
4  28 
4  3i 

4  34 
4  37 
4  39 
4  42 
4  45 

4  47 
4  5o 
4  52 
4  54 
4  56 

4  58 

5  0 

14° 

/  // 
3  9 
2  58 
2  49 
2  43 
2  39 

2  37 
2  35 
2  34 
2  33 
2  34 
2  35 
2  37 
2  89 
2  4o 
2  42 

2  44 
2  46 
2  48 
2  5o 
2  53 

2  55 

2  58 

3  0 
3  3 
3  6 

3  9 
3  II 

3  14 

3  17 
3  20 

3  23 
3  26 
3  29 
3  32 
3  35 
3  38 
3  4i 
3  44 
3  47 
3  5o 

3  52 
3  55 

3  57 

4  0 
4  2 
4  5 
4  7 
4  10 
4  12 
4  i4 
4  16 
4  18 
4  20 
4  22 
4  24 

16° 

/  // 
3  24 
3  10 
2  58 
2  5o 
2  44 
2  4i 
2  3q 
2  37 
2  35 
2  34 
2  33 
2  34 
2  35 
2  36 
2  38 

2  39 
2  4i 
2  42 
•2  44 
2  46 

2  48 
2  5o 
2  52 

2  54 
2  56 

2  58 

3  0 
3  2 
3  5 

3  7 

3  9 
3  12 

3  i5 

3  17 
3  20 

3  22 
3  24 
3  27 
3  29 
3  3i 
3  33 
3  36 
3  38 
3  4i 
3  43 

3  45 
3  47 

3   5i 
3  53 

3  55 
3  57 
3  58 

16° 

18° 
/  // 
3  39 
3  22 

3  9 
2  58 
2  5i 

2  46 
2  43 
2  4o 
2  38 
2  36 
2  35 
2  34 
2  33 
2  34 
2  35 

2  36 
2  37 
2  38 
2  4o 
2  4i 

2  43 
2  44 
2  46 
2  47 
2  49 

2  5o 
2  52 

2  54 

2  56 

2  58 

3  0 
3  2 
3  4 
3  6 
3  8 

3  10 
3  12 
3  i4 
3  16 
3  18 

3  20 
3  22 
3  24 
3  26 
3  28 

3  3o 
3  32 
3  34 
3  35 
3  36 
3  38 

18° 

20° 

/  II 
3   55 
3  35 
3  20 
3  8 
2  59 

2  53 
2  48 
2  44 
2  4i 
2  39 

2  37 
2  35 
2  34 
2  33 
2  33 

2  34 
2  35 
2  36 
2  37 
2  38 

2  39 
2  4o 
2  42 
2  43 
2  45 
2  46 
2  47 
2  48 
2  5o 
2  5i 

2  53 
2  55 
2  57 

2  58 

3  0 
3  I 
3  3 
3  5 
3  6 
3  8 
3  10 
3  12 
3  i3 
3  i5 
3  17 
3  18 
319 
3  20 
3  21 

20° 

22° 

4   II 

3  48 
3  3i 
3  18 
3  7 
3  0 
2  54 
2  49 
2  45 
2  42 
2  4o 
2  38 
2  36 
2  35 
2  34 
2  34 
2  34 
2  34 
2  35 
2  36 

2  37 
2  38 
2  39 
2  4o 
2  4i 
2  42 
2  43 
2  44 
2  45 
2  46 
2  48 
2  49 
2  5i 

2  52 

2  54 
2  55 
2  56 
2  58 

2  59 

3  I 

3  2 
3  4 
3  5 
3  7 
3  8 

3  9 
3  10 

22° 

24= 

/  // 
4  27 
4  2 
3  42 
3  28 
3  16 

3  7 
3  0 

2,54 
2  49 
2  46 

2  43 
2  4o 
2  38 
2  37 
2  36 

2  35 
2  35 
2  34 
2  34 
2  34 
2  35 
2  36 
2  37 
2  38 
2  38 
2  39 
2  4o 
2  4i 
2  42 
2  43 

2  44 
2  45 
2  46. 
2  47 
2  49 
2  5o 
2  5i 

2  52 

2  53 
2  55 
2  56 
2  58 

2  59 

3  0 
3  I 

24° 

26° 

1  11 
4  43 
4  i5 
3  54 
3  38 
3  25 
3  i5 

3  7 
3  0 

2  54 
2  5o 

2  46 
2  43 
2  4i 
2  39 
2  38 

2  37 
2  36 
2  35 
2  35 
2  34 

2  34 
2  35 
2  35 
2  36 
2  36 

2  37 
2  38 
2  39 
2  4o 
2  4i 
2  42 
2  42 
2  43 
2  44 
2  45 

2  46 
2  47 
2  48 
2  49 
2  5o 

2  5i 

2  52 

2  53 
26° 

28° 
/  II 
4  59 
4  28 
4  6 
3  48 
3  34 
3  23 
3  i4 
3  6 
2  59 
2  54 
2  5o 
2  47 

2  44 
2  42 
2  4o 
2  39 
2  38 
2  37 
2  36 
2  35 

2  35 
2  34 
2  34 
2  35 
2  35 

2  36 
2  37 
2  37 
2  38 
2  39 

2  4o 
2  4o 
2  41 
2  42 
2  43 
2  43 
2  44 
2  45 
2  46 
2  47 
2  47 

28° 

30° 

/  /' 
5  i5 
4  4i 
4  17 
3  58 
3  43 
3  3o 
3  20 
3  12 
3  5 
2  59 

2  54 
2  5o 
2  47 
2  45 
2  43 

2  4i 
2  4o 
2  39 
2  38 
2  37 
2  36 
2  35 
2  35 
2  35 
2  35 
2  35 
2  36 
2  3Q 
2  37 
2  37 
2  38 
2  38 
2  89 
2  40 
2  4o 

2  4i 
2  4i 
2  42 
2  43 

30° 

10° 

11° 

12° 

14° 

— — '                                                                                 ^"' "^ 

! 

TABLE  XLVIII.                                     t^'''="'" 

Third  Correction.     Apparent  Distance  108°. 

D's 
A  pp. 
All. 

Apparent  Jlltitudc  of  the  Sun,  Star  or  Planet. 

App 

32^ 

34° 

36° 

38° 

40° 

42° 

44° 

46° 

48° 

50° 

52° 

54° 

56° 

58° 

62° 

66° 

Alt. 

o 

f  '/ 

f  /' 

/  // 

/  // 

/  /' 

/  /^ 

r   // 

/  // 

/  // 

/   ti 

/  // 

/   u 

/  // 

/  // 

/     // 

/  // 

0 

6 

5  3o 

5  45 

5    0 

3  i5 

6  29 

5  44 

5  58 

7  II 

7    23 

1  M 

7  45 

7  56 

8    6 

8  16 

8  35 

i  53 

6 

7 

4  55 

5    8 

5  21 

5  34 

5  46 

5  58 

6  10 

6  22 

6  33 

j  43 

6  53 

7     2 

7  II 

7  20 

7  35 

7  47 

7 

8 

4  29 

4  4i 

4  52 

5     3 

5  i3 

5  23 

5  34 

5  A^ 

5  54 

6    4 

6  i3 

6  22 

6  3o 

5  38 

6  5i 

7     I 

8 

9 

4    8 

4  18 

4  28 

4  38 

4  48 

4  57 

5    6 

5  i5 

5  23 

5  3i 

5  38 

5  45 

5  53 

6     I 

b  i4 

9 

10 

3  52 

4    0 

4    9 

4  18 

4  27 

4  36 

4  AA 

4  52 

4  59 

5    6 

5  i3 

5  20 

5  27 

3  33 

5  45 

10 

1 1 

3  38 

3  AiS 

3  54 

4    2 

4  10 

4  18 

4   25 

4  33 

4  40 

4  47 

4  53 

4  59 

5     5 

5  II 

5  20 

II 

12 

3  27 

3  34 

3  41 

3  48 

3  56 

4    4 

4  II 

4  18 

4  24 

4  3o 

4  3b 

4  42 

4  47 

4    02 

5     0 

12 

i3 

3  18 

3  24 

3  3o 

3  3- 

3  AA 

3  5i 

3  58 

4    4 

4  10 

4  i5 

4  21 

4  26 

4  3o 

4  34 

i3 

i4 

3  10 
3    4 

3  16 

3  22 

3  28 

3  34 

3  40 

3  46 

3  52 

3  58 

4    3 

4    8 

4  12 

4  16 

4  20 

i4 

i5 

3     9 

3  i4 

3  20 

3  25 

3  3i 

3  36 

3  42 

3  47^ 

3  52 

3  56 

4    I 

4    5 

4    8 

i5 

ifi 

2  59 

3    3 

3     8 

3  i3 

3  18 

3  23 

3  28 

3  -62, 

3  38 

3  A'i 

3  47 

3  5i 

3  55 

3  58 

16 

17 

2  54 

2  58 

3     3 

3     7 

3    72 

3  17 

3  21 

3  26 

3  3o 

3  35 

3  39 

3  /^i 

3  46 

17 

18 

2  5i 

2  54 

2  59 

3     3 

3    7 

3  II 

3  i5 

3  20 

3  24 

3  28 

3  32 

3  35 

3  38 

18 

'9 

2  48 

2  5i 

2  55 

2  59 

3     3 

3     6 

3  10 

3  i4 

3  18 

3  22 

3  25 

3  28 

! 

19 

20 

2  46 

2  49 

2   52 

2  56 

2  59 

3     2 

3    6 

3    9 

3  i3 

3  16 

3  19 

3  22 

20 

21 

2  44 

2  47 

2  5o 

2  53 

2  56 

2  59 

3     2 

3     5 

3     9 

3  12 

3  .4 

21 

22 

2  42 

2  45 

2  48 

2  5o 

2  53 

2  56 

2  59 

3     2 

3     5 

3     8 

3  10 

22 

23 

2  4i 

2  43 

2  46 

2  48 

2  5o 

2  53 

2  56 

2  59 

3     2 

3    4 

23 

24 

2  4o 

2  42 

2  44 

2  46 

2  48 

2  5i 

2  53 

2  56 

2  59 

3     I 

24 

25 

2  39 

2  4o 

2  42 

2  44 

2  46 

2  49 

2  5i 

2  53 

2  56 

25 

26 

2  38 

2  39 

2  4i 

2  43 

2  45 

2  47 

2  49 

2  5i 

2  54 

2b 

27 

2  37 

2  38 

2  4o 

2  42 

2  44 

2  45 

2  47 

2  49 

27 

28 

2  3b 

2  38 

2  39 

2  4i 

2  42 

2  44 

2  4b 

2  47 

28 

29 

2  36 

2  37 

2  38 

2  4o 

2  4i 

2  42 

2  44 

29 

3o 

2  35 

2  36 

2  37 

2  39 

2  4o 

2  41 

2  43 

3o 

3i 

2  35 

2  36 

2  37 

2  38 

2  39 

2  4o 

3i 

32 

2  35 

3  36 

2   37 

2  38 

2  39 

2  4o 

32 

33 

2  36 

2  36 

2  36 

2  37 

2  38 

33 

34 

2  36 

2  36 

2  36 

2  37 

2  38 

34 

35 

2  36 

2  36 

2  36 

2  37 

35 

3f) 

2  37 

2  36 

2  36 

2  36 

36 

37 

2  38 

2   37 

2  36 

37 

38 

2  38 

2   37 

2  36 

38 

39 

2  39 

2  38 

39 

40 

2  39 

2  38 

4o 

4i 

2  4o 

4i 

42 

2  4o 

42 

43 

Ai 

44 

44 

45 

46 

45 
46 

47 
48 

47 

5o 

TahXt  P.     £j£c«  0/  Suv?s  Par. 

5i 

To  be  subtracted  from  the  Third 

52 

53 

Correction. 

D's 

Sun's  Apparent  Altitvuie. 

54 
55 

56 



Alt. 
5 

5 

1 

10  20  3 
I    2    1 

1  40 

50  ( 
3 

0  65  SO 
3    3 

Iso 

57 

10 
15 

3 

2  3    3 

3  3    4 

4 
4 

5 

58 

20 

4 

4    4    5 
4    5    6 

6 

b 
6 

59 

30 

5    6    6 

7 

60 

3o 

40 

6 

S 

7    7    8 

61 

45 

7 

7    8!) 

60 

8 

8'  8 

62 

65 

8 

8    9 

63 

60 

H 

9 

65 

9 

64 

70 

60 

39° 

34° 

36° 

38° 

Is^ 

50° 

40° 

40° 

440 

46° 

52° 

_ 



P^s«=^-^-l               TABLE  XLVIII. 

Third  Correction.  Apparent  Distance  112°. 

D's 
App. 
Alt. 

o 
6 

8 
9 

10 

II 

12 

i3 
i4 
i5 

17 
i8 

19 
20 

•21 

22 

23 

24 

25 

26 

27 
28 

^9 

Jo 

3i 

32 

33 
34 
35 

36 

37 
38 
39 
4o 

4i 
42 
43 
44 
45 

46 
4i 
43 

5o 
5i 

52 

53 
54 
55 

56 

57 
58 

60 

6[ 
62 
63 
64 
65 

1 

Jlpparent  Mtitude  of  the  Sun,  Star  or  Planet. 

App. 

Alt. 

0 
6 

7 
8 

9 
10 

II 
12 
i3 
i4 
i5 

16 

17 
18 

19 

20 

21 
22 

23 

24 

25 

26 

27 
28 

3o 
3i 

32 

33 
34 
35 

36 

37 
38 
39 
40 

4i 
42 
43 
44 
45 

46 
47 
48 
49 
5o 

IT 

52 

53 
54 
55 

56 

57 
58 

60 

61 
62 
63 
64 
65 

6° 
/  II 

1  4o 

2  42 
2  46 
2  5i 

2  57 

3  3 

3  9 
3  16 
3  23 
3  3i 
3  39 
3  47 

3  55 

4  3 
4  II 

4  19 

4  27 
4  35 
4  4-i 

4  52 

5  0 
5  8 
5  16 
5  24 
5  32 

5^40 
5  48 

5  56 

6  4 
6  II 

6  19 
6  26 
6  33 
6  4i 
6  48 

6  55 

7  2 
7  8 
7  i5 
7  22 

7"^ 
7  35 
7  42 
7  48 

7  55 

8  I 
8  7 
8  i3 
8  ,9 
8  25 
8  3o 
8  35 
8  4o 
8  45 
8  5o 

8  54 

8  58 

9  2 
9  5 

T 
1  II 

1  42 

2  4o 
2  42 
2  45 
2  49 
2  54 

2  59 

3  4 
3  10 
3  16 
3  22 
3  29 
3  35 
3  4i 
3  48 

3  54 

4  I 
4  8 
4  i5 
4  22 

4  29 
4  37 
4  44 
4  5i 

4  57 

5  4 
5  10 
5  17 
5  24 
5  3i 

5  37 
5  4^ 
5  5o 

5  56 

6  2 

6  8 
6  i4 
6  20 
6  26 
6  32 

6  38 
6  44 
6  49 
6  54 

6  59 

7  4 
7  9 
7  i4 

7  19 
7  23 

7  28 
7  33 
7  38 
7  43 
7  48 
7  52 

7  56 
759 

8° 

1  II 

2  45 
2  42 
2  4i 
2  43 
2  45 

2  48 
2  52 

2  56 

3  0 
3  5 

3  10 
3  i5 
3  20 
3  26 
3  32 

3  38 
3  44 
3  5o 

3  56 

4  3 

\.l 

4  21 
4  27 
4  33 
4  39 
4  45 
4  5i 

4  56 

5  2 

5  7 
5  i3 
5  18 
5  24 
5  29 

5  35 
5  4o 
5  46 
5  5i 

5  56 

6  2 
6  7 
6  12 
6  16 
6  21 

6  25 
6  29 
6  34 
6  38 
6  42 

6  47 
6  5i 
6  55 

6  59 

7  2 
7  5 
7  8 

8° 

9° 
/  II 
2  49 
2  45 
2  43 
2  4i 
2  43 
2  45 
2  48 
2  5i 
2  54 

2  58 

3  2 
3  6 
3  10 
3  i5 
3  20 

T^ 

3  3o 
3  35 
3  40 
3  46 

3  5i 

3  56 

4  2 
4  7 
4  12 

4  17 
4  22 
4  28 
4  33 
4  38 

4  43 
4  48 
4  53 

4  58 

5  3 

5  8 
5  i3 
5  18 
5  23 
•5  28 
5  33 
5  37 
5  4i 
5  45 
5  49 
5  53 

5  57 

6  I 
6  5 
6  8 
6  12 
6  i5 
6  19 
6  22 
6  25 
6  28 

9° 

10° 

/  // 
2  54 
2  49 
2  45 
2  43 
2  42 
2  43 
2  45 
2  47 
2  5r). 
2  53 
2  56 

2  59 

3  2 
3  6 
3  10 

3  i5 
3  20 
3  24 
3  28 
3  33 
3  38 
3  42 
3  47 
3  52 
3  57 

11° 

1  II 
3  0 

2  53 
2  48 
2  45 
243 
2  42 
2  43 
2  45 
2  47 
2  49 

2  5i 
2  54 

2  57 

3  0 
3  3 

3  7 
3  II 
3  i5 
3  19 
3  23 
3  27 
3  3i 
3  36 
3  4o 
3  45 

12° 

/  11 

3  7 

2  58 
2  52 

2  48 
2  45 

2  43 
2  42 
2  43 
2  45 
2  47 

2  48 
2  5o 
2  53 
2  56 

2  58 

3  I 
3  5 
3  8 
3  II 
3  i5 

3  18 
3  22 
3  26 
3  3i 
3  35 

14° 

'  // 
3  21 
3  8 
3  0 
2  54 
2  49 
2  46 
2  44 
2  43 
2  43 
2  44 
2  45 
2  46 
2  48 
2  5o 
2  52 

2  54 
2  56 

2  58 

3  I 
3  3 

3  5 
3  8 
3  II 
3  14 

3  17 

1G° 
/  // 
3  36 
3  20 

3  9 
3  I 

2  55 
2  5o 
2  47 
2  45 
2  44 
2  44 
2  43 
2  44 
2  45 
2  46 
2  48 
2  49 
2  5i 

2  52 

2  54 
2  55 

2  57 

2  59 

3  2 
3  5 

3  7 

3  12 
3  i4 
3  17 
3  19 

3  21 
3  24 
3  26 
3  29 
3  32 

3  35 
3  38 
3  4o 
3  42 
3  45 

3  47 
3  5o 
3  52 
3  55 

3  57 
359 

4  I 
4  3 
4  4 

1G° 

18° 
/  // 
3  52 
3  33 
319 

\    9 
3  2 

2  56 

2  52 

2  49 

2  47 
2  46 

2  45 
2  44 
2  44 
2  45 
2  45 
2  46 
2  47 
2  48 
2  49 
2  5i 

2  52 

2  54 
2  56 

2  58 

3  0 
3  2 
3  4 
3  6 
3  8 
3  10 

3  12 
3  14 
3  16 
3  18 
3  20 

3  -11 
3  25 
3  27 
3  29 
3  3i 
3  33 
3  36 
3  38 
3  4o 
3  42 

3  44 
3  46 

18° 

20° 
/  // 
4  8 
3  46 
3  3i 
319 
3  10 

3  3 
2  58 
2  54 
2  5i 
2  49 

2  47 
2  46 
2  45. 
2  44 
2  44 
2  44 
2  45 
2  46 
2  47 
2  48 
2  49 
2  5i 

2  52 

2  53 
2  55 

2  57 

2  58 

3  0 
3  2 
3  4 
3  5 

\    9 

3  10 

3  12 
3  i3 
3  i5 

3  17 
319 
3  21 

3  23 
3  25 
3  26 
3  28 
3  29 

20° 

22° 
/  // 
4  24 
3  59 
3  42 
3  29 
3  19 
3  II 
3  5 
3  0 
2  56 
2  53 
2  5o 
2  48 
2  47 
2  46 
2  45 

2  44 
2  44 
2  45 
2  45 
2  46 

2  47 
2  48 
2  49 
2  5o 

2  52 

2  53 
2  54 
2  55 
2  57 

2  59 

r~^ 
3  I 
3  3 
3  4 

3  6 

3  7 

3  8 
3  10 
3  12 
3  i4 
3  i5 
3  16 
3  17 

22° 

24° 

/  It 
4  4o 
4  i3 
3  54 
3  40 
3  28 
3  19 
3  12 
3  6 
3  I 
2  57 

2  54 
2  5i 
2  49 
2  48 
2  47 

2  46 

2  45 
2  44 
2  44 
2  45 

^45 
2  45 
2  46 
2  46 
2  47 

2  49 
2  5i 

2  52 

2  53 
2  55 
2  56 
2  57 
2  58 

2  59 

3  I 
3  2 
3  3 
3  5 
3  6 
3  8 
3  9 

24° 

26° 
/  // 
4  56 
4  26 
4  6 
3  5o 
3  37 
3  27 
3  19 
3  12 
3  6 
3  I 

2  57 
2  54 

2  52 

2  5o 
2  49 

2  48 
2  47 
2  46 
2  45 
2  44 

2  44 
2  45 
2  46 
2  46 
2  47 
2  48 
2  49 
2  5o 
2  5i 

2  52 

2  53 
2  54 
2  55 
2  56 
2  57 

2  58 

2  69 

3  0 
3  I 

26° 

28° 

1  II 
"  11 
4  40 
4  18 
4  0 
3  46 

3  35 
3  26 
3  18 
3  12 
3  6 
3  2 

2  58 
2  55 
2  53 
2  5i 
2  5o 
2  49 
2  48 
2  47 
2  46 

2  45 
2  45 
2  45 
2  45 
2  46 

2  47 
2  47 
2  48 
2  49 
2  5a 
2  5i 
2  52 

2  53 
2  54 
2  55 

2  55 
2  56 

28° 

30° 
It 
5  28 
4  54 
4  3o 
4  II 
3  56 

3  44 
3  34 
3  25 
3  17 
3  11 

3  6 
3  2 

259 
2  56 
2  54 

2  52 

2  5i 
2  5o 
2  49 
2  48 

^47 
2  46 
2  46 
2  46 
2  46 
2  46 
2  46 
2  47 
2  48 
2  49 

2  49 
2  5o 
2  5i 

2  52 

2  53 
30° 

4  2 
4  7 
4  12 
4  16 
4  21 

4  25 
4  29 
4  33 
4  37 
4  4i 
4  45 
4  49 
4  53 

4  58 

5  3 

5~8 
5  12 
5  16 
5  20 
5  23 

5  27 
5  3o 
5  34 
5  37 
5  4i 
5  44 
5  47 
5  5o 
5  53 
5  56 

10° 

3  49 
3  5i 

3  58 

4  2 
4  6 
4  10 
4  i4 
4  17 
4  21 

4  25 

4  28 
4  32 
4  36 
4  40 
4  44 

4  47 
4  5i 
4  55 

4  58 

5  2 

5  5 
5  8 
5  12 
5  i5 
5  18 
5  21 
5  24 
5  27 
5  29 

ir 

3  39 
3  43 
3  46 
3  5o 
3  53 

3  57 

4  I 
4  4 
4  8 
4  II 
4  i5 
4  18 
4  22 
4  25 
4  28 
4  3i 
4  34 
4  38 
4  4i 
4  44 

4  47 
4  5o 
4  53 
4  56 

4  59 

5  I 
5  4 
5  6 

12° 

3  20 
3  23 
3  26 
3  29 
3  32 

3  35 
3  38 
3  4i 
3  44 
3  47 
3  5o 
3  53 
3  56 

3  59 

4  2 
4  5 
4  8 
4  n 
4  i4 
4  17 

4  19 
4  22 
4  24 
4  26 
4  28 

43^ 
14° 

TABLE  XLVIII.                                     t^'^sessi 
Third  Correction.     Apparent  Distance  112°. 

5's 

0 

6 

7 
8 

9 

10 

II 

12 

i3 

i4 
i5 

i6 

17 
i8 

19 

20 

21 
22 

23 

24 

25 

26 

27 
28 

L' 

3i 

32 

33 
34 
35 

36 

37 
38 
39 
4o 

4i 
42 
43 

45 
4(5 
47 
48 
49 
5o 

~5~ 

52 

53 
54 
55 

56 

57 
58 

L' 

61 
62 
63 
64 
65 

Apparent  Altitude  of  the  Sun,  Star  or  Planet. 

D's 
App. 
All. 

0 

6 

7 
8 

9 

10 

11 
12 
i3 
i4 
i5 

16 

17 
18 

19 
20 

21 
22 

23 

24 

25 

26 

27 
28 
29 

3o 

~3r 

32 

33 
34 
35 

36 
37 
38 
39 
4o 

4t 
42 
43 

44 
45 

46 
47 
48 
49 

32° 

(    '/ 
5  Ai 
5    7 
4  42 
4  21 
4    5 
3  52 
3  4i 
3  3i 
3  23 
3  16 
3  10 
3    6 
3    3 
3    0 
2  57 

2  55 

2  53 

2    52 

2  5i 
2  5o 

2-49 
2  48 
2  47 
2   47 
2    47 
2    47 
2    47 
2    47 

2  48 
2  48 

2    49 
2    49 

2  5c 

34° 

/   // 
6    0 
5  2. 
4  54 
4  32 
4  i4 
4    0 
3  48 
3  38 
3  29 

3    21 

3  i5 
3  II 
3     7 
3    4 
3     I 

2  58 
2  56 
2  55 
2  53 

2   52 

2  5i 

2  5o 
2  49 

2  48 
2  48 

2  48 
2  48 
2  48 
2  48 
2  48 

2  48 

36° 

1  II 
6  16 
5  34 
5    5 
4  42 
4  23 

4    8 
3  55 
3  AA 
3  35 
3  27 
3  21 
3  16 
3  12 
3    8 
3     5 
3     3 
3    0 

2  58 
2  56 
2  54 
2  53 

2    52 

2  5i 
2  5o 
2  49 
2  49 
2  49 
2  49 
2  49 

38° 
/  II 
6  3i 

547 
5  16 
4  52 
4  32 
4  16 
4    2 
3  5o 
3  4i 
3  33 

3  26 
3  20 
3  16 
3  12 
3    8 

3     5 
3     3 
3     I 

2  59 
2  57 

2  55 
2  54 
2  53 
2  52 
2  5i 
2  5o 
2  5o 

40° 

/  // 
6  46 
6    0 
5  27 
5     I 
4  4o 

4   23 

4    9 

3  57 
3  47 
3  38 

3  3i 
3  25 
3  20 
3  16 

3    19 

3    9 
3    6 
3    3 
3     I 
2  59 
2  57 
2  56 
2  55 
2  54 
2  53 

40° 

42° 
/  II 

1  0 
6  i3 
5  38 
5  II 
4  49 
4  3i 
4  16 
4    4 
3  53 
3  44 
3  37 
3  3o 
3  25 
3  20 
3  16 
3  12 

3    9 
3    6 
3    4 
3     I 

2  59 

2  57 
2  56 

42° 

44° 

(  // 
7  14 
6  25 
5  49 
5  21 
4  58 
4  39 
4  24 
4  10 
3  59 
3  5() 
3  42 
3  35 
3  29 
3  24 
3  20 

3  16 
3  12 
3    9 
3     7 
3    4 
3     I 

44° 

46° 
/  // 
7  27 
6  37 
6    0 
5  3i 
5     7 

4  47 
4  3i 
4  17 
4    6 
3  56 

3  47 
3  4o 
3  34 
3  28 
3  23 
3  19 
3  i5 
3  12 
3    9 

46° 

48° 
/  // 
7  4" 
6  49 
6  10 
5  40 
5  i5 

4  55 
4  38 
4  23 
4  12 
4     I 

3    5'2 

3  45 
3  38 
3  32 
3  26 
3  22 
3  18 

48° 

50° 

/   /' 
7  53 
7     0 
6  20 
5  48 
5  23 

5     2 

4  A\ 
4  29 
4  17 
4    6 

3  57 
3  49 
3  42 
3  35 
3  29 

50° 

52° 

/  // 
8     5 
7  10 
6  29 
5  56 
5  3. 

5    8 
4  5o 
4  35 
4  22 
4  10 

4     I 
3  53 
3  46 

52° 

54° 

/  II 
8  18 
7  20 
6  38 
6    4 
5  38 

5  i4 
4  56 
4  4o 
4  26 
4  i4 
4    4 

56° 

/   " 
8  3o 
7  3o 
6  47 
6  11 
5  44 
5  19 
5     I 
4  45 
4  3o 

58° 
/  // 
8  4i 
7  39 
6  55 
6  18 
5  5o 

5  24 
5     5 

G0° 

1     II 

8  5o 
7  47 
7     2 
6  24 
5  55 

62° 

J  II 
8  58 
7  55 
7     8 

32° 

34° 

36° 

38° 

Table  P.     £^f cJ  6/  Sun'i  Par. 

To  be  subtracted  from  tlie  Tljird 

Correction. 

App. 
All. 

5 
10 
15 

20 
25 
30 
35 
<0 
45 
50 
55 
60 
03 

Sun's  Apparent  Altitude. 

5 

1 
2 
3 
4 
4 
5 
6 
7 
7 
6 
8 
9 
9 

0  :iO  3( 

2    2    3 
i    3    4 
i    4    4 

4  4    5 

5  5    6 

5  6    7 

6  7    7 

7  7    8 

7  8 

8  8 
9 

9 

40 

4 
5 
6 
6 
7 

50  { 

4 

5 

6 

0  65 

i    4 
3 

80 

90 

41 


P-^'^^^^i                                    TABLE  XLVIII. 

Third  Correction.  Apparent  Distance  116°. 

5's 
App. 
All. 

o 

6 

7 
8 

9 

10 

II 

12 

i3 

1 4 
i5 

i6 

17 
i8 

19 
20 

21 
22 

23 

24 

25 

26 

27 
28 

11 

3i 

32 

33 
34 
35 

36 

37 
38 
39 
40 

4i 
42 
43 
44 
45 

46 
47 
48 

5o 
5i 

52 

53 

54 
55 

'56 

57 
58 
59 
60 

62 

63 
64 
65 

Jlpparent  Altitude  of  the  Sun,  Star  or  Planet. 

App. 

Alt. 

0 
6 
7 
8 

9 
10 

II 
12 
i3 
i4 
i5 

16 

17 
18 

19 

.  20 

21 
22 

23 

24 

25 

26 

27 
28 

=9 
3o 

3i 

32 

33 
34 
35 

36 

37 
38 
39 
40 

4i 
42 
43 
44 
45 

46 

47 
48 
49 
5o 

5i 

52 

53 
54 
55 

56 

57 

59 
60 

61 
62 
63 
64 
65 

6° 
1   II 
2   5o 
2  52 

2  56 

3  I 

3  7 
3  i3 
3  20 
3  27 
3  35 
3  43 
3  5i 

3  59 

4  7 
4  16 
4  24 
4  33 
4  4i 
4  49 

4  58 

5  6 

5  i5 
5  23 
5  3i 
5  3<J 
5  47 

5  55 

6  3 
6  12 
6  20 
6  29 

6  37 
6  45 

6  53 

7  I 
7  8 
7  i5 
7  22 
7  3o 
7  37 
7  45 

7  52 

7  59 

8  6 
8  12 
8  18 

8  24 
8  3o 
8  36 
8  42 
8  48 

8  54 

8  59 

9  3 

6- 

70 

1   II 
2  62 
2  5o 

2  52 

2  55 

2  59 

3  ^ 

3  i5 
3  21 

3  27 
3  33 
3  4o 
3  47 

3  54 

4  1 
4  8 
4  i5 
4  22 
4  29 
4  36 

4  43 
4  5o 

4  57 

5  4 
5  11 

5  18 
5  26 
5  33 
5  40 
5  47 

5  55 

6  2 
6  9 
6  16 
6  23 

6  29 
6  35 
6  4i 
6  47 
6  53 

6  59 

7  5 
7  10 
7  i5 
7  20 

7  26 
7  3. 
7  37 
7  42 
7  47 
7  52 
7  57 

7° 

8° 

1  II 

2  55 

2  52 

2  5o 

2  52 

2  55 

2  58 

3  2 
3  6 
3  II 
3  16 

3  21 

3  27 
3  33 
3  39 
3  45 
3  5i 

3  57 

4  4 
4  10 
4  16 
4  22 
4  28 
4  34 
4  4o 
4  46 
4  52 

4  58 

5  5 
5  II 
5  17 
5  23 
5  29 
5  35 
5  4i 
5  47 
5  53 

5  58 

6  4 

6  ID 

6  20 
6  25 
6  3o 
6  35 
6  4o 

6  45 
6  5o 
6  55 

6  59 

7  3 
7  7 

8° 

9° 

1  II 

2  59 
2  55 

2  52 

2  5o 

2  52 

2  54 

2  57 

3  I 
3  5 
3  9 
3  i3 

3  17 
3  22 
3  27 
3  32 

3  37 
3  43 
3  49 

3  54 

4  0 
4  5 
4  II 
4  16 
4  21 
4  26 

4  32 

4  37 
4  42 
4  47 
4  53 

4  58 

5  4 
5  9 
5  i5 
5  20 

5  25 
5  3o 
5  35 
5  4o 
5  45 

5  49 
5  54 

5  59 

6  4 
6  8 

6  12 
6  16 
6  20 
6  24 
6  28 

9° 

10° 

1  II 
3  4 

2  58 
2  54 

2  52 

2  5i 
2  52 

2  54 

2  57 

3  0 
3  4 

3  7 
3  XI 
3  i4 
3  18 
3  22 

3  27 
3  32 
3  37 
3  42 
3  47 
3  52 

3  57 

4  I 
4  6 
4  10 

4  i5 
4  20 
4  25 
4  3o 
4  35 

4  4o 
4  45 
4  5o 
4  54 

4  58 

5  3 

5  7 
5  12 
5  16 
5  21 

5  25 
5  29 
5  33 
5  37 
5  4i 
5  45 
5  49 
5  53 
5  56 

10° 

11° 

1  II 
3  10 
3  2 

2  57 
2  54 
2  52 

2  5i 
2  53 
2  55 

2  57 

3  0 

3  3 
3  6 

3  9 
3  12 

3  16 
3  20 
3  24 
3  28 
3  32 
3  37 

3  4i 
3  45 
3  49 
3  54 

3  58 

4  2 
4  6 
4  II 
4  i5 
4  20 
4  24 
4  29 
4  33 
4  37 
4  4i 
4  45 
4  49 
4  53 

4  57 

5  I 

5  5 
5  9 
5  i3 
5  16 
5  20 

5  23 

5  27 
5  3o 

11° 

12° 

/  // 
3  17 
3  8 
3  1 
2  57 
2  54 
2  53 

2  53 

2  53 
2  55 

2  57 

3  0 
3  2 
3  5 
3  8 
3  II 

3  i4 
3  17 
3  21 
3  24 
3  28 

3  32 
3  35 
3  39 
3  43 
3  47 
3  5i 
3  55 

3  59 

4  2 
4  6 
4  lo 
4  i4 
4  18 
4  22 
4  26 
4  3o 
4  33 
4  37 
4  4i 
4  44 
4  48 
4  52 
4  55 

4  58 

5  I 

5  7 
12° 

13° 

/  // 
3  25 
3  i3 
3  5 
3  0 
2  57 

2^5 

2  53 

2  52 

2  53 
2  55 

2  57 

2  59 

3  I 
3  4 
3  7 

3  9 
3  12 

3  i5 

3  18 

3  21 

3  24 
3  27 
3  3i 
3  34 
3  38 

3  42 
3  45 
3  48 
3  52 
3  55 

3  59 

4  2 
4  6 
4  10 
4  i3 

4  17 
4  20 

4  23 

4  27 
4  3o 
4  33 
4  37 
4  40 
4  43 
4  46 

4  49 
13° 

14° 

1  II 
3  33 
3  19 
3  10 
3  4 
3  0 

2  57 
2  55 
2  53 
2  52 
2  53 
2  55 
2  57 

2  59 

3  I 
3  3 
3  5 
3  7 
3  10 
3  i3 
3  16 
3  18 
3  21 
3  24 
3  27 
3  3o 

3  34 
3  37 
3  4o 
3  43 
3  47 
3  5o 
3  53 

3  57 

4  0 
4  3 
4  6 
4  9 
4  12 
4  i5 
4  18 
4  21 
4  24 
4  27 
4  3o 
4  33 

15° 

/  // 
3  4i 
3  25 
3  i5 
3  8 
3  3 
3  0 
2  57 
2  55 
2  54 
2  53 

2  54 
2  55 
2  56 

2  58 

3  0 
3  2 
3  4 
3  7 
3  10 
3  12 

3  i4 
^  16 
3  19 
3  21 
3  24 
3  27 
3  3o 
3  33 
3  36 
3  4o 

3  43 
3  46 
3  49 
3  52 
3  54 

3  57 

4  0 
4  3 
4  6 
4  9 
4  II 
4  i4 
4  17 
4  19 

16° 
/  // 
3  49 
3  32 
3  21 
3  i3 
3  7 
3  3 
3  0 
2  58 
2  56 
2  54 
2  53 
2  54 
2  55 
2  56 

2  58 

3  0 
3  2 
?  4 
3  7 
3  9 
3  II 
3  i3 
3  i5 
3  17 
3  19 

3  22 

3  25 
3  27 
3  3o 
3  33 

3  36 
3  39 
3  42 
3  45 
3  47 
3  5o 
3  53 
3  55 

3  58 

4  I 
4  3 
4  6 
4  8 

18° 
/  II 
4  5 
3  46 
3  32 
3  21 
3  14 
3  9 
3  5 
3  2 
2  59 
2  57 
2  55 
2  55 
2  54 
2  55 
2  56 

2  57 

2  59 

3  0 
3  2 
3  4 
3  6 
3  '8 

3  II 
3  i3 
3  i5 

3  17 
3  19 
3  21 
3  24 

3  26 
3  20 
3  3i 
3  33 
3  36 
3  39 
3  4i 
3  43 
3  45 
3  47 
3  49 

18° 

20° 

/  // 
4  22 
4  0 
3  44 
3  3i 
3  22 

3  16 
3  10 
3  6 
3  3 
3  0 
2  58 
2  57 
2  56 
2  55 
2  55 
2  56 
2  57 
2  58 

2  59 

3  I 
3  2 
3  4 
3  5 
3  7 
3  8 

3  10 
3  12 
3  i3 
3  i5 

3  17 

3  19 
3  21 
3  23 
3  25 

3  27 

3  29 
3  3i 
3  33 
3  34 

20° 

22° 

/  // 
4  39 
4  i4 
3  56 
3  42 
3  3i 
3  23 
3  16 
3  II 
3  7 
3  4 
3  I 
2  59 
2  58 
2  57 
2  57 
2  57 
2  56 
2  57 
2  58 

2  59 

3  0 
3  2 
3  3 
3  4 
3  5 

3  6 
3  8 
3  9 
3  10 
3  12 

3  i4 
3  i5 
3  17 
319 
3  20 

3  21 
3  23 

22° 

24° 

1  II 
4  56 
4  28 
4  8 
3  53 
3  41 
3  3i 
3  23 

3  17 
3  12 
3  8 
3  5 
3  3 
3  I 
3  0 

2  59 

2  59 

2  58 
2  58 
2  57 
2  58 

2  59 

3  0 
3  I 
3  2 
3  3 

3  4 
3  5 
3  6 
3  7 
3  9 
3  10 
3  II 
3  i3 
3  i5 
3  ,6 

24° 

26° 

1  n 
5  i3 
4  42 
4  20 
4  4 
3  5o 

3  39 
3  3o 
3  23 
3  18 
3  14 
3  10 
3  7 
3  5 
3  3 
3  2 

3  I 
3  0 
3  0 

2  59 
2  59 
2  58 
2  59 

2  59 

3  0 
3  I 

3  2 
3  3 
3  4 
3  5 
3  6 

3  7 
3  8 

3  9 

26° 

14° 

15° 

16° 

TABLE  XLVIII.                                     [Page  323 

Third  Correction.     Apparent  Distance  116°. 

D's 
A  pp. 
Alt. 

Apparent  Altitude  of  the  Sun,  Star  or  Planet. 

App. 

Alt. 

28° 

.30° 

32° 

34° 

36° 

38° 

40° 

42° 

44° 

46° 

48° 

50° 

52° 

54° 

56° 

58° 

0 

1  II 

1  II 

/  // 

/  // 

/  II 

/  II 

(  II 

1  II 

/  II 

/  If 

1  II 

/  II 

/  // 

/  » 

1  II 

/  II 

0 

6 

5  3o 

5  46 

6    3 

6  19 

6  36 

6  52 

7     7 

7  22 

7  36 

1  5i 

8    5 

8  18 

8  3o 

8  4: 

8  53 

9    3 

6 

7 

4  56 

5  10 

5  25 

b  4o 

5  55 

6    9 

6  22 

6  34 

6  46 

6  58 

7     9 

7  20 

7  3l 

7  4: 

7   52 

7 

8 

4  33 

4  45 

4  58 

b  II 

5  24 

6  3b 

547 

5  58 

6    8 

6  18 

6  28 

6  38 

6  48 

6  5^ 

7     8 

8 

9 

4  i5 

4  26 

4  37 

4  47 

4  58 

5     81'  19 

5  29 

5  39 

5  49 

5  5q 

6     8 

6  16 

6    2Z 

9 

10 

4    0 

4  10 

4  20 

4  29 

4  39 

4  4f|4  58 

5     7 

5  16 

5  25 

5  33 

5  4i 

5  49 

5  5e 

) 

10 

II 

3  48 

3  57 

4    6 

4  i5 

4  23 

4   32 

4  4i 

4  49 

4  57 

5     5 

5  12 

5    iq 

5  25 

II 

12 

3  38 

3  46 

3  54 

4    2 

4  10 

4  18 

4  26 

4  34 

4  4i 

4  48 

4  54 

5     I 

5     7 

12 

i3 

3  3o 

3  37 

3  M 

3    b2 

4    0 

4    7 

4  i4 

4  21 

4  27 

4  33 

4  3q 

4  45 

i3 

i4 

3  24 

3  3o 

3  37 

3  44 

3  5i 

3  b7 

4    4 

4  10 

4  16 

4  21 

4  27 

4  33 

14 

lb 

3  19 

3  25 

3  3i 

3  37 

3  43 

3  49 

3  55 

4    I 

4    6 

4  II 

4  17 

i5 

16 

3  i5 

3  20 

3  26 

3  3i 

3  37 

3  42 

3  47 

3  53 

3  58 

4    2 

4    8 

16 

17 

3  12 

3  16 

3  21 

3  26 

3  3i 

3  36 

3  4i 

3  46 

3  5i 

3  55 

17 

i« 

3    9 

3  i3 

3  17 

3  22 

3  26 

3  3i 

3  36 

3  40 

3  45 

3  49 

18 

19 

3    7 

3  10 

3  i4 

3  18 

3  22 

3  27 

3  3i 

3  35 

3  39 

'9 

20 

3    5 

3    8 

3  II 

3  i5 

3  19 

3  23 

3  27 

3  3i 

3  34 

20 

21 

3    4 

3    6 

3     9 

3  12 

3  16 

3  20 

3  23 

3  27 

21 

22 

3    3 

3     5 

3     7 

3   10 

3  i4 

3  17 

3  20 

3  23 

22 

23 

3    2 

3    4 

3    6 

3     9 

3  12 

3  i5 

3  18 

23 

24 

3     I 

3    3 

3    5 

3     8 

3  10 

3  i3 

3  16 

24 

25 

26 

3    0 
3    0 

3     2 
3     2 

3    4 
3    4 

3     7 
3     6 

3    9 
3     7 

3  II 
3     9 

25 

26 

27 

3    0 

3     I 

3    3 

3     5 

3    6 

27 

28 

2  69 

3    0 

3     2 

3    4 

3    5 

28 

29 

2  59 

3    0 

3     I 

3     3 

29 

Jo 
3i 

3    0 
3     I 

3    0 
3    0 

3     I 
3     I 

3     3 



3o 
3 1 

32 

3     2 

3     I 

3     2 

32 

33 

3     2 

3     I 

33 

34 

3     3 

3     2 

M 

3b 

3    4 

35 

36 

3     5 

3"?r 

37 

37 

38 

38 

39 

39 

40 

40 

4i 

4i 

42 

42 

43 

43 

M 

M 

4b 

45 

46 

46 

47 

47 

48 

48 

i9 

49 

5o 
5i 

52 







1 

Table  P,     iJ^ecJ  0/  SiHi's  Pat 

b3 

To  be  subtracted  from  the  Third 

54 
55 

"56 





Correction. 

D'3 
App. 
Alt. 

Slinks  Apparent  Altitude. 

5    I 

0  20  30 

40 

50  6 

0  70 

80  S 

0 

57 

■• 

58 

5 

2 

>.    2    3 

4 

4    . 

10 

2 

I    3    4 

5 

.■) 

bQ 

15 

3 

)    4    5 

5 

6 

60 

20 
23 

4 

5 

5    6 
)    fi    6 

tj 
7 

61 









30 

5 

6    7 

35 

6 

7    3 

62 

40 

7 

8 

63 

1 

45 
SO 

7 

8 

i    8 
i 

64 

1 

6.5 

9 
9 

i 

tib 

1 

i 

1 

28° 

30° 1 32° 

34° 

36° 

38° 

40° 

42° 

44° 

46° 

48= 

1 

^"^'^'^                                    TABLE  XLVIII. 

Third  Correction.  Apparent  Distance  120°. 

])'s 
App. 
Alt. 

o 
6 

7 
8 

9 

lO 

II 

12 

i3 

i4 
i5 

i6 

17 
i8 

19 
20 

21 
22 

23 

24 

25 

26 

27 
28 

11 

3i 

32 

33 
34 
35 

36 

37 
38 

39 
40 

4i 
42 
43 

44 
45 

46 

47 
48 

5o 
5i 

52 

53 
54 
55 

56 

57 
58 

60 
61 
62 

63 
64 
65 

Ajjparent  Mtitude  of  the  Sun,  Star  or  Planet. 

5 

's 

6= 

/  // 
3  I 
3  3 
3  7 
3  12 
3  j8 

3  25 
3  33 
3  4i 
3  49 

3  57 

4  6 
4  i4 

4  23 

4  32 
4  40 

4  49 

4  58 

5  7 
5  16 
5  25 

5  34 
5  42 

5  5i 

6  0 
6  8 

6  17 
6  25 
6  34 
0  43 
6  5i 

6  59 

7  8 
7  16 
7  24 
7  32 

7  4o 
7  47 

7  55 

8  3 
8  II 

8  18 
8  25 
8  32 
8  39 
8  45 
8  5i 

8  57 

9  3 
9  9 

7° 
1  II 
3  3 
3  2 
3  4 
3  8 
3  12 

3  17 
3  23 
3  28 
3  34 
3  4i 
3  48 

3  55 

4  3 
4  10 
4  17 
4  24 
4  3i 
4  39 
4  46 

4  53 

5  I 
5  8 
5  16 
5  24 
5  3i 
5  39 
5  46 

5  54 

6  2 
6  9 

6  16 
6  23 
6  3o 
6  37 
6  44 
6  5o 

6  56 

7  2 
7  9 
7  i5 

7  21 

7  27 
7  33 
7  39 
7  45 
7  5i 

7  57 

8  3 

8° 
,  // 
3  6 
3  4 
3  3 
3  5 
3  8 

3  12 
3  16 
3  20 
3  25 

3'3o 

3  36 
3  42 
3  48 

3  54 

4  I 

4  7 
4  i4 
4  21 
4  27 
4  33 

4  4o 
4  47 

4  53 

5  0 
5  6 
5  12 
5  18 
5  25 
5  3i 
5  38 

5  44 
5  5o 

5  56 

6  2 
6  8 

6  i4 
6  19 
6  25 
6  3i 
6  36 

6  4i 
6  46 
6  52 

6  57 

7  2 
7  8 
7  i3 

9° 
/  // 
3  II 
3  7 
3  5 
3  4 
3  6 

3  8 
3  II 
3  i5 
3  19 
3  23 

3~^ 
3  32 
3  37 
3  42 
3  48 

3  53 

3  58 

4  4 
4  10 
4  i5 
4  20 
4  25 
4  3i 
4  37 
4  43 

4  48 

4  54 

5  0 
5  6 
5  12 

5  18 
5  23 
5  28 
5  34 
5  39 

5  44 
5  5o 

5  55 

6  0 
6  5 
6  10 
6  i5 
6  20 
6  25 
6  3o 

6  34 

10° 

/  // 
3  17 
3  11 
3  8 
3  6 
3  5 

3  6 
3  8 
3  II 

3  i4 
3  18 

3  22 
3  25 
3  29 
3  33 
3  38 

3  42 
3  47 
3  52 

3  57 

4  2 

4  7 
4  12 

4  17 
4  22 

4  27 

4  32 

4  37 
4  42 
4  47 
4  52 

4  57 

5  2 
5  7 
5  12 
5  17 
5  22 
5  27 
5  32 
5  37 
5  42 

5  46 
5  5i 
5  55 

5  59 

6  3 

10° 

11° 

/  ti 
3  24 
3  16 
3  u 
3  8 
3  6 
3  5 
3  6 
3  8 
3  II 
3  i4 
3  17 
3  20 
3  23 
3  26 
3  3o 

3  34 
3  39 
3  4i 
3  47 
3  5i 

3  56 

4  I 
4  5 
4  10 
4  i5 

4  19 
4  23 
4  27 
4  32 
4  37 
4  42 
4  46 
4  5o 
4  55 

4  59 

5  4 
5  8 
5  i3 
5  17 
5  22 

5  26 
5  3o 
5  M 

5  37 

11° 

12° 

1  II 
3  32 
3  22 
3  i5 
3  II 
3  8 

3  6 
3  5 
3  6 
3  8 
3  n 

3  i3 
3  i5 
3  18 
3  21 
3  24 

3  28 
3  32 
3  36 
3  39 
3  43 

3  47 
3  5i 
3  55 

3  59 

4  3 

4  7 
4  II 
4  i5 
4  19 
4  24 
4  28 
4  32 
4  36 
4  4o 
4  44 
4  48 
4  52 

4  56 

5  0 
5  4 
5  8 
5  II 
5  i4 

12° 

13° 

/  // 
3  39 
3  28 
3  20 
3  i4 
3  10 

i     8 
3  7 
3  6 

3  7 
3  9 

3  II 
3  12 

3  i4 
3  17 
3  20 

3  23 
3  26 
3  3o 
3  33 
3  36 

3  39 
3  43 
3  47 
3  5o 
3  54 

3  57 

4  I 
4  5 
4  9 
4  12 

4'T5 
4  19 

4  23 

4  27 
4  3i 

4  35 
4  39 
4  42 
4  46 
4  49 
4  53 
4  56 

13° 

14° 

/  // 
3  47 
3  34 
3  25 
3  18 
3  i4 
3  II 

3  9 
3  b 

3  7 
3  8 

3  9 
3  10 
3  12 
3  i5 
3  17 
3  19 
3  22 
3  25 
3  28 
3  3i 

3  34 
3  37 
3  4o 
3  43 
3  46 

3  49 
3  52 
3  56 

3  59 

4  2 

4  5 

4  9 
4  i3 
4  16 

4  20 

4  24 
4  28 
4  3i 
4  34 
4  37 

4  40 

15° 

/  /^ 
3  55 
3  40 
3  29 
3  22 
3  17 
3  i3 
3  n 

3  9 
3  8 

3  7 
3  8 
3  9 
3  11 
3  i3 
3  i5 

3  17 
3  19 

3  2i 

3  23 
3  26 
3  29 
3  32 
3  35 
337 
3  4o 

3  43 
3  46 
3  49 
3  52 
3  55 

3  58 

4  2 
4  5 
4  8 
4  II 
4  i5 
4  18 
4  21 
4  24 
4  27 

16° 
/  // 
4  4 
3  47 
3  35 
3  26 
3  20 

3  16 
3  i3 
3  II 
3  9 
3  8 
3  8 
3  9 
3  10 
3  II 
3  i3 
3  i5 
3  16 
3  18 
3  20 
3  23 
3  25 
3  28 
3  3o 
3  33 
3  36 
3  38 
3  4i 
3  44 
3  47 
3  5o 

3  53 
3  56 

3  59 

4  I 
4  4 

4  7 
4  10 
4  i3 
4  16 

17° 

/  // 
4  12 
3  54 
3  4o 
3  3i 
3  24 
3  19 
3  i5 
3  i3 
3  II 
3  9 
3  9 

I    9 
3  10 
3  12 

3  i3 
3  i4 
3  16 
3  18 
3  20 

3  22 
3  25 
3  27 
3  29 
3  32 

3  34 
3  36 
3  39 
3  42 
3  45 

3  47 
3  5o 
3  53 
3  55 

3  58 

4  I 
4  3 
4  6 

18° 
/  // 
4  21 
4  I 
3  46 
3  36 
3  28 
3  22 
3  18 
3  i5 
3  12 
3  II 
3  10 
3  9 

I    9 

3  10 

3  II 

3  12 
3  i3 
3  i4 
3  16 
3  18 

3  20 
3  22 
3  24 
3  26 
3  28 

3  3o 
3  32 
3  35 
3  37 
3  40 
3  42 
3  45 
3  47 
3  5() 
3  52 

3  55 
3  57 

19° 

/  II 
4  3o 
4  8 
3  52 
3  41 
3  33 

3  26 
3  21 

3  17 
3  14 
3  12 

3  11 

3  10 
3  10 
3  9 
3  10 
3  II 
3  12 
3  i3 
3  i5 
3  17 
3  18 
3  20 
3  22 
3  23 
3  25 

3  27 
3  29 
3  32 
3  34 
3  36 

3  38 
3  4; 
3  43 
3  45 
3  47 
3  49 

20°  2 

/  //  ( 
4  394 
4  i5  4 
3  594 
3  473 
3  38  3 
3  3o3 
3  243 
3  203 
3  173 
3  i4  3 

3  12  3 
3  11  3 
3  ii3 
3  103 
3  10  3 
3  103 
3  11  3 
3  123 
3  143 
3  i53 

3  163 
3  183 
3  203 
3  21  3 
3  23  3 

3  253 
3  273 
3  293 
3  3i3 
3  33  3 
3  35  3 
3  373 
3  393 
3  4i 
3  43 

Apt*. 
2°  Ah. 

/'    0 

57  6 
3o   7 
12   s 

58  9 
47  10 
38  II 
3i  12 
26  i3 
22  :4 
18  ]5 
16  lb 
i4  17 
i3  18 
12  19 
12  20 

11  21 

11  22 

12  23 
i3  24 
i4  25 

1 5  26 

16  27 

17  s8 

18  29 

19  3o 

20  3 1 
22  32 
24  33 
26  34 
28  35 
3o  l6" 
3i  37 
33  38 

39 

4o 

4i 
42 
43 
44 
45 

46 
47 
48 

49 
5o 

14° 

15° 

16° 

Tahle  P.  Effect  of  Sun's  Par. 

To  be  sub'.racled  from  Ihe  Third 

Correction. 

D's 

App. 
Alt. 

5 
10 

15 
20 

25 
30 
35 
40 
4S 
50 
55 

Sun's  Apparent  Akitupe. 

5 

2 
•2 
3 
4 
5 
6 
6 

8 
8 
9 

lu  20  3 

2  3  3 

3  3  4 

4  4  5 

4  5  6 

5  6  7 

6  7  8 

7  8 

8  8 
8 

9 

40  50J55 

4  5  5 

5  6 

6  7 
7 

7 

70  80  90 

6° 

7° 

8° 

9° 

TABLE  XLVIII.               t^^so325 
Third  Correction.  Apparent  Distance  120°. 

D's 

o 

6 

7 
8 

9 

lO 

II 

12 

1 3 

i4 
i5 

i6 

I? 
i8 

19 
20 

21 
22 

23 

24 

25 

26 
27 
28 

=9 
3o 

3i 

32 

33 
34 
35 

36 

37 
38 
39 
4o 

4i 
42 
43 
44 
45 

46 

47 
48 

49 
5o 

5i 

52 

53 
54 
55 

56 
57 
58 

61 
62 
63 
64 
65 

Apparent  Altitude  of  the  Sun,  Star  or  Planet. 

D's 
App. 

All. 

0 

6 

7 
8 

9 
10 

II 

12 

i3 
i4 
i5 

16 

17 
18 

19 
20 

21 
22 

23 

24 

25 

26 
27 
28 

3o 
3i 

32 

33 
34 
35 

36 

37 
38 

39 
4o 

4i 
42 
43 
44 
45 

46 
47 
48 

49 
5o 

"57 

52 

53 
54 
55 

56 
57 
58 

60 

61 
62 
63 
64 
65 

24° 

/  '/ 
5  i5 
4  45 
4  25 

4  9 

3  57 

3  47 
3  39 
3  32 
3  27 
3  23 

3  20 
3  18 
3  16 
3  i5 
3  i4 
3  i3 
3  12 
3  12 
3  12 
3  12 

3  i3 
3  i4 
3  i5 
3  16 
3  17 

26° 

!     II 

5  32 
5  0 
4  38 
4  20 
4  7 
3  56 
3  47 
3  39 
3  33 
3  29 

TI5 
3  22 
3  20 
3  18 
3  16 
3  i5 
3  i4 
3  i3 
3  i3 
3  i3 

3  i4 
3  i4 
3  i4 
3  i5 
3  16 

28° 
/  // 
5  49 
5  i5 
4  5i 
4  3i 
4  17 
4  5 
3  55 
3  46 
3  39 
3  34 
3  3c. 
3  27 
3  24 
3  21 
3  19 

3  17 
3  16 
3  i5 
3  i5 
3  i5 

3  i5 
3  i5 
3  i5 
3  i5 
3  16 

30° 

/  // 
6  6 
5  3o 
5  4 
4  43 
4  27 

4  i4 
4  3 
3  53 
3  46 
3  4o 

3  36 
3  32 
3  28 
3  25 
3  23 
3  21 
3  19 
3  18 
3  17 
3  17 
3  16 
3  16 
3  16 
3  16 
3  16 

32° 

/  // 
6  23 
5  45 
5  17 
4  55 
437 
4  23 
4  II 
4  0 
3  52 
3  46 

3  4i 
3  36 
3  32 
3  29 
3  27 

3  24 
3  22 
3  21 
3  20 
3  19 

3  18 
3  18 
3  18 

34° 

1  II 
6  4i 
6  0 
5  3o 
5  7 
4  47 
4  32 
4  19 
4  8 
3  59 
3  52 

3  46 
3  4i 
3  37 
3  33 
3  3i 
3  28 
3  26 
3  24 
3  23 
3  21 

3  20 

36° 

/  n 
6  58 
6  i5 
5  43 
5  18 
4  57 
4  42 
4  28 
4  16 
4  6 
3  58 
3  52 
3  47 
3  42 
3  38 
3  35 
3  32 
3  29 
3  27 
3  26 

38° 
/  II 
7  i4 
6  29 
5  56 
5  28 
5  7 
4  5i 
4  36 
4  24 
4  i3 
4  4 
3  58 
3  52 
3  47 
3  43 
3  39 

3  36 
3  33 

40° 

/  II 
7  3o 
6  43 
6  8 
5  38 
5  17 
4  59 
4  44 
4  3i 
4  20 
4  II 

4  4 
3  58 
3  52 
3  47 
3  43 

40° 

42" 

/  // 
7  46 
6  56 
6  20 
5  49 
5  27 

5  8 
4  52 
4  38 
4  27 
4  18 
4  10 
4  3 
3  57 

42° 

44° 

/  '/ 
8  2 
7  8 
6  3i 
6  0 
537 

5  17 
5  0 

4  46 
4  34 
4  24 

4  16 

44° 

46° 
/  // 
8  17 
7  21 
6  42 
6  10 
5  46 
5  25 
5  7 
4  53 
4  40 

46° 

48° 
/  II 
8  3i 
7  33 
6  53 
6  20 
5  55 
5  33 
5  i4 

50° 

/  II 
8  M 
7  45 
7  3 
6  29 
6  3 

52° 

/  // 
8  57 
7  57 
7  i3 

54° 

/  II 
9  9 

3  18 
3  19 
3  21 
3  22 
3  24 
3  26 

24° 

3  17 
3  18 
3  19 
3  20 

26° 

3  17 
3  i8 

28' 

30° 

32° 

34° 

36° 

38° 

48° 

50° 

52° 

54° 

1 

*^se  326]                                               TABLE  XLIX. 

To  find  the  correction  of  the   apparent  distance  of  the  moon  from  any  planet,  on  ac- 

count of  the  parallax  of  the  planet,  supposing  its  horizontal  parallax  to  be  35".     This  is 

to  be  reduced  to  the  actual  horizontal  parallax  by  m.eans  of  Table  L. 

Apparent  Distance. 

Apparent  Distance. 

* 

D 

o 

o 

0 

V,             o 

o 

0 

0 

0 

0 

0 

* 

» 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

Alt. 

Alt. 

2i) 

30 

40 

50 

60 

70 

80 

90 

100 

110 

120 

Alt. 

Alt. 

20 

30 

40 

50 

60 

70 

80 

90 

100 

no 

120 

0 

0 

0 

0 

10 

10 

+  1 

—  1 

2 

2 

—  3 

—  4 

—  5 

—  6 

—  7 

—  9 

-10 

25 

10 

+24 

+14 

+  9 

+  5 

+  2 

—  1 

—  3 

—  6 

—  9 

—12 

-15 

15 

—  8 

—  7 

—  6 

—  6 

—  7 

—  7 

—  8 

—  9 

—10 

—12 

—14 

15 

+15 

+  8 

+  4 

+  1 

—  2 

—  4 

—  6 

—  9 

—12 

-15 

-19 

20 

—17 

—13 

—11 

—10 

—10 

—10 

—11 

—12 

—13 

—15 

-17 

20 

+  7 

+  3 

—  1 

—  3 

—  5 

—  7 

—  9 

—12 

—15 

—18 

—22 

25 

—26 

—18 

—15 

—14 

—13 

—13 

—14 

—15 

—16 

-18 

—20 

25 

—  1 

—  3 

—  5 

—  7 

-8 

—10 

—12 

—15 

—17 

—21 

-26 

30 

—34 

—24 

—20 

—17 

—16 

—16 

—17 

—17 

—19 

—21 

—23 

30 

—  9 

—  9 

—  9 

—10 

— U 

—13 

—15 

—17 

—20 

—24 

—29 

35 

—29 

—24 

—21 

—19 

—19 

—19 

—20 

—21 

—24 

—26 

35 

—17 

—14 

—13 

—14 

—14 

—16 

—18 

—20 

—23 

27 

—32 

40 

—34 

og 

—24 

—22 

22 

22 

—23 

—24 

—26 

-29 

40 

—25 

—19 

—17 

—17 

—17 

—18 

—20 

^22 

— 25 

—29 

45 

_?1 

—27 

—25 

—24 

—24 

—25 

—26 

—29 

—32 

45 

—32 

—23 

—21 

—20 

—20 

-21 

—22 

—25 

—28 

—32 

50 

—34 

— ::o 

—27 

—26 

—26 

—27 

og 

—31 

—34 

50 

—28 

—24 

^22 

22 

-23 

—24 

—27 

—3(1 

55 

—33 

—30 

—28 

-28 

—"9 

—30 

—33 

55 

_30 

_07 

—25 

—"4 

—"5 

— ''6 

-99 

3') 

GO 
65 
70 

75 
80 
85 
90 

—34 

—31 
—33 
—34 

—30 
—32 
—33 
—34 
—34 

—30 
—31 
-32 
—33 
—3^ 
—34 
34 

—30 
—32 
—33 
—34 
—34 

—32 
—33 
—34 

—34 

GO 
G5 
70 
75 
80 
85 
10 

—29 
—32 

—27 
—29 
—31 
—32 

—26 

^28 

—29 
—30 
—31 
32 

—27 
—28 
—30 
—31 
—31 
—32 

—28 
—30 
—31 
—32 

—30 
—32 

15 

10 

+  9 

+  5 

+  2 

0 

—  2 

—  3 

—  4 

—  6 

—  8 

—10 

12 

30 

10 

+30 

+19 

+12 

+  7 

+  3 

0 

—  3 

—  6 



—13 

—17 

15 

0 

—  1 

—  3 

—  4 

—  5 

—  6 

—  7 

—  9 

—11 

-13 

—15 

15 

+22 

+13 

+  7 

+  3 

0 

-  3 

—  6 

—  9 

—12 

—16 

— 20| 

20 

—  9 

—  7 

—  7 

—  8 

—  8 

—  9 

—10 

—12 

^14 

—16 

—19 

20 

+14 

+  7 

-f  3 

—  1 

—  4 

—  6 

—  9 

—12 

— 15 

—19 

—24 

25 

—17 

—13 

—12 

—11 

—12 

—12 

—13 

—15 

—16 

-19 

22 

25 

+  (i 

+  2 

2 

—  4 

—  7 

—  9 

—12 

—15 

—18 

22 

-27 

30 

—26 

-19 

—16 

—15 

—15 

—15 

—16 

—17 

—19 

22 

-25 

30 

2 

—  4 

—  6 

—  8 

—10 

—12 

— M 

—17 

—21 

-25 

.-30 

35 

— 34 

—24 

—20 

—18 

—18 

-18 

—19 

-20 

22 

—25 

—28 

35 

—10 

—  9 

—10 

—11 

-13 

—15 

—17 

—20 

—23 

—28 

40 

—29 

—24 

22 

—20 

—20 

—21 

—22 

—24 

-27 

—31 

40 

—17 

—14 

—14 

-14 

—16 

—17 

—20 

—23 

— 2G 

—30 

45 

—34 

—25 

-23 

-23 

—23 

-25 

— 26 

—30 

—31 

45 

—24 

—19 

—17 

—17 

—18 

—20 

^22 

—25 

—28 

50 

31 

27 

''u 

05 

26 

•''7 

oq 

3.0 

nO 

on 

.0" 

—20 
—23 

—24 

—26 

—27 
—29 

—30 

55 

. .. . 

-34 

—30 

—23 

oy 

-T-27 

—29 

—30 

—34 

55 

07 

-24 

-23 

-24 

GO 

—32 

—30 

—29 

—29 

—30 

30 

GO 

-30 

—26 

—25 

—25 

-20 

—28 

—30 

(i5 

34 

31 

30 

31 

3'"' 

34 

G5 

-28 
-30 

—27 
—28 

—27 
—29 

—29 
—30 

70 

-33 

—32 

-32 

—33 

70 

-28 

75 
80 
85 
!)0 

—34 

—33 
—33 
—34 

-33 
—33 
—3-! 

—34 

75 
80 
85 

no 

-29 
—30 

—29 

—30 

—30 

30 

—30 
—30 

:o 

!0 

+16 

+  9 

+  5 

+  a 

0 

—  4 

—  6 

—  8 

—11 

—14 

35 

10 



+23 

+15 

+  9 

+  5 

+  1 

—  3 

—  6 

—10 

—14 

—19 

15 

+  B 

+  4 

+  1 

—  1 

—  3 

—  5 

—  7 

—  9 

—11 

-14 

—17 

15 

+29 

+17 

+10 

+  5 

+  1 

—  2 

—  5 

-  9 

—13 

— 1: 

-22 

20 

-1-2 

—  4 

—  5 

—  7 

—  8 

—10 

—12 

-14 

—17 

—21 

20 

+21 

+12 

+  6 

+  2 

2 

—  5 

—  8 

—12 

—16 

—20 

—25 

25 

—  9(—  8 

—  8 

-  9 

—10 

—11 

—13 

—15 

—17 

—20 

—24 

25 

+13 

+  6 

+  2 

—  2 

—  5 

—  8 

—11 

—15 

—IS 

23 

20 

30 

-17 

-14 

—12 

—13 

—13 

—14 

—16 

—17 

—20 

—23 

—27 

30 

+  5 

+  1 

—  3 

—  6 

-  8 

—11 

—14 

—17 

—21 

—26 

35 

—2,5 

—19 

—16 

—16 

—16 

—17 

—18 

—20 

22 

-25 

—30 

35 

—  3 

—  5 

—  7 

—  9 

—11 

—14 

—17 

—20 

—24 

—29 

tu 

—33 

24 

—20 

—19 

—19 

—19 

—21 

—23 

—25 

—28 

—33 

40 

—10 

—10 

—11 

—12 

—14 

-16 

-19 

—22 

0(; 

Ih 

—29 

—24 

22 

—21 

22 

—23 

-25 

07 

—31 

45 

—16 

-M 

—14 

—15 

—17 

—19 

22 

-25 

—29 

;)!) 

—33 

-27 

—25 

—24 

—24 

—25 

07 

—29 

—33 

50 

—23 

—18 

—17 

—18 

—19 

—21 

—24 

27 

bo 

—30 

—27 

—26 

— 2i5 

27 

-29 

—31 

55 

—29 

^22 

-20 

—20 

—21 

—23 

—26 

—29 

GO 

.... 

—33 

—29 

—28 

-28 

-29 

—30 

-33 

GO 

— 26 

—23 

22 

—23 

05 

27 

().'. 

31 

30 

oq 

30 1     in 

G5 

—29 

70 
75 

, ... 

-33 

—31 

-31 
3'> 

—31 
3^ 

—33 

70 

75 

—27 
—29 

—26 
-27 
—28 
-29 

-27 
— 2S 
-28 
—29 

—28 
—29 

i 

80 
85 
90 

-33 

—32 
-33 
33 

-33 

SO 
85 
90 

— 







0 

O    i     0 

0 

o 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

2i) 

30  1  40 

i.)0 

tiO 

70 

80 

•SO 

100 

110 

120 

20 

30 

40 

50 

60 

70 

80 

90 

100 

110 

120 

TABLE  XLIX.                                                [i-agess? 

To  find  ilie  correction  of  the  apparent  distance  of  the  moon  from  any  planet,  on  ac- 

i  count  of  the  parallax  of  the  planet,  supposing  its  horizontal  parallax  to  ':)e  35".     This  is 

to  be  reduced  to  the  actual  horizontal  parallax  by  means  of  Table  L. 

Apparent  Distance. 

Apparent  Distance. 

1  * 

Alt. 

Alt. 

0 

20 

311 

0 
40 

0 
50 

0 
60 

0 
70 

0 
80 

0 
90 

0 
100 

0 

no 

0 
120 

* 

Alt. 

D 

Alt 

0 
20 

0 
30 

0 
40 

0 
50 

0 
60 

0 
70 

^0 

0 
90 

0 
100 

0 
110 

1?5 

1 
'    0 

40 

0 

10 

+27 

+18 

+11 

+  6 

+  2 

_  2 

—  6 

—10 

—15 

—20 

0 

55 

0 

10 

+16 

+10 

+  4 

—  1 

—  6 

—11 

—17 

—20 

15 

+21 

+13 

+  7 

+  3 

—  1 

—  5 

-  9 

—13 

-18 

—23 

15 

+20 

+12 

+  6 

+  1 

—  4 

—  9 

-14 

-20 

1 

20 

+27 

+16 

+  9 

+  3 

—  1 

—  4 

—  8 

-12 

-16 

—21 

— 27 

20 

+16 

+  8 

+  3 

2 

—  7 

—12 

-17 

I 

25 

+19 

+10 

+  4 

0 

—  4 

—  7 

-11 

-15 

-19 

—24 

25 

+20 

+11 

+  5 

—  1 

—  5 

—10 

—15 

-20 

30 

+11 

i? 

0 

—  4 

—  7 

—10 

—14 

-17 

-22 

o~ 

30 

+  15 

+  7 

+  1 

—  4 

—  8 

—13 

—17 

35 

+  4 

—  4 

—  7 

—10 

—13 

—16 

-20 

—24 

35 

+20+10 

+  3 

—  2 

—  7 

—11 

-15 

—20 

40 

-  3 

—  5 

—  8 

-10 

—13 

-10 

—19 

—23 

-27 

40 

+13  +  5 

—  1 

—  5 

—  9 

—13 

-18 

45 

-10 

-10 

—11 

—13 

-16 

-18 

—21 

-25 

45 

+  7+  1 

—  4 

—  8 

—12 

—16 

-20 

50 

-IG 

-14 

—15 

—16 

-18 

—20 

—23 

—27 

50 

+  1-4 

—  8 

—11 

—14 

-18 

55 

—22 

-18 

-IS 

—18 

-20 

22 

— 25 

55 

-4-7 

—11 

—13 

—17 

—20 

CO 

-27 

—21 

-20 

—21 

—22 

—24 

—27 

60 

-  9J-11 

—13 

—15 

—18 

Co 

—24 

0.3 

22 

—24 

-26 

65 

-14 

—14 

—15 

—17 

-20 

70 

-27 

-24 

-24 

—25 

27 

70 

—17 

—16 

—17 

—19 

75 

-26 

— 25 

-26 

75 

-20 

—18 

—19 

-20 

80 

-27 

-26 

—27 

80 

—19 

—19 

85 

— 27 

85 

—20 

—20 

i 

90 

07 

90 

1 
146 

10 

+20 

+13 

+  7 

+  3 

—  2 

—  6 

—10 

—15 

—21 

GO 

10 

+17 

+11 

+  5 

—  1 

—  6 

—11 

—17 

j 

15 

+25 

+15 

+  9 

+  4 

0 

—  5 

—  9 

-14 

—19 

—25 

15 

+14 

+  7 

+  2 

—  4 

—  9 

—15 

1 

20 

+19+11 

+  5 

+  1 

—  4 

—  8 

—12 

—17 

—22 

20 

+17 

+10 

+  4 

2 

—  7 

—12 

—17 

25 

+2.5 

+  141+  7 

+  2 

-  3 

—  7 

—11 

—15 

-19 

— 25 

25 

....j+13 

+  6 

0 

—  5 

—10 

—15 

i 

30 

+n 

+  8+3 

—  2 

—  6 

—  9 

—13 

—17 

—22 

30 

+17 

+  9 

+  3 

—  3 

—  8 

—13 

—17 

35 

+10 

+  3 

—  1 

—  5 

—  9 

—12 

—16 

—20 

—25 

35 

+13 

+  5 

—  1 

—  6 

—10 

—15 

40 

+  3 

0 

-5 

—  8 

—11 

-15 

—19 

-23 

40 

+17 

+  8 

+  1 

-4 

—  8 

—13 

—17 

45 

—  4 

-  6 

-9 

—11 

—14 

—17 

—21 

—25 

45 

+11 

+  3 

-2 

-7 

—11 

-15 

50 

-ID 

—10!— 12 

-14 

—17 

-19 

-23 

50 

+  5 

-1-6 

—10 

—13 

—17 

55 

-15 

-HJ-lo 

—17 

—19 

—21 

— 25 

55 

0 

-5-9 

—12 

—16 

CO 

-20 

-17-18 

—19 

—21 

-23 

GO 

-  5 

—  8 

—11 

—14 

-17 

05 

—25 

—20—20 

—21 

—23 

—25 

65 

—  9 

-11 

—13 

—16 

70 

-23-5« 

—22 

-24 

70 

-13 

-13 

-15 

-17 

75 

-25-23 

—23 

■  ■25 

75 

—16 

-15 

—16 

80 

-24 

—24 

80 

—17 

-16 

—17 

85 

.... 

—25 

-2-1 

85 

...  1 

—17 

90 

90 

—17 

50 

10 

+23 

+  18 

+15 
+11 

+  9 
+  5 

+  3 
0 

2 

g 

jj 

IQ 

23 

65 

10 

+11 

+  8 

+  5 

+  2 

1 

—  6 

—  9 

|0 

15 

—  5 

—  9 

-14 

-19 

15 

+15 

—  4 

-15 

20 

.... 

+23;+14j 

+  7 

+  2 

—  3 

—  8 

-12 

—17 

—23 

20 

+11 

+  5 

—  1 

—  7 

-12 

25 

.... 

+17+  9 

+  3 

—  2 

—  6 

-10 

—15 

-20 

25 

+15 

+  7 

+  1 

—  4 

—10 

-15 

30 

+22;+l2i+  5 

0 

—  5 

—  9 

—13 

—17 

—23 

30 

+11 

+  4 

2 

—  7 

—12 

35 

+  15:+  7  +  1 

-  4 

—  8 

—11 

-16 

-20 

35 

+15 

+  7 

+  1 

—  5 

—10 

—15 

40 

+  9+2-3 

-  7 

—10 

—14 

—18 

-23 

40 

+10 

+  3 

—  3 

—  8 

—12 

45 

+  2-.3-6 

-10 

—13 

-16 

—21 

45 

+15 

+  6 

-  1 

-  6 

-10 

—15 

50 

-4-  71-10 

-12 

-15 

-19 

—23 

50 

+  9 

+  2-4] 

-  8 

—13 

55 

-  9  -10;-12 

-15 

-18 

—21 

55 

+  ^i 

—  2 

-7 

—11 

—15 

60 

-Hj-u!  -15 

-17 

—19 

-23 

60 

-  1 

—  5 

-  9 

-13 

65 

-19-17i-17 

-19 

—21 

65 

-  5 

—  8 

-12 

-15 

70 

-23—191-19 

—20 

—23 

70 

-  9 

—11 

-13 

"5 

-21  -21 1 

22 

75 

-12 

—13 

-15 

^■0 

-23 

—22 

-23 

80 

-14 

—14 

85 

—2a 

85 

—15 

—15 

90 

....  -23 

90 

•fo 

0   1    0 
30    40 

0 
50 

0 
60 

0 
70 

0 
80 

0 
90 

0 
100 

0 
110 

0 
120 

0 

20 

0 

30 

0 
40 

0 
50 

0 

60 

7°0 

0 
80 

0 
90 

100 

1?0 

0 
115 

Page  3281 


'se  3281  TABLE  XLIX. 

To  find  the  correction  of  the  apparent  distance  of  the  moon  from  any  planet,  on 
account  of  the  parallax  of  the  planet,  supposing  its  horizontal  parallax  to  be  35". 
This  is  to  be  reduced  to  the  actual  horizontal  parallax  by  means  of  Table  L. 


Apparent  Distance. 


75 


20 


+12 
+ 

—  G 
—10 


+  9- 
8 


30    40    50 


+12 

+  8 

+  3 

0 

—  4 

—  9 
—12 


+  2 


+12 
+  8 

+  1 

-  2 

-  5 

-  8 
-12 


+  9 

+  5 

+  2 

—  1 

4 

—  7| 


+12 
+  8 

+  2 

—  2 

—  5 
7 
9 

—12 


CO    70    80    90 


+  9 
+  5 
+  2 

—  1 

4 

—  7 

—  9 


+12 
+  9 
+  5 
+  2 

—  1 

—  4 

—  7 
—10 
—12 


+ 
+  G 
+  2 

—  1 

—  4 
6 
9 


+  2 

—  1 

—  4 
7 

—  9 
—12 


+  e 

+  3 
0 

—  s 

—  6 

—  9 


—  1 

—  4 

—  7 

—  9 
—12 


0 

—  3 

—  G 

—  9 


100 


Apparent  Distance. 


80 


83 


Alt.    20    30    40   50     60   70    80   90 


+  6 

—  1 

—  G 


+  G 

+  3 

—  1 

—  6 


+  3 
0 


+  6 

+  3 

—  I 

—  3 

—  6 


+  3 
—  1 


+  3 
1 
—  3 


+  3 

0 

—  3 


+ 
+  3 
0 
—  3 


+  3 
0 
—  3 


+  6 

+  3 

0 

3 

6 


+  3 
0 
—  3 


100 


TABLE  L 

To  reduce  the  numbers  in  Table  XLIX.,  so  as  to  correspond  to  the  actual  horizontal  parallax 

of  the  planet. 


J^om 

ontal  Parallax  of  the  Planet. 

a>< 

// 

// 

// 

// 

II 

// 

// 

/,' 

/^ 

// 

II 

II 

II 

II 

// 

// 

// 

II 

// 

// 

II 

// 

II 

// 

// 

// 

// 

// 

// 

II 

^^, 

n 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 

21 

22 

23 

24 

25 

26 

27 

28 

30 

35 

1 

0 

0 

0 

0 

0 

n 

0 

0 

0 

0 

0 

0 

0 

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16 

17 

18 

19 

20 

21 

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23 

24 

26 

30 

30 

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10 

11 

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12 

13 

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17 

18 

19 

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20 

21 

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24 

2,5 

27 

31 

31 

32 

y 

3 

4 

5 

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9 

10 

11 

12 

13 

14 

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17 

18 

10 

20 

21 

oo 

23 

24 

25 

26 

27 

32 

32 

33 

3 

4 

5 

G 

7 

8 

8 

9 

10 

11 

12 

13 

14 

1.5 

IG 

17 

IS 

19 

20 

21 

22 

23 

24 

2.5 

25 

26 

28 

33 

33 

34 

2 

3 

4 

5 

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7 

8 

9 

10 

11 

12 

13 

11 

1,5 

If. 

17 

17 

18 

19 

20 

21 

oo 

23 

24 

25 

26 

27 

29 

34  34 

35 

2 

3 

4 

6 

G 

7 

8 

9 

10 

11 

12 

13 

14 

1.5 

16 

17 

18 

19 

20- 

21 

22 

23 

24 

25 

26 

£7 

2b 

30. 

35  35 

TABLE   LI. 

TABLE 

LIL 

[f 

age.  329 

To 

change    mean 

solar 

time 

into 

To 

change  sideral 

time 

into  mean 

sideral  time. 

solar  time. 

Solar 

Solar 

Sidcr.il 

Sideral 

Add. 

Min- 

Add. 

Sec- 

Add. 

Sideral 
Hours. 

Subtract. 

iMin- 

Subtract 

See- 

Subtract 

utes. 

onds. 

utps. 

on  Is. 

M.      S. 

s. 

s. 

M.     S. 

S. 

s. 

1 

0      9.9 

I 

0.2 

I 

0.0 

I 

0     9.8 

I 

0.2 

I 

0.0 

2 

0  19.7 

2 

0.3 

2 

0.0 

2 

0    19.7 

2 

0.3 

2 

0.0 

3 

0  29.6 

3 

0.5 

3 

0.0 

3 

0    29.5 

3 

0.5 

3 

0.0 

4 

0  39.4 

4 

0.7 

4 

0.0 

4 

0  39.3 

4 

0.7 

4 

0.0 

5 

0  49.3 

5 

0.8 

5 

0,0 

5 

0   49-1 

5 

0.8 

5 

0.0 

6 

0  59. 1 

6 

1 .0 

6 

0.0 

6 

0    59.0 

6 

1 .0 

b 

0.0 

7 

I     9.0 

7 

1 .2 

7 

0.0 

7 

I     8.8 

7 

1 .1 

7 

0.0 

8 

I  IS.9 

8 

1.3 

8 

0.0 

8 

I   18.6 

8 

1.3 

8 

0.0 

9 

I  28.7 

9 

1.5 

9 

0,0 

9 

I   28.5 

9 

1.5 

9 

0.0 

10 

I  38.6 

10 

1.6 

10 

0.0 

10 

I  38.3 

10 

1.6 

10 

0.0 

II 

I  48.4 

11 

1.8 

II 

0.0 

II 

I  48.1 

1 1 

1.8 

II 

0.0 

12 

I  58.3 

12 

2.0 

12 

0.0 

12 

I  58.0 

12 

2.0 

12 

0.0 

i3 

2     8.1 

i3 

2.1 

i3 

0.0 

i3 

2    7.8 

i3 

2.1 

i3 

0.0 

i4 

2  18.0 

i4 

2.3 

14 

0.0 

i4 

2  17.6 

i4 

2.3 

i4 

0.0 

i5 

2  27.8 

i5 

2.5 

i5 

0.0 

i5 

2  27.4 

i5 

2.5 

i5 

0.0 

i6 

2  37.7 

16 

2.6 

16 

0.0 

16 

2  37.3 

16 

2.6 

16 

0.0 

17 

2  47-6 

17 

2.8 

17 

0.0 

17 

2  47-1 

17 

2.8 

17 

0.0 

i8 

2  57.4 

18 

3.0 

18 

0.0 

18 

2  56.9 

18 

2.9 

18 

0.0 

19 

3     7.3 

19 

3.1 

19 

0.1 

'9 

3    6.8 

19 

3.1 

19 

O.I 

20 

3  17. 1 

20 

3.3 

20 

O.I 

20 

3  16.6 

20 

3.3 

20 

O.I 

21 

3  27.0 

21 

3.5 

21 

0.  I 

21 

3  26.4 

21 

3.4 

21 

O.I 

22 

3  36.8 

22 

3.6 

22 

0.  I 

22 

3  36.2 

22 

3.6 

22 

O.I 

23 

3  46.7 

23 

3.8 

23 

O.I 

23 

3  46.1 

23 

3.8 

23 

O.I 

24 

3  56.6 

24 

3.9 

24 

O.I 

24 

3  55.9 

24 

3.9 

24 

25 

O.I 

25 

4.1 

25 

O.I 

25 

4.1 

«0.I 

26 

4.3 

26 

0.  I 

26 

4.3 

26 

O.I 

27 

4.4 

27 

O.I 

27 

4.4 

27 

0.1 

28 

4.6 

28 

O.I 

28 

4.6 

28 

O.I 

29 

4.8 

29 

O.I 

29 

4.8 

29 

O.I 

3o 

4.9 

3o 

O.I 

3o 

4.9 

3o 
3i 

O.I 

3 1 

5.1 

3i 

O.I 

3i 

5.1 

O.I 

32 

5.3 

32 

O.I 

32 

5.2 

32 

O.I 

33 

5.4 

33 

O.I 

33 

5.4 

33 

O.I 

34 

5.6 

34 

O.I 

34 

5.6 

34 

O.I 

35 

5.3 

35 

0.  I 

35 

3.7 

35 

O.I 

36 

5.9 

36 

O.I 

36 

5.9 

36 

O.I 

37 

6.1 

37 

O.I 

37 

6.1 

37 

O.I 

38 

6.2 

38 

O.I 

38 

6.2 

38 

O.I 

39 

6.4 

39 

O.I 

39 

6.4 

39 

O.I 

40 

6.6 

4o 

0.  I 

4o 

6.6 

40 

O.I 

4i 

6.7 

4i 

O.I 

4i 

6.7 

4i 

O.I 

42 

6.9 

7-1 

42 

O.I 

42 

6.9 

42 

O.I 

43 

43 

O.I 

43 

7.0 

43 

O.I 

44 

7.2 

44 

O.I 

44 

7.2 

44 

O.I 

45 

7-4 

45 

O.I 

45 

7.4 

4b 

O.I 

46 

7.6 

46 

O.I 

46 

7.5 

46 

O.I 

47 

7-7 

47 

O.I 

47 

7-7 

47 

O.I 

48 

7-9 

48 

O.I 

48 
49 

7-9 

48 

O.I 

49 

8.1 

49 

O.I 

8.0 

49 

O.I 

5o 

8.2 

5o 

O.I 

5o 

8.2 

bo 

0.  I 

5i 

8.4 

5i 

O.I 

5i 

8.4 

5i 

O.I 

52 

8.5 

52 

O.I 

52 

8.5 

52 

O.I 

53 

8.7 

53 

0.  I 

53 

8.7 

53 

O.I 

54 

8.9 

54 

O.I 

54 

8.8 

54 

O.I 

55 

9.0 

55 

0.2 

55 

9.0 

55 

0.2 

56 

9.2 

56 

0.2 

56 

9.2 

56 

0.2 

57 

9-4 

57 

0.2 

57 

9.3 

57 

0.2- 

58 

p. 5 

58 

0.2 

5?> 

9.5 

58 

0.2 

59 

9-7 

59 

0.2 

59 

9-Z 

59 

0    2 

60 

9.9 

60 

0.2 

60 

9.8 

60 

0.2 

Page  330] 

TABLE  LIII. 

This  table  g 

ives 

the  Variation  of  the  Compass  for  1858,  very 

nearly 

as 

in 

the  chart  of  F.  J.  Evans,  master,  R.  N. 

T3 

West  Longitude. 

T3 

o 

0 

0 

0 

0 

0 

0 

0 

0 

0    1    0       0   1    0       0 

0 

0 

° 

0 

0 

O 

180 

170 

160 

1.50 

140 

130 

120 

110 

100 

90  1  80     70  j  60     .50 

40, 

30 

20 

10 

0 

I 

Variation  of  the  Compass. 

o 

o 

0 

0 

0 

0 

0 

0 

0 

0    1    0    1    0   1   0    1    0 

0 

0 

0 

0 

0 

0 

boN 

i5N 

20E 

25^57 

30^33^35^ 

19^ 

7Tr39TF 

53  1^1561^55  TP' 

5iTF 

45  F 

3qTF 

3ir 

24  TF 

6oiV 

58 

i5 

19 

24 

28 

3i 

32 

oTr3i 

45  1^" 

5l      :5l 

48 

43 

37 

3o 

24 

58 

56 

U 

i8 

22 

26 

28 

29 

2E  22 

46   47 

45 

41 

35 

29 

23 

56 

54 

U 

i8 

21 

24 

27 

27 

2bE 

17 

62  W 

42     44 

43 

40 

34 

28 

22 

34 

D2 

i3 

17 

20 

23 

23 

25 

24 

12 

27 

36     41 

41 

38 

33 

27 

22 

32 

5o 

1 3 

•7 

19 

22 

23 

23 

22 

8TF 

22 

3i 

37 

38 

36 

3i 

26 

21 

5o 

48 

.3 

i6 

IP 

20 

22 

22 

21 

No  rth 

18 

27 

33 

36 

34 

3o 

25 

20 

48 

46 

i3 

i6 

i8 

19 

20 

21 

14 

Ame  rica. 

i3 

24 

3o 

Si 

32 

29 

24 

20 

46 

44 

i3 

i5 

17 

18 

19 

19 

18 

i3 

21 

27 

3o 

3o 

27 

23 

19 

44 

42 

1 3 

ID 

17 

17 

18 

18 

17 

II 

18     |25 

28 

28 

26 

23 

i9_ 

19 

42 

40 

i3 

i5 

16 

17 

17 

17 

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oE 

9 

16 

22 

26 

27 

25 

22 

40 

;i8 

i3 

i4 

i5 

16   , 

16 

16 

I 

8 

14 

20 

24 

26 

23 

22 

17 

38 

36 

i3 

14 

14 

i5 

l5 

i5 

i5 

I 

6 

12 

18 

22 

24 

24 

21 

I7TF 

36 

34 

i3 

14 

14 

i4 

14 

14 

14 

i3E 

tiE 

2 

4 

II 

16 

21 

23 

23 

21 

34 

32 

i3 

i3 

i3 

i3 

i3 

ij 

i3 

12 

8 

3 

3 

9 

14 

19 

22 

22 

20 

32 

3o 

12 

i3 

12 

12 

12 

12 

12 

u      io£ 

8 

3 

2 

8 

i3 

18 

21 

22 

20 

3o 

28 

12 

12 

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II 

II 

II 

II 

II      10 

8 

4 

iW 

6 

12 

16 

20 

21 

20  IF 

28 

26 

12 

12 

II 

II 

II 

11 

10 

10       9 

7 

4 

0 

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10 

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19 

20 

26 

24 

12 

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10 

10 

II 

10 

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10 

9 

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b  ■ 

lE 

4 

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19 

20 

24 

22 

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10 

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17 

20 

Ajri 

ca. 

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7       5 

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21 

26 

24 

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10 

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38 

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5 

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17 

23 

38 

40 

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10 

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10 

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18     119 

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2 

4 

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17 

22 

40 

42 

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12 

10 

10 

10 

11 

i3 

16 

19     ,20 

18 

14 

9 

2 

4 

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16 

22 

42 

44 

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i4 

12 

II 

10 

10 

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i3     !i7 

20     20 

18 

14 

9 

3 

3 

9 

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21 

44 

46 

i6 

14 

i3 

II 

II 

II 

12 

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21     ,21 

19 

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10 

4 

2 

9 

14 

20 

46 

48 

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i5 

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12 

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21      22 

20 

16 

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5 

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8 

14 

20 

48 

5o 

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i6 

14 

i3 

12 

12 

i3 

16     20 

22      ,23 

21 

16       12 

6 

iTF 

7 

i3 

19 

5o 

52 

i6 

i5 

i3 

i3 

i3 

14 

17      21 

18  ;22 

23     ;23 

21 

17        12 

7 

0 

6 

12 

18 

62 

54 

i8 

>7 

i5 

14 

i3 

i3 

i5 

24    ,24 

22 

18 

i3 

8 

lE 

5 

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17 

D4 

56 

'9 

•7 

16 

i5 

14 

14 

16 

19      23 

25       25 

23 

19 

14 

9 

2 

4 

10 

16 

56 

58 

20 

i8 

17 

16 

rb 

16 

17 

21        25 

26     26 

24 

21       'l5 

10 

4 

3 

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ID 

38 

6o,S' 

21^- 

jgE 

18^ 

l■^E 

11 U 

1 7  A' 

19^^ 

23E  26E, 2-] E  21  E'7bE 

21^  ibE 

11^ 

4E 

3Tr 

gW 

iDir 

6o^' 

Hi 

ISO'^ 

170^ 

160"^ 

■150'-' 

140*^ 

130-^ 

120'^ 

110'^  100'^.90^   SO'^  170^^  loO'^   50^ 

40^^ 

30" 

30*^ 

10^ 

0^ 

1 

West  Longitude. 

Page  331]                                                       TABLE    LIIL 

This  table  gives  the  Variation  of  the  Compass 

for  1858,  very 

nearly 

as 

n 

the  chart  of  F.  J.  Evans,  master,  R.  N. 

■o 

East  Longitude. 

" 

8 

O     1     o 

0       010        010       0    1    0 

0   1    0    1    0 

0 

0 

0       0 

0 

0 

0 

'1 

0  1  10 

20     30  1  40     50  1  60     70  1  80 

90  1 100 1 110 

120 

130 

140   150 

160 

170 

180 

Variation  of  the  Compass. 

o 

O    ]    o    1    o 

0 

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24 

17 

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10 

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58 

56 

23 

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3 

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10 

14 

56 

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22 

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2 

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35 

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34 

29 

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12 

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54 

17 

23 

20 

33 

36 

36 

37 

37 

38     36 

32 

24 

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3 

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19 

18 

54 

56 

16 

22 

28 

33 

36 

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39 

40     39 

34 

26 

4 

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14 

18 

20 

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58 

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0*^  110"  20"^   30-" 

37  Tf  39  W'ai  TF,45  Tr  47  ir  46  Tf;42  W\ 
40'-'  ,o0>^   60*^  ;?0"    SO"   00"  |100"110" 

7F 

bE  \5E 
140"  150" 

7lE 

160" 

2  2  A' 
170" 

21A' 

180" 

6o^' 

^ 

120*^  130" 

Hi 

1^ 

East 

Lo> 

roiTt 

roE. 

Page  332] 


TABLE   LIV. 

Latitudes  and  Longitudes. 


This  table  contains  the  Latitudes  and  Longitudes  of  the  most  remarkable  Harbors,  Islands, 
Shoals,  Capes,  &c.,  in  the  world,  founded  on  the  latest  and  most  accurate  astronomical  observa- 
tions, surveys,  and  charts. 

The  longitudes  are  reckoned  from  the  meridian  of  Greenwich. 


I.  Coast  of  the  United  States  of  America. 


ENTRANCE  of  St.  Croix 
River 

Island    of    Campo    Bello, 
(N.  point',) 

Wolf    Islands,   (northern- 
most,)   

Quoddy  Head  light 

Grand  Manan,  N.  E.  head 

S.  W.  head 

Libby     Island    lirfit,    ( 
trance  Machias  Bay . . 


Titmanan  light 

Mount  Desert  Rock, (light 

house,) 

Isle  au  Haut 

Castine  Fort, 

Matinicus  Island  lights  . , 

Wooden  Bald  Rock 

Manhegan  Island  light. . , 
Penmaquid  Point  light. .  . , 

Bantum  Ledge 

Seguin  Island  light 

Brunswick   College 

Cape  Small-Point 

Cashe's    Ledge,    (shoalest 

part  26  feet,) 

PORTLAND  light-house 
Cape  Elizabeth,  (W.  light) 
Wood  Island  light,  en- 
trance Saco  River.... 
Agamenticus  Hills, Tri.Pt. 

Cape  Porpoise 

Bald  Head   

Cape  Neddock  Nubble... 

Boon  Island  light 

PORTSMOUTH^''i?lh'r'"'( 
Isle     of    Shoals,     (White 

Island    light,) 

Portsmouth,  Ft.  Constitution 

Great  Boar's   Head 

NEWBURYPORT  W.  It. 

on  Plum  Island 42 

Ipswicli  entrance  light...  42 

Squam    light 42 

CAPE    ANN   (Thatcher's 

laland)  N,  light 

Eastern   pomt    Cape  Ann 

Harbor  light 

Liglit-house     on     Baker's 

Island 

Beverly  Spire 

SALEM,  Tall  spire 

Mnrblehoad  Black-top  ch'h 
Nahant  Point,  N.  E.  point 

of  Boston  Harbor,  Hotel 

Boston  light-housef 

BOSTON,  State-House.. 
Cambridge,  Observatory.. 

Scituate   light 

Plymouth  lights,  south.  .. 
Rac(^  Point  lin-ht 


Lat.        Loner. 


D.  M. 


45  00  N 


57 

57.5 
47-5 
45 
34 

32.5 

22 

58 
59 
22.5 

47-1 
5o 

44 


56 

37-4 
33.8 

27.4 
i3.4 
2(  -4 

l3'2 
10 

07 -3 
o3.5 

58 

o4.5 

55.1 

48.4 
4t.i 
39.7 


42  38.3 
42  34.8 


32.2 
43.0 
3l.2 

3o.4 


D.  M. 


02W 

55 

43 
58 
45 
53 


52 

08.5 

34.5 

48.5 

5i 

46 

j5 

29 

35 

45 

57 


2 

4 
50.4 

5i.5 
12.2 
II. 8 

19.4 
4i  .2 

25-2 

34-4 

35 

28 


6 
3 
4i-5 

37.5 
42.2 

47-4 

48.8 
45.8 
4o.6 


70  34.2 
70  39.5 


25.1  70 
19.6  70 


46.8 
52.4 
53.6 
5o.5 

54.0 
53.1 
o3.5 
07.4 
42.6 
35.7 
14.3 


CAPE  COD  light... 

Chatham  South  light.. 

Monomoy  Point  light 

Shoal  of  George's. 

Great  Shoal,  S.  E.  point 

N.W.  point 

From 


Western  Shoals 


To 


North  Shoal.. 

Third  Shoal 

East  Shoal 

NANTUCKET    light, 

(Great    Point,) 

Sancoty  Head 

Nantucket  South  Shoal* . . 
Cape  Poge,  (Vineyard,)Lt 
Cutterhunk  Island  liglit.. 
Gay-Head  light-house. . . . 
Noman's  Land  Triang  Pt. 
New  Bedford  Court-House 
liirht-house. 


Seaconnet  Point 

NEWPORT,  Spire 

Rhode  Island  light-house, 

(Beaver  Tail  light,) 

Providence  Baptist  Church 

Point  Judith  light 

Block  Island  light 

S.  E.  point.. 


Watch  Hill  light-house 
Little  Gull  Island  light. . . 
New  London  light-house. 
MONTAUK  POINT  (E. 
end  Long  Island)  light-h. 
Falkner's  Island  light. . . . 
NEW  HAVEN  light.... 

Stratford  Point  light 

Old  Field  Point  light 

Eaton's  Point  light 

NEW  YORK,  City  Hall. 

Sandy  Hook  light 

Neversink  lights 

Barnegat  l>ight  Ho 

Great  Egg  Harbor 

Cape  May  light 

Cape  Henlopen  light-house 

Egg  Island  light 

PHILADELPHIA  St.  Ho 

Smith's  Island  light 

Cape   C^harlcs 

Cape  Henry  light 

Norfolk 

Old  Point  Comfort 

Yorktown 

Petersburgh 

RICHMOND,  Capitol... 
WASHINGTON  City... 
BALTIMORE  W.  Ml.... 

Annapolis,  Md.  St.  Ho 

Currituck   Inlet 

CAPE  HATTERAS.... 
Deep  soundings  off  ditto.. 
Ocracock  Inlet 


LMt. 


D.  M. 

42  02. 


40.2 
33.5 

33 

44.7 
39.6 
35.9 
46.4 
46.5 
43 

23.4 
17.0 
04.2 

25.2 
24.8 
20.9 
l5.2 

38.1 
35.5 
27 
29.2 

26.9 
49.6 
21.6 
i3.4 


12.3 
19.0 


Lonar. 


41  i4. 
41  09, 
40  58, 
40  57 
40  42. 
40  27, 
40  23 
39  46. 
39  19 
38  55, 

38  46 

39  10. 
39  56. 
37  07. 

37  07. 
36  55. 

36  5o. 

37  00. 
37  i3 
37  i4 

37  32- 

38  53. 

39  '7- 
38  58, 
36  23 
35  i5. 
35  06 
35  06 


D.  M. 
70  o3.3 
69  56.6 

69  59.3 

67  39 
67  47-5 
67  49.4 
67  53 
67  48 
67  28.2 
67  22.2 

70  02.4 
69  57.6 

69  5 1 .4 

70  26.7 
70  56.7 
70  49. 8 
70  48.5 
70  56.2 

70  53.7 

71  i3.5 
71  18.5 

71  23.6 
71  24.2 
71  28.6 
71  34.2 
71  34 

71  5r  .2 

72  06.1 
72  o5.i 

71  51.1 

72  38.9 

72  53.9 

73  o5.9 
73  06.8 

73  23.4 

74  00. I 
73  59.8 

73  58.8 

74  06.0 
74  35 

74  57.3 

75  o4-7 
75  08.0 
75  08.7 

75  52.2 

75  57.9 

76  00. 2 
76  17 
76  18. I 

76  34 

77  24 
77  25-8 
77  CO. 2 
76  36.6 
76  29.1 
75  55 
75  30.9 

5  75  58.9 


*  New  South  Slinal.  4ii"  .""i 
+  Minot's  l.eilee  Light,  S.  E.  ] 


»'  0  N..   0!)°  51' 5  W. 
L  R,  iVom  Roston  Light. 


TABLE  LIV. 

Latitudes  and  Longitudes. 


[Page  333 


CAPE    LOOKOUT,  Lt.. 
Deep  soundings  off  do.. . , 

Old  Topsail  Inlet 

Beaufort 

VViliuington 

Brunswick 

Siiiithville 

New  Inlet  South  point. . . 
CAFE  FE  All  Bald  lid  Lt 
Deep  soundings  off  do..  . . 
GEORGETOWN  Clmrcli 
light-house 


.'ape  Roman 

;harleston, 

Pinckney,) 

lioht-liouse 


(Fort 


North  Edisto  River 

BEAUFORT,  (S.C.).... 

Port  Royal  Entrance, 

Tybee  light 

SAVAiNNAH  Exchange. . 

St.      Catharine's      Island, 
North  point 

Sapello  Bar  St.  Cath,  Is.. 

Doboy  Bar    Light 

Light  on  St.  Sunon's   Isl- 
and, S.  point 

Brunswick 

St.  Andrew's  Bar 

Cumberland    Island 

S.  point 

Amelia  Island,  S.  pt 


Lot. 


River  St.  John's  Light. 
St.  Augustine   light-house 
Cape  Carnaveral  Light. 
Breakers,  S.  E.  point  . . 
I^as  Tortulas,orHummocks 
iliUaborouii-li  Island,  N.  p. 

S.  p. 

Mount    Pelado,    or    Bald 

Head 

Grenville's  Inlet 

Cooper's  Hill 

Sand  Hills 

Neu-  Inlet 

Middle  River 

CAPE   FLORIDA  light. 

Carry.sfort  Light 

Key  Tavcrnier 

Old  Metacumbe,  S.  W.  pt. 

Cayo  Sombrero 

Looe  Key 

Samboes  Keys,  (eastern,). 

Key  West,  Light 

Sand  Key,  Light  (old).. . . 
Tortu^as       Islands       and 

Banlcs,  N.  E.  point 

N.W.  point 

S.   E.  point 

S.  W.  point 


Bush  Key  light 

Key  Vacas 

Key  Axi 

Cape  Sable 

Cape  Romano 

Boca     Grande,     entrance 
Bay   ('arlos 


D.  M. 

34  37. 
34  28 
34  4i 
34  43 
34  i4 

34  02 
33  54 

35  4r. 
33  52. 
33  35 
33  22 
33  i3. 
33  01. 

32  46. 
32  4i. 
32  33. 
32  27 
32  o4. 

32°OI . 

32  o5 

3i  4i. 
3i  3i 
3i  21 


J I  07 
3 1  06 
3o  53 


Lona-. 


3o  43 
3o  3o 

3o  20. 
29  52. 
28  28 
28  24 

27  35 
27  32 

27  i4 

27  01 
26  47 
26  42 
26  32 

26  18 
26  08 

25  4o 

5  i3. 

24  59 

24  52 

24  37 
24  34 

24  29 
24  33. 
24  27, 

24  4i 
24  4o 
■j4  33. 
24  3i 
24  36- 
24  42 

24  57 

25  06 

25  5i 

26  43 


D.  M. 

76  30.7 

76  4o 

76  4o 

77  58 

77  58 

78  01 
75  28.5 
77  59-8 

79  '8 
79  IO-7 
79  22  2 

79  54.4 

79  52.5 

80  10.7 
80  4o 
80  37.7 

80  5o-6 

81  o5.2 

8i  II 
81  i3 
81  18,6 

26 
Bi  3i 
81  20 

81  28 
81  26 


24.5 

20 

34 

3o 

3o 


1 1 

02 

o3 

o3 

00 

00 

09-4 

12.7 

3o 

4i 

07 

24 

40 


li  52-7 


82  23 


Tampa  Bay,  Egmont  Kej 

Anclote  Keys 

St.  Mark's  light-house . 

South-west  Cape 

Dog  Island  light 

Cape  St.  George,  Light. . . 

Cape  St.  Bias 

St.  Joseph's  Bay,  entrance 
St.    Andrew's      Bay,     en- 
trance Main  Pass. .... 
St.  Rosa's  Bay,  entrance 
PENSACOLA,  town.., 
liffht..., 


Mobile  Point,  light. 

bar,  outer 


MOBILE  Barton's  Acad'y 
Massacre  Island,  W.  pt. .. 

Ship  Island,  W.  Light 

Chandeleur      Islands,    N. 

point,  Liglit 

S.  pt.  Palos  Island 


Key  Breton,  N.  E.  pt 
MISSISSIPPI  River,  Pass 

a  rOutre    

Balize 

S.  E.  Pass.. 

S.  Pass  .... 

S.  W.  Pass. 


NEW   ORLEANS 
Barataria 

Bayou  la  Fourche 

Timbalier  Island,  (Tonba- 

lier,)  N.  W.  point 

Racoon  point 

Bayou    Decartes,  entrance 

Point  au  For  Light 

Rabbit  Island 

Sabine  River,  entrance. .. 
Galveston,  entrance 


Lat. 


Long. 


M.      D.  M. 

36N82  45VV 


17.5 
o4.5 

52 

46 
35 

39.6 
5 1 

o3 
24 
25 
21 
i3.8 


09 
41.4 
12 
12.9 

o3 
44 
29 

i4 

08.5 

06 

59.7 

58.5 

57.5 

17.5 

06 

o5 

o3 

10 

19.5 

29 

40.6 

20.5 


82  54.3 
84  10.6 
84  22 

84  34 
84°58.5 

85  16 
85  23 

85  37.7 

86  3 1 

87  11-5 

87  16.9 
00.5 
01 
01 . 
22 
57.0 

5i 

88  5i 

89  07 

89  00 

01 .4 

57 
89  07.4 

89  20 

90  00 
90  10 
90  09 

90  23 

90  57 

91  o4 
91  20 
91  36 

93  49 

94  45 


II.  Islands  in  the  West  Indies. 


TRINIDAD 

Spanish  Town, 

(fort,) 


Icacque    Point 

Pomt  Galiote 

Point  Galera 

Tobago,  N.  E.  point 

S.  W.  point 

Grenada,    Point    Salinus, 

S.  W.pt , 

Grenada  Bank, 

Barbadoes,  S.  point , 

Engineers'  wharf, 

-  N.  point. 


St.  Vincent's,  Kingston 
S.  point  . . 


St.  Lucia,  T^T,  point 
S.  point 


Martinico,  S.  point 

Diamond  Rock 

Port  Royal . 


—  Macouba  Pt. 


Dominica,  Roseau  . 
N.  point 


Lat. 


D.  M. 

39  N 
o4 


0  5o 

1  20 

1  06 

2  00 
I  55 

3  o3 
3  M 
3  20 
3  12 


3 

4 
3 
4 
4 
4 
4 
5  18 
5  38 


09 

06 

4i 

27 

26.6 

36 

55 


Loner. 


D.  M. 


57 
00 
56 

27 
46 

49 

57 

37 

38 

4r 

16 

14 

57 

54 

55 

02.7 

o4.2 

09 

25 

26 


Page  334] 


TABLE  LIV. 

Latitudes  and  Longitudes. 


The  Saint's  Island,  W.  pt. 

Mariegalante,  S.  point  . 

Guadaloupe,  S.  W.  pt.  . 

N.  W.  pt.  . 

N.  E.  pt.  .. 

Point  Chateau, 

S.  £.  pt 

Deseada, 

Antigua,  E.  point 

Fort  James 

Montserrat,'N.  E.  point  .. 

Redondo  Island 

Nevis,  Charlestown 

St.    Christopher's     or    St. 

Kitt's,  N.  point 

Basse  Terre 

St.  Eustatia,  Town 

Saba 

Aves  or  Birds'  Island  .... 

Barbuda,  N.  pt 

St.  Bartholomew,  S.  pt.  . . 

St.  Martin's,  Marigot  Fort 

Anguilla,  S.  W.  pt 

Anguilleta,  N.  E.  pt 

Prickly  Pear 

Sombrero 

St.  Croix  or  St.  Cruz,  ob- 
servatory   •■ 

S.  W.  pt.  ._ 

Anegada,  S.  point  of  shoal 
W.  point 

Virgin  Gorda,  E.  pt 

Tortola,  E.  point 

W.  point 

St.  John's 

St.  Thomas,  Fort  Christian 

Bird  Key 

Serpent  Island,  E.  part. . . 

Crab  Island,  E.  part 


Cape  St.  John,  or  N.  E.  pt. 

PORTO  RICO,  St.  Au- 
gustine's Battery,  west- 
ern turret  

Point  Bruquen,  or  N.  W.  p. 

Point  St.  Francisco 

Cape  Roxo,  or  S.  W.  point 

Caxa  de  los  Muertos 

Point  Coamo 

Cape  Mala  Pasqua,  or  S. 
E.  pt 


Mona  Island,  E.  pt 

Monito  Island 

Zacheo  or  Dessecho  Isl.. 


Cape  Engano  

Saona  Island,  E.  pt 

St.  Catherine's  Island.., 
St.  Domingo,  Light. . . . . 

La  Catalina 

Cape  Beata 

AltaA'ela  Rock , 

Cape  Jaquemel , 

Island  Vacca  (a  Vache) 

Point  Gravois 

Cape  Tiberon 


Lat. 


D.  M 

5  5iN 
5  52 
5  57 


7  24 
7  17-7 
7  29 
7  4i 
5  40 
7  47 

7  53.5 

8  o5 
8  10 
8  18 
8  20 
8  38 

7  44.5 

7  42 

8  32 
8  44 
8  3o 
8  27 
8  25 
8  18 
8  21 
8  i5 
8  19 
8  10 

8  24 


8  29 
8  3[ 

8    21 

7  57 
7  5o 
7  55 

7  59 

8  07 

3     !I 

S  24 

8  35 
8  12 
8  18 
8  28 
8  08 
7  39 

7  28 

8  10 
8  o4 
8  ot 
8  20 


Lous'. 


M 

38W 
18 
44 
5i 

29 


06 
45 

52 

12 

25 

37.9 

5o 

42.2 

00 

i4 

39 

02 

56-9 

o3 

i3 

58 

23 

27.4 

40.7 

48 

i3 


65  52 


67  2- 


68  20 
3o 
00 

52. 


Navaza  Island 

Cape  Donna  Maria 

Jeremie 

Caymito 

Petit  Guave 

Leogano 

PORT-AU-PRINCE  . . . . 
Isle  Gonave,  S.  E.  part  . . 

N.  W.  part. . 

Point  St.  Mark 

St.  Nicola  Mole 

Tortugas,  E.  point 

CAPE  HAYTI  CITY,.. 
Shoal  off  Monte  Christe . . 

Monte  Christe 

Grange  Point 

Point  Isabella 

Old  Cape  Franqois 

Cape  Samana 

Cape  Raphael 

Morant,  E.  point 

KINGSTON 

Port  Royal,  Fort  Charles  . 

Portland  Point 

Pedro  Bluffs 

Black  River 

Savannah  la  Mar 

Cape  Negril,  S.  point 

N.  point.... 

Montego  Bay 

Falmouth ; . 

St.  Ann's 

Port  Maria 

Arnatta  Bay 

N.  E.  point 


Morant  Keys,  or  Las  Panas 
Pedro  Shoals. 

Portland  R.,  N.  E.  p. 

South  Key 

Rock  5  feet  above  water. . 

N.  pt.  Pedro  Shoal 

Formigas  Shoal,  N.  E.  p.. 
S.    W.  p. 


Little  Cayman,  S.  W.  p. 

Caymanbrack,  E.  p 

Grand  Cayman,  E.  point , 

Fort  George,  W.  end 

Swan  Islands,  E.  pt 

New  shoal,  (Sandy  Key,) . 


Cape  Mayze 

Port  Negra 

Point,    entr.    Cumberland 

Harbor    

ST.   JAGO  DE    CUBA, 

Light...... 

Tarquin's  Peak 

Cape  Cruz 

Manzanillo 

Key  Breton 

Trinidad  River 

Bay  Xagua.  River  Vigia . . 
Stone  Keys 


Lat. 


D.  M. 

18  24  N 

18  37 

18  37 

18  39 

18  24 

18  3o 

18  33 

18  4o 

18  57 

19  02 

19  49-5 

20  02 

19  46.4 

20  0^ 

19  54 

19  54 

19  59 

19  4o 

19  10.2 

19  o4 

18 


56 

58 

56 

43 

52.5 

02 

12.3 

i5 

22.5 


18  29 
18  28 
18  27 
18  22 
18  16 
18  09 

17  25 

17  07 
16  57 

16  48 

17  36 

18  35 

18  27 

19  36 
19  44 
19  20 
19  i4 

17  25 
5  52 


20  1 5 
20  06 

19  54 

19  58 

20  02 

19  5o 

20  20 
4 

43 
22  02 
5-- 


Long. 


M. 
00  VV 

23 

o3 

43 

5o 

33 

16.3 

45 

i3 

47 

61 
II  . 

42 

34 

36 

10.5 

53 

i5. 

52 


46 

5o.5 

II 

45 

5i 

08. 

24 

23 

56 
4i 
i5 
54 
45 
20.5 


75  59 

,577  28 

77  53 
5 

78  54 

75  5o 

76  00 


06 

74  i3 

75  16 

75  52 

76  5i 

77  45 
77  II 

79  22 

80  00 

80  42 

81  1 5' 


TABLE   LIV. 

Latitudes  and  Loniiitudes. 


[Page  335 


Los  Jardinellos,  S.  E.  point 
of  the  Bank 

Canal  del  Rosario 

Isle  Fines,  E.  pt 

S.  W.  pt 

Indian  Keys,  N.  AV.  pt.  .. 

Key  St.  Philip,  E.  pt 

Point  Piedras 

Cape  Corrientes 

Cape  St.  Antonio 

Saiicho  Pedro  Shoal 

Shoal  discovered  in  1797  . 

Los  Colorados,  S.  W.  pt.  . 

N.  E.  pt... 

Hill  Guajibon 

Bay  Honda 

Port  Cabanas 

Mariel 

HAV^ANA  (the  More)... 

Cape  Escondido 

Point  Guanos 

Pan  of  Matanzas 

MATANZAS  

Point  Yeacos 

Stone  Key  off  do 

Key  Cruz  del  Padre 

Las  Cabezas 

Nicholas  Shoal 

Key  Verde 

Key  Carenero 

Key  Francis,  E.  p 

Key  William,  northern . . . 

Pt.  St.  Juan 

Centre  of  Key  Coco,  S. 
side  Bahama  channel  . . 

Key  Point  Paredon,  do.  . . 

The  Barrel 

Cayo  Confites 

Cayo  or  Key  Verde 

Guajara,  N.  W.  pt 

Point  INIaternillos 

Neuvitas 

Point  de  Mulas 

Tanamo 

Key  iNIoa 

Point  Guarico 

Baracoa  

N.  pt.  Nativity  Bank,  or 
E.  Reef. 

Superb  Shoal 

Silver  Key,  S.  E.  end .... 

iN.  E.  do 

. N.  do 

Square  HandkercJiief 

N.  E.  point 

S.  E.  point 

— S.  W.  point 

Turk's  Island,  N.  p.  Grand 
Turk 

Salt  Key 

• —  Sand  Key   

Endymion's  Rocks. 

Great  Caycos  Isl.,  Swim- 
mer's Shoal 

N.  E.  p.,  or  Shoal  St. 

Philip    

N.  W.  part 

*  See  Prefaf'f!. 


Lat. 

Long. 

D.  M. 

D.  M. 

21  3iN 

81  17W 

21  35 

81  5i 

21  4o 

82  23 

21  22 

83  00 

21  52 

83  i3 

21  58 

83  22 

22  01 

83  55 

21  44 

84  33 

21  5o 

84  59 

22  OI 

85  02 

22  06 

85  02 

22  09 

84  48 

23  00 

83  08 

22  48 

83  24 

23  01 

83  i3 

23  02.5 

82  59 

23  o3 

82  47 

23  09 

82  22 

23  08 

81  5i 

23  08 

81  44 

23  02 

81  46 

23  o3 

81  4o 

23  i3 

81  10 

23  i4.5 

81  07 

23  18 

80  55 

23  16 

80  36 

23  i4 

80  19 

23  09 

80  i4 

22  52 

79  49 

22  4o 

79  '3 

22  34 

78  45 

22  19 

78  57 

22  29 

78  20 

22  3o 

78  o5 

22  25 

77  56 

22  II 

77  4i 

2  2  06 

77  38 

21  55 

77  3o 

21  4i 

77  08 

21  36 

77  06 

2  1  o5 

75  3i 

20  44-5 

75  12.2 

20  43 

74  47 

20  39 

74  4i 

20  21 

74  24 

20  12 

68  46 

20  58 

69  00 

20  i4 

69  32 

20  35 

69  17 

20  55 

69  52  , 

21  07 

70  26 

20  49 

70  23 

20  55 

70  56 

21  32 

71  10 

21  20 

71  i4 

21  II. 5 

71  16 

21  07 

71  19 

21  o5 

71  32 

21  42.5 

71  25 

21  53 

72  22 

North  Caycos,  middle . . 
Booby  Rocks,  oft'  do. . . . 
Providence      Caycos, 

N.  VV.  p 

West  Caycos,  S.W.  p.. 

French  Key 

Soutli  Point  Shoal 

Great  Inagua,  or  Heneaga, 

N.  E.  p 

S.E.p 

S.  W.  p.... 

N.  W.p.... 

Little  Heneaga,  E.  p 

W.p.... 

Hogsties,  or  Corrolaes  .  . . 
Lookout  Bank,  (Cuidado) 

Mayaguana,  E.  reef. 

N.  do 

S.  W.  do. .  . . 

E.  point  French  Keys,  or 

Isle  Planas 

Miraporvos,  S.  Key 

Castle  Island,  or  S.  Key  . , 
Fortune  Island,  S.  W.  pt. 
North  Key,  Bird  Island  . , 
Crooked  Island,  W.  pt.  . , 
Acklin's     Island,     N.    E. 

pt 

Atwood's  Keys,  or  Island 
Saniana,  E.  p 

W.  p.  . . . 

Rum  Key,  E.  pt 

Walland's  Island,  N.  E.pt 

S.  W.  pt 

Conccption,or  Little  Island 

St.  Salvador,  or  Guanhari, 
S.  E.  pt 

N.  pt 

Little  St.  Salvador,  N.  pt. 

Eleuthera,  or  Hctera  Isl- 
and, S.  pt 

N.  pt 

Point  Palmeto. 

Harbor  Island 

New  Providence  light-h., 
NASSAU 

E.  pt.  . 

W.  pt.. 

Andros  Islands,  S.  pt. .. 

N.  pt.  . 

Berry  Islands,   

Stirrup  Key 

Blackwood's  Bush 

Ijittle  Isaac,  (eastern). . 

Great  Isaac 

Bemini  Island,  southern 
fresh  water  Key 

Gun  Key  light 

Cat  Key 

Riding  Rocks,  Soutli  . . 

Orange  Keys,  N 

S 

Key  Guinchos, 

Key  Lobos,  beacon,  20  ft. 

Las  Mucaras,  Diamond 
Point 

South  edge  of  the  Bank 

Brotlicrs'  Rocks 


Lat. 


Long. 


D.  M. 

D.  M. 

21  56N 

72  00  W 

21  58 

72  00 

21  5o 

72  20 

21  37.5 

72  3o 

21  3o 

72  i4 

21  02.5 

71  5o 

21  20 

73  00 

20  55 

73  08 

20  55 

73  38 

21  09 

73  40 

21  29 

72  55 

21  29 

73  06 

21  4o 

73  48 

21  57 

72  55 

22  20 

72  40 

22  32 

73  09 

22  22 

73  II 

22  4i 

73  27 

22  OD 

74  3i 

22  07 

74  20 

22  32 

74  23 

22  49  .*5 

74  24 

22  48.5 

74  23 

22  44 

73  5i 

23  o5 

73  37 

23  o5 

73  48 

23  4i 

74  46 

24  08 

74  25 

23  55 

74  32 

23  5o 

75  o5 

24  09 

75  18 

24  42 

75  43 

24  37 

75  55 

24  37 

76  08 

25  34 

76  43 

25  09 

76  08 

25  3o 

76  38 

25  05.2 

77  21.2 

25  02 

77  16 

25  01 

77  35 

23  44 

77  38 

25  )0 

78  02 

25  25 

77  44 

25  49 

77  53 

25  27 

78  o3 

25  58.5 

78  5t.3 

26  02 

79  c6.3 

25  43 

79  19 

25  34 

79  18.4 

25  3i 

79  17 

25  14.5 

79  09 

24  57 

79  08 

24  54 

79  08.5 

22  46 

78  08 

22  i 7  J 

77  33 

22  II 

77  i4 

22  o5 

76  22 

22  02 

75  42 

Page  336] 


TABLE   LIV. 
Latitudes  and  Longitudes. 


Key  San  Domingo  .... 

Key  Verde  Island 

Key  Sal,  (Ragged  Island,) 
Yuma,  or  Long  Island,  S.p 

N.  p 

Exuma,  N   VV.  p 

THE    HOLE    IN    THE 

WALL 

Light  on  do 

N.  E.  point  of  Abaco 

Elbow  Reef. 

Man-of- War  Key 

Great  Guano  Key 

Little  Bahama  Bank,  N.  p 

Memory  Rock 

Sand  Key 

Wood  Key,  or  Cape  Leno, 

Great  Bahama,  W.  p 

E.p 

Dog  Keys,  N.  W.  p 

Water  Key 

Double-headed  Shot  Key, 

(elbow,)  light 

Salt  Kej  cent  beach  W.  side 

Anguilla,  E.  p 

GEORGETOWN 

Wreck   Hill,  westernmost 

land 

Best  latitude    to  run    for 

Bermuda 


Lat. 


D. 


M. 

42  N 

02 

12 

5o 

45 

42 

5i 

5i.5 

18 

34 

37.5 

42 

35 

57 

49 

45 

42 

4o 

04 

59 

56.4 
4i.8 
29 
22.2 


Lon£ 


D.  M. 

75  45W 
75  10 
75  42 

74  5o 

75  18 

76  00 


80 


10. o 

57 

52 

57.5 
04 


06 

02 
02 
01 

48 
5o 
17 

27-7 
24.3 
26 
37.6 


64  5o 


liJ,  East  Coast  of  Jlmerica,  from  Gulf  of 
Mexico  to  Cape  Horn. 


Galveston  Inlet 

W.  p.  Galveston  Island  . . 

Rio  Brazos 

Pasa  del  Cavallo 

Aranzas  Inlet 

Corpus  Christi 

Braso  de  Santiago 

Rio  Bravo  del  Norte 

River  St.  Fernando,  entr . . 
Inlets  to  Laguna  Madre.. 
Bar  de  la  Marme,  entrance 

River  St.  Ander 

Bar  del  Tordo 

JVIount  Commandante 

Bar  de  la  Trinidad 

Bar  Ciega 

River  Tampico 

Point  de  Xeres 

Cape  Rojo 

Tamiagua  City 

River  Tuspan,  entrance  . . 

Point  Piedras 

River  Cazones  

Tenestequepe 

Boca  de  Lima 

River  Tocoluta,  entrance  . 

Mount  Gordo 

River  Nauta,  entrance  . . . 
River  Palina,  entrance  . . . 

Point  Piedras 

River  de  Santa  Nos 


Lat.       Lonor. 


D. 

M. 

29 

17N 

29 

4 

29 

3 

28 

24 

27 

49 

27 

36 

26 

6 

25 

56 

25 

22 

25 

2 

23 


45 

52 

48 
39 
34 
16 
55 
45 
16 
58 
46 
42 
4o 
34 
27 
16 
i3 
10 
00 
55 


D.   M. 

94°45'W 
95  26 

95  33 

96  18 

97  4 
97  16 
97  12 
97  12 
97  32 
97  4i 

97  58 
97  57 
97  58 
97  57 

97  58 

98  2 
97  45 
97  22 
97  29 
97  18 
97  12 
97  12 
97  9 
97  4 
97  o 
97  01 
96  47 
96  45 
96  35 
96  3o 


Point  Delgada 

Point  M.  Andrea 

Point  de  Bernat 

River  St.  John  Angel  • 

Xalapa 

Peak  de  Orizaba 

Point  de  Sampola 

River  St.  Carlos 

River  Antigua 

Point  Gorda 

VERA  CRUZ 

St.  John  de  Ulloa  .... 

Xamapa 

River  Medellin,  entrance  . 

Point  Anton  Lizardo 

Bar  de  Alvarado 

Tlacotalpan 

Vigia 

Point  Roca  Partida 

Point  Morillos 

Pic  de  San  Martin 

Point  Olapa 

Point  St.  John 

Barilla 

Bar  Guazacoalcos 

River  Tonato 

River  St.  Ann 

River  Cupilco 

Dos  Bocas 

River  Chittepeque 

River  Tabasco 

River  St.  Peter  and  Paul  . 
Island  Carmen,  W.  pt. . . . 

Point  Escondido 

Tavinal 

Point  Morros 

CAMPECHE  

Point  Desconocida 

Point  Gorda 

Point  Piedras 

Igil 

St.  Clara 

Bocas  do  Silan 

El  Cuyo 

Island  Jolvas,  N.  p 

Island  Contoy,  N.  p 

L^s  Areas  Islands  S.  W.  Id 

Bank  Obispo  centre 

Triangles  Islands  N.  W.  Id 

New  Shoal 

Bajo  Neuva  Island  centre. 

Island  Arenas 

Island  Bermeja,  or  N.  W. 
Shoal 

Sisal  Fort 

Alacranes 

N.  part  of  Bank   oiF  this 
coast 

N.  E.  do 

Isle  de  Mugeres,  or  Wo- 
men's Island 

Island  Cawkun,  S.p 

New  River 

River  Bacales 

Bay  Ascension,  entrance  . 

Island  Cosumel,  N.  p. . . , 
S.W.  p 


Lat. 


D.  M. 

9  49 
9  43 
9  40 

9  32 
9   32 

92 

9  3o 


26 

20 

i5 

12 

12. 

4 

6 

4 

46 

35 

8  38 

8  43 

8  4o 

8  3o 

8  34 

8  21 

8  II 

8  II 


Point  Tanack 


8  20 
8  26 
8  26 
8  24 
8  34 
8  38 
8  38 

8  58 

9  10 
9  45 
9  49 

20  46 
6 

9 

21  20 
21  22 

24 

21  3o 
21  3o 
21  36 
20  1 3 
20  3o.5 
20  57.8 

20  53 

21  5o.2 

22  7 

22  33 

21  10 

22  32.3 

23  43 
23  27 


20  42 
20  26 
5 

19  26 

20  36 
20  10 
18  54 


Loner. 


D.  M. 
N  96  26W 
96  21 
96  21 
96  20 

96  5o 

97  9 
96  j6 
96  1 5 
96  i4 
96  4 
96  9 

596  8 

95  58 

96  4 
95  58 
95  45 
95  36 
95  18 
95  II 

94  54 

95  10 
94  5o 
94  38 
94  35' 
94  22 
93  59 
93  49 
93  26 
93  6 
93  02 
92  40 
92  32 
91  5i 

i5 
90  58 
90  43 
90  33 
90  26 

i3 


88  56 
87  43 

87  II 
86  52 
9i°59. 
92  1 3 
92  18.9 
91  5o 
91°  4. 
91  25 

91  22 
90  2 

89  43 

88  43 
86  37 

86  42 

86  58 

87  i5 
87  34 

3 

86  45 

87  00 
87  42 


TABLE   LIV. 

Latitudes  and  Longitudes 


[Page  337 


x\.  Triangle,  N.  Key 

Sandy  Key,  S.  p 

S.  pt.  Ambergris  Key  Isl. 

BALIZE 

Turneff  Reef,  N.  pt 

S.  pt 


Englisli  Key 

Half-Moon  Key  light-house 

Hat  Key 

Tobaco  Key  Island 
Santanilla,  or  Swan  Island 
Glover's  Raef,  N.  p 

S.p 


Renegade  Key 
Sapotilla's  Kej's,  S.  E.p. 

Rattan  Island,  E.  p 

W.  p 


Guanaja,  or  Bonacca  Isl- 
and, S.  pt 

Cape  Three  Points 

Omoa 

Point  Sal 

Triunfo  de  la  Cruz 

Utilla,  N.  p 

Truxillo 

Cape  Delegado,  or  llondu 
ras 

Cape  Cameron 

Cape  False 

Cape  Gracios  a  Dios  .... 

Caxones,  W.  p 

S.E.  p 


Lat. 


D.  M. 

8  44  N 

8    22 

7  52 
7  29 
7  39 


Cayman,  or  Vivonlla. 

Key  St.  Thomas 

Alao-arte  Alia,  iN.  W.  p.  . 
S.  p 


Scranilla,  N.E. breaker. 

W.  breaker  . . 

Sarrana,   N.  p 

S.p 


Guana  Reefs,  N.  p. 
S.  p. 


Roncador . 

Musketeers,  centre 

Providence  Island,  N.  p.  . . 
Ned  Thomas's  Keys,  S.  p. 

Bracman's  Bluff* 

Man-of-War  Keys 

Little  Corn  Island 

Great  Corn  Island 

Bluefields,  entrance 

\Ac  St.  Andrew,  middle.. 

E.  S.  E.  Keys 

S.    S.  W.   Key,    or  Albu- 
querque   

Paxoro  Bovo 

River  St.  John,  S.  pt 

Port  Boco  Toro 

Isle  Escudo,  N.p 

River  Chatrre,  entrance  .. 

PORTO  BELLO 

Point  Jlanzanillo 

Point  St.  Bias 

Point  Moschitos 

Isle  of  Pines 

Cape  Tiburon 

River  Suniquilla,  entrance 
Point  Carabana 


7  19 
7  i3 
7  10 
6  57 

23 

55 


4i 


6  00 
6     2 


4 
i4 

9 
58 
33 
24 


I  20 
o  57 
9  25 

9  i4 

9  19 
9  34 
9  39-i 
9  35 
9  8 
9  I 
8  4i 

7  55 

8  38 


Long. 

D.  M. 

87  i5W 

87  i8 

88  I 

88  12 

87  4 1 

87  56 

efi 

88  2 

s 

87  34 

87  4i 

88  4 

a 

83  5i 

u 

87  40 

87  48 

88  n 

88  i4 

86  i5 

86  5i 

86  00 

88  34 

88  I 

, 

87  48 

45 

87  38 

>• 

87  2 
86  2 

?3 

86  6 

fi, 

85  i4 

83  21 

83  12 

83  18 

83  8 

83  26 

81  49 

82  27 

82  25 

79  4' 

, 

79  58 

80  16 

0 
0 

80  23 

80  44 

a 

80  41 

79  46 

80  3 

81  20 

82  21 

83  so 

82  39 

82  58 

83  3 

82  54 

81  43 

81  28 

81  52 

82  48 

83  37 

82  12 

80  57 

79  59 

79  4o 

79  32 

"9  J 

rt 

77  58 

a 

77  5o 

s 

s 

77  27 

76  56 

w 

76  58 

Point  Arboletcs 

Island  Fuerte 

Isle  St.  Barnard,  N.  W.  p. 

CARTAGENA 

Punta  de  la  Galera  de 
Samba 

West  entrance  River  Mag- 
dalen  

St.  Martha. 

Cape  Aguja 

Bank  Navio  quebrador  .  . . 

Hacha 

Cape  La  Vela 

Point  Gallinas 

Mongcs  Islands,  N.  p 

Cape  Chichibacoa 

Point  Espada 

St.  Carlos 

MARACAYBO 

Coro 

Point  Cardon 

Point  Macolla 

Cape  St.  Roman 

Island  Oruba,  N.  W.  p.... 

S. E.  p 

Point  Aricula 

Point  Zamuro 

Point  Soldado 

Key  Borracho 

Point  Tucatas 

PORTO  CABELLO  .... 

Point  St.  John  Andres. . . . 

Point   Oricaro 

Point  Trinchera 

LAGUIRA 

CARACCAS 

Centinella Island,  or  White 
Rock 

Cape  Codera 

Curacoa  Island,  N.  p.  .. 

S.  E.  p 

Little  Curacoa 

Buenayre,  N.p 

S.p 

Birds'     or     Aves     Island, 


western 
eastern 


Los  Roques,  W.  p 

S.  E.  p 

Orchilla  Island,  middle.  . 
Blanca  Island,  middle... 
E.  point  Tortuga  Island  . 
Seven  Brothers,  middle. . 

Margarita,  W.  p 

E.p 


Island    Cuagua,    or    Pearl 

Island , 

Friar's  Island 

Island  Sola 

Testigos  Island 

Morro  de  Unare 

New  Barcelona 

Island  Borracho 

Cumana 

Pta.  de  Araya 

Morro  Chocopata 

pjscondido,  or  Hidden  Port 
Cape  INIalapasqua 


Lat.        Lon 


M. 

55  N 

24 

49 
26 


0  47 

1  5 
I  i5 
I  20 
I   26 

1  33 

2  1 1 
2  25 
2  28 
2  i5 
2  4 
o  57 

0  39 

1  24 

1  36 

2  4 


24 
56 
26 
i4 
57 
5i 
28 
3o 
34 
37 
36 
3o 

5o 

36 

24 

2 

59 

00 

57 

5o 

47 

48 

5i 

55 

4-h 

59 

59 


49 


23 

6 
10 

19 
28 
38 
42 
4o 
42 


D.  i\I. 

76  3oW 
76  16 

75  56 
75  38 

75  3o 

74  56 
74  18 
74  16 
73  i5 
72  59 
72  16 
71  44 
71  3 
71  20 
71  i3 
71  44 
71  45 

69  5o 

70  23 
70  22 
70  9 
70  12 
70  I 
69  56 
68  59 
68  4o 
68  22 
68  21 

68  7 
67  5o 
67  18 
67  8 

67   Q 

67  li 

66  i5 
66  J  2 

69  17 

68  49 
68  45 
68  3 1 


67  46 
67  32 
67  I 
66  38 
66  i3 

64  41 

65  18 
64  3i 
64  3o 

63  52 

64  18 
63  49 
63  40 

63  i3 

65  22 

64  48 
64  5i 
64  16 
64  3o 
63  54 
63  29 
63  7 


43 


Page  338] 


TABLE   LIV. 

Latitudes  and  Longitudes. 


Cape  Three  Points 

Point  Galera 

Point  Pena,  or  Salina 

Dragon's  Mouth 

River  Gaurapiche, entrance 

Point  Redondo 

Mouth  of  Oronoco  River. . 

Cape  Nassau 

Essequebo  River  

DEMERARA  River  en- 
trance  

Corrobano  Point 

River  Berbice,  entrance.. 

Surinam  River,   entrance. 

Paramaribo 

River  IMarouri,  entrance.. 

CAYENNE 

Mouth  of  Oyapock  River  . 

Cape  Orange 

River  Cassipour,  entrance 

Cape  North 

Nortliern  mouth  of  River 
Amazon 

Southern  do.  do 

Cape  Magoany 

Point  Tagioca 

Para 

Bay  Maracuno 

Caite  harbor 

Cape  Gurapi 

Slioal  off  do    

Fi.  point  of  Island  of  St. 
Joao 

Vigia,  fell  in  with  by  Mr. 
Du  Sylvia,  officer  of  the 
Brazilian  marine,  in  1824 
or  1S2.J 

Vigia  of  Manuel  Luis, 
Westerly  Rock 

Mondrain  Itacolomi. . . . 

Mt.  Alegre,  (the  summit,) 

Alcantara,  (west  church,), 

Rock  E.  of  Isle  Medo  .... 

City  of  San-Luis  de  Mar 
anham,  (Cathedral,) .... 

Fort  Sant  Antonio  das 
Areias,  (the  flag-stalF.) 

Fort  San  Marcos 

Isle  Maranham, (white  sand 
hills,  N.  part,) 

Breakers  of  Coroa  Grande, 
the  North  one 

North-west  one 

West  one. . , 


Isle  St.  Anne,  N.  E.  point. 
Breakers  of  Isle  St.  Anne 

E.  point 

Morro  Alegre 

Lancoes  Grande,  E.  point 
River  Perguicas,  E.  point. 
River  Tnloya,  entrance  . . 

Pedra  de  Sal 

River  Tapuyu.  entrance.. 
Mt.  Tapuyu,  W.  summit  . 
Mt.  Ticondiba,  summit.  .. 
Point  de    Jericacoara,    the 

highest  sand-hill 

S:ind-hill  near  the  shore 


Lat.        Lons- 


M. 

45  N 
43 
43 
43 

12 

5o 
5o 

32 


M. 

46  W 

34 

56 

5i 

43 

43 

GO 

58  4o 
58  26 


60 


loN 
5S 
12 

32 

28 

33 

46 
39 
36 

19 


O    32 

o  5i 
2    9 

2  17 
2  24 
2  3o 

2  3i 

2  29 
2  28 

2    25 


2     l3 

2     17 
2     l5 


2    26 

2  4i 
2  4i 

2    47 

2  5o 

2  58 

3  II 

2  47 
2  5o 


58  Hi 
57  II 
55  3 
55  00 
53  49 
52  i3 
5 1  26 
5i  1 1 
5 1  00 
5o    6 

5o  00 
49  45 
48  29 

47  58 

48  29 
47  4i 
47  6 
45  56 
45  56 

44  5o 


^i   17 

U   i5 

i^A  25 
^4  20 
44  23 
44  19 

44  16 

44  17 
U   16 

44  04 

43  58 
AA  4 
M  5 
43  38 

43  3o 
43  i3 
43  00 
42  27 
42  12 
4i  42 
4o  5o 
4o  5 1 
40  37 

4o  27 
4o  39 


Mount  Memoca 

Roccas,  (dangerous.) 

Pernambuquinho 

Morro  Melancia 

Sand-hill  of  Parati  . « 

Mountains  of  Ciara,  1st  .. 

2d  summit. 

3d     do.   , . 

4th   do.  .. 

5th    do.  . . 

Ciara,  steeple  in  the  city  . 

Point  Macoripe 

Morro  Aracati,  summit.  .. 

Point  Reteiro  Grande 

Reteiro  Pequeno,  remarka- 
ble sand-hill 

Morro  Tib.ao 

Point  de  Mel 

Point  du  Tubarro 

(Breaker)  das  Ureas 

dela  Lavandela. 

Pt.  Calcanliar,  summit  . . . 

Point  Petetino-a,  low 

CAPE  ST.  ROQUE  .... 

Fort  of  Rio  Grande 

Point  Negra,  mountain . , . 

Point  Pipa,  sand  mount  . . 

Bahia  Fermosa,  S.  point.. 

Bahia  da  Traicao.  N.  pt. . . 

Church  of  St.  Theresa.... 

Fort  Cabcdello 

Paranahyba  de  Norte 

Cape  Blanco,  steep  part  . . 

Point  de  Guya 

Point  das  Pedras 

Village  of  Pilar 

Fort,  entrance  of  Rio  Ay . . 

Nossa  Senora  Farinha. .  , , 

Olindo,  west  tovi'er 

Tower  de  Recife,  Periiam- 
buco 

Nossa  Senhora  de   Rosario 

CAPE   ST.  AUGUSTIN 

River  Ipojuca,  entrance  . 

Mount  Sellada,  S.  peak  . 

Islands  of  St.  Alexio  .... 

Fort  de  Tamandarc 

San  Bento 

Village  of  Quinta 

La  Forqviilha,  (hill.) 

Frenchmen's  Port 

Village  at  the  point  of 
River  Alagoas 

Morro  Sant  Antonio. . . 

River  San  Francisco.  . 

Tabayana  IMountain,  sum- 
mit   

Rio  Vasa  Barris 

Rio  Real,  S.  point 

Torre  de  Garcia  de  Avila. 

River  Jacuipe 

Rock  of  Itapuan 

Itapuanzinko,  the  point.  . . 

ST.  ANTONIO,  N.  W 
tower 

Point  Caxo  Pregos,  Isle 
Itaporica 

Point  Aratuba,  do.. . 


Lat. 

D.  J>1. 

3  18  S 

3  5i 

3  2 

3  12 

3  24 

3  58 

3  53 

3  5o 

3  46 

3  39 

3  43 

3  42 

4  42 

4  36 

4  48 

4  49 

4  55 

5  2 

4  52 

4  55 

5  8 

5  22 

5  28 

5  45 

5  53 

6  i3 

6  23 

6  4i 

6  57 

6  58 

7  6 

7  8 

7  26 

735 

7  36 

7  47 

7  57 

8  I 

8  4 

8  9 

8  21 

8  23 

8  25 

8  36 

8  43 

9  5 

9  16 

9  10 

9  4o 

9  4o 

9  22 

10  29 

10  47 

II  II 

II  28 

12  32 

12  42 

.2  58 

i3  I 

i3  0 

i3  8 

i3  5 

TABLE   LIV. 
Latitudes  and  Longitudes. 


[Page  339 


Point  laburn,  Isle  Itaporica 

Blount  Conceicao,  do 

Morro  Sant  Amarro,  do. . . 

Morro  de  San  Paulo 

Isle  Boypeda 

Isle  Quiepi 

Point  of  Sluta 

Villa  of  Contas 

Os  Itheos,  the  largest  rock 
Villa  de   San   George  dos 

Itheos 

Rio  Cachoeira,  S.  point.. 

Villa  of  Unha 

iNIorro  de  Commandatuba, 

S.  E.  summit 

Village  of  Commandatuba 

Village  of  Belmont 

Santa  Cruz,  steeple 

Porto   Seguro,   steeple    of 

the  Cathedral 

I:?olated  Mount 

Mount  Pascal,  summit... 

Mount  Joao  de  Siam 

River  Cramimuam 

Columbiana 

Villa  Prado,  fort 

Abrolhos  Islands,  (the  lar- 
gest island,) 

Rio  de  San  Mattheo 

Rio  Doce,  entrance 

Serra  dos  Reis  Magos,  the 

S.  summit 

^Nlorro  Alme}'da 

Mestrc  Alvaro,  summit. 

Cape  Zubarro 

"  Piton,"  at  the  N.  of  the 

city  of  Victoria 

Nossa  Senhora  da  Penha, 

church 

Mount  Morena 

Pacotcs  Rocks 

Point  Jicu 

Isles  Piasas.. 

Isle  Calvada 

Guarapari 

Morro  Bo,  (isolated  moun- 

tnin,)  

Morro  de  Benevento  . . , 

Serra  de  Guarapari 

Mt.  de  Campos,  S.  summit 
Mountains      of      Furado, 

highest 

CAPE  ST.  THOMAS.. 
Isle  St.  Anne,  the  largest. 
Pic  do  Frade  de  Macahe . . 
iNIorro  San  Joao,  summit  . 
Cape  Buzios,  S.  point.... 
Isles  Ancora,  easternmost. 
CAPE  FRIO,  S.  point... 

Cape  Negro 

Isles  Maricas,soutliernmost 

Redondo . 

RIO    JANEIRO,   (Sutrar 

Loaf,) 

La  Gabia 

Isle  Georgi  Grego 

O.  Pakagaio,    top   of  Isle 

Grande 


Lat.       Long- 


M. 
57  S 

3 

I 

22 

38 
5i 
53 
i8 

47 

49 
49 
59 


58 
37 
37 

5o 

57 

9 

i6 


20  19 

20  21 

20  26 

20  43 

20  AA 

20  AA 

20  43 
20  55 

20  5o 

21  23 

21  5o 

22  3 
22  25 
22  12 
22  32 
22  46 

22  46 

23  I 

22  57 

23  I 

23  4 


56 
59 
i5 


D.  M. 

38  36W 
38  4i 
38  45 
38  54 
38  57 
38  57 

38  57 

39  00 

38  59 

39  00 
38  59 

38  58 

39  8 
38  56 

38  54' 

39  2 

39  3 
39  3i 
39  25 
39  37 
39  9 
39  12 
39  12 

38  42 

39  45 

39  5 1 

40  22 

40  2-0 

4o  22 
4o  17 

4o  23 

40  2-0 

40  19 
40  17 
4o  22 
4o  25 
4o  27 
40  33 

4o  4 1 

40  49 
ii    8 

41  28 

41  43 

41  00 
4>  46 

42  9 
42  6 
41  56 
4t  5r 

41  59 

42  35 

42  5 1 

43  9 

43     9 

43   23 

4  19 


Ilha  Grande,  Pt.  Acaya. . . 

Point  loatinya 

Pic  de  Parati,  summit .... 

Isle  Couves,  largest 

Isle  Victoria 

Isle  Buzios,  S.  E 

Isles  dos  Porcos,  S.  sand 
hill 

Isle  St.  Sebastian, 

highest    mountain 

Pt.  Pirasonungo 

Mouton  de  Trigo 

Lage  de  Santos 

Point  Grossa 

Taypu 

Isle  Queiniada  Grande. 

Isle  Queiniada  Pequena. 

Poin  Jurea 

Mount  Cardoz 

Isle  Bon  Abrigo 

Ptochcr  Castello 

Rochcr  Figo 

Isle  de  Mel,  S.  top 

Roc  Coral 

Roc  Itascolomi ...-..,.. 

Point  Joao  Diaz 

Isles  Tamboretes 

Isles  Remedies 

Point  Itapacoroya 

Isle  Arvooredo,  top  .... 

Isle  St.  Catharine,  E.  pt 

Pt.  P^apa 

Steeple  of  Nossa 

Senhora  do  Desterro . . . 

Morro  de  Sta  Marta 

Porto  St.  Pedro 

Cape  St.  Mary 

Island  Lobos 

Maldonado  harbor 

Point  Piedras , . 

MONTE   VIDEO,  Ratls. 

BUENOS   AYRES 

Cape  St.  Antonio 

Cape  Lobos 

Cape  Corientes 

Point  de  Ncuva 

St.  Helena 

St.  George's  Bay,  Cape 
Cordova 

Cape  Blanco 

Point  Desire,  ruins 

Port  St  Julian,  Cape  Curi- 
oso 

St.  Cruz  harbor 

Cape  Fair^yeather 

Cape  Virgin, northern  point 
of  entrance  to  INIagellan 
Straits 

Cape  Espirito  Santo,  (ex- 
treme point  of  do.) 

Terra  del  Fuego ;  Cape 
Penas 

Cape  St.  Diego 

Staten  Island,  Cape  St. 
John,  easternmost  land 
near  Cape   Horn 

Cape  St.  Bar- 
tholomew   


Lilt. 

Long. 

D.  M. 

D.  M. 

23  i5S 

AA   29W 

23  18 

AA  39 

23  19 

AA   54 

23  26 

AA  58 

23  48 

45  i4 

23  AA 

45  6 

23  34 

45  10 

23  48 

45  22 

23  58 

45  20 

23  5i 

45  52 

24  18 

46  17 

23  59 

46  24 

24  I 

46  3o 

24  28 

46  47 

24  21 

46  54 

24  33 

47  19 

24  59 

48  12 

25   7 

47  58 

25  16 

48  3 

25  22 

48  10 

25  33 

48  26 

25  46 

48  3o 

25  5o 

48  33 

26  7 

48  40 

26  21 

48  39 

26  29 

48  42 

26  47 

48  AA 

27  17 

48  29 

27  26 

48  29 

27  .23 

48  32 

27  36 

48  4o 

28  39 

48  5i 

32  07 

52   9 

34  39 

54  10 

35  01 

54  54 

34  53 

55  00 

35  27 

57  5 

M   53 

56  i3 

34  36 

58  22 

36  19 

56  46 

36  55 

56  47 

38  06 

57  27 

42  53 

64  8 

AA  3i 

65  22 

45  46 

67  21 

47  12 

65  43 

47  45 

65  54 

49  II 

67  37 

5o  09 

68  19 

5i  32 

68  55 

52  20 

68  ai 

52  38 

68  35 

53  5i 

67  33 

54  4i 

65  7 

54  43 

63  43 

54  54 

64  45 

Page  340] 


TABLE   LIV 

Latitudes  and  Longitudes. 


Staten  Island;  Cape  del 
Medio,  entrance  to  Le 
Maire's   Straits 

New  Island,  E.  part 

Evout's  Island 

Barnevelt  Island,  E.  part. 

CAPE  HORN,  S.  part  of 
Hermit's  Island 


Lat. 


55  59 


Long. 


D.  M. 

D.  M. 

54  48  S 

55  17 
55  33 
55  48 

64  45 W 
66  26 
66  45 
66  45 

67  16 


rV.   West  Coast  of  America,  from   Cape 
Horn  to  Icy  Cape. 


CAPE    HORN 

Isl.  Diego  Ramires,  S.  part 

^ N.rock 


Island  St.  Ildefonso,  S.  p.. 
Terra  del  Fuego 

False  Cape  Horn . 

York  Minster,  S. 


E.  extremity 

Cape  Gloucester. 

Cape    Pillars,    N. 


ontr.  Magellan's  Stpaits. 
Evangelist  Island,  W.  ent. 

Magellan's  Straits 

Cape  Victory 

West  Cliff  Cape 

Cape  Three  Points 

Mount  Corso,  S.  pt 

Isl.  Cainpana,  N.  point. .. 

Cape  Tres  Monies 

Cape  Taytao 

Island  Huafo,  N.  W.  part. 

Point  Quilan 

Point  St.  Carles 

Point  Quedal 

Point  de  la  Galera 

VALDI'VIA,     near     Fort 

Coral  ..... 
Point  Tirua. 
Isl.  de  la  Mocha,  N.W.  part 
St.  Maria  Islands,  N.  p 
S.  p. 


Lat.        Lons. 


5qS 

27 

22 

53.5 


55  43 


52  43 


CONCEPTION,  cit^ 
Talcahuano,  Fort    . . . 

Topocahno  Pt 

VALPARAISO,  Fort 

Point  Rallena i3i 

Coquimbo 29 

Huasco 28 

Copiapo 27 

Sugar  Loaf  Islet,  summit.  26 
Island  St.  Felix,  Eastern  .  26 
Western  26 
24 

23 
23 


Pt.  dos  Reyes 

Morro  Moreno,  summit 
Mexilones  Hill,  summit 

Point  Tames 

Pt.  Francisco 

Pavellon  de  Pica 

Point  Piedras 

Pisagua,  Point  Pichalo  . 

Arica,  Head 

Point  de  Coles 

Ilo,  rivulet  mouth 

Point  Cornejo 


16 
34 

28 

6 

45 
56 

58 

3? 

28 

42 
37 

52 


67  16W 

68  36 

68  37 

69  17 

68  6 


73  02 

74  38 

75  3 

74  55 

75  32 
75  21 
75  32 
75  23 

75  28 

75  8 
74  49 
73  33 

73  52 

74  I 
73  47 

73  29 

73  33 

74  I 
73  35 
73  34 
73  5 
73  10 
72  5 
71  4i 
71  36 


71  19 
71  19 
71  2 
70  47 
9  47 
u  3 
70  4o 
70  38 
70  35 
70  23 
70  1 5 
70  14 
70  i4 
70  19 

70  24 

71  26 

71  24 

72  21 


Point  Pescadores  .... 

Atico  cove 

St.  Juan,  needle  hummock 

Mount  Carreta 

Pisco,  middle 

Point  Frayle 

Point  Chilca 

Isl.  St.  Lorenzo,  N.  pt.  . . 

LIMA 

CALLAO  Bay,  flagstaff. 
Island  Pescador,  summit  of 

largest,  W.  pt 

Los  Hormigas  Rocks. 

Island  Pelada 

Island  Don  Martin   . . 

Point  Santander 

Rock  seen  in  1 792 .  . . 
Ferrol  Baj^ent. (Blanco  Is.) 

Truxillo,  church 

Malabrigo,  (port,) 

Island    Lobos    de    Amera 

fishing-cove '. 

Island  Lobos  de  Tierre . . . 

Eten 

Point  de  Ajugo 

Point  Payta 

Cape  Blanco 

Point  Malpelo 

GUAYAQUIL,  city 

Island  Puna,  S.  W.  p 

Point  St.  Helena 

Island  Pelade 

Point  del  Callo 

Island  de  la  Plata,  W.  p. . . 
Cape  St.  Lorenzo .... 

Manta 

Cape  Pasado 

Quito 

Arbol 

Cape  St.  Francisco  . . 
Point  de  la  Galera..  . 
River  Esmeraldas,entrance 

Point  Mangles 

Island  Tumaco 

Point  Guascama 

Island  Gorgona,  middle 
River  Cajambrie,  entrance 
Island  de  Malpelo  . . . 

Island  de  Palmas  . 

Point  Chirembira. . . . 

Cape  Corientes 

Limones 

Point  St.  Francisco  Solano 
Point  Garachine  .... 
PANAMA  Ft.  N.E. Bastion 

Point  Mala 

Puercos  Point 

Island  Quibo,  N.  p.  . . 

Los  Ladrones  

Point  Burica 

Gulfe  Dulce,  W.  p.. . 
Isl.  Cano,  ent.Engl.  Harbor 

Cape  Herradura 

Cape  Blanco 

Nicoya,  wat'g  pi.  on  E.  side 

Morro  Hermoso 

Point  Culebra.  Gorda  Pt. . . 
St.  John's  Harbor 


Lat. 


Lonff. 


M.      D.  M. 

24  S73  20  W 
i3     73  45 

75  i3 

76  20 
76  16.5 
76  35 

76  53 

77  19 
77  6 
77   i3 


I  47 


I  27 
I  2 
o  38 


9    b 

8     8 
7  43 


56 

23 

18 

4 

5? 
27 


77  20 
77  5o 
77  53 
77  43 

77  56 

78  48 

78  39 

79  4 

79  28 

80  ^/\ 

80  53 

79  54 

81  10 
8r  10 
81   16 

80  3o 

79  53 

80  8 
80  48 
80  36 
80  34 
80  57 
80  43 
80  32 
80  20 


i5N 

79  48 

39 

80  6 

48 

79  5i 

58 

79  39 

■66 

79  7 

47 

78  5o 

4o 

78  32 

59 

78  26 

19 

77  i5 

55 

Si  36 

37 

77  7 

i3 

77  26 

34 

77  26 

3 

77  23 

49 

78  fO 

4 

78  3o 

57 

79  3i 

24 

80  2 

i3 

80  27 

3i 

8r  46 

52 

82  37 

0 

83  0 

23 

83  29 

AA 

84  4 

37 

84  37 

32 

85  2 

38.5 

84  36 

8 

85  3o 

32 

85  43 

i5 

85  5o 

TABLE   LIV. 

Latitudes  and  Loncritudes. 


[Page  341 


l^oint  Dcsolado 

I.l'Oll 

Iloalejo .  entr 

Aserailores 

i'oiiit  Cosignina 

i\-jiiit  Candadillo 

SacaU'Coluca 

i\  iat  llemedios 

Point  (iuatimala 

I'uerto  V'eutosa 

Aijualco 

ACAFULCO 

Cape  Coriciites 

St.  Bias 

Tres  Marias 

St.  Joseph 

Cape  St.  Lucas 

Morro  ilermosa 

lledondo  Island 

Port  Sau  Quentiu    ..... 

Bay  Todos  Santos 

Port  Diego , 

Point  Conception 

Monterey ■ 

Port  St.  Francisco 

Cape  Mendocino 

Port  Trinidad 

Cape  Blanco,  or  Orford 

Cape  G  regory   

Cape  Foulweather 

Cape  Rond 

Cape  Disappointment  • , 

Cape  Flattery 

Breakers'  Point 

NOOTKA,  N.  pt 

Woody  Point , 

Bay  St.  Louis , 

Isles  de  Sartine,  or  Scott, 

Cape  Scott 

Cape  Caution 

Cape  Hector,  or  James.. 

Bay  de  la  Touche 

Cape  Henry , 

Bay  de  Clouard , 

Point  North   

Cape  St.  Bartolomo 

Caj)e  Omnianey 

Port  Guibert 

Port  Neckar 

Cape    Engano,  or  Edge 

coinb 

Port  Gaudaloupe 

Port  de  los  Remedios  .  . , 

Cape  Cross 

Port  des  Francais , 

Cape  Fairwcather 

Behring's  Bay , 

Point  de  la  Boussole  .... 

Mount  St.  Elias , 

Cape  Hinchinbroke .... 

Cape  Elizabeth 

Barren  Isles 

Point  Banks 

Cape  Douglass 

Cape  Whitsunday 

Cape  Grenville 

Trinity  Islands 

Foggy  Island 


Lat. 


D.  M. 

12  22  N 

12  26 

12  28 

12  35 

12  53 
.3  7 
i3  26 
i3  35 
i3  54 
16  8 
16  2 
16  55 

20  26 

21  3o 

21  28 
23  4 

22  52 
27  4.6 

29  49 

30  22 
3i  49 
32  4i 
34  27 

36  37 

37  48 
4o  28 
4i  3 
42  53 
^3  26 
44  52 

5  43 
46  16 

7 
49  24 

49  36 

50  6 
5o  34 
5o  56 
5o  48 
5i  12 
5i  57 
52  42 

52  53 

53  52 

54  20 

55  12 

56  12 
56  38 

56  43 

57  2 
57  10 
57  24 

57  57 

58  37 

58  55 

59  18 

59  5o 

60  23 
60  1 5 
59  9 
59  00 


58  4i  ! 

58 

56 

58 

i5 

57 

33 

56 

36 

56 

10 

Long. 

D.  M. 

86  58  W 

86  49 

87  8 
87  20 
87  37 

87  57 

88  32 

89  43 

90  53 
93  o 
96  52 
99  48 
o5  35 

05  i3 

06  29 
09  38 
09  52 

14  4i 
i5  10 
i5  57 

16  43 

17  II 

20  26 

21  5i 

22  21 
24  20 
24  7 
24  37 
24  21 
24  5 

23  48 

24  I 
24  43 
26  3o 

26  35 

27  43 

28  i4 
28  5o 
28  20 
27  52 
3i  7 

32  10 

32  27 

33  21 
33  i5 

33  38 

34  35 

35  00 
35  4 

35  5o 
35  43 

35  43 

36  24 

37  20 
37  52 

39  00 

40  55 
40  45 
46  16 
5i  28 
5i  46 
52  6 
52  5o 
5i  46 

52  00 

53  4o 
56  45 


Halibut  Head  Island 

Ounalashka  Island,  N.  p. 
Bristol  River,  entrance . . 

Round  Island 

Cape  Newnham 

Shoalness 

Cape  Stephens 

Cape  Denbigh 

Cape  Rodney 

Cape  Prince  of  Wales. .  . 

Cape  Mulgrave 

Cape  Lisburne 

ICY  CAPE  


Lat. 

D. 

M. 

54 

27  N 

53 

55 

58 

12 

58 

29 

58  34 

60 

00 

63 

33 

64 

17 

64  34 

65  45 

67  45 

69 

5 

70 

29 

Long. 

D.  M. 

162  3oW 

166  12 

1 57  33 

159  53 

161  55 

161  52 

162  17 
161  53 
166  37 
168  17 
i65  12 
i65  22 
161  42 


From  the  River  St.  Croix  to   Cape 
Ca7iso. 


Entrance  St.  Croix  River. 

Macgone's    Isl.    (entrance 

of  St.  John's  River)  .... 

Cape  Spencer. 

Cape  Chignecto 

Haute  Island 

Annapolis  Gut 

Breyer's  Island  light 

St.  Mary's  Cape 

Cape  Fourchu 

Seal  Island  light 

CAPE  SABLE 

Sable  Island,  E.  pt 

W.  pt 


Cape  Ftoseway,  Shelburne 
licrhts 

LIVERPOOL,  Coffin's 
Island  lights 

Lunenburg,  Cross  Island 
lights 

Sambro  light-house 

HALIFAX  Obs.  D.  Yard 

Sheet  Harbor,  entrance . . . 

Sherbroke  

Wiiite-Head  Island 

Torbay,  Berry  Head 

CAPE  CANSO,  Cran- 
berry Island  light 


Lat. 


45  00  N 


67     2W 


Loner. 


45  i3 

66  5 

45  12 

65  55 

45  18 

64  58 

45  i5 

65  0 

U  42 

65  45 

44   16 

66  22 

44    6 

66  II 

43  5o 

66  7 

43   24 

65  58.5 

43  24 

65  36 

43  59 

59  47 

43  57 

60  i4 

43  38.565  i5.5 


44     3 


45  19.5 


64  36 


44  20 

64  7 

44  26.6 

63  33.3 

44  39.4 

63  35 

44  52 

62  29 

45  8.5 

62  0 

45  12 

6j  id 

45  II 

61  20 

60  55.3 


VI.    The  Gulf  of  St.  Lawrence. 


Chedabucto  Bay 

Gut  of  Canso,  S.  entrance 

Cape  Hinchinbroke 

Cape  Portland 

LOUISBURGH 

CAPE  BRETON 

Scatari  Island,  N.  E.  pt... 

Flint  Island 

Spanish  Bay,  Sidney  light 

Port  Dauphin 

Cape  Egmont 

Cape  North 

Chetican  Harbor,  entrance 

Seal  Island 

Cape  Mabou 

Port  Hood,  entrance 

Just  au  Corps  Island 


Lat. 

45 

29N' 

45 

3o 

45  34 

45  49 

45 

53.5 

45 

57 

46 

2 

46 

12 

46 

18 

46 

24 

46 

53 

47 

2 

46  4o 

46 

23 

46 

12 

46 

00 

46 

00 

Long. 

61  00  W 
6r  i3 
60  42 
60  5 
60  00 
59  48.5 
59  4i 

59  47 

60  9 
60  3 1 
60  22 
60  24 

60  59 

61  i5 
61  26 
61  34 
61  37 


Page  342] 


TABLE   LIV 

Latitudes  and  Longitudes 


GUT    OF     CANSO,    N. 
entrance 

Cape  St.  George,  N.  end  . 

Pictou  Island,  E.  pt 

Pictou  light   

Cape  Tonnentin,  S.  E.  pt. . 
Richibucto  Harbor,  entr.. 


Cape  North 

Cape  West 

Egmont   or  'Halifax   Bav, 

Red  Head '. 

Hillsborougla  Bay,  St.  Pe 

ter's  Isl 

Bear  Cape 

Cape  East 

Richmond  Bay , 


D.  M. 

45  42  N 
45  53 
45  49 

45  4i.5 

46  o5 
46  43 

4?  o3 
46  4i 

46  26 


Cape  Esquiminac 

Miscou   Island,    (entrance 

of  Chaleur  Bay) 

Cape  Despair 

Bonaventure  Island 

Flat  Island 

Cape  Gaspo 

Cape  Rosier 

Magdalen  River 

Cape  Chatte 

Bio  Island,  Riv.  St.  Law. 

E.  pt 


Anticosta  Island,  E.  pt.  . . 

West   pt, 

S.  W.  pt 

S.  point. 

N.    point 


Deadman's  Island 

Entry  Island 

Amherst  Isl.  S.  W.  pt.. 
Magdalen  Isles,  E.  pt. . . 

Byron  Island,  E.  pt 

Bird  Island 

St.  Paul's  Island 


Lat. 


46  07 
46  00 
46  28 

46  34 

4?  o4 

48  01 
48  25 
48  3o 
48  38 
48  45 

48  5i. 

49  i5 
49  06 

48  25 

49  o5 

49  52 
49  24 
49  o4 
49  58 

47  16 
47  17 
47  '3 
47  37.6 
4?  48 
47  5r 
47  i4 


Lons. 


D.  M. 

61  29W 

61  52 

62  33 

62  4o 

63  5o 

64  5o 

64  01 
64   23 

64  08 

63  i4 

62  29 
6r  59 

63  44 

64  46 

64  3i 
64  21 
64  10 
64  II 
64  12 
64  14.8 


65 

22 

65  48 

68 

53 

61 

45 

64  35 

63 

36 

62 

18 

64 

12 

62  1 5 

61  42 

62  o4 
61  26 
61  25 
61  10 
60  II 


VIL    JVeiofoundland. 


Limits  of  the  Great  Bank  of 
Newfoundland,  N.  point 
S.  point 


Outer  Bank 

Cape  Norman 

Green  Island   

Point  Ferrole 

Point  Riche 

Ingorneclioix     Bay, 

Saunders 

Bay  of  St.  Paul's  . . . 

Bon  Bay  

Cape  St.  Gregory  .. 

South  Head 

Red  Island 

Cape  St.  George  . . . 

Cape  Anguile 

Cape  Ray  

Connoire  Bay 


Port 


Lat. 


M. 

i5N 

56 

00 

38.1 

24 

02.4 

42 

39 

5c 

33 

22 

06 

34 

28 

54 

36.9 

4o 


Lour, 


D.  M. 

5i    loW 
5o  00 
45  00 

55  56.3 

56  37 

57  o5.6 
57  27 

57  21 

57  5i 

58  oc 
58  16 

58  21 

59  16 
59  )5 
59  27 
59  20.2 
58  00 


Burgeo's  Isles 

Rainea  Islands 

Penguin's  Islands 

Fortune  Head 

Brunet  Island,  W.  H 

Great  Miguelon,  Cape  M. 
Langley's  Island,  Cape  L. 
St.  Peter's  Island,  S.  E.  pt. 

Point  May - 

Cape  Chapeau  Rouge 

Mortier  Rocks 

Red  Island,  S.  pt 

Virgin  Rocks 

Point  Breem 

Cape  St.  Mary 

Cape  Pine 

CAPE  RACE 

Cape  Race,  (Virgin)  Rocks 

Cape  Ballard 

Cape  Broyle 

Bay  of  Bull 

Cape  Spear 

St.  John's  Harbor 

Cape  St.  Francis 

Breakheart  Point 

Trinity  Harbor 

Cape  Bonavista 

Funk  Island 

Cape  Frecls 

WadJiam  Islands 

Gander  Bay 

Fago  Islands,  cape   .  — 

Snap  Rock  

Tuolinguet  Islands 

Cape  St.  John,  N.  Bill  . 
Horse  Islands,  E.  pt  . . . 
White  Bay,  entrance   . . 

Hooping  Harbor 

Belle  Isle,  southern  .... 
Groais  Island,  N.  pt. . . . 
Hare  Bay,  entrance. . . . 

St.  Anthony's  Cape. 

St.  Lunaire  Bay 

Cape  Bauld 

Belle  Isle,  northern  .... 
Oroque  Harbor 


Lat. 


Loner. 


D.  M. 

D.  M. 

47  33  N 

57  43W 

47  32 

57  25 

47  22 

57  01 

47  o5 

55  5i 

47  16 

56  00 

47  08 

56  26 

46  48 

56  27 

46  45 

56  10 

46  54 

56  o4 

46  53 

55  27 

47  02 

54  57 

47  23 

54  i5 

47  10 

54  II 

46  59 

54  16 

46  5o 

54  i3 

46  38 

53  35 

46  39.4 

53  04.6 

46  26.3 

5o  55 

46  47 

52  59 

4?  o5 

52  52 

47  18 

52  47 

47  3o.5 

52  39 

47  34 

52  4^ 

47  48 

52  5i 

48  09 

52  59 

48  22 

53  24 

48  42 

53  08 

49  45 

53  12 

49  18 

53  3o 

49  34 

53  55 

49  28 

54  26 

49  4i 

54  00 

49  55 

53  44 

49  42 

54  44 

5o  00 

55  3i 

5o  i3 

55  43 

5o  i3 

56  21 

5o  37 

56  14 

5o  49 

55  29 

5o  58 

55  35 

5i  16 

55  4i 

5i  23 

55  3i 

5i  29 

55  29 

5i  39.7 

55  27.4 

52  01 .3 

55  19. I 

5i  o3 

55  5o 

QUEBEC  

Coudras  Isl.  N.  W.  part. . 

Bay  of  Rocks 

Green  Island  light    

Point  Mille  Vache 

Bersimis  Point,  S.  E.  pt. . 

Manicougan  Point 

Cape  Nicholas 

Cape  Montpelles  light-h. 

Egg  Island     

Seven  Islands  Bay,  Store, 

Point  Moisic 

Lobster  Bay 

Point  Manitou 

St.  Jolin's  River   


D. 

M. 

46 

49 

I 

47 

24 

b 

47 

57 

48 

o3.4 

48  34 

48  54 

I 

49 

06 

2 

49 

i5 

9 

49 

19.7 

49 

38 

3 

5o 

i3 

5o 

II . 

4 

49  49. 

5 

Long. 

D.  M. 

71    16W 
70  28 
69  5o 
69  28.2 
69  II 
68  4t-6 
68  i5 
67  53.2 

67  25 
67  i3 
66  25 

66  07.7 

67  06 
5o  17.7J65   17. 1 
5o  18     64  23 


TABLE   LIV. 

Latitudes  and  Longitudes. 


[Page  343 


Mino-an  Island 

Esquimaux  Island 

Clear-water  Point,  S.W.  ex. 

Appectetet  Bay 

Mount  Joli,  Natashquan  Pt. 

Cape  Whittle 

Boat  Islands 

St.  Mary's  Islands,  S.  pt.. . 

Hare  Harbor 

Great  Mecatina  P.,S.E.p. 

Mistanoque  Island 

Grand  Point 

Forteau  Bay  Point 

Red  Cliffs 

Red  Bay  

York  Point 

Cape  Charles 

Battle  Island,  S.  E.  pt.  . . . 

Cape  St.  Lewis 

Cape  Harrison 

Enchanted  Cape 

Cardinal's  Island 

Button  Islands 


Lat. 

Long. 

D.  M. 

D.  M. 

5o  12.9 

(^     K. 

5 

5o  1 3 

63  4 1 

5o  12.6  63  28 

5o  16.7 

63  01 

5o  06 

61  46 

5o  10.7 

60  09. 

8 

5o  17 

59  A6 

5o  i3 

59  45 

5o  36.5 

59  20. 

I 

5o  U 

58  53 

5i  i5.8 

58  i5. 

I 

5i  25 

57  i4 

5i  25.6 

56  59. 

4 

5i  33 

56  47 

5i  Ai 

56  28. 

4 

5i  58 

55  55. 

9 

52  i4 

55  22 

52  16 

55  33 

52  21 

55  4i 

54  54 

58  o5 

56  40 

60  55 

58  5o 

63  00 

60  45 

64  53 

IX.    HudsoTi's  Bay  dnd  Straits,  and 
Davis^s  Straits. 


Cape  Resolution 

Saddle-Back  Island 

Upper  Savage  Island, E.pt. 

North  Bluft" 

Capes  Charles 

Cape  Dorset 

Cape  Pembroke 

C:ipe  Walsingham 

Cape  Digges,  W.  ex 

Salisbury  Islands,  E.  pt 

Mansfield  Island,  N.  part  . 

S.  part  . 

Cape  Southampton 

North  Sleepers 

West  Sleepers 

Portland  Point 

Baker's  Dozen 

Belcher's  N.  point 

James's  Bay, 

Cape  Henrietta 

Cape  Jones. 

X.  IBear  Isle. 

North  Cub. . 

The  Twins. 

Albany  Fort 

Moose  Fort 

Charlton  Island 

YorkF.)rt 

Cape  Churchill 

Prince  of  Wales's  Fort  . 

Marble  Island 

Cape  Dobbes 

Cape  Walsingham 

Dyer's  Cape 

Sanderson's  Hope 

Cape  Bedford 

Wayirate  Island 


Lat. 


M. 

29  N 
1 1 

32 

34 
46 

32 

37 
3o 

37 
27 

23 

3i 
6 
3 

8 

48 
5 


Long. 

D. 

64 
67 


M. 

3o\V 
43 
o 

25 


74 
78 
82 

77 
-8 
76 
79 
80 
84 
80 
81 
79     2 

79  3() 

80  1 5 


X.    Greenland. 


Musquito  Cove 

Gothaah,  ent.  of  River  Bal 

Bear  Sound 

Maab 

Cape  Farewell 

Whale's  Island 

Herjoisness 

Bontokoe  Island,  S.  E.  pt.  . 

Gael  Hamkes  Bay 

John  Mayen's  I.,  N.E.  Cape 


Lat. 


D.  M. 

64  55  N 

64  10 
63  20 
62  5 
59  49 
62  3o 

65  3 
73  29 
75  00 
71  10 


Long-. 


D.  M. 

52  57W 
5i  47 
49  10 
48  27 
43  54 
43  i5 
29  5o 
20  4o 

6  5i 

7  26 


XI.    Iceland. 


Cape  Reikiancss 

Bessesled 

Mount  Suaesell 

Patrixfiord 

Straumness 

North  Cape 

Hola 

Grim's  Island,  N.  pt.   . . . 

Rikefiord  

J^ongnose,  Cape 

Encliuisen  Island 

Wreeland       do 

Cape  Hecla,  Mt 

Westman's  Island,  S.  pt. 


Lat. 

Long. 

D.  31. 

D.  M. 

63  48  N 

22  42W 

64  6 

21  54 

64  52 

23  54 

65  36 

24  10 

65  4o 

24  29 

66  28 

22  26 

65  44 

19  44 

66  34 

18  o4 

66  3o 

17  35 

66  23 

i4  3i 

64  20 

i4  i5 

63  55 

18  19 

63  58 

19  4i 

63  20 

20  23 

XII.    Spitzhergen. 


Lm. 


Lonir. 


South   Cape 

Fair  Foreland 78  53 

Amsterdam  Island,  (Hack 

liiyt's  Head.) J79  46 

Smeerenburg  Harbor [79  44 

Verlegen  Hook 180     2 

Hope  Island,  W.  pt. 76  20 

Bear  or  Cherry  Island. ..  .I74  3o 


I).  31.     ID.  3r. 

76  32NI17  23  E 


10  35 

10  57 

11  II 
16  37 
20  5o 
20  00 


XIII.    Ens;h'sh  Coast,  from  London  to 
St.^Mar^'s  Light,  {Scilly.) 


LONDON 

GREENWICH  Observ 

Woolwich 

Purfleet 

Gravesend 

Rochester 

Sheerness   

Nore  light 

North  Foreland  light. . . 
South  Foreland  lights  . . 

Deal  Castle 

DOVER 

Dunseness 

Hastings  lights 


Lat. 


5o 


M. 

3iN 

II 

3o 
28 

23 

27 
29 


LniiiT. 


D 


31. 

6W 
o 

4E 
'9 


O  22 
O    32 

o  44 

0  48 

1  27 
I  22 
I  24 
I  19 
o  58 
o  36 


Page  344] 


TABLE   LIV.  ♦ 

Latitudes  and  Lonsitudes. 


Beachy  Head  light 

Brighton  lij^ht 

Shoreliam  liglits 

Arundel 

Selsey  Bill 

Owers  lio-ht 

PORTSMOUTH,  town 
Isle  of  Wight, 

Cowes,  Castle. . . 

Bembridffe     Ledge 

or  Point,  Ft.  light 

Dunnose 

St.  Calli'ne's  Pt,  It. 

Needle's  light.s. . . . 


Hurst  Ught 

Poole  light 

St.  Albau's  Head    

Weymouth  light 

Portland  lights 

Exmouth  Bar 

Torbay,  Berry  Head 

Dartnaouth 

Start  Point 

Praul's  do 

Bolt  Head 

Eddystone  liglit 

Hand  Deeps 

Ram  Head 

PLYMOUTH,  Mt 

Fowey 

Deadman's  Point 

Falmouth  light 

Manacles  Rocks 

Black  Head 

LIZARD  Point 

Mount's  Bay 

Penzance  light   

Bundle's  Stone,  beac 

Wolf  Rock 

Land's  End 

St.  Agnes'  light,  (Scilly,)  . 

St.  Mary's    

St.  Martin's 


Lat. 


M. 

44  N 

5o 

5o 

53 

43 

4i 

47 


5o  46 


Lonsr. 


D.  M. 

o  i3E 

o    8W 
o  i5 
o  35 
o  48 

0  4o 

1  G 


i8 
34 
33 
56 
3 

26 
27 


4  10 
4  38 

47 


XIV.  French  Voast  from  Calais  to  Ushant. 


CALAIS 

Cape  Griz  Nez 

Ambleteuse 

BOULOGiNE 

Etaples  Bay,  Lornet  light . . 

Montreul  

La  Rochelle 

Abbeville 

Grotoy 

St.  Vallery,  River  Somme 

Dieppe  light 

St.  Valley,  River  Cau.x. 

Fecamp  light   

Cape  de  Caux 

Ca])e  de  le  Heve  lights  . 
HAVRE  DE  GRACE 
PARIS  Observatory... 

Mouth  of  Seine 

Harfleur 


Lat.       Lon 


D.  M. 

5o  58  N 
5o  52 
5o  48 
5o  M 
5o  33 
So  28 
5o  19 
5o  7 
5o  i3 
5o  I  r 
49  56 
49  52 
49  46 
49  4i 
49  3 1 
49  29 

48  5o 
.   27 

49  3o 


D. 


31. 

5t  E 

35 
36 
37 
39 
45 
4o 
5o 
38 
38 
5 
43 


Honfleur  lights 

Caen 

Bayeux 

Carentan 

St.  Marcouf  Island  hght     . 

Cape  Barfleur  licrht 

CHERBOURGH  

Pelee  Island 

Cape  la  Hogue 

Alderncy  Island,  N.  point. 

Caskets  lights 

Guernsey,  Pier  Hd.  light  ... 

Sark  Island,  N.  point 

Jersey  Island, 

Cape  Grosness  . . . 

St.  Aubin 

St.  Clement's  Point 

Isle  de  Chausey  hghts  . . . 

Coutances  

Granville,  Mole  Hd.  light . . 

Avranches 

Mount  St.  Michael 

Pontorson 

St.  Malo,  New  Mole  light . . 

Cape  Frehel  light 

St.  Brieu.x,  Oath 

Brehat  Island,  Centre 

Tregucir 

Morlaix  light  on  T.  la  Lande 

St.  Pol  de  Leon 

Isle  de  Bas  light 

Roche  Blanche 

St.  Anthony's  lights 

USHANT,  N.E.  point, light 


Lai. 


M. 

25  N 

I  r 

16 

18 

3o 

42 

38 

4o 

46 
43 
27 
26 

i5 

i3 

9 

52 

3 


38 
33 
39 
4i 
3i 
5i 
47 
38 
4i 
45 
I 
40 
29 


Loner. 


D.  M. 

o  i4E 

o  21W 

0  43 

1  i5 

I  9 

I  16 

I  37 

I  36 

1  56 

2  12 

2  23 

2  33 

2  23 

2  16 

2  I  I 

2  00 

I  49 

I  27 

I  36 

I  22 

I  3i 

I  32 


46 
00 
i5 
53 
00 

I 
58 
29 

3 


XV.  .F)-om  the  JVorth  Foreland  to  Dun- 
canshy  Head. 


North  Foreland 

Kentish  Knock,  Ft.  light . 

Long  Sand  Head 

Galloper,  N.  point 

S.W.  point  Ft.  It. 

Shipwash,  N.  point,  Ft.  It. 

S.  point 

Gaberd,  outer 

Orfordness  lights 

A:.ldboro'  Steei^le 

Southwold 

LoestofF  lights 

Yarmouth 

W^interton  Ness  lights . . . , 
Smith's  Knowl,  S.  pt.  ... 
Hasborough  Sand,  S.  p.  . 

N.  p.  . 

Sherringham  Shoals  .  . . . , 

Hasborotigh  lights , 

Cromer  lights 

Lemon  and  Ower,  N.  p.  . , 

-^ S.p... 

Cromer  light 

Dudgeon  light • 

Outer  Dowsing ■ 

Inner  Dowsing 

Lynn  Knock 


Lat. 


31. 

23  N 

40 

45 

52 

45 
02 
53 
58 
5 

9 
20 
29 

37 
43 
48 
5i 

2 

3 

49 
56 


Long'. 


n.  M 

27  E 
39 
38 
5 
56 
38 
33 
59 
34 
36 
4i 
46 
A4, 
4i 
14 
48 
35 
20 

32 

19 

58 


o  33 
o  29 


TABLE   LIV. 

Latitudes  and  Longitudes 


[Page  345 


Spurn  lights 

Flamborough  Head  light. . 

Filey  Biig 

Scarborough  liglit 

Robin  Hood's  I3ay 

Wliitby  light 

River  Tees',  Seatou  lights. 

Stockton 

River  Tyne's  Mouth  lights 

Coquet  Island 

Staples  light 

Farn  liglits 

Sunderland  Point 

Holy  Island,  Castle 

BERWICK,  Pier  light  . . . 


St.  Abb's  Head 

DUiNBAR 

May  Island  lights 

The    Bass 

N.  Ber-\vick 

FDIiNBURGH 

Fllyness , 

Fife  Ness 

.St.  Andrew's, Pier  Head. 

Mouth  of  Tay  light 

Bell  Rock,  off  do  light... 

Buddonness  liglits 

Red  Head...-. 

Montrose  light 

Tod  Head 

NEW  ABERDEEN.... 

Newburgh 

Peter  Head,  Pier  light 

Buclian  Ness  light 

Ratrie  Head 

Kinnaird's  Head  light  . . . 

Barnff  light 

Fort  St.  George 

Inverness' 

Croinartie,  Pt  light 

Tarbet  Ness  liglit 

Clythness 

Noss  Head  hght 

Diincansby  Head 


Lat. 


D.  M. 

53  35  N 

54  7 
54  i5 
54  17 
54  27 
54  3o 
54  4o 

54  34 

55  I 
55  20 
55  39 
55  37 
55  36 
55  40 
55  46 


Lonnr. 


D. 


M. 

7E 

5 

iW 

23 

20 

37 

IS 

19 

25 
32 

40 
39 

38 
47 
59 


2  33 
2  38 

2  4i 

3  12 
2  M 

1  35 

2  47 
2  38 

2    23 

2  45 
2  29 
2  27 
2  i4 


46 
46 

49 


XVI.     The  Orkney  Islands. 


Pentland  Skerries  light . 

Slromo 

South  Ronaldsha,  S.  p.. 

Copinsha 

Lamb's  Head  on  Stromsa 

Island 

North  Ronaldsha,  N.  p.. 
Mould      Head,    on     Papa 

Westra  Island 

Nnup  Head, on  Westra  Isl. 
Marwick  Head,  on  Pomc-na 

Island 

Stromness 

Hoy  Head,  on  Hoy  Wells 

Island 

Slue  Skerry 

Fair  Island 

44  ~ 


Lat.        Lon<r. 


D.  M. 

58  41  N 
58  43 
58  44 

58  54 

59  4 
59  23 

59  23 

59   20 

59    6 

58  57 

58  55 

59  3 
59  33 


D.  M 

2  55W 

3  i4 
3  5 
2  4o 

2  32 
2  3i 

2  53 

3  04 

3  28 
3  18 

3  3i 

4  16 
I  38 


XVII      The  Shetland  Islands. 


Sumbro  Head,  S.  point  It. . 

Rose  or  Plangclilf 

Brassa  Sound,  Lerwick . . . 

Out  Skerries 

Whalsey  Isle 

Llnst  Island,  N.  E.  p 

Foul  Island [60 


Lat. 

D. 

M. 

59 

5iN 

00 

i3 

60 

II 

60 

37 

60 

32 

6t 

7 

60 

9 

Lon<: 


D.  M. 

I   16W 

0  4o 

1  00 
o    8 

O    32 

o   i5 

2  06 


XVIII.     Ferro  Islands. 


The  Monk  Rock  appears 
like  a  ship  under  sail. 

Fucloe  Island.  (N.  E.  part 
of  Ferro,)  E.pt 

East  point  of  Mygencs  Isl- 
and, (N.  W.  part  of  Fer- 
ro Islands,) 


I 

D. 

61 

jit. 

M. 

20  N 

62 

20 

62 

3 

Long. 

I).  M. 

6  4iW 

6  i3 

7  32 


XIX.     From   Duncanshy   Head  to   the 
Land's  End. 


Duncansby  Head 

Dunnet  Head  light 

Farout  Head 

Cape     Wrath,     or     Barre 

Head  light 

A  Rock  seen  at  %  ebb 

Rona  Island,  Sum 

Rochal 

St.  Kilda 

Butt  of  the  Lewis 

Gallen  Head 

Flannen  Islands,  N.  W.  . . . 

Hyskere  Island 

South  Uist  Island,  E.  pt.  . . 

Mingalay  Island 

Rea  Head 

Cana  Island 

Helsker  Island 

Rum  Island,  W.  p 

Tirey  Island,  S.  p 

Coll  Island,  N.  p.. 

Skerry vore,  Rock  light. . . . 

Ilia  Island,  S.  W.  p 

S.p 


Mull  of  Cantire  light-house 
Pladda  I.  light  off' S.  E.  pt, 
Arran  I. 

Little  Cumbrae  light 

GLASGOW 

Elsa  Island 

Irvine 

Air  light 

Loch  Ryan  light 

Port  Patrick  light 

Mull  of  Galloway  hght . . . 

Great  Scar  Island 

Burrow  Head 

Solway  Firth  light 

CARLISLE 


Lat. 


ni. 

40  N 

4o 

38 

37 

45 

7 

35 

49 
3i 

14 
i3 
37 
i3 
48 
54 
3 

56 
59 

27 
42 

19 

47 
39 

19 

26 
4^ 

52 

20 

37 
28 
58 
5o 
38 
40 
4i 
48 
54 


Long. 

I).  M. 
3  iVV 

3  21 
5  00 

4  59 

5  21 

5  49 
1 3  40 

8  35 

6  i4 


33 
33 
6  38 
6  38 
6  3o 
6  56 

6  27 

7  7 
6  24 
6  10 
5  49 


3  7 

4  5i 
4  36 
4  23 
3  32 
2  56 


Page  ;U6" 


TABLE   LIV. 

Latitudes  and  Longitudes. 


St.  Bee's  Head  light 

Wliite  Haven 

Selker  Rock 

Lancaster 

Forinby  Yiixht 

LIVERPOOL,  Obs 

Point  of  Air  light 

Great  Orms  Head 

Point  Linas  light 

Skerries  light 

Holyhead,  Pijer  Hd.  light . . 

Branchy  Pool  Head 

Bardspy  Island  light 

Barmouth 

Aberiswilh  light 

Cardigan  Harbor 

Struiiible   Head 

St.  David's  Head 

Raaisay  Island 

Small's  light-house 

St.     Ann's     do.,     Milford 

Haven 

St.  Gowan's  Head 

Caldy  Island,  South  light . 

Worm's  Head 

Mumble's  Point  and  light. 
Nash  Point,  two  lights  . . . . 

BRISTOL 

Flatholm  light 

Lundy  Island,  entrance  of 

Bristol  Ciiannel 

Mort    Point,    entrance    of 

Bristol  Channel 

Hartland  Point 

Padstow 

Cow  and  Calf 

To  wan  Head 

St.  Ive's  Bay,  Pier  Hd.  It. . 

Cape  Cornwall 

The  Seven  Stones  light  . . . 

The  Wolf  Rock 

The  Land's  End 


Lat. 


M. 

3iN 

33 

i6 

2 

3i 

25 


5i   lo 


Lons- 


D. 
3 
3 
3 

2 

3 
3 
3 
3 
4 
4 
4 
4 
4 
3 
4 
4 
5 
5 
5 
5 

5 
4 
4 

4    20 

3  58 
3  33 

2  35 

3  7 

4  4o 


M 

38W 
36 

19 

48 

10 

o 

5i 
17 
37 
37 
37 
48 

52 

o5 

38 

4 


5  48 
5  42 


XX.    Ireland. 


CAPE  CLE  AR  light  ... 

Fastnet  Rock  light 

Croolv  Haven  light 

Mizen  Head 

Slieep's  Head 

Bantry  Bay 

Grelagh   Rocks 

Dursey  Island,  W.  p 

Bull  Rock 

Cow    do 

Cod's   Head 

Kenmare  Bay 

Lamb's  Head 

Scara  Islands 

Hog's  Head 

Bolus  Head 

Skellig's     Rocks  lights. . . 

Lemon  Rock 

Bray  Head 

Dinole  Bay 


Lat. 


M. 

26  N 

24 

29 

27 

33 

35 

3i 

35 

36 

35 

40 

A3 

45 

A^ 

Ai 

47 

46 

48 

53 

00 


Long. 
D. 
9 
9 


U. 

29VV 

36 
.  43 
9  5o 

9  52 

9  53 
[o  10 
[o  i4 

[O  18 
[o  17 
10  08 

[O    I  I 

10  08 
10  i5 
10  i3 
to  20 

[O    32 

10  26 

[0    25 

10  27 


Foze  Rock 

Inishmakillaan 

Tiraght  Rocks 

Great  Blasket,  N.  pt 

Inishluiskero 

Dunmore  Head 

Dunorling  Head 

Brandon  Head 

The  Seven  Hogs  Rocks  . . 
Kerry  Head,    S.  entrance 

of  Shannon  River 

Loop  Head,  Light 

LIMERICK  Bridge 

Clare 

Hog's  Head 

North  Arran,  orlvillaney.lt. 

Galway  light 

Slyne  Head  Light 

Ennis  Shark  Island 

Ennis  Turk  Island 

Clare  Island  Light 

Achil  Head 

Black  Rock  outer 

Urris  Head  Eagle,   Id ... . 


Broad  Haven 

Tuns  Rocks,  off  Broad 
Haven 

Down  Patrick  Head 

KiUala 

Sligo  Bay  Light,  Black  R. . 

Ennis  Murray  Island 

Donnegal 

Tillon  Head 

Arran  N.  P 

Bloody  Foreland  Hill 

Tory  Island  lights 

Hoar  Head 

Mulroy 

Loch  Swilley,  Fannet  pt. . . 

Malliny  Head  light 

Ennistrahul  Rocks  light.. 

Inishone  Head,  entrance  of 
Londonderry  lights. . . 

LONDONDERRY  Brid; 

Giant's   Causeway,  pt.. . 

Rathlin    Island,  light  .. . 

Fair  Head 

Torr  Point 


The   Maid's  Rocks  light  . . 

Black  Head 

Carrickfergus 

BELFAST  

Belfast  L.  Hollywood  B.  Its. 

Mew  Island  lights 

South  Rock  light 

Dundrum 

Carlingford  Loch  light . . . 

Dundalk 

Drogheda  Bar 

St.  Patrick's  Island 

Lambay  Island 

Howth  Farb.  E.  Pier  Hd.  It. 

DUBLIN  observatory 

WICKLOW  lights 

Arklow  Light 

Glasscarrick 


Lat. 


M. 
iN 

2 

3 
6 
7 
5 
12 

17 
20 

23 

34 
4o 
5i 

5 

8 
i5 
29 
A6 

52 

5o 

59 

4 

16 


54  26 

54  3: 
54  20 
'54  It 
54  18 
54  26 
54  39 

54  41 

55  I 
55  8 
55  16 
55  i3 
55  17 
55  17 
55  23 
55  26 

55  i4 
55  00 
55  i5 
55  18 
55  i3 
55  12 

54  56 
54  46 
54  43 
54  35 
54  39 
54  45 
54  24 
54  17 
54  2 
54  o 
53  U 
53  2,6 
53  29 
53  24 
53  23.: 
52  58 
52  42 
52  34 


TABLE   LIV. 

Latitudes  and  Longitudes. 


[Page  347 


WEXFORD,  Raven  pt.. 

Tusker  Rock  light 

Carnsore  Point 

The  Saltees  Rocks  Hght. . 
Hook    hght,    VVatertbrd 

harhor 

Dungarven 

Hehviclc  Head  point 

Youo-hall  hght 

CORK  harbor  hght 

Kingsale  iiarborhght . . . , 
Old    Head    of   Kingsale 

hghts   

Seven  Heads , 

Dundedy  Head 

The  Stacrs,  ofTToe  Head 
BALTISIORE  harbor  . . 


Lat.        Lons. 


D.  IVI. 

52    20  N 
52    12 
52    II 
52       2 

52  7 
52  o5 
52  o3 
5i  56 
5i  48 
5 1  42 

5i  37 
5i  34 
5i  32 
5i  28 
5 1  29 


D.  M. 

6  21W 
6  12 
6  23 

6  40 

6  56 

7  38 
7  33 

7  52 

8  i5 
8  3o 

8  32 
8  AA 

8  58 

9  i4 
9  22 


XXI.     The  Isle  of  Man. 


Calf  of  Man  hghts , 

Douglass  hghts • 

Ramsey  harb.  light  S.  side 

Point  of  Air 

Peel  havb.  light  E.  side  . . . 
Castletown  harb.  light . . . . 


Lat.        Lous'. 


D.  M. 

54  3  N 
54  9 
54  20 
54  25 
54  i3 
54    5 


D.  M. 

4  5oW 
4  28 
4  23 
4  22 

4  42 
4  37 


XXII.     F}-om  Calais  to  the  Scaio. 


CALAIS 

Gravelines 

DUNKIRK  Pier  hd.  light. 

Nieuport 

OSTEiND 

Sluys  

ANTWERP 

Walcheren  Island,  W.  p.. 

FLUSHING  . 

Middleburgh  . 

Goeree  Island 

Schowen  Island  light 

Holland's  Hook 

The  Hague 

L^-yden  obs 

Haerlem 

ROTTERDAM 

AMSTERDAM 

Alkmaer 

Texel,  S.  point 

Harlino-en 

Ter  Schelling,  W.  end 

Gottinorcn,  obs 

EMBDEN 

Borcum  light 

Wranirer-oog  ligiit 

BPvEMEN 

Brenierlehe 

HAMBURGH 

Stade 

Glukstadt 

Cuxhaven  light 

Nework 


Lat. 


D.  M 

5o  58  N 

5o  59 
5i  3 
5i  8 

5i  14 

5i  19 

5i  i3 

5i  32 

5i  27 

5i  3o 

5i  46 

5i  43 

5i  56 
52  4 
52  9 

52    22 

5i  54 

52    22 

52  38 

53  2 
53  10 
53  22 
5i  32 
53  22 
53  36 
53  48 
53  5 
53  32 
53  33 
53  36 
53  48 
53  54 
53  55 


M. 

5iE 

7 
22 
45 
55 

23 

24 


4  if 


29 
38 
29 
53 
45 
33 


Elbe  River,  entrance 
Heligoland  light. .. . 

Tonningen 

Horn  Point 

Holmen 

Robsnout 

SCAW  light 


Lat. 

D.  M. 

54  00  N 

54  12 

54  19 

55  34 

57  8 

57  25 

57  43 

Long. 

D.  M.^ 

8  20* 


7  53 
9     5 

7  4o 

8  34 

9  34 
10  37 


XXIII.     CaUegat  and  Sound. 


SCAW  LIGHT 

Fladstrand 

Scbye  

Halls 

Grenaa 

Aarhus  oath 

Sleswick 

The  NAZE  light 

Christiansand 

Arendal,  Torungen  hght. . . 

Frederickavern 

Ferder  light 

CHRISTIANA  obs 

Frederickstadt 

Stronstadt 

Salo  Beacon 

Paternosters 

Marstrand  light 

GOTHENBURGH 

Wingo  Beacon 

Tislarne 

Niddingen  lights 

Warberg 

Falkenburg 

Halmstadt  fort 

Laholni 

Wadero  Island,  S.  end  . . . 

Engelholm 

Koll  light 

Helsinborg 

Landskrone  

Malmo 

Falsterbo  light 

SlefFen's  Head  hght 

Kioge 

COiPENHAGEN 

ELSINEUR. 

Cronenburg  light 

Nakke  Head  lights 

Nykoping 

Callundborg 

Corsor  lights 

Wordingborg 

Huen  Island,  Uraniberg.. 

Amag  Island,  Drago 

Hasel  Island 

Spro  Island  light 

Falster   Island,  Geedesbye 

or  Trindelen  light 

Moen  Island,  Speil  Cliff.. 

Anholt  light 

Little  Middle  Ground 

Lessee  Island,  E.  end  .... 


Lat. 


M. 

43  N 

26 

20 

00 

26 

3? 

58 

9 
23 
59 

2 
55 
12 
55 
21 
55 
53 
42 
38 
3o 
18 

6 
55 
40 

32 

26 

i4 

18 

2 

52 

36 

23 


55  18 
55  28 

55  41 

56  2 
56  3 
56  7 
55  55 
55  4i 
55  20 
55     I 

55  55 

55  35 

56  12 

55  20 

54  34 
54  58 

56  44 

56  57 

57  19 


Lons\ 


D. 


M. 

o  37E 

O    32 

o  3i 
o  19 
o  53 
o  i3 
9  33 
7  02 

7  59 

8  53 
o  12 
o  32 

0  43 

1  o 
I     12 

I  i4 
I  27 
I  35 
I  55 
I  36 
I  44 

1  55 

2  i5 
2  3o 

2  52 

3  00 
2  35 
2  52 
2  28 
2  42 

2  5o 

3  I 
2  49 

2  27 
2  12 
2  34 
2  37 
2  37 
2  22 
I  4o 
I     6 

'  r9 

1  59 

2  43 

2  38 
I  44 

0  57 

1  59 

2  34 
I  39 
I    59 

I     9 


Page  348J 


TABLE   LIV. 

Latitudes  and  Longitudes. 


Lessee  Island,  W.  end 
Trindelen  lifflit    


Lat. 


D.  M. 

57  i5N 
57  26 


Long. 


M, 

5oE 
16 


XXIV.     The  Baltic. 


Funen,  Odense. 
Nyborg. 


Langeland  S.  pt 

Aero,  Kiop 

Alsen,  Sondei-borg. . . . 

Laaland,  Naskou 

Nysted 

Falster  Nikioping 

Stubbekioping . 


Moen,  Stege  E.  j^t. 

Fermeren,  Borg 

Tralleborg 

Cimbrisliamn 

Ahus 

CARLSCRONA  .... 

Torum  Point 

Calmar 

Westervvyck 

Soderkoping  

Nykoping 

Trosa 

Landsort  liglit 

STOCKHOLM 

Kiel 

LUBECK 

Wismar 

Rostock 

Dars  Head 

Geblen  light 

Stralsund 

Grifswaldce  light 

Usedom 

Wollin 

Stettin 

Camniin 

Colberg  fort   

Rugenvalde 

Stolepemunde 

Heel  light 

DANTZIGobs.  ..,-.. 

Pillau  Ijo-ht 

KONIGSBERGobs. 

Brusterort  lights 

Jlemel 

Libau 

Windau 

Lyserort 

Domesness  lights . . . . 
Runo  Island  light.  — 

RIGA 

Pernau  

Dago,  Simperncss  . . . 
Danrerort  liofiit. 


Osel,  Palmerort 

llundsort 

Swasvcort  light 

Arensburgh. . . . 

Gottska  Sando  W.  pt. 

Faro,  N.  E.  end 

Gotland,  N.  E.  end  . . 


Lat.        Long. 


M. 

25  N 

19 

42 
54 
56 
5i 
42 
47 
54 
57 
28 
22 

56 

10 
5 
4o 
46 
3o 
46 
54 
44 
21 
21 

52 

54 
6 
28 
28 
18 
i5 
53 
49 

25 

57 
II 
23 

3o 

37 
22 

38 
43 
58 
44 

32 

24 
35 
46 

48 
57 

23 

6 
56 
39 
02 
55 
i5 
22 
56 
5i 


).  M. 

o  24E 
o  48 
o  42 

0  28 
9  52 

1  i5 
I  48 

1  54 

2  8 

2  33 
I  17 

3  II 

4  21 
4  18 


36 
53 
20 
38 
20 

3 

33 
53 

4 
o  10 

0  42 

1  28 

2  9 

2  3i 

3  12 
3  6 

3  56 

4  5 
4  41 
4  34 

4  5o 

5  38 

6  25 
6  5o 
8  46 

8  4i 

9  54 

20  3o 

19  59 

21  o 

20  57 

21  34 

21  45 

22  37 

23  II 

24  6 
24  3o 
22  32 
22  12 
22  28 

21  5o 

22  5 
22  28 
19  19 
19  26 
19  2 


Gotland,  WISBY 

Hoburg 

Great  Carlso 

Oland,  N.end 

Borgholms  Slott 

S.  end  light  .... 

Eartholms 

Bornholm,  N.  W.  end, 

light 

Hasle 

S.  E.  end... 

Svaneke  . . . 

Rugen, N.  end 

BERGEN 

S.    E.     end    New 

Deep 


Lat. 

D. 

M. 

57  39 N| 

56 

b7 

57 

19 

37 

22 

56 

52 

56 

12 

55 

19 

55 

18 

55  10 

54  58 

55  S 
54  4o 
54   25 

5.4   1 6 


Lon<r. 

D.  M. 

18  20E 

18  9 
18  2 
17     7 

16  37 

16  26 

i5  12 

i4  47 

:i4  47 

i5  12 

1 5  i3 

(3  3o 

i3  28 

i3  5o 


XXV.     Gulfs  of  Finland  and  Bothnia. 


Odensholni  light. 

Packerort  light 

Surep  Head  liglit .  .■ 

Nargen  Island,  N.  point.. 

REVEL 

Kokskar  liglit 

Chalk  Ground 

Stone  Skar  beac 

Little  Tyters  Island 

Great  Tyters  Island 

Lavanscar,  N.  end 

Soskar  light 

Narva 

Dolgoinos 

Tolbakon  light 

CRONSTADTcath 

PETERSBURG  

Stirsudden 

Wiburg 

Fredericksham 

Aspo  beac 

Hogland  Island,  N.  light. . 
Orrenground's  Beacon  . . . 

Lovisa 

Borgo  

Helsingfors 

Great  Jussari 

Hanaro  Beacon 


Lat. 

D.  M. 

59  19N 

59  24 

59  28 

59  36 

59  26 

59  42 

59  41 

59  49 

59  48 

59  5o 

60  2 

60  02 

59  20 

59  54 

60  2 

5959 

59  56 

60  12 

60  43 

60  34 

60  18 

60  5 

60  17 

60  27 

60  2  1 

60  10 

59  5o 

59  45 

Lon^. 


D.  M. 

23  22  E 

24  o 
24  23 
24  32 

24  46 

25  2 

26  9 
26  21 

26  58 

27  i5 

27  52 

28  23 

28  4 

29  o 

29  34 
99  47 

30  19 
29    o 

28  46 
27  16 
27  i3 
27  00 
26  37 
26  16 
25  45 
■p4  58 
23  33 
22  57 


XXVI.     Gulf  of  Bothnia. 


Lat.        LoviT. 


|D.  M. 

Uto  light ;59  47-^ 

Abo ;6o  27 

Wasa 63     4 

TORNEA 65  5i 


1).  i\I. 

2!     22  E 

22    17 
2  1    43 

24  i4 


XXVII.     From  the  J^aze  to  Archans;el. 


The  NAZE. 
Lister  Land . 


Lat.        Lnvs 


D.  M. 

57  58  N 

58  6 


D.  M. 

7     3E 
6  36 


TABLE   LIV. 

Latitudes  and  Longitudes. 


[Page  349 


Judder,  or  Walbert's  Head 
Great  Wylingose  light- 
house   

Stavanger  

Bomniel  Island,  S.  end  . 

BERGEN 

Askwold 

Ronde  light 

Ciiristiansund  light 

Droutheiui 

Werro  Island 

NORTH  CAPE  

Wardhuus  Island 

River  Kola 

Naorel  Island 


Sviatoi  Noss  Tower. .... 

Cape  Orlogenose 

Cross  Island 

Onega  Churcli 

Cape  Donega  

ARCHANGEL 

Bluenose,  or  Cape  Katness 

Cape  Good  Fortune 

Morshovet's  Island,  S.  pt. . 

Cape  Candinose 

Welgate's  Straits 

Nova  Zembla .♦ 


Lat. 


D.  31. 

58  36  N 

59  4 

58  59 

59  35 

60  24 

61  22 

62  25 

63  7 
63  26 

67  42 
71  10 
70  23 

68  52 
68  32 
68  10 

67  i^ 
66  28 

63  54 

65  8 

64  32 

65  26 

66  3i 

66  4o 

68  39 
70  5o 
76  34 


L07l£. 


M. 

4oE 

26 

45 


10  23 

ir  4i 

25  46 
7 
33  I 

38  o 

39  47 
4i  22 

40  28 

38  8 
36  47 
4o  33 

39  54 

42  53 

43  27 

44  33 
57  45 
62  45 


XXVIII.     From  Ushani  to  Gibraltar. 


Lat. 


USHANTlight  

BREST 

St.  Mattliew's  light 

Point  Raz  light 

Saints'  Rocks 

Point  L'Abbe 

Quimper 

Glenan  Islands 

Quiinperlay 

L  ORIENT 

Port  Louis 

Isle  de   Groas  It.  N.  W.  pt, 

Quiberon,  S.  point 

Belie  Isle,  N.  end 

S. end  

Vanues 

Houat  Isle 

Dumet  Isle 

NANTES 

Croisic 

St.  Gildas  Point 

Noirnioutier  Island,  S.  W. 

Isle  DTeu  light 

St.   Gilles 

Roclies  Bonnes  W 

Isle  of  Rhe  li<rht  N.  W.pt. 
ROCHELLE  light....:.. 

ROCHEFORT  

Oleron  Isle  liglit 

Island  Ai.\- hglitat  S.  pt.  . 

CORDOUAN  light 

Medoc 

BORDEAUX 

Cape  Feret  light 

BAYONNE  


Long. 


D.  M. 

D.  M. 

48  29  N 

5  3W 

48  23 

4  29 

i^i   20 

4  44 

48  2 

4  44 

48  4 

5  5 

47  49 

4  12 

47  58 

4  8 

47  44 

4  00 

4l  52 

3  34 

47  45 

3  21 

47  43 

3  21 

47  39 

3  3o 

47  26 

3  4 

47  23 

3  i4 

47  17 

3  5 

47  39 

2  46 

47  24 

2  56 

47  22 

2  36 

47  i3 

1  33 

47  18 

2  3r 

47  10 

2  16 

47  00 

2  i5 

iQ  43 

2  23 

46  4i 

I  56 

46  i5 

2  24 

46  i5 

I  34 

46  9 

I  9 

45  56 

0  58 

46  3 

I  24 

46  I 

I  II 

45  35 

I  10 

45  6 

0  45 

U   5o 

0  34 

44  39 

I  i4 

43  29  ) 

I  29 

ht. 


Lat. 


St.  John  de  Luz. , 

St.  Sebastian  light. 

Cape  Machichaco, 

BILBAO 

Santona  

SANTANDER  Ik 

St.  Vincent 

Villa  Viciosa 

Cape  Penas  

Ribadeo,  entrance .... 

Cape  Burcla 

Cape  Vares 

Cape  Ortcgal 

Cape  Prior 

FERROL  

CORUNN A  liglit.... 

Cape  Villano. 

Cape  Turiana 

Cape  Finisterre 

Point  Corrobedo 

Vigo  light  in  castle. . . 

Cape  Fasalis 

OPORTO  light 

I  Averios 

, '  Coiinbra 

■Jj  I  Cape  Mondego 

Sf  I  Cape  Fiseraon 

-  Tiie  Burhngs  light . .  • 

^  Cape    Carboeiro  hglit. 

"*  jThe   Rock  of  Lisbon  . 

LISBON   

Cape  Espichcl  light . . 

St.  Ubcs 

Cape  Sines  fort    

Cape  St.  Vincent  light 

Lagoa  

Cape  Carbonera 

Cape  St.  Mary 

Point  Arenilla 

St.  Lucar 

SEVILLE 

CADIZ  light 

Cape  Trafalgar 

Tarifa  Island  light . . . 

Point  Carnero 

Algesiraa  mole 

GIBRALTAR  mole |36 


D.  M. 

43  23  N 
43  20 
4'i  28 
43  i5 
43  27 
43  3o 
43  3o 
43  28 
43  42 
43  35 
43  42 
43  47 
43  45 
43  34 
43  3o 
43  22 
43  10 
43  3 
42  54 
42  35 
42  i5 
4i  59 
4i  09 
4o  39 
4o  i3 
4o  12 
39  24 
39  25 
39  21 
38  46 
38  42 
38  24 
38  32 
37  57 
37  3 
37  8 

37     7 

36  57 

37  8 
36  44 
36  59 
36  32 
36  10 


Lon/r. 


D. 


M. 

38  W 
00  5 

^9 
54 
26 

47 
16 
18 
46 
o5 
21 
4i 


3 
3 

4 
5 
5 
7 
7 
7 

7  56 

8  19 
8  i3 

8  24 

9  i5 
9  21 
9  21 

9  7 
8  4o 
8  45 
8  37 
8  38 
8  24 

8  54 

9  18 
3o 
24 
29 

9 
i3 


5 


8  5o 

8  53 

9  o 
8  39 
8  19 

7   52 

6  5o 
6  24 

5  58 

6  18 
6  00 
5  37 
5  23 
5  26 
5  21 


XXIX.    J\rorlh  Coast  of  the  Mediterranean 
Sea,  from  Gibraltar  to  Constantinople. 


GIBRALTAR, 

Europa  Point 

Cape  fllorat 

MALAGA  mole 

Corchuna  castle 

Almeria 

Cape  de  Gait, 

Coralletes  tower  .... 

Cape  Tinosa 

CARTHAGENA,  obs.. . . 

Escombrera  Island 

Cape  de  Palos 

Cape  Cervera, 

Torre  Vieja 

Cape  Santa  Pola |38   12 


Page  350] 


TABLE  LIV. 

Latitudes  and  Longitudes. 


ALICANTE 

Cape  St.  Martin 

Cape  Cullera  tower 

VALENCIA,  city 

Cape  Oropera  tower 

River  Ebro, 

Buda  Island. . 

Cape  Salou 

TERRAGONA 

BARCELONA -.. 

Cape  Tosa • 

Cap       t.  Sebastian 

Medas  Isles,  S.  end 

Gulf  of  Rosas,  Cape  Norfeo 
Cape  Creux 

Port  Vendre 

Perpignan 

Narbonne 

Ajrde  liarb 

Cetta  light 

Montpellier,  obs 

St.  Louis's  tower, W. mouth 

of  the  Rhone . , 

Cape  Couronne  

MARSEILLES,  obs 

Planier  Island  light 

Cape  Roux 

Ciotat 

Cape  Sicie 

TOULON  

Hycres  Islands, 

Porquerolles, 

,  Fort  Langoustier 

Portcross,    Gabi- 

niere  Rock  . . . 

Titan,  E.  point. . 

Cape  Taillat 

St.   Tropcz 

Frejus '. 

Antibes '. 

Nice 

Villa  Franca  light 

Ventimiglia  Point 

Port  Maurizio,  mole 

Cape  de  la  Melle 

Ca-peNoli 

Savona,  mole 

PollaRock 

GENOA  light 

Cape  Porto  Fine 

Cape  dell  Mcsco 

Port  Vcncre 

Pisa,  obs 

FLORENCE  

Melora  Slmal  tower 

LEGHORN  light 

Mai  di  Vctro  Shoal 

Piombino,  mole 

Castigliono  tower 

Mount  Argentaro 

Civita  Vecchia  light 

Cape  Linaro  

Fiumicino  li(rht,  R.  Tiber. 

ROME,  St.  Peter's 

Port  Anzo,  mole 


Lat.       Lons- 


D.  M. 

38  20  N 

38  47 
89  12 

39  28 

40  6 


4o  43 

4i  5 

I  II 

4i  9 

I  18 

4i  23 

2  II 

4i  43 

2  58 

4i  54 

3  i3 

42  3 

3  i3 

42  i5 

3  17 

42  20 

3  20 

42  32 

42  42 

43  II 

43  17 
43  24 

43  36 

43  21 
43  19 
43  18 
43  12 
43  i3 
43  10 
3 
7 


43  o 

42  59 

43  2 
43  8 
43  i5 
43  26 
43  35 

43  4i 
43  40 
43  4i 
43  53 

43  58 
U   II 

44  18 
44  25 
M  24 
44  18 
44  8 
44  4 

43  43 
4i  46 
43  33 
43  32 
43  20 
42  55 
42  45 
42  22 
42  5 
42  I 
4i  46 
4i  54 
^■\   27 


D.  M. 

o  26VV 
o  loE 
o  i3W 
o  24 
o  loE 


6 
54 

o 
28 
4i 
53 


4  4i 

5  2 
5  22 
5  i4 
5  20 
5  36 
5  5o 
5  56 


6  12 


7  17 
7  20 

7  43 

8  o 
8  II 
8  23 
8  28 
8  46 

8  53 

9  i5 
9  4o 

9   52 

TO  24 
1 1  16 
10  i3 
ic.  18 
I )  22 

I  3    32 

10  55 

II  12 

11  45 

1 1  5o 

12  12 
12  27 
12  42 


Cape  Monte  Circello. 


Gaeta,  mole 

Cape  Miseno 

NAPLES  light 

Salerno 

Cape  Licosa  tower. . . . 

Policastro 

St.  Euphemia 

Cape  Vaticano 

Scylla  castle 

Cape  dell  Armi  tower. 
Cape  Spartivento  . . . .  , 


Cape  Stilo 

Cape  Rizzuto 

Cape  Nau  or  Collone 

Point  del  Tronta  tower. . . 
Tarento,  St.  Pietro  Island. 
Gallipoli,  St.Andrea'Island 
Cape  St.  Maria,  convent 


Cape  Otranto,  tower. . . . 

Brindisi,  fort 

Polignano 

Bari,  mole 

Barletta  light 

Manfredonia,  mole 

Viesti  Groce  Island  .... 

Termoli 

Point  Penna 

Ortona,  mole 

Pedaso  

ANCONA  light 

Fano  light 

Pesaro  light 

Rimino  light 

RAVENNA,  tower.... 
Point  della  Maestra  .... 
VENICE,  St. Mark's  tower 

Port  Lijrnano , 

TRIESTE,  castle 

Point  Salvore  light 

Rovigno , 

Cape  Promontore , 

Fiume 

Segna,  mole , 

Karlopago,  mole 

Zara  . . ; 

Sabenico 

Spalatro,  Paulino  tower 

RAGUSA,mole  

Kattaro,  Point  Ostro  . . . 

Dulcigno,  mole , 

Cape  Rodoni 

Cape  Pah , 

Durazzo,  mole 

Aulona,  or  Valona 

Cape  Linguetta , 


Butrinto 

Parga,  town 

Previsa,  Fort  Pantokratera 
O.xia  Island,  S.  end. . . , 

Missolonghi  fort 

Roumelia  castle 

Lepanto  

CORINTH 


Lat. 

D.  31. 

4i  12H 

4i  12 
40  46 
4o  5o 
4o  4o 
4o  i4 
4o  2 
39  3 
38  37 
38  i4 
37  58 

37  56 

38  28 

38  57 

39  6 
3935 

40  26 
4o  2 

39  49 

40  8 

40  39 
4i  o 
4i  9 

4 1  20 
4i  38 
4i  53 

42  00 
42  10 

42  21 

43  6 
43  38 
43  5i 

43  56 
M  5 
U  25 

44  59 

45  26 
45  4i 
45  39 
45  3o 
45  5 

44  46 

45  20 
45  00 
44  32 
44  7 
4^  44 
43  3o 
42  38 
42  23 
4i  54 
4i  38 
4i  23 
4i  18 
4o  27 
4o  27 

39  47 
39  17 
38  56 
38  17 
33  21 
38  19 

38  22 

37  54 


TABLE   LIV. 

Latitudes  and  Longitudes. 


fPage  3bl 


Patras,  mole 

Cape  Pa2:)as 

Toruese  castle 

Cape  Catakolo 

Cape  Konello 

Navariuo  castle 

Coroii,  or  Koron 

Capo  Matapan 

Cape  St.  Angclo  

Nauplia,  or                ) 
Napoli  cli  Eomauia  j 

ATHENS,  PbUopapus 

Cape  Colonna 

Cape  Marathon 

Kcgropont,  fort , 

Cape  Doro 

Cape  Kill 

SALONICA 

Cape  Drcpaiio 

Cape  Pailiouvi  

Mount  Atlios 

Contcssa 

Ca  pc  I'axi , 

The  Dardanelles 

Gallipoli  light 

CONSTANTINOPLE, 

St.  Sophia 

Scutari 

Prince's  Isles,  westernmost 
Marmora  Island,  S.  W.  end 
E.  end,  lisrht 


Lat.        Long. 


D.  M. 

38  i4N 
38  i3 
37  54 
37  38 
37  12 
36  53 
36  47 
36  22 

36  26 

37  34 


37  58 

37  39 

38  07 
38  28 
38  10 

38  40 
4o  39 

39  57 

39  55 

40  9 
4o  5o 
4o  37 
4o  2 
4o  25 


4i  I 
4i  I 
4o  52 

4o  37 
4o  38 


D.  M. 

21  46  E 
21  26 
21  10 
21  20 
21  35 

21  4i 

22  00 

22  28 

23  i3 

.2  3  48 


23  44 

24  2 
24  5 

23  35 

24  36 
24  10 

22  57 

23  57 

23  46 

24  20 

23  52 

26  5 
26  12 

26  4o 


28  59 

29  I 
28  59 
27  35 
27  46 


XXX.     The   Black   Sea   and   Sea  of 

Azof. 


Bosphorus,  European  light 

Houriras  City , 

VARNA,  S.E.  bastion. 

Cape  Calaghi-iah 

Moutiis  of  the  Danube, 
Soulineli  liirht 


Ackonnan  

ODESSA  

Kherson 

Tendra  Island  lijrht 


Cape  Tarkan  liglit 

Koslof 

Cape  Chersonesus  light. 

Sevastopol 

Cape  Karak 


Tnjranrok,  Church . 

AZOF 

Sougoudjak 


Poti,  or  Phaz,  new  fort. 

Trebizonde 

Cape  Vona 

SLNOPE 

Heraclea  light 

Bosphorus,  Asia  light  . . 


Lat. 


Lon(r. 


XXXI.     The  East  and  South  Coast  of 
the  Mediterranean. 


Cape  Janissary  . . . . 

Cape  Baba 

Adramytti 

SMYRNA 

Cape  Karabouroun. 

Cape  Koraka 

Cape  St.  Mary 

Cape  Crio 

Gulfof  Makry, 


Cape  Iria. 


Seven  Capes 

Cape  Khilidonia 

Cape  Karaboornoo 

Cape  Anamour 

Cape  Cavallere 

Point  Lissan  e!  Kabeh. . . 
Karadash  Boornoo 


Alcxan 


Scanderoon, 

dretta 

Cape  Khynzyr , 

ALEPPO 

Latakia , 

Tortosa 

Tripoli 

Cape  Bairout 

Acre 

Jaffa , 

El  Arish 

Damietta 

Cape  Bourlas 

CAIRO 

Rosetta , 

Aboukir,  tower , 

ALEXANDRIA  light 
Ras  al  Kanais 


Tifarh  Rocks , 

Cape  Luko 

Bomba,  port  of, 

Bhurda  Island 


Cape  Razatin 

Derna 

Cape  Razat  . . . 
Bengasi,  castle 
Gharra  Island  . 

Kudia 

Boosaida 

Shaiusha 

Cape  Mesurata, 


N.  extreme 


Ziliten 

TRIPOLI,  coftle 

Zoara 

Jerba  Island,  Ziig  castle  . , 

Kabes 

Karkenna  Islands, 

Kusha  Island . 

Cape  Burdj  Kadija , 

Soussa,  mole 

Cape  Bon,  N.  point 

Zeinbra  Island,  middle.. 
TUNIS,  city 


Lat. 

D. 

4o 

39 
39 

38 
38 
38 
37 
36 


M. 

I  N 
29 
36 
26 
4o 

6 
39 
42 


38 


33 


36  35 
36  16 

36  II 
35  3i 
34  5o 
34  26 
33  5o 
32  54 
32  3 
3i  6 
3t  25 
3i  35 

30  3 
3i  25 
3i  21 
3i  12 
3i   16 

3 1  36 
3i  52 

32  23 

32-34 
32  46 
32  56 
32  7 
3o  47 
3o  44 
3t  00 
3i  10 

32  25 

32  3o 
32  54 

32  55 

33  53 

33  53 

34  49 

35  10 

35  4S 

37  6 
37  9 

36  47 


Long. 

D.  IM. 

26  i3  E 

26  5 

27  2 

27  7 
26  03 

26  37 

27  4 
27  21 


29  2 

29  10 

30  2O 
3[  43 

32  49 

33  43 
33  59 
35  21 


36  i5 
35  52 

37  10 
35  46 
35  5o 
35  49 
35  28 
35  6 
34  44 
33  56 
3i  47 

3i  18 
3o  28 
3o  6 
29  53 
27  52 

26  16 
25  3 

23  t6 

23  1 3 

22  4i 

21  39 
20  3 

19  57 

18  18 

17  39 

17  10 

i5  10 
i4  34 
i3  II 
12  4 
10  53 


I  19 

I  10 

10  39 

3 

10  49 
6 


Page  302] 


TABLE   LIV. 

Latitudes  and  Longitudes. 


Point  Farina  . . 
Piana  Island  . . 
Cape  Bianco  . . 
Cape  Serrat. . . 
Tabarca  Island 


Cape  Ross 

Cape  Mavera 

Bona,  town 

Cape  Ferro 

Capes  Bugar'oni 

Cape  Carbon 

Cape  Dellys  or  Tedilles 

Cape  Bingiit 

Cape  IMatafou 

ALGIERS  light 

Cape  Tennez 

Cape  Kulnieta 

Cape  Ferrat 

Cape  Falcon 

Cape  Figalo 

Cape  Guardia 


Lat. 


D.  M. 

37  iiN 
37  II 
37  20 
37  i4 
36  56 

36  55 

36  58 

3e  54 

37  6 
37  7 
36  47 
36  55 
36  57 
36  5i 
36  47 
36  33 
36  16 
35  55 
35  48 
35  3 1 
35  18 


Loner. 


Cape  Tres  Forcas '35 

Al  Buzenia,  I 

Garrison  Rock '35 


Pescadores 135 

Cape  T'jtuan = 35 

^  {  Cape  Negro '35 

^   Ccuta,  Alniina  Point 

TANGIER 

Cape  Spartel 


D.  M. 

10  i5E 
10  18 

9  48 
9  10 

8  43 

8  i3 
7  5o 
7  48 
7  II 
6  3.9 
5  5 
4  9 
3  55 
3  12 
3  4 
I  21 
o  27 
o  18W 

0  48 

1  10 
I  4i 


35  54 
35  47 
35  48 


Lat. 


iD.  M. 

Cape  Falcone Uo  59 ; 

Cape  Caccia '40  33 

Mai  di  Ventri  Island 39  69 

Cape  St.  Marco 39  5i 

39  46 
39  9 
38  58 
38  52 

38  52 

39  12 


Cape  Frasco 
(f  St.  Pietro  Island,  Carlo  fort 
■3  ;  St.  Antioco  Island,  S.  pt.  . 

'^  I  Toro  Rock 

j]  I  Cape  Teulada 

!»   CAGLIARI,  mole 

Cape  Carbonara, 

—  Cavoli  Island,  tower. . . 

Montorio  Island,  N.  E.  pt. 

Madelaine  Island,  N.  pt... 


Giraglia  Island,  tower ....  43     2 

Cape  Corso,  N.  pt 43     i 

St.  Fiorenzo 42  4i 

Calvi 42  33 

Cape  Turghia,  tower 42   i4 

Ajaccio '41   55 

Bonifacio,  tower 4i   23 

Port  Vecchio 4i  35 

BASTIA '42  42 


XXXII.     Islands  in  the  Mediterranean, 
Gidf  of  Venice,  and  Archipelago. 


Lat. 


A  Iboran  Isle 

Formentera, 

Point  del  Agulla  . 

•  La  Mola,  or  E.  pt 

IVICA, 

Point  den  Serra,  N.  p. 

Cape  Nono 

Bedra  Island 

Cabrera  Island, 
Point  Anciola 


MAJORCA, 

—  Point  Piobagada,  W.  pt. 

—  Cape  Cala  Figuera  . . . , 

—  Palina,  town 

—  Cape  Salinas,  S.  point, 

—  Cape  de  Pera,  E.  point, 

—  Cape  Formenton,  N.  pt 

MINORCA, 


Cape  Dartuch 

Cape  Minorca 

PORT  MA  HON, 

Cape  Mola. 

—  S.  E.  point. . . 


Cape  del  Testa   or  Longo 

Sardo,  N.  end 

Asinara  Island,  N.  E.  end. 


D.  IVI 

35  59  N 

38  38 

38  39 

39  8 
39    3 

38  5i 

3o    5 


9  M 
39  28 
39  34 
39  i4 
39  42 
39  57 


39  56 

40  3 

39  53 
39  47 


4i  i4 
4i     8 


Lon^. 

D.  M. 
3     iW 


23  E 
35 


32 
23 


2  53 


23 

3i 

39 

5 

27 
i5 


Gorgona,  tower i43  25 

Capraja,  middle 43     3 

I  ELBA.  .| 

'-  Port  Ferragio ....\4'J-  49 

Cape  Dorana  ....  [42  48 

;  Pianosa,  N.  point |42  35 


4  23 
4  20 


9    9 
8  18 


Africa    Rocks,  or    West 

Formiches 

Monte  Christo,  middle  . . 

East  Formiches 

Giglio  Island,  middle. . . . 

Gianuti,  middle 

Palmarola,  N.  pt 

Ponza,  S.  pt 

Botte  Rock 

Vandotena,  N.  end 

Iscliia,  S.  pt 

Capri,  Point  Carena 


Faro  light,  N.  E.  pt 38  16 

MESSINA  light 33  12 

Taormina 37  48 

Catania,  mole 37  28 

Syracuse  light 37     3 

Cape  Passaro,  S.  E.  pt. . . .  36  4o 
Cape  Scalambra,  town  ...  36  A& 

Alicata  castle 37     4 

Cape  Bianco 37  22 

Cape  St.  Marco 37  29 

Cape  Granitola 37  34 

Cape  Boco,  W.  point 37  48 

TRAPANl  lio-ht 38     3 

Cape  St.  VitoV  N.  W.  pt..  38  12 

Cape  di  Gallo 38  1 5 

PALERMO  light 38     8 

Cape  ZafFarana 38    6 

Cfefau,  cathedral 38     o 

Cape  Orlando 38     8 

Melazzo  liffht 38  16 


42  21 
42  20 
42  37 
42  20 
42  i4 
4o  57 
4o  53 
4o  5o 
4o  48 
4o  40 
4o  3 1 


EOLIAN   ISLANDS, 

Stromboli 

Panaria,  S.  W.  pt. 


38  48 
38  38 


Long. 

D.  M. 

•  8  1 1  J 

8  5 
i  8  16 
8  26 
I  8  27 
8  17 
8  26 
8  25 

8  39 

9  7 

9  32 

9  36 
9  25 

9  24 

9  23 

9  18 
8  45 
8  33 

8  Ai 

9  9 
9  20 

9  27 

9  52 
9  5c. 

10  20 
6 
7 


o  20 
o  53 

0  58 

1  9 

2  52 

2  58 
6 
26 
56 


i3 
t3 
i3 
f4  12 

i5  4i 


35 


^  17 
5  8 
4  3o 
3  56 
3  16 
3  00 
2  37 

2  25 
2  3o 

2  4G 

3  19 
3  22 

34 

4 

45 

i4 


TABLE  LIV. 

Latitudes  and  Longitudes. 


[Page  353 


EOLIAN  ISLANDS, 

Lipari,  castle ■ 

Vulcano 

Salina,  N.  part. . . . . 

Felicudi 

Alicudi 

Ustica 

.EGADIAN  ISLES, 

Marilimo,  castle  . . . 

Levanso , 

Favignana,  castle  . 

Pentellaria,  fort 

Liiiosa 

Lainpcdusa 

Lampion 

Goza,  N.  W.  point 

MALTA,  Valetta...... 

Point  Benhisa 

Tefola  Rock . . 


Esquerque  or  Skirki  Rocks, 

middle 

Koith's  Reef  

Galila 

Sorelli  Rocks 


Unie 

Prenmda 

Ulbe,  mole 

Grossa,  or  Lunga, 
Punta  Bianchi 


Brazza.  W.  end 

Porno  Rock 

Busi,  signal 

Lissa 

Lessina,  Fort  Imperial. . 

Curzola, 

St.  Giovanni  di  Bl. 


Cazza 

Lagosta, 

.Mount  St.  Geor<rio 

.Mclcda,  Point  Grui .... 

Pelagosa,  signal 

Cajola  Rock 

Planosa,  signal 

Tremili,  St.  Nicola 


Fano,  N.  W.  pt 

Samotraki,  N.  W.  pt 

CORFU,   Cape    Draste, 

N.  W.  pt. 

Cape    Bianco    light, 

s.  E.  pt : 

Paxo  light 

Leucadia, 

Cape  Ducato,  S.  pt, 

CEPHALONIA, 

Cape  Viscardo,  N.  pt 

Cape  .Aterra,  N.W.pt 

Cape  Skala,  S.  E.  pt 

ITHACA,    Point    Agiani 

S.  end 

Zante,  Capo  Skinari,  N.  pt. 
Cape  Kieri,  S.  pt 


Stamphanes, 

Convent  Island 


Cerigo,  Cape  Spati,  N.  pt 

Kapsali,  S.  end 

Porri 


iMt. 


D.  M. 

38  28  N 
38  23 
38  36 
38  34 
38  33 
38  43 

38  o 

38  2 
37  57 
36  5i 
35  52 
35  29 

35  33 

36  4 
35  54 
35  49 

35  47 

37  46 
37  5o 
37  3i 

37  2  5 

44  38 
44  20 
44   23 

44  9 
43  ig 
43  5 

42  58 

43  3 
43  II 

42  58 
42  46 

42  45 
42  4i 
42  24 

42  23 
42  i4 
42  8 

39  52 

39  46 

3948 

39  21 
39  12 

38  34 

38  29 
38  22 
38  3 

38  19 
37  57 
37  39 

37  1 5 

36  23 
36  7 
35  58 


Loner- 


D.  M. 

4  58E 
4  56 
4  48 
4  3o 
4  17 
3  II 


54 

52 

35 

19 
8 
3o 
33 
29 


o  47 
o  56 
8  55 

8  37 

4  i5 

4  37 
4  47 

4  5o 
6  27 

5  28 

6  I 
6  12 
6 

6 
6 

6 

7 
6 
6 
5 
5 


9  19 

9  27 

19  38 

20  7 
20  9 

20  33 

20  33 
20  24 
20  47 

20  46 
20  42 
20  5o 


22  57 

23  o 

23  i5 


45 


Cerigotto,  S.  point. 


Cape  Crio,  W.  pt 

Cape  Spada 

Cape  Maleka 

Cape  Retymo 

Cape  Sassoso 

CANDIA  

Cape  St.  John 

Cape  Sidero,  E.  end. . . 
Cape  Salimon,  E.  end. 

Gaidronisi  Island 

Cape  Malata 

Gozzo  Island,  W.  pt.. . 
Anti  Gozzo 


HYDRA,  summit 

Egina  Peak 

Zea,  port  entrance 

Thermia,  S.  point 

Andros,    Cape    Guardia, 

N.  W.  pt 

Tinos,  St.  Nicola  Road... 
Miconi,  N.  W.  mount.... 

Syra,  summit 

Siphanto 

MILO,  town 

Nio,  summit 

Naxia,  town 

Amorga,  E.  end 

Santorin,  Mt.  St.  Elias.  .. 

Skyros,  Mt.  Cochila 

Scopelo,  or  Scopoli, 

Mount  Delphi 


St.  Estrati,  summit 

LEMNOS,  N.  W.  pt 

Cape  Stala,  S.E.  pt. 

Point  Blava,N.E.pt. 


TENEDOS,  Peak 

Samothraki ,  summit 

MYTELEN,    Cape   Sigri. 

W.  end 

Port  Longoni,  cnt. .. 

PortOIiveir. entrance. 

S.  E.  pt '. .' 

ftlytelen 

SCIO, 

Cape  Mastico,  S.  end 

Scio  lights 

Nicaria,  S.  W.  mount 

SAMOS, 

Port  Vathi,  N.  E.  end 

Patino,  or  PATMOS, 
S.  mount 


Calymnos,  summit., 
Cos,  town,  N.  E.  pt. 

Scarpanto,  S.  pt 

-  N.  point  . 


RHODES,  mole 

St.  Catherine's  Island 

Cape  St.  John 


Cape  Salizano,  VV.  pt 

Cape  Cormachiti 

Cape  St.  Andrea,  N.  E.  pt. 

Cape  Grego 

Cape  Gatte 

Baffa,  or  PAPHOS 


Lat. 


D.  M. 

35  49  N 

35  16 
35  4i 
35  35 
35  25 
35  25 
35  21 
35  19 
35  18 
35  9 
34  53 
34  55 
34  52 
34  56 

37  20 
37  42 
37  40 
37  18 

37  58 
37  33 


37  29 

25  21 

37  29 

24  55 

36  58 

24  43 

36  42 

24  3o 

36  43 

25  21 

37  6 

25  22 

36  54 

26  6 

36  22 

35  29 

38  5o 

24  37 

39  8 

23  42 

39  3 1 

25   2 

39  59 

25  3 

39  48 

25  24 

4o  2 

25  28 

39  5o 

26  4 

4o  27 

25  3b 

39  II 
39  5 

39  \ 
39  6 

38  8 
38  22 
37  3i 

37  46 

37  17 

37  o 
36  53 
35  23 

35  5o 

36  27 
35  52 

38  4 

i5  6 
35  24 
35  42 
34  58 
34  33 
34  47 


Long. 

D.  M. 

23  18  E 

23  32 

23  44 

24  8 

24  4i 

25  o 
25  8 

25  47 

26  18 
26  19 
25  43 

24  45 

24   2 

23  59 

23  28 

23  3o 

24  19 
24  23 

24  43 


25  5o 

26  5 

26  33 
26  35 

25  59 

26  5 

26  3 

27  o 

26  35 

26  59 

27  17 
27  10 

27  II 

28  i3 

27  46 

28  4 

32  17 

32  57 

34  37 

34  6 

33  I 
32  26 


Page  354] 


TABLE   LIV. 

Latitudes  and  Longritudes. 


XXXin.     The  Coast  of  4fnca,  from 
Cape  Spartd  to  Cape  Verde. 


Cape  Spartel 

Araish  

Salee 

Azamor 

Cape  Blanco 

Cape  Cantin 

Saffi '. 

MOGADORE  ISLAND. 

Cape  Geer 

Cleveland  Shoal 

Santa  Cruz 

Cape  Noon 

River  Non,  entrance 

Cape  Blanca 

False  Cape , 

Cape  Bojador 

Seven  Capes  

Cape  Barbas 

Cape  Blanco  

Cape  Mirik 

Portendik 

Barbara  Point 

SENEGAL,  Fort  St.Louis 

CAPE  VERDE, 

North-west  Pitch .... 


Lat. 


M. 

D.  M. 

48  N 

5  54W 

i3 

6  10 

o3 

6  46 

i8 

8  i5 

7 

8  39 

JJ 

9  21 

i8 

9  12 

26 

9  32 

38 

9  52 

4b 

lO  22 

3o 

9  40 

i4  M     17  32 


Lons- 


XXXIV.     The  Western  Islands. 


Corvo,  N".  pt 

Flores,  N.  pt 

Faj'al,  S.  E.  point 

Pico,  Point  de  Espertal. . . 

summit  of  Peak  .... 

St.  George,  S.  E.  point  . . . 
Graciosa,  Villa  da  Praya.. 

Terceira,  Angra 

St.  Michael,  P.  Delegada  . 

Point  Ferraria 

-; N.  E.  point  . . 

P^ormigas,  or  Ants 

St.  Mary,  town 

W.  point 


Lat. 


M, 
44  N 

32 

3o 
27 
27 
3i 
2 

39 

45 
54 
5o 

17 
59 


Loner. 


D.  M 

3 1  07 W 
3i  12 
28  42 
28  36 
28  28 
27  5i 
27  59 
27  12 
25  4o 
25  56 
25  10 

24  47 

25  i3 
25  16 


XXXV.     Madeira  Islands. 


Porto  Santo,  town 

Madeira,  Lorenzo  Point  . 
Madeira,  Tristram  Point. 

FUNCHAL . . . 

S.  Dosortos,  S.  point  . . . . 

Great  Salvage,  W.  pt 

Piton,  Great 


Lat. 


Lonsr. 


n.  M. 

D.  M. 

33  4  N 

16  19W 

32  43 

16  38 

32  54 

17  17 

32  33 

16  55 

32  28 

16  3o 

3o  8 

i5  5i 

3o  I 

16  0 

XXXVI.     Canary  Islands. 


Pabna.  town 


Lat. 

I).  M. 

28  39  N 


Lon<r. 


D.  i\r. 

17  56W 


Palma,  N.  point 

S.  point 

Ferro,  N.  point 

Goniero,  St.  Sebastian  . . . 
TenerifTe,  Hidalgo  Point . . 

Orotava 

Tena  Point 

Peak  

Port  Christianos 

SANTA  CRUZ 

Canary,  N.  E.  point 

Palmas  mole 

S.  point 

Fuerteventura, 

N.  point 

—  S.  W.  point 

Lanzarote,  S.  point 

Puerto  de  Naos  . 

Punta  del  Farion 

Graciosa. 

St.  Claire 

Alegranza  


Lat. 

Long. 

D.  M. 

D.  M. 

28  52  N 

17  55W 

28  27 

17  5o 

27  5o 

17  55 

28  6 

17  8 

28  36 

16  21 

28  25 

16  32 

28  20 

17  I 

28  16 

16  39 

27  57 

16  /^A 

28  28 

16  j6 

28  II 

i5  25 

28  7 

i5  25 

27  44 

1 5  34 

28  46 

i3  54 

28  4 

i4  3o 

28  5i 

i3  46 

28  58 

i3  M 

29  i4 

1 3  29 

29  i4 

i3  3i 

29  17 

i3  32 

29  25 

i3  3i 

XXXVII.    Cape  Verde  Islands. 


St.  Anthony,  N.  W.  p.  . 

N.  E.  point.  . . 

— SANTA   CRUZ 

St.  Vincent 

St.  Lucia 

St.  Nicholas,  N.  point  .... 

E.  point  .... 

Salt  Island 

Bonavista,  W.  pt 

Leton  Rock 

Isle  of  May,  S.  pt 

St.  Jago, 

PORTO  PRAYA 

N.  point 

Fogo,  N.  point 

Peak  

Brava,  S.  point 


Lat.       Long. 


M 
12  N 


59 
46 
42 
U 
45 
2 

49 
6 

54 


56 

47 


31 
19W 

8 
i5 

6 
55 
21 

o 
56 


XXX  Vm.     F)-om  Cape   Verde  to  the 
Cape  of  Good  Hope. 


CAPE  VERDE 

Goree  Island,  town 

River  Gambia,  entrance 

Cape  Roxo 

Bissao,  fort 

Bijooga  Islands, 

—  Tombelly,  N.  point. . 

—  Galina  Island,  W.  point 

—  Orango  Island,  S.  E.  pt 
Rio  Grande  Shoals, 

South  Breakers  . 

Pullam  Islands,  S.  one  .  . . 

Nunez  River,  entrance 

Cape  Verga  

Delos  Islands 

Mataconjr  Island 


Lat.        Loner. 


M. 

44  N 

4o 

3o 

20 

5i 

29 

28 

3 

42 

52 

36 
12 
29 
i4 


D.  JM 

7  33W 
7   25 

6  4i 
6  46 
5  37 

5  3o 
5  47 

5  55 

6  18 
5  45 
4  42 
4  28 
3  48 
3  26 


TABLE  LIV. 

Latitudes  and  Longitudes. 


[Page  355 


SIERRA  LEONE,  Cape. 

False  Cape 

Cape  Schilling 

Sherbro  Island, 

Cape  St.  Ann 

Turtle  Islands,  N.  end  .  . . 

Cape  Mount 

Cape  Mesurado 

Scslros  River 

Cape  Palmas 

St.  Andrew's  River 

Lahou,  town 

Cape  Apollonia 

AXIiAI 

Cape  Three  Points 

Dix  Cove,  fort 

Elniina  castle 

Cape  Coast  castle 

Anamaboo 

Tantumquery  Point 

Accra 

jN'ingo  Fort 

Cape  St.  Paul's 

Grand  Popoe 

Whyda 

Lagos,  entrance 

Benin  Pi-iver,  N.  pt 

Cape  Formosa, 

New  Calebar  River,  W.  pt. 
Old  Calebar  Ptiver, 

Tom  Shot's  Point. 

Cameroon  River, 

Suellaba  Point. . . . 

Cape  St.  John 

Gaboon  River,  S.  point  . 

Cape  Lopez 

Settee  River,  entrance  . 
Loango  River,  entrance 
River  Congo, 

Cape  Padron 


.'imbriz  Bay. 

Dande  Point 

St.  Paul  de  Loando  . . . . 

Cape  Lcdo 

Nova  Redonda  

St.  Philip  de  Bcnguela, 
Fort  Flacrstaff. . . 


Cape  Alary 

Little  Fish  Bay,  entrance 

Cape  Negro 

Great  Fish  Bay, 

Tiger  Island,  N.  pt. 


Cape  Frio 

Cape  Cross 

Walwich  Bay, 
Pelican  Point. 


Angra  Pequena 

Cape  Voltas 

Cape  Donkin 

Cape  Deseada 

St.  Helena  Bay, 

Paternoster  Point 


Saldanha  Bay,  N.  pt. 


Lat. 

Long. 

D.   M. 

D.  M. 

8  3oN 

i3  18W 

8  26 

i3  18 

8  10 

i3  10 

7  34 

12  57 

7  4i 

i3  4 

6  45 

II  23 

6  19 

10  48 

5  27 

9  25 

4  22 

7  44 

5  0 

6  3 

5  12 

4  36 

4  57 

2  33 

4  55 

2  18 

4  45 

2  4 

4  48 

I  57 

5  5 

I  23 

5  6 

I  i5 

5  10 

I  7 

5  i3 

0  47 

5  32 

0  i4 

5  45 

0  2E 

5  5o 

0  58 

6  16 

I  54 

6  19 

2  5 

6  26 

3  25 

5  46 

5  3 

4  i5 

6  10 

4  23 

6  59 

4  35 

8  19 

3  5i 

935 

I  10 

9  22 

0  22 

9  23 

0  36S 

8  43 

2  23 

9  26 

4  39 

II  44 

6  8 

12  9 

7  5i 

i3  4 

8  28 

i3  18 

8  48 

i3  i3 

9  46 

i3  17 

11  12 

i3  54 

12  34 

i3  25 

i3  26 

12  33 

i5  i3 

12  7 

i5  42 

II  58 

16  3o 

II  46 

18  23 

12  2 

21  5o 

i3  57 

22  5[ 

i4  27 

26  38 

i5  8 

28  44 

16  32 

3i  54 

18  19 

32  i5 

18  22 

32  42 

17  54 

33  2 

17  54 

Point  Isser 

Dassen  Island 

Robben  Island 

TABLE  BAY, 

Cape  Town,  obs. . . 

Green  Point  light. 


Cape  of  Good  Hope 
Bellows  Rock , 


FALSE  CAPE,  or  Hang- 
klip  


Lat. 


D.  M. 
33  22 
33  26 
33  47 

33  56 

33  53 

34  22 
34  24 

34  24 


Lons. 


D.  M. 
18  II  E 

18  7 
18   23 

18  29 
18  25 
18  3o 
18  3o 

18  5o 


XXXIX.     Islands  between  Cape  Verde, 
the  Cape  of  Good  Hope,  and  Cape  Horn. 


St.  Paul's 

Ferdinand  Noronha 

The  Roccas,  (dangerous,) 
Ferdinand  de  Po,  N.  pt.  . 

Prince's  Island 

St.Thomas,  (Man-of- War's 

Bay,)K  pt 

Man-of-War's 


Eay,S.pt 

Annabona,  N.  pt 

Trinidad,  S.  pt 

Martin  Vas,  (largest,)  . . . . 

ASCENSION 

ST.    HELENA,   James 

Town,  observatory 

Saxemburgh* 

Tristan  d'Acunha,  N.  pt. . . 

Inaccessible  Islands,  

Nightingale  Island 

Hibernia  Rocks,  (doubtful,; 
Diego  Alvarez,  (doubtful,) 
Gouffh's  Island 


Lat.       Long. 


M. 

55  N 
53  S 
5i 

47  N 
39 


o  24 


24  s 
3i 
29 
56 


Island  Raza,  N.  W.  point 
Salvages'  Islands,  N.  point 

The  Sisters 

Port  Egmont. 

Island  Concha 

Cape  Leal 

Point  de  la  Barra,N.E. point 

Cape  Corientes 

Port  Soledad 

Cape  St.  Philip,  E.  p.  . . . 
Beauchenes  Isl.,  S.  point 

Porpus  Point 

Cape  Meredith 

Cape  Orford 

Cape  Percival 

Aurora  Isles, 

northernmost 

southernmost 


Eagle  Reef 

Alexander's  Island. 
Peter's  Island 


Island  Georgia, 

Cape  Buller  53  58 


5o  59 

50  59 
5i  7 
5i  22 
5i  i5 
5i  21 
5i  28 

5 1  24 
5i  33 
5i  43 

52  55 
52  28 
52  16 
5i  56 
5i  47 

52  43 

53  25 
5i  5i 
68  5i 
68  57 


D.  M. 
29  22W 

32  25 

33  49 

8  43E 
7  27 

6  38 

6  3o 

5  37 
29  16W 
28  5o 
i4  25 

5  45 
24  00 
12  2 
12  35 
12  20 

4  42 
II  2 

9  44 

61  28 
61  18 
60  26 
60  I 
59  00 
58  57 
57  4i 

57  52 

58  00 
57  4o 

59  12 

59  28 

60  39 

61  00 
61  II 

48  10 
47  59 
64  32 
73  10 
90  46 


38  i3 


*  The  existence  of  this  island  is  considered  doubtful ;  though  the  appearnnce  of  land  is  said  to  have 
been  seen  by  several  vessels  in  various  situations,  from  30°  8'  S.  to  30°  45'  S.,  and  from  20°  50'  E.  to 
28^^  20'  E.    The  island  St.  Matthew  does  not  exist,  being  the  same  as  Annabona. 


Page  35C] 


TABLE  LIV. 

Latitudes  and  Longitudes 


Island  Georgia, 

Cape  Disappointment 

Willie's  Isle 

Clerk's  Islands 

Sandwich    Land,    Cape 

Montague 

Candlemas  Isles 

Southern  Thule 

Isle  of  Circumcision 


Lat.       Lons- 


D.  M. 


D.  M. 


54  58  S 

54  oo 

55  I 

36  i5W 

38  26 
34  48 

58  27 

57  ID 

59  34 
54  16 

26  44 
1-j   i3 

27  45 

6  i4E 

XL.     The  Coast   and  adjacent  Islands 
from  the  Cape  of  Good  Hope  to  Canton. 


CAPE  OF  GOOD  HOPE 
False  Cape,  or  Hangkllp.. 

Danger  Point 

Dyer's  Island 

Quoin  Point 

CAPE  LAGULLAS  1.  h. 

Cape  Infanta 

Cape  Vaches 

Moselle  Bay, 

Cape  St.  Blaize 


Knysna  River,  E.  pt 
Plettenburg  Bay, 

Cape  Seal 


Cape  St.  Francis 

Algoa  Bay,  commandant's 

house  

Cape  Recif 

St.  Croix  Island,  peak 

Doddington  Rock. . . 

Bird  Island,  eastern  . 

Point  Padrone 

Great  Fish  River,  mouth  . 


Keiskama  River,  entrance 

Point  Hood,  extreme 

Buffalo  River, 

Cape  Morgan 

Hole-in-the-Wall 

St.  John's  River,  entrance 

('ape  Natal,  e.xtreme 

Fisher's  River,  entrance  . . 

Point  Durnford 

Cape  St.  Lucia 

(^;ipe  Vidal 

Delagoa  Bay, 

—  Cape  Collatto 

—  Cape  Inyack 

—  English  River,  flag-staff 
liihampura  River, entrance 

Cape  Corrientes 

Inliamban  Bay,  town.  . . 

Cape  Lady  Gray 

Bdzarouta  Island,  N.  pt. 

Inverarity's  Shoal 

Chiiluwan  Island,  N.  pt 

Sofila,  fort 

Quillimane  River,  town 
David's  Shoals 

I  Fogo  or  Fire  Island  .... 

I  Raza  Island 

Macalonga  Point 

I  Caldeira  Island,  centre  .  . . 


Lat. 


M. 

22  S 

24 

42 

44 

49 

498 

3i 

20 


Loner- 


34  10 
33  58 


33  17 
33     4 

32  42 
32  3 
3i  34 
9  53 
29  16 

9  o 
28  33 
28  10 

26  4 
25  58 
25  58 
?5  12 
24  8 
23  52 
22  56 
21  3i 
20  43 
20  38 
20  1 1 
17  52 
17  32 
17  i4 
17  7 
16  59 
16  39 


D.  M. 
18  3oE 

18  5o 

19  22 
19  28 

19  42 
o  7 

20  53 

21  57 

22  12 

23  8 

23  ?3 

24  53 

25  4o 

25  4i 
5  47 

26  1 1 
26  18 

26  2  5 

27  8 

27  32 

27  58 

28  25 
9  I 
9  29 

3i  2 
3i  33 
3i  52 
32  28 

32  38 

33  I 
33     3 

32  3- 

33  32 
35  3i 
35  25 
35  4f 
35  33 
35  10 

34  54 
34  46 

37  I 

38  32 

38  55 

39  6 
39  6 
39  46 


Mafamale  or  Mafamede  Isl. 
Mogincale  Shoals,  middle. 
MOZAMBIQUE,  St.  Jago 

Island,  centre 

St.  George's  Isl.,  fort 

Melamo  Point 

Penda  Shoal,  E.  end 

Maunbane,  or  Devil's  Pt.. 
Querimbi  Islands, 

—  Querimba  Island,  N.  pt. 

—  Matemo  Island,  E.  pt. . . 
Vumba  Island,  E.  point. . . 

Cape  Delgado 

Lindy  River,  fort 

Keelwa,  or  Quiloa, 

Pagoda  Point 

Fort . 

Monfeea,  W.  point 

Poana  Point 

Latham's  Island,  or  Sand 

Bank 

Zanzibar  Island,  S.  point . 

N.  point . 

Pemba  Island,  S.  point . . . 

Mombas  Island,  fort 

Maleenda  Port 

Formosa  Bay, 

Ras  Gomany 

Patta,  town 

Dundas  Island, 

Port  Durnford,  N.  p. 

Dedalus  Shoals 

Juba  River 

Brava,  town 

Torra 

Mukdeesha,  or  Magadosha 

Ras  Asooad 

Ras  Awath 

RasUl  Khyle 

Ras  Mabber 

Pias  Hafoon,  or  Orfui  .... 

Cape  Guardafui 

Ras  Mett 

Mette  Island 

Burnt  Islana 

Burburra 

Zeyla , 

Ras  Bir 

Abdul  Koory  Island,  W.  pt 

N.  E.  pt. 

Salte's  White  Rocks 

Socotra  Island, 

Ras  Rarby,  W.  pt.  . . 

Tamarin  Bay,  town  . 

Ras  Shoorguy,  E.  pt 

Babelmandel  Island 

Babelinandel  Cape 

Panther's  Shoal 

Cape  Ratta 

Dhalak  Island,  D.  town  . . . 

Massowa  Bay,  lights 

Port  Mornington 

Suakin 

Mirza  Sheik  Baroud 

Salaka 

Cape  Calmez 

Cape  Ras  Elans 


Lat. 

Long. 

D.  M. 

D.  M. 

16  21  S4o  4E 

i5  35 

40  34 

i5  3 

4o  48 

i5  I 

4o  4i 

i4  25 

4o   5i 

i4  i5 

4o  5 1 

12  57 

4o  38 

12  24 

4o  39 

12  i4 

40  4o 

II  9 

40  43 

10  4i 

4o  40 

10  0 

3945 

9  2 

3937 

8  57 

39  34 

7  56 

39  38 

7  3 

39  37 

6  54 

4o  I 

6  28 

39  33 

5  43 

39  21 

5  29 

39  42 

4  4 

39  43 

3  i3 

4o  II 

3  0 

40  19 

2  9 

4i  7 

I  i3 

4i  54 

0  24 

42  36 

0  i5 

42  39 

I  7N 

44    3 

I  26 

44   21 

2  2 

45  25 

4  34 

48  6 

5  33 

48  40 

7  A4 

49  46 

9  29 

5o  5o 

10  28 

5i  22 

II  5o 

5i  21 

II  55 

5o  47 

II  22 

48  53 

II  17 

47  21 

10  22 

45  6 

u  17 

43    0 

12  17 

43  17 

12  i3 

52  8 

12  12 

52  23 

12  26 

52  i4 

12  33 

53  18 

12  37 

54  01 

12  34 

54  3o 

12  38 

43  29 

12  40 

43  3i 

12  56 

43  8 

i4  56 

40  52 

i5  46 

4o  6 

i5  36 

39  21 

18  16 

38  32 

19  7 

37  20 

19  35 

37  24 

20  28 

37  27 

21  28 

37  25 

23  56 

35  48 

TABLE  LIV. 

Latitudes  and  Lonsriiudes. 


[Page  357 


St.  Jolin's  Island 

Reef  of  breakers 

Tlirce  Islands 

llei'f  of  breakers 

Dedalus  Shoal 

Centurion  Island,  doubtful 

Koseir 

Tlie  Brothers 

SUEZ 

Cape  Jehan  Peak    

Tor  Harbor 

lias  Mahomed 

Shaduau  Island,  S.  point.. 
Kareedy  Harbor  Cape  .... 

Yanibo 

Juddai) 

Canitidia 

iMarabia  Reefs, 

Western  part 

Doorhal  Island 

Loheia 

Cape  Israel 

Gebol  Tor 

Gebel  Zebayr 

Hodeida 

Gebel  Zeghir,  K  I 

Great  Arroe 

MOCHA 

Cape  St.  Anthony 

Cape  Aden 

Cajie  Booratshua 

Kisseen  Point 

Cape  Fartak 

Uas  Morebat,  extreme. . . . 

Ras  Noss,  S.  pt 

Curia  Maria  Isles, 

—  JiblyPeak 

Hallanny,  N.  E.  pt. 

Soda  

Hasky 

Ras    Garwow,    or    Cape 

Chancilly 

Ras   Madrake,    or    Cape 

Isolette , 

Massera  Island,  S.  point.. 

N.  point.. 

Ras  Jibsh 

Ras  al  Had,  or  Cape  Rasal- 

gat 

Muscat 

Burka 

Debhali,  towii 

Ormus,  fort 

La-nek  Hill 

Kishina  Island, 

Kishma,  town 

■  Luft 

Anijar  or  Anjruam  Island, 

IN .  point . . . 

S.  point  . . . 

Great  Tumb  Island 

liombosa  Island 

Polior  or  Belior  Isl.,  middle 

Kaez  or  Kyen  Island 

Hinderabia , 

Busheab,  W.  point 


Lat. 


Lous'. 


D.  M. 

D.  M. 

23  36N 

36  10  E 

24  4 

36  16 

24  25 

35  26 

24  54 

35  49 

24  56 

35  5i 

25  20 

35  48 

26  8 

34  i5 

26  21 

34  49 

29  59 

32  34 

28  33 

33  20 

28  j5 

33  36 

27  43 

34  i5 

27  28 

34  5 

24  17 

37  33 

24  4 

38  I 

21  29 

39  1 5 

19  7 

4o  5o 

19  1 1 

4o  5 

16  i5 

42  8 

i5  42 

42  39 

i5  i5 

42  4 1 

i5  32 

42  00 

i5  3 

42  i3 

i4  48 

42  54 

i4  5 

42  44 

t3  4i 

42  52 

1 3  20 

43  12 

12  4i 

44   10 

12  46 

45  3 

'4  49 

5o  3 

i5  20 

5i  48 

i5  38 

52  16 

16  58 

54  42 

17  12 

55  i8 

17  29 

56  19 

.73. 

56  3 

17  28 

55  5 1 

17  28 

55  35 

17  52 

56  21 

18  58 

57  46 

20  8 

58  33 

20  43 

58  52 

2r  26 

59  12 

22  23 

59  55 

23  37 

58  35 

23  42 

57  57 

2  5  36 

56  18 

27  5 

56  29 

26  52 

56  28 

26  57 

56  19 

26  55 

55  55 

26  4i 

55  57 

26  37 

55  54 

26  i5 

55  04 

25  54 

55  8 

2b  18 

54  35 

26  29 

54  2 

26  4o 

53  39 

26  48 

53  7 

Crescent  Shoal,  about 

Cape  Budistan 

Zezarini  Island 

Keyn  Island 

Busheer 

Karack  Island 

BASRA,  or  BUSSORA.. 
Phelechi  Island,  S.  E.  end 

Graen 

Khubber  Island 

Garwow  Island 

Malmaradam  Island 

Ras-ul-Lur 

Ras-ul-Zoor 

Durable  Shoal  

KatifBay 

Kore  Hussan 

Ras  Reccan  

Sandy  Island 

Hawlool 

Sherarow 

Daeny 

Seir  Beni  Yass 

Dalmy,  S.  end 

.'Xrzenie 

Jernain 

Dauss 

Zlrcooa,  or  Zara 

Seir  Abonaid,  N.  pt 

Ras   Luffan 

Ras-el-Allarch 

Jezurab-ain-Lassart 

Ras  Boogmais 

Goodwin's  Islands 

Ras-el-Machereeb 

Jiljbnb  Hadwareah 

Stannu's    Shnal,  N.  end  .. 

Mount  Jibbul  Alii 

Abotliubbce 

Debai 

Sliarga 

Aymaun 

Red  Island,  town 

Raa-e!-Khyma 

Rtiumps 

Shaum,  towers 

Boukha  Point 

Cape  Jedda  or  Yedda 

Ras  Sheik  Mumoud 

Perforated  Rock 

Great  Quoin 

Cape  Mussendom 

Cape  Jask 

Ciiurbar 

Cape  Gvvadel 

Cape  Arid)ah 

Cape  Monze 

Point  Jigat 

Diu  Head 

Scarbett  Island 

Cambav 

SURAT  Castle 

Vaux's  Tomb 

Demaun 

OmeriTon 

St.  John's  Highland  . , . 
Basseen  Fort 


Lat. 


31. 

44  N 

58 

59 

45 

00 

16 

3o 

23 
23 

4 

49 

40 


25  38 
25  19 
25  4 
25  7 
24  5i 
22  1 3 
20  42 

20  56 
22  17 

21  1 1 
21  5 
20  22 
20  II 

30      3 

19  19 


Long. 


M. 

43  E 

19 

8 

7 
5o 

19 

00 

'9 

58 

24 
42 
35 
5 
16 


57  48 
60  35 
62  1 5 
64  3o 
66  36 

69  01 

70  5i 

71  A4 

72  36 
72  47 

72  38 

73  3 
72  55 
72  43 
72  49 


Page  3581 


TABLE  LIV. 

Latitudes  and  Longitudes. 


BOMBAY  rflag-staff,) 
liffl: 


rht-house . 


Henery  and  Kenery  Islands 

Coullaba  Island 

Cliaoul 

Radjapour  Harbor 

Bancoot  River 

Sevendrooo- 

Dabul . . .  r. 

Argenwell  Fort 

Boria  Point  .' 

Zughur  Point 

Retina- Geriah 

Radjapour  Fort 

Geriah  Point  and  flag-staff' 
Angrias  Bank,  N.  p 

S.p 


D.  M. 

8  56N 
8  54 
8  42 
8  37 
8  34 
8  i6 
7  57 
7  47 
7  46 
7  34 
7  25 
7  i6 


Dewghur  Harbor 

Atchera  River 

Melundy,  (fortified  island,) 

Newtee  Point 

Vingorla  Rocks,  or  Burnt 

Islands  . . .. . 
Raree  Point.. , 
Cliiracole  Fort 
Chapra  Fort  . 
Alguada    Pt.,  N.  entrance. 

Goa  Bay 
GOA  .... 
St.  George's  Isl.  (western) 
Cape  Ramas 
Oyster  Rocks,  (outermost,) 

Carwar  Head 

Anjedwa,  (island,) 

Merjee  River 

Fortified  Island 

Onore 

Pigeon  Island 

Barcalore  Peak 

St  Mary's  Rocks,  N.  p. 
S.  p. 


Lat. 


Molky  Pyramid 
Premeira,  or  Molky  Rocks 

MANGALORE 

Mount  Dilly 

Canonore  Point  and  fort. 
Telliclierry  flag-staff" .... 

Mahe  fort 

Sacrifice  Rock 

Calicut 

Beypore  River 

Paniany  River 

Cliilwa  ciiurch 

Cranganore   or   Aycotta 

River 

Cochin 

Alippee 

Porca ♦. . . 

Iviker,  or  Aybicka 

Quilon 

Angenga  fort 

Rultera  Point 

Cadiapatam  Point 

CAPE  COMORIN 


47 
3i 
38 
i8 

23 

II 

3 

56 

53 
44 
4i 
36 


6 
6 
6 
6 
6 
5 

5 
5 
5 
5 

5  29 
5  a8 
5  22 
5  5 
4  48 
4  47 
4  44 
4  3o 
■  19 


Ceylon,  Point  Pedro  . 

— —  Columbo . 

Adam's  Peak 


4 
4 
3 
3 
3 
3 

3  II 
2  5i 
2  02 

I     52 

I  45 
I  4i 
I  3o 
I  i5 
I  10 
o  47 
o  33 

10  12 

9  58 

9  3o 

9  20 

8  54 

8  53 

8  39 

8  23 

8  9 

8  5 

9  49 
6  57 
6  52 


Lons- 


D.  M. 

72  54  E 
72  52 
72  52 

72  54 
72  54 

72  58 

73  I 
73  5 
73  12 
73  i3 
73  12 
73  10 
73  19 
73  22 
73  22 
71  43 
71  43 
73  3i 
73  38 
73  39 
73  42 

73  39 

73  49 
73  52 

73  54 

73  46 
73  52 
73  45 

73  55 

74  o3 
74  08 
74  o5 
74  20 
74  24 
74  27 
74  18 

74  52 

74  54 
74  54 
74  5i 

74  38 
49 

75  II 
75  21 
75  28 
75  36 
75  3o 
75  46 

75  52 
76 

76  o3 

76  12 
76  1 4 
76  24 
76  27 
76  35 
76  33 
76  45 

76  58 

77  20 
77  3o 

80   23 

79  5o 

80  29 


Ceylon, 


D.  M. 


Point  de  Galle.. .. 

Matura 

Dondra  Head 

Grand  Bassas  .... 

Little  Bassas 

Elephant  Point,Rk, 

Agaus,  or  Aganis  . 

Battacola,  Fort 

Vendoos  Bay,ISr.  pt 

Trincomaley,  Flag- 
staff" Point 

Molewal,  or  Mola- 

teeva  House  , . . . 

■  Point  Palmyra.. . . 


Lat. 


iN 
58 
55 
II 

25 

24 
53 
4i 
57 


Loner. 


Manapa  Point 

Trinchindere  Pagoda 

Punnecoil 

Tutacarine 

Point  Ramen 

Deviapatam , 

Tondy , 

Point  Calymere , 

Pagodas - 

Negapatam  Fort 

Five    White    Pagodas    of 
Nagore 

Tranquebar  

Devicotta,  Coleroon  River 

Porto  Novo 

Cuddalore 

PONDICHERRY 

Sadras 

M  ADRAS,Fort  St.George 

Ennore 

Pulicat 

Armegon 

Point  Pennar 

Gondegam 

False  Point  Divy 

Point  Divy 

MASULIPATAM 

Narsapour  Point 

Point  Gordeware 

Coringa 

Jaggernautporam 

Wattara 

Vizagapatam 

Biinlipatam 

Cliicacole  River 

Ganjam  flag-staff' 

Manikpatam 

Jaggernaut  Pagodas 

Black  Pagoda  

False  Point 

Point  Palmyras 

BALLASORE 

Ingerlee  Pagoda 

Kedgeree 

CALCUTTA,   Fort   Wil- 
liam  

Chandernager 

Sangor  I.  \V.  pt 

Liglit-house  Point 

Tail  Western  Brace,  S.  p. 

Tail    Western    Sea    Reef, 

SP 


8  33 

9  i3 

9  49 

8  22 
8  3o 
8  4i 

8  48 

9  17 
9  29 
9  45 
0  18 

0   23 

0  45 

0  49 

1  I 
I  27 
I  3i 
I  43 
I  56 


4 
i5 

25 

58 
3o 
20 
45 
59 

9 
19 

48 

49 

6  56 

7  26 
7  42 
7  53 


40 

48 

52 


9 

9 

9 

9 
20  20 

20  4i 

21  3o 
21  44 

21  5i 

22  34 
22  5i 
21  37 
21  3o 
21   10 


D.  M. 

80  i4E 
80  36 

80  38 

81  32 
81  52 
81  32 
81  56 
81  42 

34 

81  i5 

80  5o 
80  i4 

78  3 
78  7 
78  6 

78  i3 

79  22 
79  00 
79  12 
79  5i 
79  58 
79  5i 

79  5o 
79  5i 
79  47 
79  43 
79  46 

79  5o 

80  9 
80  16 
80  24 
80  18 
80  2 
80  17 
80  6 

li  I 
ii  10 

;i  8 

ii  4i 

82  27 

82  17 

82  17 

82  55 

83  17 
83  37 
83  54 
85  3 
85  39 

85  54 

86  8 

86  59 

87  II 

87  10 

88  00 

87  56 

88  20 
88  27 
88  01 
88  27 

87  47 

88  3 


TABLE   LIV. 

Latitudes  and  Lonjiitudes. 


[Page  359 


u 


Tail  Eastern    Sea   Reef, 

S-P 

Floating  light-vessel 
Tail  of  Sanger  Sand,  S.  p. 
Codja  Deep,  (island,) 
Islamabad,  or  Chittagong 

Red  Crab  Island 

Donibucli    or    Elephant 

Point 

St.  Martin's  Reef,  S.  pt. 
Mosque     Point,  entrance 

Aracan  

Terribles,  W.  ... 
Cheduba  Pagoda 
Tree  Island  .... 

Foul  Island 

Cliurch,    (or    St.    John's 

Rocks,) 

Calventura  Rocks,  KW. 

Buffalo  Rocks 

Cape  Negrais  ....,..,. 

Diamond    Island 

Sunken    Island,    or    La 

Guarda 

Rangoon   or  Pegu  River 

entrance 

PEGU 

Martaban 

Tavay  Point 

Tavay  Island 

Cabossa  Island 

West  Canister  Island. 

Tanasserim  Island 

Mergui 

Tores  Islands,  western 

Black   Rock 

Dojnel  Island , 

St.  Matthew's  Island., 
Seyer's  Islands,  N.  p.  . 
S.p.., 


Junkseylon  Island,  N.p.. 

S.  p.. 

I'arlis  Piiver 

Elt>pliant's  Mount 

Queda 

Prince  of  Wales's   Island, 

Fort  Cornwallis 

Cape  Caran 

Salangore  Ilill  and  fort. . 
Palo  Callam   or   Colonjr, 

sp .; 

Parcelar  Hill 

Parcelar  Point 

Tanjong  Tuan,  (Cape  Ra- 

chado,) 

Tanjong    Clin,    or    Peer 

Punjab 

Fisher's  Island 

Malacca  fort 

Water  Islands,  southern  . 
Mount  3Iora  or  Moar. . . . 

Mount  Formosa 

Mount  Battoo  Ballo 

Pulo  Pisang 

Pulo  Cocob 

Sincapore 

Little  Hill, or  False  Johore 

HiH 


Lot. 


D.  M. 

20  58: 

2£  2 

21  00 

21  27 

22  21 
22  26 

21  10 

20  34 

20  7 

9  22 

8  5l 

8  26 


52 


541 

16  29 

18  00 

16  32 
32 

6 


42 

34 

27 
I  48 

I  23 

I  10 
9  58 
8  43 
8  28 
8  9 
7  4o 
6  21 
6  10 
6  G 


2  56 

2  52 

2  42 

2  26 

2  17 
2  i3 
2  II 
2  4 
I  59 
I  49 
I  39 
I  28 
I  19 
I  17 

I  26 


IjOns^. 


D.  M. 

88  II  E 
88  25 
88  37 
88  M 
91  48 

91  52 

92  4 
92  20 

92  54 

93  16 
93  44 

93  56 

94  6 

94  23 
94  1 5 
94  12 
94  i3 

94  17 

94  i3 

96  25 

96  52 

97  35 

98  9 
98  i4 
97  55 
97  42 

97  49 

98  3fi 
97  26 
97  38 

97  57 

98  10 
97  4o 

97  4o 

98  18 
98  18 

100  1 3 


100  21 
loi  8 

101  22 

loi  16 

101  25 

101  32 

loi  5o 

102  8 
102  12 
102  i5 
102  20 
102  4o 
102  54 
io3  1 1 
io3  i3 
io3  25 
io3  5o 

io4  4 


Lat. 


D.  M. 

I  23: 


4  47 

5  i5 


Johore  Hill 

Barbucet  Hill 

POINT  ROMANIA....!  i 

False  Barbucet  Hill i 

Romania  Reef j    i 

Eastern  Bank,  (outer  part) 

Pulo  Tingy 

Blair's  Harbor 

Pulo  Varela 

Palian  Road 

Tingoram 

Howard's  Shoal  .... 

Pulo  Brala,   or  Capas  de 
Mer 

Pulo  Capas  de  Terra. 

Tringany  Piiver,  entrance 

Great  Redang  Island.. 

Pulo  Printian 

Calantan  Road 

Cape  Patani 

Pulo  Lozin 

Pulo  Cara 

Slam  River,  E.  entrance 

JUTHIA,  or  SIAM.. 

Cape  Liant  

Pulo  Way 

Pulo  Oby  False 

Pulo  Oby 

Cambodia  Point 

Cambodia  River,  W.  ent. 

Cape  St.    James,   (E.  en- 
trance Saigon  River,)  . 

Cape  Trivoane 

Point  Babeck 

Brittos  Bank,  N.  E.  p 

Cow  Island 

Point  Kega 

Point  Viiiay 

Mui-guio,  or  Little  Cape. 

Point  Lagan 

Pulo  Ceicer  de  Terre .... 

Cape   Padaran 

Padaran  Bay 

Cape  Varela  False 

Carmaigne  Harbor,  ent.  . 

Water  Islands 

Tre  Island 

Pyramid  Island 

Nhiatrang 

Three  Kings  Rocks 

Hone  Colie  Harbor 

Cape  Varela,  or  Cape  Pa- 
goda   

Perforated  Rock 

Phuyen  Harbor,  entrance 

Coumong  Harbor,  ent... 

Pulo  Cambir 

Cape  Sanho 

Quinhone  Harbor 

Buffalo  Island 

Point  Nuoc  Ngol 

TamqiL-xi  River 

Pulo  Canton 

Port  Qui-quick,  ent.  .... 

Cham  Callao 

Ca])e  Turon  or  Tienchu. 
Callaohanne  Island,  (N.' 
entrance  Turon,) I  iG  11 


i3  23 

i4  55 
12  34 
9  58 
8  56 
8  25 

8  35 

9  M 

o  17 
o  21 
o  3o 

o  32 

o  39 
o  4i 
o  54 

4 

9 
i3 
21 
35 

A4 

49 
3 
16 
21 
26 
37 
45 

55 
59 

23 

29 
33 


4  19 

4  39 

5  23 
5  28 

5  59 

6  8 


Lonsr. 


D.  M. 

04  6  E 

04  TI 
04  16 
04  16 

04  25 
04  35 
o4  II 
o3  4o 
o3  47 

o3  18 
o3  3 1 

o3  4i 
o3  12 
o3  I 
02  56 
02  4o 

02  i4 
01  o5 
01  59 
00  35 
00  34 

00  o 

01  1 1 

03  48 

04  38 
04  54 
04  56 

06  20 

07  4 
07  16 
07  33 
07  48 

07  52 

08  4 
08  19 
08  3i 
08  40 

08  48 

09  00 
09  4 
09  12 
09  12 
09  19 
09  19 
f)9  23 
09  10 
09  25 
09  12 

09  28 
09  23 
09  1 4 
09  1 3 
09  18 
09  i4 
09  II 
09  16 
09  7 

08  56 

09  6 
08  5o 
08  4o 
08  19 

108  12 


Page  360] 


TABLE  LIV. 

Latitudes  and  Longitudes, 


Cape  Chcuvay  

Hue  or  Huesso  River, W.e 

Tiger  Island 

Hainan   Island  and  adja- 
cent Islands, 

—  Yaitchew  Bay 

—  Yulenken  Bay,  Zenby 

—  South  Point  of  Hainan 

—  Galong  Bay 

—  Brother's  Islands,  east- 

ern ....'. 

—  Luengso}'^  Point,  S.  p. 

—  Sail  Rock 

—  Saddle  Island 

—  Point  of  land 

—  Nankin  Island 

—  Tinhosa  Island 

—  False  Tinhosa 

—  Toongean  Mount,  pt. . 

—  Hainan  Head,  N.  E.  p. 

—  South  Taya  Island  .  . . 

—  North  Taya  Island  . . . 

Nowchou  centre 

Ty-foong-kyoh    Island, 

(Tienpak  Harbor,) 

Ty-Chook-Chow  Island  . 

Song-yue  Point 

Mamee-Chovv,      or      the 

Twins,  near  S.W.  p.  of 

Hai-ling-shan 

Ty-oa  Point ;2 1 

Nampang  Island , 

Mandarin's  Cap 

Mong-Chow  Island  .  , . . 
Haw-Cheun,  S.  W.  end 
Passage      Island,      (near 

S.  W.  p.  Haw-Cheun,) 
Wy-Caup    Island,    (neat 

S.  point  St.  John's,),  . , 

Lieu-Chew  Island 

Wizard  Piocks 

Ty-katn  Island 

Cou-cock  Island  S,  W.  Pt 

Tyloo  Island,  S.  p 

Great  Ladrone 

Potoe  or  Passage  Island . . 

Laft-Samee  Peak 

Typa,.., 

Macao,  city 

Lantoa  or    Tyho    Island. 

S.  W.  p 

Lintin  Island,  peak 

Asses'  Ears 

Great  Lema  Isl.,  N.E.  p.. 

Nine  Pin  Rock 

Wlmmpoa  anchorage. . . . 
CANTON 


Lat. 


M. 

21  N 

35 

10 


8  24 
8  II 
8  10 

8    12 

8  ir 

8    22 

8  26 
8  35 
8  40 
8  38 


49 
35 
00 

49 
59 

52 


34 
43 
34 
28 
39 
35 

35 

34 
36 
46 
5i 
5o 

52. 

56 


24 
53. 

4 
16 

6 

7 


Lons- 


D    31. 

07  59  E 
07  41 
07  22 


08  52 

09  35 
09  34 
09  39 

09  4i 
o  00 
o    8 

O    II 

o  24 

O    2[ 

o  28 

0  34 

1  2 

0  57 

1  12 

I   i5 

0  3a 

1  i3 

I     25 

I  4o 


I  5o 


29 

32. 


35 


53-5 


i4 

39 

49 
33 
33 

51 


3 

3  48 

4  02 -5 
4  19 

4  22 
3  22 
3  i4 


XLT.  hkinds  and  Skoals  in  the  KVDMJV 
OCEAJ\r,  between  the  meridians  of  the 
Cape  of  Good  Hope  and  Sumatra,  inclu- 
ding those  jr.  andJV.  IV.  ofMw  Holland. 


Dutch   Bank,  Stot  Van 
Capelle,  various  (  from 
situations, >  to  . , 


Lat. 


Lon^ 


D.  M. 

D.  M. 

4o  00  S 

38  5oE 

36  00 

43  3o 

very 


Telaiuaque  Shoal,  doubt- 
ful, various  sit-  ^  from 
uations, >  to  , . 

Brunswick  Bank, doubtful 

French  Shoal,  doubtful, , 

Atlanta's  Rock,  doubtful 

Wellington    Shoal 
doubtful 

Prince  Edward's  Islands 

southernmost 

northernmost 

Kerguellan's  Land,  or 
Isle  of  Desolation, 

Bligh's  Cap,  N.  p.. , 

Christmas  Harbor,  , 

Port  Paliser 

Cape  Digby,  or  E.  p. 

Cape  George,  or  S.  p. 

Island   Solitaire , , , , 

Cape  Louis 

St.  Paul's  or  Amsterdam 
Island 

Amsterdam  or  St.  Paul's 
Island,         

Danish  Rock,  doubtful  . , 

Cloate's  Island,  (longitude 
uncertain,) 

Tryal  Rocks 

Rosemary    Island 

A  reof  10  miles 
N.  W.  of  Rose- 
mary Island . , ,    j 

Abrohlos  Shoals.  J 

Christmas  Island .10 

Cow  Isles, 

Northern , , . . 

Southern  . . , 


Very 
near 
New 
Hol- 
land. 


Lat. 


D.  M. 


9S 
00 

25 

8 
43 

53 

53 

40 


37  52 


46 
17 

7 
40 

27 


28 


Clark's   Reef,  S.  E,  point 

Imperieu.se  Shoal 

Dampier's  or  Scott's  Reef, 

N.  W.  end 

N.  E.  end . 


Coral    Bank 

Coral  Bank,  9  fathoms.. 
Coral  Bank,  7  fathoms  .. 
Cartier's  Sandy  Island  or 

Bank 

Red    Island,    (very   near 

New   Holland,) 

Coral   Bank,    10  fathoms 

or  less 

Hibernia's  Shoal 

Sahul    Shoal,    S.    W.    p., 

12  fathoms 

Echo's   Soundings, 

Rock 

Coral  7  fathoms  Bank . . , 


Fortune  Shoal 

Union  Shoal 

Dutch  Bank 

Otter's  Shoal,  doubtful,. 
Princess  Augusta's  Shoal, 

doubtful 

Union  Rocks,  doubtful,. 
Swallow  Rocks  and 

Breakers,  doubtful . , , . 
Belliquese  Shoal,  doubtful 


i5 


3o 
3i 

5o 

23 

28 
35 

52 

I 

32 
25 

46 
28 
i3 

25 

56 

35 

16 
56 


33  8 
35  25 
3 1  U 
33  56 

33  44 
35  23 

28  20 
98  43 


Lonff. 


1).   M, 

21  57  E 
23  24 

36  19 
43  6 
52  00 

71  43 

37  46 

38  8 


■  68  44 
69  4 

69  37 

70  34 
70  10 
68  5 
68  18 

77  35 

77  36 
98  25 

112  3o 
io5  3o 
116  3o 

116  23 

ii3  35 
io5  33 

97  4 

97  i5 

119  20 

118  56 

121  59 

122  16 
124  29 
124  12 
124  32 

123  56 

124  18 
124  1 1 

123  28 

124  i4 

126  00 
129  35 

43  5 
4i  12 

44  00 
36  00 

36  16 
4t  20 

42  10 
42  33 


TABLE  LIV. 

Latitudes  and  Longitudes. 


[Page  361 


CAPE  ST.  MARY..,, 

Star  Reefs,  S.  end 

St.  Augustine  Bay, 

Sandy  Island 

Cape  St.  Vincent 

Mourondava , 

Cape  St.  Andrew 

Boyanna  Bay,  entrance. 
Benibatooka  Bay, 

Majunga  Point 

Majunbo  Bay,  entrance  . . 

Nareenda  Bay 

Sancasse  Island,  N.  pt. . .. 
Passandava  Bay, 

Nine  Pin  Island 


Dalrymple  Bay 

Nos  Bell  Island,  N.  pt.  . . . 

Minow  Island,  N.  pt 

Cape  St.  Sebastian 

CAPE  AMBER,  N.E.pt. 
Britisli  Sound,  entrance  . . 
Port  Levon, 
Nosh  How  Island 


Cape  East,  town 

A  ntongil  Bay  ,Port  Choiseul 
Cape  Bellones 


St.  Mary's  Island,  N.  pt.. 
-  S.pt.. 


Lat. 


D.  M. 

25  39 
25  24 

23  38 

21  54 

20  18 

6  II 

5  59 

43 
12 
4o 
3i 

28 
3o 
3  12 
2  5o 
2  26 

1  58 

2  I A 


Foul  Point. . 

Tamatave  Point 

Fong  Isles 

fllanooroo 

Rangazarah 

Mananibatoo 

St.  Luce  Bay,  N.  Isle 
FORT   DAUPHIN.. 

St  ir  Bank 

Bassas  de  India 

Europa  Rocks,  S.  pt.  . 


Sussex  Rocks 

Bazaruto  Islands,  Cape  . . . 

Barren  Islands,  western  .. 

English  Bank 

Juan  de  Nova  or  St.  Chris- 
topher's Island 

Coffin  Island 

Chesterfield  Shoal 

Mayntta  Island 

Moliilla  Island,  E.  pt 

Johanna  Island,  peak  .... 

Comoro,  S.  E.  pt 

Portuguese  Slioals 

John     Martin's     Island, 
doubtful 

Rover  Shoal  

Aldabra  Islands,  N.  W.  p.. 

Assumption  Island,  Hura'k 

Cosmoledo  Island,  N.  pt.    . 

Marquis  of  Huntley's  Bank 

St.  Peter's  Island 

Natal  Island,  doubtful 

Sandy  Island 

St.  L-Twrence  Island 

Zanzibar  Island,  S.  p 

N.  p 


.'Xmirante  Island,  N.  W.  p. 
S.  E.  p.. 


8  10 

8  27 

9  55 

20  58 
24  17 

24  45 

25  I 

25  7 
25  25 

2  2  23 

21  3r 

21  25 
21  3l 

18  4i 
17  40 

17  3 

17  3o 

16  17 

12  54 

12  20 

12  i5 

11  54 

12  3o 

10  i5 
12  22 

9   23 

9  46 
9  38 
9  55 
9  20 

8  26 

9  10 


Lonn. 


6  20 


D.  M. 

45     7' 
AA  18 

43  38 

43  20 

44  19 

44  3i 

45  23 

4G  20 

46  59 

47  26 

47  35 

i5 

48  2 
48  19 
48  39 

48  46 

)<;  19 

49  23 

49  53 

50  3o 

49  52 
9  54 

50  5 
49  5i 
49  37 
49  28 

49  26 
48  52 
48  33 

25 

47  i4 
47    2 

44  16 

4o  24 

39  36 
42  36 
35  33 

44  3 

40  1 5 

42  47 

43  47 

43  55 

45  i4 

44  o 

44  3o 
43  33 

46  5o 

43  5o 

46  25 

45  5o 

46  34 

47  36 

50  1 5 
5o  5o 

47  12 

48  10 
5o  23 
39  33 
39  21 

53  45 

54  3o 


MaheBank,  N.  W.  p 

S.  E.p 

Seychelle    or 

Mahe  Island 
-W.  pt.  Praslin  Island 


French  Shoal 

African  Islands 

Alphonso  Island 

Sandy  Island  or  Bank. 
Isle  Bourbon  St.  Denis 
jMauritius,or  Isle  of  France, 
Port  Louis 


Lat. 


D.  M. 

3  20 
5  3o 


Diego  Rais  or  Rodrique  . 
St.   Branden   or   Cargados 
Garajos, 

N.  part  of  the  Bank 

Low  Sandy  Island 

Islet  with  huts.  — 

Soutii  Islet 


Nazareth  Bank,  S.  VV.  p.  . 

N.  E.  p... 

Sandy  Island 

Galega,    or    S.    Roquepiz 

middle 

Saya  de  Malha  Bank. 
limits . 


::{ 


46 


Fortune  Bank,  10  fath 
Joim  do  Nova,  N.  pt.  . . . 
Providence  Island,  N".  pt. 
Coetivy  Island,  N.  pt.  • . 
Chagos  Archipelago, 

DietTO  Garcia 


37 
17 
58 
55 
o 
12 


4 
4 
3 
4 
7 
7 
20  52 


20  10 
19  4o 


i3  4i 
16  5 
16  27 
16  47 
16  47 
i3  4i 
i5  52 

10  25 

11  3o 
8  18 

7  16 
o    7 

7     6 

7 
7 
7 
6 

7 


Pitt's  Bank 

Centurion's  Bank.  . . 

Ganges  Bank I  7  22 

Owen's  Bank |  6  46 

Egmont's  or  Six  Isl-| 

ands. .  .  .<. 6  " 

Danger  Island 6 

Eagle  Island 6 

Tiiree  Brothers 6 

Peros,  Banhos  Islands    5 
Saloman's  Isl's,  S.  W.    5 

Sandy  Islands 5 

Speaker's  B'k,  N.E.pt.    4 


Pona  Molubque  Atoll,  S.  p. 

N.  VV.  p 

—  N.  E.p.. 


Addon  Island,  middle 
Suadiva,  southern  oroup, 
South  Reef 

South  Island .... 

S.  W.  Island..., 

N.W.  Island..., 

N.  Island 

Northern    group, 

S.  W.  Island  . 

N.W.  Island..., 

N.  E.  Island 

Adoumatis  Atoll, 

—  S.  \V.  extremity. , . . 

—  Southernmost  Island 

—  Island 

—  N.  W.  Island 

-N.  E.  Island 


Long. 

D.  M. 

54  40  E 

56  59 

55  3i 

55  44 

54  42 
53  3o 
52  43 
52  43 

55  29 

57  3o 
63  24 


4i 
34 
33 


9N 


o  28 
o  34 


o  5r 

0  58 

1  5o 
I  47 

1  5i 

2  7 

2     7 


61  i5 
59  47 
59  4o 
59  24 
59  3 1 

61  i5 
34 

56  39 

62  20 
59  58 

57  o 
5i  8 
5i  7 
56  22 


72  22 

71  18 

70  57 

71  2 

70  20 

71  24 
71  i3 
71  18 
71  35 

71  48 

72  10 
72  37 

72  24 

73  6 
73  12 
73  25 
73  35 

73  i5 

73  12 

3  4 

73  2 

73  8 

73  19 

73  20 
73  33 

73  27 
73  22 
73  38 
73  35 

73  35 


Page  3C2] 


TABLE  LIV. 

Latitudes  and  Lonofitudes. 


Collomandous  Atoll, 

South  Island... 

Long  Island. . . . 

N.  W.  extremity 

West  entrance  of 

Coll.  Channel 

Molucque  Atoll,  S.  ex 

Nillandos  Atoll 

Poulisdous  Atoll 

Ari  Atoll,  N,  Is.  S.  pt 

Male  Atoll, 'or  Maldivia 
S.E.  p 

Gafer  Island 

Todu  Island 

Cordivia  Island 

Maloss  Madoll,  S.  pt 

Padipolo  Atoll,  E.  p 

MillaDoue  Atolls,  E.pt.   . 

Tilla  Dou  Matis,  or  Head 
of  the  Islands,  northern 
limit 


Lat. 


Long- 


Minicoy,or  Malicoy. 


Seuvelli  Islands, 

Southern    

Northern 

Southern     ex- 
treme Reef. 

Kalpeni  Islands,  S.  p 

N.  p 


Courutee  Island 

Pittie  Sand  Bank 

Underoot  Island,  E.  pt. . 

Aucutta  Island 

Bingaro  Island 

Tingaro  Island 

Ameni  Island 

Permulpar  Island 

Cardamuin  Island, 

Elicapeni  Bank,  E.  pt.... 

Kittan  Island,  S.  pt 

Betrapar  Island, N.  ex... 

Cliittae  Island 

Cherbaniano     Bank,    (not 
explored,)  S.  pt, 


Angrias  Bank,  N.  p.  ... 

Bale    of    Cotton    Rock, 

(doubtful,) 

Le  Meme's  Reefjfdoubtful) 
Prp])ari3  Islraid,  N.  p 
-  S.  p 


Great  Coco  Island,  N.  p. 
S.p.. 


Little  Coco  Island 

Landfall  Island 

Great  Andaman, 

Cape  Price,  N.  end 

S.  E.  point 

Port  Cornwallis. . . 

Port  Chatham  .... 

Port  Campbell.... 

Rutland  Island,  S.  p 

Interview  Island,  N.  p.. .. 

S.p.... 

North  Centinel 


21 

3o 

lO 

46 
4o 
36 
3o 

27 

46 
26 
58 
00 

25 

5i 


7  6 

8  17 


9  56 
o  4 
o  10 
o  3i 
o  45 
o  48 
o  5i 
o  55 

0  55 

1  6 

I  9 

I  i4 
I  i3 

I  25 

I  35 
I  4o 


2  i5 

6  38 
6  18 

5  18 
I  20 
4  56 
4  49 
4  II 
4  2 

3  58 
3  39 


34 
3o 


D.  M.  D.  M. 

2  iSNyS  21  E 

73  8 
73  8 

73  21 
73  23 

72  54 

73  M 

72  5o 

73  42 
73  4o 

72  58 

73  26 

72  58 

73  38 
73  27 


I  43, 
I  56 

1  24 
3  I 

2  47 
I  33 


72  53 

73  3 


72  12 
72  i5 

72  9 

73  35 
73  35 
72  36 

72  32 

73  42 
72  10 
72  16 
72  18 
72  41 
72  o 

72  ^i 

73  56 
73  o 
72  II 
72  42 

72  o 

71  43 
71  43 


94  20 
93  4o 
93  4o 
93  21 
93  21 
93  1 5 
93  4 

93  4 
92  56 


South  or  Little  Centinel. 

Five  Islands,  S.p 

Sisters,  southern 

Brothers,  northern 

Little  Andaman,  N.  p. .  • . 
: S.  E.  p 

Invisible  Bank,  N.  p 

S.p 

Flat  Rock 

Barren  Island 

Narcondam 

Car  Nicobar 

Batty  Maloe 

Chowry  Island 

Terressa  Island,  N.  p.  . . . 

S.  p.  ... 

Katchall,  W.  end 

Noncowry  Island  and  har- 
bor  

Comorta,  N.  p 

Tillangchong  Islands,  N.p 

S.p 

Meroe  Island 

Little  Nicobar,  N.p 

S.p 

Great  Nicobar,  N.p 

S.p 


D.  M. 

II  ooN 

17 
II  10 


Lat. 


II 

00 

10 

53 

10 

26 

II 

27 

10 

56 

II 

8 

12 

16 

i3 

26 

9 

10 

8  A6 

8 

28 

8 

22 

8 

12 

7 

54 

8  00 
8  i5 
8  33 
8  22 

7  29 
7  26 
7  i3 
7  8 
6  45 


Long. 

D.  M. 

92  22  E 
92  55 
92  46 
92  4i 
92  38 

92  40 

93  41 
93  40 
93  34 
93  54 


92  46 

92  5i 

93  3 
93  17 
93  6 
93  23 

93  46 
93  42 
93  40 
93  40 
93  34 
93  42 
93  34 
93  55 
93  54 


XLII.  The.  Islands  of  Sumatra,  Java, 
Billington,  Caspar,  Banco,  loith  the 
adjacent  Islands  and  Straits. 


Acheen 

Golden  Mountain .... 

Pedir  Point 

Elephant  Mountain.. 
Tooloo-Samwoi  Point 

Diamond  Point 

Tanjong  Bou 

Batacarang  Point  . . . . 

Fourth  Point 

Third  Point 

Second  Point 

First  Point 

Hog  Point 

Flat  Point 

Billimbing  Ba)' 

Bencoonat  


Lat. 


Cawroor '  4 


Manna  Point 

Buffalo  Point 

BENCOOLEN,    (Fort 

Marlborough,) 

Caytone 

Moco-Moco 

Indrapour  Point 

Padang  Head  

Priaman 

Natal 

Tappanooly,  P.  Kaeheel. 

Tappoose 

Sinkel  Point 

Bulo  Samah 


M. 

35  N 
22 

1 

i3 
i4 

5S 

o 
20 

23 

4i 
00 
54 
00 
54 
35 
56 
33 
58 

48 

34 
10 
56 
40 
33  N 

44 
00 
i5 
33 


Long. 

D.  M. 

95  19E 

95  45 

96  5 

96  5o 

97  14 

97  38 
io4  3o 
io4  5i 
io5  i5 
io5  32 
io5  4i 
106  3 
io5  45 
io4  36 

io4  27 
io3  34 
102  49 
102  19 

102  19 
102  14 
loi  20 

100  48 
roo  20 
100  10 

99  " 

98  45 

97  57 
97  46 
97  54 


TABLE  LIV. 

Latitudes  and  Longitudes. 


[Page  363 


I    Lat. 


Troumon 

Pulo  Duas 

Baccoongung 

Pulo  Munkie 

Oujong  Coomoowung. .. 

Oujong  Cluet 

Qualali  Bahoo 

Qualali  Assehahn 

Tainpat  Tuan 

Batto  Plyeer 

South  Tallapou 

Pulo  Sooroodung 

Muckie 

Laboun  Iladjie 

Mungin 

North  Tallapou 

Soosoo 

Pulo  Kio 

Qualali  Battoo 

Oujontr  Se  Mium 

Cape  Fflix 

Oujong  Trlpah 

Senaligun   

Analaboo 

Oryong-Booboon    or    Ba^ 

hoo 

Pulo  Rungass,   off  Rigas 

Bay 

Oujong  Chellung 

Rigas  

Tellow  Goolumpung.. 

Pulo  Cass 

Pulo    Riah   and  Pulo  M. 

centre 

Barbce  Wee 

Diah 

Oujong  Dahway 


Pulo  Rondo 

Pulo  Way 

Pulo  Brasse 

Pulo  Rajah 

Cocos  Islands  . . . 

Hog  Island,  N.  p. 
S.p. 


(  from 
(to  .. 


Flat  Islands. 


Pulo   Assayo 

Coral  Bank 

C  from 
(to 

Burgh  Rock 

Shoal,  10  feet 

Castlcreagh  Siioal  .... 

North  Pulo  Dua 

Passage  Island 

Bird  Island 

Pulo  Lucotta 

Londise  Shoal,  (N.  N.  E. 
^  E.  from  Lucotta,  dis- 
tant 2^  leagues.) 

Mcnsular  Island,  N.W.  pt. 

Pulo  Dua 

Pulo  Nyas,  S.  p 

Pulo  Tamong 

Pulo   Panjang 

Clappes  Island,  middle.. 

Pulo  Mintaon,  or  Batao  . 

Pulo  Ayer  Besar 


M. 

49  N 

54 

57 

55 

5? 

4 

6 

8 
i6 


4  i3 


38 
38 
39 

42 

Ao 

52 

55 


6  4 
5  49 
5  42 
4  4o 

2  59 

3  2 
2  57 
2    21 

2  4i 

3  3i 
2  4 
2  i3 
2  47 


I  57 
I  40 
1  27 
o  36 
o  54 
o  i3 
o    o 

0  25 

1  24 


Lono-- 


D.  M. 

97  5iE 
97  44 
97  42 
97  39 
97  38 
97  32 
97  3i 
97  3o 
97  23 
97  19 
97  18 
97  16 
97  i4 
97  II 
97  4 
97  3 
96  58 
96  57 
96  56 
96  46 
96  42 
96  3i 
96  24 
96  18 

96    5 

95  38 
95  4o 
95  4o 
95  37 
95  34 

95  3o 
95  3o 
95  28 
95  25 

95  i4 
95  23 
95  6 

95  33 
95  33 

95  58 

96  38 
96  34 
96  42 
96  47 

96  54 

97  26 
97  33 
97  6 
97  39 

97  5o 

98  7 


98  3o 
98  20 

97  56 

98  40 
98  3o 
98  3o 
98  7 

100  17 


D.  M. 


G.  Fortune 


Se-beero, 

Island,  N.p 

S.  W.  p 

Se-pora,  or  South  Pora, 
N.  W.  p 


-S.p 

North  Poggy  Island,  N.  p, 

S.  p. 

South  Poggy  Island,  N.  p. 
S.  p, 


Laage  or  Larg  Islands. 

Rat  Island 

Trieste  or  Reefs  Island 

Pulo  Pisang 

Little  Fortune  Island . . 
Encrano  or  Deceit  Island, 
N.p 

—  E.  point 

—  S.  E.  point  ... 

—  S.  point 

—  W.  point 


Java  Head  

First  Point 

Second  Point 

Third   Point , 

Anger 

Bantam  or    St.  Nicholas 

Point 

Bantam   

BATAVIA  obs 

Carawang  Point 

Sedary  Point 

Point  Pamanoekan.  • . , 
Woerden  Castle  Rock, 
Princess  Charlotte    Shoal 

Indramaye  Point 

Pulo  Rackit 

Bumkin's  Island,  or  outer 

Shoal 

Cheribon  Mountain  . . , 

Taggal , 

Rock 


Samarang  flagstaff, 

anchorage  . . . . 


Mandalique  Island 

Lerang  Point 

Rambang 

Point  Panka  or  Panco. . . 

Sour.abaye,  fort 

Cape  Sandana 

Balainbonang     Bay,     Ft 

Goonog  Ikan 

■  E.  point 


Turtle  Bay. 

Tulan  or  Dirck  Vrie's  Bay 

Wine  Cooper's  Point 


Noesa  Baron  Island,  S.  p. 
Tangala  Islands,  largest. 
Clappe's  Island,  about. . . 


Mew  Island 

Peak  on  Prince's  Island  . 
Peak  on  Crocatoo  Island. 
Peak  on  Tamarind  Island, 

or  Pulo  Bessy , 

Pulo  Sebooko 


Lat.        Loner 


56  S 
47 

00 

25 
32 
52 

5o 
20 
3o 
5i 

3 

8 
54 

i5 
22 
3o 
3i 


6  48 
6  44 
6  36 
6  27 
6  3 

5  53 

6  2 
6 
5 
5 
6 
5 
5 
6 
5 


5  47 

6  55 
6  5o 


8  23 
8  46 
7  48 
7  5o 

7  25 

8  32 
8  26 

7  I 


D.  M. 

98  38  E 

99  2 

99  33 
99  58 

99  37 
00  1 3 
00  1 5 

00  4 1 

01  3 

02  1 5 

01  6 
04  6 
o4  3o 

02  25 
02  40 
02  38 
02  20 


o5  1 3 
f>5  12 
o5  21 
o5  4o 

05  56 

06  4 
C16  10 

06  5o 

07  3 
07  27 
07  49 
07  58 

07  54 

08  20 
08  22 

08  23 

08  26 
04  i4 

10  27 
10  26 

10  5i 

1 1  27 

11  17 

12  32 

12  45 

i4  22 

i4  25 
i4  33 

09  48 
08  1 2 
06  26 


o5  i5 
o5  i5 
o5  20 


Page  3S4] 


TABLE   LIV. 

Latitudes  and  Longitudes. 


Cap 

Button 

Tliwart-tlie-way 

Zutphen  Islands, (largest,) 

N.  p 

South  W.M-'her 

Man-eater's  Island 

Pulo  Baby 

Thousand  Islands,  N.  . . . 
Pruysen's  Droogte  Shoal 

Annuyden  Bank 

North  Watcher 

Three  Sisters 

North  Island 

Two  Brothers,  northern  . 

Lynn  Shoal 

Shabunder  Shoal 

Brouwer's  Shoal 

Lueepera,  S.  entrance  St. 

Banca , 

Nanka  Islands 


Lat. 


Banca  Island, 

South  Point 

Tanjong  Panjong,  or 

Point  Lalary   .... 

Monopin  Hill 

Tanjong  Goonting  . 

Tanjong  Muncooda 

N.  of  Banca 

Tanjong  Tuan 

Songy  Leat  Bay .... 

Tanjong  Ryah 

Goonong    JVIarass 

Mount 

Tanjong  Breket. . . . 

Rocky  Point 

Entrance    Point,    or 

S.  E  p 

Essex    Shoal,   or    Fairlie 
Rock 


Vansittart's  Shoals 5 

Pulo  Leat  or  Middle  Isl.. 

Alceste  Shoal 

Shoal  Water  Island 

South   Island 

North  Island 

General  Hewitt's  Rock . . 

Discover}'  Rock 

Pulo  Glassa  or  Gaspar  Isl. 

Tree  Island 

Warren  Hastings's  Shoal 
Belvidere's  Shoal,  N.  pt. 

Vansittart's  Shoal 

Hillsborough  Slioal 

Magdalen's  Shoal 

Severn's  Shoal 

Billiton  Island.  S.  E.p... 

S.  W.  point 

N.p 

N.  W.  Island,  off  Billiton 

Shoe     Island,     (formerly 

Bird  Island  and  White 

R) 

Fo.T  Shoal 

Pulo  Mancap 

Shoal,  S.  p. 


D.  M. 

5  59S 
5  53 
5  57 

5  5o 
5  4i 
5  54 
5  48 
5  32 


5  17 
5  i3 
5  12 
5  M 
5  4i 


3  i3 
2  25 


3    8 

2  49 
2  00 
I  43 


I  28 
I  38 
I  5o 
I  55 

1  53 

2  36 
2  56 


27 


3  5 
2  5i 

2  46 

3  20 
3  00 
2  58 
2  53 
2  54 

2    25 

2  28 

2    23 


2  3 

I  56 

1  4o 

3  22 
3  i5 

2  33 
2  3i 


3  47 
3  3o 
3  5 
3  22 


Lono-. 


D.    M. 

o5  57  E 
o5  57 
o5  5i 

05  47 

06  43 
06  3o 
06  i4 
06  35 
06  47 
06  48 
06  3o 
o5  48 

05  49 

06  3 
06  1 3 

05  56 

06  i4 

06  8 
o5  48 


06  28 

06  4 
o5  12 

o5    2C 

05  53 

06  6 
06  9 
06  i4 

05  52 

06  5o 
06  54 

06  52 

07  2 
07      2 

07  8 

07  5 

07  2 

07  1 3 

07  1 5 

07  1 5 

07  i4 

06  56 

07  4 
06  58 

06  57 

07  00 
06  42 

06  22 

07  o 

06  3o 

08  10 

07  3i 
07  53 
07  33 


08  o 

10  6 

10  7 

10  7 


Discovery 'sWesternBank 

Eastern  Bank 

Reef 


Osterly's  North  Shoal. . 
Cirencester's  Sand  Bank 
Shoal  .... 


Montaran  Islands,  South 

Eastt  n .< . 

Toekoekemou, 

(highest  island,) 

Minto  Rocks 

Ontario's  Shoal 

Rendezvous  Island,S.W.p 

Souroutou,  W.  p 

Carimata  Island  Peak, 

Pulo  Papan 

Pulo  Panumbangan. . , 
Massa  Teega  Isles  ... 
Greiff's  Shoal , 


The  Seven  Islands,  N.  W 
Pulo  Varela  or  Barallah  . 

Pulo  Taya 

The  Calantigas 

Ilchister   Shoal 

Ungin,    Tanjong   Eang, 

S.  E.  extremity 

East  Domino  Island  . . 
Geldrias    Bank,  same 

Dogger  Bank 

Rhio 

Eastern  Island,  off  Pulo 

Panjang 

Island  Laage,  E.  pt.   . . 
Three  Brothers,  south . 

Pedro  Branco  

Islands  off  P.  Romaine 
Bintang  Island,  (the  hill,) 
-  N.  W.  p 


Johore  Shoal 
Shoal  ent.  Rhio  Straits 
Sincapore  Island,  E.  p. 
Pulo  Battain,  N.  E.  p.. 
St.  John's  Island,  S.  p. 

Rocky  Reefs 

Middle  Island 

Coney  Island 

Buffalo  Rock 

Rocks 

Red  Island 

Tree  Island 

Alligator  Island 

Rocks 

Little  Carimon 

Great  Carimon,  S.  pk. 

The  Brothers 

Pulo  Cocob 

Pulo  Pisang 

Water    Islands,    or   Four 

Brothers,  S.  p 

Fisher's  Island 

Bambeck  Shoal 

Pulo   Callam  or   Colong, 

S.p 

Two  and  a  half  fathoms 

Bank 

Round  Arroa 

Blenheim's  Shoal 


Lat. 


M. 

39  S 

33 

36 

19 

17 
2  54 
2  35 
2  3i 

2  3i 
2  i4 
2  I 
2  44 
I  42 
I  36 
I  28 
I  12 
o  55 

0  55 

1  8 
o  5o 
o  45 
o  35 
o  26 

o  20 

O    10 

o  48N 
o  57 


46 
3i 
20 

23 


2  i3 

2  37 

2  56 

2  54 

2  49 

3  3 


Loner, 


D.    M. 

108  43  E 

109  10 
108  48 

108  40 

109  o 
n.8  58 

108  52 

io8  36 

109  5 1 
108  39 
no  9 
108  38 

108  5i 

109  26 
109  12 
109  18 
108  35 

io5  12 
104  25 
104  55 
io3  5i 
io4  57 

io5  o 
io5  o 

io4  58 
io4  3o 

io4  49 
104  45 
io3  44 
io4  23 
104  16 
io4  26 
104  16 
104  4 
104  II 
104  00 
104  4 
io3  5i 
io3  55 
io3  46 
io3  4 1 
io3  48 
io3  45 
U.3  38 
io3  36 
io3  4o 
io3  36 
io3  22 
io3  19 
io3  2t 
io3  25 
io3  i3 

102  20 
102  12 
loi  4i 

loi  16 
100  OJ 


TABLE  LIV. 

Latitudes  and  Longitudes. 


[Page  365 


Long  or  Great  Arroa. . . . 

Two  Brotliers,  Pulo  Pan- 
dan  

Pulo   Salanama 

Pulo  Varela 

Pulo  Jarra 

Sainbilap.g  Isl.,  southern. 

Dinding  island,  W.  p. . . . 

Prince  of  Wales's  Island, 
Fort  Cornwallis 

Pulo  Pera 

Boonting  Island,  southern 

Pulo  Bonton,  (dome,) . 

Pulo  Ladda,  S.  p 

Trotto  Island,  N.  p 

Sangald  or  Guilder  Rock 

Pulo  Tolibon,  S.  W.    .... 

The  Brothers 

Pulo  Rajah,  or  P.  Taya.. 

Juiikseylon,  S.  p 


Lat. 


M. 

52  N 


24 

99  54 

21 

99  52 

47 

99  36 

00 

loo  lo 

3 

100  3o 

i6 

loo  35 

Lonff. 


D. 

100 


M. 

44  E 


21 

57 
i8 

I? 

42 

39 

45 

24 


98  18 


XLIII.     Islcmds  and  Shoals  in  the 
CHIMA  SEj]. 


Si  Barbc  Island 

Direction  Island 

Pulo  Datoo 

Welstead's   Rock 

St.  Esprit  Islands,  E 

Green  Island 

St.  Julian  Island 

Tanibolan  Islands,  East  or 

Great  Island 

(Jap  Rock 

Europe  Shoal 

Rocky  Island 

Camel's  Hump 

Saddle  Island 

French  White  Rock  .... 

Victory  Island 

Acasta  Rock     

White  Rock 

Macedonian  Reef 

South  Anambas,  limits  < 

Pulo  Domar 

JMiddle  or  G.  Anambas, 
W.  limit 

North  Anambas 

PuloTingy 

Ex.  IsletoffP.Tingy.... 

Pulo  AOR  or  Wawoor  . . 

Pulo  Pisang  or  Pambee- 
lan 

Pulo  Tinioan,  S.  p.  . . . 

—  N.  p 

—  Bay  on  S.  W.  side  . 

—  N.  Islet  ofl'N.W.  side 

Pulo  Varela 

Pulo   Brala,  or  Capas  de 

Terre 

Pulo  Capas  de  Terre  . . 
St.  Pierre  Islands 

Ledge  of  Rocks. . 

Larkin's  Reef 


Lat.        Lons- 


M. 

7N 
i5 
7 

32 

34 
4o 
54 


9 
10 
16 

32 

34 
39 


2   25 
2    18 

2  4o 

2    45 


9 

27 

17 

8 

29 

37 
44 
54 
48 
56 
16 

47 
i5 
54 
53 


[o5  4i 
[06  i5 
[o4  II 
104  i4 
[o4  35 

[o4  i3 
[o4  1 5 
[o4  i5 


[o3  47 


South  Haycock  Island  . . 
South  Natunas  Islands, 

—  South  Island,  or  Sapata 

—  East  Island 

—  West   Island 

—  North  or  Flat  Island . . 

Low  Island 

Hutton's  Shoal 

Diana  Shoal 

North  Haycock  Island. .. 
Grand  or  Great  Natuna  C 

Island,  limits \ 

Mt.  Peaked  Island 

Pyramidal  Rocks 

N.  W.  Island 

Coral  Reef 

Coral  Reef 

North  Natunas  Islands, S.p 

N.p 

Rock  above  water . . 

Saddle  Island 

Success  Slioal 

Pulo  Oby 

The  Brothers,  (eastern,). 

PuloCONDORE 

Charlotte's  Bank 

Phaeton  Bank 

Royal  Bishop's  Bank,  S.p. 

Britto's  Bank 

Holland's  Bank,  S.  W.  p. 

N.  E.  p. 

Pulo  SAPATA 

Pyramid  Rock,  or   Little 

Catwick 

Round   Island,   or    Great 

Catwick 

Pulo  Ceicer  de  Mer 

Minerva's  Bank 

Investigator's  Coral  Patch 
Triton's  Island   or  Bank, 

S.  W.  part 

Passoo  Keah,  (Sandy  Isl.) 
Bombay  Merchant's  Shoal, 

E.  p. 

S.  p. 

Discovery  Shoal,  W.  p.. . 

E.p.  .. 

Jehangire's  Coral  Bank.. 

Vulador's  Shoal,  E.  p 

W.  p... 

Crescent  Chain, 

Money's  Island .... 

Robert's  Island  .... 

Battle's  Island 

Drummond's  Island 

Governor    Duncan's 

Island 

Antelope's  Shoal.  . . 

Observation  Bank,  N.p.. 

Pyramid  Rock 

Lincoln  Island 

Rocky  Island 

Woody  Island 

Amphitrite  Islands,  W.  p. 

—1 E.  p. 

North  Shoal,  W  p 

E.  p 


Lat. 

Long. 

D.  M. 

D.  M. 

2  9N 

109  10 E 

2  26 

109  8 

2  42 

109  18 

2  5o 

108  28 

3  3 

108  54 

3  0 

107  45 

3  0 

107  57 

3  9 

107  44 

3  i5 

107  18 

3  40 

108  26 

4  16 

108  11 

4  01 

108  10 

4  7 

107  26 

4  7 

107  5o 

4  I 

107  5o 

3  57 

107  47 

4  42 

107  58 

4  5i 

108  0 

4  39 

107  57 

4  3i 

107  M 

4  23 

107  54 

8  25 

104  54 

8  35 

106  1 5 

8  40 

106  42 

7  5 

107  37 

7  0 

107  29 

9  4o 

108  21 

10  32 

107  48 

10  36 

108  32 

10  48 

iu8  47 

10  I 

109  2 

TO  2 

109  00 

10  6 

108  52 

10  32 

108  53  , 

10  37 

110  18 

i4  12 

112  52 

i5  45 

III  II 

16  3 

in  45 

t6  4 

112  38 

i5  59 

1 12  26 

16  11 

11 1  32 

16  16 

I II  46 

16  )8 

112  35 

i6  19 

112  7 

16  18 

112  0 

16  28 

III  3o 

16  3i 

III  34 

16  33 

I II  36 

16  29 

III  44 

16  27 

III  4') 

16  27 

III  35 

16  37 

III  4i 

16  35 

112  37 

16  4o 

1 12  42 

16  52 

1 12  20 

16  5o 

112  18 

16  59 

1 12  12 

16  54 

112  23 

17  5 

III  26 

17  6 

III  32 

Page  366] 


TABLE  LIV. 

Latitudes  and  Longitudes. 


\ 


Macclesfield  Bank,  C 
limits ( 

Scarborough  or  Mar-  C 
singola  Shoal,  limits  (^ 

St.  Esprit  Shoal,  (by  Lt 

Ross,) 

•  (by  As- 


seveido,). , 
Pratas  or  Prater's  Shoal, 

N.  E.  p.  . . , 

N.  W.  p.., 

•  Anchorage 

Island  .... 


Great  Ladrone 

[The  Islands  near  Canton 
are  given  in  No.  XL. 
and  in  No.  XLVI.] 

Pedro  Branco 

Lamock  Islands,  outer- 
most   


Lat. 


M 

17N 
21 
4 
i3 

3o 


(very 


Andrade    Rock, 

doubtful,) .... 
Luconias  Shoals, 

Hard  Rocks  . .' 

Two  Fathom  Shoal. 

Dry  Sand 

Sea-Horse  Reef 

Half-Moon  Breakers 

•  Bank  . 

Paraquas,  5  or  6  leagues 

from  Palawan 
Euphrates  Shoal 

Kirton's  Shoals. 


Louisa's  Breakers . . . 

Mantannane  Isles 

Barton's  Shoals 

Royal  Charlotte's  Rocks. 
Sands. 


Swallow  or  Investigator's 

Rocks  

Viper's  Bank 

Breakers 


Ardasier's  large  coral  flats 
and  Taps, 

—  W.  p.  (Walpole,  Corn 

wallis  and  A.) 

—  N.  E.  p.  (Walpole  and 

A.) 

—  E.  p.  (Ardasier) .  . 

—  S.    p.      (Pennsylvania 

and  A.) 

Gloucester  Shoal 

Stag's  Shoal 

Prince  of  Wales's  Bank, 

limits 

London  Breakers 

Reef,  western 

— —  Reef,  eastern  .... 

Breakers 

Breakers 


Ganges  Breakers. . . . 
Investigator's  Shoal,  W.p 


9  56 

5  24 
5     5 

4  57 

5  35 
8  46 

10  57 


9 

5 
5 
5 
6 
6 
6 
6 
10  47 

7   23 

7  3o 

8  00 


7  56 

7  54 
7  4o 

7  3o 

7  5o 

8  24 
8    5 

8  i3 

9  36 
8  55 

8  48 
7  33 

7   25 

9  25 

10  3o 
5 


Lons- 


D.   M. 

I i3  44  E 
ii4  59 
117  4,i 
117  53 

ii3    6 

ii3     5 

116  54 
116  42 
116  42 
116  45 
ii3  A^ 


ii5  8 
117  19 

111  4 

112  3o 
112  24 
U2  3o 
112  28 

1 16  3o 

117  53 

117  28 
ii3  3o 
ii3  i5 
ii3  2 
ii3  18 
116  7 
116  i3 
ii3  38 
ii4  29 

ii3  49 
ii5  o 
ii5  25 


ii3  12 

ii4  24 
ii4  47 

ii4  34 
ii4  14 
112  57 
no  27 
no  34 
n2  26 
n2  00 
112  24 
A 


14 


ii3 

n3  3 

ii4  10 

n5  ID 

n4  35 


Investigator's  Shoal, 

Shoal 

Shoal , 


Coral  Rocks. 


Cavallo  Marino's  Shoal ) 

Black  Rocks 

Bank  

White  Sand 

Low  Black  Island . . 

Friendship's  Shoal .... 

Hardwicke's  Reef*  (or 
Dolphin's) , 

Breakers  *  (ditto)  . , 

Royal  Captain's  Shoal... 

Bombay's  Shoal , 

Dolphin's  Reef*  (or  Hard 
wicke's) 

Breakers  * 

Breakers  *  (ditto)  . , 

Great  Reef,  N.p.*.. 

Long  Island  * 

Breakers  * 

First  Island* 

Ledge  * 

■  Breakers  * 

Breakers  * 

Falmouth's  (or  Essex)  low 
Island* 

Bank,   or    Gossard's 

Bank, 

Essex  (or  Falmouth)  low 
Island* 

Gossard's  Reef  (or  Mid- 
djeburgh  R.) 

Small  Island 


Lat. 


Cornwallis  Breakers 

Sabut  Jung  low  Island 
Bank 


■\ 


Gaspar  Shoals 

South  Sea  Castle's 
Sandy  Islands  and 
dangers,  limits  (by 
Lieut.  Ross) 

Two  Islands 

An   Island   (Investigator) 

An  Island,         ditto  . . . . 

A  Reef 


Discovery's  Reef. . 
York  Breakers,  W. 


(Viper's) 
Pennsylvania   ,     p-^  V 

Krpn  rpr«  _  _  ^  -z  ' 


D.  M. 

8  10: 

9  12 

10  ^/i 
9  4o 
9  42 
5  54 

8  3i 

9  39 


5  52 

6  00 

9  54 
10    2 

9  4 
9  27 

9  59 
9  45 
o  8 
o  7 
o  17 
o  22 
o  35 
o  4o 

0  4Q 

1  10 

0  58 

1  25 
I       2 

8  58 
o  42 

0  00 

8  52 

1  32 

I  34 

I  36 

I  29 
I  21 

I  27 
I  8 
o  A^ 
o  i5 
o  00 
o    8 

9  55 
8  17 
8  5o 

8  58 

9  4 
10  00 

9  48 

9  32 

9  49 

9  52 
10  25 
10  52 


Long. 

\).    M. 

n4  5i  £ 

!  16    32 

ri4  U 
n3  4 
n3  i5 
\\4  18 
1 14  21 
n4  58 
n5  7 
n5  i3 
n5  17 
n2  34 
n2  49 

n2  17 
n2  12 
116  40 

1 16  57 

n2  17 
n  2  3o 
n2  i5 
n2  9 
n2  35 
n2  3i 
n2  38 
n2  47 
n2  47 
112  54 

112  4o 

i\A  i3 

n2  4o 

in  5 
ii3  26 
1 14  22 
ii4  12 
ii3  29 
ii3  5i 
n3  5i 

ii4  20 
1 14  16 

wi^i  22 
wA  18 
1 14  26 

11 3  40 

n3  5o 

1 17  55 
ii4  43 
ii5  17 
ii5  21 
ii5  20 
n5  12 
ii4  4o 

6  28 
116  47 
116  48 
116  37 
116  55 


The  longitudes  of  these  places  ought  probably  to  be  increased. 


TABLE  LIV. 

Latitudes  and  Longitudes 


Page  367 


XLIV.  Islands  and  Shoals  between 
Batavia  and  J\ew  Guinea,  South  of 
the  Celebes. 


Carimon  Java,  W.  ex.   . . . 
Lubeck  or  Babian  Island, 

Arrogant's  Shoal , 

Madura  Island,  N.  W.  p., 
N.  E.  p. , 


Pondy  Island 

Great   Solombo  Island, 
(hill  on  S.  E.  p.) 

Little  Solombo  Island  . . . 

Arentes  Island 

Little  Pulo  Laut,  (middle) 

Four     Brothers,     sunken 
Islands 

Urk  Island.. . . 

Kansrelang  or  Cangayang 
Island,  N.  p 

S.p 

S.  E.  Island,  or  Hast- 
ings's Island 

Kalkoon    Islands,    north- 
ern, about . 

Four  small  islands,  middle 

Great  Paternoster  Islands 
W.  p 

S.  W.  Island  . . 

S.  Island 

Two  low  Islands 

E.p 

Postilion's      Islands. 
N.  W.  p... ■ 

Eastern  Island 

S.  post 

Nocsa  Sera  Islands. 

Noesa  Comba,  about 

Sd.  Bank  ofF  Noesa  Comba 

Caloeoliij     or    Rotterdam 
Island  . 

Hen  and  Chickens,  S.  p.. 

Zalinaff,  or    Saflanaff,  or 
Lacr's  Island 

Coral  Bank  ofF  ditto. 

S.p  .... 

ditto  E.  p., 

ditto  W.  p 

Five  Fatlioms  Bank 

Tonyn  Islands,  S.  W.  Isl. 
E.   Island 


D.  M. 

5  5oS 
5  49 

5  12 

6  53 

6  53 

7  I 


Siioal 

Taiiakeka  or  Tunikik  Isl. 
Brill  Shoal,  N.  p 
. S.  p. 


Mansfield  Shoal 

Middle  Island 

Salayer  Island,  N.  p.  . . . , 
Cambyna  Island,  S.  p.. . 

Peak   

South  Island 

Hegadis  Island 

Bouton  Island,  S.p 

Town 

N.  E.  point , 

Calansoese   Harbor 


Lat. 


5  33 

5    21 

5  lo 
4  5i 


6  54 


i5 

32 

34 
36 

42 
32 

45 
58 

2 

i5 

52 


28 


Loner. 


D. 


5  3i 

5  54 


45 
i3 
5  42 
5  27 
4  23 
4  55 


E.  point I  5  i5 


M. 

o    3E 

2  48 

3  00 

2  45 

3  58 

4  4 

4  24 
4  25 

4  32 

5  53 

4  5o 

5  16 

5  17 

5  25 

6  II 

5  46 
5  5o 

7  00 
7  16 
7  3o 

7  55 

8  3o 

8  ^6 

9  i5 
8  56 
7  9 
7  9 
7  10, 

7  38 

7  54 

8  M 


8  26 

7  58 

8  20 
8  36 
8  5o 

5 

24 


9 
9 
9    2 
9    o 

20  17 

20  28 

20  28 

21  57 

22  28 
22  4o 
22  4i 

22  48 

23  4 

23    II 

23  i5 


Lat. 


D.  M. 


Token  Bessy's  Islands, 

—  Wangiwangi,N.VV.  Isl. 

—  Pinnunko,  S.  lim 

—  Velthoens  or  Koko  C 
Island \ 

St.  Matthew's  Islands, 
(middle) 

Mamalakjee  Island,  (N.W. 
Tonin  Island,) 

Scliiedam  Islands,   N.  W. 

S.  E.. 

Shoal. 

Kalatoa  Island 

Alfred's  Shoal 

Jagger's  Reef,  or  Banga- 
lore Shoal,  about 

another  estimate. 

Angelica's  Shoal 

another  estimate . . 

Rusa  Raji  or  Lusardy  Isl. 
Rusa  Linguette  or  Rosa- 

galet  Island 

Tiie  Three  Bastards 

Bally  Island, 

—  Table  Point,  or  S.  p. . . 

—  Volcano 

—  N.  E.p 

Bally  Straits,  S.  entrance 
A  shoal  near  the  anchor- 
age at  Balariibuing, 
bears  S.  W.  h,  W.  from 
the  flagstaff,  distant  % 
mile  from  shore. 

Mynder's  Rocks 

Banditti  Island,  S.  E.  pt. 
Lombock  Isl.,  S.  p.  about 

Peak,  near  N.  E.  p.. 

N.  E.  pt 

Lombock  Island, 

Isles  near  N.  W.  p.. 

Ampannan  River, 

entrance 

Loboagee    or    Bally 

Town ■. 

Selonda  Island 

Pulo  Majo  or  Mayo,  N.  p. 

Flat  Island 

Sandbuy's  Four  Shoals 

limits 

Sumbava  Island,  S.  W.  p 
Timor  Yung  Island, 

(off  N.  W.  p  ) . . . . 

Sumbava  Bay 

Tumbora  Mountain. 

Biema  Bay,  rugged 

point 

ditto,  rocky  point . . . 

Sapy  Bay, anchorage 

S.  E.  Point 

Goonong  Apee  Isl.  Peak. 

Comodo  Island 

Flores   or  Mangerj'e   Isl- 
and, S.  W.  p.  about 

—  S.  p.  about 

—  Lobetobie   Volcano. 

—  N.  p.  Flores  Head, 

Iron  Cape 


18  S 
5 
58 


5  18 


4i 
I 
12 
27 
12 
9 

7  40 

7  55 

7  4o 

8  17 

8  5 
8  i4 

8  5o 
8  21 
8  18 


Lens. 


8  5o 

7  4i 

8  5i 
8  5o 
8  26 
8  19 

8  i3 

8  33 

8  42 
8  8 
8  7 
8  7 

7  42 
7  56 

9  2 

8  21 

8  27 
8  i5 

8  II 
8  8 
8  3o 
8  42 
8  i; 
8  22 

8  48 

9  00 
8  35 

8  I 

D.  M. 

123  33  E 

123  56 

124  43 


124  16 

120  1 4 
120  28 

120  56 

121  i3 
121  43 
121  39 

121  13 
121  46 

121  25 

122  18 

121  36 

122  3 

122  4i 

ii5  2 

ii5  27 

ii5  43 

ii4  40 


ii4  22 
ii5  29 
116  00 
116  26 
1x6  4^ 

II 5  59 


6  33 


6  57 

7  34 

7  55 

8  4i 
8  36 
93 


i4 

5 

37 

54 
3o 


I'age  368] 


TABLE  LIV. 

Latitudes  and  Longitudes. 


Straits  of  Florcs,S.  ent.  Id. 
Sandal  Wood  Isl.,  N.  p.  . 

BlufFor  W.  p 

S.  extremity 

E.  end 

Padewawy   or    Bar- 
ing's Bay 

Savu  Island,  W.  pt 

New  Island,  S.  pt 


Lat. 


Polo  Comba  or  Cambay 
Lomlilen  Island  Peak  (on 

N.  W.  p.) 

E.p 

P-intnr  Island,  N.  E.  p 
East  Island,  Strait  of  Aloo 
Middle  Island,  ditto... 
Oinbay  or  Mallao  Island, 

N.  W.  p 

E.p 

RottoorRottelsl.,  S.W.  p. 

—  Booca  Bay,  on  S.  side. 
Timor  Island,  S.  W.  p.  .. 

—  Copang,FortConcordia 

—  Peak , 

—  N.  W.  point 

—  Tulycaon  Bay 

—  Batto-gady 

—  point  nearest  Ombay 

—  Dilly,  or  Diely 

—  E.  end 

Pulo  Batto 

Pulo  Cambing  or  Passage 

Island,  S.  p 

N.  p.  . . . 

Wetter  Island,  E.p 

Pulo    Baby,    near 

S.W.  p 

Goonong  apy  or  Burning 

Island 

Dog  Island 

Kisser  Island 

Pulo    Jackee,    or     Noosa 

Nessing 

Lettee  Island,  W.  p 

Roina  Island 

Lucapin-ha}"-  or  Lucepera 

Island 

Turtle  Islands,  eastern  . . 
Cerowa  Island,  about. . . . 
Babber  Island,  about  .... 
Timor  Laut,  S.  &  W.  end 
Arroe  Island,  S.  extr.... 


D.  M. 

8  38S 

9  i5 

9  42 

10    22 
10   OO 

9  4o 

lO    32 

10  49 

7  49 

8  12 

8  i4 
8  10 
8  20 
8  23 

8  9 

8  i5 

11  2 
TO  53 

10  23 

10  9 

9  4i 
9  24 
9  12 
8  57 
8  39 
8  33 

8  2T 

9  i4 


Lons. 


8  II 

7  46 

8  o5 

6  35 

7  4[ 

8  6 

8  21 
8  i4 
7  42 


5  40 

5  25 

6  10 

7  25 

8  27 

9  00 


D.    M. 

122  58  E 

19  00 

20  35 
20  5 1 

20  18 

21  35 

21  10 

23  38 

23  47 

23  35 

24  20 
24  00 

23  55 

24  27 

25  i5 

22  55 

23  5 
23  3o 

23  35 

24  1 1 

23  55 

24  23 

24  5o 

25  i3 
25  34 
27  1 3 

23    52 

25  29 

25  43 

26  54 


126  4o 
125  56 

127  7 

127  i3 
127  4o 
127  26 

127  21 
127  38 
129  53 
i3o  40 
i3i  7 
1 35  00 


XLV.  Borneo,  Celebes,  Luconia,  wilh 
the  adjacent  Islands  and  Shoals,  as  far 
East  as  JYew  Guinea. 


Tanjong  Sambar,  S.  W.  p 

Succadana  

Tanjong  Factie 

Pontiana  or  Lewa  R.,  ent, 

Point  Mampava 

Slackoo  Road 

River  Sambas,  entrance  . 


Lat. 


I   i3 


Lons- 


X 

M. 

D  M 

1 

53  S 

109  58 E 

I 

16 

1 10  0 

I 

16 

109  35 

0 

2N 

109  10 

0 

17 

109  GO 

108  58 


Tanjong  Apee 

Tanjontr  Datoo 

BORNEO  Read 

Pulo  Teega 

Abai  Harbor 

Keeney  Balloo  Mountain 
Tanjonc   Sampanmangio, 

N.  p.° 

Point  Unasang 

Point  Kanneeoongan 

River  Passier,  entrance . . 

Ragged  Point 

Shoal  Point 

Point  Salatan,  S.  p 


Lat. 


D.  M. 

I  58  N 


Lonff. 


Point  Layk,  S.  W.  p... 
macassar:  Town,  fort 

Cape  Mandhar 

Cape  William 

Cape    Temoel   or  Samsa 

S-P 

N.  W.  p 


Cape  Donda 

Cape  Rivers 

Manado,  fort , 

Cape  Coffin , 

Isle  Banca,  E.  pt 

Kema  Village 

Castican  Baj' 

Goonong  Telia  River 
Cape  Talabo,  E.  pt.  , . 
Weywongy  Island,  about 
Waxway   Island,  middle 
Cambyna  Island  Peak.  . 

Middle  Island 

Boele-comba  Hill 


Waller's  Shoals  and  C 
Laurel  Rocks,  limits  ( 

Noesa  Sera  Islands 

Noesa  Comba 

Shoal  off  Noesa  Comba. . 

Little  Pulo  Laut  Isl.,  mid. 

Moresses  or  Manevessa 
Island 

Dwaalder  Island 

Royal  George  Shoal 

Two  Brothers 

Great  Pulo  Laut,  N.  E.  p 

—  N.  p 

—  S.  Isl.  off  the  S.  E.p.. 
The  Three  Alike   Islands 

Dry  Sand  Bank 

Triangle  Islands,  middle 
Little  Paternosters,  S.  p. 

N.  E.  p. 

N.  W.  p. 

Pamaroong  or  Dondrekin 

Island,  S.  p 

Seven  Islands 


Banguey  Peak 

Balanibang  Isl.,  N.  Harb. 
Balabac  Island,  (hill,). 

Mangsee  Islands* 

St.    Michael's    Islands, 

(Bangcawang,) 

Toob-Bataha  Slioal,  S.  extr 


3 

00 

5 

00 

5 

45 

6 

21 

6 

8 

7 
5 
I 

I 

17 
3 

I  44  s 

2  10 

2  33 

4  10 

5  37 
5  9 

3  35 
2  37 


o  iN 

0  48 

1  20 
I  29 
1  42 
I  43 
I  19 
o  48 
o  28 

o  55S 

4  3 

3  34 

5  21 
5  40 
5  33 

4  3o 

4  37 

5  2 
5  i5 
5  26 
4  5i 

4  25 

4  12 

4  17 

4  26 

3  23 


D. 

[09 
1 10 

5 
ii5 

6 
116 

116 
119 
118 
116 
116 
ii6 
ii4 

119 


M. 

19E 

36 

00 

35 

20 

36 

46 
2 
5o 
26 
33 

25 

42 

25 


i3 

5 

39 

37 

3 

5o 


o  5i 

o  32 

7  19N 
7  16 
7  59 
7  32 

7  48 

8  00 


iig  23 
119  2 
1x8  5o 


119  37 

119  57 

120  45 

124  5o 

125  II 

125  12 
125  4 
125  00 

123   o 

123  3o 

123  i4 

121  57 
120  28 
120  9 

117  7 
1 17  i5 
117  9 
117  9 
1 17  o 
ii5  53 

ii5  5o 
116  5 
1 16  i4 
1 16  II 
1 16  20 

116  II 

116  37 

117  48 
117  53 

117  M 
117  28 

117  33 
119  4o 

117  6 
116  58 

116  56 

117  19 

118  ^G 

119  5o 


*  See  Table  on  pnse  451 


TABLE   LIV. 

Latitudes  and  Longitudes. 


[Page  3G9 


Palawan,  W.  end 
N.p.... 


Ragged  Island 

Cagayan  Soolo 

Soolo  Island,  town*. 

Takoot      Paboonoowan 

Shoal 

Pangootaran  Island  .... 
Belawn  Island,  E.  p.  . , . 
Tapeantana  Island,  E.  p. 

Taniook  Island , 

Mataha  Island,  S.  p.  . . . . 

Peelas  Island,  N.  p , 

Ballook  Ballook , 

Basilan  Island,  E.  p 

Santa  Cruz  Island 

Sangboj's  or  Hare's  Lips, 
Teynga  Island,  N.  ex.  . . . 


Catanduanes  Island,  S.  p. 

Cape  del  Espiritu  Santo. 
N.  E.  p.  Saniur  Island. 

St.  Bernardino  Island . . . . 

Ticao  Island,  Port  St.  Ja- 
cinto   


Lat. 


D.  M 

8  24N 

II  3o 

II  i5 

7  00 

6  3 


6  i5 
6  i5 
6  00 
i4 
28 

32 

4i 
47 
4i 
5o 
46 


Manilla 

Cavite 

Entrance  Manilla  Bay. 

Point  Caponcs 

Two  Sisters  Islands. . . 

Point  Boliano 

Cape  Bajador 

Point  Cavnaion 

Cape  Enganno  

Mauban 

Cape  St.  Ildefonso. . . . 


Samboongan 

Point  Balagonan 

Suriago  Village,  near  N 

point 

Cape     St.     Augustine, 

S.  E.  p 

South  Point 

Mindanao 


Negros,  S.  point 

Point  Sojoton. 


Cagayancs  Islands,  middle 
Panay  Island,  Point  Na- 

sog,  S.  p 

-  Aslomau  village 

—  Point  Potob  or  N.p... 

Dry  Sand  Bank 

Sombrero  Rock 

White  Rock 

Cuyos  Islands, 

—  Quiniluban    (Northern 

Island)  W.  ex 

—  Grand  Cuyo 

—  Southern  Island 

Caravos  or  Buffalos 

Betsey's  Bank,  .5  fatlioms 
Ylin  "Islands,    S.    p.,    off 

S.p.  Mindoro 

Coral  Shoal,  W.  of  ditto, 
about 


6 
6 
6 
6 
6 
6 
6 
6 
6  52 

i3  38 

12  34 
12  46 

12  34 

i4  36 
i4  20 
i4  28 
i4  52 

1 5  5o 

16  27 
18  42 
18  48 
18  39 
i4  8 
1 5  27 

6  55 

7  5i 

9  47 

6  4 
5  39 

7  10 

9  5o 
9  M 

10  25 

10  32 
n  A(J 

11  24 
10  45 
10  28 


II  3o 
10  52 

10  4o 

11  53 

11  42 

12  9 
12  II 


Lone 


D.  M. 

17  141 
19  37 
19  21 

18  28 


21  32 

20  40 

22  8 
22  8 

21  56 
21  5o 
21  45 

21  5o 

22  17 

22  12 
21  3o 
21  27 

24  2 

25  i5 

24  14 

23  46 


20  55 

20  3 
19  49 
19  5i 

21  00 

21  i4 

22  16 
21  44 

21  46 

22  8 
22  6 

25  25 

26  i3 
25  18 
24  35 

22  56 
22  24 

21  23 

22  6 
22  6 
21  56 
21  34 
21  i5 
21  5 


20  47 

21  i5 
21  i3 
21  48 

20  57 

21  i5 

20  57 


Apo  Bank,  S.  p 

-E.p 

—  N.p 

—  S.  W.  p.  Islet 

—  West  or  Great  Islet.. 

—  Discovery  Bank 

Coron  Island,  Is.off  N.E.p, 

Green  Island 

Haycock 

Pinnacle  Rock 

N.  W.  Rock 

Sail  Rock 

Busvagnon  Island,  N.  p.. 
Calavite  or  High  Island. . 
Group  of  Islands,  S.p..,. 

-N.  p... 


Turret  Island 

North  Rock 

Mindoro  Island,  S.  p 

Point      Dongan     or 

Pandan  

Point  Calavite 

Luban,  N.  pt 

Goat  island 

Babuyan  Islands, 

Lapurip  or  Daluperi 

Island 

Fuga  or  New  Babu- 
yan Island 

Caniiguin  Island. .. 

Guinapac  Rocks  .  . . 

Didicas  Rocks 

Claro  (or  Old)  Ba- 
buyan   

-  —  Calayan  Island 

Bashee  Islands, 

Balintang   or    Rich 

niond  Isles,  N.  cue 

Sabtang  Island,  S.  pt 

Bashee  Island 

Goat  Island  . . 

Batan  or  Monmouth 

Island,  S.  p 

ditto  Mount,  N.  p. . , 

Grafton   or    High 

Round  Island. . . . 

Bayat  or  Orange  Isl 

North  Bashee,  High  Isl.. 
northernmost  Isl 


Lat. 


!).  M. 

2  36] 
2  4o 
2  45 
2  4o 
2  39 
2  4o 

1  59 

2  3 
2  9 
2  18 
2  23 


2  17 


2  27 
2  II 


2  46 

3  28 
3  52 
3  55 


Gadd's  Reef 

Cumbrian's  Reef,  doubt- 
ful ;  probably  the  same 
as  Gadd's  Reef 

Little  Botel  Tobago  Xima 

Botel  Tobago  Xima 

Vele-rete  Rocks 

Formosa  Island,  South 
Cape 


Gomano  Island 

Lissamatula  Isl.,  S.  E.  p 
Xulla  Bessey,  S.  &  E.  p.    . 

N.  E.  p.... 

N.  W.  p... 

Xulla  Mangola,  W.  end  . 

Greyhound  Straits ....  5 


9  i5 

9  I 
9  o 
9  5 
9  12 

9  37 
9  28 


9  58 
20  17 
20  1 4 
20  21 

20  17 
20  28 

20  4i 

20  47 

21  3 
21  9 
21  43 


21  35 
56 
5 
21  42 

21  56 


I  46  S 

1  46 

2  28 
I  58 
I  58 
1  43 
I  4o 

I  56  I 


Lonff. 


D.  M. 

20  33  E 
20  36 
20  3i 
20  29 
20  28 
20  43 
20  36 
19  49 
19  5i 
19  54 
19  55 
19  56 
19  56 

19  56 

20  23 


21  22 

20  47 
20  26 
20  8 
20  3 


21  10 

21  20 

21  53 

22  5 
22  12 

22  12 

21  46 


122  12 

121  53 

122  9 
121  48 


122 
122 


121  57 
121  53 
121  57 
121  59 
121  4i 


121  43 
121  4i 
121  38 
120  49 

120  56' 


127  27 
126  27 
126  7 

125  48 
125  21 

124  3o 


*  See  Table  on  oace  451. 


Page  370 


TABLE   LIV. 

Latitudes  and  Longitudes- 


Haycock  Island,  off  S.  W. 

p.  Xulla  Talaybo 

Skelton's  Island,  on  N.W. 

p.  ditto  

Middle  Island 

Albion's  Island 

Bouro  Island,  N.  W.  p... 
N.  extr 

—  N.E.  p 

Cajeli  on  Bouro  Bay. 

S.  point 

Amblaw  Island 

Manipa  Island,  E.  pt 

Bonoa  Island,  about 

Ceram   Island,   Seeal,   or 

S.  W.  p 

Kessing,  or  E.  p 

Waroo  Bay 

Old  Lamata  or  Flat  P. 

Sawa  Bay 

Leeuwarden  Island,  S.  pt. 

Shoal 

Goram  Island 

Matlabella  Islands 

AMBOYNA  Island,  Fort 

Victoria 

Noesa  Laut  Island,  E.  pt. 
Banda  Island,  anchorage . 
Lookisong   or  Landscape 

Island,  S.  p 

Pulo  Gasses,  S.  p 

Kekik 

Pulo  Pisang 

Horsburg's  Rocks 

Boo  Islands 

Weeda  Islands 

Kanary  Islands,  Grand  K. 

Ef be  Harbor 

Pulo  Popo,  S.  E.p 

Battanta     Island,      Cape 

Cambo,  W.  p. 

Fisher's  Island 

VVaygecooe    Island,  S.  E. 

p.,  or  Point  Pigot 

Offiik  Harbor 

Boni  Road 

Amsterdam  Island 

Fow  or  Faux  Island 

Gagy  Island,  N.  pt 

Geby  Island,  N.  W.  end. 

S3'ang  Island 

E3'e  Island  -^ 

Islet  E.  of  PuloMoar 

Catharine's  Islands 

Canton  Packet  Shoal 

'Ornisbce's  Shoal,  N.  pt.  . 
Ditto  soundings,  15  fath.. 
Yowl  or  Aiou  Islands, 

—  Aiou,  the  largest  Isle  . 

—  N.  W.  Island 

—  N.  E.  Island 

—  Reef  N.  part 

Asia's  Islands,  S.  W.  Isle 

N.  E.  Island... 

Gillolo  Island,  N.  end  ... 

—  Ossa  village 

—  Maba  village 

—  Islet  near  Pulo  Moar. . 


Lat. 


D.  M. 

I  47  S 

I  45 
I  45 

1  53 
3  4 
3  2 
3  49 

3    22 

3  54 
3  49 
3  24 
3  00 

3  33 

3  55 

3  25 

2  53 

2  5l 

3  20 

2  56 

CO 

3o 
4i 

42 

3i 

39 
4i 
33 

23 


32 


56 
56 


19 
6 

24 
2N 
22 

23 

35 

46 
42 


38 
36 
4i 
o4 
4 

23 

45 

53 

9 


Lon^. 


D.    31. 

124  24 E 

24  36 
24  28 

24  19 

25  57 

27  10 
27    6 

26  37 

27  10 
27  4o 
27  56 

27  5i 
3i  10 
3o  45 
29  42 

29  6 

30  48 
3o  43 
3i  A^ 
3i  47 

28  10 
28  49 
3o  00 


28  4 
28  i5 
28  35 
28  53 

28  20 

29  20 

28  35 

29  42 

29  52 

30  25 
3o  23 

3i  18 
3o  43 
3i  5 
32  9 
29  28 
29  53 
29  19 
29  55 
29  5i 

28  58 

29  II 
28  55 

30  4 
3o    3 


3i  o 
3i  8 
3i   i5 

3i  21 
3i  23 

28  22 

28  58 


Gillolo  Island,   point  en- 
trance Straits  Patientia 

—  Cocoa-nut  Point,or  S.p. 
Batchian  Island,  S.  E.  p.. 

Amsterdam  Island 

Kayo  or  Cayo  Island,  S.  p. 

N.p. 

Negory  Kalam,  N.  p 

Wolf  Rock 

Tidore  Island,  S.  extr. .. 

Mountain 

N.  E.  end 

Ternate  Island 

Tyfore  Island 

Meyo  Island 

Morty  or  Mortay  Island, 

(N.  cape,) 

Bangay  Island,  peak 

Tagalondo 

Bejaren  Island,  peak. . . . 
Siao  Island,  S.  point  .... 

peak  

Sangir  Island,  S.  end. . . . 

—  Watering  place  on  the 

W.  side 

—  N.  end 

Glatton's  Rock 

Sallibobo  or  Toulor  Isls. 

Kabruang,   S.  p.  .  . . 

Tulour     or     Karka- 

lang,  N.  p 

Meangis  or  Menangus  Isl. 

Serangi  Islands,  S.p 

peak  on  W.  Island  . 

N.p 


Lat. 


D.  M. 


i3S 
5i 
48 
20 

I 

7N 
28 
20 
34 
40 
A& 
49 


2  M 

1  52 

2  23 
2       6 

2  4o 

2  43 

3  21 


3  47 

4  28 

5  00 
5  20 

5  3i 


Loner. 


D.   M. 

127  45 E 

128  22 
128  3 
127  53 
127  23 

127  37 
127  9 
127  24 
127  25 
127  34 

127  19 

[26  12 

126  25 

128  25 
125  24 
125  36 

125  25 

125  35 
125  35 
125  46 

125  44 

125  44 

126  4 

127  o 

126  55 

127  17 
125  35 
125  32 
125  43 


XL  VI.  The  Coast  from  CAJVTOJVto 
KAMTSKATKJl,  with  the  adjacent 
Islands  and  Shoals. 


CANTON 

Mir's  Bay  Paint 

Single  Island,  or  Chueng 

Chow 

Mendoza's  Island 

Fokoi  Point 

Pedro  Branco 

Point  Chelang  

Point  Tongmi , 

Point  Cup-chi 

A  black  conical  Mount . . 

Breaker  Point 

Cape  of  Good  Hope 

Fort  Island 

Lamo  or  Namolsl.,  W.  p 
N.  pt.... 


Lamock  Island,  S.  W.  Rk. 
The  Brothers,  southern.. 

Chapel  Island 

Amoy 

.Chin  Chew  Bay,  mid 

Lamyet  Islands,  E.  Peak. 

Ting-hae   harbor 

Hiesham  Oroup  Saddle  Id 
Quesan  Isl's  Patahecock . . 


Lat. 


M 

7N 
27 

24 
3i 
33 
18.5 

39 
44 

52 

56 
14 

25 

26 
29 
II 

32 

10 

28 


Lone 


D.    U. 

3  i4E 

4  3o 


7    » 
7  i4 

7  42 

8  i4 
8    4 

8  47 

9  35 
9  5o 

22°l4 

22  i4 


*  White  Doss  S.  P.  255  57  120°  01 


TABLE   LIV. 

Latitudes  and  Longitudes. 


[Page  371 


Clmsan  Island,  Tingbae. . 

Saddle  Group  Is;  Is 

Chiu-san  Island 

Amherst  Eoclis 

bliaug-TuDg  Prom.  S.  p. 

—  N.p 

Cape  Zeu-ou-Tau 

Ten-choo-Foo  City 

Tchoo-san  Island 

Keusen  Islands,  northern 
Pekin    River,    anchorage 

at  Peiho  Entrance 

Alceste  Island,  S.  W. 

extr.  Corea 

Cape  Clouard 

Sanpon 

Ternai  Bay 

Suffren  Bay 

Cape  Lesseps 

Castries'  Bay 

Vanjuas  Point 

Bay   de  Langlc 

B*ay  d'Estaing 

Monneron   Island 

La  Dangereuse  Rock. 


Lat. 


M 

iN 
5o 

25 

lO 
00 
25 

36 
48 


38  58 


Cape    Crillon,    (entrance 
Perouse's   Straits,) .... 

Cape  Aniwa 

Cape  Lowenorn 

Bay  MordwinofF 

Cape  Tonyn 

Point  Siniavin 

Mount  Spenberg  or  Ber 
nizet  

Point  MulofTsky 

Cape  Alexander  Dalryin- 
pie 

Ca])e  Soinsonoff 

River  Nova,  entrance .... 

Gulf  Patience,  N.  p 

Robber     Island     Reef, 
N.  E.  p 

—  S.  W.  p 

Cape  Patience 

Cape  BiUinghausen 

Mount  Tiara 

Cape  RatmanofF 

Cape  Croyere 

Downs  Point 

Slioal 

VVurst   Point 

Cape  Klokatschef 

Cape  Lowenstern 

Cape  Elizabeth 

North  Bay 

Cape  Maria 

Espenberg  Peak 

Cape  Golowgtscheff 


Cape  Romberg 

Cape  ChavarofF 

Jonas   Island 

Ochotsk 

Yamsk 

Bolcheretsk 

C.  Lopatka,  Kamtskatka 
St.  Peter  and  St.  Paul... 


45  54 

46  2 
46  23 
46  48 

46  5o 
4?  i6 

47  33 

47  58 

21 

48  52 

49  i5 
49  19 

48  36 

48  28 

52 

49  35 

50  3 

50  48 

5 1  00 
5i  53 

52  3o 

52  57 

53  40 

54  3 
54  24 
54  16 
54  17 
54  4 
53  32 

53  26 
53  38 
56  25 

59  20 

60  46 

52  54 
5i  2 

53  o 


Lonrr. 


D.  M. 

22  6E 
22  4i 
22  21 
22  22 
22  4i 
22  45 
21  28 

20  4o 

21  2 
20  43 

17  48 

25  21 
29  45 
28  55 
36  00 
38  44 
4i  3o 
4i  o 
42  42 
4i  57 
4i  27 
4i  II 
42  9 


4i  58 
43  3o 
43  4o 
43  i4 
43  33 
43  00 

42  20 
42  44 

42  5o 

43  2 
43  2 


44  33 
44   lo 

44  46 
44  26 
43  37 
43  53 
43  43 
43  i3 
43  29 
43  18 
43  7 
43  i3 
42  47 
42  37 
42  18 
42  5o 
4i  55 


4i  45 
4i  26 
43  16 
43  12 
54  3o 
56  5o 
56  46 
58  44 


Shipunskey-noss,  Cape  . . 

Nisjui  Kamtskatka 

Cape  Tschulkolskoi 

East  Cape 

Cape  Serdze  Kamen.... 
North  Cape 


Formosa  Island,   S.  cape 

N.  W.  point. 

N.  point 

N.  E.  point. . 

Lamay  Island 

Pehoe  or  Pescadore  Isles, 

Southern  limit 

High  Isl.,  S.W.  limit 

Pachan  Island 

Northern  limit 

Treble  I.  S.E.p 

ditto,  nine  feet  reef  . 

Pat-chow  or  Madjicose- 
mah  Islands, 

Southernmost  Island 

Bluff  Point,  W.  ext. 

Great  Island 

Kumi  Island 

Eastern  Island,  Ty- 

pin-san 

Providence  Reef. . . 

Lew  Chew  Islands, 

Great    Lew  )  From 

Chew,  f    To 

ditto,  adjacent  Isl- 
and, N.  p 

Western  Island .... 

Hoapinsu  Island 

Ty-ao-yu-su  Island 

Sulphur   Islet 

Island 

Ousima 


Lat. 


D.  M. 

53    6  N 
56  16 
64  i3 

66  6 

67  12 

68  56 

21  56 

35    II 

25  18 

25    I  1 

22  19 

23  1  I 

23  19 

23    32 

23  47 
23  3i 
23  28 


Group  of  seven  Islands, 
limits 


Pinnacle  Islands 

Ormsbee's  Peak 

A  Rock 

South  Island 

Gotto  Island,  S.  end  .... 

Ashes'  Ears 

Quelpacrt  Island,  S.  p... 
Kiusiu  Island, 

Cape  TschirikofT. . . 

Cape  Danville 

Cape  NagaefF. 

Mount  Schubert  . , . 

Mount  Horner,  peak 

CapeTschitschagofF, 

S-  p 

CapcTschesma,W.p. 

Cape  Kagul,  N.  p.  . 

MountUnga,voIcano 

Nangasaky  harbor, 

entrance 

Cape  Nomo,  S.  p.  of 

Bay  Nan 

Cape  Seurote 

Sanao-sima  Island,  N.  p.. 

S.  p.. 

Tenegasima  Isl.,  (middle) 


24  6 

24  17 
34  25 

24  42 

25  6 

26  o3 

26  53 

27  M 

26  20 
25  47 
25  57 

27  5 1 
24  48 

28  16 

29  25 

30  06 

29  52 

29  4o 

30  45 

3 1  3o 

32  35 

32  3 

33  8 


3i  4t 
3i  9 

3o  57 
24 
3i  42 
3i  43 

32  44 

32  35 
32  58 
3o  42 
3o  24 
3o  23 


Lons- 


D.  M. 

60  4  E 
62  00 
71  24W 
69  4o 
71  49 
79  57 

20  56  E 

21  6 
21  34 
21  56 
20  27 

19  23 
19  16 
19  26 
19  32 
19  39 
19  4i 


23  52 

23  45 

23  00 

25  29 
25  6 

27  34 

28  25 


27  17 

23  29 
23  40 

28  1 4 
4i  20 

so  21 

29  38 

30  04 

29  52 

4o  20 

23  46 
4o  00 
28  44 

28  37 
26  19 

3i  4i 
3i  27 
3i  II 
3i  12 

30  28 

3o  36 
3o  2 
3o  7 
3o  i4 

29  46 

29  42 
29  35 
3i  00 


|i3o  So 


Page  372] 


TABLE   LIV. 

Laiitudes  and  Longitudes. 


Volcano  Island 

Seriphos  Island 

Apollo  Island 

Julie  Island 

St.  Claire  Island 

Symplegados      Islands, 

i\.  E.p 

—  S.  W.  p 

Meac-Sima  Isl.,   S.  W.  p. 

,—  N.  E.  p. 

Nadeshda  Rocks 

Tsus  Island,  S.  end 

Cape    Fida-Buen 

gono 


Lat. 


N.  p. 


Colnett's  Island 

Dagelet  Island,  N.  E.  p.. 
Niphon  Island, 

SP 

Cape  Noto 

A  Rock 

Jootsi-Sima 

'  Jedo 

Cape  Kennis 

Zach's  Mountain . . . 

Russian's  Promonto- 
ry, S.  p 

N.  E.  p 

Town  

Cape  Gamally 

Peak  Tilesius. ..... 

Cape  Greig 

Cape    Sangar,    (ent. 

Straits  of  Sangar,) .... 

Osima  Island 

Kosima  Island 

Okosir  Island,  (middle,)  . 

Jesso  Island, 

Cape  Nadeshda, (ent. 

of  Straits  of  Sangar,) . . 

Cape  Sineko 

Matzumay  Town . . . 

Cape  Oota-Nizawu. 

Cape  KutusofF 

Cape  Rayten 

Cape  Okaraay,  S.  p. 

Cape  Taka-sima 

Mount  Rumoifsky  . 

Cape  Malespina. . . . 

Cape  SchischkofF  . . 

Pallas  Mountain  . . . 

Cape    Romanzoff, 

N.  p 

Cape  Soya 

■  Cape  Shaep 

Peak    de    Langle,    Rios- 
chery  Island 

Cape    Guibert,    Reifuns- 
chery,  N.  E.  p 

Jeurire  Island 

Janikesseri  Island 


D.  ftl. 

3o  43  N 
3o  A'i 
3o  U 
3o  27 

30  45 

3 1  3o 
3i  26 
3t  35 
3i  49 
3 1  .fa 
34    6 

M  1; 

34  40 

34  16 
37  25 

33  25 
37  36 
37  36 
37  5i 

35  40 
37  10 
35  25 

39  46 

40  00 
4o  5o 
4o  38 

40  40 
4i    9 

41  16 
I  3i 

41  21 

42  9 


Staten  Island,  S.  W.  end 

Cape   Vries,    (Vries 

Straits,) 

Company's  Island, 

Cape  Slionten 

N. end 


4i  25 
4i  38 
4i  32 
42  18 
42  38 

42  57 
^3  II 

43  21 
f2  5o 

43  42 

■i4  20 
^^  00 

45  26 
45  3i 
45  21 

45  II 

45  28 
^i\   28 

44  29 

44  26 

45  26 

46  18 
45  28 


Loner. 


D.  M. 

i3o  17 '. 
i3o  44 
i3o  24 
i3o  1 3 
129  54 


129 

42 

129 

37 

129 

40 

129 

5i 

129 

33 

.29 

17 

129 

3o 

129 

29 

129 

56 

i3o 

56 

1 35  47 
i37  20 
i36  5o 
i36  56 
i4f>  00 
i4i  3o 
i32  20 

139  44 

i4o  6 
139  48 
i4o  II 
i4o  8 

i4o  i4 

39  19 

139  46 

139  3o 


i4o  9 

i33  53 

i4o  4 

139  46 

i4o  I 

i4o  16 

i4o  i3 

i4o  3i 

i4i  1 1 

i4i  18 

i4i  37 

i4i  54 

i4i  34 

i4i  5[ 

42  12 

i4i  12 

i4i  4 

i4i  17 

i4i  22 

147  28 

149  43 

i5o  58 

i5i  20 


Marikan  Island,  N.  end 

S.  end,(Bousole  Sts.) 

SarytschefF  Island  Peak. 

Raakok  Island 

Mussir  Island 

'  Trap  Rocks 

I  Charamukatan    Isl.  Peak 

I  Poromuschir  Island,  S.  p. 

' Peak  Fuss,  (S.W.p.) 

E.  p 


Lat. 


D.  M 

47  10  N 
46  46 

48  6 
48  8 
48  16 

48  36 

49  8 

50  o 
5o  i5 
5o  28 


Long. 


D.  31. 

i53    6E 

152  32 

1 53  12 
i53  i5 
i53  i5 
i53  44 
i54  39 
i55  24 
i55  10 
i56  9 


XLVII.    NEW  HOLLAND  and  the 

adjacent  Islands  and  Slioals. 


Pedro  Branco,  (Rubrick,) 

South-west  Cape 

Mew  Stone 

South  Cape 

Eddystone 

Sidmouth's  Rock 

Tasman's  Head 

D'Entrecasteaux's  Chann 

Adventure  Bay 

Frederick  Henry  Bay  . . . 

Cape  Pillar 

Oyster  Bay 

St.  Patrick's  Head 

Cape  Portland 

Port  Dalrymple 

Circular  Head 

Cape  Grim,  N.  W.  prom. 
West  Cape,  or  Sandy  Pt. 

Macquarie's  Harbor 

Rocky  Point 

Port  Davey 


Lat. 


Ent.  to  Banks's  Straits.. 
Furneaux  Islands, 

Barren  Isl.,  S.E.  ext. 

Clarke's  Isl.,  S.  ext. 

N.  Sister,  near  N.  p. 

of  Great  Island 

Endeavor  Rock 

Kent's  Group  Light 

The  Pyramid 

Waterhouse  Island 

Hunter's  Islands, 

Black  Pyramid.    W. 

Albatross  Isl.,  N.  W. 

King's  Island,  C.  Wickam 
W.  ent.  Bass's  Straits  . 


Cape  Albany  Otway 

Port  Philip, "pt.  Nepean.. 
Western  Port,  Phillip  I.  . 
Wilson's     Promontory, 

S.ext 

Ram  Head 

Cape  Howe 

Cape  Dromedary,  Mt 

Jervis  Bay,  N.  pt 

Rod  Point 

Botany     Bay,     entrance, 

(Cape  Banks,) 

Port  Jackson,  entrance.. 


M. 

59  S 

37 
46 
40 
5i 
46 
3o 

32 

17 
58 
12 
4o 
42 


4o  4o 

4o  26 

4o  36 

39  38 
39  37 
39  3o 

39  49 

40  48 

4o  28 
4o  22 
39  35 
39  23 

38  52 
38  18 

38  27 

39  8 
37  39 
37  3o 
36  16 
35  7 
34  29 

34  02 
33  5o 


Lonff. 


D.  M. 

47  43  E 
46    o 
46  3i 

46  59 

47  4 
47  " 
47  10 
47  12 

47  32 

47  42 

48  6 
48  2 
48  17 

47  57 
46  48 
45  17 
44  42 

44  35 

45  18 
45  3r 
45  59 

48  20 
48  3 1 


48  01 
47  6 

47  21 

47  16 
47  37 

44  2  2 

44  39 
43  53 
43  4o 

43  34 

44  4o 

45  18 

46  24 

49  45 

50  o 
5o  9 
5o  58 


5i  i3 
5i  18 


TABLE  LIV. 

Latitudes  and  Longitudes. 


f  Page  373 


Broken  Bay 

Port  Stephens,  pt 

Cape  Hawke 

Smoky  Cape 

Solitary  Islands  .... 

Cape  Byron 

Point  Danger 

Slioals  oti"  ditto 

Cape  Morton    

Shoal,  Dry  Eocks.. . . 

Sandy  Cape  Sli'l,  9  ft.  rock 

(JronpCapricorn, N.  AV.  Is. 

Keppel  I 

Barrier  Reef,  S.  extreme 

Cape  Townsend 

Cape  Palmerston 

Cape  Hillsborough 

Cape  Conway 

Cape  Gloucester. ....... 

Cape  Cleveland 

Cape  Sandwich 

Cape  Grafton 

Cape  Flattery 

Cape  York 

New  Year's  Island 

Van  Uieman's  Cape 

Red  Island,  off  P.  Vulcan 

Minstrel's  Shoal,  N.W.  pt. 

Greyhound  Shoal 

Clarke's  Reef,  north  of 
Rosemary  Island 

Eastern  Rosemary  Island. 
N.E.p .' 

Western  ditto,  N.  p 

Doubtful  Shoal 

Piddinn-ton's  Islands.... 

Shoal  (land  of  N.  Holland 
in  sio-ht  from  the  mast- 
head)  

North-west  Cape 

Dirk  Hartog's  Road,  eiit. 
to  Sharks'  Bay 

Houtinan's  or  Abrohlos 
Shoals,  N.  I 

Rottenest  Island 

Cape  Leuwen  or  S.  W. 
Cape,  S.  pt 

Cape  Chatham    

Cn  pe  Howe 

King  George  III,  Harbor 

Point  Hood,  Ptocks  off  . . . 

Termination  Island 

Endeavor,  small  island. . 

P'>rt  Lincoln 

Nepean  Bay  

Cape  Jaffa 

C.  Nortbuniberland.  Eocks  off 


iMt. 


M. 

34  6 
4i 
1 5 
56 
i3 
56 
38 
7 

3 

56 
36 
i8 
II 

23 

i4 
3o 
5/i 


20    .)4 
20    32 


20 


20     17 
26 

35 

37 

36 


i5 
5o 

25    22 


Lons- 


D. 

M. 

i5i 

20  E 

l52 

14 

l52 

3i 

i53 

5 

i53 

18 

1 53 

38 

1 53 

3i 

i53 

39 

i53 

27 

i53  3i 

1 53 

22 

i5i 

43 

i5i 

08 

I  52 

37 

i5o 

29 

1 49 

28 

1 49 

06 

i49 

0 

i48 

25 

i46 

59 

1 4b 

19 

i45  57 

145 

i6 

142 

33 

1 33  o3 

i3o 

18 

124 

18 

119 

10 

ii4  40 

1 15  4o 
112  25 
ii4  56 


ii4  o4 

1 12  57 

ii3  35 
n5  3i 


ii5 

6 

116 

25 

117 

38 

118 

2 

119 

33 

121 

58 

127 

2 

i35 

55 

i37  55 

139  4o 

i4o 

4i 

XL VI II.     Islmids,  Rocks,  and  Shoals,  in 
the  JVORTH  PACIFIC  OCE.IjY. 


I  Aleootskia  Islands, 

I westernmost,  W.p. 

I Ounalashka 


Lat. 


D.  M. 


Lons- 


D.  M. 


52  52 N  173  24 E 

53  54      166  32W 


Bank  (G4  fathoms) 

Rica  de  Plata  or  Crespo. . 

Reef 

Island 

Weeks's  Reef,  36'  N.  E 

and  S.  W 

Island 

Ganges  Island 

Bank  of  Soundings 

Island 

Island 

Island 

Island 

Island 

Roca  de  Oro 

Island,  Rica  de  Oro 

Island 

Island 

Island 

Calunas  Island 

ditto  (another  account) 

Island 

Patrocinio  Island 

Disa])])ointment  Island  .. 

St.  Juan 

Bassiosos  Island 

Island 

Reef 

Copper  island 

Tree  Island 

Laskcr's  Island 

Island 

Island 

Reef 

Bishop's  Rock 

North  Island 

Island 

Grampus  Island 

Sulphur  Island 

Kendnck's  Rock 

Marcus  Island 

Weeks's  Island 

Dextcr's  Island 

Island 

Reef 

Jardines 

Parel  or  Peru  Island . 

Abregoes  Shoal 

Reef 

Douglas  Reef 

Lamira  Island 

Island 

Bishop's  Rock 

Weeks's  or  Wilson's  Isl. 

Reef 

Halcyon  Island 

Folcrer's  Island 

Reef 

Tarquin  Island 

Reef 

Island 


Lat. 


Guy  Rock  

Urracas,  about 

Assumption  Island. 
Almagan  Island  . . . 

Bird  Island 

Tinian 


D.  M. 

Z3  22  N 
32  44 
32  00 
3 1  3o 

3i  i5 
3 1  00 
3o  45 
3o  5o 
3o  00 
3o  00 
3o  00 
3o  00 
3o  00 
29  54 
29  25 
29  33 
29  3o 
29  35 
28  55 
28  53 
28  3o 
28  10 
27  14 
27  3o 

25  58 

26  6 
26  3 
20  o3 
26  00 
26  o3 
25  53 
25  42 
25  3o 
25  22 
25  i4 

25  12 
25  10 

24  48 
24  35 
24  18 
24  00 
23  24 
23  3 
6 
4o 


20  42 

20  32 
o  20 
20  3o 
20  16 
19  II 
19  28 

19  6 

18  22 
17  9 

17  00 
7  36 
6  oc 

20  3o 
20  10 

19  4i 

18  5 
16  I 
i5  00 


Lonsr. 


D.  M. 

78  3oE 
70  8 
47  00 

40  00 

53  9 
47  6 

54  25 
77  3o 
37  00 

39  00 

4 1  3o 

43  00 

44  24 

57  3 

65  55 
37  00 
43  00 
l4  4i 

58  00 
62  00 
76  5o 
75  48 

40  57 

42  48 
73  3i 
54  36 
60  00 
3i  48 

45  44 
73  42 
3i  17 
3i  i3 

52  5o 
32  00 
4i  i4 
3i  36 

46  4o 

4 1  20 
34  00 

53  42 

54  00 
62  58 

62  57 

42  28 
5i  35 
4 1  40 
36  43 
53  00 
36  6 
64  1 5 
52  5o 
36  53 

66  55 
66  29 

63  33 

55  i5 

56  i3 
60  00 
69  3o 
71  42 

45  3o 
45  25 
45  27 

45  54 

46  o3 
45  37 


Page  3741 


TABLE  LIV. 

Latitudes  and  Longitudes. 


Guam,  Umatac  Bay. 


Radack  chain  of  islands 
viz.  : — 

Aour,  circular  group  of 
32  islands,  extending  13 
miles  N.  W.  and  S.  E., 
anchorage , 

Kaven  group,  33  miles 
N.  W.  and  S.  E. 

—  Araksheef      Island, 

(largest  island,). . 

—  Southern  Island. . . . 
Chatham,  circular 

group  of  islands,  N.  W. 
and    S.    E.    24    mile 
Eregup 

Chatham  Is.circular  group 
of  Co  islands,  E.  and  W. 
30  miles,  and  10  miles 
wide,  enclosing  a  sea 
12  miles  wide  and  2< 
miles  long. 

— Otdia    Island,    eastern 
(anchorage,)        

Legiep  or  Hayden  group 

Ailou  group,  15  mile 
long,  5  miles  wide, 

—  Krusenstern  Capenius 

Island,  (northern,) 
Isle  Du  Nouvel  An ... 
KutosofForUdirick  group, 
separated  by  a  channel 
from  a  southern  group 
called  Souvoroff  or  Ta 
gay,  extending  N.  and 
S.  25  miles, 

—  Channel 

Group  south  of  KutosofF, 

Mille 

Medjuro 

Arno 

Bigar,  north  of  KutosofF. 


Pescadores  Isls,*  eastern, 
western. 


Ralick  chain  of  islands 
extend  nearly  N.  and  S. 
about  one  degree  west 
of  the  Radack  chain, 
viz. : — 

Ebon  group 

—  Noamureck  Island  . 

Kuli  group 

Helut  group 

Odia  group 

Namou  group 

Litel  Island 

Tebot  Island 

Quadelon  group 

Oudia-Milai  group.... 

Radogala  group 

Bigini  (northern) 

Johannes  

tiion's  Island 

St.  Andrew's  Island . . . 

Pulo  Anna 


Lat. 


D.  M. 

i3  17N 


19 


54 
29 


27 


25 


23 

8 


Loner. 


D.  M. 

i44  4oE 


171  12 


170  49 

171  II 


170    4 


5  5o 

167 

i5 

5  3o 

6  4o 

7  3o 

8  i5 

9  00 

8  55 

8  3o 

9  20 

10  45 

II  00 

II  20 

167 

i5 

6  55 

l32 

3o 

5  16 

l32 

i3 

5  20 

l32 

16 

4  38 

l32 

3 

170  iG 
169  i3 


170  00 
170  55 


169  5o 


170  07 

167  37 
167  22 


Pulo  Mariere 

Lord  North's  Island. . . 
Ganges  Shoal,  S.  W.  p 

N.  E.  p. 

Helen's  Shoal 

Freewill  or  St.  David's 
Islands,  limits 


Pelew  Islands, 

—  Baubelthouap,  E.  p. . 

—  Northernmost,    Kyan 

gle , 

—  Large  Reef,   part  dry, 

—  Southernmost,  Angour 
Matelotes,  N.  extr.  

Southernmost 


Yap  or  Hunter's  Isl.,  N.  p 
-S.p 


Philip  Islands  . 
Thirteen  Islands,  S.W.  ex. 

Haweis's  Island 

Strong's  Island 

Islands 

Islands 

Islands 

Islands 

Hope's  Islands 

Baring's  Islands* 

Teyoa  Island 

Providence  Islands 

Ditto  

Brown's  Range, 

Arthur's  Island,  N.. 

Parry's  Island,  S. . . 

Margaret's  Island ....... 

Lydia's  Island 

Catharine's  Island 

Arrecife's  Island 

Mosquito    group,     loi 

and  dangerous 

Peterson's  Island  .... 
Chatham  Island,  E.  pt.  . . 

Reef 

Calvert's  Islands,  S.  extr. 

Ibbetson's  Islands 

Elmore  Islands, S.I.  .... 
Mulgrave's  Islands, mid'le 
Banham's  Island,*  E.  jit.. 

Cook's  Island 

Hall's  Island! 

Reef 

Pitt's  Island  *  N,  pt 

Matthew's  Island* 

Simpson's  Island 

Macasgill's  Islands 

St.  Bartholomew 

Cornwallis  or  Smyth's 

Isles 

Wake's   Island* 

Lamira,  \V.  pt 

Gaspar  Island 

Gaspar  Rico  Island 

Wake's  Rocks 

St.  Peter 

Barbadoes  

Krusenstern's  Rock 

Necker  Islandt 

French  Fritrate's  Shoalt. 


Lat. 


D.  M. 

D.  M. 

4  19N 

i32  28 E 

3  3 

i3i  4 

2  52 

i3i  7 

3  6 

i3i  23 

3  0 

i3i  55 

0  49 

i34  17 

I  2 

1 34  3o 

7  4i 

8  8 
8  18 

6  53 
8  4i 

8  19 

9  4o 

9   25 

8    6 

7  18 
7  3o 
5  12 
5  28 
5  47 


II  4o 
1 1  19 

8  52 

9  4 
9  i4 
9  3i 

7  46 

8  10 

8  54 

9  3o 
10  00 

8  3o 


7  i5 
6  16 
6  01 
I   18' 

0  5i 

1  00 
3  20 

2  3 
o  3o 
6  12 

i5  10 

16  5o 

19  17 

20  20 
i4  55 
i4  42 

17  20 
II  i3 

8  54 

22  II 

23  34 
23  45 


Long. 


1 34  58 

1 34  5o 
i34  4i 
1 34  21 
1 37  40 
i37  33 

i38  I 
i4o  52 
143  53 
1 46  28 
162  58 
1 53  24 
1 57  42 
160  5i 

159  12 
i65  9 
168  26 
162  29 

160  58 


162 

i5 

162 

25 

166 

i5 

i65 

58 

166 

2 

161 

8 

168 

23 

168 

00 

166  35  1 

170 

i5 

179 

21 

171 

II 

171 

8 

168  45  1 

171 

55 

169 

48 

171 

57 

173 

4 

179 

34 

172 

57 

•73 

26 

173 

53 

160  48  1 

1 63 

53 

169  4o 

166  32 

164 

i5 

176 

20 

169 

3 

172 

40 

179 

ooW 

178 

00 

175  42 

164  43 

1 65 

59 

*  See  Table  on  page  451. 


t  See  Table  on  page  450 


TABLE   LIV. 

Latitudes  and  Lonaritudes. 


[Page  375 


Lisiansky's  Island  . . 

Owhyhee,  N.  point* 

E.  point  . 

S.  point.. 

• KarakakoaBay 

Mowee,  E.  point*. . , 

S.  point. . . . 

W.  point. . , 

Talioorowa 

Ranai,  S.  point 

Morotoi,  W.  point     . 

Woaiioo*  N.  pt 

Attoi,  Whymoa  Bay 

Talioora 

Oneelieow 

Oreehoua 

Bird's  Island* 

Gardner's  Island,  discov- 
ered 1820 

Maro's  Reef,  ditto  . 

Gallego  Island  .... 
Cliristmas  or  Noel  Island 
Sidney  or  Tanning's  Isl 

Island 

New  York  Island* 

Cocoss  Islands,  or  Chat- 
ham Bay 

Palmyra  Island,  S.  pt. . 

Island 

Barbary  Island 

Reef 

Clipperton's  low  Island 
Manuel  Rodriguez  .... 

Island 

Island 

Shoal 

Slioal 

Cluster  of  Islands 

Island 

Passion  Rock 

Cornwallis  Island 

New  Blada 

Clarion  Island 

Island 

Shoal  . .-. 

Socora  Island 

Wilson's  I 

St.  Benedicto  

Freshwater 

Roca  Partida 

Mallon  Island 

Cloud's  Island 

Copper  Island 

Island 

Shovel    Island 

Massachusetts  Island . . 

Island 

Henderson  Island 

another  account. . 

Gardner's  Reef 

Polland's  Island 

Allen's  Reef 

Cooper's  Island 

Maro's  Reef,  W.  pt 

Island 


Lat. 

D.  M. 

26  3  N 

20  23 

19  34 

18  54 

19  28 

20  4? 

20  32 

20  54 

20  3i 

20  43 

21  6 

2t  43 

21  57 

21  4o 

21  5o 

22  2 

23  5 

25  8 

25  3i 

I  42 

I  58 

3  52 

4  3o 

4  42 

5  33 

5  48 

6  4i 

8  55 

10  00 

10  28 

10  67 

II  33 

i3  4 

i3  32 

1 4  42 

16  00 

17  00 

16  3o 

16  54 

16  57 

18  12 

18  21 

18  22 

18  27 

18  48 

19  i3 

19  18 

19  22 

19  6 

19  20 

19  43 

20  6 

21  0 

22  6 

22  28 

24  6 

24  12 

24  26 

25  8 

24  52 

25  00 

25  43 

25  3i 

22  3 1 

Long. 

D.  M. 

173  4oVV 

1 55  54 

1 54  54 

1 55  49 

1 55  56 

i56  7 

1 56  25 

1 56  48 

1 56  39 

1 57  0 

1 57  24 

i58  7 

159  42 

160  35 

160  i5 

160  8 

161  49 

ce 

(« 

168  09 

M 

170  46 

0 

a 

io4  5 

-^ 

i57  32 

i58  22 

169  33 

160  i3 

87  I 

162  23 

166  2 

178  00 

179  28 

109  19 

i53  45 

164  00 

168  35 

170  26 

170  23 

1 33  00 

1 36  00 

i63  54 

109  9 

i6q  36 

ii4  5 

. 

ii4  23 

i55  i5 

CS 

170  3o 
no  56 
166  55 

I1 
a? 

Hi 

110  45 
ii5  8 

III  52 

i65  23 

ii4  57 

i3i  43 

178  3o 

112  1 4 

176  36 

167  4i 

128  6 

168  9 

168  20 

167  57 

i3i  35 

170  46 

i3[  0 

A  Rock 

Laysan's  Island 

Liscanskey's  Island 

Neva  Island 

Maro's  Reef,  N.  pt.  E.   ... 

Island  and  Rock 

Pearl  and  Kermes  group. 
Clarke's  Reef,    60   miles 

N.W.  and  S.  E 

Bunker's  Hill 

Island 

Ocean   I.,  N.  i)t 

A  Bank    

Culpepper's  Island 

Wenman's  Island 

Redondo  Rock  

Abington    Island, 

Mid 

Albemarle    Island, 

N.  pt 

S.  W.  Point  

James  I.  Harbor 

Charles  Island,  S.  p 

Chatham  Island,  N.  E.  p 
Stephen's  Bay 


Lat. 


31. 

3oN 
46 
o3 
54 

24 

43 

43 


I  4o 

I     23 

o  i5 
o  32 


09 
00  S 


20 
45 
53 


Lor.s- 


D.    31. 

174  3W 

171  49 
173  41 

172  20 
170  32 
170  54 

175  48 

175  48 

173  20 

176  5o 
178  25 
118  49 

92  4 
91  53 
91  4o 

90  48 

91  25 
91  32 
90  56 
90  33 

89  37 


XLIX.     Islands,  Rocks,  and  Shoals,  in 
the  SOUTH  PACIFIC  OCEAjY. 


New  Guinea, 

—  Middleburg  Island  . . . 

—  Cape  of  Good  Hope . . 

—  Flat  Point 

—  Cape  Valshe 

—  Cape  Rodney 

—  King  William's  Cape. 
Torres  or  Endeavor  Straits 
Eastern  Fields  or   Reefs, 

N.  E.  end .' 

—  N.  W.  part 

Murray's  Islands 

Wamvax  or  Darnley  Isl.. 
Pandora's  Shoals,  N.  p... 

Wreck  Reef,  S.  p.  . 

Portlock's  Reef .... 

cnt.  Torres  Straits  . 

Boot  Reef. 

Indefatigable's  ent.  ditto. 

Halfway  Island 

Booby  Isle 

York  Island 

West  I 

Prince  of  Wales's  Is. 

S.  pt. 

Kangaroo  Coral  Reef. . . . 
Providence  Islands, 
Little  Providence  or 

Dann-er  Island 

N.  W.  ext.  of  Shoal 

off  ditto 

Louisiade  Isles, 

Cape  Deliverance . . 

Stephen's  Island 

Durour's  Island 


Lat. 


D.  31. 


o  20  S 
o  20 
o  46 
8  22 
10  i5 
6  40 


59 

56 
35 
55 

25 

48 
54 
59 
5o 
7 
37 


9  45 

10  46 
i3  22 


Lon<i 


34 


D.    31 

32    9  E 
32  3i 
34  25 

37  40 
48  3o 
48  3i 


45  43 
45  26 
44    3 

43  49 

44  i4 
44  00 
44  45 
44  42 
44  4o 
44  10 
43  19 
4i  56 

43  27 

42  12 

43  47 


35  12 

35    8 

54  26 
37  48 
43  12 


See  Table  oa  page  4."/0. 


Page  376] 


TABLE   LIV. 

Latitudes  and  Lonofitudes. 


Admiralty  Island,  Is.  N.  C 
limits ^ 

Sydney  Shoal 

Active's  First  Reef,  (dis- 
covered 1811,) 

Second  Reef,  (do.) 

New  Ireland, 

—  Cape  St.  George 

—  Carteret's  Harbor  . 

New  Hanovej,  W.  end.. 
New  Britain, 

—  Cape  Palliser 

—  Cape  Orford 

—  Port  Montague 

—  South  Cape 

Cocos  Islands 

Shoals    W.    of    Bougan- 

ville's  Strait 

Bouganville's  Strait 

Laughlan's  Islands,  S.  E. 
ext 

Bridgewater  Shoal 

Cape  Deception 

Cape  Nepean 

Cape  Marsh 

Deliverance,  small  Islands 

Indispensable  Strait,  S 
ent 

Bellona  Island 

Bellona  Shoal 

Pandora  and  Indispensa- 
ble Shoal,  N.  p 

S.p 


Lat. 


).  M. 

1  54  s 

2  24 

3  20 


4  5i 
4  42 

2   32 


37 
24 
10 
3o 
40 


9  20 
8  54 
8  42 
8  5i 

10  5i 

10  i5 

11  12 


Lonff. 


Wells's  Shoal 

Port  Praslin 

Stewart's  Island 

Bradley's  Shoal  ..... 
Lord  Howe's  group.  . . 

Hunter's  Islands 

Shank's  Island 

Blanoy's  Island 

Dundas  Island 

Drummond's  Islandt.. 

Byron's  Island 

Hope  Island 

St   Augustine  Island*. 

Sherson's  Island 

Ellice's  group*  N.W.  one.  8  26 
Mitchell's  group,  S.pt.  ..    9 

Plaskett's  Island 9 

Independence  Island 10  25 

Mitchell  Island 10  27 

Island 10  45 

Onaseuse  or  Hunter's  Is).  i5  3i 
De  Peyster's  Isrs.*N.  one .  7  56 
Ocean's  High  Island  ....    o  48 

Pleasant  Island 0  25 

Gardner's  Island i  00 

Duff's  group 10  00 

Ganges'  Island 9  44 

Stewart's  Island 8  24 

Egmont  or  Santa  Cruz  Isl 

Cape  Byron 10  4i 

Pitt's  or  Alderney  Island  11   5o 

Cherry  Island 11   37 

Volcano  Island |io  23 

Mitre  Island 11   55 

Bar  well  Island 12  21 


12  6 

12  46 

12  20 

7  25 

8  27 
6  52 
5  3o 

4  48 
o  28 
o  39 

0  i5 

1  i4 

1  18 

2  23 

5  35 
5  56 


D.    M. 

46  5iE 

48  10 
46  5o 

46  53 
46  37 

52  55 
52  46 

49  5o 


52  16 

52  4 
5o  3o 
49  48 

56  5o 

54  22 

55  55 

53  42 

57  12 
57  18 

57  32 
59    7 

62  27 

61   i5 

59  54 

59  48 

60  3o 

60  42 

58  i3 

58  20 

63  00 

61  06 

59  3i 
57  00 
63  00 
74  i5 

73  58 

74  53 

77  45 
76  59 
76  6 
76  33 
79  i4 
79  48 
79  5o 
79  00 

79  22 
7935 
76  II 

78  29 
70  49 

67  20 

68  40 
66  5o 
66  43 
63  00 

66  10 
66  46 

69  Ai 
65  49 

70  9 
68  48 


Pandora's  Reef. 
Charlotte  Bank. 


Sir  J.  Banks's  Island 

Espiritu  Santo,  Cape  Lis- 

burne  

Cape  Cumberland . . 

Bay    St.  Philip   and 

St.  James 

Cape  Quiros 

Lepar's  Island 

Maskelyne's  Island 

Mallicolo,Cape  Sandwich 
Port   Sandwich 


Lat. 


D.  M 

2  II  S 

I  5o 

3  27 

5  4i 

4  43 

5  10 


St.  Bartholomew's  Island 
Aurora  Island,  N.  pt  . 

Table  Island 

Wjiitsuntide  Island. . 
Ambrym  Island,  E.pt. 

Paoom  Island 

Three  Hills 

A^ae  Island,  N.  jot 

Sheppard's  Islands 

Monument 

ftlontague  Island 

Hinchingbroke  Island  , . . 
Sandwich  Island,  S.E.  pt. 
Erromango,      Traitor's 

Head 

Immer  Island 

Tanna,  Port  Resolution. 

Erronam   

Enatum,  W.  pt 

Durand's  Reef. 

Walpole  Island* 

Matthew's     or    Hunter's 

Island*  Rock, 


•1 


56 

23 
32 

28 

25 

42 

56 
38 
26 
i4 
6  26 

7 
6 
6 

7 
7 

7 
7 


Long. 


Diana's  Bank,  about 

Bougainville's  Reefs 

Alert's  Reef. 

Mellish  Reef,  sand  cay. . 

Bampton  Reef,  N.  pt 

Avon  Island,  S.  W.  islet. . 
Chesterfield  G.  Loop  Is*. 

N.  W.  point  of  reef. . 

Bellona  Reef,  Booby  R.,  K. 

W.  Hn 

N.  horn  of  N.W.  Reyf 

Minerva's  Shoal. 
Barina's  Shoals  . 


Sandy  Island 

Kciin  Reef,  N.  pt 

Saumarez  R.,N.E.  sand  cay 
Small,  low,  woody  Island. 

Huon  Island 

Reef,  about 

N.  W.  p 

Balleabea  Island 

Pudyonn,  K.  W.  p 

Cape  Colnett 

Cape  Coronation 

Qu.  Charlotte's  Foreland. 
Isle  of  Pines 


8  46 

9  21 

9  32 

9  3i 
20  10 
22  6 
22  27 

22  27 

i5  4i 
i5  35 
i5  17 

17  25 
19  01 

19  32 

19  59 

19  37- 

20  57 

20  48 

20  5o 

21  22 

20  4o 

21  5o 

21  24 
21    06 

21  38 

i8     3 

18  II 

19  00 

19  58 

20  7 
20  6 
20  3o 

22  2 
22  i5 
22  38 


D.    M. 

72  7E 

73  12 

67  24 

66  /\A 
66  40 


67  5 
67  5 
67  54 
67  59 
67  59 
67  46 

67  17 

68  6 

67  7 

68  10 
68  24 
68  29 
68  19 
68  10 
68  42 
68  35 
68  17 
68  38 

68  33 

69  1 5 
69  3 1 

69  29 

70  8 
69  42 
69  2 
69  7 

72  lO 

5o  3o 
48  00 
47  57 
5i  4q 
55  53 
58  27 
58  i5 
58  3o 
58  i3 

58  32 

58  28 

59  23 
59  10 

58  4o 

59  3o 
58  34 
55  46 
53  47 
62  5i 
62   52 

62  52 

63  3o 

64  07 
64  7 
64  A^ 
67  47 

66  55 

67  25 


•  See  Table  on  page  451. 


t  See  Table  on  page  450. 


TABLE   LIV. 

Latitudes  and  Longitudes. 


[Page  377 


Botany  Island 

Prince  of  Wales's  Fore- 
land, S.  p 

Port  St.  Vincent 

f .lOyally  Island 


Lat. 


Wreck  Reef,  West  cay  . 

Cato's  I.  and  Bank 

Reef 

Reef 

Ray's  Island 

Reef 


Sir  C.  Middleton's  Island 

Middleton's  Shoals 

Elizabeth  Reef*  K  E.  pt. 

Island 

Lord  Howe's  Island 

Norfolk  Island,  (Mt.  Pitt,) 
Rosavetta  Reef , 


end 


North  Cape 

Cape  Rren 

Cape  Colville  . . . 
Mercury  Bay. . . . 

Cape  East 

Tolaga  Bay 

Table  Cape 

Cape  Kidnappers 
Cape  Turnagain. 
Banks's  Island,  E 
Cape  Saunders . . 
Molineaux  Harbor,  N.pt. 

The  Snares,  E.  one 

Knight's  Island 

Cape  South 

South-west  Bay 

Solander's  Island 

West  Cape 

Dusky  Bay,  N.  pt.  en,  . . . 

Open  Bay 

Cape  Foulweather 

Cape  Farewell 

Queen  Charlotte's  Sound 

Cape  Campbell 

Cape  Palliser 

Cape   Egmont 

Gannet  Island 

Macquarie's  Island* S.pt. 
The  Judge  and  his  Clerk 
The  Bishop  and  his  Clerk 
Auckland's  group*S.cape. 

Campbell's  Island 

Bounty  Islands 

Antipodes  Islands 

Chatham  Island,   Cape 

Young  

Cornwallis  Islands 
Macauley  Island 
Sunday  Island 
Vasques 

Nicholson's  Shoals 


Rottunah    or    Grenville's 
Island,  E.  Sum 


D.  M. 

22  27 

22  3o 
22  10 
20  54 


22  12 

23  l5 

23  40 
23  48 

25  00 

26  4 
26  12 

28  i3 

29  20 
29  34 
3i  19 
3i  37 

29  o 

30  3o 

34  24 

35  10 

36  28 

36  48 

37  42 

38  22 

39  6 

39  4i 

40  32 
43  46 

45  53 

46  25 
3 

48  i5 

4?  17 
46  3o 
46  32 
45  56 
45  43 
43  5i 
4t  46 
4o  3i 
4i  5 
4i  4o 
4i  38 
39  20 
37  57 
54  44 

54  55 

55  i5 
5o  56 

52  32 

i7  44 

49  35 


Lons. 


12  3i 


D.  M. 

167  lE 

66  5o 

65  55 

66  3o 


55  II 

55  34 
60  1 4 
64  i4 

66  21 
60  00 

60  3 1 

58  53 

59  24 

60  42 
59  1 4 

67  46 
73  28 

73  I 
75  o 
75  20 

75  43 
78  40 
78  26 

78  7 

77  9 

76  43 

73  i4 

70  5o 
69  55 
66  45 

66  44 

67  32 

67  25 
66  54 
66  6 
66  27 

68  43 

71  3o 

72  47 

74  27 

74  27 

75  21 

73  39 

74  32 
59  49 
59  10 
58  56 
66  7 

69  1 3 

79  7 
79  2 

76  58W 

75  27 

78  32 
78  i3 
74  56 
78  20 

77  52 
68  36 


i5E 


Solitary  Island 

Duke  of  Clarence'slslandt 
Duke  of  York's  Islandt  . 

Quiros  Island 

Jesus  Island 

Lctticus  Island 

Suwarrow's  Islands. . .  J 

Wallis  Islandt 

Proby's  Island 

Gardner's  Island 

Keppel's  Island 

Boscawen's  Island 

Navigator's  Islands, 

Opoun,  E.  p 

Leone,  S.  p 

Tanfoue,  W.  p 

Maoune,  S.  E.  p.  f . . 

Oyolava,  S.  E.  p 

Otatuelah 

Calinasse,  N.  p 

Islet  Plat 

Amargura 

Vavaoo  (Howe's)  Island. 
Lati  or  Bickerton  Islftnd. 

Savage  Island,  S.  pt 

Toofoa 

Haanho .' . 

Bouhee 

Annamoka. 

Hoonga-hapee 

Tongataboo, 

Van  Dieman's  Road 

Eoaa,  E.  p 

Pvlstaart's  Island 

[N.E.pt, 

Pearl  and  Herme's  Reef. 

King  Geoi-ge's  Reef 

Palmerston  Island 

Whytootaeke 

flervey's  Island 

Wateoo  Island 

Maria  Island 

Mangea  Island 

Pioxburirh  Islands 


Lat. 


Scilly  Island 

Lord  Howe's  Island    . . . . 

Maunura  Island 

Bnlabola  Island 

Ulictea 

Oliameneno  harbor. 

Ilualieine,  Owharre  Bay. 
Sir  C.  Sanders's  Island.. 
Eimeo,  (Taloo  harbor,)  . . 

Tethuroa 

Otaheite,  Point  Vcnust. . 

Oailipeha  Bay 

Osnaburg  or  Miatea 


Prince  of  Wales  Isl.,  N.  p. 

Palliser's  Island 

Chain  Island 

Duke  of  Gloucester  Is.  E.I. 

Ohetiroa 

Remitura  Island 

Toobouai 

Hiirh  Island 


D.  31. 

o  4o 
9  5 
8  36 
o  4o 
6  46 


9 
8 

4  II 
4  19 
4  3 
4  3o 
3  45 
3  5i 

7  58 

8  39 

8  49 

9  10 
9  46 
9  4t 
9  34 
o  1 4 
o  36 


6 

24 


27  4» 
19  56 
18  4 

18  54 

19  17 

20  o 

21  45 
21  57 
21  36 


6 

6 

6 

6 

6 

6 

6 

7 

7 

7  I 

7  29 

7  46 

7  52 

4  58 

5  38 

7  25 

20  42 

22  34 

22  4o 

23  25 

23  42 


Lonar. 


D.  M. 

76  ooW 

71  38 

72  4 
70  00 
66  00 

62  00 

63  23 
63  3i 
76  9 
75  5i 
75  17 

73  58 

73  48 

69  2 
69  16 

69  38 

70  37 

71  21 

70  4 1 

71  5i 
71  48 

74  16 
74  00 
74  35 
60  5o 


75  28 

75  5 

74  57 
7(1  o4 

75  36 
67  3o 
63  10 
59  32 
58  54 
58  6 
55  10 

58  o 

59  4o 


1 55 

10 

1 54 

21 

l52 

12 

5i 

46 

i5i 

3i 

5t 

35 

5i 

8 

5o  58 

49 

47 

49 

27 

49 

29 

49 

i4 

48 

6 

47 

5o 

46  3o 

45  3o 

4f 

8 

DO 

iJ 

52 

5o 

49 

24 

48 

3 

48 


*  See  Table  on  page  451. 


f  See  Tiilile  on  [kijio  4j(' 


Page  373] 


TABLE  LIV 

Latitudes  and  Longitudes. 


Byron's  Islands, 

Taoukaa  Island 

Disappointment  Islands* 

Adventure  Island 

Furneaux  Island 

Resolution  Island*  S.  E. . 

Island 

Island 

Bird  Island , 

Bow  Island , 

Prince  Henry's  Island. 
Cumberland  Island, S.E.pt 
Gloucester  Island,  N.E.pt 
Queen  Charlotte's  Island 
Whitsunday  Island  .  . . 
Lagoon  Island 


Lot. 


Osnabur^  Island,S.  W.  pt. 
Biig'li's  Lagoon  Island.  . . 
Carysfoot  Island,  N.  E.  pt. 

Lord  Hood's  Island 

Gambier's  Islaii'd,  mt.  . . , 
Crescent  Island,  S.  pt.. . . 

St.  Juan  Baptista 

Pitcairn's  Island 

Oparo  Island 


Nukahiwa  Isl.,  (T'ederal,) 

—  Port  Tocliitschagoff. . . 

—  Port  Anna  Maria,  ent. 

—  Cape  Martin, S.  E.  p.. 

—  S.  point 

—  N.  W.  point 

Uahuga  Island, (Wasliing- 

tonTsland,)  W.  p 

Uapoa  Island,  (Adams,). 

Level  Island,  (Lincoln,). 

Mottauity  Islands, (Frank- 
lin,) 

Hiau  Island,  (Knox,  Rob- 
erts,)   

Small  sandy  Island 

Fattnuhu  Island,  (Han- 
cock,)   


D.  M. 

i4  3o 
i4  4 
17  3 
17  3 
17  23 
i6  00 
17  00 

17  49 

18  17 

18  43 

19  i3 
19  8 
19  iS 

19  26 
18  43 

21  54 
21  38 

20  45 
3i 

23'  8 

23  20 

24  26 

25  4 
27  36 


Hood's  Island 

Hiva-Oa,  N.  pt. 

Oliitahoo,  Resolution  Bay 

^  I  Onateaya  Island 

Magdalena  Island ..... 


Bunker's  Siioal 

Marcus  Island 

island 

Brock's  Island 

Island , 

Hero  Island , 

Island 

A  Rock 

Pcnnryhn's  Island,  Kpt., 

Tienhoven  Island 

Groningue  Island   

Reirscu's  Island 

Huinplirey's  Island 

A  Reef 

Pescado  Island 

Roggewein's  Island 

Tiburone's   Island 


8  57 
8  57 
8  57 
8  59 
8  53 

8  58 

9  21 
9  29 

8  43 

7  59 
7  57 

7  5o 

9  26 
9  34 
9  55 
9  58 

10  27 

o  17 

0  26 

1  5 
I  i3 
3  32 

5  40 

6  34 

7  5i 

8  55 
10  5 
10  5 
10  2 
10  38 
10  46 
10  33 
10  5i 
ID  58 


Long. 


D.  M. 


i45 
i4i 
i44 
1 43 
i4i 
139 
i38 
1 43 
i4o 
i4r 
i4i 
i4o 
i38 
i38 
1 38 


9W 
22 
14 

7 
35 
00 
00 

7 
43 
42 
II 
37 
42 
36 
43 


139  37 
i4o  38 
i38  22 
i35  32 
i34  55 
1 34  35 
i35  6 
i3o  8 

^■^  II 


1 39  42 

i4o  6 
139  32 
139  ^^ 
139  49 

139  33 
i4o  6 


i4o  43 

i4o  48 
i4o  3o 

i4o  6 

i38  57 
>39  4 
139  9 
i38  5i 

1 38  49 

160  4o 
159  5o 
1 33  54 
159  3o 
173  45 
i55  55 
1G6  3o 

139  54 
i58  6 
1 56  57 
i56  5o 

161  10 
161  2 
166  6 
159  25 
i56  7 
i43  o 


E.p. 


Flint  Islan  * 

Baunian's  Islands. 
Eng  George's  Is. . . 

Tiokea,  Oura, 

Isle  des  Chiens* . . 
Isle  RomanzofF. . . 
Isles  de  Krusen- 

stern,   extend- 
ing  N.   N.  E.  )-  centre 
and  S.  S.  W 

15  miles __ 

Chaine  du  Rurick,  N.E.  p 

—  E.p 

—  W.  p* 

Dageraad  Island 

Dean,  or  Prince  of  J  r^ 

Wales,  or  Oan-  '*^ 
na  Island ^ 

Island 

Island 

Island 

Elizabeth  Island  . . . 

Eunice  Island 

Armstrong's  Island 

Anderson's     Island,     (or 
Elizabeth  Island, )N.E.p. 

Ducie's  Islandj  N.E.  pt. . . 

Island 

St.  Ambrose  Island  IST.  p. 

ISr.  pt.  of  St.  Felix..- 

Gray  s  Island 

Sales  y  Gomez 

Easter  Island,  Peak 

Island 

Group  of  Islands 

Massafuera 

Juan  Fernandez,  S.  W.  p. 

E.p.... 


Lat. 


NEW    SOUTH    SHET- 
LAND. 
Clarence    Island,  Floyd's 

Promontory 

Cape  Bowles 

Cornwallis  Island 

Seal  Islands 

Cape  Valentine 

Sarah  Island 

Obrien's  Islands 

Bridgeman's  Islands  ... 

Cape  Melville 

Sheriff'  Cape 

Ditto,    (another    ac 

count,) , 

Yankee  Straits , 

Ragged  Island 

Ditto,    (another    ac 

count,) 

Ditto,  the  harbor,  (by 
another  person,) . 

New  Plymouth 

Monroe's    Island,    Presi- 
dent's Bay 1 . . , 

Castle  Rock,  (W.  of  Mon 

roe's  Island,) 

Mount  Pisgali 

Ditto,  (another  ac- 
count.)  


D.  M. 

II  26 

II  52 

i4  22 

14  U 
i4  5o 
i4  57 


1 5  00 


i5  II 

i5  20 
i5  20 
i5  45 

1 5  o5 
i5  17 

16  00 

17  00 

20  00 

21  6 
21  8 
21  21 

24  22 

24  4o 

25  i3 

26  20 
26  17 
26  24 

26  28 

27  8 

28  6 
3i  3 
33  45 
33  49 
33  4i 


60  57 

61  20 
61  o4 
61  00 
61  3 
61  22 

61  32 

62  06 
62  00 

62  28 

62  2  1 

62  3o 
6?.  4o 

62  42 

62  55 
62  45 

62  46 

62  5o 

63  00 


Long. 


62  57 


D.  M. 

i5i  48W 
i55  12 
1 44  58 
(45  20 
1 38  47 
1 44  35 


48  4i 


1 46  47 
1 46  3o 

1 46  56 

147  59 
147  i4 

139  00 
1 38  00 
167  5o 
178  36 
178  47 
i6i  4 

128  19 
124  48 
i3o  28 

80  GO 
80  21 
92  24 

io5  26 
109  17 

95  12 

129  24 

80  47 
79  6 
78  53 


54  6 
54  8 

54  28 

55  32 

54  4o 

55  3o 

55  52 

56  4o 

57  3o 

60  28 

61  47 

60  22 

62  10 

62  20 

63  5 

61  37 

62  20 

62  3o 

63  00 

63  4o 


See  Table  on  page  450 


TABLE  LV.  [I^'ige  379 

This  Table  shows  the  Times  of  High  Water  at  the  Full  and  Change  of  the  Moon,  at  the 
principal  Ports  and  Harbors  of  the  world,  and  the  vertical  rise  of  the  Tide,  in  feet.  [Those 
marked  with  a  star  arc  corrected  establishments.] 


Abbeville 

Aberdeen*  

Aberystwith 

Acapulco 

AchiUHead*.... 

Aden 

Ayre,  Point  of .  . . . 

Aix,  Isle  d' 

Albau's  Head,  St.* 

Algoa  Bay  

Amazon  River.  .  .  . 

Anibleteuse 

Anicland* 

Amhviek  Point . . . 
Anioor  Strait  .  . . . 

Anioy 

Amsterdam 

Amsterdam  Island 
Andrew's  Bay,  St.* 

Ani^ra  Bay 

Anliolt  Island.  .  . . 

Ann,  Cape* 

Annapolis*    

Anticosta     Island, 

W.  end 

Antwerji* 

Annamooka 

Areliangel 

Arklow* 

Arran  Island 

Arundt-l 

Astoria* 

Augustine,  St.*. . . 
Aug'stine's  Bay,St. 
Avranches 

Babelmandel  Strs. 

Bali.sore 

Ballingskellings 

Bay* 

Baltimore 

Baltimore* 

Baliia 

Banff 

Bantry  Bay* 

Bardsey  Island .. . 

Barfleur*  

Barmouth 

Barnstable  Bay  .  . 

Batavia 

Baudsey  Clift' 

Bay  of  Islands  . . . 

Bayonne  

Beachy  Head  .  . .  . 

Bear  Islantl 

Beaumaris 

Bee's  Head,  St.... 
Belfast  (entrance) 

Belle  lAe 

Bembridge  Point* 

Bergen 

Bermuda  Island. . 

Berwick* 

Bilboa* 

Blakeney 

Blanco,  Cape  .  . .  . 


Fi-ai)ce   

Scotland 

Wales 

Mexico   

Ireland 

Arabia 

Isle  of  Man. . . 

France 

England  

Africa 

America 

France  

Xorth  Sea     . . 

Anglesea 

Asia 

China 

Holland 

Indian  Ocean. 

Scotland 

Terceira 

Cattcgat 

America 

America,  U.  S . 


America , 

Belgium , 

Facilic  Ocean  .  . , 

Russia 

Ireland , 

Scotland 

England 

Oregon 

America , 

Madagascar  . . . . 
France , 


Red  Sea 
India  . . . 


Ireland 

Ireland 

America 

Brazil 

Scotland 

Ireland 

Wales 

France  

Wales 

England 

Java 

England 

New  Zealand  . . . . 

France 

England 

Hudson's  Bay. . . . 

Wales 

England 

Ireland 

Bay  of  Biscay. . . . 
Isle  of  Wight  . . . . 

Norway 

Atlantic  Ocean  . . 

England 

Spain 

England 

Africa 


h.  in. 

lo  3o 

o  48 

7  3o 

3  6 

4  56 
9  45 

lo  3o 


46 

CO 
CO 
00 
00 

3o  24 


00:  6 

25  i5 

I 
00; 

28'   2 

45J  2 

49  10 

25  16 

42  7 
20  5 
3o  i3 
00 1 

3o 
00  1 5 

46  12 
23  10 

33:     I 

3o[  8 
28  10 
2  8 
40  1 5 
45  17 
4o  17 
3o  19 
00  2 
3o' 
i5    9 

45  [2 
20  20 
00 
32  21 

i5 

43  9 

°°'  / 
II  i4 

3o'  4 

i4   4 

i5  i5 

20'  9 

3o  i5 

46  6 


(Sibyl 


Blaskets 

Head) 

Block  Island*... 

Bojador,  Cape. .  . 

Bolt  Head 

Bombay 

Borkum  Island  .. , 
Boston  Light*  . . 
Botany  Bay  .... 

Boulogne* 

Boi'deaux 

Brassa  Sound  . . . 
Bray  Head*  .... 

Bremen 

Brest* 

Bridgewater  .... 

Bridport 

Brighton* 

Bristol 

Broad  Haven  .  .  . 
Burnt  Island .... 
Button's  Islands. 


Cadiz* 

Caen 

Caernarvon  

Calais* 

Caldy  Island 

Calf  of  Man 

( 'allao 

Camj)bel!  Town  .  . 
Canary  Isl.,  Pt.  de 

la  Luz 

Causo,  Cape 

Cantire,  Mull  of.  . 
Canton  II.  (ent). . 
Capricorn,  Cape . . 

Cardiff 

Cardigan  Bar. . . . 

Carlingford* 

Carlisle 

Caimarthen 

(baskets 

Catherine's  Pt.,  St. 

Catness 

Cayenne  

Cedar  Keys*  . . . . 
Charente    R.,  Ro- 

chefort 

Charles,  Cape 

Charleston,  S.  C* 
Charlottetown  .. , 

Chatliam 

Chepstow 

Cherbourg*  

Chester  Bar 

Chicht'ster  Harbor 
Cliristmas  Sound.. 
Churchill,  Cape. . . 
Clear,  Cape*  . . . . 

Cod,  Cape* 

Condore  Pulo. . .  . 

Conway 

Copeland  Island.  . 

Coringa  Bay 

Coquet  Island  .  .  .  . 


Ireland 

America 

Africa 

England 

India 

Hrf)lland 

America 

New  Holland 

France  

France  

Shetland 

Ireland 

Germany 

France 

England 

England 

England 

England 

Ireland 

Scotland 

Hudson's  Bay. . . . 


Spain 

France 

Wales 

Fiance 

Wales 

St.  Georg.  Chan'l. 

Peru 

Gulf  St.  Lawrence 

Atlantic  Ocean. . 

America 

Scotland 

China 

New  Holland  .  .  . 

Wales 

Wales 

Ireland 

England 

Wales 

English  Channel. 
Isle  of  Wight  .  .  . 

White  Sea 

South  America. . 
Florida 


France  

America 

America 

Prince  Edw'd'slsl. 

England 

England 

France  

England 

England  

South  America. . . 
Hudson's  Bay.  .  ,  . 

Ireland 

America 

China  Sea 

Wales , 

Ireland 

India 

Ensrland 


ft. 


3o  12 

37 
00 
55 

4o  i5 
3o!  9 


16 
35 
5  II 
i5  i5 
56  44 
00  10 
5o'io 
5o 


I  4o 

10  57 
9  33  14 

11  3219 

6  00  34 

11  17  16 

5  4?!  4 
4  00  10 

i2  5o  10 

8  3o'  6 
10  3o  5 
10  00 

8  00    7 

6  59'38 

7  00  1 4 
10  53, i5 

12  1020 

3o'2Z 

45  i5 

00 

i5 

45 
5o 


4617 
7  45 
7   t3 
10  45 

1  00^17 
7  3o|38 
7  33,17 

10  30,26 

11  3o 

2  3o 
7  20 
4  00 

II     25 

3  00 
10  i5 
;o  49 

9  i5 
3  00 


^age380]  TABLE  LV. 

This  Table  shows  the  Times  of  High  Water  at  the  Full  and  Change  of  the  Moon,  at  the 
principal  Ports  and  Harbors  of  the  world,  and  the  vertical  rise  of  the  Tide,  in  feet.  [Those 
marked  with  a  star  are  corrected  establishments.] 


Cornwall,  Cape  . . 
Cornwallis,  Port.  . 
Cork  Harbor  (en- 
trance)* .... 

Corunna  

Coutance  

Cowes 

Croeotoa  Island 
Cromartie*  .... 

Cromer* 

Crookhaven. . .  . 
Cross  Island  . .  . 
Cuxhaven  


Dartmouth 

David's  Head,  St. 
Deadman's  Point. . 

Deal 

Dee,  River 

Delaware    Bay* 
(breakwater) . . . 

Demerara 

Diamond  Point. . . 

Diego,  San* 

Dieppe* 

Dingle  Bay* 

Donegal* 

Dover*     

Douglas 

Downs 

Droglieda 

Drontheim 

Dublin* 

Dudgeon  Lights. . 

Dunbar* 

Duncansby  Head  . 
Dundalk  Bay  .  . . . 
Dundedy  Head  .. . 

Dundee 

Dungaroon 

Dungeness* 

Dunkirk* 

Dunnose 


Eastern  Brace  . . . . 

EastporL* 

Eddystone 

Elbe  R.,  red  buoy. 

Embden 

Exmouth  Bar*  . . . 

Exuma  Bar 

Eyder  River 

Eyemouth  Harbor. 


Fair  Head , 

Falmouth , 

Fayal  Road 

Fear,  Cape*  . . . , 

Fecamp 

Fernandina*  . .  . . 
Fernando  Po. . . . 

Ferrol 

Ferriters 

Fifeness 

Filey 

Finisterre,  Cape. 


SITUATION. 


England   

Prince  of  Wales'  Is. 

Ireland 

Spain  

France  

Isle  of -Wight 

Strait  of  Sunda  . . 

Scotland 

England 

Ireland 

White  Sea 

Germany 


England 
Wales  . . 
England 
England 
Scotland 


America , 

S.  America 

Malacca  Strait  . . 

California 

France  

Ireland 

Ireland 

England  

Isle  of  Man 

England 

Ireland 

Norway 

Ireland 

North  Sea 

Scotland 

Scotland 

Ireland 

Ireland 

Scotland 

Ireland 

England 

France 

Isle  of  Wight . . . 

Bay  of  Bengal  . . 
Maine,  America  . 
English  Channel. 

North  Sea 

Germany 

England 

Bahamas 

Germany 

Scotland 


Ireland  . 
England 
Azores  . . 
America 
France. . 
Florida  . 
Africa  . . 
Spain  . .  . 
Ireland  . 
Scotland 
England 
Spain  . . . 


TIME.    R. 


h.   m.  ft. 

4    3o  22 

I  3o 


4  37 
3  00 


6  00 

II     l5'l2 

7  oo[  3 
II  43  1 3 

6  43  i6 
4  9|io 
4  i5 


I    00  10 


6 

i6 

i4 

6 

00 

5 

3o 

1 1 

i5 

i6 

II 

00 

23 

8 

00 

4 

4  45 

9 

[2 

oo 

9 

9 

38 

5 

1 1 

6 

27 

3 

4o 

9 

b 

8 

II 

II 

00 

i8 

II 

12 

21 

II 

OO 

i5 

10 

45 

2 

i5 

II 

II 

i3 

6 

00 

2 

oo 

i4 

10 

i4 

10 

10 

56 

i3 

4 

00 

II 

2 

32 

i4 

4 

3o 

lO 

54 

21 

II 

55 

i6 

9 

i5 

9  45 

11  i3  i8 

5  5o[i8 

12  oo 
12    00 

6  21 

7  20 
12    OO 

2     l5 


9  OO 

4  57 

II  45 

7  19 


10  44  23 


7  53 
4  00 
3  00 

3  3o 

2  00 

4  20 

3  00 


Finmark     

Fishguard  Bay  . . . 
Flamborough*  . . . 

Flushing 

Fly,  or  Vlie   Gat- 
way  

Fly,  or  Vlie  Road. 
Foreland,  North. . 
Foreland,  South.  . 
Formby  Point. . . . 

Fox  Island 

Fowey* 

Francisco,  San*  . . 
Fuuchal 


Gal  way  Coast*. . , 
Galloway,  Mull  of 
Gambia  River  (en 

trance) , 

Gay  Head 

Georgetown  Bar*. 

Gibraltar 

Gloucester* 

Goa 

Good  Hope,  Cape* 

(St.  Simon'sBay) 
Good  Hope*  (Ta 

ble  Bay) 

Goree  Gatway. . . 

Granville* 

Gravelines 

Gravesend 

Grizness,  Cape  . . . 

Haerlem 

Hakodadi 

Halifax 

Hamburgh 

Hartland  Point  . . 

Hartlepool 

Harwich 

Hastings 

Hatteras,  Cape*. . 
Havre  de  Grace*. 

Helena,  St 

Helen's,  St 

Helvoetsluys*  . . . 
Henlopen,  Cape*  . 

Henry,  Cape 

Hobarton 

Hogue,  Cape  La. . 
Holy  Isl'd  Harbor. 
Hongkong  Road. . 

Honfieur  

Hoogley  R.  (ent.). 

Hull 

Humber  R.  (ent.) . 
Hurst  Castle 


SITUATION. 


Lapland 
Wales  . . 
England 
Holland. 


Holland  . . 
Holland . . 
England  . 
England  . 
England  . 
America  . 
England  . 
California 
Madeira.  . 


Ireland  . 
Scotland 


Ice  Cove  

Ichaboe  

Ipswich 

Isle  Dieu 

Isle  of  Man,  South 

side 

Ives,  St 


Africa  . . 
America 
America 
Spain . . . 
America 
India  . . . 


Africa  , , 


Africa  . . . , 
North  Sea. 
France  . . .  , 
France  . . . , 
England  .  . 
France .  . . , 


Holland , 

Japan 

Nova  Scotia  . . 
Germany  . . . . , 

England , 

England 

England 

England 

America 

France 

Atlantic  Ocean , 
Isle  of  Wight . , 

Holland 

America 

America 

Tasmania 

France   

England 

China 

France 

India 

England 

England 

Entrland 


Hudson's  Bay. 

Africa 

England 

France 


St.  George's  Chan. 
England 


7i.    m.  ft. 

i5 
6  56  II 
4  3o  12 
I   20  i5 


45 
3o 
i5  16 

6  i5 
35'28 
45 
i4i5 

6    4 
7 

24 
r5 


7  37 
7  56 
2  20 
:i   00 

1  3o 

2  48 

2  29 
I  3o 

5  54 


37 


00.19 
10  17 
27  21 


00 
00 

49 
29 
00 
28 

6 
53|24 

4\  2 

3622 

3 


II  45 
2  3o 
8  00 

7  14 

8  00 

8  45 
2  3o 

10  i5 

9  3o.23 

10  00  II 

6    29'2I 

5  i5'i8 

11  00!  7 

10  00 

1  oo    6 

2  35ji3 

3  00  i4 

10    20| 

4  44!2t 


TABLE  LV.  [i^'^ge  831 

This  Table  sho-ws  the  Times  of  IIigu  Water  at  the  Full  and  Change  of  the  Moon,  at  the 
principal  Ports  and  Harbors  of  the  world,  and  the  vertical  rise  of  the  Tide,  in  feet.  [Those 
marked  with  a  star  are  corrected  establishments.] 


Jackaon,  Port . . . . 

Janeiro,  Rio 

John's,  St 

John's,  St  ,  River* 

John's,  St 

Jutland  Coast. . . . 


Kedgeree 

Kenniaie,  River. , 

Kennebec 

Kentish  Knock . .  , 

Key  West* 

Killibecrs 

King's  Channel. . 
King's  Road  .... 

Kinsale* 

Kinnaird's  Head. , 


Lambaness 

Lancaster 

Land's  End 

Leitli  Pier 

Lemon  andOwer. 

Lerwick* 

Lewis  Islands  .  .  .  . 
Lewis,  Butt  of  . . . 

Limerick 

Lisbon 

Liverpool 

Lizard  

Loch  Swilly 

Loire  River 

London  

Londonderry  .  .  .  . 
Long  Sand  Head . 

Longsliips 

Lookout,  Cape*  . . 
Loop  Head 


SITUATION. 


N^ew  Holland  . . . 
South  America.  . 
N'ew  Brunswick. 

Florida 

Newfoundland  .  . 
Denmark 


India 

Ireland 

America 

River  Thames.  . 

Florida 

Ireland 

River  Thames. . 
Bristol  Channel 

Ireland 

Scotland 


Shetland 

England 

England 

Scotland 

North  Sea.  . . . 
Shetland  .  .  .    . 

Scotland 

Scotland 

Ireland 

Portugal 

England 

England 

Ireland 

France 

England 

Ireland 

River  Thames. 

England 

America 

Ireland 


L'Orient 'France 

Lundy  Island  .  . .  .!  Bristol  Channel  . 


Lyme  Regis 
Lynn  Deeps 


Macao 

Machias 

Madeira 

Madras 

Malacca  Roads .  . . 

Mulo,  St 

Manilla 

Marblehead 

Margate  Road.  .  . , 

Marks,  St.* 

Martin  Vas  ...... 

Mary's,  St 

Maulmain 

May,  Cape* 

Melbourne 

Milford  Haven  .  . 

Miramichi 

Mizzen  Head. .  .  . 

Monrovia 

Monterey* 

Montrose* 

Morocco  Coast  . . 
Mount's  Bav*.  . . 


England 
England 


China 

Amei"ica 

Atlantic  Ocean . . . 

India 

India  ....    

France , . . 

Philippine  Islands. 

America 

River  Thames. . . . 

Florida 

Atlantic  Ocean. . . 
Scilly  Islands  . . . . 

India 

America 

Australia 

England 

Canada  

Ireland    

Africa 

California 

Scotland 

Afiica 

England  


/t.  m. 

8  i5 

3  00 

[I  24 

7  28 

7  3o 


9  3o 
I  i5 
4  3o 


7  00 

ID  3o 

6  00 


2  3o 

II     26[26 

5  GO  i4 

6  3o 

3  4o[i5 
2  7 
8  00 

II  3o 
3o 
10 


i5 


6  21 
6  00 


10  00 

11  00 

12  48 
7  34 

7  3o 
6     5 

10  4o 

11  3o 
II  45|i5 

1  i3    3 

3  45 

4  II 

2  o 

8  19!  6 
I  20  3 
6  00124 


5 

00 

4 

2 

6 

0 

0 

22 

I 

4o 

2 

i5 

4 

19 

Mount  Desert. . . 
Mozambique. . . ., 


Xamjasaki 

Xantucket* 

Xantes 

River  Loire. 

Xassau 

N^atal,  Port 

Needles 

Newcastle 

New  Bedford* .  . . 
Xewburyport* . .  . 
New  Haven*  . . .  . 
New  London*  . . . 

Newport 

Newport* 

New  York* 

Nootka  Sound . . . . 

Nore  Light 

North  Cape 


Olonne 

Oporto  (Bar)*. . 

OrforJness 

Orkney  Islands. 
Ornis  Head . .  . . 
Ortegal,  Cape. . 

Ostend 

Owers 


America 
Africa  . . 


Japan   

America     , 

France 

France 

New  Providence. . 

.Africa 

Isle  of  Wight .  . . 

England 

America 

America 

America 

America 

Wales 

America 

America 

North  America. . 
River  Thames. . . 
Lapland 


Padstow  * 

Panama  Road.  . . . 

Para 

Passamaquoddy 

River 

Passier  Roads. . . . 

Penmarks 

Penobscot  River. . 
Pentland  Frith... 

Penzance 

Pernanibuco 

Peter  Head* 

Philadelphia*  .. .  . 

Phillips  Port 

Pictou 

Plymouth  Sound.. 

Plymouth* 

Pouit  de  Galle- . . . 
Pol  de  Leon,  St. . . 

Poole 

Port  Glasgow. . . . 

Port  Hood 

Port  Howe 

Port  Jackson 

Portland  Bill 

Portland  Race  . . . 

Portland* 

Port  Louis 

Port  Louis 

Porto  Pra3-a 

Port  Roseway  . . . 
Port  Royal  Island . 
Portsm'th  Harbor 

Portsmouth* 

Pulo  Pinang 


SITUATION. 


France   

Portugal 

England 

North  Sea 

Wales 

Spain 

Belgium 

English  Channel. 


England 

New  Grenada  . 
South  America. 


America 

Borneo  

France 

America 

Scotland 

England 

Brazil 

Scotland 

America 

Australia 

Nova  Scotia  .  . .  . 

England 

America 

India 

France 

England 

Scotland 

Cape  Breton . . . . 
Nova  Scotia  . . . . 
Nova  Scotia  . . . . 

England 

England 

America 

France 

Mauritius 

Cape  Verde  Isl. , 
[Nova  Scotia  . . . , 
I  North  America. , 

England 

America 

India 


TIME.     R, 


\.  m.  ft. 
I  10  i3 
4  i5  12 


4524 
45  5 
i3  5 
20  7 
3oi5 
00 


3  So 

2  3o 

11  i5 
10  00 
10  i5 

3  00 

12  21 
6  3o 


l4 


i5 


4  56  20 

3  23'i8 

12  00' 1 1 


3o 


25 


7 
3 
6 

37|i5 
19  II 


4  i5 
I  00 

0  18 

9  ° 
8  3o 

8  00 

7  i5 

9  i5 

1  25 

3  II 
:2  3o 
:i  ? 

8  3o 
8  i5 

ti  36 

[I  23 

2  i5 


Page  382]  TABLE  LY. 

This  Table  shows  the  Times  of  High  Water  at  the  Fall  and  Change  of  the  Moon,  at  the 
principal  Ports  and  Harbors  of  the  world,  and  the  vertical  rise  of  the  Tide,  in  feet.  [Those 
marked  with  a  star  are  corrected  establishments.] 


Quebec 

Queda  Roads. 


Rachlin's  Island*. 

Ram  Head 

Ramsey 

Ramsgate* 

Rangoon  (entr.). . 

Rhe  Island 

Rio  Janeiro.. . . '. . 
Robin  Food's  Bay 

Rochef  -i  u 

Rochelle 

Rochester 

Rodrigues  Island. 

Roman,  Cape 

Roseness 

Rotterdam 

Rye  Harbor 


Sable,  Cape 

Sable  Island 

Salem* 

Salvador,  St 

Sandwich 

Sandwich  Bay.. . 
Sandy  Hook*..  . . 
Savannah  (entr.)* 
Scarborough .... 

Scaw 

Scilly  Islands*  . . 

Seal  Islands 

SelseaBill* 

Senegal  R.  (entr.) 
Seven  Islands. . . 

Shanghae 

Shannon  R.  (entr. 

Sheerness 

Sheepscut  

Shetland  Island, 
(south  end)  . . . 

Shields 

Shoreham 

Sierra  Leone .... 

Simoda 

Simon's  Bar,  St.* 

Sincapore  

Skerries 

Skerries 

Sky  Island 

Sligo* 

Slyne* 

Smalls 

Somme  River. . . 
Southampton  . . . 
Southwold 


Canada. 
India  . . 


Ireland 

England 

Isle  of  Man 

England 

India 

Bay  of  Biscay . . . 
South  America. . 

England 

France  

France  

England 

Indian  Ocean. . . . 

America 

Orkneys 

Holland 

England 


TIME. 


Nova  Scotia , 

America 

America 

South  America . . 

England 

Nova  Scotia 

New  Jersey 

America 

England 

Denmark 

Ijlnglish  Channel. 
Bay  of  Fundy.  . . 

England 

Africa 

Lapland 

China 

Ireland 

England 

America • 


h.  in. 
6  38 

12    00 


ft. 


53 

45 

44  i3 
i5  21 
00 
00 
45 
6 
3i 
00 
45 
00 
3o 
3o 
20 


8  3o 
lo  3o 


North  Sea  . 
England  .... 
England  .... 

Gruinea 

Japan 

America  .... 

Asia 

Wales 

Scotland  .... 
Scotland  .... 

Ireland 

Ireland 

Wales 

France    

England  .... 
England 9 


i3 
3o 
00 
00 
29 

7  20 

4 


Spurn  Point  . 
Start  Point . . 
Stockton  . . . . 
Stonehaven. . 
Stromness* . . 

Suez 

Sunbury  .  . .  . 
Sunderland* . 

Surinam 

Swansey 

Sweetnose  . . 

Sydney  

Sydney  


England 

England 

England 

Scotland 

Orkneys 

Red  Sea 

North  America. 

England 

South  America. 

Wales 

Lapland 

Cape  Breton  I. 
Australia 


8  20 
I  40 

4  12 

o  .37 
10  45 

10  3o 

3  23 
II 

7 
5 

7 

9 

10  00 

11  00 
6  00 

5  25 

4  32 

5  5o 
II 
II 


16 


16 


Tees  River. . . . , 
Telling,  Cape. . , 

Terceira 

Texel  (entrance  of) 
Texel  Road... 
Thames  River 
(mouth)    ... 
Tynemouth.  . . 

Todhead 

Torbay 

Tory  Island.  . . 
Tuscar  Rock. . 
Typa  Roads  . . 


England 
Ireland  . 
Azores. . 
Holland  . 
Holland. 


Ushant* 


Valparaiso 

Yannes 

Vincent,  Cape  St. 


Wardhuys 

Watchet 

Waterford  Harb.* 
Weser  River  (ent.) 
Western  Brace. . . 
Wexford  Harbor. . 

Weymouth 

Whitby* 

Whitehaven 

Wicklow 

Winterton 

Woolwich 

Wrath,  Cape 


Yang-tse-Kiang 

(entrance)  .... 
Yarmouth  Roads 

Yarmouth 

Yorkshire  Coast. 
Youfrhall* 


Zanzibar. 


England  .... 
England  .... 
Scotland  .... 
England  .... 

Ireland 

Ireland 

River  Canton 


France 


Chili. . , 
France . 
Spain  . 


Lapland 

Britisli  Channel 

Ireland 

Germany 

Bay  of  Bengal  . 

Ireland 

England 

England 

England 

Ireland 

England 

England 

Scotland 


China 

England  .... 
Isle  of  Wight 
England  .... 
Ireland 


i5i4 
00  5 
38    5 


3  45  i5 
6  00 
12   32 

6  45 

7  45 


12  00  17 
3  20' 1 5 

12  45 
6  00  i3 

6  00 

7  00 
10  00 

3  39  19 


9  82 
4  3o 
2  3o 


Africa 


II   1 5  23 


29  9 
5o  10 
3718 
3oi5 


i5 


9 

i5 

6 

II 

00 

7 

4 

3o 

5 

i4 

10 

4  20 


TABLE  LVI. 


[Page  383 


The  following  table  contains  extracts  from  the  Nautical  Almanac  for  the  year 
183G,  in  those  parts  which  are  used  in  this  work,  to  accommodate  those  who 
may  not  have  a  copy  of  that  Almanac  to  refer  to. 


Lunar  Distances  and  Proportional  Logarithms. 


0;iy  of  the 
Month. 


183G. 
January  G 
April  1 
May  11 
June  20 
Oct.      30 


Aldebaran  W- 
Antares  ..E. 

Sim E. 

Venus. .  ..W. 
Sun E. 


0  Hours. 


Di.-tances.    P.  L 


66 
6i 


4o 


46  34 
3o  58 
112   54 


59 

1 3  2348 
3o97 
3(>35 
3458 


49 


3  Hours. 


Distances.     P.  L 


6j  4i  43  2872 

59  55  24  2337 

45  5  493108 

32  28  lb  3019 

III  32  59I3459 


G  Hours. 


Distances.    P.  L. 


69    l4    38  2864 
58    ID    19  2326 


43    37    49 

33    58     7 

no    II    49 


3i  17 

3oo2 

3460 


9  Hours. 


70  47  43 

56  24  58 

42  10  00 

35  28  17 

108  5o  4o 


P.  L, 


2856 
23i7 
3127 
2985 

3460 


Day  of  tlie 
Month. 


12  Hours. 


15  Hours. 


18  Hours. 


21  Hours. 


S\Iean'riine. 


183G. 
Feb.      12 

Auir.    2G 


Sun E. 

Mars E. 


5 1    28    10 
:i4    55      6 


Distances.     P.  L. 


2552 

12455 


49   48     9  255i 
1 13    12   49  2467 


Distances.     P.  L. 


48     8     7 
III    3o    49 


Distances.     P.  L. 


255i 
2479 


46 
109 


28 
49 


255i 
2492 


Moon^s  Semi-diameter,  Horizontcd  Parallax,  S^'C. 


g 

tin's 

Latitude. 

JVoon. 

II 

N. 

0.43 

0.66 

0.34 

S. 

O.IO 

0.23 

0.33 

N. 

0.86 

S. 

0.48 

0.44 

JN. 

0.17 

o.o5 

S. 

0.32 

N. 

0.32 

Day  of  tlie 
Month. 


January  6 
April  1 
Oct.  30 
May  11 
Feb.  12 
13 

June  20 
Aus.  26 
°  27 
June  2G 
27 
Sept.  26 
Nov.     29 


Sun's  Longi- 
tude. 


Lo^'.  Radius 
Vector. 


285  16 

II  47 

217  7 
5o  45 

322  5l 

323  52 
89  5 

i53  12 
i54  10 

94  49 

95  46 
i83  24 
247  22 


27.6 
34.6 
36.5 
58.8 

54.9 
33.0 
53.5 
57.6 
55.6 
8.9 
20.0 
26.6 
1 1.8 


9926712 

,0000753 
.9965769 
.0046545 
,9945669 
.9946561 
.0070882 
.0042613 
.oo4i6i4 
.0071787 
.0071880 
,0007178 
,9937878 


Moon's  Semi-di- 
ameter. 


1 5  4-8 
i5  58.7 
i4  45.6 
i5  16.8 

16  16.2 
16  17.6 

1 5  9.6 

16  i4-5 
16  4.7 
16  30.9 
16  39.4 
i5  32.4 
i4  48.5 


Miiln. 


8.5 

3.5 
45.7 
12.5 
17.2 
17.3 

l5.2 

1 0.0 
58.9 

35.7 

4i.8 

26.9 

4  5i.4 


Moon's  Horizontal 
Parallax. 


20.3 
38.1 
10. 1 

44 
42.5 
47-4 
38.1 
36.3 

0.3 


55  33.8 

58  55.9 

54  10.3 

55  48.6 

59  46.0 
59  46.3 
55  58.6 
59  19.5 
58  39.0 

36.4|6o  53.8 
7.461  16.4 
i.7|56  41.4 

20.6:54  3 1. 1 


Examples  I.  IX.  X 
p.  232,  241,  242 
Ex.  H.  p.  233. 
HI.  Vin.234,240 
Ex.  IV.  p.  235. 

I  Ex.  V.  p.  23G. 

Ex.  VI.  p.  237. 

I  Ex.  VII.  p.  238. 

I  Ex.  I.  p.  172. 

Ex.  II.  p.  172. 
Ex.  p.  214. 


Swi's  Right  Ascension,  ifc. 


Dav  of  the 
Month. 


Nov. 


29 

30 

20 

27 

5 

G 

8 

9 

16 

17 

24 

25 

March  10 
11 

Oct.  30 
May  11 
Feb.     12 


IMay 

Jan. 

Sept. 

April 

July 


THE   SUN'S 


Riirht  Ascension.       Declination.       Pemi-diam 


h.  m. 

16  22 

16  26 

4  l3 

4  17 

19  I 

19  6 

II  7 

II  II 

I  38 

I  42 

8  i5 

8  19 

23  23 

23  26 

i4  19 

3  i3 

21  4o 


i4.56 
32.98 

2.5l 

5.56 

55.40 

18.78 

47-32 

23.47 

20.06 

2.38 

5.79 

3.18 

10.85 

51.37 

6.64 

18.24 

5i.5o 


S.  21  33 

21  43 

N.  21  II 

21  21 
S.  22  4i 

22  35 
N.  5  35 

5  i3 

10  i4 

10  35 

19  5o 

19  37 

S.  3  58 

3  34 

i3  54 

N.17  57 

S.  i3  54 


J9.7 
28.7 
i4-o 
21.2 
53.8 
10.5 
55.7 
14.6 
7-4 
16.0 
18.5 
27.4 
17.9 
45.2 
18.2 
AU 
27.3 


6  i4.8 
6  14.9 
5  48. o 

5  47-8 

6  17.3 
17.3 
54.5 
54.8 
56.6 
56.4 
46.2 
46.3 

6.7 
6.5 
8.5 
5o.7 
i3.o 


Equation  of 
Time,  to  be 
applied  *  to 

Mean  Time. 


+  1' 
-f-io 

+  3 

+  3 

—  5 

—  5 
+  2 
+  2 
+  0 
+  0 

—  6 

—  6 


4-16 
+  3 

+t4 


21.56 
59.70 
17.61 

II. 12 

26.1  I 
52.93 

3i.3o 

51.70 

17.81 

32.o5 

8.74 

9.57 

25.45 

9.41 

12.78 

53.53 

33.06 


*  Tliosewith  the  sign  -(-are  to  be  added  to  the  mean  time  ;  those  with  the  sign  — are  to  be  subtracted,  to  oMain 
the  apparent  time.    These  signs  must  be  changed  if  we  wish  to  obtain  the  mean  time  from  the  apparent  time. 


Page  384] 


TABLE   LVI. 


The  following  table  contains  extracts  from  the  Nautical  Almanac  for  the  year 
1836,  in  those  parts  which  are  used  in  this  work,  to  accommodate  those  who 
may  not  have  a  copy  of  that  Almanac  to  refer  to. 


Sun's  Declination,  Sfc. 


Day  of  the 
Month. 


jjpp.  l^ime. 


183G. 
May       9 

10 
March  25 
July     25 

20 
Nov.     25 

20 
April    11 

28 


THE    SUN'S 


Ri^lit 
Ascension. 


h.  ni.  s. 

3  5  2^.3i 

3  9  23.16 

0  17' 56.79 
8  19  4.19 
8  23  0.98 

16  5  5.68 

16  9  21.29 

1  19  53:95 

2  23  i5.i6 


Dilf.for 
\  hour. 


9-744 
9.769 
9.0S1 
9.866 
9.84 1 
io.65o 
10.681 
9.189 
9-484 


Declination. 


N.  17  26  27 

17  42  i3 

I  56  41 

19  37  24 

19  24  i3 

S. 20  5o  i4 
21     I  4o 

N.  8  26  o 
i4  i5    I 


Diir.  for 
1  hour. 


39.43 
38.69 
58.82 
32.95 
33.75 
28.58 
27.59 
54.^0 
46.69 


Sid.  Time 
of  the 

S.  Diam. 
passing 
the  me- 
ridian. 


in.  s. 


6.64 
6.72 
4.39 
7.16 
7.08 
9.68 
9.78 
4.74 
5.77 


Equation  of 
Time,  to  be 
applied  to  the 
Apparent  Time. 


—  3  48.73 

—  3  51.42 

-\-  6  2.10 

+  6  9.57 

+  6  9-79 

— 12  42.11 

12  23.12 

+  o  58.69 

2  40.94 


D  iff.  for 
1  hour. 


O.112 

0.088 
0.774 
0.009 
o.oi5 
0.791 
0.823 
0.665 
0.371 


I  Ex.  p.  220. 
Ex.  V.  p.  157. 
I  Ex.  I.  p.  247. 

\  Ex.  II.  p.  247. 

Ex.  p.  250. 
Ex.  p.  248. 


Moon^s  Declination,  fyc. 


Day  of  llie 
Month. 


Mean  Time. 


1836. 
April    18 
June    26 

Sept.    26 

Nov.    29 

April    23 


h. 

7 
i5 
16 

7 


3 
12 
i3 


THE    MOON'S 


Ri-'iht  Ascension. 


h.     ni.  s. 

3  52  47.56 

16  29  28.69 

16  32  8.61 

I  38  36.i6 

I  4o  35.44 

9  24  38.95 

9  26  39.34 

12  32  52.81 

12    34  57.22 


21   1 3  5 1. 7 

23  87  43.2 

23  46  3 1. 1 

8  47  27.3 

9  I  56.2 
20  4i  6.1 
20  3 1  3o.i 

o  20  5o.8 

o    4  54.5 


Diff.  declination 
for  10  minutes. 


87.98 

86.33 

144.82 

i44-27 

96.00 

96.90 

159.38 

159.58 


Ex.  p.  171 
I  Ex.  I.  p.  172. 

I  Ex.  n.  p.  172. 

Ex.  III.  p.  173. 
Ex.  p.  213. 

I  Ex.  p.  248. 


Moon's  Passage  ever  the  Meridian,  fyc. 


Day  of  the 
Month. 


Mean  Time. 


1836. 
April    18 

19 
June     26 

27 
Sept.    25 


Nov. 


28 

29 

March  17 

May     23 


JMoon's  Longitude. 


57  4  35.8 

69  I  3o.o 

240  r  4  [-7 

254  59  4i.8 

8  5o  46.8 

22  12  43.5 

124  8   16.4 

i36  2    4-2 

358  27     2.5 

149  27  19.5 


63     3  59.5 

74  57  30.9 

247  28  26.0 

262  34  28.9 

i5  34  19.6 

28  45  53.9 

i3o    4  34.6 

142     I  16. 1 

5     I  33.1 

i55  45  12.4 


Moon's  Latitude. 


N.o  38  .36.9 

1  42  46.2 
S.  o  58    3.9 

2  i4  35.9 
2  47  10.5 
I  42 


N.  5 

5 

S.  4 

N.  5 


7.3 
16.5 
53.0 

54.9 
19.7 


N. 


9.5 
9.0 
0.0 
4.6 


2  i3 

S.  I  37 

2  5o 

2  i5  23.7 

I  7  52.9 

N.5  i3  15.9 

5  7  6.6 

S.  3  4i  i4.o 

N.5  4  46.3 


Age. 

JVoon 


d. 
2.5 

3.5 
12.3 
i3.3 
i4.5 
i5.5 
19.4 
20.4 

0.1 

7-9 


Meridian 
Passage. 


h.  m. 

1  55.6 

2  43.0 
9  55.9 

10  59.8 
12  42.8 
i3  28.0 

16  33.1 

17  18.6 
o  21. 1 
6  21. 1 


|Es.  p.  170. 
I  Ex.  I.  p.  172. 
\  Ex.  II.  p.  172. 

1  Ex.  III.  p.  173. 

Ex.  I.  p.  121. 
Ex.  II.  p.  121. 


Declinations,  Right  .Ascensions,  and  Time  of  passing  the  Meridian  of  Jupiter,  Venus,  Sf'C. 


Mean  T. 


1836. 
Oct.    22 

23 
Sept.  16 

17 
May   2(5 

27 

Jan.     5 

G 

April  28 

29 


GEOCENTRIC 


Noon. 


h.  m.  s. 
9  11  ."53.47 
9  12  2-2.92 
8  41  5."). (50 
8  45  14.25 
7  8  6.,55 
7  8  57.01 
14  10  11.19 
14  10  215.19 
0  415  47.41 
0  47  27.73 


»  (  /( 
N.15  47  17. 
16  43  18. 
14  49  33. 
14  44  22. 
22  49  18. 

22  4-i    0, 
S.  10  35  2S, 

10  36  33. 
N.23  16  40, 

23  15  5S 


Log.ufd  St. 

from 
the  E.ir-h. 


0.7390451 
0.7378510 
9.7458961 
9.7516468 
0.7760038 
0.7767647 
1.0021683 
1.0014895 
0.749.5943 


0.750G908I  4 


m. 

5.4 
2.0 
59.5 
58.9 
51.4 
43.3 
10. 
7.1 
20.3 
17.0 


HELIOCENTRIC 


If 


124  56  35.6 

125  1  24.9 

26  19    7.0 

27  51  ,59.6 
112  52  34.8 
112  57  29.1 
208  31  44.1 
203  33  39.2 
110  31  53.1 
110  39  48.6 


N.O  34  45.9 
0  34  51.9 

S.2  33  25.3 
2  29  38.0 

N.O  19  13.6 
0  19  20.2 
2  28  28.7 
2  28  28.2 
0  16  9.3 
0  16  15.9 


0.7936939 
0.7237210 
9.8.599886 
9.8.599080 
0.7193880 
0.7194179 
0.989.5685 
0.9895807 
0.7185405 
0.7185707 


)  Example  I.    p.  174. 
j     Jupiter, 
i  Example  II.  p.  175. 
\      Venus. 

Example  I.   p.  215. 
\     Jupiter. 

Example  II.  p.  216. 
(      Saturn. 
\  Example,    p.  249. 
\     Jupiter. 


[Page  385 

TABLE   LVIL 

«j 

X.  i 

Latitude. 

j^   c 

=; 

Q 

IB 

^ 

0° 

5° 

10° 

15° 

20° 

25° 

30° 

35° 

40° 

45° 

50° 

55° 

60° 

G5° 

70° 

75° 

O 

0 

1 

/ 

/ 

/ 

; 

/ 

1 

/ 

/ 

1 

1 

/ 

/ 

/ 

1 

0 

0 

10 

IIO 

.4 

.4 

.4 

.5 

.5 

.6 

.7 

.8 

1 .0 

1.3 

1.8 

2.9 

no 

IC 

20 

.4 

.4 

.5 

.6 

•7 

.8 

1  .u 

1 .2 

1.6 

2.6 

20 

3o 

.4 

.5 

.6 

•7 

•9 

I .  I 

1.5 

2.3 

■ 

3o 

4o 

.5 

.6 

.8 

1 .0 

1.3 

4o 

bo 

.7 

•9 

1 .2 

5o 

bo 

•9 

60 

10 

io5 

.3 

.3 

.3 

.3 

.4 

.4 

.5 

.G 

.8 

•9 

1.2 

1.8 

3.0 

ro5 

10 

20 

.3 

.3 

.4 

.4 

.b 

.6 

•  7 

•  9 

1 .2 

t.6 

2.7 

20 

3o 

.3 

.4 

.5 

.6 

•7 

.8 

1 . 1 

1.5 

2.4 

3o 

4o 

.4 

.5 

.6 

•7 

I.O 

1.3 

4o 

bo 

.4 

.6 

.8 

1 .2 

5o 

6o 

.6 

•9 

60 

i5 

too 

.2 

.2 

.2 

.3 

.3 

.4 

.4 

.b 

.6 

.8 

I.I 

1.6 

2.9 

100 

i5 

20 

.2 

.2 

.3 

.3 

.4 

.5 

.b 

•7 

.9 

I.I 

1.6 

2-7 

20 

Jo 

.2 

.3 

.3 

.4 

.b 

.6 

.8 

r .  I 

1 .5 

2.4 

3o 

4o 

.2 

.3 

.4 

.6 

•7 

•9 

1.3 

2. 1 

4o 

be. 

.3 

.4 

.6 

.8 

1 .2 

5o 

6o 

.3 

.6 

•9 

60 

i5 

95 

.1 

.1 

.2 

.2 

.3 

.3 

.4 

.5 

.6 

.8 

I.I 

1.7 

3.0 

"95 

i5 

20 

.1 

.2 

.2 

.3 

.3 

.4 

.b 

.b 

.8 

I.I 

1.6 

2.8 

20 

3o 

.2 

.2 

.3 

.4 

.5 

.6 

.8 

I.O 

1.5 

2.5 

3o 

4o 

.2 

.3 

.4 

.b 

•7 

•9 

1.3 

2.1 

40 

5o 

.3 

.4 

.6 

.8 

I .  I 

5o 

6o 

.2 

.3 

.6 

•9 

■ 

60 

20 

90 

.0 

.0 

.1 

.1 

.1 

.2 

.2 

.3 

.4 

.6 

•  7 

I.I 

1.6 

3.0 

9c 

20 

Jo 

.0 

.1 

.1 

.2 

.2 

.J 

.4 

.b- 

.7 

I.O 

1.5 

2-7 

3o 

4o 

.0 

.1 

.2 

.3 

.3 

.5 

.6 

•9 

1.3 

2.2 

4o 

5o 

.0 

.  I 

.2 

.4 

.5 

.8 

I .  I 

5o 

(x> 

.0 

.2 

.3 

.5 

•9 

Co 

70 

.0 

.2 

.6 

I.I 

70 

20 

85 

.  1* 

.1* 

.0 

.0 

.0 

.  I 

.  I 

.2 

.3 

.3 

.5 

.7 

I.e. 

1.6 

3.1 

85 

20 

Jo 

.  I* 

.0 

.0 

.  1 

.  I 

.2 

.2 

.4 

.b 

•  7 

1 .0 

1.5 

2.7 

3o 

4o 

.  1* 

.0 

.0 

.  I 

.2 

.3 

.4 

.6 

•9 

1.3 

2.3 

4o 

bo 

.  1* 

.0 

.  I 

.2 

.0 

.b 

•7 

1 .1 

5u 

6o 

.7* 

.0 

.  I 

.3 

.b 

•9 

60 

70 

.3* 

.0 

.2 

.6 

I .  I 

70 

20 

80 

.2* 

.2* 

.1* 

.1* 

.1* 

.0 

.0 

.0 

.1 

.1 

.2 

.4 

.5 

•9 

1.5 

3.1 

«0 

20 

Jo 

.2* 

.2* 

.  I* 

.0 

.0 

.  I 

.1 

.2 

.3 

.4 

.6 

•9 

1.5 

2.8 

3o 

4o 

.2* 

.2* 

.  I* 

.0 

.1 

.2 

.3 

.4 

.6 

•  9 

1.3 

2.4 

40 

bo 

.3' 

.2* 

.  I* 

.  I 

.2 

.3 

.5 

•7 

I .  I 

5o 

6o 

.4* 

.2* 

.0 

.1 

.3 

.b 

•  9 

fio 

70 

.6» 

.3" 

.0 

,.2 

.6 

1 .2 

70 

30 

75 

.3' 

.3* 

.2* 

.2* 

.2* 

.  I* 

.  1* 

.  I* 

.  I* 

.0 

.0 

I 

.2 

.3 

.fi 

1 .2 

75 

20 

Jo 

.3' 

.3* 

.2' 

.2* 

.  I* 

.  1* 

.0 

.  I 

.1 

.2 

.4 

.6 

.   0 

1.5 

3.0 

3o 

4o 

.4* 

.3* 

.2* 

.1* 

.1* 

.0 

.1 

.2 

.4 

.5 

.8 

1.3 

2 . 5 

4o' 

bo 

.4* 

.3" 

.2* 

.1* 

.0 

.1 

.3 

.5 

•  7 

r.  I 

5o 

bo 

.6* 

.4* 

.2* 

.1* 

.1 

.3 

.5 

•9 

fio 

70 

1 .2* 

.6* 

.3* 

.0 

.2 

.6 

1 .2 

70 

20 

70 

.4* 

.4* 

.3* 

.3* 

.3* 

.3* 

.2* 

.2* 

.2* 

.2* 

.2* 

.2* 

.2* 

.2* 

.2* 

.2* 

70 

20 

Jo 

.4* 

.4* 

.3* 

.3* 

.2* 

,2* 

.1* 

.  I* 

.0 

.0 

.1 

.2 

.6 

.8 

1.5 

3.T 

3o 

40 

.5* 

.4* 

.3* 

.3* 

.2* 

.1* 

.0 

.  I 

.2 

.3 

.5 

.8 

1.3 

2.6 

4n 

bo 

.6** 

.5* 

.3" 

.2* 

.2* 

.0 

.1 

.3 

.4 

.7 

I .  I 

5n 

bo 

•9* 

.6* 

.4* 

.3* 

.1* 

.1 

.2 

.5 

•9 

fin 

70 

1 .2* 

.6* 

.3* 

.1* 

.2 

.6 

1 .2 

70 

— 









T.  ^ 

0^ 

5^ 

10^ 

15° 

20° 

25° 

30° 

35° 

40° 

45^ 

50° 

55° 

G0° 

65° 

70° 

75° 

I.  S 

c; 

Tj~. 

Latitude. 

"'c 

■^1 

49 


Pace  386] 

TABLE   LVII. 

Table  LVII.  shows  nearly  the  error  hi  longitude,  in  miles  and  tenths  of  a  mile, 
occasioned  by  an  error  of  one  mile  in  the  latitude. 

Tiais,  when  the   sun's  altitude  is  30°,  the   latitude  30°,  and  the  polar  distance 
100°,  the  error  is  8  tenths  of  a  mile. 

The  error  affects  the  longitude  as  follows :  — 

When  in  west  long.,  ^  A.  M.  ^                       C  decreased;  ^  when  the   correction 
and    the    time    is  >             <  the  long,  is  <                       i      '^   marked    *  ,    the 
found  in  column  )  P.  M.  (                       (  increased  ;  )      longitude  is 

'  increased. 
*  decreased. 

When  in  east  long.,  ^  A.  M.  ^                       T  increased;  ^  when    the  correction 
and    tlie  .time    is  >              <  the  long,  is .?                      V      is    marked    *,    the 
found  in  column  ^P.  M.  (                      (decreased;)     longitude  is 

C  decreased. 
'  increased. 

CATALOGUE   OF   THE   TABLES, 


EXAxMPLES  OF  THE  USES  OF  THOSE  WHICH  ARE  NOT  EXPLAINED  IN  OTHER 
PARTS  OF  THIS  WORK. 


TABLES  I.  and  II.  Difference  of  Latitude  and  Departure. — The  first  table  contains  the 
difference  of  latiluilo  and  departure  corresponding  to  distances  not  exceeding  300,  and  for 
courses  to  every  quarter-point  of  the  compass.  Table  II.  is  of  the  same  nature  and  extent, 
but  for  courses  consisting  of  whole  degrees.  The  manner  of  using  these  tables  is  particu- 
larly explained  under  the  article  of  Inspection,  in  the  different  Problems  of  Plane,  Middle 
Latitude,  and  Mercutor's  Sailing. 

TABLE  III.  Miridional  Parts. — An  explanation  of  this  table  may  be  found  in  pages  78 
and  79,  and  the  uses  of  it  are  shown  in  all  the  Problems  of  Mercator's  Sailing. 

TABLE   IV.     Tke  Sans  Declination. — This  table  is  explained  in  page  I5G. 

TABLE  IV.  A.  This  table  contains  the  equation  of  time  for  every  noon  at  Greenwich, 
and  is  to  be  reduced  to  any  other  hour  by  means  of  Table  VI.  A.  Thus,  suppose  the  equa- 
tion of  time  was  required  for  May  2,  IStiG,  sea  ticcount  at  10  A.  M.  apparent  time,  corre- 
sponding to  May  Id.  2'2.\\.  by  the  N.  A.  Table  IV.  A.  gives  the  equation  May  ),  at  noon. 
snh.  3m.  Cs.  and  daily  increase  7s.  Find  this  at  the  top  in  Table  VI.  A.  and  22h.  at  the 
side,  the  corresponding  correction  Gs.  increases  the  equation  3m.  Gs.  to  3m.  12s.  which  is 
the  equation  at  the  proposed  time.  This  Gs.  would  have  been  sulitractive  if  the  equation 
had  been  decreasing,  as  it  is  in  JNIarch.  The  equation  of  time  being  thus  found,  sub.  3m.  12s. 
is  to  be  subtracted  from  the  apparent  time  22ii.  as  in  the  table  to  get  the  mean  time  21h. 
56ni.  48s.  If  the  turan  time  21h.  SGm.  48s.  had  been  given  to  find  the  apparent,  it  must  be 
applied  differently  from  the  direction  in  the  table,  and  in  this  example  must  therefore  be 
added  to  21  h.  5Gm.  48s.  to  obtain  the  apparent  time  22h. 

TABLE  V.  For  reducing  the  Sun's  Declination  given  for  JVoon  at  Greenwich  to  any  other 
Time  under  any  other  Meridian. — The  manner  of  using  this  and  the  preceding  Table  IV.  is 
explained  in  pages  15G  and  157. 

TABLE  VI.  The  Sun's  Right  .Ascension. — The  Sun's  mean  right  ascension  given  in 
this  table  may  be  used  when  a  Nautical  Almanac  cannot  be  procured,  and  no  great  accuracy 
is  required.  The  table  is  to  be  entered  at  the  top  with  the  month,  and  at  the  side  with  the 
day  of  the  month. 

TABLE  VI.  A.  is  explained  in  the  precepts  for  the  use  of  Table  IV.  A. 

TABLE   VII.     Amplitudes. — This  table  is  explained  in  page  159. 

TABLE  VIII.  Right  .Ascensions  and  Declinations  of  the  principal  fixed  Stars. — This 
table  contains  the  right  ascensions  and  declinations  of  the  principal  fixed  stars,  adapted  to 
the  1st  of  Janunry,  1830,  and  the  annual  variations  in  right  ascension  and  declination  ;  by 
means  of  which  the  right  ascensions  and  declinations  of  any  of  these  stars  may  be  obtained 
for  any  time  before  or  after  the  year  1830,  by  the  rule  at  the  end  of  the  table.  '  To  illustrate 
the  method  of  doing  this,  we  shall  here  give  the  following  examples  : — 

To  find  the  right  ascension  of  a  star  at  any  time. 


EXA.MrLE  r. 

Required  tl)e  right  ascension  of  Aklcbaran,  Janu- 
ary 1,  1834.  , 
■^    '  h.  m.  s, 

R.  A   liy  the  T.-ihle  in  1830 4  2611 

Variation  in  4  years,  add M 

R.  A.  in  January.  1834 4  26  2o 

EXAMPLE  III. 

Required  the  rij-ht  ascension  of  Snica,  Utay  20, 
1836. 

Ii.  m.  s. 

R.  A.  hy  the  Table  in  1830 13  16  1.5 

Variation  in  6  years  4§  niunths,  add 20 

R.  A.  May  20,  1830 13  16  35 


EXAMPLE  IL 
Required  the  right  ascension  of  Aldeliaran,  Janu- 
ary 1,  1810. 

h.  m.  s 

R.  A.  by  the  Table  in  ia30 4  26  II 

Variation  in  20  years,  sulilract 1    9 

R.  A.  on  January  1,  1810 4  2.\   2 

EXAMPLE  rV. 

Required  the  right  ascension  of  Sirius,  November 
C,  1817.  , 

'  h.  m.  s. 

R.  A.  by  the  Table  in  1830 6  37  39 

Variation  in  13  years,  subtract 34 

R.  A.  in  January,  1817 6  37    ."i 

Variation  for  10  ukmiiIjs  aiftl  6  days,  add..  9 

R.. A.  November  6,  1817  ,,...  6  37    7 


The  sun's  right  ascension  for  any  time  may  be  found  accurately  by  the  Nautical  Almanac, 
by  taking  proportional  parts  of  the  daily  difference,  as  will  be  explained  m  the  precepts  of 
Table  XXX.  .XXXi.  But  in  cases  where  no  great  accuracy  is  required,  the  right  ascension 
may  be  obtained  within  2  or  3  minutes,  by  means  of  Table  Vi. 


388 


CATALOGUE   OF  THE  TABLES. 


To  find  the  declination  of  a  star  at  any  time. 


EXAMPLE  I. 
Required  the  declination  of  Aldebaran,  January  1, 
1834. 

Declination  by  tlie  Table  in  1830 1C°  10' N. 

Variation  in  4  years  32",  add  nearly. ...  1 

Declination  in  1834 16' 11' N. 

EXAMPLE  II L 
Required  the  declination  of  the  star  Spica,  May  20, 
1836. 

Declination  by  the  Table  in  1830 10°  16' P. 

Variation  in  U  years  4^  months 2 

Declination  May  20,  1836 10°  IS*  S. 


EXAMPLE  II. 

Required  the  declination  of  Aldebaran,  January  L 
1820. 

Declination  by  the  Table  in  1830 1G°  10' N. 

Variation  in  10  years  1'  20",  subtract  ...  1 

Declination  January  1,  1810 16°    9' N. 

EXAMPLE  IV. 
Required  the  declination  of  Sirius,  November  6, 
1807. 

Declination  by  the  Table  in  1830 1G°  29'  S. 

Var.  in  22  years  1  month  24  days,  is  sub.  2 

Declination  November  6,  1807 16°  27'  S. 


The  right  ascensions  and  declinations  obtained  by  the  preceding  calculations,  are  the 
mean  values,  to  which  must  be  applied  the  corrections  for  the  Nutation  and  Aberration 
Tables  XLII.  XLllI.  in  cases  where  great  accuracy  is  required,  as  is  now  done  in  the 
Nautical  Almanac  for  100  of  the  brightest  stars  for  every  10  days  in  the  year;  and  the 
numbers  in  tlie  Nautical  Almanac  are  to  be  preferred.  We  must  neglect  the  correction 
Part  III.,  Table  XLllL,  when  the  mean  equinox  is  used,  as  is  the  case  with  the  improved 
Nautical  Almanac. 

To  find  when  a  star  will  be  on  the  meridian,  '  • 

Rule.  Find  the  riglit  ascension  of  the  sun  and  star  in  the  preceding  Tables  VL  and 
VIIL;  subtract  the  sun's  right  ascension  from  tlie  star's,  having  previously  increased  the 
latter  by  24  hours  when  the  sun's  right  ascension  is  the  greatest;  the  remainder  will  be  the 
time  of  the  star's  coming  to  the  meridian.  If  the  remainder  be  greater  than  12  hours,  the 
star  will  come  to  the  meridian  after  midnight ;  but  if  less  than  12  hours,  before  midnight 


EXAIMPLE   I. 

At  what  time  will  Aldebaran  be  on  the  meridian, 
January  1  i  ,,  ^ 

Aldebaran's  right  ascension 4  26 

Add ^4 

28  26 
Sun's  right  ascension 18  46 

Aldebaran  souths  in  the  evening 9  40 

EXAMPLE   IIL 

At  what  time  will  the  star  Regukis  be  on  the  me- 
ridian, December  12?  jj  ^ 

Resulus's  right  ascension 9  59 

Add ^4 

33  .59 
Sun's  right  ascension 17  17 

After  midniglit 16  42 

Subtract ." ^2 

In  the  morning 4  42 


EXAMPLE   II. 
At  what  time  will    Pollux   be  on  the  meridian, 

March  31  ?  , 

li.  m. 

Pollux's  right  ascension 7  35 

Sun's  right  ascension 38 

Comes  to  the  meridian  in  the  evening 6  57 


EXAMPLE   IV. 

Required  the  time  when  the  star  Fomalhaut  comes 
on  the  meridian,  June  1.  j^ 

Fomalhaut's  right  ascension 22  48 

Sun's  right  ascension 4  36 

After  midnight 18  12 

Subtract J2 

In  the  morning 6  13 


To  find  what  star  will  come  upon  the  meridian  at  any  given  time. 

Rule.  Add  the  time  from  noon*  to  the  right  ascension  of  the  sun,  tlie  sum  (rejecting 
24  hours  when  it  exceeds  24)  will  be  the  right  ascension  of  the  star  required  to  be  known  ; 
with  which  enter  the  table  of  the  star's  right  ascension,  and  find  wh.at  star's  riglit  ascension 
agrees  with,  or  comes  the  nearest  to  it,  and  tJiat  will  be  the  star  required,  if  tlie  declination 
of  .the  star  agrees  witji  the  table,  which  may  be  ascertained  by  observing  the  meridian 
altitude  of  the  star,  the  latitude  of  the  place  being  given. 


EXAMPLE   I. 
What  star  will  be  on  the  meridian  about  10  at  night, 
January  26?  ^  ,„_ 

Sun's  right  ascension  January  2G 21)  34 

Given  time  10  hours  P.  M 10 

30  34 
Subtract .' 94 

Nearly  answers  to  Sirius 6  34 


EXAMPLE    II. 
What  star  will  be  upon  tiie  meridian  30  minuteil 
past  four  in  the  morning.  May  10? 

h.m. 

Sun's  right  ascension  May  10 3    8 

Given  time  16  hours  30  minutes 16  30 

Right  ascension  of  mid.  heaven 19  38 

Answers  nearly  to  Athair  in  the  Eagle. 


*  The   time    from    noon    must    be    reckoned    from    the    preceding  noon,  so  that  4h.  A.  M    irnst  be 
called  16h. 


CATALOGUE   OF  THE   TABLES. 


389 


EXAMPLE   Hi 

What  star  will  be  on  llie  meridian  at  Gli.  6"Jin.  P.  M 

AP"'^-'  h.m. 

Sun's  rijht  ascension  April  1 42 

Given  time 6  53 

Right  .iscension  of  the  meridian 7  35 

Answers  nwirly  to  Pollux. 


EXAMPLE   IV. 
What  star  will  be  on  the  nieridiar,  September  1,  at 
5h.  37m.  P.  M.?  ,|  ^^ 

Sun's  right  ascension  Sept.  1 10  41 

Given  time 5  37 

Right  ascension  of  the  meridian 16  18 

Answers  nearly  to  Aniares. 


In  all  the  preceding  examples,  the  right  ascension  of  the  sun  ought  to  iiave  been  calculated 
for  the  moment  of  llie  star's  passing  the  meridian,  as  will  be  more  fully  explained  in  the 
precepts  of  Tables  XXX.   XXXL 

TABLE  IX.  Sciui-diurnul  and  Semi-nocturnal  arches. — This  table  exhibits  half  the  time 
that  a  celestial  object  continues  above  the  horizon  when  the  latitude  and  declination  are  of 
the  same  name,  or  below  when  they  are  of  a  contrary  name  ;  the  former  time  being  usually 
called  the  semi-diurnal  arch,  the  latter  the  semi-nocturnal  arch  ;  whence  the  time  of  rising 
and  setting  may  be  computed  by  the  following  rules  : —  ^ 

Tojind  the  time  of  the  sun's  rising  and  setting,  and  the  length  of  the  dan  ^^"'^  night. 

Rule.  Find  tlie  sun's  declination  at  the  top  of  the  table,  and  the  latitude  in  eitlier  side 
column  ;  under  the  former,  and  opposite  the  latter,  will  be  the  time  of  the  sun's  setting  if  the 
latitude  and  declination  are  of  the  same  name,  but  the  time  of  rising  if  of  different  names. 
The  time  of  rising,  subtracted  from  12  hours,  will  give  the  time  of  setting;  or  the  time  of  setting, 
subtracted  from  \2  hours,  will  give  the  time  of  rising.  The  time  of  rising,  being  doubled,  will 
give  the  length  of  the  night;  and  the  time  of  setting,  being  doubled,  will  give  the  length  of 
the  day. 

EXAMPLE  I. 

Let  it  be  required  to  find  the  time  of  the  sun's  rising  and  setting,  with  the  length  of  the 
day  and  night,  in  latitude  51°  north,  the  Dth  of  July,  1837. 

The  sun's  declination  on  the  given  day  is  20"  52'  north,  or  21"  nearly,  under  which,  and 
against  the  latitude  ol",  stand  7h.  53m.,  the  time  of  the  sun's  setting  on  the  given  day,  in 
lat.  51°  noitli,  w'hich  doubled,  gives  15h.  4Gni.,  the  length  of  the  day  ;  and  by  subtracting 
7h.  53m.  from  12h.,  the  remainder,  4h.  7m.,  is  the  time  of  the  sun's  rising,  which  doubled  gives 
6h.  14m.  tlie  Icngtli  of  the  night. 

But,  when  the  sun  has  21"  south  declination  in  this  latitude,  tlie  time  of  sun-sgtting  be- 
comes 4h.  7m.,  the  time  of  rising  7h.  53m.,  the  length  of  the  day  8h.  14m.,  and  the  length 
of  the  night  J5h.  4Cm.,  as  was  the  case  nearly  on  the  2(Jth  of  November,  1837. 

EXAMPLE  IL 

Let  it  be  rci]Mireii  to  find  the  time  of  the  sun's  ris- 
ing, setting',  anil!  tlie  Iciigtli  of  the  day  and  night,  at 
Boston,  tlie  I2ili  of  July,  1833. 
Under  ^22",  which  is  nearlv  the  declination  on 

that  (lay,  and  against  42°  23'  or  42='  N.,  the 

laliliiile  of  Boston,  stands  the  time  of  the    h.  m. 

sun's  setting 7  25 

Subtracted  rrom  I2h.  leaves  sun-rising 4  35 

Suti-setting  clu.ililed  is  the  length  of  day 14  50 

Sun-r.siiig  douliled  is  the  length  of  night. ...     9  10 


EXAMPLE   III. 

Required  the  time  of  the  sun's  rising  and  setting, 

and  length  of  day,  in  latitude  34"  29'  S.,  iNlay  15lh,  183ti. 

Under  the  declination  18° 57'  or  19°  N.  h.m. 

and  against  the  lat.  34°  S.  stands  the  12    0 

sun's  rising 6  54 

Time  of  sun's  setting 5    G 

2 


The  length  of  the  day 10  19 

And  lih.54in.  doubled  is  length  of  night 13  49 


When  a  gr(?at  degree  of  accuracy  is  required,  proportional   parts   may  be  taken   for   th( 
minutes  of  latitude  and  declination. 


To  find  the  time  of  rising  and  setting  of  stars  lohosc  declination  docs  not  exceed  23 '28'. 

Enter  Table  IX.  and  find  the  star's  declination  at  the  top, and  the  latitude  at  tlie  side  ;  under 
the  former,  and  opposite  to  tlie  latter,  will  be  the  semi-diurnal  arch,  when  the  latitude  and 
declination  are  both  north  or  both  south;  but  if  one  be  north  and  the  other  south,  the  difference 
between  the  Tabular  number  and  12  hours  will  be  the  semi-diurnal  arch.  Find  the  time  of  the 
star's  coming  to  the  meridian  according  to  the  precepts  of  Table  VIII.,  and  subtract  therefrom 
tlxe  semi-diurnal  arch  ;  the  difference  will  be  the  time  of  rising  ;  or  by  adding  together  the 
semi-diurnal  arch,  and  the  time  of  passing  the  meridian,  the  time  of  setting  will  be  .obtained. 


,   EXAMPLE  IV. 
Required   when   the  star  Arcturus  rises  an 
December  1,  in  latitude  51°  N. 
The  time  of  the  star's  coming  to  the  meridi- 
an, or  siwithiuL',  in  the  morning,  is  nearly. 
Then   under  star's  declination  20°  nearly, 
and  against  latitude  51°,  stand 

Time  of  star's  rising  in  the  morning 

Added  gives  the  time  of  the  star's  setting. .. 

Star  sets  2r>  minutes  after  5  in  the  evening  . 


d  sets 

h.  m. 

9  30 

7  47 

1  .52 

17  2S 
12 

5  2G 

EXAMPLE  V. 

What  time  will  the  Dog-star  Sirius  rise  and  set  at 

Philadelphia,  Feb.  1.' 

Under  the  declination,  which  is  near-  h.  m 

ly  1G°  S.,  and  against  the  latitude,  12    0 

which  is  nearly  40°  N.,  stand 6  56 

Subtracted  from  ]2h.  leaves  half  the  time 
the  star  is  above  the  horizon 5    4 

The  star  conies  to  the  meridian  in  the 
evening  nearly  at 9  40 

Sum,  rejecting  12  hours,  is  the  time  of  set- 
ting in  the  morning 2  44 

Dilference  is  the  time  of  ri.sing  in  the  evening    4  3U 


390  CATALOGUE   OF   THE   TABLES. 

In  like  manner  may  the  rising  and  setting  of  any  planet  be  found  when  tlie  declination 
does  not  exceed  23"^  28',  and  the  time  of  the  passage  over  the  meridian  is  iinown. 

Suppose  it  was  required  to  find  the  time  of  Jupiter's  rising  and  setting,  August  7,  1836, 
civil  account,  in  the  latitude  of  52'--'  N. 

In  the  Nautical  Almanac  for  1836,  I  find  that  Jupiter  passes  the  meridian,  August  6d.  23h. 
llm.,or  August  7d.  llh.  11m.  A.  ]VL,  civil  account,  his  declination  being  20'-^  17'  N.,  or  nearly 
20"^.  Under  tlie  declination  20°,  and  opposite  to  the  latitude  52^,  stand  7h.  51m.,  wliich  is 
half  the  time  Jupiter  is  above  the  horizon  ;  this  subtracted  from  12h.  leaves  half  the  time 
that  he  is  below  the  horizon,  4h.  9m.  j  subtracting  7h.  51m.  from  llh.  11m.  A.  M.  leaves 
3h.  20m.  A.  M.,  August  7,  for  the  time  of  Jupiter's  rising;  and  added  to  llh.  llin.  gives 
7h.  2m.  P.  M.,  August  7,  for  the  time  of  Jupiter's  setting,  nearly. 

Suppose  it  was^  required  to  find  the  ti^ne  of  the  moon's  setting,  Maj' 2, 1836,  civil  account, 
in  the  latitude  of' 52°  N. 

In  the  Nautical  Almanac,  pages  iv.  v.,  ■we  find  that  the  moon  passes  the  meridian  May 
Id.  12h.  35)n.,  or  May  2d.  Oh.  35m.  A.  M.,  civil  account ;  her  declination  being  about  18°  S. 
Under  the  declination  18°,  and  opposite  lo  the  latitude  52°,  stand  7h.  38m.,  half  the  time  the 
moon  is  below  the^horizon.  Subtracting  this  from  12h.  we  get  half  the  time  she  is  above  the 
iiorizon,  4h.  22m.  ;  adding  this  to  Oh.  35in.  we  obtain  the  time  of  the  moon's  setting  May 
2d.  4J1.  57m.,  civil  account.  If  we  subtract  4h.  22m.,  from  Oh.  35m.  -f-  24h.,  we  get  the  time 
of  rising  May  Id.  20h.  13m.  or  May  Id.  8h.  13m.  P.  M. 

If  greater  accuracy  is  required,  you  must  find  the  time  at  Greenwich  corresponding  to 
this  approximate  time  of  her  rising  and  setting;  then  find  the  moon's  declination,  and  the 
right  ascensions  of  the  sun  and  moon  for  that  moment  of  time.  The  former  subtracted  from 
the  latter  leaves  the  corrected  time  of  the  moon's  passing  the  meridian.  With  these  data 
repeat  the  operation.  In  this  way  we  may  obtain  tlie  time  of  rising  and  setting  to  any  de- 
gree of  accuracy.  Instead  of  taking  the  difference  of  the  right  ascensions  of  tlie  sun  and 
moon,  you  may  take  the  daily  diflerence  in  the  time  of  her  coming  to  the  meridian  of 
Greenwich,  and  take  a  proportional  part  for  the  longitude  of  the  place  of  observation  (by 
means  of  Talile  XXVIII.)  and  another  proportional  part,  for  the  interval  between  the  hour 
of  passing  tlie  meridian,  and  the  time  of  rising  or  setting.* 

It  may  be  noted,  that  tlie  numbers  of  Table  IX.  were  calculated  for  the  moment  the  sun's 
centre  appears  in  the  true  horizon  ;  allowance  ought  to  be  made  for  the  dip,  parallax,  and 
refraction,  by  whicli  tlie  sun  and  stars,  when  near  the  horizon,  appear  in  general  to  be  ele- 
vated above  half  a  degree  above  their  true  place,  and  the  moon  as  much  below  her  true  place. 

TABLE  X.  For  Jinding  the  Distance  of  any  Terrestrial  Object  at  Sea. — The  explanation 
and  use  of  tliis  table  is  given  in  Problems  useful  in  Navigation,  VIII. — XII.,  pages  95,  96. 

TABLE  X.  A.  For  the  planets  is  similar  to  Table  XIV.  for  the  sun.  The  parallax  is 
found  by  entering  at  the  top  with  the  planet's  horizontal  parallax,  and  at  tlie  side  with  the 
altitude  of  the  planet ;  the  corresponding  number  is  the  parallax  of  the  planet  in  altitude. 

TABLE  XI.  Tafile  of  Projwrlional  Parts. — The  method  of  using  this  table  is  given  in 
the  prejiarations  necessary  for  working  a  lunar  observation  page  229. 

TABLE   XII.     7////cr;/AV/mc^:o/i.— Explained  in  page  154. 

TABliE   XIII.     Dip  of  the  Horizon.— Explained  in  page  154. 

TABLE   XIV.     Sun's  Parallax  in  Mltitudc. — Explained  in  page  153. 

TABLE  XV.  Augmentation  of  tlie  Moons  Semi-diameter. ^The  moon's  semi-diameter 
given  in  tlie  Nautical  Almanac  is  the  same  as  would  be  seen  by  a  spectator  su])poscd  to  be 
placed  at  tlie  centre  of  the  earth,  or  nearly  the  same  as  would  be  seen  b}'  a  spectator  on  the 
surface  of  the  eartli,  when  the  moon  is  in  the  horizon.  Now,  when  the  moon  is  in  the  zenith 
of  the  spectator  placed  at  the  surface,  her  distance  from  him  is  less  than  when  at  the  horizon 
by  a  semi-diameter  of  the  earth  ;  consequently  iier  apparent  semi-diameter  must  be  aug- 
mented in  proportion  as  the  distance  is  decreased,  that  is,  about  one  sixtieth  part,  or  16". 
At  intermediate  altitudes  between  the  horizon  and  zenith,  the  augmentation  is  proportional 
to  the  sine  of  the  altitude,  and  the  value  for  every  5°  or  10°  of  altitude  is  given  in  Table 
XV.  The  augmentation  corresponding  to  the  altitude  being  found  in  the  table,  must  be 
added  to  the  semi-diameter  taken  from  the  Nautical  Alnianac  for  the  time  of  observation 
reduced  to  Greenwich  time,  as  is  explained  in  the  preparations  necessary  for  working  a 
lunar  observation. 

TABLE   XV 1.     Dip  ofj/te  Sea  at  Different  Distances  from  the  Observer. — Explained  in- 
page  155. 

TABLE  XVII.  For  finding  the  Difference  hdioccn  60'  and  the  Correction  of  the  JlUitude 
of  a  Star' or  Planet,  fur  Parallax  and  Refraction ;  also  the  corrcsjionding  Logarithm.— The 
first  pag(!  of  tliis  table  is  to  be  used  for  a  star,  or  for  the  planets  Jupiter  and  Saturn,  whose 
parallax  is  small.  In  other  cases,  that  page  of  the  table  is  to  be  used,  wliich  contains,  at  the 
top,  the  horizontal  parallax  of  tiie  planet,  or  comes  the  nearest  to  it;  the  tables  being  cal- 
culated for  every  5"  of  iiorizontal  parallax,  from  0"  to  35". 

TABLE  XV III.  For  finding  the  Difference  beticecn  the  Correction  of  the  Snn's  Altitude 
for  Parallax  and  Refraction  and  60',  also  a  Logarithm  corresponding  thereto. — The  manner 
of  tailing  the  numbers  from  the  two  preceding  tables,  and  the  uses,  to  wliich  they  may  be 
applied,  are  explained  in  the  preparations  necessary  for  working  a  lunar  observation, 
page  230.  &c. 

TABLE  XIX.     For  finding  a  Correction  and  Logarithm  vscd  in  the  First  Method  of  work 

*  In  strict  iie-ss,  llils  liisl  correctiim,  found  by  the  tahle,  o'lglit  to  he  deireased  in  the  ratio  of  2!li.  to  211i.  it" 
creased  Uy  llie  daily  diiieieiite  of  llie  lime  of  tlie  moon's  jjassing  tlie  meridian. 


CATALOGUE   OF   THE   TABLES. 


a9i 


tng  a  Lunar  Observation. — The  correction  found  in  this  table,  being  subtracted  from  5!)'  42" 
will  leave  a  remainder  equal  to  the  correction  of  the  moon's  altitude  for  parallax  and  re- 
fraction. It  will  be  unnecessary  here  to  point  out  the  method  of  taking  out  tliis  correction, 
as  it  is  fully  explained  in  the  first  pages  of  the  table.  It  may  not.,  however,  be  amiss  to 
observe,  that,  after  constructing  the  logarithms  of  this  table,  it  was  concluded  to  subtract 
therefro)n  the  greatest  correction  of  the  Table  C  corresponding,  in  order  to  render  those 
corrections  additive.  Thus  the  logarithm  corresponding  to  the  alt.  30"-'  and  her.  par.  54', 
was  found  at  first  to  be  2372  ;  and  for  the  hor.  par.  54'  10'  the  correction  was  2358  ;  so  that 
if  these  numbers  had  been  published,  the  correction  for  seconds  of  parallax  would  have  been 
subtractive  ;  but  as  this  would  have  been  inconvenient,  it  was  thought  expedient  to  subtract 
from  each  of  the  numbers  thus  calculated,  the  greatest  corresponding  correction  of  Table  C, 
which  in  the  preceding  example  is  12;  by  this  means  the  above  numbers  were  reduced  to 
23t30  and  234(J  respectively,  and  the  corrections  of  Table  C  were  rendered  additive.  In  a 
similar  manner  the  rest  of  the  logarithms  of  the  table  were  calculated.  It  is  owing  to  this 
circumstance  tliat  the  corrections  in  Table  C  for  0"  of  parallax  are  greater  than  for  any  other 
number.  Similar  metliods  were  used  in  calculating  the  other  numbers  of  this  table,  and  in 
arranging  the  Tables  A  and  B. 

TABLE  XX.  Third  Correction  of  the  Apparent  Distance. — The  manner  of  finding  the 
correction  from  this  table  is  explained  in  the  first  method  of  correcting  the  apparent  distance 
of  the  moon  from  the  sun,  page  231 ;  and  also  at  the  bottom  of  the  table. 

TABLE  XXI.  To  reduce  Longitude  into  Time,  and  the  contrary. — In  the  first  column  of 
this  table  are  contained  degrees  and  minutes  of  longitude,  in  the  second  the  corresponding 
hours  and  minutes,  or  minutes  and  seconds  of  time ;  the  other  columns  are  a  continuation 
of  the  first  and  second  respectively.  The  use  of  this  table  will  evidently  appear  by  a  few 
examples. 


EXAMPLE  I. 
Required  tlie  time  (.orresiiondin;;  to  .50'  3!'. 

Ii.  in.  s. 

Opposite  50°  in  rol.  I  is 3  •i^  i) 

31' 2   1 


Soiijriil  time 3  2>  4 


EX.\.MI'LE  II. 

Required  the  de^'rees  and  minutes  corresponding 
to  till.  33jn.  203. 


Opposite  Ch.  32m.  Os in  col.  4  is. 

1     20    in  cut.  2  is. 


98°  0 
20 

98  20 


TABLE  XXII.  Proportional  Logarithms. — These  logarithms  arc  very  useful  in  finding 
'Jie  mean  time  at  Greenwich  corresponding  to  the  true  distance  of  the  moon  from  the 
sun  or  star,  as  is  explained  in  the  examples  of  working  a  lunar  observation.  They  may  be 
also  used  like  common  logarithms,  in  working  any  proportion  where  the  terms  are  given  in 
degrees,  minutes,  and  seconds  ;  or  in  hours,  minutes  and  seconds,  as  in  the  example  of 
taking  a  lunar  observation  by  one  observer.  The  table  is  extended  only  to  3°  or  3h  .;  and  if 
iny  of  the  terms  of  a  given  proportion  exceed  3°  or  3h.,  you  may  take  all  the  terms  one  grade 
lower  ;  that  is,  reckon  degrees  as  minutes,  minutes  as  seconds,  &c.,  and  work  the  proportion 
as  before  ;  observing  to  write  down  the  answer  one  grade  higher ;  that  is,  3-ou  must  esti- 
mate minutes  as  degrees,  seconds  as  minutes,  &c.  Instead  of  taking  all  the  terms  one  grade 
lower,  3'ou  may  change  two  of  the  terms  only,  viz.  one  of  the  middle  term?  and  one  of  the 
extreme  terms  ;  thus  the  1st  and  3d  or  the  1st  and  2d  may  be  taken  one  grade  less,  and  the 
fourth  term  v^fill  be  given  correctly  ;  but  if  the  fourth  term  be  taken  one  grade  less,  you 
must,  after  working  llie  proportion,  write  it  one  grade  higher,  as  is  evident.  To  illustrate 
'.his,  we  shall  give  the  following  examples: — 

EXA.MPLE   II. 
If  the  sun's  declination  i  lianKcs  Ifi'  13"  in  24  hours, 
liow  much  will  It  chiuige  in  81).  2m.  ? 
Here  the  1st  and  3d  terms  must  he  taken  one  grade 

less. 
As       21m.  Os...  Aiith.Comp....Proii.  Log.  9.121'J 

Is  to     W  Id" Prop.  Loir.   1.042fi 

So  is     8m.  23 Prop.  Log.   1 .3.^04 

To         5*  28" ". Prop.  Log.  1.5 1 79 

EXA.MPLE  IV. 
ir  in  Ifim.  the  sun  rises  3°  27',  how  much  will  it  rise 
in  3m.  10s.  ? 
Here  the  2d  and  4tli  terms  must  be  taken  one  grade 
less. 

As       lOm.  Os Arith.  Comp Prop.  Log.  8.94S*f 

Is  to     a*  27" Prop.  Log.  1.7173 

So  is    3m.  10s Prop.  Log.   1.7547 

To        0'  41" Prop.  Log.  2.421C 

Which,  taken  one  grade  higher,  is  41',  the  answer 
required. 

TABLE  XXIII.  For  finding  the  Latitude  Inj  two  Altitudes  of  the  Sun. — The  manner  of 
using  this  table  is  explained  in  the  examples  of  double  altitudes  given  in  pages  165 — 189. 

TABLE  XXIV.  JS'atural  Sines. — This  table  contains  the  natural  sine  and  cosine  for 
every  minute  of  the  quadrant  to  the  radius  lOOOOD,  and  is  to  be  entered  at  the  top  or  bottom 
with  the  degrees,  and  at  fhe  side  niarkcd  M.  with  the  minutes:  the  corresponding  numbers 


EXAMPLE  I. 

If.in   I5in.  10s.  of  lime  the  sun   rises  2°  40',  how 
nuch  will  It  rise  in  3iii.  10s.  at  the  same  rate? 
As      15m.  lO.s... Arith.  Comp.... Prop.  Log.  8.02.^ 

Is  to    2°  40' Prop.  Log.     .05 12 

So  is    3m.  10s Prop.  Log.   1.7.i47 


To 


33'  24" Pro)).  Log. 


.731.' 


EXAMPLE  III. 
If  in    I2li.  the  moim's  hmgitiide  vanes  7°  1',  what 
vill  it  vary  in  4!i.  20m. .' 
Here  all  the  terms  must  be  taken  one  grade  less. 

As      12m.  Os Arith.  Comp Prop.  Log.  8.82.39 

l.-<  to    T   I" Prop.  Log.  1.4091 

So  is   4m.  203 Prop.  Log.  1.0185 

To       2' 32"  2'" Prop.  Log.  1.8515 

Whii  h,  taken  one  grade  higher,  is  2°  32'  2",  the  an- 
swer reipiired. 


392 


CATALOGUE   OF  THE   TABLES. 


will  be  the  natural  sine  and  cosine  respectively,  observing  that  If  the  degrees  are  found  at 
the  top,  the  name  sine,  cosine,  and  M.,  must  also  be  found  at  the  top,  and  the  contrary  If 
tlie  degrees  are  found  at  the  bottom.  Thus  4336G  is  the  natural  sine  of  25°  42',  or  the 
cosine  of  64°  18'.  . 

We  have  given  in  tnis  edition  of  the  present  table,  in  the  outer  columns  of  the  margin, 
tables  of  proportional  parts,  for  the  purpose  of  finding  nearly,  by  inspection,  the, proportional 
part  corresponding  to  any  number  of  seconds  in  the  proposed  angle ;  tlie  seconds  being 
found  in  the  marginal  column  marked  M.,  and  the  correction  In  the  adjoining  cohnnn. 
Thus,  if  we  suppose  that  it  were  required  to  find  the  natural  sine  corresponding  to  25°  42'  19" ; 
the  difference  of  the  sines  of  25° 42'  and  25°  43' is  26;  being  the  same  as  at  the  top  of  the  left- 
hand  column  of  the  table  ;  and  In  this  column,  and  opposite  to  19",  In  the  column  M.,  is  the 
correction  8.  Adding  this  to  the  above  number  4336G,  because  tiie  numbers  are  iiicrcusiiin-, 
we  get  43374  for  the  sine  of  25°  42'  19".  In  like  manner,  we  find  the  cosine  of  the  same, 
angle  to  be  90108  —  4  =  90104,  using  the  riglit-hand  columns,  and  subtracting  because 
the  numbers  are  decreasing ;  observing,  however,  that  the  number  14  at  tlie  top  of  this 
column  varies  1  from  the  difference  between  the  cosines  of  25°  42'  and  25°  43',  wiiich  is 
only  13 ;  so  that  the  table  may  give  in  some  cases  a  unit  too  much,  between  tlie  angles 
25°  42'  and  25°  43' ;  but  this  is,  in  general,  of  but  little  Importance,  and  wlien  very  great 
accui"  V  is  required,  the  usual  method  of  proportional  parts  is  to  be  resorted  to,  using  tlie 
actuai  labular  difference.  Similar  tables  of  proportional  parts  are  Inserted  in  this  edition  of 
Tables  XXVI.  XXVII.  for  the  like  purpose. 

TABLE  XXV.  Logarithmic  Sines,  Tangents,  and  Secants  to  every  Point  and  Quartet- 
Point  of  the  Compass. — This  table  is  to  be  used  instead  of  Table  XXVII.  when  tlie  course 
is  given  In  points.  The  course  is  to  be  found  In  the  side  colunm,  and  opposite  tlierelo  will 
be  tlie  log.  sine,  tangent,  &c. ;  the  names  being  found  at  the  top  whon  the  course  Is  less 
than  4  points,  otiierwise  at  the  bottom. 

TABLE  XXVI.  Logarithms  of  Numbers. — The  explanation  and  uses  of  this  table  are 
given  in  tlie  article  treating  on  logarithms  in  the  body  of  the  work,  pages  28 — 33. 

TABLE  XXVII.  Logarithmic  Sines,  Tangents,  and  Secants. — Th}s  table  is  explained  in 
the  corresponding  article  In  the  body  of  the  work,  pages  33 — 35. 

TABLE  XXVIII.  Fur  reducing  the  Time  of  the  Moon's  Passage  over  the  Meridia.ji  of 
Greemcich,  to  the  Time  of  her  Passage  over  any  other  Meridian. — The  manner  of  doing  this 
Is  explained  In  the  corresponding  part  of  the  body  of  the  work,  page  170. 

TABLE  XXIX.  Correction  of  the  Moons  Mtitude  for  Parallax  and  Refraction. — The 
mean  correction  of  the  moon's  altitude  is  given  In  this  table  for  every  degree  of  altitude 
from  10°  to  90°.     The  manner  of  using  this  table  Is  explained  in  pages  172,  173. 

TABLES  XXX.  XXXI.  For  finding  the  Suns  Right  Ascension  and  Declination,  the 
Equation  of  Time,  and  the  Moon's  Right  Ascension. — The  uses  of  these  tables  will  be  seen 
by  the  following  examples,  the  values  for  apparent  noon  being  taken  from  the  Nautical 
Almanac,  together  with  the  horary  motions. 

EXAMPLE  I. 

Required  tlie  sun's  right  ascension  in  1836,  May  Id. 
6h.  35in.,  apparent  time,  astronomical  account,  at 
Greenwich. 

Here  the  horary  motion  by  N.  A.  is  9s.551. 

h.  m.   s. 

R.  A.  May  1,  at  noon,  by  N.  A 2  34  39.6 

Hor.  motion  (Sh.  X  9s.55l 57.3 

For  35m.  in  Table  XXX 5.5 


R.  A.  May  Id.  6h.  35m 2  35  42.4 

EXAMPLE  in. 

Required  the  moon's  right  ascension  in  1836,  Sept. 
lOd.  8h.  '20m.30s.,  J«can  time,  astronomical  account,  at 
Greenwich.  h.  m.    s. 

By  N.  A.  Rt.  As.  Sept.  ind.  9h.  is  ll'  16    9.96 
Sejit.  10d.8h.  isll  14  12.90 
Horary  motion  in  Rt.  Ascen.  ..        1  57.06=117".06 
Proportional  part  fur  20ni.  30s. 

Table  XXX 40" 

Add  to  R.  A.  Se|)t.  lOd.  8h 11  14  13  nearly. 

Gives  D's  Rt.  Asc.  Sept.  lOd. 
8h.  20m.  30s 11  14  53 

EXAMPLE  V. 

Required  the  sun's  declination  in  1836,  May  Id.  Gh. 
3om.  apparent  time,  astronomical  account,  at  Green- 
wich. 

Here  the  honn'  motion  by  N.  A.  is  44".85. 

Declination  May  l,at  noon,  by  N.  A.  15°  10'  19"  N. 

Hor.  motion  (ill.  X  44".85 4   29 

For  35m.  in  Table  XXX 26 

15°  15'  14" 
If  the  declination  had  been  decreasing,  the  horary 
motion  would  be  .tiditractice  instead  oi additive,  as 
in  the  above  example. 


EXAMPLE  n. 

Required  the  equation  of  time  in  1836,  July  9d.  8h 
20m.  apparent  time,  astronomical  account,  at  Green- 
wich. 
Here  the  horary  motion  by  N.  A.  is  Os.364. 

ni.  s. 
Equation  of  time  July  9,  at  noon,  by  N.  A.  -\-  \  49.3 

Hor.  motion  8h.  x  Os.364 2.9 

For  20m.  in  Table  XXX .1 

Right  Ascension,  1836,  July  9d.  8h.  20m.   +  4  52.3 

EXAMPLE  IV. 

Required  the  moon's  right  ascension  in  1836,  May 
lid.  17h.  35m.  36s.  incan  time,  astronomical  account 
at  Greenwich.  ,,  ^   ^ 

By  N.  A.  Rt.  As.  May  lid.  18h.  is  0  55  40.89 
May  lid.  17h.  is  0  53  48.72 
Ilor.  motion  in  Riglit  Ascension        1  52.17=:]  12    J'' 
Proportional   part  for  35m.  36s. 

Table  XXX.,  66"  = 106 

Add   to  Right  Asc.   May   lid. 

17h.  by  N.  A 0  53  49  nearly. 

Gives    C  's   Rt.  Asc.  May  lid. 

17h.  35in.  36s 0  54  55 

EXAMPLE   VI. 

Required  the  moon's  declination  in  1836,  Sept.  lOd. 
8h.  20m.  30s.    mean  time,  astronomical  account,   at 
Greenwich. 
Here  the  motion  in  declination  for  10m.  is  by  N.  A 

140i'.07. 
Motion  for  20m.  is  2  X  140i'.07=2SO".14 
Table  XXX.  with  140"  at  top, 
and  30s.  at  side,  in  col.  M. 
the  correction,  divided  by  10,  is  7     0 

Motion  in  declina.  in  20m.  303.  287".  1  =  4' 47".  1 
Sub.  from  declination  Sept.  lOd  8h.  9  32  13".3 
!■ '.-,  dcclinn.  Sept   lOd.  8h.  20m.  30s.   9°  27'  26".2  N 


CATALOGUE  OF  THE  TABLES. 


393 


EXAMPLE  VII. 
R'-n)i"ired  tlie  moon's  declination   in  1S36,  May  Ud.  171i.  3jni.  3os.  mean  time,  astrononiital  aico.inl,  a( 
Greenwich. 
Here  llie  motion  in  declination  for  10m.  is  by  N.  A.  143".02. 

Motion  for  30m.  is  14;;'i.(i2  X  3  = 429''.  I 

5ni.  is  143'i.02x  0.5 71  .5 

Tab.  XXX.  143"  at  top,  and  3Gs.  at  side  in  col.  M.  the  corr.  divided  by  10  is      8  .6 


Mocion  in  declination  is 509". 2 

Add  to  declination  May  Ud.  17h.  by  i\.  A 

j)'s  declination  May  I  Id.  17h.  3jin.  Sis 

Here  the  correction  8'  29".2  is  added,  because  the  declination  is  increasing. 


=   &  29i'.2 
2     19  25  .9 

2    27'  55"  1  N. 


If  we  wish  to  find  accurately  the  time  that  any  star  comes  to  tlie  meridian,  or  the  time  of 
rising  or  setting,  we  must  talie  the  sun's  right  ascension  for  noon  at  Greenwich,  from  the 
Nautical  Almanac  ;  then  the  star's  right  ascension  from  Table  VIIL,  and  with  these  find 
the  appro.ximate  time  of  rising,  setting,  or  coming  to  the  meridian,  by  the  method  already 
given  in  the  precepts  for  using  Tables  VIIL  and  IX.  Then  calculate  the  sun's  right  ascen- 
sion for  this  approximate  time,  and  repeat  the  operation  till  the  assumed  and  calculated 
times  agree,  and  we  shall  have  the  true  time  required. 

To  explain  this  method,  we  shall  give  the  following  examples : — 

To  Jind  the  time  tvhen  a  star  comes  to  the  meridian. 


EX.\MPLE  I. 
At  what  time  was  Aldcharan  on  the  merid 
a  place  in  the  longitude  of  70°  50'  W.,  Jan.  2, 
sea  arconnt? 
Jan.  2,  sea  account,  is  Jan.  1,  N.  A.,  on 
which  day  the  sun's  R.  A.  at  noon  at     h. 

Greenwich  was 18 

Aldeliaian's  R.  A 4h.  2Gm.  32s. 

Add 24 


ian  of 

183G, 


m.  s. 
44  19 


28  26  32 


D.'tTerenre  is  the  appro.vimate  time 9  42  13 

Now,  calculating  tlie  sun's  R.  A.  for  this 

time  in   the  long,  of  70°  50'  W.  from  h.  m.  s. 

Greenwich,  we  find  it  was 18  4t>  58 

Aldebaran's  R.  A.-|-24h 28  26  32 

App.  time  of  coming  to  the  meridian 9  39  34 


EXAMPLE  IL 

At  what  time  was  Pollux  en  the  meridian  of  a  place 
in  the  longitude  of  70°  46'  VV.,  March  31,  1836,  sea 
account .-' 

March  31 ,  sea  account,  is  March  30,  N.  A., 
on  which  day,  at  noon,  the  sun's  right    h.  m.  s. 

ascension  was 0  36    G 

This,  subtracted  from  R.  A.  of  Pollux 7  35  17 

Gives  the  approximate  time  of  southing...  6  59  II 
R.  A.  for  this  time  in  long.  70°  46'  VV. 

from  Grecnw ich 0  37  53 

Right  ascension  of  Pollux 7  35  17 

Diff.  is  app.  time  of  comingto  the  meridian.  6  57  24 


To  Jind  the  time  of  rising  or  setting  of  a  star. 

Rule.  Enter  Table  IX.  with  the  declination  of  the  star  at  the  top,  and  the  latitude  of  the 
place  at  the  side ;  tlie  corresponding  number  will  be  the  time  of  the  star's  continuance  above 
the  horizon,  when  the  latitude  and  declination  are  of  the  same  name  ;  but  if  they  are  of  dif- 
ferent names,  the  tabular  number  subtracted  from  12h.,  will  be  tiie  timeof  continuance  above 
the  horizon.  Add  this  time  to  the  star's  right  ascension,  if  we  wish  to  find  the  time  of  set- 
ting ;  but  subtract  the  former  from  the  latter  if  we  wish  the  time  of  rising.  From  this  sum 
or  difference  subtract  the  sun's  right  ascension*  corrected  for  tlie  longitude  of  the  ',)lace;  the 
remaijider  will  be  the  approximate  time  sought.!  Enter  Table  XXXI.  with  the  distance  of 
this  approximate  time  from  noon,  and  the  horary  variation  of  the  sun's  right  ascension  :  the 
correction  corresponding  is  to  be  added  to  the  approximate  time  in  the  forenoon,  but  sub- 
tracted in  the  afternoon,  and  we  shall  have  the  corrected  time  of  rising  and  setting. 


EXAMPLE  L 
At  what  time  did  the  star  Aldebaran  set  May  24, 
'336,  sea  account,  in  the  latitude  of  38°  53'  N.  and  the 
longitude  of  77°  VV.,  or  oh.  8m.  VV.? 

The  star's  declination  was  1G°  10'  N.,and  the  lati- 
tude 38°  53'  N.,  corresponding  to  which  in  Table 

IX.  is Gh.  54m. 

Star's  right  ascension 4    26 

Sum 11    20 

May  24,  sea  ace,  or  May  23  by  N.  A.  at 

noon,  sun's  R.  A 4h.   Im.  Hor.  var.  10s. 

Corr.  for  long.  5h.  8m.  VV..  1 

Sum,  subtract 4      2 

Remains  approximate  time  of  setting 7     18 

Corr.  in  Tab.  XXXI.  for  7h.  20m.,  sub....  1 

Corrected  time  of  setting,  P.  M 7     17 


EXAMPLE  II. 

At  what  time  did  the  Dog-Star  Sirius  rise  in  tlit; 

latitude  39°2U'N.,  and  tlie   longitude  of  76°  50'  VV 

=  ,'.h.  7m.  20s.  VV.,  Jan.  2,  1836,  sea  account.' 

The  star's  declination  is  16°  29'  S.,  and  the  latitude 

is  39°  20'  N.,  corresponding  to  which  in  Table  IX. 

is  nearly 6h.  56m 

Which  subtracted  from 12      0 


Leaves  the  time  of  the  star's  being  above 

the  horizon 5      4 

Subtract  from  star's  R.  A 6    38 

Remainder 1     34 

Add 24 


Sum 25  34 

Jan.  2,  sea  ace.  or  Jan.  1,  by  N.  A.  at 

noon,  sun's  R.  A 18Ii.  44m.   Hor.  var.  lis 

Corr.  for  long.  5h.  7m.  20s. VV.      1 

Subtract  the  sum 18  45 

Remains  approxim.  time  of  rising 6  49 

Corr.  in  Tab.  XXXI.  for  Gh.  49m.,  sub..  1 

Corr.  time  of  rising  in  the  afternoon 6  48 


icreasing  the  number  from  which  the  subtraction  is  to  be  made,  by  24  hours,  when  necessary. 

ejecting  2!  Iioiirs  when  it  exceeds  24  hcurs.     If  the  time  of  rising  or  setting  be  more  than  12'h.,  it  wll  bo 

inidniirht :  but  if  less  than  12h..  it  v^i'l  Ke  before  midnirlit. 


*  Increasi 

T  I^ejectins  ^-i  iniuis  ^vum  ii.  ca*.  ecus  i;t  iiu'ird.     11  me  nine  oi 
fiRer  midnight ;  but  if  less  than  12h.,  it  v^i'!  Ke  before  midnight. 

50 


;j«J4  CATALOGUE   OF   THE   TABLES. 

TABLE   XXXIL      Variation  of  the  Sun's  Allitvdc  in  one  J\Ihtutefroin  JVoon. 

TABLE  XXXllL  To  reduce  the  JVumbcrs  of  Table  XXXII.  to  other  given  Intervals  oj 
Time  from  A"oo7i. 

The  method  of  using  the  two  preceding  tables  is  explained  in  the  examples  of  finding  the 
latitude  by  one  altitude  taken  near  noon,  given  in  the  body  of  the  work,  pages  201 — 20X 

TABLE  XXXIV.  Errors  arising  from  a  Deviation  of  V  in  the  Surfaces  of  the  Central 
Mirror.  This  table  shows  the  error  arising  in  measuring  an  angle  by  an  instrument  of 
reflection  from  a  deviation  of  1'  in  the  parallelism  of  the  surfaces  of  the  central  mirror,  the 
line  of  intersection  of  those  surfaces  (produced  if  necessary)  being  perpendicular  to  the  plane 
of  the  instrument.  If  the  line  of  intersection  be  inclined  to  that  plane,  the  numbers  in  the 
table  must,  in  general,  be  decreased  in  proportion  to  the  sine  of  the  angle  of  inclination. 

The  second,  third,  and  fourth  columns  of  the  table  are  calculated  upon  the  supposition 
that  the  surface  of  the  horizon  mirVor  is  inclined  80°  to  the  axis  of  the  telescope,  or  that  the 
angle  intercepted  between  the  raj'  incident  on  the  horizon  glass  and  the  corresponding  re- 
flected ray  passing  through  the  telescope  is  20^,  which  is  the  case  in  circular  instruments 
of  De  Boiida's  construction,  and  on  this  supposition  the  errors  of  an  instrument  in  measur- 
ing different  angles  may  be  ascertained  by  the  rules  in  pages  136  and  143  ;  when  the  inter- 
cepted angle  is  greater  or  less  than  2(y-^,  which  is  the  case  in  most  sextants  and  quadrants, 
the  error  in  any  measured  angle  corresponding  to  an  inclination  of  the  surfaces  of  1',  may 
be  obtained  as  follows  : — 

Find  in  the  first  column  the  intercepted  angle,  and  the  sum  of  that  angle  and  the  observed 
distance  ;  take  the  corresponding  corrections  from  column  5th,  and  their  difference  will  be 
the  sought  correction. 

In  a  circular  instrument  yon  must  find  in  tl.'e  side  column  the  sum  and  the  difference  of 
tbe  intercepted  angle  and  observed  angle,  and  taiie  out  the  corresponding  corrections  from 
column  5th  :  half  their  difference  will  be  the  sought  correction.  Having  thus  found  the 
correction  corresponding  to  1',  you  may  find  the  correction  for  other  angles  as  in  pages  1"36 
and  143. 

TABLE  XXXV.  Correction  for  a  Deviation  of  the  Telescope  of  an  Instrument  of  Re- 
flection from  the  Parallelism  to  the  Pinnc  of  the  Instrument. — The  uses  of  this  table  are 
explained  in  pages  135,  and  143. 

TABLE  XXX  VI.  Correction  of  the  Mean  R  fraction  for  Various  Heights  of  the  Barome- 
ter and  Thermometer. — The  use  of  this  table  is  explained  in  page  154. 

TABLE  XXXVII.  Latitudes  and  Longitudes  of  the  Fixed  Stars. — This  table  contains 
the  latitudes  and  longitudes  of  the  principa:!  fixed  stars,  adapted  to  the  beginning  of  the 
year  1830,  with  the  annual  variations  for  precession  nud  the  secular  equation,  by  which  the 
mean  values  at  any  time  may  be  obtained,  in  lilie  manner  as  the  right  ascensions  and  decli- 
nations are  from  Table  VIII.  ;  by  adding  the  corrrction  of  longitude  after  1830,  subtracting 
before  1830,  and  applying  the  correction  of  latitude  with  the  same  sign  as  in  the  table  after 
1830,  but  with  a  contrary  sign  before  1830. 

EXAMPLE  I. 
Required  the  longitude  and  latitude  of  a  Pegasi,  July  IG,  1828. 

La!)g.  by  Table  XXXVII Us.  21°  07' 05"   |   Latitude  by  Table  XXXVII "lO"  21'  45"  N. 

Vajialion  1  year,  54ni.,  sub 1   13        Variation  1  year,  5.\ni.,  sub 0 

Loig.  July  16,  1828 II    21    05  52     |  Latitude  July  16,  1828 19    24   45   N. 

■     EXAMPLE   II. 
Required  the  longitude  and  latitude  of  a  Pegasi,  July  1,  1832. 


Long  by  Table  XXXVII lis.  21°  07'  05" 

Variation  21  years,  add 2     5 

Long.  July  1,  1832 11    21    09    10 


Latitude  by  Table  XXXVII 19'  24'   45"  N 

Variation  2.|  years,  add 0 

Latitude  July  I,  1832 19    24    45   N 


The  latitudes  and  longitudes,  thus  obtained,  are  the  mean  values.  When  great  accuracy 
is  required,  the  corrections  for  the  equation  of  the  equinoxes,  Table  XL.  and  aberration, 
Table  XLI.  must  be  applied. 

TABLE  XXXVIII.  Reduction  of  Latitude  and  Horizontal  Parallax. — This  table  con- 
tains the  corrections  to  be  subtracted  from  the  latitude  of  the  place  of  observation,  and  from 
the  horizodtal  parallax  of  the  moon,  given  in  the  Nautical  Almanac,  in  calculating  eclipses 
of  the  sun  or  occultations.  Thus,  if  the  latitude  of  the  place  was  40'-',  and  the  moon's 
horizontal  parallax  57',  the  correction  of  latitude  would  be  nearly  — 11'  18",  and  that  of 
parallax  — 4".7,  so  that  the  reduced  latitude  would  be  39^  48'  42",  "and  the  reduced  parallax 
56  55". 3.  These  values  are  to  be  used  in  occultations  ;  but  in  eclipses  of  the  sun,  this 
parallax  is  to  be  further  decreased  by  8". 6  for  the  sun's  parallax.  When  the  latitude  is  not 
given  exactly  in  the  table,  the  two  nearest  numbers  must  be  found,  and  a  proportional  part 
of  their  difference  is  to  be  applied  to  one  of  the  numbers,  as  usual.  In  calculating  this 
table,  the  ellipticity  of  the  earth  was  supposed  equal  to  ^otj-,  as  in  the  third  edition  of 
La  Lande's  Astronomy,  and  in  Vince's  Astronomy.  This  value  differs  but  little  from 
-^-^■-^  and  ^(j"5'(7"5>  deduced  by  La  Place  from  two  lunar  equations  in  the  third  volume  of 
his  immortal  v/ork.  La  Micanujne  Ciliste.  In  the  second  volume  of  the  same  work,  he 
calculated  the  ellipticity  to  be  ^-^g  from  the  lengths  of  pendulums  observed  in  different  lati- 
tudes :  this  calculation  corrected  for  a  small  mistake  in  the  numerical  co-efficient  of  i/  in  the 


CATALOGUE    OF   TilC    TAJ5LES.  3<)5 

tenth  of  his  equations  A"  becomes  3-^5,  which  does  not  clitrer  very  miicli  from  tlie  value 
assumed  in  tiiis  tal)le. 

TABLE  XXX IX.  Merration  of  the  PUincts. — This  table  contains  the  aberration  of  the 
planets,  to  be  applied  to  tlie  true  longitude  or  latitude,  with  the  same  sign  as  in  tiie  table. 
The  argument  at  tb.e  side  is  the  elongation  of  the  planet  from  the  sun  ;  that  is,  the  dilference 
of  their  geoc-entric  longitudes,  or  its  supplement  to  3(J0  \  Thus,  on  July  1'.',  IS'JO,  the  longi- 
tude of  the  sun  was  3s.  2G°  38',  the  geo.  long,  of  Venus  4s.  13-^  23',  their  difii-icnce  lU^  45' 
is  the  elongation  or  distance  from  the  inferior  conjunction,  corresponding  to  which  is  the 
aberration  -|-  3"  to  be  aiiplied  to  tlie  true  longitude  given  by  the  tables  to  obtain  the  ajiparent 
longitude.  The  aberration  of  IMercury  is  given  at  its  greatest,  least  and  mean  distances 
from  tlie  sun.  At  the  intermediate  places,  a  proportional  part  of  the  differences  of  the 
nearest  tabular  numbers  must  be  applied. 

TABLES  XL.  and  XLl.  Equation  of  the  Equinnzes  and  Merration  in  Longitude. — 
Table  XL.  contains  the  equation  of  the  equino.xes  in  longitude  common  to  all  the  heavenly 
bodies.  The  argument  is  the  longitude  of  the  moon's  ascending  node;  the  signs  of  longitude 
being  found  at  the  top  or  bottom,  and  the  degrees  at  the  side,  the  corresponding  number 
with  its  sign  is  the  equation  of  the  equinoxes  in  longitude. 

Table  XLI.  contains  the  aberration  of  the  stars  in  longitude  and  latitude,  to  be  calcu- 
lated by  the  rules  at  the  bottom  of  the  tables ;  the  signs  of  the  argument  being  found  at 
the  top,  and  the  degrees  at  the  side,*  taking  proportional  parts  for  minutes.  The  corrections 
of  longitude  found  in  these  tables  are  to  be  applied,  with  their  signs,  to  the  mean  longitude 
found  in  Table  XXXVII.,  and  the  correction  of  latitude.  Table  XLI.,  is  to  be  applied  to  the 
mean  latitude  deduced  from  Table  XXXVII.  Thus,  on  July  IG,  1830,  by  the  e.xamples  at 
the  bottom  of  Tables  XL.  XLI.,  the  equation  of  the  equinoxes  was  —  5". 3,  and  the  aberration 
lu  longitude -(- 1 1".3 ;  tiiese  corrections  being  applied  to  tlie  mean  longitude  of  the  star 
deduced  from  Table  XXXVII.,  lis.  2F  7'  32'',  gives  its  apparent  longitude  lis.  21"  7'  38". 
In  a  similar  manner  the  aberration  in  latitude,  — 5".G,  found  at  the  bottom  of  Table  XLI., 
applied  to  the  mean  latitude,  19°  24'  4.j"  N.,  deduced  from  Table  XXXVII.,  gives  the  ap- 
parent latitude  of  the  star  19"  24'  30"  N. 

TABLIOS  XLII.  XLIII.  Merration  and  JVutation  in  R'ght  ^Isccnsion  and  Drdination. — 
Table  XLII.  contains  the  aberration,  and  Table  XLIII.  the  nutation  in  right  ascension  and 
declination,  to  be  found  by  the  rules  at  the  bottom  of  the  tables,  and  applied,  with  their 
signs,  to  the  mean  values  deduced  from  Table  VIII.  Thus,  by  Table  VIII.,  the  riglit  ascen- 
sion of  a  Pegasi,  July  IG,  1830,  was  221i.  Sfmi.  20s.,  and  its  declination  14"  18'  N.  The 
aberration  of  right  ascension  in  time  was  nearly -|-0s. 8,  in  declination  —  0".8  ;  the  nuta- 
tion in  right  ascension  in  time  — Os.l,  in  declination  -[-  0''.-'>,  as  appears  i»y  the  examples  at 
the  bottom  of  the  tables.  The.se  corrections  being  applied  to  the  mean  values,  give  the 
apparent  right  ascension  22h.  r)Gm.  2!s.,  and  tlie  apparent  declination  14"  18'  N.  The  equa- 
tion of  tlie  ohliquity  of  the  eclijjtic  may  be  calculated  by  the  rule  at  the  bottom  of  the  table. 
Thus,  on  July  IG,  1830,  the  equation  was  —  9". I,  which,  applied  to  the  mean  obliquity  23° 
27'  42".0,  gives  the  apparent  obliquity  23°  27'  32". 9. 

TABLE  XLIV.  j-inirnientntionoftkeMoonsScmi-diamctcr. — This  table  is  divided  into 
four  parts,  and  is  useful  in  finding  the  augmentation  of  the  moon's  semi-diameter  by  means 
of  the  altitude  and  longitude  of  the  nonagesimal  when  the  moon's  altitude  is  unknown. 
The  precepts  for  this  calculation  are  given  at  the  bottom  of  the  table,  and  Ibr  further  illus- 
tration anr)ther  example  is  added,  in  which  it  is  required  to  find  the  augmentation  at  the 
commencement  of  the  occultation  calculated  in  Problem  VII.  of  the  Appendix,  when  the 
D's  S.  D.  by  the  Nautical  Almanac  was  IG'  13". 9,  her  true  latitude  I"  5.7  IJ"  S.,  parallax 
in  lat.  40'  2>".G,  altitude  of  the  nonagesimal  81"  17' 32",  and  the  moon's  apparent  distance 
tVom  the  nonairesimal  51"  38'  2G",  as  in  Example  III.  Prob.  V.  Appendix.  In  this  case  the 
arguments  of  Part  1.  are  81"  17'  32"  -f  51"  38'  2G",or  nearly  4s.  12°  5G'  and  Os  29"  39',  and 
the  corresponding  corrections -)- G". 00, -f- 4". 05,  whoso  sum  is  10". 05.  This  in  Part  II. 
gives -|- 0".  10.  In  Part  III.,  with  the  moon's  true  latitude,  1"  5.5'  11"  S.,  and  her  par.  in  lat. 
10'  23".(>.  t!ie  correction  is  —  0".10.  The  sum  of  these  three  parts  is  -f-  10". 05,  wliich  bein2 
found  at  the  side  of  Part  IV.,  and  the  moon's  horizontal  S.  D.  IG'  18". 9  at  the  top,  gives  the 
corresponding  correction -(- 0" .40.  This  connected  with  the  th.-ce  former  parts -f- 10" .05, 
gives  the  sought  augmentation  10". 45,  or  10". 4,  as  in  the  example  Prob.  VII.  Apjjendix. 
It  may  be  observed  that  the  calculation  by  Problem  IV.  will  sometimes  produce  the  supple- 
ment of  the  altitude  of  the  nonagesimal;  but  this  requires  no  alteration  in  the  rule,  since  tJie 
result  is  tli"  same  whether  the  altitude  or  its  sujijjlement  is  used. 

TABLi'i  .XLV.  Equation  of  Second  Differences. — This  table  contains  tlie  equation  of 
the  second  differences  of  the  moon's  motion,  or  the  correction  to  be  made  on  account  of  her 
unequal  velocity  between  the  times  marked  in  the  Nautical  Almanac.  The  manner  of  ap- 
plying this  correction  is  taught  in  Problems  I.  II.  III.  of  the  Appendi.x. 

TABLE  XLVI.  Variation  of  the  Altitude  of  an  Object,  arising  from  a  Chatige  of  100 
Seconds  in  the  Declination. — This  table  is  useful  in  finding  the  latitude  by  double  altitudes 
of  the  sun,  or  any  other  object.  It  is  explained  in  the  precepts  for  such  calcul.iiions,  pages 
189,  190,  191,  &c.     The  table  is  to  be  entered  at  the  top  with  tlie  latitude  of  the  place,  and 

*  The  ilc!:rv<'.-!  'n  tlrs  and  the  following  tables  are  to  be  foiinfl  in  the  column  marked  \)  on  th?  same  hori- 
(^ontril  linr  \v  !h  the  s^^mis.  'I'lms  if  the  signs  are  at  the  top  of  the  table,  the  dcsrrces  mn.^t  be  foiiiiil  n  ;Iu 
k-rt  r  iliiiiiii,  olhiTWiSc  in  the  rifcht. 


396  CATALOGUE   O^    THE   TABLES. 

at  the  side  with  the  declination  and  altitude  of  the  body  ;  the  corresponding  number  is  the 
variation  of  the  altitude,  in  seconds,  for  a  cliange  of  100"  in  the  declination. 

TABLES  XLVn.  XLVIIL  are  used  in  finding  the  First,  Second  and  Third  Corrections 
in  Lyons  Improved  Method  of  ^corking  a  Lunar  Observation. — The  first  of  these  tables  gives 
the  first  and  second  corrections.  The  first  correction  is  always  taken  out  with  the  degrees 
and  minutes  marked  at  the  top  of  the  table.  The  second  correction  is  also  taken  at  the  top 
when  the  apparent  distance  exceeds  90°,  but  at  the  bottom  when  the  apparent  distance  is 
less  than  90-'. 

TABIjLS  XLIX.  L.  are  used  in  finding  the  Correction  for  Parallax  in  Lyons  Improved 
Method  of  xcorking  a  Lunar  Observation. — The  first  of  these  tables  gives  the  correction,  sup- 
posing the  parallax  to  be  35".  It  is  to  be  entered  at  the  top  with  the  apparent  distance,  and 
at  the  side  with  the  altitudes  of  the  object ;  the  corresponding  number  is  the  correction  fo: 
the  horizontal  parallax,  35".  This  is  to  be  found  in  the  side  column  of  Table  L.,  and  the 
horizontal  parallax  at  the  top  ;  the  corresponding  number  is  the  actual  parallax  in  altitude, 
which  is  to  be  applied,  with  the  same  sign  as  in  Table  XLIX.,  to  the  apparent  distance. 
Thus,  if  the  app.  dist.  =  C0°,  *'s  alt.  =25^^,  D's  alt.  =45°,  the  correction  in  Table  XLIX. 
is  — 20"  ;  and  if  the  planet's  horizontal  parallax  be  15",  the  corresponding  correction  in 
Table  L.  will  be  —  D"  ;  to  be  applied  as  a  third  correction  to  the  apparent  distance. 

TABLE  LI.     To  change  mean  solar  time  into  sideral  time. 

TABLE  LI  I.     To  change  sideral  time  into  mean  solar  time. 

TABLE  Lill.  Gives  the  variation  of  the  compass  very  nearly  as  in  the  chart  of 
P,  Barlow. 

TABLE  LIV.  Table  of  Latitudes  and  Longitudes. — This  table  (as  observed  in  the 
Preface)  has  been  completely  revised  for  this  edition,  and  the  latitudes  and  longitudes  of  a 
great  number  of  places  are  added  to  those  given  in  some  of  the  former  editions  of  this  work. 

TABLE  LV.  Tide  Table. — The  explanation  and  uses  of  this  table  are  given  in  the  body 
of  the  work,  in  treating  of  the  manner  of  computing  the  times  of  the  tide,  page  121,  «S:c. 

TABLE  LVI.  Extracts  from  the  Nautical  Almanac  of  the  numbers  used  in  the  exam- 
ples of  lunar  observations  &c. 

TABLE  LVII.  shows,  nearly,  the  error  in  Longitude  in  miles  and  tenths  of  a  mile, 
occasioned  by  an  error  of  one  mile  in  the  Latitude.- 

Thus,  when  the  sun's  altitude  is  30°,  the  Latitude  30°,  and  the  Polar  distance  100°,  the 
error  is  8  tenths  of  a  mile. 

The  error  affects  the  Longitude  as  follows  : 

When  in  West  Long,  and  J  A.  M.  t  ,.     t  „„„  j ,   \  decreased.  \  When  the  cnrrection  ia  mark-  (  increased.   ) 
the  time  is  found  in  Ool.        J 1*.  M.  <  o-  »   j  increased.  J     ed  X  tlie  Longitude  ia  )  decreased.  \ 

When  in  East  Long,  and   >  A.  M.  i  ..     t  „_„  i.    j  increased,  >  When  the  correction  is  mark-  \  decreased.  ) 
the  tsjuc  is  found  in  Col.         JP   M. )      e  jjon^.  jS    (  decreased,  j     ed  X  the  Longitude  ia  )  increased.  \ 


APPENDIX, 


CONTAINING 


METHODS    OF    DETERMINING    THE    LONGITUDE    BY    OBSERVATIONS    OP 
ECLIPSES,  OCCULTATIONS,  &c. 


The  longitude  of  a  place  may  be  determined  in  a  very  accurate  manner,  by  observing  th'. 
beginning  or  end  of  a  solar  eclipse,  or  occultation  of  a  fixed  star  by  the  moon,  or  tlie  differ 
ence  between  the  times  that  the  moon  and  a  known  fixed  star  pass  the  meridian.  These 
observations,  when  made  on  land  with  a  good  telescope  and  well-regulated  time-keeper, 
furnish  by  far  tlie  most  accurate  method  of  determining  the  longitude,  and  when  made  on 
board  a  ship  without  a  telescope,  will  in  general  give  it,  with  a  greater  degree  of  accuracy  than 
any  other  method.  For  this  reason  we  have  inserted,  in  this  Appendix,  the  usual  rules  of 
calculating  such  observations,  by  means  of  the  Nautical  Almanac.  The  first  thing  to  be 
taken  notice  of,  is  the  method  of  determining  the  longitude,  latitude,  &c.  of  the  moon  or 
other  object,  having  regard  to  the  unequal  velocity  between  the  times  for  which  these 
quantities  are  given  in  the  Nautical  Almanac.  This  calculation  is  rendered  much  more 
simple  by  making  use  of  the  signs  -|-  and  — ,  and  performing  addition  and  subtraction  as  in 
the  introductory  rules  of  algebra ;  and  as  it  is  possible  that  these  rules  may  not  be  familiar 
to  some  readers  of  this  work,  we  have  given  an  explanation,  as  far  as  will  be  necessary,  in 
the  present  problems. 

Quantities  witkuut  a  sign,  or  with  the  sign  -\-  prefixed,  are  called  positive  or  affirmative, 
as  7  or  -}-  7  ;  and  those  to  which  the  sign  —  is  prefixed,  are  called  negative,  as  —  7.  Mdi- 
tion  of  quantities  liavrng  the  same  sign,  that  is,  all  affirmative  or  all  negative,  is  performed  by 
adding  them  as  in  common  arithmetic,  and  prefixing  the  common  sign.  Thus  the  sum  of -f-  4 
and  -p  3  is  -f-  7.  The  sum  of —  4,  —  3,  arid  —  5,  is  —  12.  When  the  quantities  have  not  the 
same  sign,  the  positive  quantities  must  be  added  into  one  sum,  and  the  negative  into  another, 
as  ahore  ;  the  difference  of  these  two  sums,  with  the  sign  of  the  greater  sum  prefixed,  will  be 
the  sum  of  the  proposed  quantities.  Thus  the  sum  of  -f-  14,  —  7,-^5,  and  —  2,  is  found  by 
adding  -\-  14,  -\-  5,  whose  sum  is  -f-  19 ;  and  then  —  7  and  —  2,  whose  sum  is  —  9  ;  the  dif- 
ference of  19  and  9  is  10,  to  which  must  be  prefixed  the  sign  of  the  greater  number,  19, 
which  is  -f-,  so  that  the  sought  sum  is  -|-  10.  The  ibllowing  examples  will  illustrate  these 
rules: — 

Add 


--4 
--3 
--7 
—  2 

Add  +  4'  Iff' 
+  2    5 

Add  —  4'  10' 
—  2    5 

Add  —  4'  10" 
+  2     5 

Add  +  1 
—  1 

Add  +  6'    0" 
-2  15 
-1-4  13 
—  3    7 

Sum  +  a  15 

Slim  —  6  15 

Sum  —  2    5 

Sum       0 

+  12 

Sum  +  4  51 

Subtraction  is  performed  by  changing  the  sign  of  the  number  to  be  subtracted  from  -f-  to  — , 
or  from  —  to  -{-  ;  and  then  adding  the  numbers  by  the  preceding  rule.  Thus  to  sr.btract  -\-  3 
from  -}-  7,  the  sign  of  -f-  3  must  be  changed,  and  the  numbers  —  3  and  -f-  7  added  together 
as  in  algebra,  which,  by  the  preceding  rule,  gives  -j-  4  ;  and  if  it  were  required  to  subtract 
—  3  from  7,  the  sign  of  —  3  must  be  changed,  and  -j-  3,  -|-  7  added  together  ;  the  sum  -\-  10 
represents  the  souglit  difference.  It  is  not  usual  to  make  an  actual  change  of  the  sign  in 
any  proposed  question,  it  being  sufficient  to  suppose  the  number  to  be  subtracted  to  have  a 
different  sign  from  tliat  prefixed  to  it,  and  to  perform  the  operation  accordingly.  To  illus- 
trate this,  the  fallowing  examples  are  added  : — 

From +  4' 10"       From  +  4' 10"       From  —  4' 10"       From  —  4' 10"       From  +  1        From  —  1        From  +  1 
Sub.    -1-2    5        Sill).   —2    5        Sub.   —25        Sub.   +2    5         Sub.   —1        Sub.    —1        Sub.   +  1 

Rem.  +25         Rem.  +  6  15        Rem.  —  2    5        Rem.  —  G  15        Rem.  +  2        Rem.       0        Rem.       0 

From       108  From  —  108  From       108  From  —  103  From— 201 

Sub.        201  Sub.   —201  Sub.   —  201  Sub.        201  Sub.         108 

Rem.    —  93  Rem.    +  93  Rem.  +  309  Rem.  —  309  Rem.  —  309 

Observing  that  when  no  sign  is  annexed  to  a  quantity,  the  sign  -|-  is  always  understood 
to  be  prefixed. 

PROBLEM   I. 

To  find  the  hvgifude,  latitude,  S,"c.  of  the  moon  at  any  given  time  at  Greemvich,  having 
regard  to  the  wiequdJ  vdociti)  between  the  times  marked  in  the  J^aulical  Almanac  ;  ilif 
inJervuls  of  these  limes  being  12  hours. 


398 


TO   FIND   I'lIE   LONGITUDE    &c.  OF   THE   SUN,  MOON,  &*,. 


RULE. 

Take  from  the  Nautical  Almanac  the  two  longitudes,  latitudes,  &c.  next  preceding  the 
iriven  time  at  Greenwich,  and  the  two  immediately  following  it,  and  set  them  down  in  suc- 
cession below  each  other,  prefixing  the  sign  -|-  to  the  southern  latitudes  or  declinations,  and 
the  sign  —  to  the  northern.  Subtract  each  of  these  quantities  from  tlie  following  for  the 
first  differences,  and  call  the  middle  term  arc  A ;  subtract  each  first  difference  from  the  fol- 
lowing for  the  second  differences,  and  take  the  half  sum  or  mean  of  them,  wliich  call  the 
arc     B,  noting  the  signs  of  the  quantities  as  in  algebra. 

Find  the  difference  between  the  given  time  and  the  second  time  taken  from  the  Nautical 
Almanac,  wliich  call  T ;  then  to  its  logarithm  add  the  log.  of  A  and  the  constant  logarithm 
5.3G452 ;  the  sum,  rejecting  10  in  the  index,  will  be  the  logarithm  of  the  proportional  part,* 
to  which  prefix  the  sign  of  the  arc  A  ;  observing  to  express  all  these  quantities  in 
seconds. 

Enter  Table  XLV.  with  the  arc  B  at  tlie  top  and  the  time  T  at  the  side  :t  opposite  to  this 
will  be  the  correction  of  second  differences,  to  which  prefix  a  different  sign  from  tliat  of  the 
arc  B,  and  plate  it  under  the  proportional  part  found  above,  and  the  second  quantity  taken 
from  tlie  Nautical  Almanac,  and  connect  these  tliree  quantities  together  as  in  addition  in 
algebra  :  tiie  sum  will  be  the  sought  longitude,  latitude,  &c. ;  the  latitude  or  declination 
being  south,  if  it  has  the  sign  -(-  ;  north,  if  it  has  the  sign  — . 

EXAMPLE   I. 

Required  the  longitudes  and  latitudes  of  the  moon,  December  12,  1808,  at  15h.  48m.  20s. 
and  17h.  Im.  29s.  app.  time  by  astronomical  computation  at  Greenwich,  which  correspond  to 
the  immersion  and  emersion  of  Spica,  calculated  in  Problem  VII. 


1808.  Dec. 

D  long.  N.  A. 

I) 

s.     °     '    " 

12  noon. 

0     10  45  20 

12  niiiln. 

6     17  51  30 

13  noun. 

6    25    2  54 

13  luidn. 

7      2  18  59 

1st. 

aiff.    1 

° 

( 

II 

7 

fi 

16 

A  7 

11 

18 

7 

It) 

5 

2.1  diff. 


+  2  40  58 

-f5    2 

+  2    6  37 

+  4  47 

-f- 1  29  52 

==  +  4  51.5 

-f  0  51  18 

IMMERSION. 


T  =  3h, 

A  =  7_ 

+       2 
-l-C  17 


Constant  5.36452 

48m.  29s.  =  13709s Log.  4.13701 

11       18     =25878    ....Log.  4.41293 


IG      52.2=  8212.2.... Log.  3.91446 

51      36    Second  longitude. 

31.9  Table  XLV.  B  =  4'  51".5 


f)   20 


56.3  D's  longitude. 


D  lat:t.  S. 


1st  d;ir. 
/    II 

—  34  21 

A  —  36  45 

—  38  34 


2d  diff 


—  2  24 
-1  4.9 
B=  — 2  06.5 


5.30452 

4.13701 

•22i)o" Log.  3.34341 


699.7 Log.  2.84494 


r 


37     Second  latitude. 

13.7  Table  XLV.  15  =  —  2'  6l'.5 


-\- 1.   55     11.0  D's  latitude  south 


EMERSION. 


Constant  5.36452 

:.51i.  lin.29s.=  180893 Log.  4.25742 

7    11       18=25878 Log.  4.41293 


-f       3      0      36.0  =  10836 Log.  4.03487 

-1-6  17    51      36      Second  longitude. 

—  3.-..9  Table  XLV.  B.  =  4'  54 '.5 


6  20   51     36.1  D's  longitude. 
These  quantities  are  made  use  of  in  Problem  VII, 


■  3G'    45"  =  —  2205"  , 


5.36452 

4.2.i742 

.Log.  3  34341 


—       15    23.3        =  —  923.3.... Log.  296535 
6 


+  2 
+ 


37     Second  latitude. 
15.4  Table  XLV.  B.  =  - 


-f- 1   51     29.1  D's  latitude  south. 


EXAMPLE  II. 

Piequired  the  longitudes  and  latitudes  of  the  moon,  June  IG,  1806,  at  2h.  49m.  bDs.l,  ana 
r)li.  3'lm.  ()s.G,  app.  time,  astronomical  account  at  Greenwich,  which  correspond  nearly  to  the 
beginning  and  end  of  the  total  eclipse  of  the  sun  as  observed  at  Salem. 


18J6. 

June. 

D  long.  N.  A. 

15d 

midn. 

2    14  48  .58 

16 

noun. 

2    22    6  19 

16 

mid  11. 

2    29  27  12 

17 

ni'on. 

3      6  50  47 

1st  diff. 
<>     I    II 

7  17  21 

A  7  21)  53 

7  23  35 


—  1    14     6 

-}-3  32 

—  0  34   13 

4-2  42 

+  n     (i  33 

=  +  3     7 

-f  0  47  28 

D  lat.  N.  A. 


-1-  39  ,'■.3 
-[-  40  46 
-i-40  55 


2d  diff 


-f53 

+   9 
D  =  -I-  31 


*  This  corrcition  may  also  be  found  by  proportion,  by  saying,  As  12  hours  ;ire  to  tin?  time  T,  so  is  the  arc 
A  to  the  soiiirht  proportional  part;  and  "this  nictlMd  is  the  shorte.'st  when  T  is  an  alii;uol  part  of  12  hoiws. 
Thus,  if  T  be  3,  6,  or  9  hours,  the  proporlimial  part  will  be  ^,  i,  or  ^  of  the  arc  A  respectively.  This  method 
is  made  use  of  in  Problem  XVII.  in  interixilatiu!:  the  distance  of  the  moon  and  sun. 

t  If  t!ie  arc  B  consists  of  minutes  and  seccuuls,  the  correction  fir  minutes,  tens  of  seconds,  and  units  of 
:'econds,  must  be  found  separately  :  the  sum  of  these  three  parts  will  be  the  sought  correction.  Proportional 
parts  for  the  m'nutes  of  the  time  T  may  be  taken   in  finding  the  correction  of  this  table,  wtiiui  necessary.     In 

this  rule,  part  of  the  correction  of  the  th'rd  difference  is  neglected.     Th':s  part  never  exceeds 1 of  the  third 

difference,  and  rarely  amounts  to  a  small  fracli(ui  of  a  second. 


TZS 


TO   FIND    THE   LONGITUDE,  &c.  OF   THE    SUN,  MOON,  &c.        399 


BEGINNING   AT  2h.  49m.  50s.l  =  T 


fleoond  lonsitiide 

\  7°  20'  5;{i'    Prop.  p;irt + 

l(  3     7     Table  XLV — 


2s.  22°     O   19' 

1    43    59.8 

16.8 


D 's  longitude 2    23    50     2.0 


Second  latitude  N 

A  40'  4G"    Prop.  part... 
B         31      Table  XLV. 

D's  latitude  N.., 


—  0°  34' 
+         9 


13" 

37.0 

28 


END  AT  5h.  34m.  Gs.G  =  T. 


Second  longitude 2s.  22°    fi'   19" 

A  7°  20'  53"    Prop,  part +      3    24    35.3 

U         3     7     Table  XL,V — 23_2 

D's  longitude 2   25    30    31.1 


Second  latitude  N —0°  34'  13" 

A  40'  4(,"    Prop,  part +       18    55.0 

B  —  31"    Tat)le  XLV —  3.& 


])'s  latitude  N. 


—  0    15    21.8 


The  proportional. parts  of  tlie  arc  A  were  calculated  in  this  e.^ample  by  arithmetic  with- 
out logarithms.  By  observations  of  the  eclipse  on  that  day,  it  was  found  that  the  moon's 
longitude  was  too  great  by  fiS".."),  and  her  Kititude  too  great  by  11".4.  These  corrections 
are  "applied  to  tlie  above  longitudes  and  latitudes,  in  calculating  the  eclipse  in  Problem  VL 

Remark  1.  It  will  not  be  necessary  to  take  notice  of  the  second  differences  in  calculating 
the  paralla.x  or  semi-diameter  of  the  moon,  or  any  of  the  solar  elements  useful  in  calculating 
an  eclipse  or  occultation.  In  this  case,  the  quantities  immediately  preceding  and  following 
the  proposed  time  at  Greenwich,  must  be  taken  from  the  Nautical  Almanac ;  and  their  dif- 
ference will  be  the  arc  A  ;  also  the  difference  between  the  proposed  time  and  that  taken 
first  from  the  Nautical  Ahnanac  is  to  be  called  the  time  T.  Then,  by  proportion,  as  the 
interval  between  the  times  taken  from  the  Nautical  Almanac  is  to  the  time  T,  so  is  the 
arc  A  to  tiie  correction  to  be  applied  to  the  first  quantity  taken  from  the  Nautical  Almanac ; 
additive  if  increasing,  subtractive  if  decreasing.  This  correction  may  also  be  found  by 
logarithms  as  above,  using  the  constant  logarithms  5.30452  if  tlie  interval  of  the  times  in  the 
Nautical  Almanac  is  12  hours,  and  5.00349  if  the  interval  is  24  hours.  The  proportional 
part  of  the  moon's  paralla,\  and  semi-diameter  may  also  be  found  by  Table  XL,  and  that  of 
the  solar  elements  by  Tables  XXX.  XXXL,  as  taught  in  the  explanation  of  these  tables; 
these  calculations  being  sometimes  much  facilitated  in  the  new  form  of  the  Nautical  Alma- 
nac, by  means  of  the  horary  motions,  which  are  given  for  several  of  the  elements.  To 
exemplify  this,  the  rest  of  tlie  quantities  requisite  in  calculating  the  eclipse  and  occultation 
(Problem  VI.  VII.)  are  here  found. 


EXAMPLE   III. 


1808. 

Dec.  12,  midnight 

Dec.  n,  noon 

Difference  .\ 

Pro.  part  T  =  3h.  48m.  29s. 
Corrrspomlin;^  values.... 
Pro.  part  T  :=51i.  lin.  29s.. 
Corresponding  values 

180G. 

June  IG,  noon 

IG,  midnight 

Differences  A 

Pro.  partT=2li  49in.  SOs.l 

Corresponding  values 

Pro.  part  T  =5h.  34m.  6s.6 
Corresponding  values 


D  S.  D. 

Hi'     17" 


1.9 
18.9 

2.5 
19.5 


1)  S.  D. 

IG'    27" 
IG 

t 

IG 

+ 
16 


D  H.  P.  • 

59'   4G" 

60      6 

20 

6.3 
50    52.3 

8.4 
59    54.4 


Dec.  12,  noon 

13,  noon 

Difference  A 

Pro.  part  T=  loll.  48ni.29s. 
Corresponding  values.... 
Pro.  partT=17h.  lui.29s. 
Corresponding  values.... 


EXAMPLE  IV. 


])  H.P. 
60'  21" 


30 

GO 

34 

3 

+ 

13 

0.7 

+ 

3.1 

27.7 

GO 

21.1 

1.4 

+ 

6.0 

28.4 

GO 

27.0 

1806. 

June  16,  noon 

17,  noon 

Differences  A 

Pro.  partT=  21i.  49in..'')0s.l 
Corresponding  values.... 
Pro.  part  T=5h.  34m.  6s. 6 
Corresponding  values.... 


O  long. 

©R. 

A. 

8s.  20°  22'  4" 

17h.  18m.  4j 

4 

8  21  23  10 

17  22 

29 

.5 

1   1  6 

4 

25 

.1 

40  15 

o 

54 

.6 

8  21  2  19 

17  20 

59 

.0 

43  21 

3 

•  8 

.1 

8  21  5  25 

17  21 

12 

.5 

0  long. 

OR. 

A. 

.84°  34'  18" 

5h.  36ni 

.  2i)s 

6 

85  31  35 

5  40 

30 

.0 

57  17 

4 

9 

.4 

-f   6  45.4 

+ 

29 

.4 

84  41   3.4 

5   36 

.50 

.0 

+    13  17.5 

+ 

57 

.9 

84  47  35.5 

5  37 

18 

.5 

e  sun's  semi-diameter  by  the   Nautical   Almanac,  June   13,  130G,  was   15'  4G".3,  and 
1!),  1806,  was  15'  45".9.     Hence,  at  the  above  time,  it  was  15'  40". 1.     This,  in  eclipses 


The  semi-diameters  thus  found  must  be  decreased  2"  for  inflexion,  and  augnwnted  by  the 
correction  Table  XLIV.  in  calculating  an  eclipse  or  occultation  by  Problem  XIII. ,  or  in 
deducing  the  longitude  from  observations  by  Problems  VL  VII.  VIII.  or  IX.  We  may 
however,  observe,  that  some  astronomers  neglect  the  correction  of  2"  for  inflexion. 

The  "  .     .     -.  ... 

June 

of  the  sun,  must  be  decreased  3.^"  for  irradiation. 

Rcviiirk  2.  The  above  rule  for  calculating  the  second  differences  of  the  lunar  motions 
where  the  intervals  in  the  Nautical  Almanac  are  12  hours,  may  be  made  use  of  whe«  the 
intervals  are  any  number  of  days,  as  is  the  case  with  the  elements  of  the  motions  of  the 
planets,  by  taking  two  longitudes,  latitudes,  &c.  before,  and  two  after,  the  given  time  at 
Greenwich,  and  thence  deducing  the  arcs  A.  B,  and  the  longitudes,  latitudes,  &c.,  and 
then  making  use,  instead  of  T,  cf  the  qunlient  of  the  difference  between  the  given  time  and 
that  marked  in  the  Nautical  Almanac  against  the  second  longitude,  &c.  divided  by  the  num- 
ber of  lialf  days  in  the  given  interval.  Thus,  if  the  interval  is  1  day,  the  divisor  is  2;  if  the 
interval  is  4  days,  the  ciivisor  is  8 ;  and  if  the  interval  is  5  days,  the  divisor  is  10.     In  like 


shall  give  the  following  examples  : — 


400        TO  FIND  THE    HORARY   JVlOTlOiN    OF  THE   SUiN',  MOON,  &.c. 

EXAMPLE   V. 

Required  the  right  ascension  of  Venus,  1836,  August,  23d.  ICh.  40m.  mean  time,  astronomi 
cal  account,  at  Greenwicli. 


Times. 

Right  Ascen. 

h.   in.      s. 

iigust  29 

7    44    23.51 

^       23 

7     45    24.05 

24 

7     4(j    32.84 

25 

7     47    49.  C5 

1st  diff. 


00.54 
08.79 
16.81 


8.25 

8.02 

B=8.13 


Second  right  ascension 7h.  45ni.  24s.05 

A  =  lin.08s.79    Proportional  part...  47.77 

B=          &S.13    Table  XLV —.83 

Venua's  right  ascension 


71i.  4eni.  lOs.99 


In  this  example,  the  intervals  in  the  Nautical  Almanac  being  1  day,  we  must  divide  the 
time,  ICh.  40m.,  by  2,  to  get  T  =  8h  20m. 

EXAMPLE   VL 

Required  the  declination  of  Mars,  1836,  June,  14d.  13h.  30m.,  mean  time,  astronomical 
account,  at  Greenwich. 


Times. 

Declinations. 

"       1  ■     II 

June  13 

15    14    19.2 

14 

15    27    56.9 

15 

15    41    25.0 

16 

15    54    43.4 

1st  diff. 

13    37.7 

=  13    28.1 

13    18.4 


—  9.6 

—  9.7 
B=— 9.6 


Second  declination 15°  27'   5G''.9  N. 

A  =  13'28".l    Proportional  part....  7    34.6 

B=    —  9".6    Table  XLV 1  .2 

Mars's  declination 15°  35'  32".7  N 


In  this  example,  as  in  the  last,  we  divide  the  time,  13h.  30m.,  by  2,  to  get  T: 

EXAMPLE   VII. 


:Gh.  45m. 


Required  the  logarithm  of  the  distance  of  Jupiter  from  the  earth,  1836,  June,  2d.  8h.,  mean 
time,  astronomical  account,  at  Greenwicli. 


Times. 

June  1 

2 

3 

4 


Log.  Dist. 
0.7803725 
0.7810545 
0.7817232 
0.7823787 


1st  djff. 

G820 

A  =  6687 

6555 


2d  diff. 

—  133 

—  132 
!  =  — 132 


Second  distance 0.7810545 

A=:    6687    Proportional  part 2229 

B=  — 132    TableXLV 15 


Log.  distance  Jupiter  and  Earth 0.7812789 


In  this  example,  we  also  divide  the  time,  8h.,  by  2,  to  get  T  =4h. 

EXAMPLE  VIII. 

Required  the  moon's  declinatioij,  1836,  January,  16d.  9h.  45m.  50s.,  mean  time,  astro- 
nomical account,  at  Greenwich. 


Times. 

Declination  S 

d.    h. 

»       /        II 

Jan. 16    8 

26    26    58.3 

9 

28    08.4 

•10 

29    06.4 

11 

29    52.3 

70.1 

A  =  58.0 

45.9 


2d  diff. 


—  12.1 

—  12.1 
r— 12.1 


Second  declination 26°  28'   08''.4  S. 

A  =      58«.0    Prop,  part  Tab.  XXX.  44  .3 

B=  — 12".l    TableXLV 1.1 


Moon's  declination 26°  28'   5.3".8  S. 


In  this  example,  the  time,  45m.  50s.,  divided  by  5,  and  changing  minutes  into  hours,  Slc. 
gives  T  =  !'h.  10m.,  which  is  used  in  entering  Table  XLV.  with  B= — 12". 1,  to  find  the 
corresponding  correction, l".l.  We  may,  however,  remark,  that  the  second  differences  of  the 
right  ascensions  and  declinations  of  the  moon  may  generally  be  neglected  as  insensible, 
because  these  quantities  are  given  in  the  Nautical  Almanac,  for  every  hour,  and  their  second 
differences  are  quite  small.  The  same  is  to  be  observed  relative  to  Die  sun's  longitude,  right 
ascension,  the  equation  of  time,  &c.  The  second  difference  of  the  sun's  declination  may 
sometimes  be  3"  or  4",  but  is,  in  general,  insensible.  The  second  differences  of  the  log. 
radius  vector  must  be  taken,  if  we  wish  to  obtain  the  logarithm  correct  in  the  seventh  deci- 
mal place.  We  can  always  judge  of  the  necessity'  of  using  the  second  differences,  by  observ- 
ing that  the  greatest  error  from  neglecting  them  altogether  is  equal  to  J  B.  Thus,  in  the 
last  example,  the  greatest  error  from  neo-lecting  the  consideration  of  the  second  differences 
is  J  B  =  I  X  12".l  =  1".5. 


PROBLEM  II. 

To  find  the  Iwrarxj  motion  of  the  moon  in  longitude,  latiiiide,  fyc.  at  any  given  time  at 
Greemoicli ;  supposing  the  intervals  of  the  times  i?i  the  JVautical  Almanac  to  be  12 
hours. 

RULE. 

Tuke  from  the  Nautical  Almanac  the  four  longitudes,  latitudes,  &c.,  two  immediately 
preceding  the  given  time  at  Greenwich,  and  two  immediately  following.  Prefix  the  sign  -j- 
to  the  southern  latitudes  or  declinations,  and  the  sign  —  to  the  northern.  Then  find  the  first 
and  second  differences,  the  arc  B,  and  the  time  T,  as  in  Problem  I.  The  mean  of  the  two 
first  differences,  noticing  the  signs  as  in  algebra,  will  be  the  approxijnate  motion  in  12  hours. 

To  tlie  proportional  logarithm  of  one  fourth  part  of  the  time  T,  add  the  proportionai 
logarithm  of  the  arc  B  :  the  sum  will  be  the  proportional  logarithm  of  the  correction  of  the 
approximate  motion,  to  be  applied  to  it  with  the  same  sign  as  the  arc     C,  and  the  corrected 


TO  FIND   THE    HORARY  MOTIONS  OF  THE  SUN,  iMOON,  &c.       401 

motion  of  the  moon  in  12  hours  will  be  obtained,"  which,  being  divided  by  12,  will  give  the 
horary  motion. 

EXAMPLE   I. 

Required  the  horary  motions  of  the  moon  in  longitude,  Dec.  12,  1803,at  loh.  4Sm.  298., 
and  17h.  Im.  2'>s.,  apparent  time,  at  Greenwich. 

This  corresponds  to  Example  I.,  preceding,  in  which  T  is  3h.  48m.  29s.,  or  oh.  Im.  299. 
The  two  Jiisi  differences  in  longitude  are  T^  (i'  IG",  and  7"  11'  18";  their  mean,  7"^  8'  47",  is 
the  appro.ximate  motion  in  12  hours,  and  the  arc  B  is  4'  54".5.  The  rest  of  the  calculation 
is  as  follows  : — 


At  151».  48in.  29s.    T  =  3li.  48m.  2Ds. 

Ardi  B     4'  54" .5     I'n>ii.  Lo?.  l..')G44 
\     T  57     7  Prop.  Log.      4985 

•  Corr.  -f     1    33  Prop.  Log.  2.0529 

Approx.  million    7     8    47 
Motion  12  hours  7    10   20 


In  1  hour 35   51.7 

In  a  similar  manv 
two  first  differences 


At  17Ii.  Im.  293.    T  =  51i.  Im.  29s. 

1.5G44 

\  T  U\.  15m.  223.     I'rop.  Log.     3781 

Corr.    +23    Prop.  Log.  1.9425 
Approx.  motion  ..7      8    47 
Motion  12  liours..  7     10    50 


In  1  hour 35    51.2 


In  a  similar  manner,  if  the  horary  motion  in  latitude  was  required  at  12d.  17h.  /{3m.,  the 
iw'o  firsL  dff'creaies  in  latitude  arc  — 34'  21",  and  — 3G'  45"  ;  their  mean,  —  35'  33",  is  the 
approximate  motion  in  12  hours.  The  correction  found  by  the  above  rule  with  the  time 
T,  5h.  33m.,  and  the  arc  B  =  —  2'  G''.5,  is  —  59",  whence  the  true  motion  in  12  hours  is 
—  3G'  32",  which,  divided  by  12,  gives  the  horary  motion  — 3'  2". 7.  The  negative  sign  — 
indicates  that  the  north  polar  distance  is  decreasing,  the  positive  sign  -(-  that  it  is  increasing. 
In  the  present  example,  the  north  polar  distance  was  decreasing,  and  as  the  latitude  was 
south,  it  was  also  decreasing,  as  is  evident. 

EXAMPLE  II. 

Required  the  horary  motions  of  the  moon  in  longitude,  June  IC,  1806,  at  2h.  49m.  50s.l, 
and  .^>h.  34m.  (is.G,  apparent  time,  by  astronomical  computation,  at  Greenwich. 

This  corresponds  In  Example  II.  preceding,  in  which  T  is  2h.  49m.  50s. 1,  or  5h.  34in 
Gs.6;  the  two  first  differences  are'7°  17'  21",  and  7°  29'  53",  the  mean  of  which,  7"  19'  7' 
is  the  approximate  motion  in  12  hours.    The  arch  B  is  -j-  3'  7". 


At  21i.  49m.  50s.l  =T. 

Ar(hB=-t-  3'  7"   Prop.  Log.  1.7616 

^TinieT=      42  27    Prop.  Log.     C274 

Correction  +    0  44    Prop.  Log.  2.3893 

Approx.  motion 7    19     7 

Motion  In  12  hours..  7    19  51 

Motion  in  1  hour 3G  39.2 


At5h.  34m.  Cs.G^T. 

1.7616 

\  T  =  111.  23m.  32s.  Prop  Log.     33,34 

Correction  +    1      27     Prop.  Log.  2.0950 


Motion  in  12  hours  7 
Blotion  in  1  hour.. 


REMARKS. 

1.  When  it  is  required  to  find  the  motion  of  the  moon  in  longitude  or  latitude,  for  any 
given  interval  of  time,  the  motion  in  12  hours  must  be  found  for  the  middle  of  that  interval; 

2.  In  calculating  an  nccultation  of  a  star  by  the  moon,  the  relative  horary  motion  in  longi- 
tude is  the  same  as  the  horary  motion  of  the  moon,  because  the  star  is  at  rest ;  but  in  calcu- 
lating a  solar  eclipse,  the  sun's  horary  motion  must  be  found  from  the  Nautical  Almanao 
in  the  maiincr  mentioned  below,  and  subtracted  from  the  moon's  horary  motion  in  longitude: 
the  remainder  will  be  the  horary  motion  of  the  inoon  from  the  sun  in  longitude.  Thus,  on 
the  Kith  of  JvuK',  180(1,  tiie  sun's  horary  motion  was  2'  23".],  which,  being  subtracted  fron. 
the  horary  motions  found  in  Exaniple  II.,  3u'  39". 2,  and  36'  42" .8,  leaves  the  correspondino 
horary  motions  of  tiic  moon  from  the  sun  in  longitude  34'  10". 1,  and  34'  19". 7. 

As  the  sun  has  no  sensible  motion  in  latitude,  the  relative  horary  motion  of  the  moon  from 
the  sun  in  latitude,  is  the  same  as  the  true  horary  motion  of  the  moon  in  latitui'o. 


*  Tlie  motion  in  12  hours  tlius  olitained,  which,  for  distinct  ion,  will  be  called  tlie  arc  M,  is  not  porfectly 
u', curate,  sin  i-  Ihi-lh  rd  ami  hiL'lier  orders  of  diflVrences  are  neglected  ;  but  the  horary  motion  deduced  there- 
from is  abundantly  siiiii  -.ent  for  tlie  purpose  of  projecting  an  eclipse  or  occultation.  "  \\'hL-n  creater  accuracj- 
is  required,  the  ill  ril  d.tiVrences  maybe  taken  into  accomit  in  the  following  manner: — Having  found  the 
second  ilijf'n-eiici:<  as  .-ibove  directed,  subtract  the  first  of  them  from  the  second,  not'nsr  the  sijrns  sw  in  algebra, 
and  call  the  reuiaiwlir  the  arc  6.  ICnter  Table  .\LV.  with  this  arc  at  the  top,  and  the  time  T  at  the"  side, 
and  take  out  ilie  i  orrespoiul  ng  correition,  which  is  to  be  imreased  by  one  sixth  part  of  tlie  arc  U,  w'JhoiU 
noting  the  sgns.  To  the  i|uanti'ty  thus  found  is  to  be  prefi.ved  a  sign  different  from  that  of  the  arc  h,  and  then 
it  is  to  be  appl  ed  to  the  arc  M,  w  th  its  sign,  to  obtain  the  true  motion  in  12  hours.  This,  in  the  above 
example,  llie  .-ecnnd  diji'crcnccii  of  long  tudeare+  5'  2"  +  4'  47".  Si'litrai  ting  the  former  from  the  latter,  leaves 
the  third  d  If  rem  e  or  arc  6  =  —  1.5".  Correspond  ng  to  this  and  the  time  T  3h.  48rn  2;!s.  in  Table  XLV.. 
is  I'M),  wliii  h,  in  reased  by  one  sixth  of  i=z2".5,  gives  the  sought  correction  4".l  or  4",  to  wlikh  must  bie 
prefixed  the  s  gn -|- (bei  ause  the  s  gn  of  6  is  negative),  making  it  +  4".  Th  s,  connei  ted  w  th  the  are 
M  =  -f-7''  10' 20',  g  ves  the  true  motion  in  12  hours,  7°  10'  24",  whence  the  horary  motion  is  3.5' 52".  In  a 
similar  manner,  if  the  th  rd  d  (Terences  were  noticed  in  the  above  example  for  find.ng  the  horary  motion  in 
latitude,  III';  two  sccmid  diff'errnce.i  — 2'  21"  and  —  1'  49",  the  arc  4;=  4- 35",  the  corteclioa  of  the  motion  iii 
taluKirs  —  3i;i  32"   is  —  10"  ;  making  it  —  36'  42",  or  3'  3".5  per  hour. 

51 


accu 


402         TO  FIND  THE  ECLIPTIC  CONJUNCTION  OF  THE  SUN,  &c. 

3.  The  hor.iry  motions  of  the  sun  in  longitude  were  formerly  given  in  page  iii.  of  the 
Nautical  Almanac  ;  but  they  are  discontinued  in  its  new  form,  so  tiiat  we  mubl  now  deduce 
the»horary  motion  from  the  daily  difference  of  longitude,  by  dividing  it  by  ^4. 

EXAMPLE   III. 

Thus,  if  it  were  required  to  find  the  sun's  horary  motion  in  longitude,  in  the  interval  be- 
tween July  1  and  July  2,  1S36,  mean  time,  astronomical  account,  at  Greenwich;  we  should 
have  the  longitude  at  noon,  July  1,  99"  35'  03".0;  July  2,  100"  32'  13". 7.  Their  difference 
is  57'  10". 7 ;  dividing  it  by  24,  we  get  the  sun's  horary  motion  in  longitude  2'  22'  .9. 

The  same  method  may  be  used  in  finding  the  horary  motions  of  the  planets,  neglecting 
the  second  difterences ;  but  if  we  wish  to  notice  the  second  diff'erences,  we  may  proceed  as 
in  the  three  preceding  examples,  making  use  of  the  arc3  A,  13,  T,  found  as  in  Remark  2. 
Problem  I. 

EXAMPLE  IV. 

Required  the  -horary  motion  iS  Venus  in  right  ascension,  1836,  August  23d.  IGh.  40m., 
mean  lime,  astronomical  account, .'  t  Greenwich. 

Here  we  have,  as  in  Example  V.  of  tlie  nreceding  prolilein,  T:=8li.  20ni.  ;   aiul  the 
mean  of  llie  two  first  dillerences,  lin.  00s..54,  and  Ini.  08s.79,  is  llie  apprii.xiiiiate 

motion,  lm.0-ls.G6;   also  the  arch  B  =  + 8s.  13 Prop.  Log.    3.124 

JT=2h.  5ni I'rtip.  Log.        158 

,  Correction  5s.66 Prop.  Log.    3.282 

Appro.ximate  motion 1       04  .66 

Motion  of  Venus  in  24  hours. ...      Im.  lOs.32 

Dividing  it  by  24,  we  get 2s. 93,  which  represents  the  horary  motion  of  Ve- 

nus in  riglit  ascension,  corresponding  to  August  23d.  Ibh.  40m. 

The  hnrarij  motion  of  the  moon  in  right  ascension  or  declination  is  fouml,  by  inspection,  in 
the  JVaullrnl  JJhaanac,  taking  the  differences  of  the  two  successive  numbers  in  the  Kautical 
Almanac,  the  one  before,  the  other  after,  the  time  for  which  the  horary  motion  is  wanted. 

EXAMPLE  V. 

Required  the  horary  motion  of  the  moon  in  right  ascension  and  declination,  between  the 
hours  of  10  and  11,  on  the  4th  of  August,  1830,  mean  time,  astronomical  account,  at 
Greenwich. 

1833,  August  4d.  lOh.     Moon's  right  ascension    3h.   07m.20s.l5  Declination    17°  55'  44".2  N. 

4      11  3      09       19  .57  18     03    30  .4 

The  differences  !*  the  horary  motioi.^      !n  R.  A.  Im.  59s.42        In  deilnation  10'  4C".2 

These  horary  motions  correspond  very  nearly  to  the  middle  of  the  time  between  lOh.  insl 
Ilh.,  that  is  to  say,  lOh.  30m. 

PROBLEM   IIL 

To  Jlnd  (he  lime  of  the  ecliptic  conjunction  or  opposition  of  the  moon  tvith  the  sun,  ii 

planet,  or  a  fixed  star. 

The  time  of  the  ecliptic  conjunction  of  the  sun  and  moon  is  the  same  as  the  time  of  nevr 
moon  given  for  the  meridian  of  Greenwich  in  page  xii.  of  the  month  of  the  Nautical  Alma- 
nac. Thus,  in  January,  1836,  the  ecliptic  conjunction  is  on  the  I7lh  day,  at  20h.  27m.8, 
mean  time,  at  Greenwich.  The  time  of  the  ecliptic  opposition  of  the  sun  and  moon  is  the 
same  as  at  the  time  of  full  moon  given  in  the  same  page  of  the  Nautical  Almanac.  Thus 
the  full  moon  or  ecliptic  opposition  in  May,  1836,  was  3dd.  3h.  59ni.7,  at  Greenwich. 

The  time  of  the  ecliptic  conjunction  i«  easily  coiuputed  from  the  geocentric  longitudes  of 
the  objects  ;  and  we  have  here  inserted  the  rule,  adapted  to  the  calculation  of  the  conjunc- 
tion of  the  sun  and  moon,  which,  with  a  slight  modification,  will  answer  for  any  planet,  or  a 
fixed  star. 

RULE. 

Take  from  the  Nautical  Almanac  the  two  longitudes  of  the  sun  and  moon  at  the  noon  and 
midnight*  preceding  the  time  of  the  conjunction,  and  the  two  immediately  following.  Sub- 
trait  the  Inngiludes'of  the  sun  from  uiose  of  the  moon,  noting  the  signs  as  in  algebra ;  the 
remainders  will  represt^nt  the  distances  of  the  sun  from  t*'.e  moon  on  the  ecliptic.  Subtract 
each  of  these  from  the  following  to  obtain  the  first  differences,  and  call  the  middle  term  the 
arch  A  ;  subtract  each  of  these  differences  from  the  following  for  the  second  differences,  and 
take  their  half  sum  or  mean  for  the  arc     B,  noting  the  signs  as  in  algebra. 

To  the  constant  logarithm  4.63548,  add  the  arithmetical  complement  of  the  log.  of  the 
arch  A  in  seconds,  and  the  log.  of  the  second  of  the  above-found  distances  in  seconds ;  the 

•  The  sun's  longitude  at  midnight  is  the  mean  of  the  longitudes  on  the  preceding  and  following  noona 
nearly 


TO  FLND  THE  ECLIPTIC  CONJUNCTION  OF  THE  SUN,  &c. 


403 


Kurn,  rejecting  10  in  the  index,  will  Be  the  logarithm  of  the  approximate  value  of  T  in 
seconds. 

With  this  time  T  at  the  side  of  Table  XLV.,  and  the  arc  B  at  the  top,  find  the  equation 
of  second  diflerences,  the  logarithm  of  which,  added  to  the  two  first  logarithms  used  in  find- 
ing T,  will,  in  rejecting  10  in  the  index,  give  tlie  logarithm  of  the  correction  of  the  approxi- 
mate time  T  in  seconds,  to  be  applied  to  it  with  the  same  sign  as  the  arc  B,  and  the 
mean  time  of  the  conjunction  at  Greenwich,  counted  from  the  second  noon  or  midnight, 
taken  from  the  Nautical  Almanac,  will  be  obtained.  From  which  the  time  of  conjunction 
under  any  other  meridian  may  be  easily  obtained,  by  adding  to  it  the  longitude  in  time  when 
cast,  or  subtracting  when  irest. 

Remark  1.  When  the  time  of  the  ecliptic  conjunction  of  the  moon  and  a  planet  is  re- 
quired, the  longitudes  of  the  planet  must  be  found  by  Problem  I.  for  the  noon  and  midnight 
immediately  preceding,  and  those  immediately  following  the  time  of  the  conjunction,  and 
these  are  to  be  used  in  the  above  note  instead  of  the  sun's  longitudes.     If  the  ecliptic  con- 

i' unction  of  the  moon  with  a  fixed  star  is  required,  its  longitude  must  be  found  in  Tabh- 
^XXVII.,  and  corrected  for  the  ecjuation  of  the  equinoxes  and  aberration  by  Tables  XL 
XL!.,  as  shown  in  the  explanation  of  tliose  tables.  This  longitude  is  to  be  used  instead 
of  the  sun's,  in  the  above  rule.  Tlie  longitude  and  latitude  of  tlie  star  may  also  be  com- 
puted more  accurately,  from  the  right  ascension  and  declination,  given  in  the  Nautical 
Almanac,  by  tlie  method  in  Problem  XIX.  of  this  Appendix,  whenever  the  star  used  is  one 
of  the  100  stars,  whose  places  are  given  for  every  10  days  in  the  Nautical  Almanac. 

Remark  2.  By  the  same  rule,  tlie  time,  when  the  moon  is  at  any  distance  from  the  sun. 
may  be  found,  by  increasing  the  sun's  longitudes  given  in  the  Nautical  Almanac,  by  the 
quantity  tiie  moon  is  supposed  to  be  distant  from  the  sun,  counted  according  to  the  order  of 
the  signs ;  tlicn  supposing  a  fictions  sun  to  move  so  as  to  have  these  increased  longitudes 
at  the  corresponding  times,  and  finding  by  the  above  rule  the  time  of  conjunction  of  the 
moon  with  \.\\\s  fictions  sun,  which  will  be  the  sought  time  when  the  moon  is  at  the  proposed 
distance  from  the  sun.  Thus,  to  find  the  time  of  the  first,  second,  or  third  quarter  of  the 
moon,  the  sun's  longitudes  must  be  increased  3,  G,  or  9  signs  respectively  (rejecting,  as  usual, 
12  signs  when  the  sun  exceeds  that  quantity).  Thus,  if  the  first  quarter  of  the  moon  which 
happened  in  the  afternoon,  July  21,  183G,  was  required  :  The  sun's  longitudes  increased  by 
3  signs  give  the  longitudes  of  the  fictions  sun,  July  20d.  12h. ;  21d.  Oh.;  21d.  12h.,  and 
22d.  Oh.  respectively,  208°  11'  10".0;  208^  3D'  4S".8;  209°  OS'  27".7,  and  209°  37'  0G".7. 
The  longitudes  of  the  moon  corresponding  are  200'^  22'  15".8  ;  207°  03'  18".4 ;  213°  49' 
32" .4,  and  220°  41'  13" .8.  Hence  the  time  of  the  conjunction  of  the  moon  with  the  fictions 
sun  found  by  the  above  rule,  was  July  21  d.  3h.  5m.  at  Greenwich,  which  is  the  time  of  the 
♦vst  quarter  required.  In  a  similar  manner,  by  increasing  the  longitudes  of  a  planet  or  a 
star,  the  time  may  be  found  when  the  moon  is  at  any  proposed  distance  from  it. 


EXAMPLE. 

Required  the  mean  time  of  the  ecliptic  conjunction  of  the  sun  and  moon  in  January,  1836 


1838,  Jan. 


17d. 

Oh 

17 

IQ 

18 

0 

18 

12 

J)  long. 


0  long 


(I 


284  41  2,1.1—296  28  53.5 

292  07  35.4  — 29G  59  2i;.G 

299  31  33.0  —  297  29  59.8 

30(3  52  13.2  —  298  00  32.5 


Distances. 
»      (       // 

—  11    47    28.4 

—  4  51  51.2 
2  01  33.2 
8    51    40.7 


1st  difference. 

o         I  II 

6    55    37.2 

A  =  6   53    21.4 

6    50    07.5 


2(1  difference. 


—  3 
I5  =  — 2 


19  8 
lfi.9 

44.8 


Constant  4.G3.")48 
A  =  G°  53'  24''.4  =  24804".4  Arith.  Conip.  Log.  5.U0547 
2ddis.  4   51  51  .2  =  17511  .2 Log.  4.24332 

3049S3. 


:8Il. 


23m.  18s Log.  4.48427 

—  30 


4.63548 

5.60547 

.1.0''.  1.23300 


Table  XLV.  Corr.  17".l.. 

Correction  303 Lo".   1 .47395 


T 
Correction 

Conjunction  6h.  27m.  4Ss.  past  midnight,  on  January  17d.  20h.  27m.  4Ss.,  mean  time  at 
Greenwich  ;  being  the  same  as  in  the  Nautical  Almanac.  The  time  of  conjunction  under 
any  other  meridian,  as  for  example,  30°  W.,  is  found  by  subtracting  the  longitude  2h.  from 
20h.  27m.  4Ss.,  which  leaves  Ih'h.  27m.  48s.  If  the  longitude  had  been  30°  E.,  the  time  of 
conjunction  would  have  been  22h.  27m.  48s. 

The  usual  method  of  calculating  the  parallaxes  in  eclipses  of  the  sun  or  occultations,  is  that 
by  using  the  longitude  and  latitude  of  the  nonagesimal  or  ninetieth  degree  of  the  ecliptic  above 
the  horizon ;  or,  in  other  words,  the  longitude  and  complement  of  the  latitude  of  the  zenith, 
relative  to  the  ecliptic.  Several  methods  have  been  proposed  for  calculating  the  altitude  and 
longitude  of  this  point,  which  are  required  at  eacli  of  the  phases.  The  following,  which  is  an 
improvement  I  have  made  on  that  given  in  La  Lande's  Astronomy,  seems  well  adapted  to 
the  purpose,  since  several  of  the  logarithms  are  the  same  at  each  of  the  phases,  which  much 
abridges  the  calculation,  and  on  this  account  it  admits  of  considerable  simplifications,  by  a 
table  hke  that  on  page  403.  The  method  of  making  these  calculations  will  first  be  given  at 
full  length,  and  then  in  the  abridged  form,  by  means  of  the  proposed  table.  The  process  of 
calculating  the  parallaxes  with  the  right  ascensions  and  declinations,  instead  of  the  longi- 
tudes and  latitudes  of  the  bodies,  adapted  particularly  to  the  new  form  of  the  Nautical 
Almanac,  will  be  given  towards  the  end  of 'his  Appendix. 


404  rO   FIND   THE   ALTITUDE,  &c.  OF   THE   NONAGESLMAl.. 

PROBLEM  IV. 

Gtveii  the  apparent  time  at  the  place  of  observation,  counted  from  noon  to  noo7i,  according 
to  the  manner  of  astronomers,  the  snn^s  right  ascension,  and  the  latitude  of  the  place, 
reduced  on  account  of  the  sphej-oidal  fgure  of  the  earth,  by  subtracting  the  reduction 
of  latitude,  Table  XXXVIII. ;  to  fnd  the  altitude  and  longitude  of  the  nonagesimal 
degree  of  the  ecliptic. 

RULE  NOT  ABRIDGED. 

Add  G  hours  to  the  sum  of  the  sun's  right  ascension  and  the  apparent  time  of  observation, 
and  call  the  sum  the  time  T,  rejecting  24  hours  when  it  exceeds  that  quantity.  Seek  for 
this  time  in  the  column  of  hours  of  Table  XXVH.,  supposing  that  marked  A.  M.  to  be 
increased  by  12  hours,  as  in  the  astronomical  computation.  The  corresponding  log.  co- 
tan  o-ent  being  found,  is  to  be  marked  in  the  first  and  second  columns,  as  in  the  following 
examples. 

If  the  reduced  latitude  is  north,  subtract  it  from  90-^ ;  if  south,  add  it  to  90°  ;  the  sum  or 
difference  will  be  the  polar  distance.  Take  half  of  this,  and  half  the  obliquity  of  the  ecliptic, 
and  find  their  difference  and  sum.  Place  the  log.  cosine  of  the  difference  in  the  first  column 
its  \oa.  sine  in  the  second  column ;  the  log.  secant  of  the  sum  in  the  first  column,  its  log. 
.•osecant  in  the  second  column,  and  its  log.  tangent  in  the  third. 

The  sum  of  the  logarithms  in  the  first  column,  rejecting  20  in  the  index,  will  be  the  log. 
tangent  of  the  arc  G  ;  the  sum  of  these  in  the  second  column,  rejecting  20  in  the  index, 
wilfbe  the  log.  tangent  of  the  arc  F ;  these  arches  being  less  than  90°  when  the  time  T  is 
found  in  the  column  A.  M..  otherwise  greater.  This  rule  is  general  except  in  places  situated 
within  the  polar  circles.  Within  the  north  polar  circle,  the  supplement  of  F  to  300°  instead 
of  F,  must  be  taken ;  within  the  south  polar  circle,  the  supplement  of  G  to  180°  must  be 
taken  instead  of  G ;  the  other  terms  remaining  unaltered.  In  all  cases,  the  longitude  of  the 
nonagesimal  is  equal  to  the  sum  of  the  arcs  F,  G,  thus  found,  and  90° ;  rejecting  300"^ 
when  the  sum  exceeds  that  quantity. 

Place  in  the  third  column  the  log.  cosine  of  G,  and  the  log.  secant  of  F;  the  sum  of  the 
three  logarithms  of  this  column,  rejecting  20  in  the  index,  will  be  the  log.  tangent  of  half 
the  altitude  of  the  nonagesimal. 

EXAMPLE. 

Required  the  altitude  and  longitude  of  the  nonagesimal  at  Salem,  in  the  reduced  latitude 
42°  22'  4"  N.,  June  15,  1806,  at  22h.  Cm.  18s. 1,  apparent  ti)ne,  or  22h.  Cm.  21s.5,  mean  time, 
by  astronomical  computation,  when,  by  the  Nautical  Almanac,  the  sun's  right  ascension  waa 
5h.  30m.  50s.,  and  the  obliquity  of  the  ecliptic  23°  27'  4S". 

The  sum  of  the  apparent  time,  sun's  right  ascension,  and  G  hours,  rejecting  24  hours,  is 
9h.  43m.  8s.l  =T.  The  polar  distance  is  47°  37'  50"  ;  its  half  is  23°  48'  58",  and  the  half 
obliquity  11°  43'  54"  ;  hence  their  difference  is  12°  5'  4",  their  sum  35°  32'  52".  The  rest 
of  the  calculation  is  as  follows : — 


Column  1. 

Diff.        la*     5'     4"  Cosine  9.99027 

Sum        35    32    52     .Secant  10.089.57 

T    91i.  43m.8s.l  P.  M Cotang  9.4882J 

G.     159°  42'     0"                            Tang  9.5G810 

F      173    40    31  

90 


Column  2. 
Sine  9.32088 

Cosecant      10.23554 

9.48823 


F    Tang        9.04468 


Column  3. 

Tangent  9.85403 
G.  Cosine  9  97215 
F.      Secant     10.00095 


33=  59'  25"      Tang.        9.82883 
67  58  50  =  Alt.  nonagesimal. 


Sum  C3    22    31,  rejecting  3C0°,  is  the  longitude  of  the  nonagesimal. 

The  two  upper  logarithms  of  the  first  and  second  columns,  and  the  upper  logarithm  of  the 
thifd  column,  vary  but  little  in  several  centuries  ;  and  as  these  numbers  occur  twice  in  cal- 
culating a  partial  eclipse  or  occultation,  and  four  times  in  a  total  or  annular  eclipse  or  transit, 
it  will  fend  considerably  to  abridge  the  calculations,  to  have  a  table  like  the  following,  con- 
taining their  values  for  various  places,  for  the  obliquity  23°  27'  40",  with  the  variations  for 
an  increase  of  100"  in  the  latitude  or  obhquity.  The  logarithms  A,  B,  C,  of  the  table,  were 
calculated  in  the  following  manner  : — 

In  north  latitudes  subtract  the  reduced  latitude  from  90°,  in  south  latitudes  add  the 
reduced  latitude  to  90°,  the  sum  or  difference  will  be  the  polar  distance  :  take  half  of 
this  and  half  of  the  obliquity  of  the  ecliptic,  11°  43'  50",  and  find  the  sum  and  difference. 
Then,  ■  ^    , 

Loo-.  A  is  equal  to  the  log.  cosine  of  the  difference  added  to  the  log.  secant  ot  the  sum, 
rejecting  20  in  the  index. 

Log.  C  is  equal  to  the  log.  tangent  of  the  sum. 

Log.  B  is  equal  to  the  log.  tangent  of  the  difference,  increasing  the  index  by  10,  less  the 

Thus,  for  Salem,  in  the  reduced  latitude  42°  22'  4",  the  half  polar  distance  is  23°  48'  58" 
Ihe  half  obliquity  11°  43'  50",  the  difference  12°  5'  8",  the  sum  35°  32'  48". 

Difference..     12°     5'     8"  Cosine    9.99027  Tangent  +  10  =  19.330C.5 

Sum 35    32    48  Secant  10.08956  Tangent  =  C  ==   9.85403 

Sum  A 0.07983  Difference  B _9£7663 


TO   FIND   THE   ALTITUDE,  &c.  OF  THE   NONAGESIMAL. 


405 


In  tills  way  the  logarithms  may  be  found  for  places  not  included  iu  the  table.  The 
changes  for  an  increase  of  100"  in  the  latitude  or  obliquity,  are  found  by  repeating  the 
operation  with  these  increased  values,  and  ascertaining  the  corresponding  changes  in  the 
values  of  A,  B,  C.  These  logarithms  are  given  to  six  places  of  figures,  thougii,  in  general, 
five  will  be  quite  sufficient,  since  the  latitude  and  longitude  of  the  nonagesimal  are  rarely 
required  to  a  greater  degree  of  accuracy  than  10". 


Table 

ca 

culated 

for  the  obliquity 

23°  27'  40 

". 

Reduced 

Var 

.A. 

Var.  B. 

Var.C. 

riaces. 

Lattu 

le 

A. 

+ 100". 

B. 

+ 

00". 

C. 

+  100  '. 

JV.irtl 

Lai. 

Oi,l. 

•l.,.,. 

01.1. 

L;U. 

Obi. 

o 

, 

II 



+ 

_ 

— 

— 

Alhanv, 

4-> 

27 

13 

0.079''7n 

53 

97 

9.47,5733 

293 

739 

9.85,3323 

223 

223 

Berlin, 

50 

20 

1 

21 

9S 

o.O:;h;o8 
fl.(i:;2i(;u 

49 

49 

75 

9.324 1:!5 

9.331054 

618 
GOO 

1099 

loao 

9.771197 
9.773925 

240 
240 

240 
210 

Caiiiliriilye,  K.. .. 

C.'iiiilir;d{;e,  A.. . . 

A-?. 

1-2 

0 

(i.()^i)!5;) 

97 

9.4783S3 

288 

733 

9.8553.55 

222 

').)0 

nul)liii  Obs 

53 

12 

7 

o.Oiionyo 

48 

73 

9.301  IGG 

G70 

1155 

9.7G3705 

2I2 

212 

ICilinbiirch, 

55 

4ii 

2 

0.0551;  18 

47 

07 

9.233401 

878 

137G 

9.741011 

249 

219 

Greenwich  Obs... 

51 

17 

2S 

0.0;i34nu 

49 

77 

9.34fi39r) 

5G2 

1038 

9.780232 

238 

2:18 

Iliivanna, 

aT 

3 

3t 

0.120000 

(14 

148 

9.597i;5S 

95 

51G 

1U.003015 

210 

210 

Kinilerliook, 

49. 

11 

37 

0.080  u;3 

52 

9R 

9.478455 

289 

733 

9.855411 

222 

222 

Lan  aster, 

39 

51 

18 

0.084(i48 

51 

101 

9.501042 

249 

C88 

9.874005 

219 

2T9 

Leon  [.  Obs 

3r, 

If) 

.52 

0.09 lf80 

55 

112 

9.529940 

202 

G34 

9.902005 

21G 

21 G 

51 
31 

19 
17 

29 

3(1 

0.0o34;)li 
0.101899 

49 

58 

77 
125 

9.345714 
9.5';i510 

564 
1.52 

1040 
577 

9.779944 
9.940447 

238 
212 

238 
212 

N'aichez, 

Oxibrd  Obs 

51 

34 

2S 

0.0ti2963 

51) 

77 

9.34058G 

.576 

1051 

9.777800 

239 

239 

Par  s, 

48 
39 

3S 
45 

51 
41 

0.0;i6207 
0.0S4828 

50 
53 

83 
104 

9.394413 
9.501872 

4.52 

2-18 

918 
687 

9.802327 
9.874738 

233 
219 

233 
219 

Pli  ladelphJa, 

Rii  hmoiid  Obs. .. 

51 

l(i 

.51 ; 

0.0i;3482 

49 

78 

9.346.576 

5n2 

1038 

9.780308 

238 

2;)8 

Kiitland, 

^3 

24 

32 

0.0778fifi 

.52 

95 

9.4G5330 

312 

760 

9.845548 

224 

224 

42 
19 

22 
52 

4 
38 

0.079832 
0.127485 

52 

GG 

98 
157 

9.476637 
9.607G02 

291 

78 

731 
500 

9.8540 IG 
10.027183 

211 

222 
211 
.... 

Plaie  Prob.  VII.. 

These  logarithms  are  calculated  for  tiic  obliquity  23*^  27'  40".  The  columns  marked  Lat. 
represent  the  variations  of  A,  B,  C,  for  an  increase  of  100"  in  the  reduced  lat.  The  column 
Obi.  represents  the  variations  of  A,  B,  C,  for  an  increase  of  100"  in  the  obliquity  of  the 
ecliptic.  The  signs  must  be  chanrred  if  the  latitude  or  obliquity  is  less  than  23°  27'  40' , 
which  is  used  in  calculating  the  table. 

EXAMPLE. 

Required  the  values  of  A,  B,  C,  for  Salem,  when  the  obliquity  is  23°  27'  48". 

Tab'ilar  numbers 0.079832  9.476G37 9.854016 

Varati(infor  +  8    obliquity +8 —.58 -f    18 

Sought  values A  =  0.079810  D  =  9.47G579 C  =  9.85 1034 

jibridged  method  of  calculating  the  altitude  and  longitude  of  the  nonagesimal  by  the  preceding 

table. 

Add  together  the  sun's  right  ascension,  the  apparent  time  at  the  place  of  observation, 
(counted  from  noon  to  noon),  and  G  hours  :  the  sum,  rejecting  24  or  48  hours  if  greater  than 
those  quantities,  is  to  be  called  the  time  T  :  this  is  to  be  sought  for  in  the  column  of  houra 
of  Tabic  XXVII.,  supposing  the  column  marked  A.  M.  to  be  increased  12  iiours,  as  in  t!ie 
astronomical  computation.*  The  corresponding  log.  cotangent,  added  to  the  log.  A  of  the 
table,  gives  the  log.  tangent  of  the  arc  G  :  this  added  to  the  log.  B  of  the  table,  rejecting  10 
in  the  inde.^,  will  be  the  log.  tangent  of  the  arc  F  ;  these  arcs  being  less  than  !)()"  whei: 
T  is  found  in  the  column  A.  M.,  otherwise  greater. \  [This  rule  is  general,  e-xcejit  in  ])laces 
situated  within  the  polar  circles,  which  is  a  case  that  very  rarely  occurs.  Within  tbe  vortli 
polar  circle,  the  supplement  of  F  to  3G0°  is  to  be  used  instead  of  F ;  within  the  so7Uli  polar 
circle,  the  supplement  of  G  to  180°  is  to  be  taken  instead  of  G;  the  other  terms  remaining 
unaltered.]  Then  the  longitude  of  the  nonagesimal  is  equal  to  the  sum  of  the  arcs  F,  G, 
and  90°,  neglecting  as  usual  360°  when  the  sum  exceeds  that  quantity. 

To  the  tabular  log.  C,  add  the  log.  cosine  of  the  arc  G,  and  the  k)g.  secant  of  the  arc 
F  :  the  sum,  rejecting  20  in  the  index,  will  be  the  log.  tangent  of  half  the  altitude  of  the 
nonagesimal.  t 

*  Thus,  if  the  t'lne  T  is  5  hours,  it  must  be  called  5h.  P.  M.  ;  if  T  is  14  hours,  it  must  be  called  2h.  A.  M. 
Ill  making  use  of  a  common  table  of  logarithms,  you  must  turn  the  lime  T  into  degrees,  and  make  use  of  the 
log  cotangent  of  its  half.  To  prevent  mistake,  it  may  be  prober  to  remark,  that,  in  finding  'I',  we  must  add 
the  appureid  t  me,  and  not  the  mean  t'lne ;  for  if  the  mean  time  be  used,  we  ought  to  use  also  the  inr.an  right 
ascension;  whereas  the  o/iparcnj  right  ascension  is  given  in  the  Nautical  Almanac;  and  this  must  be  added 
to  the  apparent  time  in  finding  T. 

t  The  arcs  F,  G,  are  acute,  when  the  time  T  is  found  in  the  column  A.  M  ,  otherwise  obtuse.  This  i.-< 
fasily  remembered  from  the  circumstan'-e  that  a  is  the  first  letter  of  acute  and  A.  M.  Some  writers  liave  not 
taken  nntii  e  of  the  cases  of  the  values  of  F,  G,  within  tlie  polar  circles. 

J  Stri'tly  speaking,  the  quant  ty  thus  obtained  is  the  distance  between  the  north  pole  of  the  ecliptic  and 

•i«  zenith  of  the  place,  whith,  in  southern  laftndes,  and  between  the  tropii  s,  is  frequently  the  supplement 

.3f  tititude  of  the  nonagesimal      The  above  form  is  made  use  of  to  simplify  the  rules  for  ajiplying  iJi 


406 


TO   FIND   THE   ALTITUDE.  &c.  OF  THE   iNONAGESIMAL. 


EXAMPLE  I. 

Required  the  altitudes  and  longitudes  of  the  nonagesimal  at  Salem,  June  IG,  1806,  at  tlte 
times  of  the  beginning  and  end  of  the  eclipse,  calculated  in  Problem  VI. 


BEGINNING   OF  THE  ECLIPSE. 

ti.  m.    s. 

5  36  .50.0  O  riglit  ascension. 
22     6  18.1  j9pparent  time. 
6 A  0.07984 

9  43     8.1    Cotang.  9.4S826 


G  159-42'    0" 
90 


.Tang.  9.56810 Cosine  9.97215 

B  9.47653  C   9.85403 


P  173  40  31 Tans 


9.04468 Secant  10.00265 

9.82883 


63  22  31  =  long.  N.  33  59  25 Tang 

Altitude  nonagesimal..  G7  58  50 


END   OF  THE  ECLIPSE, 
h.  m.    s. 

5  37   13.5  O  right  ascension. 
0  50  34.6  Apparent  time. 

6  A  0.07984 


12  27  53.1       Cotang.  8.78470 

4°  11' 13" Tang.  8.86454 Cosine  9.93834 

90  B  9.47658  C  9.85403 


15  23 Tang.  8.34112 Secant  10.00010' 

26  35=long.  N.  35  23  53 Tang.  9.85297 


Altitude  nonagesimal..  70  57  46 


EXAMPLE  II. 

Required  the  altitudes  and  longitudes  of  the  nonagesimal  at  the  times  and  places  men- 
tioned in  the  Example  of  Problem  VII. 

IMMERSION, 
h.  m.  s. 

17  20  59  O  right  ascension. 
16  57  29  Apparent  time. 
_6 A  0.12748 

T  16   18  28 Cotang.  9.8009& 


G  40-18'    7" Tang.  9.92846. 

90  B  9.607CO 


osine    9.88233 
C  10.02718 


F    18  57  48 Tang.  9.53606 Secant  10.02423 

149  15  55  =long.  N.  40  38  46 Tang.    9.93374 

Altitude  nonagesimal..  81  17  32 


EMERSION, 
h.  m.   s. 

17  21    12.5  O  right  ascension. 

18  10  29     Apparent  time. 

6  A    0.12748 


T   17  31   41.5     Cotang.    9.94622 


G  49°  50' 18" Tang.  10.07370 Cosino  9.80953 

90  B    9.60760  C  10.02718 

F  25  38  40 Tang.    9.68130 Secant  10.04504 


165  28  58  =  long.  N.  37  17  39 Tang.   9.88175 

Altitude  nonagesimal..  74  35  18 


In  these  calculations,  it  is  usuai  l^  take  the  sun's  right  ascension,  and  the  apparent  times, 
to  tenths  of  a  second,  and  to  take  proportional  parts  for  the  seconds  and  tenths  in  finding  the 
logarithms.  Thus,  in  Example  I.,  in  finding  the  log.  cotangent  of  9h.  43m.  8s. 1,  the  near- 
est logarithms  are  9.48849,  9.48804,  corresponding  to  the  times  9h.  43m.  4s.,  9h.  43m.  12s. 
These  logarithms  differ  45,  the  times  8s.;  and  the  difference  between  9h.  43m.  4s.,  and  9h. 
43m.  8s. 1,  is  4s. 1.  Hence,  8s.  :  45  . .  4s. 1  :  23,  the  correction  to  be  subtracted  from  the  first 
log.  9.48849  (because  it  is  decreasing),  tv  obtain  the  sought  log.  cotangent  9.48826. 


PR0BLE3I   V. 

Given  the  altitude  and  longitude  of  the  nonagesimal ;  the  longitude,  latitude,  and  hori- 
zontal parallax  of  the  moon,  and  the  latitude  of  the  place  of  obsenation ;  to  find  the 
moon^s  parallax  in  latitude  and  longitude. 

RULE  BY  C03IM0N   LOGARITHMS. 

From  the  horizontal  parallax  of  the  moon,  subtract  its  correction  from  Table  XXXVIII., 
corresponding  to  the  latitude  of  the  place ;  the  remainder,  in  occultations  of  a  fixed  star, 
will  be  the  reduced  parallax  ;  but  in  solar  eclipses,  tliis  quantity  is  to  be  diminished  by  the 
sun's  horizontal  parallax,  8". 6,*  to  obtain  the  reduced  parallax. 

To  the  logarithm  of  the  reduced  parallax  in  seconds,  add  the  log.  sine  of  the  altitude  of 
the  nonagesimal,  and  tlie  log.  secant  of  the  moon's  true  latitude  .;t  the  sum,  rejecting"  20  in 
the  index,  will  be  a  constant  log.  From  the  moon's  true  longitude,!  increased  by  360°  if 
necessary,  subtract  the  longitude  of  the  nonagesimal ;  the  remainder  will  be  the  vioon's 
distance  from  the  nonngesimal,  which,  if  less  than  180°,  is  to  be  called  the  arc  D,  other- 
wise its  supplement  to  SCO"  is  to  be  called  the  arc  D.  To  the  constant  logarithm  add  the 
log.  sine  of  D ;  tlie  sum,  rejecting  10  in  the  index,  will  be  the  logarithm  of  the  approzimate 
parallax  in  longitude  in  seconds,  which  add  to  the  arc  D ;  then  take  the  log.  sine  of  the 
sum,  and  add  it  to  the  constant  logarithm,  rejecting  10  m  tlie  index,  and  the  logarithm  of 
the  corrected  parallax  will  be  obtained.  This  will,  in  genera],  be  sufliciently  exact;  but  when 
great  accuracy  is  required,  the  operation  may  be  again  repeated,  by  adding  the  arc  D  to 
the  collected  parallax  ;  t  then  to  the  log.  sine  of  the  sum  add  the  constant  logarithm,  rejecting 
10  in  the  index,  and  the  logarithm  of  the  parallax  in  longitude  P  will  be  obtained.     This  is 

parallaxes.  It  is  immaterial  uhctlier  the  altitude  of  the  nonagesimal,  or  Its  supplement,  is  made  use  of  in 
Table  XLIV. 

*  This  is  nearly  the  mean  value  of  the  sun's  parallax  ;  but  it  will  be  more  accurate  to  use  the  actual  value 
D's  it  is  given  in  page  266  of  the  Nautical  Almanac. 

t  Corrected  for  the  errors  of  the  tables,  when  known. 

J  This  sum  D  -|-  cor.  par.  -s  nearly  erpial  to  !)+  P,  the  apparent  distance  of  the  mnon  from  the  nosiagesi- 
inal   to  be  made  use  of  in  Table  XLIV.,  in  finding  the  augmuntat  on  of  the  moon's  S.  U 


TO   FIND  THE   PARALLAXES  OF  THE   MOON. 


407 


to  be  added  to  the  true  longitude  of  the  moon  when  her  distance  from  the  nonagesimal  is 
kss  than  130^,  otherwise  subtracted  to  obtain  her  apparent  longitude. 

If  the  true  latitude  of  the  moon  is  south,  prefix  the  sign  -}-to  it;  i£  north,  the  sign — .  Then 
to  the  logaritlim  of  the  reduced  paralla.t  in  seconds,  add  the  log.  cosine  of  tlie  altitude  of  th« 
nonagesimal,  and  the  log.  cosine  of  the  moon's  apparent  latitude;*  the  sum,  rejecting  20  in 
the  inde.x,  will  be  the  logarithm  of  the  first  part  of  the  parallax  in  latitude  in  seconds,  to 
which  prefix  lh(^  sign  -f-  wiien  the  altitude  of  the  nonagesimal  is  less  than  iH)^,  otherwise 
the  sign  — ;  this  being  added  to  the  true  latitude  of  the  moon,  due  regard  being  paid  to  the 
signs,  will  give  her  approximate  latitude. 

To  the  logarithm  of  the  reduced  parallax  in  seconds,  add  the  log.  sine  of  the  altitude  of 
the  nonagesimal,  the  log.  sine  of  the  moon's  approximate  latitude,  and  the  log.  cosine  of 
the  sum  of  the  arcs  13  and  .^  P  ;  the  sum,  rejecting  30  in  the  index,  will  be  the  logarithm 
of  the  second  part  of  the  parallax  in  latitude  in  seconds,  to  which  prefix  the  sign  —  when 
the  arcs  D  -{-  h  P,  and  the  approximate  polar  distance,!  are  both  greater  or  both  less  than 
!)0°,  otherwise  the  sign  -\-;  this  term,  being  connected  with  the  approximate  latitude,  will 
give  the  apparent  latitude  of  the  moon,+  which  will  be  south  if -f-j  nortii  if — .  The  moon's 
true  latitude  subtracted  irom  her  apparent  latitude,  noticing  the  signs,  will  give  the  parallax 
in  latitude. 

BY  PROPORTIONAL  LOGARITHMS. 

The  above  rule  will  answer  in  calculating  by  proportional  logarithms,  with  the  following 
alterations.  When  the  log.  sine  occurs,  read  log.  cosecant;  for  log.  cosine,  read  log.  secant; 
for  log.  secant,  read  log.  cosine ;  and  for  log.  cosecant,  read  log.  sine.  The  parallaxes  may  be 
calculated  to  the  nearest  second  by  proportional  logarithms.  When  greater  accuracy  is 
required,  common  logarithms  must  be  made  use  of. 

To  illustrate  this  rule,  the  following  examples;  corresponding  to  the  timesof  the  beginning 
and  end  of  the  total  eclipse  of  the  sun,  of  June  1(3, 180G,  as  observed  at  Salem,  are  given.  The 
elements  necessary  fur  tiiis  purpose  have  already  been  calculated  in  Problems  i.  and  IV. 
For  greater  accuracy,  the  longitudes  and  latitudes  of  the  moon  are  corrected  for  the  errors 
—  58".5  in  longitude,  and  —  11". 4  in  latitude,  which  were  found  by  comparing  several 
observations  of  the  eclipse  made  at  different  places. 


EXA3IPLE  I. 

Given  the  altitude  of  the  nonagesimal  G7°  58'  50",  its  longitude  G3°  22'  31";  the  longi- 
tude of  the  moon  S'i"^  4;)'  3".5,  her  latitude  24'  27" .4  N.,her  horizontal  parallax  GO'  24". 1  ;  the 
latitftde  of  the  place  of  observation  42'-'  33'  30" ;  required  the  parallaxes  in  longitude  and 
latitude. 

The  correction  in  Table  XXXVIII.  corresponding  to  the  latitude  42°  33'  30",  and  parallax 
GO'  24". 1,  is  5''.G  ;  this, and  the  sun's  horizontal  parallax, 8". 8, subtracted  from  the  moon's  hori- 
zontal parallax,  GO' 24''.1,  leaves  the  rcrfi^cefZ  parallax  GO'  9".7  =  3G09".7.  The  longitude  of  the 
nonagesimal,  03-^22'  31",  subtracted  from  the  moon's  longitude,  83°  49'  3",  leaves  the  moon's 
distance  from  the  nonagesimal,  20°  2G'  32'',  equal  to  the  arc    D,  because  it  is  less  than  180**. 

CALCULATION   BY   COMMON   LOGARITHMS. 


Keduced  parallax              3C09' 
,\lt!tude  iKMiaKesimal     C7  58 
j)'s  true  latitude                  24 

.7 
50 
27.4 

32 

29 

1 

47 
19 

46.8 
3.5 
50.3 

Log. 
Sine 
Sec. 

Sine 
Log. 
Sine 

Log. 
Sine 

Log. 

3.55747 
9.90710 
10.00001 

Reduced  parallax 
Altitude  nonagesimal 
D's  app.  latitude 

1  part  paral.  1353".3  = 
])  's  true  latitude 

])  's  approx.  latitude 

Reduced  parallax 
Altitude  nonagesimal 
D-f  AP 

2  part  parallax 

Approx.  latitude 

5  '3  app.  latitude 

The  sun'B  parallax 
it  will  be  more  accura 

30Q9".7 
G7  53  50 

=  +  ^>'  33".3 
—  24  27  .4 

Log.  3..55747 
Cosine  9.57394 
Cosine  10.0000* 

Constant  log. 

D                                     20  20 

3.52453 
9.54315 

3.00773 

9.54970 
3.52458 

3.07428 

9.54980 
3.524,58 

3.07438 

Log.   3.13141 

.\ppr.  paralla.t       1  ICO"  =19 

—    1   54  .1 

Sine      6.743 

D  +  Appr.  parallai       20   46 
Constant  log. 

Cor.  paralla.x     =  1187'' =  19 

D  +  cor.  parallax           20  40 
Constant  log. 

Par.  long.       P   1180".8  =  19 

D's  true  longitude        83  49 

I) 's  app.  longitude         84     8 

20  3G  25 

—  1  .7 

—  I  .54.1 

—  1  55  8 

■brmerly  used 
le  to  use  8". 6, 

Log.      3.557 
Sine      9.967 
Cosine  9.971 

Log.      0.238 

or     1'  55".8  N. 

i.s  above,  is  8".8  j 
as  in  the  rule. 

*  In  solar  eclipses,  the  apparent  latitude  is  so  small  that  its  log.  cos.  may  be  put  equal  to  10.00000.     In  occul- 

Catioiis,  you st  calculate  the  first  part  of  the  parallax  in  altitude  by  approximation,  making  use  of  the  true 

latitude  instead  of  the  apparent  in  the  above  rule,  and  deducing  the  approximate  value  of  the  first  part  ol 
the  parallax;  this  applied  to  the  t^ue  latitude  will  give  the  approximate  apparent  latitude,  with  which 
the  operat.on  is  to  be  repeated,  and  the  first  part  of  the  parallax  will  be  obtained  to  a  autficient  degree  of 
exactness. 

fThe  apparent  polar  distance  is  found  by  adding -j- 90' to  the  approximate  latitude,  due  regard  being 
had  to  the  signs.  To  be  perfectly  accurate,  the  apparent  instead  of  the  approximate  latitude  ought  to  be 
made  u.=e  of  in  this  part  of  the  calculation,  and  the  logaritlnus  of  this  term  ought  to  be  increased  by  the  log. 
secant  les?  radius  of  .|  P  ;  but  these  corrections  are  too  small  to  affect  the  result.  In  calculating  the  second 
part  of  the  parallax  in  latitude,  it  will  be  sufficient  to  take  the  logarithm  to  three  or  four  places  of  the 
decimals. 

J  Th  s  rule  g-ves  the  apparent  latitude  in  all  cases  ;  but  it  may  not  be  amiss  to  observe,  that,  in  several  late 
publicat  ons,  the  cases  where  the  moon  is  between  the  zenith  and  the  elevated-pole  are  by  mistake  neglected. 


408 


TO   FIND   THE   PARALLAXES   OF  THE   MOON. 


EXAMPLE  II. 

Given  tlie  altitude  of  the  nonagesimal  70°  57'  4G",  its  longitude  05°  20'  30"  ;  the  longi- 
tude of  the  moon  85°  29'  32".0,  her  latitude  15'  10" .4  N.,  her  horizontal  j)arallas  00'  27".0  ; 
tlie  latitude  of  the  place  of  observation  42°  33'  30" ;  required  the  parallaxes  in  longitude  and 
'atitude. 

The  correction  in  Table  XXXVHI.,  corresponding  to  the  latitude  42°  33'  30",  and  paral- 
lax 60'  27",  is  5".0 ;  this,  and  the  sun's  horizontal  parallax,  8".8,  subtracted  from  the  moon's 
horizontal  parallax,  00'  27".0,  leaves  the  reduced  parallax  00'  12". 0.  Tiie  longitude  of  the 
nonagesimal,  05°  20'  30",  subtracted  from  the  moon's  longitude  increased  by  3()0°,  viz. 
445°  2'.)'  33",  leaves  the  vioon's  distavcc  from  the  nonagesimal  350°  2'  57",  the  supplement 
of  which  to  i;00°  is  9°  57'  3'',  equal  to  the  arch  D. 


Reduced  [laialliix 
Altitude  iioiKigcs. 
J>'s  true  latitude 

70 

60' 
15 

12' 
40 
10 

.G 
4 

Cciiistaiit  log. 
D 

9 

57 

3 

Approv.  purallux 

9 

50 

n-f-a|ipr.  parallax 
Constant  log. 

10 

6 

53 

Corrected  parallax 

10 

0 

D-f-C(ir.  parallax 
Constant  log. 

10 

7 

3 

Par.  long.  P 

10 

00 

D's  true  longitude 

8.5 

29 

32 

.G 

BY  PROPORTIONAL  LOGARITHMS 
Prop.  Log.  0.4753 
Cosecant   10.0214 
Cosine        lO.OD-aO 


0.5000 
Cosecant   10.7024 


Prop.  Log.  1.2024 

Cosecant   10.7.551 
0.5000 


Prop.  Log.  1.2554 

Cosecant    10.7553 
0.51)00 


Prop.  Log.  1.25.53 


Reduced  parallax         00'  12".6 
Altitude  nonages.  70  57   4G 
])'s  app.  latitude 


Prop.  Log.  0.4756 

Seiaul        10.4865 
Secant        10.0000 


1  part  par.  lat. 

Sr  19  38  .5 

Prop.  I-og.  0.9C21 

I)'s  true  latitude 

—  15   10  .4 

D's  appro.x.  lat. 

-f    4  2«  .  1 

Cosecant   12.8361 

Reduced  jiar. 
Altitude  nonages. 
D4-5  P 

10    2     3 

Prop.  Log.  0.475G 
Cosecant    10.0244 
Secant       10.0067 

2  part  par.  lat. 

-\-         4  .4 
+    4  28  .1 

Prop.  Log.  3.3928 

Approx.  latitude 

Appare.1t  lat. 

+    4  32  .5 

or     4'  32" .5  S. 

5 's  app.  longitude  85   19  32.6  I 

EXAMPLE  III. 

Required  the  parallaxes  in  longitude  and  latitude  at  the  time  of  the  occultation  of  Spica 
December  12,  1808.  at  the  times  and  places  mentioned  in  the  Example  of  Problem  VIL 


Reduced  parallax  .50'  50". 9 

Alt.  nonagesimal      81  17  32 

J's  true  latitude        1  55  11 

Constant  log. 

D                                50  52     1 

Approx.  parallax  45  55 

D  -{■  appr.  parallax  51  37  .56 
Constant  log. 

Correited  parallax  46  25 

D  -f-  cor.  par.allax    51  38  26 
Constant  log. 

Par.  long.  P              -|-  43  25 

>  's  true  longitude  200  7  56  .3 


>  's  app.  long. 

200 

54 

21 

3 

Reduced  p.arallax 
.111.  nonagesimal 
D's  true  latitude 

74 

1 

.59 
35 
51 

53 
18 
29 

.0 
1 

Coiuitant  log. 
D 

35 

22 

38 

App».  parallax 

33 

26 

D  -|-  appr.  par. 
<.\instant  Log. 

35 

56 

4 

Corrected  parallax 

33 

54 

D  -\-  cnrr.  )iar. 
-Constant  log. 

35 

56 

32 

Par.  long.  P 
1>'8  true  long. 

+ 
200 

33 
51 

54 
36 

1 

>'3  iipp.  long. 

201 

25 

30 

1 

Prop.  Lo 
Cosecant"  10.0050 
Cosi::e         9.9908 


IMMERSION. 

0.4782 


Cosecant 

4830 
10.1103 

Prop.  Log 

5933 

Cosecant 

10.1057 
4830 

Prop.  Log 

5887 

Cosecant 

10.1056 
4830 

Prop.  Log 

5836 

])  's  app.  latitude* 

1  part  par.  lat.      -}-  9'  311.3 

D's  true  lat.          -f  1  .55  11  .0 

J's  approx.  lat.   -]-  2  4    14  .3 

Reduced  parallax 
Alt.  nonagesimal 
D  +  i  P 


0.4783 

.Secant       10.8199 
Secant        10.0003 


51  15   13 

2  part  par.  lat.      -f  1  20  .3 

])'s  approx.  lat.    -f   2  4   14  .3 

D'sapp.  lat.         -|-   2  5  34  .6  South. 

5's  par.  latitude  -}-  10  23  .6 


Prop.  Log.  1.2984 


Cosecant   11.4421 

Prop.  Log.  0.4789 
Cosecant  10.0050 
Secant       10.2035 


Prop.  Log.  2.1288 


EMERSION 
Prop.  Log.  0.4780 
Cosecant   10.0159 
Cosine         9.9998 


Cosecant 

0.4937 
10.2374 

Prop.  Log 

7311 

Cosecant 

10.2315 

4937 

Prop.  Log 

72.52 

Cosecant 

10.2314 

4937 

Prop.  Log.     7251 


0.4780 

10  5755 

])  's  approx.  latitude 

Secant 

10.0003 

1  part  par.  lat.      -f- 

15 

54".2 

Prop.  Log 

1.0538 

5 's  true  lat.         -(- 

1 

51 

29  .1 

])  's  approx.  lat.   -\- 

2 

7 

23  .3 

Co.secant 

11.4313 

Eeduced  parallax 
Alt.  nonagesimal 
D-i-iP 

!5 

39 

35 

Propu  Log 
Cosecant 
Secant 

0.4780 
10.01.59 
10.0902 

2  part  par.  lat.      -f- 

1 

44  .2 

Prop.  Log 

2.0154 

])  's  approx.  lat.    -|- 

2 

7 

23.3 

B  's  apjiar.  lat.     -J- 

2 

9 

7  .5 

South. 

B  's  parallax  lat.  -|- 

17 

38.4 

*  The  moon's  true  lat'tude,  1°  5.5'  II",  must  first  be  used,  its  log.  secant  being  10. 00112,  which  give  the  Isl 
part  p:irall;>\  9'  '.V,  which,  added  to  the  true  latitude  of  the  moon,  gives  the  'aj)prox;nuite  lat  tude  nearly 
a  4'  14",  the  log.  secant  of  which  is  10.0003,  as  above.  The  calculation  for  the  emersion  is  made  in  asimilaj 
manner. 


TO  FIND  THK  LON'GITUDE  BY   AN  ECLIPSE   OF  THE   SUN.        409 

Having  thus  explained  the  method  of  calculating  the  parallaxes  of  the  moon,  it  now  re- 
mains to  give  the  rules  for  finding  tJie  longitude  by  eclipses  and  occultations.  The  main 
object  in  tliese  calculations  is  to  determine,  from  tiie  observed  beginning  or  end  of  tlie  eclipse 
or  occultation,  the  precise  time  of  the  ecliptic  conjunction  of  the  sun,  or  star  and  moon,  free 
from  the  effects  of  parallax,  counted  on  the  meridian  of  the  place  of  observation,  since  the 
difference  of  the  times  of  conjunction,  obtained  in  this  manner  at  two  places,  will  be  their 
difference  of  longitude.  If  the  lunar  and  solar  tables  were  perfectly  correct,  t!ie  longitude 
might  Le  determined  by  taking  the  difference  between  the  time  of  conjunction  given  in  the 
Nautical  Almanac,  and  that  deduced  from  the  observations  of  the  eclijise  or  occultation  ;  but 
it  is  much  more  accurate  to  compare  the  times  deduced  from  observations  actually  made  at 
the  places  for  v\-hich  the  difference  of  longitude  is  sought.  There  are  two  different  methods 
of  finding  the  ecliptic  conjunction,  according  as  the  latitude  of  the  moon  is  supposed  to  be 
accurately  known  or  not.  If  the  latitude  was  given  correctly  by  the  lunar  tables,  or  was 
accurately'  known  by  other  observations,  the  ecliptic  conjunction,  and  the  longitude  of  the 
place,  niigiit  be  determined  by  each  of  the  phases  of  the  eclipse  or  occultation,  by  tlie  method 
given  in  Problems  VIII.  and  IX.  But  the  moon's  latitude  not  being  generally  given  to  a 
sufiicient  degree  of  accuracy,  it  is  usual  to  combine  together  the  observations  of  the  begin- 
ning and  end  of  the  eclipse  or  occultation,  or  the  beginning  and  end  of  total  darkness  in  a 
total  eclip.se,  or  the  two  internal  contacts  of  an  annular  eclipse,  to  ascertain  the  error  of  the 
moon's  latitude,  by  the  method  given  in  Problems  VI.  and  VII.  In  making  the  calculations 
in  these  Problems,  it  will  be  necessary  to  know  nearly  the  longitude  of  the  place,  in  order 
to  find  tiie  supposed  time  at  Greenwich,  so  as  to  take  out  the  elements  from  the  Nautical 
Almanac  ;  and  if  the  longitude  deduced  from  the  observation  should  diller  considerably,  the 
operation  must  be  repeated  with  the  longitude  obtained  by  this  operation. 

PROBLEM  VI. 

Given  the  latitude  of  the  place,  and  the  apparent  times  of  the  beginning  and  end  of  a  solar 
eclipse,  counted  from  noon  to  noon,  according  to  the  method  of  astronomers,  tofnd  the 
longitude  of  the  place  of  observation. 

In  the  rule  for  solving  this  problem,  references  will  be  made  to  figure  12,  Plate XIII,  in 
which  DSE  represents  a  small  arc  of  the  ecliptic;  S,  the  place  of  the  centre  of  the  sun 
supposed  at  rest ;  F,  L,  the  apparent  places  of  the  centre  of  the  moon  at  the  beginning  and 
end  of  the  eclipse  respectively  5  FD,  SC,  and  AEL,  are  perpendicular  to  UE  ;  FA  parallel 
to  DE,  and  SB  perpendicular  to  FL.  Then  it  is  evident  tiiat  FD,  LE,  represent  the  apparent 
latitudes  of  the  moon,  which  fall  below  DE  if  south,  above  if  north  ;  and  SF,  SL,  represent 
the  sums  of  the  corrected  semi-diameters  of  the  sun  and  moon,  at  the  beginning  and  end  of 
tlie  eclipse  respectively. 

RULE.* 

To  tlie  apparent  times  of  the  beginning  and  end  of  the  eclipse,  add  the  estimated  longitude 
of  the  place  in  time  if  it  is  icest,  but  subtract  if  east ;  the  sum  or  difference  will  be  the  sup- 
posed time  at  Greenwich,  corresponding  to  which,  in  the  Nautical  Almanac,  find,  by  Prob- 
lem I.,  tiie  moon's  semi-diameter,  horizontal  parallax,  longitude  and  latitude,!  and  the  sun's 
semi-diameter,  longitude,  and  right  ascension  ;  also  the  moon's  horary  motion  from  the  sun, 
by  Problem  II.  Decrease  the  sun's  semi-diameter  3^"  for  irradiation,  and  the  remainder 
will  be  his  corrected  semi-diameter.  Decrease  the  moon's  semi-diameter  2"  for  inflexion,  if 
it  be  thought  necessary,  and  to  the  remainder  add  the  correction  in  Table  XLIV.;t  the  sum 
will  be  tiie  moon's  corrected  semi-diameter.  Find  also,  in  the  Nautical  Almanac,  the  ob- 
liquity of  the  ecliptic. 

With  these  elements,  and  the  apparent  time  at  the  place  of  observation,  calculate  the  alti- 
tudes and  longitudes  of  the  nonagesimal,  by  Problem  IV.  ;  the  parallaxes  in  longitude  and 
latitude,  and  the  moon's  apparent  longitudes  and  latitudes,  by  Problem  V. 

Take  tlie  difference  between  the  apjiarent  longitudes  of  the  moon  at  the  bcginninT  and 
end  of  the  eclipse,  and  subtract  therefrom  the  difference  of  the  sun's  longitudes  at  the  same 
time;  the  remainder  will  be  the  relative  motion  in  longitude  DE  or  FA.  'i'lie  relative  motion 
in  latitude  AL  is  found  by  taking  the  difference  of  the  moon's  apparent  latitudes  at  the 
beginning  and  end  of  the  eclipse,  if  they  are  botii  north,  or  both  south,  but  their  sum,  if  one 
be  north,  the  other  south.  From  the  logarithm  FA,  increasing  tlie  index  by  10,  subtract  the 
logaritliiii  of  AL  ;  the  remainder  will  be  the  log.  tangent  of  the  an<rle  of  iiirUnation  DSB; 
this  an^Ie  is  to  be  taken  greater  than  DO'^,  when  the  moon's  apparent  latitude  FD,  at  the 
beginning  of  the  eclipse,  is  greater  than  at  the  end  EL,  otherwise  less.§     Then  to  the  log. 

*  Til's  rule  is  peciirarly  adapted  to  the  use  of  the  longitudes  and  lat'tiides  of  the  tiodies.  We  shall  here- 
after "ive  the  methods  of  |)eifiirniin>;  the  same  lalculatioiis  by  jnt^aiisof  the  right  ascensions  and  dei  linations, 
adaptMijr  the  rules  to  the  new  form  of  the  Nautical  Almanac.  The  same  is  to  be  observed  relative  to  the  fol- 
lowin;;  Problems,  VH.  VIU.,  &c. 

t  Corre:  led  for  the  errors  of  the  tables  in  lonjilnde  and  latitude,  when  known. 

1  This  correition  must  be  found  after  the  altitude  and  longitude  of  the  nonaijesimal  are  calrulated. 

5  Th's  rule  is  equally  true,  whether  the  latitude  be  of  the  same  or  dfTereut  names,     [f  the  latitudes  are 
equal,  and  of  the  sajiie  name,  the  angle  DSB  wdl  be  90^.     If  they  are  equal,  but  of  d  fierent  nanu^s,  the  angle 
DSB  may  be  taken  at  ute  or  obtuse,  since,  in  that  rase,  the  ans;le  FSB  is  flO".     Striitly  speaking,  when  the 
points  F   li   fall  on  dlTerent  sdes  of  the  line  DE.  the  angle  DSB  is  greater  or  Ics-^  than  90°,  according  as  tb 
5"2 


410        TO  FIND  THE  LONGITUDE  BY  AN   ECLIPSE  OF  THE  SUN. 

cosecant  of  the  angle  of  inclination,  add  the  logarithm  of  the  relative  motion  in  longitude 
FA;  the  sum,  rejecting  10  in  trie  index,  will  be  the  logarithm  of  the  apparent  motion  of  the 
moon  FL  on  lier  relative  orbit.  Then,  in  the  triangle  SFL,  the  sides  bF,  SL,  represent  the 
sums  of  tiie  corrected  semi-diameters  of  the  sun  and  moon  at  the  beginning  and  end  of  the 
eclipse,  and  these,  with  the  relative  motion  FL,  are  given  to  find  the  angle  FSB  (^by  Case 
VI.  Obi.  Trig.)  Thus,  to  the  log.  arith.  comp.  of  FL,  add  the  logarithm  of  the  sum  of  SF 
and  SL,  and  the  logarithm  of  their  difference;  the  sum,  rejecting  ]0  in  the  index,  will  be 
the  logaritlim  of  the  difference  of  the  segments  FB,  BL ;  half  of  which,  beino-  added  to  and 
subtracted  from  half  of  FL,  will  give  the  two  segments  FB,  BL  ;  the  greater  segment  being 
contiguous  to  the  greater  side,  whether  SF  or  SL.  Then,  from  the  logarithm  ol'  the  segment 
FB,  increasing  the  index  by  10,  subtract  the  logarithm  of  SF;  the  remainder  will  be  the  log. 
sine  of  the  angle  FSB,*  which  is  always  less  than  90" ;  the  difference  betu'een  this  and  the 
angle  of  inclination  DSB  will  be  the  central  angle  DSF. 

To  the  log.  cosine  of  the  central  angle,  add  the  logarithm  of  the  sum  of  the  corrected  semi- 
diameters  at  the  begiiining  of  the  eclipse  SF,  rejecting  10  in  the  index ;  the  sum  will  be  the 
logarithm  of  SD,  tlie  ap])arent  difference  of  longitude  of  the  sun  and  moon  at  that  time. 
This  is  to  be  subtracted  from  the  longitude  of  the  sun  at  the  beginning  of  the  eclipse,  if  the 
central  angle  is  less  than  !)0",  but  added  if  greater  than  90°;  the  sum  or  difference  will  be  the 
moon's  apparent  longitude  :  to  this  must  be  added  the  moon's  parallax  in  longitude,  when 
her  distance  from  the  nonagesimal  (found  as  in  Problem  V.,  by  subtractinjr  ihe  longitude 
of  the  nonagesimal  from  the  moon's  longitude,  borrowing  3C0°  when  necessary)  is  greater 
than  180'^ ;  otherwise  the  parallax  must  be  subtracted  ;  the  sum  or  difference  will  be  the 
moon's  true  longitude  at  the  beginning  of  the  eclipse. 

Take  the  difference  in  seconds  between  the  sun's  and  moon's  true  longitudes  at  the  be- 
ginning of  the  eclipse,  to  the  logarithm  of  which  add  the  arith.  comp.  logarithm  of  the  moon's 
horary  motion  from  the  sun  t  in  seconds,  and  the  constant  logarithm  3.5r<(J30  ;  the  sum,  re- 
jecting 10  in  the  index,  will  be  the  logarithm  of  the  time  from  the  conjunction  in  seconds, 
which  is  to  be  added  to  the  observed  apparent  time  of  the  beginning  of  the  eclipse,  when  the 
sun's  longitude  at  that  time  is  greater  than  the  moon's  true  longitude,  otherwise  subtracted; 
the  sum  -jx  difference  will  be  the  apparent  time  of  the  true  ecliptic  conjunction  of  the  sun 
and  moon  at  the  place  of  observation.  The  difference  between  this  and  the  time  of  con- 
junction at  Greenwich,  inferred  from  the  Nautical  Almanac  by  Problem  111.,  will  be  the 
longitude  of  the  place  of  observation.  But  if  corresponding  observations  have  been  made  at 
different  places,  it  will  be  much  more  accurate  to  find  the  times  of  the  conjunction  at  each 
place  by  the  above  rule;  and  the  difference  of  these  times  will  be  the  difference  of  meridians, 
if  it  does  not  differ  much  from  the  supposed  difference  of  longitude.  If  there  is  considerable 
difference,  the  operation  must  be  repeated,  making  use  of  the  longitude  found  by  this  opera- 
tion ;  and  thus,  by  successive  operations,  the  true  longitude  may  be  obtained. 

The  longitude  of  the  place  of  observation  being  accurately  known,  the  errors  of  the  lunar 
tables  in  longitude  and  latitude  may  be  easily  found.  For  the  difference  between  the  moon'a 
true  longitude  deduced  by  the  above  method  from  the  observations,  and  the  longitude  found 
from  tlie  Nautical  Almanac,  Vv'ill  be  the  error  of  the  tables  in  longitude.  To  find  the  error 
in  latitude,  add  the  log.  sine  of  the  central  angle  DSF  to  the  logarithm  of  the  sum  of  the 
corrected  seini-diameters  at  the  beginning  of  the  eclipse  SF ;  the  .sum,  rejecting  lU  in  the 
index,  will  be  the  logarithm  of  the  moon's  apparent  latitude  FD  at  that  time;  which  will 
be  south,  if  the  point  F  falls  below  D,  otherwise  north.  Take  the  difference  between  this 
and  the  moon's  apjjarent  latitude,  found  by  Problem  ^ .,  if  they  are  both  north,  or  both  south ; 
but  their  sum,  if  one  be  north  and  the  other  south;  and  the  error  of  the  tables  in  latitude 
will  be  obtained.}; 

REM.4.RK. 

The  above  rule  will  answer  for  deducing  the  longitude  from  the  observed  beginning  and 
end  of  the  internal  contacts  of  a  total  or  annular  eclipse.     The  differences  consist  in  reading 

FD  EL 

expression  -—  is  greater  or  less  tlian  ■:; —  ;  but,  as  the  divisors  SL  and  SF  are  nearly  equal,  they  may  be  neg- 
lected (as  ill  the  above  rule),  exrept  in  a  ease  wliicli  very  rarely  occurs,  namely,  wlien  the  difference  of  SL, 
SF,  is  greater  thiiii  the  dtiereiice  of  tlie  two  anpareiil  latitudes  EL,  FD,  in  wiiixih  case  the  rule  in  this  note 

EL    FD 
must  be  made  use  of;  observing  that  the  fractions  , ' — ,  represent  the  quotients  of  the  moon's  ajiparent 

latitudes  divided  by  the  sum  of  the  semi-diameters  of  the  sun  and  moon. 

*  When  SF,  SL,  al'e  eijiial,  or  tlieir  difference  is  so  small  that  it  may  be  neglected,  the  log.  sine  of  the  an- 
f[le  FSI)  may  be  <il)laiiuHl  mm  h  more  expeditiously  by  subtracting  the  logarithm  of  the  sum  of  SF  and  SIj 
from  the  bignritlim  of  FL,  increasing  ihe  index  by  10.  This  method  may  almost  always  be  made  use  of  with- 
out much  error.     It  is  llie  rule  adopted  by  Doctor  Mackay  in  liis  treatise  on  longitude. 

f  When  the  horary  motion  varies,  it  must  be  taken  to  correspond  to  tlie  middle  time  between  the  begin 
ning  of  the  eclipse  and  the  coiijiini  tion  or  new  moon. 

X  When  tlie  e(  lipse  or  occiihation  is  nearly  central,  or  (in  other  words)  when  FD,  EL,  are  very  small  in 
comparison  w.tli  SF,  tlie  buitude  tliiis  found  cannot  be  depended  on,  as  a  small  error  in  the  times  of  observa- 
tion Will  produce  a  considerable  error  in  the  latitude.  Indeed,  the  case  may  occur,  when  FD,  EL,  are  less 
than  30",  that  it  may  be  iinicrta  n  whether  the  points  F,  L,  fall  above  or  below  the  Ine  DE,  because  the 
error  of  the  lunar  tables  in  lat  tiide  may  sometimes  be  equal  to  30".  In  this  case,  the  correct  hUitiule  of  the 
moon  may  be  found,  (I.)  Ry  observations  made  at  another  iilace,  where  the  eclipse  or  occiiltation  was  not  so 
central  ;  (2.1  By  tlie  number  of  dig.ts  ei  lipsed,  if  it  was  a  solar  eclipse  ;  (3.)  15y  the  d  fferen(;e  of  declina- 
tions of  the  moon  and  star,  observed  before  and  after  the  immersion  or  emersion  ;  (4.)  I5y  the  meridian  alti- 
tude of  the  niDOM  oliscrveil  the  same  day,  whence  it  may  be  found  whether  the  nicon  v  as  north  or  south  ul 
her  place  given  by  the  tables 


TO  FlISD  THE  LONGITUDE  BY  AN  ECLIPSE  OF  THE  SUN. 


4ia 


the  rule,  beginning  and  end  of  the  internal  contacts,  instead  of  beginning  and  end  of  the 
eclipse,  and  taking  SF,  SL,  equal  to  the  differences  of  the  corresponding  senii-dianieters, 
instead  of  their  sums. 

EXAMPLE. 

At  Salem,  in  the  latitude  of  42"  33'  30"  N.,  longitude  by  estimation  4h.  43m.  32s.  W. 
from  Greenwich,  tiie  beginning  of  tlie  total  eclipse  of  June,  ISOG,  was  observed  at  Ifid.  22h. 
6in.  18s. 1,  and  the  end  at  the  Kid.  Oh.  .50m.  34s. (J,  apparent  time,  by  astronomical  computa- 
tion.    Required  the  longitude  of  the  place  of  observation. 

Most  of  the  followinrr  elements  are  calculated  in  Problems  L  II.  IV.  V. 


EI.EMEXTS   OF  THE   ECLIPSE. 


Beginning. 


Apparent  times  of  observation  at  S:i!eni 

EstlmateJ  longitmie  VV.  from  Greenwicli 

Supimseil  apparent  time  at  Greeiuvicli 

0's  ri{:lit  ascension 

Lat.  of  place  40'  33'  30"  —  Keduct:on  in  Talile  XXXVIII.  11'  2i3"  , 

01)li(piity  of  the  ecliptic 

D  '3  lon>;.  hy  N.  A.  —  Err.  'I'aljle  58". 5  =  True  long.  ])  I'rob.  I 

Jjongitude  of  the  nonagesinial,  by  I'roh.  IV 

J) '3  true  long.  —  Long,  nonagesinial  =  5 's  dist.  from  nonagesimal 
This  distance,  or  its  snpiilenient,  if  greater  than  180°,  is  arch  D... 

Altitude  of  nonagesimal,  I'rob.  IV 

])'s  horizontal  parallax,  l)y  Prob.  I 

—  O's  hor.  par.  8".8*  —  Correction  Table  XXX VIII.  3".G 

Reduced  paralla.v 

D's  semi-diameter  by  N.  A.  —  Inlle.\ion  2" 

Add  correction  Table  XLIV 

5  's  corrected  semi-diameter 

O's  semi-diameter  by  N.  A.  15' 4C".l  —  Irradiation  3".5 

Snm  of  the  corrected  semi-diameters 

D's  horary  motion  in  longitude  by  I'rob.  II.  Example  II 

0's  horary  motion 

D's  horary  motion  from  the  sunf 

D's  parallax  in  longitude  P 

D  's  apparent  longitude  —  Error  Table  58".5  by  Prob.  V 

0's  buigi tilde  by  I'rob.  I 

Difference  D  ';>"  aj>p.  longitade  =  D  's  app.  motion 

Difference  Q)'^  lungiludcs  =r  ©'s  ajjp.  motion 

Difference  of  motiuns  of  Q  ]) 

D's  true  lat.  by  N.  A.  Prob.  I.  —  Error  Tal)le  11".4 

D's  app.  lat.  corr. -for  error  Table  11". 1  by  Prob.  V 

D  's  latitude  at  end  —  Latitude  at  beginning 


IG 


d.   h.  m. 

15  22  6 

4  43 
2  49 

5  36 
42»22' 
23  27 
83  49 
63  22 
20  2t) 
20  26 
67  58 

60 

60 
16 


SF  = 


16 

15 

.32 

36 

2 

34 

19 

84     8 

84  41 


s. 

18.1 
32 
50.1 
50.0 

4" 
48 

3.5 
31 
33 
32 
50 
24.1 
14.4 

9.7 
25.7 
15.2 
40.9 
42.6 
2;J.5 
39.2 
23.1 
Ifi.l 
46.8 
50.3 

3.4 


—  24  27.4 
FD=—     1   55.8 


h.  m.    s. 
0  .50  34.6 

4  43  32 

5  34     6.6 
5  37   18.5 


85  29  32.6 

95  26  36 

350     2  57 

9  57     3 

70  57  46 

60  27.0 

—  14.4 

60  12.6 

16  26.4 

16.4 

16  42.8 

15  42.6 

32  25.4 

36  42.8 

2  23.1 

34  19.7 

10     0.0 

85   19  32  6 

84   47  35.5 

1   10  42.3 

6  32.1 

64  10.2 

—  15  10.4 

EL  =  -|-     4  32.5 

Al.  =  4-     6  28.3 


SL  = 


FA 


As  the  apparent  latitude  at  the  beginning  of  the  eclipse  is  north,  and  at  the  end  south,  tho 
point  F  corresponding  to  this  example  falls  above  DE,  the  point  L  below  it.  The  rest  of 
the  calculation  is  as  follows  : — 


6.41232 

Log.    3..5S983 
Log.    0.27875 


FA64' 10".2=3S.50".2  Log.  13.58.548  3.58548 

AL  6  08.3  =  338  .3  Log.  2..58917 

Inclination  84' 14'... Tan.  10.99631  Cosecant  10.00220 

Apparent  mnlion  FL 3S69".7         Log.    3.58768 

Its  arith.  comp 

SF+SL  =  64' 48".9 3388".9 

Diff.  SF.SL 1.9  

Diff.  segments 1.91        Log.    0.28090 

Its  half 0.95 

Half  of  FL 1934.85 

Sum  is  great  segment 1935  .8 

\y\XX.  is  le.sser  segment  FB.   1933  .9  Log.  13.28644 

•5F  32' 23'i.5  = 1943.5  Log.    3.28853 

Angle  FSB 84°  19'        Sine    9.99786 

Inclination 84    14 

Diff.  is  central  angle  DSF.    ~0 
SF 

SD  =  32'23I5  = 1943".5         Log.    3.28853 


Cosine  10.00000 
Log.    3.2(<858 


O's  longitude 84°  41'  3".4 

SU _  32  23  .5 

D's  app.  longitude 84     8  30  .9  by  obs. 

D's  par.  longitude —   19  46.8 

D's  true  longitude 83  48  53.1 

O's  longitude 84  41     3  .4  Const.  3.55630 

Difference  31.30".3 =  .50    10.3  Log.  3.49555 

D  's  hor.  mot.  from  O  34'  17".l=2U57".l  A.C.  6.6»675 

h.  m.     1.  

Time  from  conj.         13118.1=5478".!  Log.  3.73863 
App.  time  obs.    15  22    6  18.1 

App.  time  conj.  15  23  37  36.2  at  Palem. 
Conjunction       16    4  19  at  Greenwich. 

Diff  Mejid.  4  41  23.8 


Sine  7  16270 

Log.  3.288.58 

App.  lat.  FD=:2".8..  Log.  0.45128 


*  The  mean  parallax  formerly  used  was  8". 8  :   it  is  now  found  to  be  nearly  8".6. 

t  This  horary  motion  increases  from  34'  16".l  to  34'  19".7,  or  3".6,  during  the  eclipse  2h.  41m.  16s,5,  which 
is  1".32  per  hour.  Now  the  ecli|)tic  conjun(  tion,  or  time  of  new  moon,  at  Creeiiwich,  by  the  N.  A.,  was 
4h.  19m.,  or  rather  4h.  20m.  473.,  corresponding  to  2.3h.  37m.  ISs.  at  Salem,  which  is  Ih.  .^Om.  .57s.  after  the 
beginning  of  the  eclipse;  and  the  increase  of  the  horary  motion  in  half  that  time  is  1",  which,  added  to 
34'  I6".l,  gives  the  horary  motion  34'  17". 1,  corresponding  in  the  middle  time  between  the  beginning  of  the 
eclipse  and  the  conjunction.  This  is  used  in  calculaling  the  correct  time  of  conjunction.  We  mav  remark 
that,  in  Ilie  above  calculations,  wehave  used  the  apparent  times  of  observation,  to  confurm  to  the  arrangement 
of  the  Nautical  .Almanac  in  1806 ;  but  in  the  present  form  of  the  Nautical  .-Mmi 
U8e  the  viean  time. 


.•\lmanac,  it  will  be  convenient 


412  TO  FIND  THE  LONGITUDE   BY  AN  OCCULTATION. 

In  finding  tlie  time  of  conjunction  or  new  moon,  at  Greenwich,  4h.  19m.,  in  the  Nautical 
Almanac,  the  longitude  of  the  moon  was  supposed  to  be  given  correct)}'  by  the  tables.  If  the 
calculation  be  made  by  Problem  111.,  after  allowing  for  the  error  — 5b". 5,  the  result  will  be 
4h.  20m.  47s.,  whence  the  difference  of  meridians  =4h.  43m.  10s. 8,  which  differs  so  little 
from  the  assumed  longitude,  4h.  43m.  o2s.,  that  it  will  not  be  necessary  to  repeat  the  operation. 
If  the  eclipse  was  observed  at  Greenwich,  the  time  of  conjunction  ought  to  be  determined 
thereby,  in  a  similar  manner  to  the  above  calculations  ;  or  by  those  of  Problem  Vlll.,  if  onl}' 
one  of  the  phases  is  observed :  by  this  means  the  errors  of  the  tables  will  be  wiolly  avoided. 
If  the  eclipse  v.'ss  not  observed  at  Greenwich,  the  observations  at  any  other  place  whose  longi- 
tude is  known  might  be  made  use  of,  and  thus  the  difference  of  meridians  accurately  obtained. 

The  moon's  true  longitude,  deduced  from  the  above  observation,  is  83P  48'  53". 1  ;  by  the 
Nautical  Almanac  it  is  83°  50'  2".0 ;  the  difference,  —  68". 9,  would  be  the  error  of  the  tables 
by  this  observation,  if  the  assumed  longitude,  4h.  43'  32",  and  the  solar  tables,  were  correct. 
By  repeating  the  operation  with  the  assumed  longitude,  4h.  43m.  lOs.8,  the  error,  68". 9,  would 
be  reduced  to  nearly  the  estimated  value,  58". 5. 

The  eclipse  was  so  nearly  central  at  Salem,  that  a  variation  of  a  minute  in  the  moon's  lati- 
tude would  hardly'alter  the  times  or  duration  of  the  eclipse ;  so  that  the  latitude  could  not 
be  determined  by  the  above  observations  to  any  considerable  degree  of  accuracy.  From  this 
cause  it  happens  that  the  apparent  latitude  at  the  beginning  of  the  eclipse  is  by  the  above 
calculation  2". 8,  instead  of  1'  55".8,  as  found  by  allowing  the  error,  11".4,  deduced  from  other 
observations  made  where  the  eclipse  was  not  so  nearly  central,  and  by  the  limits  of  the 
shadow  of  total  darkness. 

PROBLEM  VII. 

Given  the  latitude  of  the  place,  and  the  apparent  times  of  the  beginning  and  end  of  an  oc- 
cultation  of  a  fixed  star  by  the  moon,  to  find  the  longitude  of  the  place  of  observation. 

In  the  following  rule,  reference  will  be  made  to  figure  13,  Plate  XIII., in  which  DSE  repre- 
sents a  parallel  to  the  ecliptic  passing  through  the  place  of  the  star  S ;  SF,  SL,  the  corrected 
semi-diamelens  of  the  moon  at  the  beginning  and  end  of  the  occultation ;  DF,  EL,  the  dif- 
ferences between  the  apparent  latitudes  of  the  moon  and  the  star,  when  of  the  same  name, 
or  their  sums,  when  of  diff'erent  names;  either  of  these  lines  falling  ic/oio  DE  if  tlie  moon's 
apparent  latitude  is  more  southerly  than  that  of  the  star,  otherwise  above. 

RULE. 

To  the  apparent  times  of  the  beginning  and  end  of  the  occultation,  add  the  estimated  longi- 
tude of  the  place  in  time  if  it  is  west,  but  subtract  if  east :  the  sum  or  difference  will  be  the 
supposed  time  at  Greenwich  ;  corresponding  to  which,  in  the  Nautical  Almanac,  find,  h~ 
Problem  I.,  tiie  moon's  semi-diameter,  horizontal  parallax,  longitude  and  latitude,*  and  the 
sun's  right  ascension  ;  also  the  moon's  horary  motion  by  Problem  II.,  and  the  true  longitude 
and  latitude  of  the  fixed  star,  by  Table  XXXVIl.,  corrected  for  aberration  and  equation  of 
equinoxes  by  Tables  XL.,  XLI.  This  may  also  be  deduced  from  the  right  ascension  anc. 
declination  of  the  star,  if  it  be  given  in  the  Nautical  Almanac,  by  means  of  Problem  XIX 
of  this  Appendix.  Find,  also,  in  the  Nautical  Almanac,  the  obliquity  of  the  ecliptic.  To  the 
moon's  semi-diameter,  add  the  correction  in  Table  XLlV.,t  and  from  the  sum  subtract  the 
inflexion,  2",  if  it  be  thought  necessary  ;  the  remainder  will  be  her  corrected  semi-diameter. 
With  these  elements  and  the  apparent  times  of  the  place  of  observation,  calculate  the  alti- 
tudes and  longitudes  of  the  nonagesimal,  by  Problem  IV.,  and  the  parallaxes  in  longitude 
and  latitude,  and  the  moon's  apparent  longitudes  and  latitudes,  by  Problem  V. 

Take  the  difference  between  the  oppureiit  longitudes  of  the  moon  at  the  beginning  and  end 
of  the  occultation,  which  will  be  the  moon's  apparent  motion  in  longitude,  the  logarithm  of 
which,  in  seconds,  being  added  to  the  log.  cosine  of  the  meant  of  the  apparent  latitudes  of  the 
moon  at  the  beginning  and  end  of  the  occultation,  rejecting  10  in  the  index,  will  be  tlie  loga- 
rithm of  the  motion  of  the  moon  on  the  parallel  FA.  The  relative  motion  in  latitude  AL  is 
found  by  taking  the  difference  of  the  moon's  apparent  latitudes  at  the  beginning  and  end  of 
the  eclipse  if  they  are  both  north  or  both  south  ;  but  their  sum  if  one  be  north  and  the  other 
south.  From  the  logarithm  of  FA;  increasing  the  index  by  10,  subtract  the  logarithm  of  AL  ; 
the  remainder  will  be  the  log.  tangent  of  the  (mgle  of  inclination  DSB  ;  this  angle  is  to  be 
taken  greater  than  90°  when  the  difference  of  the  moon's  and  star's  apparent  latitudes  at  the 
beginning  of  the  occultation  FD  is  greater  than  at  the  end  EL,  otherwise  less.  §  Then  to 
the  log.  cosecant  of  the  angle  of  inclination,  add  the  logarithm  of  the  relative  motion  FA  ; 
the  smn,  rejecting  10  in  the  index,  will  be  the  logarithm  of  the  apparent  motion  of  tiie  moon 
in  her  orbit  FL. 

*  Correcteil  fur  tlie  errors  of  tlie  tal)les  in  longitude  and  latitude,  when  known. 

t  This  correrlidii  must  be  found  alter  the  alt  tuile  and  longitude  of  the  nonagesimal  are  calculated. 

X  Tlie  menu  hil  lude  is  half  the  sum  of  the  two  latitudes,  if  they  are  of  the  same  name,  but  their  half  differ 
cnc3,  if  of  d  tii-reiil  names.  In  solar  ellipses,  the  correi  tioii  fur  the  mean  latitude  of  the  moon  is  neglected 
as  too  small  to  he  taken  notice  of,  the  d. stance  FA  being  taken  equal  to  the  difference  of  longitude  DE 
(fig.  19.  Plate  Xlll.).  ,        .   ,     .        rvr. 

§  This  rule  IS  ei|ually  true,  whether  the  points  F,  L,  fall  on  the  same  or  on  different  sides  ot  the  line  Ut. 
If  DF,  EL,  areci|u:il,  aild  the  points  F,  L,,  fall  on  the  same  side  of  Ui:,  the  angle  DSB  will  be  90°.  If  they  are 
equal,  and  those  points  fall  on  differtMit  sides  of  the  line  DE,  the  angle  USB  may  be  taken  acute  or  t'btusa 
in  strictness,  when  the  iioints  F,  L,  fall  on  different  sides  of  DE,  the  angle  DSB  is  greater  or  'ess  than  ao* 

Fl)  EI. 

according  as  Hie  niiantilv  —  is  greater  or  less  than  — -. 

*"  '  •    SF  SI. 


TO   FIND  THE   LONGITUDE   BY   AN  OCCULTATION.  413 

Then  in  tlie  triangle  SFL,  the  sides  SF,  FL  (representing  the  corrected  seini-diumeters» 
of  the  moon  at  the  iniinersion  and  emersion),  and  the  relative  motion  FL,  are  yiven  to  find 
tlie  angle  FSB  (by  Case  VL  Oblique  Trig.).  Thus:  to  the  log.  arith.  comp.  of  FL,  add  the 
logarithm  of  ihe  sum  of  SF  and  SIj,  and  the  logarithm  of  their  difference  :  the  sum,  rejecting 
10  in  the  index,  will  be  the  logarithm  of  the  difference  of  the  segments  FB,  BIj ;  half  of  this, 
being  added  to,  or  subtracted  liom  the  half  of  FL,  will  give  the  two  segments  FB,  BL;  the 
greater  segment  being  contiguous  to  the  greater  side,  whether  SF  or  SL.  Then,  from  the 
logarithm  of  the  segment  FB,  increasing  its  index  by  10,  subtract  the  logarithm  of  SF;  the 
remainder  will  be  the  log.  sine  of  the  angle  FSB,*  wliich  is  always  less  than  !)()'-'.  The  dif- 
ference between  this  and  the  angle  of  inclination  DSB,  will  be  the  central  avglu  DSF. 

To  the  log.  cosine  of  the  central  angle  add  the  logarithm  of  the  moon's  corrected  serai- 
diameter  at  the  immersion  SF,  and  the  log.  secant  of  the  star's  latitude  :  the  sum,  rejecting 
20  in  the  index,  will  be  the  logariliini  of  the  apparent  difll-rence  of  longitude  of  the  moon 
and  star  at  that  time.  This  is  to  be  subtract! d  from  the  true  longitude  of  the  star,  if  the 
central  angle  is  less  than  i)0",  but  added,  if  greater  than  90":  'the  sum  or  difference  will  be 
the  moon's  apparent  longitude;  to  this  must  be  added  the  moon's  parallax  in  longitude, 
when  her  distance  from  the  nonagesimal  (found  as  in  Problem  V.,  by  subtracting  the  longi- 
tude of  the  nonagesimal  from  tlie  moon's  longitude,  borrowing  300°  when  necessary)  is 
rrreatcr  than  180",  otherwise  tlie  parallax  must  be  sitUracUd ;  the  sum  or  difference  will  be 
the  moon's  trve  loiigiludc  at  the  begiiming  of  the  occultalion. 

Take  the  diff"ercnce  in  seconds  between  the  true  longitudes  of  the  star  and  moon  at  the 
beginning  of  the  occultation  ;  to  the  logarithm  of  this  add  the  arithmetical  comp.  log.  of  the 
moon's  horary  motion  1  in  seconds,  and  the  constant  logarithm  3.55030  :  the  sum,  rejecting 
10  in  the  index,  will  be  the  logarithm  of  the  time  from  the  conjunction  in  seconds,  which  is 
to  be  a.dded  to  the  observed  apparent  time  of  the  beginning  of  the  occultation,  when  the  star's 
longitude  is  greater  than  the  moon's  true  longitude  at  that  time,  otherwise  sahlructcd  :  the 
sum,  or  diffi>rence,  will  be  the  a])parcnt  time  of  the  true  ecliptic  conjunction  of  the  star  and 
moon  at  the  place  of  observation ;  the  difference  between  this  and  the  time  of  conjunction, 
inferred  from  the  Nautical  Almanac  by  Problem  III.  for  the  meridian  of  Greenwich,  will  be 
the  longitude  of  the  place.  If  corresponding  observations  be  made  at  different  pl.ices,  it  will 
be  much  more  accurate  to  deduce  from  them  the  time  of  conjunction  at  each  place,  and  take 
the  diff'erence  of  those  tiuies  for  the  diff'erence  of  meridians,  if  it  does  not  diff'er  much  from 
the  supposed  difference  of  longitude.  If  there  is  considerable  diff'erence,  the  operation  must 
be  repeated,  making  use  of  the  longitude  found  by  this  operation  ;  and  thus,  by  successive 
operations,  tlie  true  longitude  may  be  obtained. 

The  longitude  of  the  place  of  observation  being  accurately  known,  the  errors  of  the  lunar 
tables  in  latitude  and  longitude  may  be  easily  found.  For  the  difference  between  the  moon  a 
true  longitude,  deduced  from  the  observations  by  the  above  method,  and  the  longitude  found 
from  the  Nautical  Almanac,  will  be  the  error  of  the  tables  in  longitude.  To  find  the  error  in 
latitude,  proceed  thus  :  To  the  log.  sine  of  the  central  angle  DSF  add  the  logarithm  of  the 
corrected  semi-diameter  of  the  moon  at  the  immersion  SF ;  the  sum,  rejecting  10  in  the  in- 
dex, will  be  the  logarithm  of  the  apparent  difference  of  latitude  of  the  moon  and  star,  which, 
being  added  to  the  true  latitude  of  tlie  star,  with  the  sign  -f-  if  the  point  V  falls  licUno  the  line 
DE,  but  with  the  sign  —  \i'  almve,  will  give  the  apparent  latitude  of  the  moon  at  tJiat  time  :  the 
diff'erence  between  this  and  the  apparent  latitude,  found  by  Problem  V.,  will  be  the  error  of 
the  tables,  alwaj's  supposing  the  sign  -\-  to  be  prefixed  to  southern  latitudes,  the  sign  —  to 
northern,  and  noting  the  signs  as  in  algebra.}: 

REMARK. 

In  the  two  preceding  problems,  the  time  of  the  true  conjunction  is  calculated  by  means  of 
the  triangle  SFD;  but  it  will  be  useful,  for  the  purpose  of  verification,  to  go  over  the  calcula- 
tion by  means  of  the  triangle  SLE.  The  process  is  nearly  the  same  in  both  methods.  The 
diflTerences  consist  in  finding  the  angle  LSB,  by  subtracting  the  logarithm  of  SL  from  the 
logarithm  of  LB,  increasing  its  index  by  10;  tl)e  remainder  will  he  tlie  log.  sine  of  the  acute 
angle  LSB,  which,  being  added  to  the  angle  of  inclination  (lound  as  before),  will  give  the 
central  angle  DSL :  with  this,  and  the  distance  SL,  corresponding  to  the  I'nd  of  the  eclipse 
or  occultalion,  maybe  found  the  apparent  difflsrence  of  longitude  between  the  sun  and  moon, 
and  moon  and  star:  this  is  to  be  added  to  the  longitude  of  the  sun  or  star  at  that  lime,  if 
the  central  angle  exceed  5)0°,  otherwise  subtracted :  the  sum,  or  difference,  will  be  the  ap. 
parent  longitude  of  the  moon  corresponding,  from  v;liich  the  time  of  the  ecliptic  conjunction 
may  be  obtained  as  before.  If  the  central  angle  exceed  180",  the  sine  and  cosine  of  the  excess 
of  that  angle  above  IdO"  must  be  found  instead  of  the  sine  and  cosine  of  the  central  angle. 

The  apparent  latitude  of  the  moon  is  found  as  in  the  preceding  rules,  by  making  use  of 
the  central  angle  DSL,  and  the  value  SL,  corresponding  to  the  end  of  the  eclipse  or.  occul- 
talion;  whence  maybe  deduced  the  apparent  latitude,  and  the  error  of  the  tables  in  latitude. 

It  is  evident  that  both  these  methods  ought  to  give  the  same  results,  and  thus  furnish, a 
proof  of  the  correctness  of  the  calculations.  All  these  calculations  may  be  made  by  propor- 
tional logarithms,  by  reading  in  the  rule,  log.  cotangent  for  log.  tangent,  log.  cosecant  for 
log.  sine,  &c.,  as  was  mentioned  at  the  end  of  the  rule  in  Problem  V.,  and  by  using  the 
constant  log'.  0.4771,  instead  of  3.55030.  ^ 

*  Wlien  SF  =  SL,  tlie  anule  may  be  romul  as  in  tlie  nnt<;  with  this  marlc  in  paje  408. 
I  Wlien   this  varies,  it  must  be  taken  to  rorrcsjiniid  tci  the  ni  (Idle  time  between  the  immersion  and  true 
riinj;iii<  tiiin.  t  Siie  nutc  with  tliis  marit  in  iia^e  41)8. 


414 


TO   FIND   THE   LONGITUDE   BY   AN    OCCULT ATION. 


EXAMPLE. 

Suppose  in  a  place  in  the  latitude  of  20°  0'  N.,  longitude  ]h.  9ra.  Os.  east  of  Greenwich, 
by  estimation,  the  occultation  of  Spica  by  the  moon  on  December  12,  1808,  was  observed, 
the  immersion  at  ICh.  57m.  29s.,  emersion  at  18h.  lOni.  29s.,  apparent  time,  by  astronomical 
computation.     Required  the  longitude  of  the  place  of  observation. 

Most  of  the  elements  in  the  following  Table  are  calculated  by  Problems  I.,  II.  and  VI. 


ELEMENTS  OF  THE  OCCULTATION. 


Apparent  times  nf  observation 

Estimated  lonj^tiide  E.  from  Greenwich 

Supposed  apparent  time  at  Greenwich 

0's  riglit  ascension 

Lat.  ofpla  e20°0'—  Reduc.  Table  XXXVIIL  7' 22" 

Obliq\iity  of  the  ecliptic 

D's  Ions;,  hy  N.  A, —  Prnb.  I 

Longitude  of  the  nonagesimal,  by  Prob.  IV 

J)'s  long.  — Long,  nonagesimal  =  D's  distance  from  nonagesimal 

Thii=  'listance  or  its  snpplement  to  3G0°  is  arch  D 

All  I     1  ■  of  nonagesimal,  Prob.  IV 

•D's  iiurizontal  parallax .' 

—  Ilednction,  Table  XXXVIII 

Reduced  parallax 

D's  semi-diameter  by  N.  A.  —  Inflection  2" 

Add  correction,  Table  XLIV 

D's  corrected  semi-diameter 

D's  horary  motion  in  longitude  by  Prob.  II.  Example  I.f 

D  's  parallax  in  longitude 

D's  apjiarent  longitude 

Difference  of  D's  apparent  longitudes 

D's  true  lat.  by  N.  A.  Prob.  I South 

D's  parallax  iii  latitude 

D's  apparent  latitude  south 

*'s  true  lat.  =  lat.  Tab.  XXXVII.  2' 2'  13".9  S.  — Tab.  XLL  0".C 

Dilleren  e  of  D  *  apparent  latitudes 

Difference  of  D's  apparent  latitudes 

*'s  true  long.  =  Loni.  Tab.  XXXVn.20r  10'  29".3  +  Tab.  XL.  ) 
11".5  — 'Jab.  XLI.^IO'M ( 


d.  h.  m. 

12  16  57 
1  9 

12  l.'i  48 

17  20 

19°  52' 

23  27 

200  7 

119  15 

50  52 

D  50  52 

81  17 

59 

59 
16 

F     16 

35 

46 

200  54 


Emersion. 


29 

0 
29 
59.0 
38" 
39 
56.3 
55 

1 

1 
32 
52.3 

1.4 
50.9 
16.9 
10.4 
27.3 
51.7 
25 
21.3 


1  55  11.0 
10  23.6 

2  5  34.6 
2  2  13.3 

FD  =   3  21.3 


d.  h.  ni.  s. 

12  18  10  29 

1  9  0 

12  17  1  29 

17  21  12.5 

o     /    II 

200  51  36.1 
165  28  58 
35  22  38 
D   35  22  33 
74  35  18 
59  54.4 
1.4 
.'•.9  53.0 

16  17.5 
13.3 

L     16  30.8 

35  54.2 

33  54 

501  25  30.1 

31  8.8 

1  51  29.1 

17  38.4 

2  9  7.5 
2  2  13.3 

C  54.2 
3  32.9 


EL  = 

AL  = 


The  difference  of  the  apparent  latitudes  of  the  moon  and  star  at  the  beginning  of  the  oc- 
cultation 3'  21".3,  beinn-  less  than  at  the  end,  0'  54".2,  the  angle  of  inclination  Is  less  than 
90^.  In  tills  example  the  moon's  latitude  is  more  southerly  than  the  star's^  hence  the  points 
F,  L,  fall  below  the  line  DE. 

Log.    3.27156 
Cosine    9.99970 


D  31'  8".8  =  1668".8 
2    7  21 


212.9 


1879.6  . 


Difference  apparent  Ion 
D  's  mean  ai)iiare;:l  lat. 

Distance  FA 
D's  difference  lat AL=:3  32.9 

Inclination 83°  30' 

Apparent  motion  FL.. 

Its  Arilh.  Comp 

gp^SL =  32  53.1=1978.1 

Difference  SF,  SL... 

Difference  segments.. 

Its  half. 

Half  FL 

FB 

SF 

FSB 

Inclination 


Log.  13.27126 
Log.    2.32818 

Tang.  10.94308 


6.72594 
3.29625 
0.541'J7 


Log. 

Log.    

Log.    0.56626 


71°  49' 
83    30 


Log. 

Log.    

Sine    9.97775 


2.97220 
2.99445 


Diff.  is  central  angle.. 
SF 
Star's  latitude. 


Cosine  9.99091 
Log.  2.99445 
Sec.  10.00027 


Diff.  apparent  long 
*'s  longitude  .... 


D  * 


967".5  =     16     7.5 
201    10  30.7 


Log.    2.98563 


D's  apparent  longitude. 
D  's  par.  longitude 


200  54  23.2  by  observation. 
—  46  25 


D's  true  longitude 200     7  58.2        Constant    3.55630 

Difference  true  longitude  3752.5  =  1     2  32.5  Log.    3.57432 

D 's  horary  motion 2153.5=     35  53.5  Ar.Co   Log.    6.66686 

Time 6273  =  lh.  44m.  33s.  Log.    3.79748 

[muiersion 16     57     29 

Conjunction 

Conjiinction 

Difference  of  meridians. . 


Log     3.27126 

Cosecant  10.00280 
Log.    3.27406 


Sine  9.3064? 

Log.  2.99445 

FD  199".9  = 

3' 

19".9    Log.  2.3008f 

*'s  latitude 

2 

2 

13.3 

D's  app.  lat. 
D  's  app.  lat. 

2 

2 

5 
5 

33  .2  hy  obs. 

34  .6  by  N.  A. 

Error  Table 

_ 

1  .4  in  latitude. 

18     42       2  at.  place  of  observation. 
17     33       0  at  Greenwich. 


Ih.  9m.  2s. 


D  's  true  Ion.  200  7  .58  .2  by  obs. 
D  's  true  Ion.  200  7  56  .3  by  N.  A. 


EiTor  Table    -f 


1  .9  in  longitsfe 


t  The  inocui's  liorary  motion  varies  from  35'  51".7,  to  35'  54".2,  during  the  occultation:  hence,  at  the  middle 
time,  17h.  49tu.  4."s.,  between  the  immersion,  16h.  57m.  29s.,  and  the  conjunction,  18h.  42m.  (deduceil  from  the 
Nautiial  Ahiiaiiai  ),  the  horary  motion  was  35'  53". 5,  as  is  ens  ly  fmirul  by  a  cahulation  similar  to  that  in  lb<> 
r.xauipU-  "f  Problem  VI. 


TO  FL\D  THE  LONGlTUfiE  BY   AN  ECLIPSE  OF  THE  SUxN.        415 

The  (liiTiTcnce  of  meridians  deduced  from  the  observation,  Ih.  9in.  23.,  differs  but  2s.  from 
the  assuaied  quantity,  Ih.  9m.  Os.  If  the  difference  had  been  considerable,  it  would  liave 
been  necessary  to  repeat  the  operation  with  the  difference  of  meridians  thus  calculated,  and 
so  on  till  the  assumed  and  calculated  longitudes  agree.  The  errors  of  the  tables  above  found, 
v/ero  deduced  upon  the  supposition  that  the  observations  were  actually  made  at  the  place 
mentioned  in  this  example,  and  that  the  true  longitude  of  the  place  of  observation  was 
Ih.  i>m.  Os.  For  it  must  be  observed,  that  the  errors  of  the  tables  in  longitude  cannot  be 
found  by  an  observation  of  an  eclipse  or  occultation,  without  knowing,  by  other  observations, 
the  [irecise  longitude  of  the  place  of  observation.  This  is  evident  by  observing,  that,  ly  re- 
peating !he  operation  till  the  assumed  and  calculated  longitude  of  the  place  of  observation 
agree  vvilh  each  other,  the  lono-itude  of  the  moon,  deduced  from  the  calculation,  will  agree 
also  with  the  longitude  by  the  tables.  The  time  of  conjunction  at  Greenwich,  17h.  133m.  Os., 
taken  from  the  Nautical  Almanac,  is  liable  to  a  small  error  from  the  incorrectness  of  the 
tables.  To  obviate  this  eiror,  it  will  be  necessary  to  deduce  (by  the  above  method,  or  by 
Problem  IX.  when  only  the  beginning  or  end  is  observed)  the  time  of  conjunction  from 
observations  actually  made  at  two  places;  the  difference  of  these  times  will  be  the  difference 
of  meridians  free  from  the  errors  of  the  tables. 

PROBLE3I  VIII. 

To  find  the  longUude  of  a  place  by  an  eclipse  of  the  sun,  lohcn  the  beginning  or  end  only 
is  observed ;  the  apparent  time  being  estimated  from  noon  to  noon,  according  to  the 
method  of  astronomers ;   the  latitude  of  the  place  being  also  known. 

RULE. 

To  the  apparent  time  apply  the  estimated  longitude  of  the  place  in  time,  by  adding  ificcst, 
subtracting  iC east ;  the  sum,  or  difference,  will  be  the  supposed  time  at  Greenwich.  Cor- 
responding to  this  time  in  the  Nautical  Almanac,  find,  by  Problem  L,the  moon's  semi-diame- 
ter, horizontal  parallax,  longitude,  and  latitude;*  and  the  sun's  semi-diameter,  longitude, 
and  right  ascension  ;  also  the  moon's  horary  motion  from  the  sun  by  Problem  II.  Decrease 
the  sun's  semi-diameter  3A"  for  irradiation.  Decrease  the  moon's  semi-diameter  2"  for  in- 
Jlexio  11,  if^\i  be  thought  necessary,  and  to  the  remainder  add  the  correction  to  Table  XLIV.t; 
the  sum  will  be  the  moon's  corrected  semi-diameter.  Find  also,  in  the  Nautical  Almanac, 
the  obliquity  of  the  ecliptic. 

With  these  elements,  and  the  apparent  time  at  the  place  of  observation,  calculate  the  alti- 
tude and  longitude  of  the  nonagesimal  by  Problem  IV.,  and  the  parallaxes  in  longitude  and 
latitude,  and  the  moon's  apparent  latitude  by  Problem  V 

To  the  sum  of  the  corrected  semi-diameters  of  the  sun  ana  moon,  add  and  subtract  the 
moon's  a])parent  latitude,  and  find  the  logarithms  of  the  svm  and  difference  in  seconds.  Half 
the  sum  of  these  two  logarithms  will  be  the  logarithm  |  of  an  arc  in  seconds,  to  be  added  t<< 
the  sun's  longitude  if  the  phase  is  after  the  apparent  conjunction,  but  subtracted,  if  before  ;§ 
the  sum,  or  difference,  will  be  the  apparent  longitude  of  the  moon.  To  tliis  add  the  moon's 
parallax  in  longitude,  when  the  moon's  distance  from  the  nonagesimal  (found,  as  in  Problem 
VI.,  by  subtracting  the  longitude  of  the  nonagesimal  from  the  moon's  longitude,  borrowing 
3G0'-'  when  necessary),  is  greater  than  180°,  otherwise  subtracted ;  the  sum,  or  difference,  will 
be  the  trite  longitude  of  the  moon. 

Take  Ihe  difference  in  seconds  between  the  true  longitudes  of  the  sun  and  moon,  and  to 
its  logarithm  add  the  arithmetical  complement  log.  of  the  moon's  horary  motion  from  the 
sun  in  seconds,  and  the  constant  logarithm  3.55G30  ;  the  sum,  rejecting  10  in  the  index,  will 
be  the  logarilhui  \  of  the  correction  of  the  given  time,  expressed  in  seconds.  This  is  to  be 
added  to  the  ajjparent  time  of  observation,  when  the  moon's  true  longitude  is  less  than  the 
sun's,  otherwise  subtracted;  the  sum,  or  difference,  will  be  the  time  of  the  true  conjunction 
at  the  place  of  ob.ser'vation.  The  difference  between  this  and  the  time  of  conjunction  inferred 
from  the  Nautical  Almanac  for  the  meridian  of  Greenwich,  by  Problem  III.,  will  be  the 
longitude  of  the  place  of  observation  in  time,  supposing  the  lunar  and  solar  tables  to  be  cor- 
rect; but  it  is  much  more  accurate  to  compare  actual  observations  made  at  different  places, 
by  deducing  the  times  of  the  ecliptic  conjunction  from  each  observation;  the  difference  of 
these  times  will  be  the  difference  of  longitude. 

EXAMPLE. 
At  Salem,  in  the  latitude  of  42- 33'  30"  N.,  longitude  by  estimation  4h.  43m.  32s.  W.  from 
Green  wich,  the  beginning  of  the  total  eclipse  of  June,  1800,  was  observed  at  15d.  22h.  Gm.  18s. 1 , 

*  Tlie  loiiiiitiuli;  ami  latitude  must  be  corrected  for  the  errors  of  the  tables,  when  known,  by  a  previous 
operaliim,  or  liy  other  observations. 

t  Th  .s  corre  linn  must  lie  found  after  the  altitude  and  longitude  of  the  nonagesimal  are  ralciilated. 

j  These  i  alculat  ons  may  be  made  in  the  same  manner  by  using  proportional  logaritlims  ;  the  only  differ 
ence  cons'sts  In  using  the  constant  logarithm  0.4771,  instead  of  3.55G:!0,  in  finding  tlie  time  of  conjunction. 

$  In  general,  the  beginning  of  an  eclipse  or  occultation  precedes  the  apparent  conjunction,  and  the  end  is 
after  the  apparent  conjunrtion  ;  but  there  is  a  case  (which  very  rarely  occurs)  where  the  contrary  may  take 
pince  ;  namely,  where  the  point  F  or  L  (Plate  XIII.  fig.  12,  13)  falls  between  C  ami  B,  which  can  happen 
only  when  thi;  lines  FD,  RL,  are  nearly  equal  to  SF  or  SL.  In  tliis  case,  it  n  ay  be  ascertaintd  whether  the 
phase  [(recedes  or  follows  the  conjunction,  by  making  the  calculation  as  in  I'roblem  VI.  or  VII.,  with  the 
limes  of  beginnin;:  and  end,  calculated  by  Problem  XIII. ;  and,  as  the  central  angle  is  greater  or  less  than  90° 
the  phase  will  follow  or  precede  the  apparent  conj:inction,  the  latitudes  given  by  the  tables  being  suppos<»<» 
correct. 


416  TO   FIND   THE   LONGITUDE   Bt   AN    OCCULTATION 

apparent  time,  by  astronomical  computation.     Piequired  the  longitude  of  the  place  from 
tills  observation. 

The  elements  must  be  calculated,  as  in  the  Example  of  Problem  VI.,  for  the  beginning  of 
the  eclipse,  except  those  marked  in  italics.  The  rest  of  the  calculation  may  be  made  b} 
proportional  logarithms,  as  follows  : — 

Sum  semi-diameter  O  ]) 32'23'i.5 

J)  's  apparent  latitude 1  55  .8 

Sum 34  19  .3 Prop.  Log.  0  /I97 

Diflerence 30  27  .7 Prop.  Log.  0.7715 

Sura 1.4912 

Half  sum Arc  32  20    corresponding  to  Prop.  Log.     7456 

O's  longitude 84  41    3.4 

D's  apparent  longitude 84    8  43.4 

J>'s  par.  longitude —  19  46.8  , 

D's  true  longitude 83  48  56.6  . 

0's  true  longitude 84  41    3.4  Constant  Log.  0.47Vi 

Difference 52    6.8 Prop.  Log.  0..5363 

D's  horary  motion  from  0 34  17.1    Arith.  Comp.    Prop.  Log,  9.2798 

Time  from  conjunrtion 111.  31m. 13s Prop.  Log.  0.2952 

Apparent  time  observation...  15  22      6      18 

Apparent  conjunrtion  Salem.  15  23    37     31 

App.  conjunction  Greenwich  16     4     19         by  Nautical  Almanac. 

Difference  of  nierid  ans 4h.  41ni.21}s. 

If  we  suppose  the  time  of  conjunction  at  Greenwich  to  be  4h.  20m.  473.  as  calculated  in 
the  Example,  Problem  VI.,  the  difference  of  meridians  would  be  4h.  4:'m.  IGs.,  agreeing 
nearly  with  the  assumed  longitude,  so  thai  it  will  not  be  necessary  to  repeat  the  operation 
The  remarks  at  the  end  of  that  example,  respecting  the  errors  of  the  lunar  tables,  and  tht 
comnaring  of  actual  observations  at  different  places,  are  equally  applicable  to  the  presen: 
problem. 

PROBLEM   IX. 

To  find  the.  longiiiule  of  a  place  hy  an  occultation  of  a  fijced  star  by  the  moon,  xchen  t'u 
immersion  or  emersio7i  only  is  observed ;  the  apparent  time  being  estimated  from  7iom 
to  noon,  according  to  the  method  of  astronomers,  and  the  latitude  of  the  place  bein^ 
known. 

RULE. 

To  the  apparent  time  apply  the  estimated  longitude  of  the  place  turned  into  time,  by 
adding  if  7/;c.9^  subtracting  if  east ;  the  sum  or  difference  will  be  tlie  supposed  time  at  Green- 
wich. At  this  time  find  In  the  Nautical  Almanac  the  sun's  right  ascension,  the  moon's  semi- 
diameter,  horizontal  parallax,  longitude,  and  latitude,* by  Problem  I.;  and  the  moon's  horary 
motion  by  Problem  II. ;  also  the  latitude  and  longitude  of  the  fixed  star  by  Table  XXXVII., 
and  correct  it  for  aberration  and  equation  of  equinoxes  by  Tables  XL.  XLI.  De'crease  the 
moon's  semi-diameter  2"  for  inflexion,  if  it  be  thought  necessary,  and  to  the  remainder  add 
the  augmentation  from  Table  XLIV.;  1  the  sum  will  be  the  corrected  semi-diameter.  Find 
also,  in  the  Nautical  Almanac,  the  obliquity  of  the  ecliptic.  With  these  elements,  and  the 
apparent  time  of  observation,  calculate  the  altitude  and  longitude  of  the  nonagesimal  by 
Problem  IV.,  also  the  parallaxes  in  longitude  and  latitude  of  the  moon's  apparent  latitude  by 
Problem  V. 

Take  the  difference  between  the  latitude  of  the  star  and  the  apparent  latitude  of  the  moon 
which  add  to  and  subtract  from  the  moon's  corrected  serni-diameter  (these  quantities  being 
expressed  in  seconds)  ;  half  the  sum  of  the  logarithms  of  these  quantities,  increased  by  the 
log.  secant  of  the  star's  latitude,  rejecting  10  in  the  index,  will  be  the  logarithm  |  of  an  arc 
in  seconds,  to  be  added  to  the  star's  longitude  if  the  moon  has  passed  the  apparent  conjunc- 
tion, but  subtracted  if  before  ;\  the  sum,  or  difference,  will  be  the  apparent  longitude  of  the 
moon.  To  this  add  the  moon's  parallax  in  longitude  when  the  moon's  distance  from  the 
nonagesimal  (found  as  in  Problem  VII.,  by  subtracting  the  longitude  of  llie  nonagesimal 
from  the  moon's  longitude,  borrowing  3G0"  when  necessary)  is  greater  than  180'^,  otherwise 
subtract  it ;  the  sum  or  difference  will  be  the  true  longitude  of  the  moon.  Take  the  differ- 
ence in  seconds  between  tlie  moon  and  star's  true  longitudes,  and  to  its  logarithm  add  the 
arithmetical  comp.  log.  of  t!ie  moon's  horary  motion,  and  the  constant  logarithm  3.55030; 
the  sum,  rejecting  10  in  the  index,  will  be  the  logarithm  t  of  a  correction  in  seconds  to  be 
applied  to  the  given  time  of  observation  by  adding  when  the  moon's  true  longitude  is  less 
than  the  star's,  otherwise  subtracting ;  the  sum  or  difference  will  be  the  time  of  the  true 

*  Corrected  for  the  errors  of  the  tables  in  longitude  or  laftude  when  known. 

f  This  corrertion  nrist  he  found  after  the  altitude  and  longitude  of  the  nonagesimal  are  cnlrulatert. 
\  Proportionnl  logar  llims  may  be  used  instead  of  coumion  logarithms,  the  C(>nstani  logarithm  being  0.4771. 
instead  of  3. .55' 3'),  and  the  log.  cosine  being  used  instead  of  log.  secant. 
5  See  note  with  this  mark  in  page  413. 


TO    CALCULATE    AN    LCLIPSE    OF   THE    MOO.N.  417 

conjunction  at  the  place  of  observation.  The  difference  between  this  and  tlie  time  of  con- 
junr.tion  inferred  from  the  Nautical  Almanac  by  Problem  III.,  for  the  meridian  of  Green- 
wich, will  be  the  longitude  of  the  place  of  observation,  if  the  tables  are  correct;  but  it  is 
much  more  accurate  to  compare  the  times  of  conjunction  deduced  from  actual  observations 
at  the  different  places  in  the  manner  mentioned  at  the  end  of  the  rule  given  in  Problem  VH. 

EXAMPLE. 

Sujvpose  in  a  place  in  the  latitude  of  20°  0'  N.,  longitude  by  estimation  Ih.  !1m.  Os.  east 
from  Greenwich,  the  emersion  of  the  star  Spica  was  observed  on  December  12.  1808,  at 
18h.  10m.  2ys.,  apparent  time,  by  astronomical  computation.  Requited  the  longitude  of  tht 
place  of  observation. 

The  elements  must  be  calculated  as  in  the  example  of  Problem  VIL,  for  the  emersion  of 
Spica.     The  rest  of  the  calculation,  made  by  common  logaritiuns,  is  as  follows  - 

P's  semi-diameter ICi  30".8  =  9f)0".3 

Uifloreiice  apjiareiit  l;U.  5  * 0  5i  .2       414  .2 

Slim ]40.').0        Loff.  3.14708 

Ui/Tereme 57G  .6        Lo''.  2.7tiU87 


5.908r«  its  half 2.9.5427 


*'s  latitude  2°  2'  13". . .  .Sec.  lO.OOlW 


Arc 15'   0".0  =  900". 6 Log.    2.9.')4.=i4 

*'s  longitude 201   10  30.7 

5 's  apparent  loiiiiitiule 201  2.5  31  .3 

D's  piir.  longitude —  33  54 

D  's  true  longitude 200  51  37  .3  Constant  3..5.5r)30 

Ditlerence  true  longitude  D  ».  18  53.4  =  1133.4 Log.  3.0.5433 

]) 's  horary  motion 35  54.7  =  2154.7  Arith.  Comp.  Log.  G.filiGlil 

Time 0h.31in.34s.  =  1894 Log.  3.27729 

Time  of  observation 18     10      29 


Conj.  at  place  of  observation.   18    42        3  by  observation. 
Conjunction  at  Greenwich...   17    33        0  by  Nautical  Almanac 

Difference  of  meridians lli.   9in.  3s. 

The  difference  of  tueridians  by  calculation,  Ih.  9m.  3s.,  differs  but  3s.  from  the  assumed 
longitude,  so  that  it  will  not  be  necessary  to  repeat  the  operation.  All  the  remarks  made  ai 
the  end  of  the  example  in  Problem  VIL  arc  applicable  to  this  problem.  It  may  also  be 
further  observed,  that  the  emersion  or  immersion  which  happens  on  the  dark  limb  of  the 
moon  can  be  observed  with  much  more  accuracy  than  on  the  enlightened  limb  ;  because  the 
light  from  this  limb  prevents  the  observer  from  perceiving  the  star's  immersion  or  emersioi» 
so  iostantaneously  as  on  the  dark  side  of  the  moon. 

PROBLEM    X. 

Tt/  calculate  an  eclipse  of  the  moon. 

The  time  of  beginning  or  end  of  a  lunar  eclipse  at  any  place  may  be  found  by  subtractincr 
or  adding  the  longitude  to  the  times  given  in  the  Nautical  Almanac  for  the  meridian  of 
Greenwich,  according  as  the  longitude  is  west  or  east.  But  as  some  readers  may  wish  to 
know  the  method  of  deducing  these  times  from  the  longitudes,  latitudes,  Ac.  of  the  moon- 
and  sun,  given  by  the  Nautical  Almanac  or  by  other  tables,  it  was  thought  proper  to  iii.sert 
the  rule  for  these  calculations. 

An  eclipse  of  the  moon  can  only  happen  at  the  time  of  the  full  moon.  If  her  longitude 
at  that  time  is  not  distant  from  cither  nodet  of  the  moon's  orbit  more  than  about  12'^,  there 
may  be  an  eclipse.  To  find  whether  there  will  be  one,  and  to  calculate  the  times  and  phases, 
proceed  as  follows  : —  • 

RULE. 

Find  the  time  of  full  moon  at  Greenwich  by  the  Nautical  Almanac  or  Problem  III.,  to 
which  add  the  longitude  of  the  place  turned  into  time,  if  east  ;  but  suhirucl  if  wi:s( ;  the  sum 
or  difll^rence  will  be  the  time  of  the  ecliptic  opposition  at  the  proposed  place. 

For  the  time  at  Greenwich,  find,  by  Problem  I.,  tlie  moon's  latitude,  horizontal  par.allax, 
and  semi-diameter  (whicii  requires  no  augmentation)  ;  also  the  sun's  semi-diameter;  then, 
by  Problem  IL,  the  horary  motion  of  the  moon  from  the  sun  in  longitude,  and  the  inoon'3 
horary  motion  in  latitude. 

Draw  the  line  ACB  (Plate  XIII.  figure  G) ;  and,  perpendicularly  thereto,  the  line  PCR. 
Select  a  scale  of  equal  parts  to  measure  the  lines  of  projection,  and  from  it.  take  C(r,  equal  to 
the  moon's  latitude,  and  set  it  on  CR  from  C  to  G,  ahnve  the  line  AB  if  the  latitude  of  the 
moon  is  north,  below  if  south,  t     Take  CO,  equal  to  the  horary  motion  of  the  moon  from  the 

t  The  long'tnde  of  the  moon's  a.?rending  node  is  given  in  the  Nautical  Almanac.  The  long'tude  of  the 
other  node  is  f  luiiii  by  adding  or  subtracting  G  signs. 

I  The  nortlfern  lat  tudes  fnurid  by  Problem  f.  have  the  sign  — ,  the  southern  +.  In  the  figure  the  latitude 
is  south      If  it  b;i  I  been  north,  tlie  point  (i  nuist  have  been  placed  on  the  continual  on  of  RC  above  C 

53 


418 


TO   CALCULATE   AiN    ECLIPSE   OF  THE   MOOK. 


sun  in  longitude,  and  set  it  on  the  line  CB  to  the  right  of  C,  from  C  to  O.  Take  CP,  equal 
to  the  moon's  horary  motion  in  latitude,  as  found  with  its  sign  by  Problem  II.,  and  set  it  on 
the  line  CR,  from  C  to  P  ;  aiore  the  line  AB  if  its  sign  is  — ,  heloiv*  if +.  Join  OP,  which 
is  equal  to  the  horary  motion  of  the  moon  from  the  sun,  and  parallel  thereto  through  G 
draw  the  relative  orbit  of  the  moon  from  the  sun  NGL,  on  which  are  to  be  marked  the 
places  of  the  moon  before  and  after  the  full,  by  means  of  the  horarjr  motion  OP,  so  that  the 
moment  of  full  moon,  or  ecliptic  opposition  at  the  proposed  place,  may  fall  exactly  on  the 
point  G.  This  may  be  done  by  making  the  extent  OP  equal  to  the  transverse  distance  of 
tiO,  CO,  on  the  line  of  lines  of  the  sector,  then  measuring  from  the  same  lines  the  transverse 
distance  corresponding  to  the  minutes  and  parts  of  a  minute  in  the  time  of  full  moon  at  the 
place  of  observation,  and  setting  it  on  the  line  GN  from  G  towards  the  right  to  the  point  x, 
where  the  whole  hour  preceding  the  full  moon  is  to  be  marked.!  Then  the  distance  OP 
set  from  x  to  the  riglit  hand  on  the  line  LGN  reaclies  to  the  hours  preceding  the  full 
moon,  and  set  to  the  left  hand  reaches  successively  to  the  following  hours.  Tliese  intervals 
are  to  be  divided  into  60  equal  parts,  representing  minutes,  if  the  size  of  the  scale  will  ad- 
mit of  it. 

Add  50"  to  the  moon's  horizontal  parallax,  +  and  from  the  sum  subtract  the  sun's  semi- 
diameter;  the  remainder  will  be  the  semi-diameter  of  the  shadow  CB,  with  which  <iescribe 
the  circle  ASB  about  the  centre  C.  Add  the  moon's  semi-diameter  to  the  radius  CB,  and 
with  that  radius  describe,  about  tlie  centre  C,the  circle  DRM  ;  which,  if  there  be  an  eclipse, 
will  cut  NL  in  the  points  E,  H,  representing  respectively  the  places  of  the  moon  at  the 
beginning  and  end  of  it.  If  there  is  no  intersection,  there  will  be  no  eclipse.  Draw  the 
line  CKST  perpendicular  to  LN,  cutting  it  in  K,  and  meeting  the  circles  ASB,  DRH  in  S, 
and  T.  With  a  radius  equal  to  tlie  moon's  semi-diameter,  describe  about  the  centres  E,  H,  K, 
the  small  circles  represented  in  the  figure  ;  of  which  that  drawn  round  K  cuts  the  line  CKS 
in  the  points  I,  F;  and  if  the  eclipse  is  total,  the  whole  of  tliis  circle  will  fall  within  ASB, 
as  in  fig.  G  ;  but  if  part  of  the  circle  falls  without  ASB,  as  in  fig.  7, Plate  XIII.,  the  eclipse 
will  be  partial.  In  either  case,  tlie  number  of  digits  eclipsed  may  be  obtained  by  saying. 
As  the  diameter  of  the  moon  FI,  is  to  the  obscured  part  FS,  so  are  12  digits  lo  the  number 
of  digits  eclipsed.  When  the  eclipse  is  total,  the  beginning  and  end  of  total  darkness  may 
be  found  by  taking  a  radius  equal  to  CB,  decreased  by  the  moon's  semi-diameter,  and  sweep- 
ing with  it  round  the  centre  C,  a  circle  d  e  h  m,  cutting  LN  in  the  points  e,  /(,  representing 
respectively  the  points  of  beginning  and  end  of  total  darkness.  Then  the  hours  and  minutes 
marked  in  the  line  NL,  at  the  points  E,  c,  K,  h,  H,  will  represent  respectively  the  times 
of  the  beginning  of  the  eclipse,  beginning  of  total  darkness,  middle  of  the  eclipse,  end  of 
total  darkness,  and  end  of  the  eclipse.  In  this  rule  no  allowance  is  made  for  the  oblate 
figure  of  the  earth,  the  correction  from  tliis  source  being  much  less  than  the  errors  of 
observation. 

EXAMPLE. 

Required  the  times  of  beginning,  end,  &c.,  of  the  eclipse  of  the  moon  of  May  9,  1808, 
at  a  place  in  the  longitude  of  30°  W.  from  Greenwich. 

By  the  Nautical  Almanac  the  time 
of  full  moon  at  Greenwich  was 
May  ijth,  at  19h.  3!)m.  From  tliis 
subtracting  the  longitude  of  the 
place  of  observation,  30°  W.,or  2h., 
tlie  remainder,  17h.  3'Jm.,  was  the 
time  of  full  moon  at  the  place  of 
observation.  Corresjionding  to  the 
time  at  Greenwich,  19h.  3iTm.,  the 
elements  in  the  adjoined  table  were 
calculated  by  Prob.  I.  and  II.,  and 
the  values  CB,  CD,  C(/,  found  by 
the  above  rule.  Upon  the  centre 
C,  with  the  radii  CB,  CD,  Cd,  ta- 
ken from  a  scale  of  equal  parts, 
describe  the  cirek's  ASB,  MRD, 
mrd.  Draw  the  line  ACB,  representing  tlie  ecliptic,  and  make  CG,  perpendicular  thereto, 
equal  to  the  moon's  latitude,  10'  44  '.8  S. ;  the  point  G  being  taken  below  C,  because  that 


ELEMENTS   OF   THE  ECLIPSE,  MAY  9,  19h.  39m. 


App.  time  of  coiijiinctioii  at  Greenwich,  May  9... 

Longitude  place  30°  W 

App.  timeof  conjiiiRtion  at  place  of  ol)servation.. 

))  's  lat.  by  Prob.  L   S.  decreasing CG 

i)  's  liorizoiUal  parallax 

i)'s  semi-diameter Bl) 

'~'s  semi-diameter 

5 's  horary  motion  in  longitude,  Prob.  II 

0's  horary  motion  in  longitude 

;)'s  liorary  motion  from  ©  in  longitude Cf) 

D's  horary  motion  in  lat.tiide,  Prob.  II CP 

D's  hor.  paralla.v -|-50"  —  Q's  semi-diam..  =  CB 

t'B-J-  J)'s  semi-dia,i)eter =  CD 

CB  —  J's  serai-diameter =  Cd 


19h.39m. 

2   0 

17  39 

+  10' 4-1' 

.8 

61  13 

..'> 

16  40 

.7 

1.^  51 

.3 

37  37 

.8 

2  24 

.8 

35  13 

.0 

—  3  23 

.2 

46  12 

_2 

62  52 

.9 

2D  31 

.5 

*  III  other  words,  the  point  P  will  fall  above  C  if  the  moon  i.s  approaching  to  the  north  pole  of  the  ecliptic, 
otherwise  below  :  Ihat  is,  the  point  P  must  fall  above  C  if  the  mo(m's  latitude  is  smith  dccrea-sin g  or  noTtIt 
iiicrra.iiiiir,  otherwse  below.  When  no  great  accuracy  is  re(iuired,  the  horary  motion  in  latitude  need  not  be 
found  by"  Problem  11.  Instead  of  which,  the  angle  COP  mav  be  taken  equal  to  5°  40',  in  eclipses  of  Oie  moon 
or  sun,  "and  tlie  line  OP  equal  lo  CO  increased  by  9"  or  10"  :"bul  this  method  will  not  answer  in  occultations 
in  whicli  the  angle  COP  varies  above  5  degrees. 

t  The  d  stance  Gi  may  also  be  found  by  common  arithmetic,  by  saying,  As  60  minutes  are  to  the  minnteg 
and  seconds  in  the  time  of  full  moon  (which  in  the  present  e.xample  is  39'),  so  is  OP  to  Gx.  After  marking 
the  hours  on  the  line  LGN,  it  is  usual  to  divide  them  successively  into  halves  and  quarters  of  an  hour,  then 
into  five  minutes  and  one  minute. 

X  The  semi-d  ameter  of  the  shadow  is  increased  by  the  earth's  atmosphere  from  20"  to  60",  accoruing  w 
the  estimates  of  ditierent  astronomers.  Mayer  supposes  this  correction  to  be  one  60th  part  of  the  shadow, 
varying  from  3:"  lo  4i;''.  Tlie  mean  of  Mayer's  correction  addfid.to  the  sun's  paralla.\  is  nearly  equal  to  SU* 
assumed  as  above. 


TO   PROJECT  AN   ECLIPSE   OF  THE   SUN.  419 

< 

latitude  is  south.  Make  CO  equal  to  the  horary  motion  of  the  moon  from  the  sun  in  lonori- 
tude,  35'  13" .0,  and  CP  perpendicular  thereto  equal  to  the  horary  motion  in.  latitude, 
—  3'  2S".2,  tlie  point  P  being  placed  above  C,  because  the  moon's  horary  motion  in  tlie  lati- 
tude lias  tiie  sign  —  prefi-ted  ;  or,  in  otlier  words,  the  latitude  was  south  decreasing.  Join 
OP,  and  parallel  thereto  draw  through  G  the  line  NGL,  and  on  it  let  fall  the  perpendicular 
CK.  Make  the  distance  OP  a  transverse  distance  of  60,  GO,  on  the  line  of  lines  of  the 
sector,  and  measure  from  the  same  lines  the  transverse  distance  39,  3!)  (corresponding  to  tiie 
minutes  in  tlie  time  of  full  moon  at  tlie  place -of  observation)  ;  this  distance,  set  on  the  line 
GN,  to  the  riglit  of  G,  reaches  to  the  point  x,  where  tlie  hour,  17h.,  preceding  the  full  moon, 
is  to  be  marked.  Take  the  extent  OP,  and  lay  it  from  ]71i.  to  the  right  hand  to  IGh.,  and 
successively  to  the  left  to  ]8h.  lL)h.,  &c.  Subdivide  these  lines  into  GO  equal  parts,  represent- 
ing minutes,  if  the  scale  will  permit,  and  the  times  corresponding  to  the  points  E,  e,  K,  A,  H, 
will  represent  respectively  the  beginning  of  the  eclipse,  15h.  oGm. ;  the  beginning  of  total 
darkness,  IGh.  54m.;  the  middle  of  the  eclipse,  171i.41m. ;  tlie  end  of  total  darkness,  Idh.  28m. ; 
and  the  end  of  the  eclipse,  IDli.  2Gm. ;  which  times  agree  nearl}'  with  those  in  the  Nautical 
Almanac,  allowing  for  the  difference  of  meridians  2  hours. 

CALCULATION  BY  LOGARITHMS. 
The  phases  of  the  eclipse  may  also  be  calculated  by  logarithms  in  a  very  simple  manner. 
Thus,  suppose  it  was  required  to  find  the  time  of  the  beginning  of  the  eclipse  in  the  above 
example.  In  this  case,  in  tiie  rirrht-angled  triangle  OCP,  there  would  be  given  CO  =  21I3".0, 
and  CP  =208".2,  to  find  OP  =  2123" .2,  and  the  angle  OPC  ==  84°  22'.  This  angle  is  equal 
to  RGE,  because  GE,  OP,  are  parallel,  and  its  supplement  gives  the  angle  CGE  =95°  38'. 
Then,  in  the  triangle  CGE,  there  are  given  the  annle  CGE  =  95°  38',  the  moon's  latitude 
CG  =  G44".8,  and  the  line  CE  (=  CD)  =3772".9,  to  find  CEG  =  90  48',  GCE=74°  34', 
and  GE  =  3G54".5.  Then  say.  As  OP  (2123".2)  is  to  1  hour  (3G00s.),  so  is  GE  (3G54"5.)  to 
the  time  (G19Gs.  =  ),  Ih.  43m.  IGs.,  between  the  beginning  of  the  eclipse  and  the  full  moon 
at  the  place  of  observation,  17h.  39m. ;  and  as  the  point  E  falls  to  the  right  hand  of  G,  that 
time  must  be  subtracted  from  17h.  39m.,  to  obtain  tlie  time  of  the  beginning  of  the  eclipse, 
15h.  55m.  44s.,  which  agrees  nearly  with  the  projection.  As  these  calculations  are  very 
simple,  it  will  be  unnecessary  to  take  notice  of  the  different  cases,  or  to  give  the  calcula- 
tions at  full  length,  the  whole  being  sufficiently  evident  from  the  figure.  'The  middle  of  the 
eclipse  is  found  by  means  of  the  triangle  GKC,  similar  to  OCP,  in  which  the  angles  and 
hypotenuse  CG  are  given  to  find  CK,  KG.  The  time  of  describing  KG  being  added  to,  or 
subtracted  from  the  time  of  full  moon  at  the  place  of  observation,  according  as  the  point  K 
falls  to  the  left  or  right  of  G,  will  give  the  time  of  the  middle  of  the  eclipse.  The  distance 
CK,  10' 41".7,  subtracted  from  the  radius  CD  or  CT  =  G2' 52".9,  will  leave  a  remainder 
equal  to  the  eclipsed  part  FS  (=  KT),  52'  11".2 ;  and  the  moon's  diameter,  33'  21".4,  is  to 
FS,52'  11". 2,  as  12  digits  to  the  digits  eclipsed,  ISJ.  In  making  these  calculations,  common 
or  proportional  logarithms  may  be  made  use  of. 

PROELEM   XL 

To  project  an  eclipse  of  the  sun  for  any  given  place. 

An  eclipse  of  the  sun  can  happen  only  at  the  time  of  new  moon.  If  the  moon  s  longitude 
at  that  time  is  not  distant  from  either  node  of  the  moon's*  orbit  more  than  17J'-',  there  may 
be  an  eclipse.  To  find  whether  there  will  be  one,  and  to  calculate  the  times  and  phases, 
proceed  by  the  following 

RULE. 

To  the  time  of  the  new  moon,  given  in  the  Nautical  Almanac  (or  calculated  by  Prob.  III.), 
add  the  longitude- of  the  proj)osed  place,  turned  into  time,  if  east;  but  subtract  if  west ;  the 
sum  or  difference  will  be  the  time  of  conjunction  at  the  proposed  place.  Corresponding  to 
the  time  of  new  moon  at  Greenwich,  find,  by  Problem  I.,  the  moon's  latitude,  horizontal 
paralla.x,  and  semi-diameter ;  also  the  sun's  longitude,  semi-diameter,  and  declination. 
Then,  by  Problem  II.,  find  the  horary  motion  of  the  moon  in  latitude,  and  the  horary  mo- 
tion of  the  moon  from  the  sun  in  longitude. 

Draw  the  line  ACB  (Plate  XIII.  fig.  10),  representing  the  ecliptic,  and,  perpendicularly 
thereto,  the  line  PCR.  Take  a  scale  of  equal  parts  to  measure  the  lines  of  the  projection  ; 
measure  from  it  an  interval  equal  to  the  moon's  latitude,  and  apply  it  on  CR  from  C  to  G  ; 
above  the  line  ACB  if  the  moon's  latitude  is  north,  below  if  south.]  Take  CO,  equal  to  the 
horary  motion  of  the  moon  from  the  sun  in  longitude,  and  set  it  on  the  line  CB,  to  the 
right  hand  of  C  to  O  ;  take  CP,  equal  to  the  moon's  horary  motion  in  latitude,  found  by 
Problem  II.,  and  set  it  on  the  line  CR,  from  C  to  P ;  above\  the  hne  ACB  if  the  sign  is  — , 
below  if -(—     •'oin  OP,  which  represents  the  horary  motion  of  the  moon  from  the  sun  on  the 

*  See  note  with  the  mark  \  in  page  415.  All  the  eclipses  that  can  happen  in  any  part  of  the  earth  are 
indicated  In  the  Nautical  Ahiianac. 

t  In  the  fig'ire,  the  latitude  is  supposed  north.  If  it  had  been  as  much  south,  the  point  G  would  have  been 
as  rnucli  below  C  as  it  is  now  aliove  it. 

X  See  note  with  the  mark  *  in  page  416 


420  TO  PROJECT  AN  ECLIPSE  OF  THE  SUN. 

• 

relative  orbit,  and  parallel  to  that  line  draw  the  relative  orbit  of  the  moon,  NGL,  on  which 
are  to  be  piarked  the  places  of  the  moon  before  and  after  tlie  conjunction,  by  means  of  tlie 
horary  motion  OP,  so  that  the  moment  of  the  new  moon,  or  ecliptic  conjunction,  at  the  pro- 
posed place  may  fall  exactly  on  the  point  G,  as  in  the  figure,  where  the  new  moon  is  at  23h. 
35^m.  This  may  be  done  by  taking  the  extent  OP,  equal  to  the  transverse  distance  of  GO,  60, 
on  the  line  of  lines  of  the  sector,  then  measuring  from  the  same  lines  the  transverse  dis- 
tance corresponding  to  the  minutes  and  parts  of  a  minute  of  the  time  of  new  moon  at  the 
place  of  observation,  and  setting  it  on  the  line  GN  from  G  towards  the  right  hand  to  the 
point  X,*  the  place  of  tJie  moon  at  the  first  whole  hour  preceding  the  conjunction  (which  in 
the  present  figure  is  23h.)  Then  the  distance  OP  being  taken  in  the  compasses,  and  set  from 
a;  to  the  right  hand,  gives  successively  the  hours  preceding  the  new  moon,  and  the  same 
distance  set  to  the  left  gives  tlie  following  hours,  as  in  the  figure,  where  they  are  marked  in 
succession  22h.,  23h.,  24h.,  Ih.  Tliese  hours  are  to  be  divided  into  60  equal  parts,  repre- 
senting minutes,  the  scale  being  taken  sufficiently  large  for  that  purpose.!  In  the  present 
figure,  the  subdivisions  are  carried  only  to  five  minutes. 

From  tlie  moon's  horizontal  parallax  subtract  tlie  sun's,  8". 6  ;  the  remainder  is  to  be  taken 
•from  the  scale  of  equal  parts  for  tlie  radius  CB,  with  which,  on  the  centre  C,  describe  the 
circle  BRA,  cutting  CR  in  R.  Open  the  sector  till  the  transverse  distance  of  00°,  00°,  on 
the  line  of  chords,  is  equal  to  the  radius  CB,  and  measure  from  the  same  lines  the  trans- 
verse distance  23°  23'  (equal  to  the  obliquity  of  the  ecliptic),  which  set  on  the  circle  ARB 
on  each  side  of  R  to  T  and  U.  Join  TU,  cutting  CR  in  Q.  On  Q  as  a  centre,  with  the 
radius  QT,  describe  the  circle  TVU,  on  which  set  off  the  arc  TV  equal  to  the  sun's  longi- 
tude. Through  V  draw  the  line  VP'  parallel  to  CR  to  cut  TU  in  P',  the  place  of  the  pole 
of  the  earth,  t  Draw  CP',  and  continue  it  on  either  side  so  as  to  cut  the  circle  ARB  in  the 
point  W,  situated  above  AB  if  tlie  latitude  of  the  proposed  place  is  north,  hclow  if  south.  In 
the  present  figure,  the  latitude  is  north.  If  it  had  been  south,  the  lower  part  of  the  circle 
ARB  ought  to  have  been  made  use  of.  Open  the  sector  so  as  to  make  the  transverse  dis- 
tance 60°,  60°,  on  the  chords,  equal  to  CB,  and  measure  off  the  transverse  distance  equal  to 
the  chord  of  the  complement  of  the  latitude  of  the  place,  which  set  from  W  on  each  side  to 
D  and  d.  With  the  same  opening  of  the  sector  measure  the  cliord  of  the  sun's  declination, 
and  set  it  on  the  same  circle  from  U  on  each  side  to  E  and  F,  and  from  d  on  each  side  to  e 
and/.  Draw  the  dotted  lines  F/,  Dd,  Ec,  cutting  CW  in  /,  q,  n.  Bisect  hi  in  r,  and  erect 
the  line  VI  r  XVIII,  perpendicular  to  CW,  and  make  r  VI  and  r  XVIII,  each  equal  to 
qD.  Open  tlie  sector  to  make  the  transverse  distance  90°,  90°,  on  the  sines,  equal  to  qD, 
and  measure  off  the  transverse  distance  corresponding  to  15°,  30°,  45°,  60°,  75°  (or  1,  2,  3, 
4,  5  hours),  which  set  on  each  side  of  the  point  r,  on  r  VI  and  r  XVIII,  to  the  points 
marked  witli    the    numbers  15°,  30°,   &c.      Through    these   points  draw  the  lines  I  XI, 

11  X,  III  IX,  &c.,  as  in  tlie  figure,  parallel  to  CW.  Open  the  sector  so  as  to  make  rn 
equal  to  the  transverse  distance  of  90°,  90°,  on  the  sines,  and  measure  the  complements  of 
the  former  degrees  as  transverse  distances  on  the  sines,  viz.  75°,  00°,  45°,  30°,  15°,  and  set 
them  on  the  above  lines  I  XI,  II  X,  &c.  from  the  points  of  intersection  with  the  line 
VI  r  XVIII,  above  and  below  that  line.  The  points  I,  II,  III,  &e.  obtained  in  this  man- 
ner, will  represent  the  situation  of  tlie  spectator  at  the  proposed  place,  at  those  hours,  and  a 
regular  curve  drawn  through  these  points  will  represent  his  path.  In  marking  tlie  hours,  it 
niust  be  observed,  that  the  place  of  noon  will  be  at  the  lower  point  n,  if  the  sun's  declination 
is  north  ;  but  at  the  -upper  point  I,  if  tlie  declination  is  south  :  the  hours  must  be  marked 
from  noon  towards  the  left  in  numerical  succession  completely  round  the  curve,  ending  at 
24h.,  according  to  the  method  of  astronomers.    In  the  present  figure,  the  declination  is  nortli, 

*  See  note  witli<th's  iiiaik  f  in  page  416. 

f  The  scale  I  generally  make  use  of,  is  one  inch  to  ten  minutes,  reducing  the  seconds  to  decimals  of  a 
minute.  Thus,  50'  I3G"  in  decimals  is  50'. G,  which  by  tills  scale  would  be  5.06  inches,  obtained  by  placing  the 
'Jecimal  point  one  figure  to  the  left. 

J  This  may  also  be  found  as  follows  : — After  drawing  TQ.U,  as  above,  open  the  sector  till  the  transverse 
distance  90°,  90°,  on  the  sines,  is  equal  to  Q.T ;  then  measure  from  that  line  tlie  e.xtent  QP'  as  a  transverse 
distance  corresjionding  to  the  sine  of  the  diflerence  between  tlie  sun's  longitude  and  90°  or  270°.  When  the 
sun's  longitude  exceeds  6  signs,  the  point  V  will  fall  in  the  semi-circle  below  TU.  This  is  not  drawn  in  the 
ligiire,  for  want  of  room.  When  the  longitude  e.xceeds  2,  4,  6,  &c.  signs,  it  will  be  convenient  to  mark  on 
llie  circle  TYU  the  points  corresponding  to  those  signs,  by  setting  olTthe  radius  Q,T  as  a  chord  from  T  to  n, 
from  n  to  ^,  &c.,  and  then  taking  from  the  sector  the  chord  corresponding  to  the  e.\cess  of  the  given  longi- 
tude above  that  of  the  point  n,  ^,  &c.  immediately  preceding.  Thus,  if  the  sun's  longitude  be  81°  44',  it 
will  be  convenient  to  set  off  60°  from  T  to  n,  and  24°  44'  from  n  to  tlie  sought  point,  V. 

In  case  of  not  having  a  sector,  an  arc,  as  RT,  may  be  marked  off  by  a  plane  scale,  even  when  the  radius 
CR  difl'ers  from  that  of  the  scale,  by  drawing,  by  Problem  VI.  of  Geometrical  Problems,  the  line  CT,  making 
an  angl^with  CR  e(pial  to  the  proposed  arc,  23°  28'.  The  intersection  of  that  line  with  the  circle  ARB  will 
give  the  sought  point,  T.  In  a  similar  manner  the  point  V  may  be  found  by  drawing  a  line,  QV,  making  the 
angle  TQ.V  equal  to  the  proposed  arc,  TV.  The  points  15°,  30°,  45°,  fee.  on  the  line  VI  r  XVI [f,  may  be 
found  by  describing  on  that  line  as  a  diameter,  and  on  r  as  a  centre,  a  semi-circle,  which  is  to  be  divided  into 

12  equal  i)arts  of  1.5°  each.  The  dotted  lines  drawn  through  these  points  perpendicular  to  the  diameter 
VI  r  XVIII,  will  cut  it  ill  the  sought  points,  15°,  30°,  &c.  This  circle  is  not  drawn  in  the  proposed  figure, 
to  prevent  confusion.  Draw  the  line  VI  k  perpendicular  to  r  VI,  and  equal  to  rn.  Join  rk  cutting  the  lines 
75°  V,  C0°  IV,  &c.  in  the  points  1,  9,  3,  4,  5.  JMake  the  lines  15°  I,  30°  II,  45°  III,  &c.  resi>ectivelv 
equal  to  75°  1,  (iO°  9,  45°  3,  &c.,  and  the  sought  points,  I,  II,  III,  &c.  will  be  obtained.  This  inethoil 
may  be  used  when  the  line  rn  is  too  small  to  be  taken  from  the  sector.  The  same  method  may  be  made  use 
of  in  projecting  an  occultation,  by  drawing  Ik  (fig.  8,  Plate  XIII.)  perpendicular  to  ri  and  equal  to  rn,  and 
joining  rk  to  cut  the  dotted  lines  drawn  parallel  to  CP'in  the  points  1,2,  3,  &c.  as  above. 


TO    rilOJECT   AN    ECLli'riL:  OF   THE    SUiN.  421 

and  the  point  ii  tlie  place  of  noon  or  0  hours.  If  it  had  been  south,  the  point  I  would  have 
been  marked  Oh.,  and  the  points  marked  XI,  X,  &C.  would  be  I,  II,  &c.  respectively. 
The  path  touches  the  circle  ARB  in  two  points,  representing  the  points  of  sun  rising  and 
setting,  which,  in  the  present  figure,  are  respectively  IGh.  2(Jm.  and  7h.  34m.  These  points 
divide  the  path  into  two  parts,  of  which  one  represents  the  path  by  day,  the  other  b}'  night, 
as  is  evident  from  the  hours  marked  on  the  curve.  Half  hours,  or  any  other  intermediate 
time,  may  be  marked  in  a  similar  manner.  Thus,  for  the  time  3h.  30m.  =  52°  30',  set  the 
sine  of  f/sJ.^"  to  the  radius  r  VI,  from  r  to  h  on  the  line  r  VI,  and  erect  the  perpendicular 
/u' equal  to  tiie  sine  of  37;^°  (whicli  is  the  complement  of  ^2^'^)  to  the  radius  rn,  and  the 
point  i  will  be  the  place  of  the  spectator  at  the  proposed  time.  In  this  way  the  Tialves  and 
quarters  of  hours  may  be  marked  on  those  parts  of  tiie  path  wliere  necessary.  The  smaller 
subdivisions  may  generally  be  obtained  to  a  sufficient  degree  of  accuracy  by  dividing  the 
quarters  of  hours  into  equal  parts. 

Take  from  the  scale  of  equal  parts  an  extent  equal  to  the  sum  of  the  semi-diameters  of  the 
sun  and  moon,  and,  beginning  near  N,  find,  by  trials,  the  point  p'  of  the  moon's  path,  and 
the  point  Z'  of  the  path  of  the  spectator,  marked  with  the  same  time  and  at  that  distance 
apart.  That  time  will  be  the  beginning  of  the  eclipse.  If  no  such  jjoints  can  be  found, 
there  will  be  no  eclipse  at  the  proposed  place.  Proceed  in  the  same  way  towards  the  point 
L,  and  find  the  points ^^",  Z",  at  the  same  distance  apart;  the  corresponding  time  will  be 
the  end  of  the  eclipse.  Find,  by  trials,  t!ie  point  p  of  the  moon's  path,  and  the  point  Z  of 
the  path  of  the  spectator,  marked  with  the  same  times  at  the  nearest  distance  from  each 
other  (which  will  in  general  be  nearly  the  middle  time  between  the  beginning  and  end  of 
the  eclipse)  ;  that  time  will  be  the  middle  of  the  eclipse.  On  Z  as  a  centre,  with  a  radius 
equal  to  the  sun's  semi-diameter,  describe  the  circle  whose  diameter  is  Ss,  representing  the 
sun's  disc  ;  and  on  the  centre  p,  v/ith  a  radius  equal  to  the  moon's  semi-diameter,  describe 
the  circle  whose  diameter  is  Mm,  representing  the  moon's  disc.  The  part  of  the  sun's  disc 
that  is  cut  off  by  tliis  circle  will  represent  the  part  of  the  sun  that  is  eclipsed.  In  the  ex- 
ample of  figure  10,  the  centre,/;,  of  the  moon's  disc  is  so  near  that  of  the  sun,  Z,  that  the 
eclipse  is  nearly  central ;  and.  as  the  moon's  semi-diameter  is  greater  than  the  sun's,  the 
eclipse  must  be  total.  Under  similar  circumstances,  if  the  moon's  semi-diameter  had  been 
least,  the  eclipse  could  hava  been  annular.  In  case  of  a  partial  eclipse,  the  sun's  disc  will 
not  be  wholly  covered  by  the  moon,  as  in  figure  11, Plate  XIII.,  where  the  circles  represent- 
ing the  discs  of  the  sun  and  moon  are  marked  with  the  same  letters  as  in  figure  10,  but  the 
objects  are  placed  in  a  di§'erent  situation.  In  this  case,  the  number  of  digits  eclipsed  may  be 
obtained  by  drawing  a  line  through  the  centres  p,  Z,  to  meet  the  discs  in  the  points  S,  M, 
s,  m,  and  by  saying.  As  the  distance  Ss  (representing  the  whole  disc)  is  to  the  obscured 
point  M*,  so  are  12  digits  to  the  number  of  digits  eclipsed.  The  beginning  and  end  of 
total  darkness  in  a  total  eclipse  are  found  like  the  beginning  and  end  of  the  eclipse,  except 
in  taking  in  the  compasses  the  difference  between  the  semi-diameters  of  the  sun  and 
moon,  instead  of  their  sum.  For  the  points  of  the  path  of  the  spectator  and  of  the  moon's 
orbit,  marked  with  the  same  time,  and  at  that  distance  from  each  other,  will  represent  the 
situations  and  times  of  the  beginning  and  end  of  total  darkness.  The  beginning  and  end  of 
the  internal  contacts  of  an  annular  eclipse  are  found  in  the  same  manner  ;  the  only  differ- 
ence is  that,  in  a  total  eclipse,  the  moon's  seiiii-diameter  is  greatest,  but  in  an  annular  eclipse 
the  least. 

In  observing  the  beginning  of  a  solar  eclipse,  it  is  of  some  importance  for  the  accuracy  of 
the  observation,  to  know  on  what  part  of  the  sun's  limb  the  eclipse  will  begin.  This  is 
easily  found  by  means  of  the  projection.  Thus  at  the  beginning  of  the  eclipse,  v;hich  cor- 
responds to  the  point  p'  of  the  moon's  path,  and  the  point  Z'  of  the  path  of  the  spectator, 
the  first  point  of  contact  g  may  be  obtained  by  drawing  about  the  centre  p',  with  a  radius 
equal  to  tiie  moon's  semi-diameter,  a  circle  representing  the  moon's  disc;*  about  Z' as  a 
centre,  witli  a  radius  equal  to  the  sun's  semi-diameter,  another  circle  representing  the  sun's 
disc,  touching  the  former  in  the  point  ff.  Draw  the  line  CZ',  meeting  the  sun's'disc  in  the 
points  a,  c,  the  point  c  being  the  most  distant  from  the  centre  C.  Then  the  circle  sr  «  c,  be- 
ing held  between  the  eye  of  the  observer  and  the  sun.  the  engraved  or  marked  side  of  th;' 
figure  towards  the  eye,  and  the  line  c  «  in  a  vertical  direction  with  the  point  c  uppermost. 
vi^ill  represent  the  appearance  of  the  sun  as  viewed  by  the  naked  eye  at  that  time;  r  will 
represent  the  upper  part  of  the  sun,  a  the  lower,  and  g  the  point  of  contact.  If  the  eclips.' 
be  observed  with  an  inverting  telescope,  the  contrary  will  be  observed  ;  that  is.  t!;e  part  it 
must  be  uppermost,  c  the  lowest,  and  g,  the  point  of  contact,  will  appear  to  the  left'  hand 
ore  a.  In  a  similar  manner  the  appearance  of  the  objects  may  be  obtained  at  any  other  part 
of  the  eclipse,  but  it  is  not  necessary  except  at  the  beginning  of  it,  where  there  is  nothing  tn 
direct  the  eye  of  the  observer. 

•  Instead  of  this  circle,  the  line  p'  Z'  may  be  drawn  cutting  the  sun's  disc  in  the  sought  point  of  contact  g. 


422 


TO    PROJECT    AiN    ECLIPSE    OF  THE    SUN. 


EXAMPLE. 

Required  the  times  and  phases  of  the  total  eclipse  of  the  snn,  June  10,  1806,  at  Salem,  Ih 
the  latitude  of  42^  33'  3U"  iN.,  and  the  longitude  of  4h.  43in.  32s.  west  from  Greenwich. 

By  the  Nautical  Almanac,  the  time  of  new 
moon  at  Greenwich  was  June  16d.  4h.  lUm., 


ELEMENTS. 


Conjunction  at  Greenwich,  June  16..., 

Salem  W.  from  Green wicli 

Ecliptic  ccmjunction  at  Salem,  June  15, 

Latitude  of  Salem 

P  's  horizontal  parallax , 

0's  horizontal  parallax , 

D  's  reduced  horizontal  parallax , 

I)'s  semi-diameter < 

0's  semi-diameter 

Sum  of  semi-diameters 

Difference  of  semi-diameters 

])  's  horary  motion  in  longitude,  Prob.  II 

0's  horary  motion 

D's  horary  motion  from  0 CO 

D  's  horary  motion  in  latitude CP 

D's  latitude  by  Prob.  I CG 

0's  longitude TV 

0's  deciinatibn DF 


h.  m.  s. 
4  19  00 
4  43  32 
23  35  28 
42^33' 30" 
CO  25.7 
8.6 
60  17.1 
16  28.1 
15  46.1 

32  14.2 
42.0 

33  41.9 

2  23. 1 

34  18.1 

3  22.5 
19  37 

84  44  36 
23  22  N. 


+ 


corresponding  to  June  15,  23h.  3.5m.  28s.,  at 
Salem.  At  the  time  at  Greenwich,  4h.  19m. 
tJie  elements  of  the  eclipse  were,  as  in  the 
adjoined  table,  calculated  by  the  above  rule. 
Draw  At;B  (Plate  XIIL  fig.  10),  and  per- 
pendicular thereto  the  line  CGPJ..  Make 
CG  equal  to  the  moon's  latitude,  19'  37"  N., 
taken  from  a  scale  of  equal  parts,  the  point 
G  being  above  C  because  the  latitude  is 
north.  Make  CO  equal  to  the  moon's  horary 
motion  from  the  sun,  34'  18". 1,  to  the  right 
hand  of  the  point  C  ;  and  CP  equal  to  the 
moon's  horary  motion  in  latitude  -j-  3'  22". 5, 
the  point  P  being  below  C  because  this  hora- 
ry motion  has  the  sign  -(-  prefixed.  Draw 
NGL  parallel  to  OP.  Make  OP  a  transverse 
distance  of  GO,  GO,  on  the  line  of  lines  of  the 
sector,  and  measure  from  the  same  lines  the 

transverse  distance  35.^,  3.3J  (corresponding  nearly  to  the  minutes  in  the  time  of  new  moon) ; 
this  distance,  set  on  the  line  GN  to  the  right  of  G,  reaches  the  point  x,  where  the  hour  pre 
ceding  the  new  moon  is  to  be  marked,  viz.  23h.  Take  OP  m  the  compasses,  and  mark  it  suc- 
cessively on  the  line  NL  from  x,  or  23h.,  to  the  right  to  22h.,  and  to  the  left  to  24h.  or  Oh., 
Ih..  &c.  These  are  subdivided  into  five  minutes,  the  scale  not  admitting  smaller  divisions. 
Take  the  moon's  reduced  horizontal  parallax,  GO'  17". 1,  from  the  scale  of  equal  parts,  and 
with  that  radius  describe  about  the  centre  C  the  circle  ARB.  Set  off  (by  means  of  the 
sector)  the  arcs  RT,  RU,  each  equal  to  23"  28'.  Join  TQU,  and  about  that  diameter 
describe  the  circle  TYU.  Make  the  arc  TV  equal  to  the  sun's  longitude,  84°  44'  36' 
which  is  done  by  setting  the  radius  QT  as  a  chord  from  T  to  D,  and  then  the  arc  nV  = 
24°  44'  36"  by  means  of  the  sector.  Draw  P'V.  parallel  to  CR,  to  meet  TU  in  the  point  P'. 
Join  CP',  and  continue  it  to  meet  the  circle  ARB  in  W.  Make  (by  the  sector)  the  arcs 
WD,  Wrf,  equal  to  the  complement  of  the  latitude  of  the  place,  47°  26.^'  nearly,  the  radius 
being  CB.  In  a  similar  manner  make  the  arcs  DF,  DE,  df,  de,  &c.,  each  equal  to  the 
sun's  declination  23°  22'.  Draw  the  lines  FIf,  Dqd,  Ene,  cutting  CW  in  /,  q,  n.  Bisect  In 
in  r.  Draw  the  line  VI  r  XVIII  parallel  to  Dqd,  and  make  r  VI,  r  XVIII,  each  equal  to 
qD.  Through  the  points  I,  VI,  n,  XVIII,  /,  draw  the  path  of  the  spectator  as  taught  in  the 
above  rule,  and  mark  the  hour  of  noon,  Oh.,  at  the  point  n  because  the  sun's  declination  is 
north.  Mark  the  following  hours  in  succession  to  the  left,  I,  II,  III,  Sec,  as  in  the  figure. 
Take  an  extent  in  the  compasses  equal  to  the  sum  of  the  semi-diarneters  of  the  sun  and 
moon,  32'  14" .2,  and,  beginning  towards  N,  find,  as  above  directed,  the  points  p'Z'  at  that 
distance  apart  and  marked  with  the  same  time,  22h.  7m.  nearly,  which  is  the  time  of  the  be- 
ginning of  the  eclipse.  Proceed  in  the  same  way  for  the  end  of  the  eclipse  corresponding 
to  the  points  p",  Z",  and  to  the  time  Oh.  53m.,  which  is  the  time  of  the  end  of  the  eclipse. 
Take  the  difference  of  the  semi-diameters  of  the  sun  and  moon,  42",  in  the  compasses,  and 
proceed  in  the  same  way  to  find  the  beginning  and  end  of  total  darkness,  23h.  27m.,  and 
23h.  31m.  The  points  corresponding  could  not  be  drawn  in  the  figure,  as  they  are  so  near 
to  p  and  Z,  and  the  scale  small.  Find,  by  trials,  the  points  p,  Z,  marked  with  the  same  time 
and  at  the  least  distance  apart;  this  will  be  the  time  of  the  middle  of  the  eclipse,  23h.  29m. 
With  an  extent  equal  to  the  moon's  semi-diameter,  IG'  28". 1,  as  a  radius,  describe  about/* 
the  circle  whose  diameter  is  Mm  representing  the  moon's  disc  ;  and  with  the  sun's  semi- 
diameter,  15'  46'  .1 ,  describe  about  Z  the  circle  whose  diameter  is  Ss,  representing  the  sun's 
disc  at  the  middle  of  the  eclipse.  The  sun's  disc  being  wholly  covered  by  the  moon,  in- 
dicates that  tlie  eclipse  was  total.  Describe,  in  the  same  way,  about  p'  and  Z'  the  discs  of  the 
sun  and  moon,  at  the  beginning  of  the  eclipse,  touching  each  other  in  g.  Draw  CZ',  cut- 
ting the  moon's  disc  in  c  and  a.  Then  the  arc  e  g  will  be  the  distance  of  the  first  point  of 
contact  of  the  sun  and  moon  from  the  sun's  zenith  towards  the  western  part  of  the  limb. 


REMARKS. 

1.  The  correction  for  the  spheroidal  form  of  the  earth,  the  augmentation  of  the  moon's 
semi-diameter,  inflexion  and  irradiation,  are  neglected  in  the  above  rule,  as  not  sensibly 
affecting  the  result  of  the  projection,  though  these  points  might  be  attended  to  by  the  follow- 
ing precepts. 

2.  From  the  latitude  of  the  place  subtract  the  correction  of  latitude  of  Table  XXXVIII., 
and  from  the  moon's  horizontal  parallax,  decreased  by  8".G,  subtract  the  correction  of  paral- 
lax in  the  same  table  ;  the  remainders  will  be  the  corrected  latitude  and  parallax  to  be  :nade 
use  of  in  the  above  rule  to  correct  for  the  spheroidal  form  of  the  earth. 


TO   PROJECT  AJN    OCCULTATION  423 

3.  Decrease  liio  moon's  semi-diameter  given  by  tlie  Nautical  Almanac  by  2"  for  inflexion, 
if  it  be  thouglit  necessary. 

4.  Decrease  the  sun's  semi-diameter  3^"  for  irradiation,  and  from  the  remainder  subtract 
a  correction  equal  to  tlie  aujrnientiition  (Table  XV.)  that  the  moon's  semi-diameter  would 
have  when  at  the  same  altitude  as  the  sun  ;  the  remainder  will  be  the  corrected  semi-diame- 
ter of  the  sun,  to  be  used  in  tlie  above  rule  in  finding  all  the  times  and  phases  of  the  eclipse. 
This  metljod  of  decreasing  the  sun's  semi-diameter  produces  nearly  the  same  result  as  that 
by  augmenting  tlie  moon's  semi-diameter,  horary  motion,  and  horizontal  parallax,  and  taking 
the  sun's  semi-diameter  as  given  in  the  Nautical  Almanac. 

5.  Besides  these  corrections,  there  are  otliers,  depending  on  the  change  of  the  moon's 
semi-diameter,  horizontal  parallax,  and  horary  motion  during  the  eclipse;  but  all  these  cor- 
rections are  usually  neglected  in  projecting  an  eclipse  or  occultation. 

6.  The  altitude  of  the  sun,  which  is  nearly  the  same  as  that  of  the  moon  during  the 
eclipse,  may  easily  be  found  by  moans  of  the  projection.  Tlius,  if  it  were  required  at  the 
beginning  of  the  eclipse,  when  the  spectator  is  at  Z'  :  Take  the  distance  CB,  and  a])ply  it  as 
a  transverse  distance  90"^,  90°,  to  the  sines  of  the  sector;  then  the  distance  CZ',  ap|)lied  in 
the  same  manner  to  those  lines,  will  give  the  zenith  distance  of  the  sun,  about  '31^,  cor- 
responding to  the  altitude  59°.  The  correction  (Table  XV.)  cofresponding  to  this  altitude 
is  14",  which  is  nearly  the  correction  to  be  subtracted  from  the  sun's  semi-diameter,  15' 42" .6 
(corrected  for  irradiation),  to  obtain  the  corrected  semi-diameter,  15'  28". G,  as  taught  in  §4. 
fable  XV.  was  calculated  for  the  mean  semi-diameter,  15'  37",  and  tlie  correction  of  the 
Table,  14",  ought  to  be  increased  in  ratio  of  the  sun's  semi-diameter,  15'  46". 1,  to  15'  37", 
when  very  great  accuracy  is  required.  The  difference  of  tlie  corrected  semi-diameters  of 
the  sun  and  moon,  15' 28" .6  and  IG'  2G".l,  is  57^",  which  is  to  be  used  instead  of  42"  in  find- 
ing the  beginning  and  end  of  total  darkness.  The  duration  of  the  total  darkness  found  by 
the  corrected  value  57.y,  is  4;'^  minutes,  but  with  the  uncorrected  value  42",  is  only  3^ 
minutes.  It  was  probably  owing  to  the  neglect  of  this  correction  that  some  of  the  Almanacs 
publislied  in  this  country,  for  180G,  mentioned  tlie  duration  as  3  minutes. 

7.  The  path  of  the  spectator,  I,  II,  III,  IV,  &.C.,  calculated  for  the  proposed  latitude 
42°  33'  30",  may  be  made  to  answer  for  any  other  latitude  by  altering  the  centre  of  projection 
and  the  scale  of  equal  parts.  By  this  means  the  trouble  of  repeatedly  describing  that  patli, 
when  tlie  eclipse  is  to  be  calculated  for  several  places,  may  be  avoided.  To  do  this,  ad(i 
the  prop.  log.  of  the  reduced  parallax  to  the  log.  secant  of  the  latitude  of  the  place;  the  sum, 
rejecting  10  in  the  index,  will  be  the  prop.  log.  of  an  arc  A.  To  this  prop.  log.  add  the 
log-,  secant  of  the  sun's  declination  (or  star's  in  an  occultation),  and  the  log.  cotangent  of  the 
latitude  of  tiie  place  ;  the  sum,  rejecting  20  in  the  index,  will  be  the  prop.  log.  of  the  arc  B. 
Take  the  radius  r  VI  (or  qD),  in  tlie  compasses,  and  make  it  a  transverse  distance  on  the 
line  of  lines  of  the  sector  corresponding  to  the  arc  A,  and  with  that  openin,'^  of  the  sector 
measure  the  transverse  distance  corresponding  to  the  arc  B,  which,  set  from  r  towards  C  on 
the  line  rC  (continued  if  necessary),  will  reacli  to  the  centre  of  the  projection  corresponding 
to  the  proposed  latitude  ;  the  transverse  distance  corresponding  to  the  redu'ied  parallax., 
measured  from  the  line  of  lines  with  the  same  opening,  will  be  the  radius  of  tlis  projection, 
and  the  transverse  distance  corresponding  to  the  horary  motion  of  the  moon  from  tlie  sun  or 
star,  in  an  occultation,  will  be  tiie  horary  distance  to  be  made  use  of  in  marking  the  hours  on 
the  lunar  orbit  LN  ;  lastly,  the  latitude  of  the  moon  at  the  conjunction  is  to  be  measured  as- 
a  transverse  distance,  and  set  from  tlie  new  centre  of  projection  on  a  line  drawn  througli  it 
parallel  to  CR,  and  the  point  where  it  reaches  will  be  the  new  point  G,  corresponding  to  the 
place  of  the  moon  at  tiie  ecliptic  conjunction.  Through  this  point  tlie  line  of  the  moon's 
path  is  to  be  drawn  parallel  to  the  line  LN  of  the  figure,  and  the  hours  are  to  be  marked  on 
it  as  before.  'Whence  tlie  times  of  beginning  and  end  of  the  eclipse  may  be  found  as  in 
the  above  rule.  An  example  of  this  method  is  not  given,  as  it  would  render  the  scheme  too 
confused. 

PROBLEM   XIL 

To  project  an  occultation  of  a  Jixed  star  by  the  moon,  at  any  given  placr. 

The  method  of  projecting  an  occultation  is  nearly  the  same  as  that  of  an  eclipse  of  the  sun ; 
but  to  save  the  trouble  of  reference,  it  was  thought  expedient  to  give  the  rule  without  abridg- 
ment. 

RULE. 

To  the  time  of  the  ecliptic  conjunction  of  the  moon  and  star,  computed  from  the  Nautical 
Almanac  by  Problem  III.,  add  tiie  longitude  of  the  proposed  place  turned  into  time,  if  east, 
jut  subtract  if  west;  the  sum  or  difference  will  be  the  time  of  conjunction  at  the  ])roposed 
place.  Corresponding  to  the  time  of  conjunction  at  Greenwich,  find,  by  Problem  I.,  the 
moon's  latitude,  horizontal  parallax,  and  semi-diameter ;  also  the  sun's  right  ascension. 
Then,  by  Problem  II.,  find  the  horary  motion  of  the  moon  in  longitude  and  latitude,  and  by 
Tables  VIII.  and  XXKVIL,  the  star's  right  ascension,  declination,  longitude  and  latitude.* 

*  In  strictness,  these  quantities  oiislit  to  be  corrected  for  aberration  and  nnlation,  by  Tables  XXXIX. 
XLIK.,  but  the  correction  is  so  small  th.-rt  it  may  always  be  neu'lected.     If  the  right  ascension  and  declina- 
tion only  are  given,  the  latitude  and  lonjiitude  may  be  fonnd  by  Problem  XIX.,  and  if  the  latter  are  given, 
the  former  may  be  calculated  by  Problem  XX.     It  will  be  found  most  convenient  to  nse  the  right  ascensions. 
and  declinations  which  are  given  In  the  Nautical  Almanac,  wlien  any  of  the  stars  n'arket'  iii  it  are  used 


124  TO    PROJECT   AN    OCCULTATJON 

Draw  flie  line  ACB  (Plate  XIII.  fig.  8),  representing  a  parallel  of  the  eclipfic  passing 
through  the  star,  and  perpendicular  thereto  the  line  CPR.  Take  a  scale  of  equal  parts  tc 
measure  the  lines  of  jjrojection,  and  from  it  take  an  interval  equal  to  the  diiTerencc  of  the 
latitudes"  of  the  moon  and  star,  and  apply  it  to  the  line  CR  from  C  to  G,abuvc  the  line  AC13 
if  the  moon's  latitude  is  north  of  the  star's,  otherwise  bcloio*  Take  CO  equal  to  the  horary 
motion  of  the  moon  in  longitude,  and  set  it  on  the  line  CB  to  the  right  hand  of  C  to  O  ; 
take  CP  equal  to  the  moon's  horary  motion  in  latitude,  found  with  its  sign  by  Problem  II., 
and  set  it  on  the  line  CR  from  C  to  P,  above  \  the  line  ACB  if  its  sign  is  — ,  below  if  -f-. 
Join  OP,  which  represents  the  horary  motion  of  the  moon  in  her  orbit,  and  parallel  to  that 
hne  draw  the  orbit  of  the  moon,  NGL,  on  which  are  to  be  marked  the  places  of  the  moon 
before  and  after  the  conjunction  by  means  of  the  horary  motion  OP,  so  that  the  moment  of 
the  ecliptic  conjunction  at  the  proposed  place  may  fall  exactly  at  the  point  G,  as  in  the 
figure  wlierc  the  conjunction  is  at  18h.  42m.  This  may  be  dune  by  making  OP  equal  to  the 
transverse  distance  (iO,  60,  on  the  line  of  lines  of  the  sector,  then  measuring  from  the  same 
lines  the  transverse  distance  corresponding  to  the  minutes  and  parts  of  a  minute  in  the  time 
of  the  ecliptic  conjunction  at  the  place  of  observation,  and  setting  it  on  the  line  Gi\  from  G 
towards  the  right  to  the  point  x,  the  place  of  the  moon  at  the  first  whole  hour  t  preceding 
the  conjunction  (which  in  Wie  present  figure  is  18h.)  Tlien  tlie  distance  OP,  being  taken  in 
the  compasses,  and  set  from  x  to  the  riglit  hand,  gives  successively  the  preceding  hours,  and 
the  same  distance  set  to  the  left  gives  the  following  hours,  as  in  the  figure,  where  they  are 
marked  17h.,  18h.,  19h.,  20h.  These  hours  are  to  be  divided  into  GO  equal  parts  representing 
minutes,  the  scale  being  taken  sufliciently  large  for  that  purpose. §  In  the  present  figure 
the  subdivisions  are  carried  only  to  five  minutes.  Take  the  moon's  horizontal  parallax  from 
ihe  scale  of  equal  parts  for  the  radius  CB,  with  which,  on  the  centre  C,  describe  the  circle 
BRA,  cutting  CR  in  R.  Open  the  sector  till  the  transverse  distance  C0°,  C0°,  on  the  line  of 
chords  is  equal  co  the  radius  CB,  and  measure  from  that  line  the  transverse  distance  23-  28' 
(equal  to  the  obliquity  of  the  ecliptic),  which  set  on  the  circle  ARB,  on  each  side  of  R  to  T 
end  U.  Join  TU  cutting  CR  in  Q.  On  Q  as  a  centre,  with  the  radius  QT,  describe  a 
circle,  TYUV,  on  which  set  off  the  arc  TYV,  equal  to  the  star's  longitude.  Through  V 
draw  the  line  VP'  parallel  to  CR.  Open  the  sector  till  the  transverse  distance  C0°,  90'^,  on 
the  sines,  is  equal  to  the  radius  CB;  then  take  in  the  compasses  from  the  same  lines  an  ex- 
tent equal  to  the  transverse  distance  corresponding  to  the  complement  of  the  declination  of 
the  star,  and  with  one  foot  in  C  sweep  a  small  arc  to  cut  the  line  VP'  in  P',  the  place  oi* 
tlie  pole  of  the  earth. ||  Draw  CP',and  continue  it  on  either  side  so  as  to  cut  the  circle  ARB 
in  the  point  W,  situated  above  AB,  if  the  latitude  of  the  proposed  place  is  north,  but  bel9w 
if  souih.  In  the  proposed  figure  the  latitude  is  north.  (If  it  had  beensouth,  the  lower  part 
of  the  circle  ARB  ought  to  have  been  made  use  of.)  Open  the  sector  as  before,  so  as  to 
make  the  transverse  distance  of  C0°,  00°,  on  the  chords,  equal  to  CB,  and  take  the  chord 
of  the  complement  of  the  latitude  of  the  place,  which  set  from  W  on  each  side  to  D  and  d. 
With  the  same  opening  of  the  sector  measure  the  chord  of  the  star's  declination,  which  set 
on  the  circle  ARB  from  the  point  D  on  each  side,  to  E  and  F,  and  from  d  on  each  side  to  e 
and/.  Draw  the  dotted  lines  F/,  Dd,  Ee,  cutting  CW  in  /,  q,  n.  Bisect  Z  w  in  r,  and  erect 
the  line  tru  perpendicular  to  CW,  and  make  rt,  ru,  each  equal  to  ^D.  Oj>en  the  sector  to 
make  the  transverse  distance  90°,  90°,  on  the  sines  equal  to  r  t,  and  on  each  side  of  /•  mark 
on  the  line  trji  the  sines  of  15°,  30°,  45°,  G0°,  75°  (equal  to  Ih.,  2h.,  3h.,  4h.,  5h.,  respective- 
ly) to  that  radius,  and  mark  tlie  points  with  those  degrees  as  in  the  figure  ;  through  these 
points  draw  the  dotted  lines  parallel  to  In  as  in  the  figure.  Open  the  sector  so  that  the 
radius  rl  may  correspond  to  the  transverse  distance  90°,  90°,  on  the  sines,  and  measure  the 
complemcnls  of  the  former  degrees  as  transverse  distances  on  the  sines,  viz.  75°,  00°,  45°, 
30°,  15°,  and  set  them  on  the  above  dotted  lines,  on  each  side  of  the  points  15°,  30°,  &c., 
respectively,  above  and  below  the  line  t  ru.  A  rcgulnr  curve,  ntlun,  drawn  through  the 
extremities  of  these  dotted  lines,  will  represent  the  path  of  the  spectator  in  the  given  lati- 
tude. Subtract  the  sun's  right  ascension  from  the  star's  (increasing  the  latter  by  24  hours 
when  necessary) ;  the  remainder  will  be  the  hour  of  the  star's  passing  the  meridian, IT  which 
is  to  be  marked  at  the  upper  point  I  of  the  path  if  the  star's  declination  is  south,  but  at  the 
lower  point  n  if  the  declination  is  north.  The  other  hours  are  to  be  marked  from  this  point 
towards  the  left,  by  marking  successively,  at  the  points  where  the  dotted  lines  meet  the 
path,  the  hour  of  the  star's  passing  the  meridian,  increased  by  Ih.,  2h.,  3h.,  &e.,  completely 
round  the  curve,  observing  to  reject  24  hours  when  the  sum  exceeds  24h.  In  the  present 
example,  the  star's  declination  is  south  ;  consequently  the  upper  point  /  of  the  path  is  taken 
for  the  hour  of  passing  the  meridian,  l!}h.  54m. ;  the  extremities  of  the  dotted  lines  to  the 
left  being  marked  successively  20h.  54m.,  21h.  54m.,  22h.  54m.,  23h.  54m.,  Oh.  54m.,  &c. 

*  In  the  figure  the  point  G  is  placed  above  ACB,  because  the  moon  is  in  a  less  southern  latitude  than  the 
star.  This  part  of  the  rule  may  also  be  thus  expressed  : — Find  the  moon's  latitude  with  its  sign  as  in  Prob- 
lem II.  Prefix  tlie  sign  +  to  the  star's  latitude  if  nortli,  the  sign  —  if  south.  Add  the  latitudes,  noticing 
the  signs  as  in  algebra,  and  the  distance  CG  will  be  obtained.  If  its  sign  is  — ,  the  point  G  is  to  be  placed 
above  C,  but  below  C  if  the  sign  is  -f-. 

t  See  note  with  tlie  mark  *  in  page  416.  ^ 

I  See  note  with  the  mark  |  in  page  416. 

^  See  note  with  the  mark  f  in  page  418. 

I'l  The  distance  of  the  line  \VV  from  the  line  CR,  the  situation  of  t(je  point  P',  and  the  path  of  the  spectator, 
may  be  found  as  in  the  note  J  page  418. 

H  Or  r.ilher  the  horary  distance  of  the  sun  and  star  at  the  time  of  the  ecliptic  coiijiinit  on  "f  the  mooin 
,  ana  star 


TO    PROJECT   a:>    Ool^ULTATION, 


425 


The  path  touches  the  circle  ARB  in  two  points,  representing  tlio  points  of  rising  and  setting 
of  the  star,  whicii,  in  the  present  figure,  are  J41i.  I'm.,  and  Ih.  3l'ni.  These  points  divide  tlie 
path  into  two  parts,  of  which  one  represents  the  path  wliile  tlie  star  is  above  tiie  horizon, 
tlie  other  when  below,  as  is  evident  from  the  hours  marked  on  the  curve.  I'he  half  hours, 
or  any  other  intermediate  time,  may  be  marked  in  a  similar  manner.  Thus,  for  the  time 
4h.  24in.,  which  is  3h.  30m.,  or  o2°  30',  from  the  time  7h.  54m.,  marked  at  the  point  ?i,  set 
the  sine  of  52A'-'  to  the  radius  rt  from  r  to  h  on  the  line  it,  and  erect  the  perpendicular  //i, 
equal  to  the  sine  of  37.;^°  (which  is  the  complement  of  52.^") ,  to  the  radius  rn,  and  the  point  t 
will  represent  the  place  of  tlie  spectator  at  the  proposed  time.  In  this  way  the  halves  and 
quarters  of  hours  may  be  marked  on  tho.se  parts  of  the  path  where  necessary.  The  smaller 
subdivisions  may  n;enerally  be  obtained  to  a  sufficient  degree  of  exactness  by  dividing  the 
(juarters  of  hours  into  equal  parts. 

Take  from  the  scale  of  equal  parts  an  extent  equal  to  the  senii-diamcter  of  the  moon,  and, 
beginning  at  the  line  NL,  towards  N,  find,  by  trials,  the  point//  of  the  moon's  path,  and  the 
point  Z'  of  tiie  path  of  the  spectator,  marked  with  the  same  time  and  at  that  distance  apart. 
That  time  will  be  the  beginning  of  the  occultation  or  immersion  at  the  proposed  place.  Pro- 
ceed in  the  same  w.ay  towards  the  point  L,  and  find  the  points  p,  Z,  at  the  same  distance 
apart ;  the  corresponding  time  will  be  the  end  of  the  occultation  or  emersion.  About  the 
points  //,  /;.  as  centres,  with  a  radius  equal  to  the  moon's  semi-diameter,  describe  the  small 
circles  meeting  the  paths  of  the  spectator  in  the  points  Z',  Z.  These  circles  will  represent 
the  moon's  disc  ;  the  points  Z',  Z,  the  places  of  the  star,  and  the  line  CZ',  CZ,  the  vertical 
circles  |>assing  through  the  star  at  the  times  of  immersion  and  emersion  respectively.  To 
render  this  part  of  tlie  scheme  more  distinct  to  the  eye,  it  is  drawn  separately  in  figure  9, 
Plate  XIII.,  in  which  the  points  C, />',Z',  are  similarl}' situated  to  the  corresponding  points 
of  figure  8,  marked  with  the  same  letters.  Through//  draw  the  line  a' p' c'  parallel  to  CZ  , 
to  meet  tlie  moon's  disc  in  a',  c'.  Then  the  circle  a'  Z'  c',  being  held  between  the  eye  of  the 
observer  and  the  sun,  the  engraved  or  marked  side  of  the  figure  towards  the  eye,  and  the 
line  CZ'  (or  a'  p'  c')  in  a  vertical  position  with  the  point  Z'  above  C,  will  represent  the  ap- 
pearance of  the  moon  and  star  as  viewed  by  the  naked  eye ;  c'  will  represent  the  upper  part 
of  the  moon,'  a'  the  lower  part,  and  Z'  the  point  of  contact.  The  contrary  will  be  ob.served 
if  the  object  be  viewed  by  an  inverting  telescope.  It  will  generally  be  conducive  to  the 
accuracy  of  an  observation,  to  estimate  in  this  manner  the  point  of  emersion,  so  as  to  keep 
that  point  of  the  moon's  limb  in  the  field  of  view  of  the  telescope,  and  the  eye  directed  to- 
wards that  point  of  the  liuib,  so  as  to  perceive  the  star  at  the  first  instant  of  its  appearance. 
The  situation  of  the  point  of  emersion  with  respect  to  the  horns  q,  0,  of  the  moon  may  also 
be  made  use  of  for  this  purpose.  The  line  (i/;  0,  connecting  the  moon's  horns,  is  nearly 
parallel  to  the  line  CR,  except  very  near  the  new  or  full  moon  ;  so  that  in  general  it  will 
be  sufficiently  correct  to  draw  tlirough  p  the  line  QpO  parallel  to  CR.  If  greater  accuracy 
is  required,  the  following  construction  may  be  made  use  of  Subtract  the  sun's  longitude 
from  the  moon's,!  make  the  arc  TYU<\  equal  to  the  remainder,  and  join  QX.  Set  on  the 
same  circle  the  arc  T;i  equal  to  the  moon's  latitude  ;  below  the  point  T  if  that  latitude  is 
south,  aliove  if  north.  Through  (i  draw  the  line  ^i  5  parallel  to  TQ  to  cut  QX  in  t  and  CR 
in  S.  Take  the  extent  QT  and  set  it  on  the  line  6Y  above  S  to  .«.  Join  u  i,  and  parallel 
thereto  through  p  draw  tlie  line  QpO  cutting  the  moon's  disc  in  the  points  qO  representmg 
the  horns,  the  figure  being  viewed  as  above  directed.  The  enlightened  part  of  the  moon  is 
that  nearest  to  the  sun  j  the  dark  part  is  the  most  distant  from  it. 

EXAMPLE. 

Required  the  limes  of  immersion  and  emersion  of  Spica,  December  12,  1808,  at  a  place  in 
the  latitude  of  20°  N.,  and  in  the  longitude  of  Ih.  9m.  east  from  Greenwich. 

By  tile  first  page  of  the  Nautical 
Almanac  for  the  month  of  Decem- 
ber, 1808,  the  time  of  the  ecliptic 
conjunction  of  the  moon  and  Spica 
(marked  ])  a  IT^)  was  December  12, 
17h.  33m.  at  Greenwich,  correspond- 
ing to  18h.  42m,  at  the  proposed 
place.  This  time  may  also  be  com- 
puted by  means  of  the  longitudes 
of  the  objects,  as  in  Problem  III. 
of  tl.is  Appendix,  At  the  time  at 
Greenwich,  17h,  33in.,  the  elements 
of  the  occultation  were,  as  in  the 
adjoined  table,  calculated  by  the 
above  rule. 

Draw  ACB,  and  perpendicular 
thereto  the  line  CGY.  Make  CG 
equal  to  the  difference  between  the 


ELEMENTS. 


Conjunction  at  Greenwich,  Dec.  12,  1808.... 

Longitude  east  from  do 

Conjunction  at  place  of  oliservation 

*'s  right  ascension.  Table  Vril 

0's  right ascen.  by  Nautical  Almanac. subtract 

*  passes  the  meridian 

Latitude  of  the  place 

D's  Iiorizontal  paral.  by  Nautical  Almanac. CB 

P'ssemi  diameter  by  Nautical  Almanac 

D's  horary  motion  in  longitude,  Prob.  II.... CO 

D's  horary  motion  in  latitude,  Prob.  II CP 

#'3  longitude,  Table  XXX  VH TYV 

P  's  latitude  by  Nautical  Almanac 

*'s  latitude,  table  XXXVII 

D  (Terence  of  latitudes  5  N.  of  * CG 

*'s  declination 


h.  m.  ,s. 

17  33  00 
1  09  00 

18  42  00 
13  15  08 
17  21  13 

19  53  50 
20°   0'   0" 

50  5.i.2 

16  19.8 

35  55.2 

—    3  02.7 

201  10  31 

1  49  ^,3  S. 

2  2  13  S. 
12  20  N. 

10  10        S. 


f  In  strictness,  the  Irng  tude  and  latitude  of  the  moon  at  tlie  time  of  immersion  or  emersion  ought  to  be 
made  use  of;  but  it  w  11  be  su(lic;ontly  e.x.-ict  to  use  the  star's  longitude  instead  of  the  moon's  (increasing  it 
by  3(30°  when  less  than  the  sun's  longitude),  and  the  moon's  latitude  at  t!)e  conjunction  tiuaritities  of  the 
same  order  as  the  moon's  parallax  are  neglected  in  the  value  of  the  arc  TYUA. 


54 


426  TO   PROJECT   AN   OCCULTATIOM. 

latitudes  of  the  moon  and  star,  12'  20",  taken  from  a  scale  of  equal  parts,  tlic  point  G  bein<? 
above  C,  because  the  moon  is  northward  of  the  star.  Make  CO  equal  to  tlie  moon's  horary 
motion, in  longitude,  35'  55".2,  to  tlie  right  of  C  ;  and  CP  equal  to  the  horary  motion  in  lati- 
tude, —  3'  2" .7,  tiie  point  P  being  above  C  because  the  sign  is  —  (or  the  latitude  is  south 
decreasing).  Draw  NGL  parallel  to  OP.  Make  OP  a  transverse  distance  of  (JO,  tiO,  on  the 
line  of  lines  of  the  sector,  and  measure  from  the  same  lines  the  transverse  distance  42,  42 
(corresponding  to  the  minutes  in  the  time  of  the  conjunction) ;  this  distance  set  on  the  line 
GJN,  from  G  towards  the  right  hand,  reaches  to  the  point  x  of  the  path  where  tlie  hour  prece- 
ding the  conjunction  is  to  be  marked,  viz.  18h.  Take  OP  in  the  compasses,  and  mark  it  on 
the  line  LN,  from  x,  or  18h.,  to  the  right  to  17h.,  and  to  the  left  to  VJh.,  20h.,  ttc.  These 
are  subdivided  into  five  minutes,  the  scale  not  admitting  of  smaller  divisions.  Take  the 
moon's  parallax,  59'  55". 2,  from  the  scale  of  equal  parts,  and  with  that  radius  desc.-ibe  about 
the  centre  C  the  circle  ARB.  Set  off  (by  means  of  the  sector)  the  arcs  RT,  RU,  each 
equal  to  23^^  28'.  Join  TQU,  and  about  that  diameter  describe  the  circle  TYUVT.  Make 
the  arc  TYV  equal  to  the  star's  longitude,  201"  10'  31",  which  is  done  by  making  the 
arc  UV  =21"  10'  31".  Draw  P'V  parallel  to  CR,  and,  with  an  extent  equal  to  the  comple- 
ment of  the  star's  declination,  79°  50',  taken  as  a  transverse  distance  from  the  sines,  with  the 
radius  CB,  and  -with  one  foot  in  C,  sweep  an  arc  cutting  P'V  in  P'.  Join  CP'  and  con- 
tinue it  to  meet  the  circle  ARB  in  W.  Set  on  each  side  of  W  the  arcs  WD,  \Yd,  equal 
to  the  complement  of  the  latitude  of  the  place,  70°.  Make  the  arcs  DF,  DE,  d f,  de,  each 
equal  to  the  star's  declination,  lO"  10',  and  draw  the  lines  Flf,  D  q  d,  Ewe,  cutting  CW  in 
/,  q,  n.  Bisect  In  in  r,  draw  t  ru  parallel  to  D  q d,  and  make  rt,  ru,  equal  to  ^D.  Through 
the  points  I,  t,  n,  u,  I,  draw  the  path  of  the  spectator,  as  taught  in  the  above  rule,  and  mark 
the  hour  of  the  star's  passing  the  meridian,  iyh.53m.  50s.,  or  19h.  54m.,  at  the  upper  point  l^ 
because  the  star's  declination  is  south.  Mark  the  following  hours  in  succession,  20h.  54tn., 
21h.  54m.,  &c.,  to  the  left,  as  in  the  figure.  Take  an  extent  in  the  compasses  equal  to  the 
moon's  semi-diameter,  16'  19".8,  and,  beginning  towards  N,  find,  as  above  directed,  the  points 
p',  Z',  at  that  distance  apart,  and  marked  with  the  same  time,  16h.  57m.,  which  is  the  time 
of  the  immersion.  Proceed  in  the  same  way  for  the  emersion  corresponding  to  the  points 
p,  Z,  at  the  same  distance  apart,  and  the  time  of  the  emersion,  l&h.  10m.,  will  be  obtained. 
With  the  same  extent  describe  about  p  and  p'  the  small  circles  representing  the  disc  of  the 
moon  at  these  times,  and  cutting  the  path  of  the  spectator  in  the  point  Z,  Z'.  Join  CZ',  Cp', 
and  parallel  to  CZ' draw  c'p' a'  cutting  the  moon's  disc  in  c',  a' (as  in  fig.  9, Plate  XIll.) 
and  the  arc  a'Z'  will  represent  the  distance  of  the  point  of  immersion  from  the  lower  part  a' 
of  the  moon.  The  line  CZ  runs  nearly  through  the  point  p,  so  that  the  toj)  part  of  the 
moon  c  and  the  point  Z  nearly  coincide ;  consequently  the  emersion  happened  near  the 
moon's  zenith.  By  subtracting  the  sun's  longitude,  261°  7',  from  the  moon's  or  star's, 
201°  10'  (increased  by  3()0"),  the  remainder  is  300°  3',  which  is  to  be  marked  on  the  circle 
TYUV  to  the  point  \.  Make  the  arc  T/J  equal  to  the  moon's  latitude,  1°  49'  53",  taking 
the  point  (9  below  T  because  the  latitude  is  south.  Draw  the  lines  X  Q,  (isi^  /lis,  opO,  as  in 
the  rule,  and  the  points  q,  9,  will  represent  the  places  of  the  moon's  horns.  The  point 
of  emersion  Z  will  be  to  the  westward  of  the  upper  horn  q,  about  C0°  measured  on  the 
moon's  limb. 

REMARKS. 

1.  When  it  is  thought  necessary  to  take  notice  of  the  spheroidal  form  of  the  earth,  the 
corrections  of  latitude  and  parallax  of  Table  XXXVIII.  must  be  subtracted  from  the  latitude 
of  the  place  and  the  moon's  horizontal  parallax  respectively,  to  obtain  the  latitude  and  paral- 
lax to  be  made  use  of  in  the  above  rule. 

2.  Subtract  2"  from  the  moon's  semi-diameter  given  by  the  Nautical  Almanac,  if  it  be 
thought  necessary ;  the  remainder  is  to  be  made  use  of,  without  augmentation,  on  account 
of  the  altitude  of  the  moon. 

3.  The  corrections  for  the  change  of  the  moon's  semi-diameter,  horizontal  parallax,  and 
horary  motion  during  the  occultation,  are  neglected  in  the  above*  rule,  as  not  materially 
affecting  the  result. 

4.  The  line  CZ'  measured  on  the  sines  as  a  transverse  distance  to  the  radius  CB,  will  l>e 
the  star's  zenith  distance  at  the  immersion.  In  a  similar  manner  it  may  be  found  at  the 
emersion  at  Z,  or  at  any  other  point. 

5.  The  curve  Itnu  may  be  made  to  answer  for  any  latitude,  as  in  Problem  XL,  Remark  7. 

Calculation  of  an  occultation  of  a  planet  by  the  moon. 

By  a  similar  process  the  times  of  immersion  and  emersion  of  a  planet  may  be  calculated 
by  finding  the  planet's  right  ascension  and  declination,  geocentric  longitude  and  latitude, 
from  the  Nautical  Almanac,  and  using  them  instead  of  the  star's;  also,  by  Problem  II., 
the  horary  motion  of  the  moon  from  the  planet  in  longitude  and  latitude,  which  are  to  be 
used  instead  of  tiie  horary  motion  of  the  moon.  In  this  projection  it  will  not  be  necessary 
to  take  notice  of  the  parallax  of  the  planet,  but  it  may  be  easily  allowed  for  by  taking  the 
radius  CB  equal  to  the  difference  of  the  horizontal  parallaxes  of  the  moon  and  planet.  The 
apparent  diameter  of  the  planet  may  also  be  neglected,  making  the  distances  pZ.  p'Z' ,  equal 
to  the  moon's  semi-diameter.  When  great  accuracy  is  required,  the  sum  of  the  semi-diame- 
ters of  the  moon  and  planet  must  be  made  use  of  for  finding  the  external  contacts,  :md  their 
difference  for  the  internal  contacts. 


//'/AXiy 


E.tcO.Wrm.T'NT. 
UUil 


TO  CALCULATE  THE  BEGINNLNG  OR  END  OF  AN   ECLIPSE.      427 

PROBLEM   XIIL 

To  calcvlaie  the  beginning  or  end  of  a  solar  eclipse. 
RULE. 

Tliis  must  be  done  by  approximation,  by  assuming  a  time  for  tiie  beginning  or  end  of  the 
ellipse,  as,  for  example,  the  time  obtained  by  projection  by  Problem  XL,  the  time  of  new 
moon  at  the  place  of  observation,  or  an  hour  before  pr  after,  according  as  it  is  the  beginning 
or  end  of  the  eclipse  that  is  sought.  With  this  time  calculate  the  elements  of  the  eclipse 
and  the  parallaxes,  as  taught  in  Uie  first  part  of  Problem  VIIL  The  parallaxes  applied  to 
the  longitude  and  latitude  of  the  moon  by  the  Nautical  Almanac,  will  give  the  apparent 
longitude  and  latitude.  Find  the  difference  of  the  apparent  longitudes  of  the  moon  and  sun, 
and'from  its  proportional  logarithm,  increasing  the  index  by  10,  subtract  the  proportional 
logarithm  of  the  moon's  apparent  latitude  ;  the  remainder  will  be  the  log.  tangent  of  an 
an°gle,  whose  corresponding  log.  cosine  is  to  be  added  to  the  proportional  logarillim  of  the 
dilference  of  longitudes  ;  the  sum,  rejecting  10  in  the  index,  will  be  the  proportional  loga- 
rithm of  the  apparent  distance  of  the  centres  of  the  sun  and  moon,  which  ought  to  be  equal 
to  the  sum  of  the  corrected  semi-diameters,  if  the  assumed  time  was  correct.  If  this  is  not 
the  case,  the  operation  must  be  repeated  with  an  assumed  time  differing  a  few  minutes  from 
the  former,  and  the  apparent  distance  of  the  centres  of  the  sun  and  moon  must  be  calculated 
in  this  new  supposition.  Then  add  together  the  arithmetical  complement  of  the  proportional 
logarithm  of  the  difference  of  the  apparent  distances  thus  calculated,  the  proportional  loga- 
rithm of  the  difference  between  the  first  calculated  distance  afid  the  sum  of  the  semi-diame- 
ters, and  the  proportional  logarithm  of  the  interval  of  time  between  the  two  suppositions ; 
the  sum,  rejecting  10  in  the  index,  will  be  the  proportional  logarithm  of  the  correction  to  be 
applied  to  the  first  assumed  time,  which,  at  the  beginning  of  an  eclipse,  is  to  be  added  to  the 
first  assumed  time,  if  the  distance  be  greater  than  the  sum  of  the  semi-diameters,  but  sub- 
tracted if  less  ;  and  the  contrary  in  calculating  the  end  of  an  eclipse;  the  sum  or  difference 
will  be  the  approximate  time  of  the  beginning  or  end  of  the  eclipse.  If  great  accuracy  is 
required,  the  operation  may  be  repeated  with  this  approximate  time,  combining  this  result 
.with  one  of  the  former  sjuppositions ;  and  thus  the  operation  may  be  repeated  till  the  apparent 
distance  of  the  centres  at  the  assumed  time  is  found  to  be  exactly  equal  to  the  sum  of  the 
corrected  semi-diameters. 

REiMARK. 

This  rule,  with  some  modification,  will  answer  for  calculating  the  time  of  an  occultation 
of  a  fixed  star  or  planet  by  the  moon.  In  this  case,  the  star's  longitude  is  to  be  found  in 
Table  XXXVII.,  and  corrected  for  the  equation.  Table  XLL*  (or  the  planet's  longi- 
tude is  to  be  taken  from  the  Nautical  Almanac ;)  the  difference  between  this  and  the  moon's 
apparent  longitude  corresponding  to  the  assumed  time  being  found,  its  proportional  loga- 
rithm is  to  be  added  to  the  log.  secant  of  the  moon's  apparent  latitude,  and  the  sum  is  to  be 
used  in  finding  the  distance  of  the  centres  instead  o&  the  proportional  logarithm  of  the  dif- 
ference of  longitude  of  the  sun  and  moon,  with  the  index  increased  by  10.  The  latitude  of 
the  star  is  to  be  found  by  Tables  XXXVII.  and  XLL,  or  the  planet's  latitude  by  the  Nautical 
Almanac,  and  added  to  the  latitude  of  the  moon,  if  of  a  different  name ;  otherwise  their 
difference  is  to  be  taken  and  made  use  of,  instead  of  the  moon's  latitude  in  the  above  rule. 
Lastly,  instead  of  the  sum  of  the  semi-diameters,  the  semi-diameter  of  the  moon  is  to  be 
made  use  of  When  very  great  accuracy  is  required  in  calculating  an  occultation  of  a  planet 
by  the  moon,  the  difference  of  tlie  parallaxes  of  the  moon  and  planet,  decreased  by  the  cor- 
rection of  parallax,  Table  XXXVllI.,  is  to  be  made  use  of  as  the  reduced  parallax,  in  finding 
the  parallaxes  in  longitude  and  latitude.  When  the  apparent  distance  of  the  centres  of  the 
moon  and  planet  is  equal  to  the  sum  of  their  semi-diameters,  their  limbs  will  just  appear  to 
touch  each  other;  and  when  that  distance  is  equal  to  the  difference  of  the  semi-diameters,  the 
planet  will  be  wholly  covered  by  the  moon. 

EXAMPLE. 

Required  the  time  of  the  beginning  of  the  solar  eclipse  of  June,  180G,  at  Salem,  supposing 
the  errors  of  the  moon's  longitude  and  latitude  in  the  Nautical  Almanac  to  be  unknown. 
To  abridge  the   present  calculation,  suppose   the  beginning  of  the   eclipse   to   be  June 


hence  the  uncorrected  values  are  84°  9'  48" .8,  and  2'  7" .2  N.     The  difference  between  this 
appatent  longitude  of  the  moon,  and  the  sun's  longitude,  84°  41'  3".4,  is  31'  14".6. 

Difference  of  longitufle...  31' 14".6; Prop.  Log.  10.7605 0.7G05 

D 's  apparent  latitude 2    7.2 Prop.  Log.     1.9289 

Tang.    8.8316  —  Corresponding  Cosine  9.9990 
Apparent  distance  O  p. ...31'  19". 0 Prop  Log.    .7595 

*  We  must  also  apply  the  correction   of  Table  XL.,  if  the   longitudes  are  counted  from   the  onparent 
'^oinox,  as  was  the  case  formerlv  m  the  NaiUical  Almanacs 


428 


TO  FIND  THE  APPARENT  TIME  AT  GREENWICH 


This  apparent  distance  differs  1'  4".5  from  the  sum  of  the  semi-diameters,  32'  23". 5.  It  is 
therefore  necessary  to  make  a  second  supposition,  as  for  example  ton  minutes  later,  or  at 
22h.  ICra.  18s. 1 ;  with  tliis  time  tlie  elements  are  to  be  again  calculated  as  in  Problem  VI., 
namely,  moon's  apparent  longitude  uAcorrected,  84°  14'17".l ;  sun's  longitude,  84"  41' 27  .2; 
their  difference,  27'  10". 1 ;  moon's  apparent  latitude  uncorrected  for  error  of  tables,  1'  58".8  N 

Difference  oflongiturte.,.   27' lO'M...,  Prop.  Log.  10.8212 0.8212 

]) -3  ai)pareiit  latitude 158.8 Prop.  Log.     1.9586 

Tang.    8.8G2G Corresponding  Cosine  9.9988 

Second  apparent  distance  Q  D 27'  14".7 Prop.  Log.     .8200 

First  apparent  distance  O  ]) 31  19.0 

Difference 4    4  .3..Arith.  Comp.  Prop.  Log.  8.3545 

Difference  first  distance  and  semi-diameters 1    4.5 Prop.  Log.  2.2238 

Interval 10    0      Prop.  Log.  1.25.53 


.Prop.  Log.  ].8338 


Correction 2      38     

First  supposed  time , lod.  22h.  6m.  18s.l 

Appro.ximate  time 15d.  221i.  3m.  40s.l 

If  the  approximate  time  differ  very  much  from  the  assumed  times,  it  will  be  necessary  to 
repeat  the  operation  till  the  last  assumed  and  calculated  times  agree. 

PROBLEM  XIV. 

Given  the  moon's  true  longitude  tojind    the  mean  time  at    Greenivich, 

IvULE. 

1.  Take  from  the  Nautical  Almanac  the  two  longitudes  immediately  preceding  the  given 
longitude  and  the  two  following,  and  find  the  first  and  second  differences,  as  in  Problem  I. 
Call   the  middle  term  of  the  first  differences  the  arc  A,  and  the  half-sum  of  the  second _ 
differences,  (noticing  the  signs,)  the  arc  B. 

2.  To  the  constant  logarithm  4.63548  add  the  arithmetical  complement  of  t!ie  logarithm 
of  A,  in  seconds,  and  tlie  logarithm  of  the  difference  in  seconds  between  the  given  longitude 
and  the  second  longitude,  taken  from  the  Nautical  Almanac  ;  the  sum,  rejecting  10  in  the 
index,  will  be  the  logarithm  of  the  approximate  time  T  in  seconds. 

3.  Enter  Table  XLV.  with  the  arc  B  at  the  top.  and  this  time  T  at  the  side,  and  find  the 
corresponding  correction ;  to  the  logarithm  of  which  add  the  two  first  logarithms  above 
found  ;  the  sum,  rejecting  10  in  the  index,  will  be  the  correction  of  tlie  approximate  time,  to 
be  applied  witli  the  same  sign  as  the  arc  B,  and  the  correct  mean  time,  counted  on  from 
the  second  noon  or  midnight,  will  be  obtained. 


EXAMPLE. 

Suppose  the  moon's  longitude,  July  12,  1836,  v.-as  98°  10'  16".0.     Required  the  corre- 
sponding mean  time  at  Greenwich. 

2d  difference. 


Mean  time.  | 

d. 

h. 

July 

11 

12 

12 

0 

12 

12 

13 

0 

])'s  lonn 

tudes. 

o 

( 

II 

89 

12 

57.4 

95 

07 

44.7 

101 

03 

20.9 

106 

59 

59.8 

1st  difference. 
»    (       It 

5   54    4/  .o  1^  4Q  Q 

A  =  5  55   36.2  TZo7 

i>   Sb    Jo.9         n— -1-55.8 

Constant  Log.  4.63548 

A =  2133fi".2 Arith.  Comp.  Log.  5.67089 

Diff.  oflong 10951".3 Log.  4.03946 

Appro.v.  lime..  Gli.  OOiii.  33s.  =  22173s Log.  4.34583 

Correction....  +14 


D's  longitude 98°  10'  ]G".0 

July  12d.  Oil 95   07   44  .7 


Diff.  longitude. 


3  02  31  .3  =  I0951".3 


4.63548 

• 5.67089 

Eq.  Tab.  XLV.  +  7".0 Log.  0.84510 

Correction,  +14s Log.  1.15147 


Wean  time....  Cli.  09in.  47s.  past  noon,  July  ISd. 

The  same  method  might  be  used  in  finding  the  time  from  the  moon's  right  ascension, 
supposing  the  Nautical  Almanacs  to  give  the  right  ascensions  at  noon  and  midnight  only, 
as  was  formerly  the  case ;  but  as  they  are  now  given  for,  every  hour,  we  may  obtain  the 
time  much  more  simply  by  the  following  rule  : — 

RULE. 

Take  from  the  Nautical  Almanac  the  right  ascensions  of  the  ngoon  which  immediately 
precede  and  follow  the  time  at  Greenwich,  of  the  proposed  observation.  Take  the  differ- 
ence, D,  of  those  two  right  ascensions,  in  seconds  of  time,  also  the  difference,  d,  in  seconds 
of  time,  between  the  given  right  ascension  and  that  corresponding  to  the  first  hour.  Then 
to  the  constant  logarithm  3.55630  add  the  arithmetical  complement  of  the  logarithm  of  D, 
and  the  logarithnrof  d;  the  sum,  rejecting  10  in  the  index,  will  be  the  logarithm  of  a  num 
ber  of  seconds  to  be  added  to  the  hour  first  marked  in  the  Nautical  Almanac,  to  obtain  the 
mean  time  of  the  observation  at  Greenwich,  nearly. 


ru   FiiND   THE   LOiNGlTUDE   OF  A   PLACE.  429 

EXA3IPLE. 

Tlie  moon's  right  ascension,  July  12, 1S3G,  was,  by  observation,  Gh.  36m.  393.35.  Required 
the  mean  time  of  observation. 

Right  Ascension.     Difference. 

July  ]Oii Observed  right  ascension,  Gli.  SCm.  39s.35     ^  _  oi,  no  Constant  Log.  3.55030 

•Julyl2d.Ch byN.A.  6      3(3      17.06     ^  =  1^3    04 XViilV  Conn.  Lo^'  787602 

Julyl2d.7h byN.A.6     38      30.70    "— ^-^J-"* Aran,  t^onip.  i>o„.  /.B/oi« 

Om. 47s. =5873 Log    2.76858 

First  hour ISd.  Gh.  0        0  — ; 

Jlean  time  of  observation,  July  12d.  6h.9in.47s. 

PROBLEM    XV. 

Given  the  distance  of  the  moon  from  afxed  star  not  marked  in  the  JVaidical  Jllmanac, 
together  with  the  altitudes  of  the  objects,  the  mean  time  of  observation,  and  the  estimated 
longitude,  to  find  the  longitude  of  the  place  of  observation. 

First  sulutlan,  using  the  tnlitudcs  and  longitudes  of  the  moon  and  star. 

RULE. 

To  the  mean  time  of  observation,  by  astronomical  computation,  add  the  estimated 
longitude  in  time  if  west,  or  subtract  if  east ;  the  sum  or  ditlefence  will  be  the  supposed 
mean  time  at  Greenwicii,*  corresponding  to  which,  find  the  moon's  latitude,  by  Problem  L, 
also  tlie  loniritude  and  latitude  of  the  star,  by  Table  XXXVIL,  and  correct  them  for  aberra- 
tion, by  Table  XLL 

VV'ith  the  apparent  altitudes  and  distance  of  the  objects,  find  the  correct  distance  by  the 
usual  rules  of  working  a  lunar  observation. 

To  the  correct  distance,  add  the  latitudes  of  the  moon  and  star,  and  find  the  difference. 
between  the  hulf-snm  and  the  distance.  Then  to  the  log.  secants  of  the  latitudes  of  the 
moon  and  star,  rejecting  10  in  each  index,  add  the  log.  cosines  o[  the  half-sum  and  differ- 
ence, if  the  latitudes  are  of  the  same  name,  or  the  log.  sines,  if  of  a  contrary  name  ;  half  the 
sum  of  these  four  logarithms  will  be  the  log.  cosine  of  half  the  difference  of  longitude,  if  the 
latitudes  are  of  the  same  name,  or  its  log.  sine,  if  of  a  different  name. 

The  difference  of  longitude  is  to  be  added  to  the  apparent  longitude  of  the  star,  if  the 
moon  is  east  of  the  star,  otherwise  subtracted,  (borrowing  or  rejecting  300°  when  neces- 
sary ;)  the  sum  or  difference  will  be  the  true  longitude  of  the  moon  ;  whence  the  mean  time 
ai  Greenwich  may  be  found,  by  Problem  XIV.  The  difference  between  this  ajid  the  mean 
time  at  the  ship,  will  be  the  longitude,  which  will  be  ivcst,  if  the  mean  time  at  Greenwich 
be  greater  than  the  mean  time  at  the  ship,  otherwise  cast. 

REMARK. 

This  method,  with  a  slight  modification,  can  be  used  in  finding  the  longitude  from  the 
observed  distance  of  the  moon  from  a  planet,  as  Jupiter,  Venus,  Rlars,  or  Saturn,  in  cases 
where  they  are  not  marked  in  the  Nautical  Almanac.  The  only  difference  in  the  rule, 
when  a  planet  is  used  instead  of  a  star,  consists  in  finding  from  the  Nautical  Almanac,  by 
Problem  I.,  the  geocentric  longitude  and  latitude  of  the  planet,  which  are  to  be  used 
instead  of  the  longitude  and  latitude  of  the  star  in  the  above  rule.  For  the  daily  variation 
of  the  longitude  and  latitude  of  a  planet  is  so  small,  that  no  error  of  moment  can  arise  from 
calculating  those  quantities  for  the  sxipposcd  instead  of  the  true  time  at  Greenwich;  and  the 
parallax  and  semi-diameter  of  the  planet  can  be  allowed  for  by  the  methods  pointed  out  in 
working  a  lunar  observation. 

The  latitudes  of  the  moon  and  the  fixed  star  or  planet,  made  use  of  in  these  observations, 
ought  not  to  differ  very  much,  on  account  of  the  decrease  of  the  relative  motion  arisint; 
from  this  source.  If  the  latitudes  are  of  a  different  name,  their  sum,  otherwise  their 
difference,  ought  to  be  found,  and  if  it  does  not  exceed  one  third  part  of  the  difference  of 
longitude  of  the  two  objects,  they  may  in  general  be  made  use  of. 

EXA.'\IPLE. 

Suppose  that,  on  the  7th  of  January,  1836,  sea  account,  at  Ilm.  57s.  past  midnight,  mean 
time,  in  the  longitude  of  127°  30'  E.,  by  account,  the  observed  distance  of  the  farthest  limb 
of  the  moon  from  the  star  Aldebaran,  was  GS°  36'  0",  the  observed  altitude  of  the  star 
32°  14',  and  the  observed  altitude  of  the  moon's  lower  limb  34°  43'.  Required  the  true 
longitude,  without  using  the  distances  marked  in  the  Nautical  Almanac,  upon  the  supposi- 
tion that  tliey  are  not  given  in  it. 

This  lunar  observation  has  already  been  computed  by  the  common  m.cthods,  in  page  232, 
where  we  have  found  that  the  supposed  time  at  Greenwich  is  Jan.  Gd.  3h.  41m.  57s.,  the 
moon's  semi-diameter  15'  15'',  the  moon's  horizontal  parallax  55'  24",  the  star's  apparent 
altitude  32°  10',  the  moon's  apparent  altitude  34°  55',  the  apparent  distance  of  the  centres  of 

*  This  time  may  also  be  obtained  from  the  chronometer,  if  you  have  one  which  is  pretty  well  rrgiilnl<  d 
•o  astronomical  time 


130 


TO   FIND   THE   LONGITUDE   OF   A   PLACE 


the  moon  and  star  G3°  20'  45".  With  these  we  find  the  true  distance  of  the -centres  of  the 
moon  and  star,  by  the  usual  rules  for  working  a  lunar  observation,  to  be  68°  3'  0'',  as  in 
page  232.  The  moon's  latitude,  deduced  from  the  Nautical  Almanac,  by  Problem  L,  is 
4°  59'  10"  N.  Tlien  the  star's  longitude  and  latitude  are  found  as  below,  by  Tables  XXX  VIL, 
XLL,  making  use  of  the  sun's  longitude,  235°  17',  as  given  in  the  Nautical  Almanac,  these 
longitudes  being  counted  from  the  mean  equinox  ;  with  these  elements  the  calculation 
is  made  in  the  following-  manner : — 


Table  XXXVII *'j^  longitude,  Jan.  6,  1836. 

'I'aljle       XLI Alieiration 


67°  29'  47".l *'s  latitude. 

-|-  15  .9 Aberration. 


'  28'  39''. 0  S. 
+    1   .2 


*'s  apparent  longitude 67  30  03 


-*'s  apparent  latitude  5  28  40  .OS. 


True  distance 68°  03'  00" 

D's  latitude 4   59  10  N Secant    0.00104 

*'s  latitude 5   28  40  S Secant    0  00199 

Sum....: 1 78   30  50 


Half-sum 39    15  25 Sine* 

Difference  of  lialf-sum  and  distance  28   47  35 Sine* 


9.80126 
9.68274 


Ualf-dilTerence  of  longitude. 


2)19.48763 
33   40  06 Sine*       9.74381 


Difference  of  longitude 67   20  12 

*'3  longitude '. •  67   30  03 

D's  longitude 134   50  15 

D's  longitude,  Jan.  Od.  Oh 132   54  51 

Difference 1   55  24  =  0024"  =difrerence. 


d.  h. 

D's  longitude... Jan.  5  12 

6  0 

6  12 

7  0 


126 

41  37.6 

132 

54  51.1 

139 

10  47.8 

145 

29  34.2 

1st  differences 

o        /  II 

6   13  13.5 

A  =  6   15  56.7 

6   18  46.4 


2d  differences, 
fi 

-f  2  43.2 
•     -f  2  49.7 


Mean  =  +2  46.5 


Constant  Log.  4.63548 

A  =  6°  15' 56".7  =  2-255Ci'.7 Log.  Ar.  Co.  5.64672 

Dirt'erence,  6924" Log.  3.84036 


Approx.  time,  3h.  4Im.01s.: 
Correction...  +34 

Time 3h.  41m.  3."8. 


:13261s Log.  4.1-2256 


....  4.63548 
....  5.64672 
Log.  1.24551 


Equation,  XLV.  -{-  17". 0. . 

Correction +  34s Log.  1.52771 


Hence  time  at  Greenwich,  Jan.  6d.    3h.  41m.  35s. 
Mean    time    at    the  sliip,  Jan.  6     12      11      57 

Longitude 8h.  30m.  223.  =  127' 35' 30"  E.  from  Grrenwich,  differing  5' 15"  from 

the  calculation  in  page  232. 

The  computed  time  at  Greenwich,  3h.  41m.  35s.,  differs  from  the  assumed  time, 
3h.  41m.  57s.,  only  22s. ;  and,  during  this  interval,  the  moon's  latitude  varies  so  little,  that  it 
will  not  be  necessary  to  repeat  the  operation  on  account  of  this  variation  ;  observing  that 
an  error  of  one  minute  in  the  moon's  latitude  affects  the  secant  of  the  latitude  about 
0.00001,  and  this  produces  in  the  difference  of  the  longitude  an  error  of  only  2"  or  3"  in 
the  present  e.xample  ;  and  as  the  latitudes  are  always  small,  it  will  hardly  ever  be  necessary 
to  repeat  tlie  operation  when  this  method  is  used. 

Second  solution,  using  the  right  ascensions  of  the  moon  and  star. 

RULE. 

To  the  mean  time  of  observation,  by  astronomical  calculation,  add  the  estimated  longi- 
tude in  time  if  west,  or  subtract  if  east ;  the  sum  or  difference  will  be  the  supposed  mean 
lime  at  Greenwich.  This  time  may  also  be  taken  from  the  chronometer,  if  you  have  one 
which  is  pretty  well  regulated  for  mean  time  at  Greenwich.  With  this  time,  enter  the 
Nautical  Almanac,  and  find  from  it  the  right  ascension  and  declination  of  the  star  or 
planet,  and  the  declination  of  the  moon. 

With  the  apparent  altitudes  and  distances  of  the  objects,  find  the  correct  distance  by  the 
usual  rules  of  working  a  lunar  observation. 

To  the  correct  distance  add  the  declinations  of  the  moon  and  star,  and  find  the  difference 
between  the  half-sum  and  the  distance.  Then  to  the  log.  secants  of  the  declinations  of  the 
moon  and  star,  rejecting  10  in  each  inde.x,  add  the  log.  cosines  of  the  half-sum  and  of  the 
difference,  if  tlie  declinations  are  of  the  same  name,  or  the  log.  sines,  if  of  a  contrary  name  ; 
half  the  sum  of  these  four  logarithms  is  to  be  sought  for  in  tlie  column  of  log.  cosines,  if  the 
declinations  are  of  the  same  name,  or  in  the  colunm  of  log.  sines,  i?  of  different  names;  and 


♦  l-'se  cosine  if  the  latitude.*  arfi  of  the  same  name. 


UY    A   TRANSIT   OF   THE   MOONS    LIMB.  431 

Uie  correspomling-   time  in  the  column  p.  m.  is  the  difference  of  tiie  right  ascensions  ol"  the 
moon  and  star.  • 

This  difference  of  right  ascension  is  to  bqi  added  to  tlie  apparent  right  ascension  of  the 
star,  if  the  moon  is  east  of  the  star,  otherwise  subtracted,  (borrowing  or  rejecting  24h.  \\'hen. 
necessary  ;)  tiic  sum  or  difference  will  be  the  true  right  ascension  of  the  moon's  limb. 

If  the  moon's  true  right  ascension  can  be  found  exactly  in  the  Nautical  Almanac,  the 
corresponding  hour  will  be  the  mean  time  at  Greenwich.  If  it  cannot  be  found  exactly,  as 
will  most  commonly  happen,  take  out  the  right  ascensions  for  the  hours  immediate!}'  pr> 
ceding  and  following,  and  note  their  difference,  D,  in  seconds  of  time  ;  take  also  the  diller 
ence,  d.  in  seconds  of  time,  between  the  moon's  true  right  ascension  and  that  right  ascension 
marked  for  the  first  hour  in  the  Nautical  Almanac.  Then,  to  the  constant  log.  3.55G30,  add 
the  arithmetical  complement  of  the  logarithm  of  D,  and  the  logarithm  of  d  ;  the  sum, 
rejecting  10  in  the  index,  will  be  the  logarithm  of  a  number  of  seconds,  to  be  added  to  the 
hour  first  marked  in  the  Nautical  Almanac,  to  obtain  the  mean  time  of  the  observation  at 
Greenwich.  The  difference  between  this  and  the  mean  tinie  at  the  ship,  will  be  the  longi- 
tude, which  will  be  jccst,  if  the  mean  time  at  Greenwiclf  be  greater  than  the  mean  time  ;il 
the  ship,  otherwise  cast. 

We  may  observe,  that  we  can,  as  in  the  first  solution,  use  a  planet  instead  of  a  star. 

We  shall  now  calculate,  by  this  method,  the  same  example  as  in  the  first  solution.  Jn 
this  case,  for  the  supposed  time  at  Greenwich,  January  (jd.  3h.  41m.  57s.,  we  find,  by  means; 
of  the  Nautical  ."Mmanac,  Aldebaran's  right  ascension  4h.  2Gm.  31s. 3,  Aldebaran's  declina- 
tion 1G°  10'  29"  N.,  and  the  moon's  declination  21°  9'  33"  N. 

True  distance,  as  in  page  232 68°  03'  00" 

J'sdetlinaiion 21   09  33  N Secant    0.03032 

*'s  declination 16  10  29  N Secant    0.01754 

Sum 105  23  02 


Half-sum 52  41  31 Cosine*  9.78254 

Difference  of  half-sum  and  distance    15  21  29 Cosine*  9.98420 

2)!9.81460 

Diflerence  of  *  and  D's  right  ascensions  4h.  48m.  56s.9 Cosine*  9.90730 

•  's  right  ascension 4     26      31  .3  

D's  riglit  ascension 9     15      28  .2    t,,.^_„,      .        «,,  Constant  Log.  3  55630 

liy  N.  A.  D  's  right  ascension,  Jan.  6d.  3h.  9     13      59  .8    D  Terence  d  =  88s.4. .     .     .  Log.  1.9-  64.1 

•'  f        b  Jan.  Od.  4li.  9     16      08.1     iJ'ffei'ence  U=:  128  .3.  .Arith.  Conip.  Log.  7.8917V 

41m.  20s.=  2480s Log.  3.30453 

Add  3h.  00     00  

'i'ime  at  Greenwich 3     41     20 

Time  al  the  sliip 12     11      57 

Longitude  8     30     37  =  127°  30'  15'    E.  from    Greenwich 

differing  1'  30|'  from  tlie  calculation  in  page  232. 

PROBLEIM    XVI. 

Given  the  intervals  of  time  between  the  passages  of  the  moon''s  bright  limb  and  a  fixed 
star  over  two  different  meridians,  to  find  the  difference  of  longitude  between  the  two 
meridians. 

This  problem  includes,  also,  the  case  wliere  one  of  the  observations  is  supposed  to  be 
made  at  Greenwich,  considering  the  time  of  the  transit  of  the  moon's  bright  limb  over  that 
meridian,  given  in  the  Nautical  Almanac,  as  an  actual  observation;  the  error  arising  from 
this  supposition  being  very  small,  on  account  of  the  great  degree  of  accuracy  of  the  lunar 
tables  used  in  the  computation  of  the  Nautical  Almanac.  We  may,  however,  observe  that, 
where  good  observations  can  be  obtained  at  both  meridians,  it  is  always  best  to  use  them 
in  preference  to  the  computed  transits  in  the  Nautical  Almanac. 

The  principle  upon  which  the  longitude  is  found  in  this  method  is  similar  to  that  which 
is  used  in  a  common  lujiar  observation,  and  depends  on  the  observed  motion  of  the  moon  ; 
but,  in  the  present  problem,  this  motion  is  ascertained  by  observing  the  time  when  the 
moon's  bright  limb  passes  the  meridian,  instead  of  measuring  the  angular  distance  of  the 
moon  from  the  sun  or  a  star.  The  variation  of  the  moon's  right  ascension,  corresponding 
to  a  change  of  15°  in  the  longitude,  is  given  very  accurately  by  the  Nautical  Almanac  for 
every  transit  of  the  moon's  limb  at  Greenwich.  This  variation  is  about  2m.  in  time  for  Ih. 
of  longitude,  and  when  the  difference  of  the  times  of  transit  under  different  meridians  haa 
been  found  by  observation,  it  is  easy  to  get,  by  proportion,  the  corresponding  longitude,  as 
we  shall  see  in  the  following  examples. 

This  method  of  computing  the  longitude  is  very  much  facilitated  by  the  new  table  of 
moon-culminating  stars,  inserted  in  pages  410 — 451  of  the  Nautical  Almanac.  To  show  the 
construction  of  the  table,  we  shall  insert  the  following  extracts  from  it,  contained  in 
page  433  of  the  Nautical  Almanac  for  1836. 

•  ■  ♦  Use  sine  if  the  declinations  are  of  different  names 


432 


TO   FIND   THE   LONGITUDE   OF   A    PLACE 


Col.  1. 

Col.  2. 

CoL.  3. 

Col.  4. 

• 

Col.  5. 

Col.  6. 

Col.  7. 

'      183fi. 

Name. 

Magnitude. 

App.  R.  Ascens. 
in  time. 

Declination. 

Var.  J'sRiglit 
Ascension  in 
Ih.  of  long. 

.Sid.  Time  J)  's 

semi-diameter 

pass,  nierid. 

h.  m.    s. 

»      1 

,_ 

5. 

Sepl.  15 

Moon  I      u.  c. 

(4.6) 

15  07  52.16 

18  25  S. 

140.92 

69.80 

15 

Moon  I.    /.  c. 

15  36  34.71 

20  50  S. 

146.22 

71.16 

16 

Moon  I.    u.  c. 

(5.6: 

16  05  21.71 

22  57  S. 

151.02 

72.52 

16 

Moon  1.    I.  c 

16  37   12.45 

24   42  S. 

156.77 

73.79 

17 

Moon  I.    u.  c. 

(6.7) 

17   09   01.65 

213  04  S. 

161.29 

74.88 

17 

Moon  I.    /.  c. 

17   41   3'J.15 

27   00  S. 

164.75 

75.69 

16 

a  Pcorpii. 

1 

16   19  23.07 

26  04  S. 

10 

r  S<'.orpii. 

3.4 

10  25   42.40 

27   52  S. 

17 

a  t^corpii. 

1 

16    19  23.05 

26   04  S. 

The  stars  whose  right  ascensions  and  declinations  arc  inserted  in  this  table,  are  called 
inoon-culnnnating  stars,  because  they  have  nearly  the  same  declination  as  the  moon,  and 
do  not  differ  much  in  right  ascension.,  so  that  they  are  conveniently  situated  for  observa- 
tions of  the  differences  of  tiie  times  of  the  transit  which  are  required  in  tliis  problem.  The 
first  colunni  of  this  table  contains  the  date  ;  the  second,  the  name  of  the  star  or  moon.  If 
the  bright  limb  of  the  moon  be  the  first  whicli  passes  the  njeridian,  it  is  marked  I. ;  but  if  it 
be  the  second  limb,  it  is  marked  II.  The  upper  culmination  of  tJie  moon  is  marked  u.  c. ; 
the  lower  culmination,  I.  c. ;  this  last  being  of  frequent  use  in  high  latitudes.  The  third 
column  contains  the  magnitudes  of  tlie  objects ;  that  of  tlie  moon  being  denoted  bv  her 
age,  expressed  in  days  and  tenths  of  a  day.  The  fourth  column  contains  the  apparent  right 
ascension  of  the  moon's  bright  limb,  at  the  time  of  the  transit  over  the  meridian  of  Green- 
wich; and  the  fiftii  column,  its  declination  at  that  time  :  the  same  columns  contain  also  the 
right  ascensions  and  declinations  of  the  moon-culminating  stars  at  their  upper  culmination. 
Tiie  sixth  column  contains  the  variations  in  the  right  ascension  of  the  moon's  bright  limb 
during  the  intervals  of  her  transit  over  two  meridians ;  one  of  these  meridians  being 
7°  30'  W.  from  Greenwich,  and  the  other  7°  30  E.  from  Greenwich  ;  so  that  the  distance 
of  these  two  meridians  is  15°,  or  Ih.  in  longitude.  For  convenience  of  reference,  we 
shall  call  this  variation  the  arc  H,  supposing  it  to  be  e.xpressed  in  seconds  of  time,  as  in 
column  G. 

The  arcs  H,  in  the  sixth  column,  are  deduced  from  the  right  ascensions  of  the  moon's 
bright  limb,  contained  in  the  fourth  column,  so  that  they  include  the  effect  produced  by  the 
changes  of  the  moon's  semi-diameter.  The  seventh  column  contains  the  intervals  of  the 
transit  of  the  moon's  semi-diameter  over  the  meridian  expressed  in  sideral  time  ;  this  time 
being  generally  used  in  making  such  observations,  and  for  this  purpcjee  it  is  usual  to  note 
the  times  of  transit  by  a  clock  regulated  to  sideral  time.  If  the  intervals  are  given  in  mean 
time,  they  may  be  reduced  to  sideral  time  by  adding  the  correction  in  Table  LI.  correspond- 
ing to  that  time.  Thus,  if  the  interval  is  Ch.  mean  time,  the  tabular  correction  in  column  1 
of  that  table  is  r)!!s.],  making  the  interval  Ch.  Om.  59s. 1,  sideral  time.  If  the  interval  be 
Gh.  58m.  mean  time,  tiie  corrections  in  Table  LI.,  columns  1,2,  are  59s. 1  -f-9s.5=  Im.  8s. 6; 
consequently  the  interval  in  sideral  time  is  Gh.  59m.  8s. G. 

The  numbers  in  columns  4,  5,  G,  7.  of  tlie  table  of  moon-culminating  stars,  correspond  to 
the  meridian  of  Greenwich,  and  may  be  reduced  to  any  other  meridian  by  the  usual  method 
of  interpolation,  as  in  Problem  I.,  page  3!JG.  Thus,  from  the  above  extracts  from  this  table, 
it  appears  that,  at  the  time  of  the  upper  culmination,  September  IG,  l^SG,  the  right  ascen- 
sion of  the  moon's  bright  limb  was  IGh.  OGm.  y]s.7L  At  the  following  lower  culmination,  it 
was  IGh.  37m.  12s. 45,  whicli  may  be  considered  as  corresiponding  to  the  upper  culmination, 
September  IG,  in  a  place  VZ\i.  iu  longitude  wpst  from  Greenwich;  and  at  the  next  upper 
culmination,  the  right  ascension  was  17h.  G9m.  Ols.GS,  which  may  be  considered  as  apjier- 
(aining  to  September  IG,  in  a  place  24h.  west  from  Greenwich;  according  to  the  ancient 
method  of  counting  tlie  longitude,  in  a  westerly  direction  completely  round  the  globe.  In 
like  manner,  in  east  longitude,  we  have,  at  the  upper  culmination  at  Greenwich,  September 
IG,  1S3G,  the  right  ascension  of  the  moon's  bright  limb  IGh.  OGm.  21s. 71,  and  we  may 
suppose  the  preceding  transit,  15h.  3Gm.  34s.71,  to  correspond  to  the  longitude  12h.  east, 
and  so  on. 

This  being  premised,  we  shall  now  proceed  to  show  how  to  find,  by  interpolation,  the 
moon's  rifjht  ascension  at  the  time  of  her  transit  over  any  meridian  in  east  or  west  longitude 
from  Greenwich.  The  process  of  calculation  is  very  nearly  the  same  as  that  in  Problem  L, 
page  39(),  but  for  convenience  we  have  reduced  it  to  the  following  form  : — 

RULE. 

Tu  find  llie  moons  right  ascension  at  Ilct  transit  over  any  meridian. 
1.  Take  from  the  fourth  column  of  the  table  of  moon-cuhiiinating  stars,  the  right  asce»- 
siona  of  the  same  limb  of  the  moon  corresponding  to  four  successive  'culminations,*  so  that 


*  Near  tlie  time  of  full  moon,  when  the  limb  marked  in  the  table  chan'j-s  IVoni  I.  to  II.,  there  may  he  on« 
or  two  of  lliese  quantities  not  marlted  in  column  4th  of  the  tahle  for  the  lunh  which  is  wanted  in  the 
calculation.     In  tliis  case,  the  reiiuired(viaMtities  can  be  obtained  from  the  corresponding  tabular  nuniliers.  by 


BY   A  TRANSIT   OF  THE   MOONS   LIiMB 


483 


two  nwy  precede  and  tico  folloio  after  the  time  of  transit  at  the  proposed  place.  Put  these 
numbers  below  each  other  in  their  regular  order ;  then  find  their  first  and  sl  cond  differences. 
Call  the  middle  term  of  the  first  differences,  the  arc  A;  the  mean  of  the  2nd  differences,  the 
arc  B  ;  and  if  the  longitude  be  west  from  Greenwich,  put  T  equal  to  that  longitude  in  time  ; 
but  if  the  longitude  be  east,  put  T  equal  to  the  difference  between  It^h.  and  that  longitude. 

2.  To  the  constant  logarithm  5.36452  add  the  logarithm  of  T  in  seconds  of  time,  and  the 
logarithm  of  A  in  seconds  of  time  ;  tlie  sum,  rejecting  10  in  the  index,  will  be  a.  proportional 
part,  which  is  to  be  added  to  the  second  right  ascension  taken  from  the  Nautical  Almanac 

3.  Enter  Table  XLV.  with  the  arc  B  at  the  top,  and  the  time  T  at  tlie  side  ;  opposite  to 
this  will  be  the  correction  of  second  differences,  to  which  prefi.x  a.  different  sign  from  that  of 
the  arc  B,  and  place  it  under  the  second  ascension  and  the  proportional  part  above  found. 
Connect  these  three  quantities  together,  as  in  addition  in  algebra ;  the  sum  will  be  the  sought 
right  ascension  of  the  moon  at  the  time  of  her  transit  over  the  proposed  meridian. 

The  same  process  may  be  used  for  interpolating  the  numbers  in  columns  5,  0,  7,  as  we 
shall  see  in  the  following  examples  : — 

EXAMPLE  1. 

■  Required  the  right  ascension  of  the  moon,  September  16,  1836,  astronomical  account,  at 
the  time  of  the  transit  over  the  meridian  of  a  place  whose  longitude  is  3h.  48m.  298.  west 
from  Greenwich;  also,  the  value  of  the  arc  H,  deduced  from  the  numbers  in  column  6,  for 
the  time  of  this  transit. 

Here  we  have  T=:3h.  48m.  29s.,  being  the  same  as  in  Example  I.,  page  3.  6  ;  this  value 
being  selected  in  order  to  show  more  readily  the  similarity  of  the  present  calculation  with 
that  in  page  396. 


183G.  Sept 

Uight  ascension. 

h.    m.     s. 

15    I.  c. 

J5    36    34.71 

16    u.  c. 

16    06   21.71 

16    I.  c. 

16    37    12.45 

17    u.  e. 

17    09    01.65 

1st  difference. 


A  =30 
31 


50.74 
49.20 


2d  difference. 


4-63.74 

-1-58.46 

B  =  -t-6I.10 


Arc.  II. 
Var.  U.  A. 

146.22 
151. 62 
156.77 
161.29 


1st  difference. 

s. 

5.40 

A  =  5. 15 

4.52 


2d  difference. 


—  0.25 

—  0.63 


Constant  Log.  5.36452 

T  =  3h.  48m.2r.s.     =  13709s Log.  4.13701 

A=        30      50.74=1850.74 Log.  3.26735 


t 


9      47  .33     =  587  .33 Log.  2.76838 

16      06     21  .71  second  right  ascension.   

6  .62  Table  XLV.     B  =  613.10. 


16h.  16ni.  02S.42  R.  A.  in  long,  of  3h.  48ni.  29s.  W. 


5.36452 

4  13701 

A=     5s,15 Log.  0.71181 

4-     1  .63 Log.  0.31334 

4-151  .62  second  value  of  H.  

-f-         .05  Table  XLV.     B  =  — Os.4!. 

H  =  153s.30  corresponding  to  long.  31i.  43ni.293.  VV 


Hence  it  appears,  that,  on  September  16, 1836,  astronomical  account,  m  a  place  8h.  dSm.  20s. 
west  from  Greenwich,  the  right  ascension  of  the  moon's  bright  limb  at  tiie  time  of  passing 
the  meridian,  was  IGh.  16m.  02s. 42,  and  that  the  arc  H,  corresponding  to  that  meridian, 
was  153s  30.  This  arc  H  represents  the  variation  of  the  moon's  right  ascension  between 
the  times  of  tlie  transit  of  her  bright  limb  over  the  two  meridians  whose  longitudes  are 
T  —  "JOm.  and  T  -|-  30m.,  corresponding  respectively  to  3h.  18m.  29s.  west,  and  41i.  18m.  29s. 
west,  from  Greenwich. 

In  the  preceding  example,  t])e  longitude  of  the  place  is  given,  to  find  the  moon's  right 
ascension  at  the  time  of  the  passage  of  her  bright  limb  over  the  meridian  of  that  place;  but 
we  may  suppose  tliat  right  ascension  to  be  given,  to  find,  by  an  inverse  process,  the  longitude 
of  the  place  of  observation,  or  the  time  T.  The  solution  of  this  problem  is  very  similar  to 
that  of  Problem  XIV.,  page  426,  changing  longitude  into  rigid  ascension,  &.C.;  and  it  may 
be  expressed  as  in  the  following  rule  : — 


RULE. 

To  find  the  longitude  of  any  place  from  the  moon's  right  ascc7isio?i  at  her  transit  over  the 

meridian  of  that  place. 

1  Take  from  column  4  of  the  table  of  moon-culminating  stars,  in  the  Nautical  Almanac, 
the  four  right  ascensions  of  tlie  bright  limb  of  the  moon,  as  in  the  above  example  ;  and 
then  compute,  as  in  that  example,  the  values  of  the  arcs  A,  B,  in  seconds  of  time. 

2.  To  the  constant  logarithm  4.63548  add  the  arithmetical  complement  of  the  logarithm 
of  the  arc  A  in  seconds  of  time,  and  the  logarithm  of  the  difi'ercnce  in  seconds  of  time 
between  the  given  right  ascension  and  the  second  right  ascension  taken  from  the  Nautical 


434 


TO   FIND   THE   LONGITUDE   OF  A   PLACE 


Almanac  ,  the  sum,  rejecting  10  in  the  index,  will  be  the  logarithm  of  the  approximate  time 
T  in  seconds. 

3.  Enter  Table  XLV.,  with  the  arc  B  at  the  top,  and  the  time  T  at  the  side,  and  find  the 
corresponding  correction  ;  to  the  logarithm  of  which  add  the  two  first  logarithms  above  found ; 
the  sum,  rejecting  10  in  the  index,  will  be  the  correction  of  the  approximate  time,  to  be 
applied  \vith  the  same  sign  as  the  arc  B,  and  the  correct  value  of  T  will  be  obtained,  which 
will  express  the  longitude  of  the  place  of  observation,  if  it  be  west  from  Greenwich;  but  if 
the  longitude  be  east,  we  must  subtract  this  value  of  T  from  12h.  to  obtain  the  true  longitudo 
in  time  east  from  Greenwich. 

EXAIMPLE  IL 

Suppose  that,  in  a  place  in  west  longitude,  on  the  IGth  of  September,  1S36,  the  moon's 
bright  limb  passed  the  meridian  in  3m.  20s.6.5,  sideral  time,  before  the  star  Antares.  Re- 
(^uired  the  longitude  of  the  place  of  observation. 

In  the  Nautical  Almanac,  column  4,  the  star  Antares  or  o  Scorpii's  right  ascension, 

Sept.  IG,  183J5,  was ICh.  19m.  23s.07 

Subtract  the  observed  difference  of  the  transits  in  sideral  time 3      20  .65 

The  remainder  is  the  right  ascension  of  the  moon's  bright  limb  at  the  transit IG     IG      02.42 

The  next  less  right  ascension  in  column  4  of  the  N.  A.,  corresponds  to  Sept.  IG,  u.  c.  IG     OG      21  .7] 

Difference  of  these  right  ascensions  is 580s.71  =  9m.  40s.71 

The  four  right  ascensions  to  be  taken  from  the  Nautical  Almanac,  are  those  corresponding 
to  September  15,  Z.  c,  September  16,  u.  c,  September  1(5,1.  c,  and  September  17,  u.  c,  being 
the  same  as  those  in  the  preceding  example,  where  we  have  found  A  =  30m.  50s.74  = 
1850s. 74,  B  ==  4"  61s. 10.     The  rest  of  the  calculation  is  as  follows  : — 


Equation  Table  XLV.   6s.58  . 
Correction  2m.  33s.6  =  153s.6. 


4.63548 

6.732G5 

..Log.  0.81823 

..Log.  2.18636 


Constant  Log.  4.63548 

A=1850s.74 Arith.  Comp.  Log.  6.732G5 

Diff.  of  right  ascension  580s.71 Log.  2.76396 

Approx.T  =  3h.  45m.  54s.7  =  13554s.7..Log.  4.13209 
Correction  =        2      33  .6  

T^Sti.  46m.  283.3^  the  longitude  of  the  place  of  observation. 

This  longitude  agrees,  within  a  fraction  of  a  second,  with  the  value  of  the  longitude 
assumed  in  Example  1.;  observing  that  the  computed  right  ascension  in  Example  L  is 
ICh.  ICm.  02s. 42,  being  the  same  as  that  which  is  supposed  to  be  observed  in  the  present 
example. 

When  the  difference  of  meridians  is  small,  we  may  compute  their  difference  from  the 
observed  difference  of  the  times  of  the  moon's  transit,  by  means  of  the  arc  H,  deduced  froiiJ 
column  6  of  the  table  of  moon-culminating  stars,  by  the  following  rule  : — 

RULE. 

To  compute  the  dijfcrcncc  of  meridians  hy  means  of  the  arc  H. 

1.  To  the  constant  logarithm  3.55630  add  the  arithmetical  complement  of  the  logarithm 
of  the  arc  H,  and  the  logarithm  of  the  difference  of  the  times  of  the  moon's  transit  over  the 
two  meridians  in  sideral  time ;  the  sum,  rejecting  10  in  the  index,  will  be  the  logarithm  of 
the  difference  of  meridians  expressed  in  seconds  of  time 

EXAMPLE  in. 

Suppose  that,  in  a  place  west  from  Greenwich,  Sept.  16, 1836,  the  moon's  bright  limb  passed 
the  meridian  in  20m.  02s. 30,  sideral  time,  after  the  star  Antares.     Required  the  longitude. 

It  appears  by  column  4  of  the  table  of  moon-culminating  stars,  that,  on  September  16,  the 
right  ascension  of  Antares  was,  16h.  19m.  23s.07.  Adding  this  to  20m.  02s.30,  we  get 
161i  3!tm.  25s. 37  for  the  right  ascension  of  the  moon's  bright  limb  at  the  time  of  its  transit 
over  the  meridian  of  the  place  of  observation.  Subtracting  from  this  the  time  of  transit 
at  Greenwich,  IGh.  37m.  12s. 45,  taken  from  column  4  of  the  table  of  moon-culminating 
stars,  we  get  2m.  12s. 02=  132s.n2,  for  the  diflerence  of  the  times  of  the  transits,  to  be  used 
in  the  above  rule.  Moreover,  the  arc  H,  corresponding  to  the  time  of  the  transit  at  Green- 
wich, is,  by  column  6  of  the  table,  H  =  156s. 77.     Then  we  have, 

Constant  Log.  3..55fi30 

Arc  H  =  15GS.77 Arith.  Comp.  Log.  7.80473 

Difference  of  times  of  transit  132s.92 Log.  2.12359 

Difference  of  longitude  50m.  52s.3  =  3052s.3 Log.  3.48462 


In  strictness,  the  value  of  H,  here  used,  ought  to  be  increased  a  little;  for,  by  column  6  of 
the  preceding  table,  its  value  for  Greenwich  is  151s.62,  and  for  a  place  in  the  longitude  of 
12h.  west,  is  156s.77.  The  difference  between  these  two  values  of  H,  is  5s. 15,  which  repre- 
oents  its  increment  corresponding  to  a  change  of  12h.  in  the  longitude,  being  at  the  rate  of 
Os.429  for  a  change  of  Ih.  in  the  longitude  ;  and  at  this  rate  the  increment  for  the  longitude, 
50m.  52s.3,  will  be  Os.364,  which  will  be  increased  to  Os.38,  if  we  notice  the  correction  of 


BY  A   TRANSIT  INSTRUMENT.  435 

second  differences  depending  on  the  arc  B,  and  compute  the  arc  H  as  in  Example  I. 
Hence  tlie  value  of  the  arc  H,  corresponding  to  the  meridian  of  the  place  of  observation,  is 
156s.77-{-0s.38=157s.l5.  If  we  take  the  mean  of  the  values  of  H  at  Greenwich,  156s.77, 
and  at  the  place  of  observation,  157s. 15,  it  becomes  H  ==  15Cs.96,  and  with  this  we  may 
repeat  the  above  calculation,  and  obtain  a  corrected  result. 

Constant  Log.  3.556^0 

Corrected  arc  H  =  15Cs.96 Arith.  Comp.  Log.  7.80421 

Dilierence  of  times  of  transit  132S.92 Log.  2.12359 

Correct  difTerence  of  long  tude  50m.  48s.6  =  30 18s.C Log.  3.43410 

In  general,  the  longitudes  of  places  where  such  observations  are  made,  are  known, 
within  a  few  seconds,  so  that  it  will  be  easy  to  find  at  once  the  value  of  the  arc  H,  corre- 
sponding to  the  estimated  meridian  which  falls  midway  between  tlie  meridians  of  the  two 
places  of  observation ;  the  meridian  of  Greenwich  being  used  as  one  of  these  places,  when 
the  times  of  transit  given  by  the  Nautical  Almanac  are  used  as  if  they  were  actual  obser- 
vations.    We  shall  give  the  following  example  of  this  method  : — 

EXAMPLE  IV. 

In  a  place  whose  longitude  was  known  to  be  3h.  3Sm.  29s.  W.  from  Greenwich,  it  wa3 
found  by  observation,  on  September  16,  1836,  that  the  moon's  bright  limb  passed  the  meridi- 
an 3ra.  46s.2,  sideral  time,  before  the  transit  of  the  star  Aldebaran ;  and  in  another  place, 
estimated  to  be  20m.  in  longitude  west  from  the  first  place,  or  in  3h.  53m.  2!)s.  W.,  the  observed 
difference  of  the  transits  was  2m.  55s. 0.  Required  the  difference  of  longitude  which  results 
from  this  observation. 

The  mean  of  these  two  longitudes  is  3h.  48m.  29s.,  and  we  have  found  in  Example  I., 
that  the  arc  H,  corresponding  to  this  meridian  on  that  day,  was  153s. 30.  Moreover,  the 
difference  of  the  two  times  of  transit,  3m.  46s.2,  and  2m.  55s.O,  is  51s.2 ;  then  we  have,  as 
in  the  last  example, 

Constant  Log.  3.55630 

Arc  H  =  153S.30 Arith.  Comp.  Log.  7.81446 

Difference  of  times  of  transit  51s.2 Log.  1.70927 

Difference  of  longitude 20m.  02s.3  =  1202s.3 Log.  3.08003 

Add  longitude  of  the  first  place 3h.  38      29  .0  

Gives  the  longitude  of  the  second  place  3h.  58m.  31s.3  W.,  as  it  is  deduced  from  tliis  observation. 


PROBLEM  XVIL 

Given  the  longitudes  of  the  sun  and  moon,  and  the  moorCs  latitude,  tojind  their  distance. 

RULE. 

Find  the  difference  of  the  two  longitudes,  and  to  its  log.  cosine  add  the  log.  cosine  of  the 
moon's  latitude ;  the  sum,  rejecting  10  in  the  index,  will  be  the  log.  cosine  of  the  sought 
distance,  which  will  be  of  the  same  affection*  as  the  difference  of  longitude. 

EXAJMPLE. 

July  20th,  1836,  at  noon,  mean  time  at  Greenwich,  by  the  Nautical  Almanac,  the  sun's 
longitude  was  117°  42'  31",  the  moon's  longitude  193°  46'  05",  and  the  latitude  2°  47'  16"  N. 
Required  tlieir  distaijce. 

0's  longitude 117°  42*  31' 

D's  longitude 193  46  05 

Difference  of  longitudes    76  03  34 Cosine  9.38186 

D's  latitude 2  47   16 Cosine  9.99949 

Distance 76  04  35 Cosine  9.38135,  as  in  the  Nautical  Almanac 

This  is  calculated  by  another  method  in  Example  III.  of  Problem  XVIII.  In  this  rule, 
the  sun's  latitude  is  neglected,  being  only  a  fraction  of  a  second. 

Tlie  distances  being  calculated  from  noon  and  midnight  by  this  (or  by  the  following) 
problem,  they  may  be  interpolated  for  every  three  hours,  by  Problem  I.  The  following 
example  will  serve  for  an  illustration : — 

EXAMPLE. 

Given  the  distances  of  the  sun  and  moon,  in  July,  1836, 19d.  12h.,  20d.  Oh.,  20d.  12h. 
and  21d.  Oh.,  respectively  70°  02'  35",  76°  04'  35",  82°  11'  29",  and  88°  23'  32".  Required 
tlie  distances,  July  20d.  at  3h.,  6h.,  and  9h. 

*  Two  arcs  are  said  to  be  of  the  same  affection  when  they  are  both  crreater  than  90',  or  both  lest  than  90°,  bul 
.<■  different  affection  when  the  one  is  greater  and  the  other  less  than  90°. 


436 


TO   FIND   THE   DISTANCE   OF  THE   MOON  AND  A   STAR 


d.  h. 

1836,  July  19  12 

20  0 

20  12 

21  0 


■•<^c 


70    02  35 

76    04  35 

11  29 

23  32 


82 


1st  difference. 


6    02   00 

A  =  6    06    54 

6    12   03 


2d  difference. 


+  4    54 
4-5    09 

B=  +  5    01 


Second  longitude 
Proportional  part. 
Table  XLV 

Distances 


+  76  04  35 
JA=    1  31  43.5 
T  =  3h.  corr.         —  28.2 

At3h.=:77  35  50 


At6h. 

»     (     1/ 

+  76  04  35 

^  A  =   3  03  27 

T  =  6h.  corr.         —  38 

At  6h.  =  79  07  24 


At9h. 


+  76  04  35 
3  A  =   4  35  10.5 
:9h.  corr.         —28.2 


These  distances  agree  with  the  Nautical  Ahnanac. 

PROBLEM   XVIII. 

Given  the  longitudes  and  latitudes  of  the  moon  and  a  star,  tojlnd  their  distance. 

RULE. 
To  the  log.  secant  of  the  difference  of  longitude  of  tlie  moon  and  star,  add  the  log.  tangent 
of  the  greatest  latitude  ;  the  sum,  rejecting  10  in  the  index,  will  be  the  log.  tangent  of  an 
arc  A,  of  the  same  affection  as  the  difference  of  longitude.  Take  the  sum  of  the  arc  A,  and 
the  least  latitude,  if  the  latitudes  are  of  a  different  name,  but  their  dfference  if  of  the  same 
name,  and  call  this  sum  or  difference  the  arc  B.  Then  add  together  the  log.  secant  of  the 
difference  of  longitude,  the  log.  secant  of  the  greatest  latitude,  the  log.  cosine  of  the  arc  A, 
and  the  log.  secant  of  the  arc  B ;  the  sum,  rejecting  30  in  the  index,  will  be  the  log.  secant 
of  the  distance  of  the  moon  and  star,  which  will  be  of  the  same  affection  as  B. 

EXAMPLE    I. 

Required  the  distance  of  ♦he  moon  from   the  star   a  Pegasi,   at  noon,  mean  time    at 

Greenwich,  July  9d.  1836,  when,  by  the  Nautical  Almanac,  the  moon's  longitude,  counted 

from  the  mean  equinox,  was  59°  40'  32",  and  her  latitude  0°  59'  15"  N. ;  the  longitude  of 

the  star,  computed  t  as  in  Problem  XIX.,  being  351°  12'  29",  and  its  latitude  19°  24'  29"  N. 

J 's  longitude 59°  40'  3=2" 

*'s  longitude 351   12  29 

Difference  of  longitudes    68  23  03 Secant..  10.43530 10.43530 

Greatest  latitude 19  24  29N Tansent     9.54C93 Secant  10.02541 


Arc  A 43  49  41... 

Least  latitude 0  59  15  N. 


Difference X  is  arc  B. . 


,  Tangent     9.98223 Cosine    9.85819 


42  50  26 Secant  10.13475 

Distance*  ])  69°  24' 00" Secant  10.45365 


This  distance  agrees  with  the  calculated  value  given  in  page  146  of  the  Nautical  Almanac. 

We  may  observe,  that  the  log.  secant  of  the  distance  is  also  equal  to  the  sum  of  the  log. 
cosecant  of  the  greatest  latitude,  the  log.  sine  of  the  arc  A,  and  the  log.  secant  of  the  arc  B, 
rejecting  20  in  the  sum  of  the  indices ;  but  the  above  rule  is  in  general  most  convenient,  on 
account  of  the  smallness  of  the  greatest  latitude,  except  when  the  difference  of  longitude  is 
nearly  equal  to  90°. 

We  may  use  the  same  method  for  finding  the  distance  of  the  moon  from  the  sun,  star,  or 
a  planet,  when  tlieir  right  ascensions  and  declinations  are  given,  instead  of  their  longitudes 
and  latitudes.  The  rule  is  the  same  as  that  we  have  given  above,  changing  longilvde  into 
right  ascension,  and  latitude  into  declination.  To  exemplify  this,  we  shall  compute  the 
same  example  by  this  second  method. 

EXAMPLE   II. 

Required  the  distance  of  the  moon  from  the  star  a  Pegasi,  at  noon,  mean  time  at  Green- 
wich, July  9d.  1836,  when,  by  the  Nautical  Almanac,  the  moon's  right  ascension  was 
57°  15'  01"  from  the  mean  equinox,  the  moon's  declination  21°  3'  55"  N. ;  tlie  star's  riglil 
ascension  from  the  same  equinox  344°  9'  20",  and  the  star's  declination  14°  19'  32"  N. 

P's  richt ascension....    57°15'01'' 
*'sriglit  ascension....  344  09  20 

Difference 73  05  41 Pecant..  10.53642 10.53649 

Greatest  declination  ..    91   03  55N Tangent     9..58566 Secant  10.03004 


.Tangent  10.12208 ....Cosine    9.779D8 


Arc  A 52  56  .56... 

Least  decl  ination 14   19  32  N. 

Difference  §  is  arc  B . . .  '38  37  24 .  Secant  10.10720 

Distance  *  <r  69°  23'  58" piecant  10.45364 


t  We  liave   preferred   these   computed   values,  as    licing   rather   more  accurate    than   the   numbers  in 
Table  XXXVII. 

J  The  suvi  must  be  used  if  the  latitudes  are  oC diffi-revt  names. 
§  The  sum.  must  be  used  if  the  declinations  are  o{  difftrcni  names. 


TO   FIND   THE   LONGITUDE,    <fec.   OF  A    CELESTIAL  OBJECT.      437 

This  differs  2''  from  the  former  method,  from  the  neglect  of  the  tenths  of  a  second  in  the 
angles,  and  from  not  taliing  the  logarithms  to  G  or  7  places  of  fiijurcs. 

EXAaiPLE  III 

July  20,  1S3G,  at  noon,  mean  time  at  Greenwich,  by  the  Nautical  Almanac,  the  sun's 
right  ascension  was  119°  47'  35",  the  sun's  declination  20°  38'  23"  N.,  the  moon's  right 
ascension  193°  45'  07",  and  the  moon's  declination  2°  52'  03"  S.     Required  their  distance. 

J)'s  right  ascension....  193°  45'  07" 
©'s  right  ascension...  119  47  35 

Diff.  of  right  ascension    73  57  32 Pecant..  lO.SS-'SS 10.55858 

Greatest  declination  . .    20  38  23  N Tangent     9.5759G Secant  10.02881 

Arc  A 53  44   II ..Tangent  10  13454 Cosine    9.77196 

Least  declination 2  52  03  Jj.  

Sam  t  is  arc 56  36   14 Secant  10.25930 

Distance  ©  d   76°  04' 35" Secant  10.61805 

This  agrees  with  the  distance  marked  in  the  Nautical  Almanac. 


PROBLEM  XIX. 

Given  the  light  ascension  and  declination  of  a  celestial  object,  with  the  mean  obliquity  oj 
the  ecliptic  E,  to  find  its  longitude  and  latitude. 

RULE. 

To  the  log.  tangent  of  the  declination  add  the  log.  cosecant  of  the  right  ascension  of  the 
object ;  the  sum,  rejecting  10  in  the  index,  will  be  the  log.  tangent  of  the  arc  A,  to  be  taken 
out  less  than  90°,  and  called  noi-tli  or  south,  as  the  declination  is.  If  the  right  ascension  is 
less  than  180°,  call  the  obliquity  of  the  ecliptic  south;  if  above  180°,  north.  If  A  a^d  E  are 
of  the  same  name,  take  their  sum,  otherwise  their  difference,  which  call  B,  and  mark  it 
with  the  same  name  as  the  greater  number,  whether  N.  or  S.  Then  add  together 
the  log.  secant  of  A,  the  log.  cosine  of  B,  and  the  log.  tangent  of  the  right  ascension;  the 
sum,  rejecting  20  in  the  index,  will  be  the  log.  tangent  of  the  longitude  in  the  same 
quadrant  as  the  right  ascension,  unless  B  be  greater  than  90°,  in  which  case  the  quantity 
found  in  the  same  quadrant  as  the  right  ascension,  subtracted  from  360°,  will  be  the 
longitude. 

To  the  log.  sine  of  the  longitude  add  the  log.  tangent  of  B ;  the  sum,  rejecting  10  in  the 
index,  will  be  the  log.  tangent  of  the  latitude,  of  the  same  name  as  B. 

Remark.  As  the  Tables  of  this  collection  are  not  marked  above  180°,  you  must  subtract 
180°  from  the  right  ascension,  when  it  exceeds  that  quantity,  and  find  tlie  log.  tangent  and 
log.  cosecant  of  the  remainder ;  and  then  the  arc,  corresponding  to  the  log.  tangent  of  the 
longitude,  is  to  be  taken  of  tlie  same  affection  as  this  remainder,  and  180°  added  thereto  ; 
the  sum  will  be  the  longitude,  unless  B  is  greater  than  90°,  in  which  case  the  supplement 
of  that  sum  to  3G0°  is  to  be  taken,  as  observed  above. 

EXAMPLE. 

From  the  Nautical  Almanac  we  find,  that,  on  the  9th  of  July,  1836,  the  right  ascension  of 
a  Pegasi  was  22h.  5Gm.  37s.35=  344°  9'  20",  its  declination  14°  19'  32"  N.,  and  the  mean 
obliquity  of  tlie  ecliptic  23°  27'  38".     Required  its  longitude  and  latitude. 

Declination..    14°19'32"N Tang..    9.40717 

Kight  ascens.  344  09  20     Cosec.    10.56380 Tang.     9.45303 

A 43  05  12    N Tang..    9.97097 Secant  10.13648 

E 23  27  33    N.  [This  ia  S.  when  R.  A.  U 

kss  than  180°.] 

.B 66  32  50    N Cosine    9.50088 Tang.  10.36208 

*'s  longitude  351°  12'29" Tang.    9.18939 Sine..  9.18425 

*'s  latitude  19°  24'  29"  N Tang.    9.54693 

This  longitude  is  counted  from  the  mean  equinox,  July  9d.  1836,  and  if  we  wish  to 
reduce  it  to  the  apparent  equinox,  we  must  apply  to  the  preceding  longitude  the  equation  of 
tlie  equinoxes  deduced  from  Table  XL.,  which  is  nearly  — 12";  so  that  the  longitude, 
counted  from  the  apparent  equinox,  is  351°  12'  17",  and  the  apparent  latitude  19°  24' 29"  N. 
We  have,  in  this  example,  taken  the  right  ascension  and  declination  of  the  star  from  the 
Nautical  Almanac,  where  they  are  given  to  fractions  of  a  second  ;  which  is  more  accurate 
than  Table  VIII.,  where  the  declinations  are  given  to  the  nearest  minute.  We  may, 
however,  use  the  numbers  in  Table  VIII.,  when  great  accuracy  is  not  required,  correcting 
for  the  aberration,  as  in  the  precepts  to  Table  XLI.  The  numbers  computed  in  tliis  problem 
agree  nearly  with  the  results  obtained  from  Table  XXXVII. 

t  The  difference  is  to  be  used  if  the  declinations  are  of  the  same  name. 


438  SPHERIC  TRIGONOMETRY. 

PROBLEM  XX. 

The  longitude  and  latitude  of  a  celestial  object  being  given,  ivith  the  mean  obliquity  of 
the  ecliptic  E,  to  find  the  right  ascension  and  declination. 

RULE. 

To  the  log.  tangent  of  the  latitude  add  the  log.  cosecant  of  the  longitude ;  the  sum, 
rejecting  10  in  the  index,  will  be  the  log.  tangent  of  the  arc  A,  which  is  to  be  called  7iorth 
or  south,  as  the  latitude  is.  If  the  longitude  is  less  than  180°,  call  the  obliquity  E  nortk  ;  if 
above  180°,  south.  If  A  and  E  are  of  the  same  name,  take  their  sum,  otherwise  their 
difference,  which  call  B,  marking  it  with  the  same  name  as  the  greater  number.  Then  add 
together  the  log.  secant  of  A,  the  log.  cosine  of  B,  and  the  log.  tangent  of  the  longitude  ;  the 
sum,  rejecting  20  in  the  index,  will  be  the  log.  tangent  of  the  right  ascension  in  the  same 
quadrant  as  the  longitude,  unless  B  be  greater  than  90°,  in  which  case  the  quantity  found 
in  the  same  quadrant  as  the  longitude,  si^jtracted  from  360°,  will  be  the  right  ascension. 

To  the  log.  sine  of  the  right  ascension  add  the  log.  tangent  of  B;  the  sum,  rejecting  10 
in  the  index,  will-be  the  log.  tangent  of  the  declination,  of  the  same  name  as  B. 

Remark.  If  the  longitude  exceeds  180°,  you  must  subtract  180°  from  it,  and  find  the 
log.  tangent  and  log.  cosecant  of  the  remainder.  The  arc  corresponding  to  the  log.  tangent 
of  the  riglit  ascension  is  to  be  taken  of  the  same  afTeclion  as  this  remainder,  and  180°  added 
thereto,  will  be  the  right  ascension,  unless  B  is  greater  than  90°,  in  which  case  the  supple- 
ment of  that  sum  to  3(J0°  is  to  be  taken,  as  was  observed  before. 

EXAMPLE. 

On  the  9th  of  July,  1836,  the  apparent  longitude  of  the  star  a  Pegasi  was  351°  12'  29  ', 
counted  from  the  mean  equinox,  the  star's  apparent  latitude  19°  24'  29''  N.,  and  the  mean 
obliquity  of  the  ecliptic  23°  27'  38".     Required  its  right  ascension  and  declination. 

Latitude....     19''24'29ii Tang.     9.54693 

Longitude..  351  12  29 Cosec.  10.81575 Tang.    9.18939 


66  32  50  N Tang.    10.3G268 Secant  10.40012 


E 23  37  38  S.   [ThU  la  N.  when  the  longitude 

ia  less  than  ISO".] 

B 43  05  12  N Cosine    9.86352 Tang.  9.97097 

*'s  right  ascension  344'  09'  20  ' Tang.     9.45303 Sine     9.43620 

«'s  declination 14°  19' 32"  iS" Tang.  9.40717 


The  assumed  longitudes  in  this  example  are  the  same  as  those  computed  in  Problem  XIX., 
by  means  of  the  right  ascension  and  declination  taken  from  the  Nautical  Almanac;  being 
rather  more  accurate  than  the  results  of  Table  XXXVII.  The  right  ascension  and  declina- 
tion computed  in  this  example,  agree  with  those  assumed  in  Problem  XIX.,  which  serves 
as  a  proof  of  the  correctness  of  tlie  calculation. 

If  the  given  longitude,  latitude,  and  obliquity,  are  the  mean  values,  the  resulting  right 
ascension  and  declination  will  be  the  mean  values;  but  if  the  proposed  quantities  are  cor- 
rected for  aberration  and  nutation,  the  resulting  quantities  will  also  bo  corrected.  This 
remark  is  equally  applicable  to  the  preceding  problem. 


SPHERIC  TRIGONOMETRY. 

Most  of  the  rules  given  in  the  preceding  problems  may  be  easily  demonstrated  by 
Spheric  Trigonometry.  As,  for  example,  that  of  Problem  XVII.  may  be  investigated  as 
follows  : — In  Plate  XIII.,  fig.  1,  let  A  be  the  place  of  the  moon,  C  that  of  the  sun,  CP  an 
arc  of  the  ecliptic,  and  AP  a  circle  of  latitude  passing  through  the  moon,  and  cutting  the 
ecliptic  at  right  angles,  at  P.  Then  the  difference  of  longitude  of  the  sun  and  moon  is 
equal  to  the  arc  CP,  and  the  moon's  latitude  is  AP ;  whence  the  distance  AC  may  be  found 
by  the  rule  of  Napier,  radius  X  cos.  AC  =  cos.  AP  X  cos.  CP.  This  in  logarithms  gives 
log.  COS.  AC  =  log.  COS.  AP -f- log-  cos.  CP  —  log.  radius,  which  is  the  formula  made  use  of. 
Want  of  room  prevents  the  insertion  of  the  demonstrations  of  the  methods  of  calculatino-  the 
other  problems. 

The  celebrated  rules  given  by  Lord  Napier,  for  solving  the  problems  of  right-angled 
spheric  trigonometry,  being  very  easily  remembered,  are  much  made  use  of  by  mathemati- 
cians. In  a  paper  commimicated  by  the  author  of  this  work  to  the  American  Academy  of 
Arts  and  Sciences,  and  published  in  the  third  volume  of  the  first  series  of  the  Memoirs  of 
that  society,  a  method  was  given  for  the  more  easy  application  of  those  rules  to  oblique 
spheric  trigonometry,  and  as  the  tables  of  this  collection  may  sometimes  be  made  use  of  in 
solving  various  problems  of  spherics  besides  those  given  in  the  former  part  of  this  work,  it 
was  thought  proper  to  insert  this  improved  method,  with  the  formulas  most  frequently  made 
use  of,  to  enable  any  person  acquainted  with  spheric  trigonometry,  to  make  use  of  tlif 
tables,  without  the  trouble  of  referring  to  another  work  for  the  rules. 

In  every  right-angled  spheric  triangle  there  are  Jive  circular  j)arls  ;  namely,  the  two  legs 


SPHERIC  TRIGONOMETRY.  439 

the  complement  of  the  hj'potenuse,  and  the  complements  of  the  two  oblique  angles,  whicl 
are  named  adjacent  or  opposite,  according  to  tlieir  positions  with  respect  to  each  other.  Th« 
right  angle  is  not  included  as  one  of  tlie  circular  parts,  neither  is  it  supposed  to  separate  tin 
legs.  In  all  cases  of  right-angled  spheric  trigonometry,  two  of  these  parts  are  given  to  fin* 
the  third.  If  the  three  parts  join,  that  which  is  in  the  middle  is  called  the  middle  part :  if 
the  tiiree  parts  do  not  join,  two  of  them  must,  and  the  other  part,  which  is  separate,  ia 
called  the  middle  part,  and  the  other  two,  opposite  parts,  as  in  Plate  XIII.  fig.  1,  2.  Then 
putting  the  radius  equal  to  unity,  tiie  equations  given  by  Napier  will  become 

Sine  of  middle  part=  Rectangle  of  the  tangents  of  the  adjacent  parts. 
=  Rectangle  of  tlie  cosines  of  the  opposite  parts. 
The  method  of  applying  these  solutions  to  the  various  cases  of  right-angled  suheric 
trigonometry,  is  very  simple,  and  is  explained  in  several  treatises.  To  apply  the  mctnoa  to 
oblique-angled  spheric  trigonometry,  it  is  necessary  to  divide  the  triangle  into  two  right- 
angled  spheric  triangles,  by  means  of  a  perpendicular  AP  (Plate  XIII.  tig.  3,  4,  5,  ]4,)  let 
fall  from  the  point  A  upon  "the  opposite  side  BC  ;  the  perpendicular  being  so  chosen  as  to 
make  two  of  the  given  things  fall  in  one  of  the  riglU-angkd  triangles  ;  or,  in  other  words,  tlie 
■perpendicular  ought  to  he  let  fall  from  the  end  of  a  given  side  and  opposite  to  a  given  angle.'* 
Each  triangle  thus  found  contains,  as  above,  five  circular  parts,  the  perpendicular  being 
counted  and  bearing  the  same  name  in  each  of  them  ;  consequently  the  parts  of  each 
triangle  similarly  situated  with  respect  to  the  perpendicular,  must  have  the  same  name. 
In  every  case  of  oblique-angled  spheric  trigonometrjr,  there  are  three  parts  given  to  find  a 
fourth;  and  in  making  use  of  the  method  of  a  solution  by  means  of  the  perpendicular,  there 
will,  in  general,  be  two  of  these  parts  in  each  of  the  triangles  ACP,  ABP,  similarly  situated 
with  respect  to  each  other.  To  each  of  these  must  be  joined  the  perpendicular  A P,  and 
there  will  be  three  parts  in  each  triangle,  wliicli  are  to  be  named  middle,  adjacent,  or  oppo 
site,  according  to  the  above  directions.  Then  the  equations  for  solving  all  the  cases  of 
right-angled,  and  all  except  two  cases  of  oblique-angled  spheric  trigonometry,  are, 


^^  »   o.  ■  1  ,,  A  =  }^  Tangents  of  tlie  adjacent  partsA 

^  ''  "       (  oc  5  Cosines  of  the  opposite  parts. 


These  equations,  when  applied  to  right-angled  spheric  triangles,  signify,  as  before,  that 
the  sine  of  the  middle  part  is  equal  to  the  rectangle  of  the  tangents  of  the  adjacent  parts,  or 
to  the  rectangle  of  the  cosines  of  the  opposite  parts  ;  but  when  applied  to  an  oblique-angled 
triangle,  they  signify  that  the  sines  of  the  middle  parts  are  proportional  to  the  tangents  of 
the  adjacent  parts;  or  that  the  sines  of  the  middle  parts  are  proportional  to  the  cosines  of 
the  opposite  parts  of  the  same  triangle  ;  observing  that  the  perpendicular,  being  common  to 
both  triangles  APR,  APC,  and  bearing  the  same  name  in  each  of  them,  must  not  be  made 
use  of  in  the  analogies,  nor  counted  as  a  middle  part.  This  can  produce  no  embarrassment, 
because  the  cases  of  oblicjue  spheric  trigonometry  may,  in  general,  be  solved  in  the  shortest 
manner,  without  calculating  the  perpendicular. 

The  first  case  not  included  in  the  above  rules,  is  where  the  question  is  between  two  sides 
and  the  opposite  angles,  which  may  be  solved  by  the  noted  theorem,  that  the  sines  of  the 
sides  are  proportional  to  the  sines  of  the  opposite  angles,  or,  as  it  may  be  expressed  in  an 
abridged  form  for  more  easy  reference, — 

(2.)  Sine  side  oc  sine  opp.  angle. 

This,  combined  with  the  above  improved  formula,  furnishes  a  complete  solution  of  the 
various  cases  of  splieric  trigonometry,  except  where  three  sides  are  given  to  find  an  angle, 
or  (which  is  nearly  the  same  tiling,  l)y  taking  the  supplementary  triangle)  three  angles  to 
find  a  side.  The  above  rules  (marked  1,  2,)  are  simple  in  their  form,  and  the  first  varies 
but  little  from  that  made  use  of  by  Napier,  so  that  it  is  extremely  easy  to  remember  them. 
The  case  not  inchided  in  these  rules  may  be  solved  by  one  of  the  formulas  of  Case  V.  or  VI., 
which  may  be  committed  to  memory  with  little  trouble.  To  illustrate  these  rules,  the 
following  examples  are  given,  which  include  all  the  cases  of  oblique  spheric  trigonometry. 

CASE  I.     Plate  XIII.,  fig.  3,  4,  5,  14. 

Given  AC,  AB,  and  the  opposite  angle  C,  to  find  BC,  and  the  angles  A,  B. 

In  the  right-angled  spheric  triangle  APC,  are  given  AC  and  C,  and  by  marking  it  as  in 
fig.  2,  CP  may  be  found  by  the  rule  sine  mid.  =  tang,  adj.,  which  gives  sine  (co.  C)  = 
tang.  CP  X  tang.  (co.  AC)  or  tang.  CP=  cos.  C  X  tang.  AC.^  Then,  in  the  triangles 
ABP,  ACP,  are'given  AB,  AC,  and'CP.to  find  BP.  If  to  these  is  joined  the  perpendicular 
AP,  it  will  be  found  that,  in  the  triangle  ACP,  the  complement  of  AC  is  the  middle  part 
(as  in  fig.  3.)  and  CP  an  opposite  part.  The  triangle  ABP  is  to  be  marked  in  a  similai 
manner.     Then  the  rule  sine  mid.  oc  cos.  opp.  gives  sine  (co.  AC)  :  cos.  CP  : :  sine  (co.  AB) . 

*  When  this  can  \e  done  in  two  different  wnys,  (as  in  Cases  II.  IV.,)  it  will  generally  produce  the 
shortest  solution  to  make  use  of  that  perpendicular  which  does  not  divide  the  required  angle  or  side  into 
segments. 

t  It  will  be  of  considerable  assistance  in  remembering  these  rules,  to  note  tliat  the  second  letters  of  the 
words  tantrent  and  cusine  are  the  same  as  the  first  letters  of  adjacent  and  opposite.  The  symbol  cc,  which  ia 
used  in  this  example,  signifies  ;»ra^urtiu/ia(;  thus,  3 z  ex:  z  signifies  that  Si  is  proportional  to  2,  z  being  any 
number  wh  itever 

X  In  puttinj;  this,  or  any  similar  expression,  in  logarithms,  the  radius  must  be  neglected  in  the  sum  of  the 
two  logarithms  of  the  second  member. 


440  SPHERIC  TRIGONOMETRY 

COS.  BP,  and  BC  =  BP+  CP.  By  marking  the  segments  as  in  fig.  4,  the  rule  sine  mid.  oc 
tang.  adj.  gives  sine  CP  ■  tang.  (co.  C)  : :  sine  BP  :  tang.  (co.  B).  Having  found  BC,  the 
angle  A  may  be  found  by  the  rule  sine  side  cc  sine  opp.  angle,  which  gives  sine  AB  : 
sine  C  :  :  sine  BC  :  sine  A. 

Othericise — If  the  side  BC  is  not  required,  the  angles  A,  B,may  be  found  in  the  following 
manner.  The  rule  sine  mid.  =  tang.  adj.  gives,  by  marking  as  in  fig.  1,  sine  (co.  AC)  = 
tang.  (co.  C)  X  tang.  (co.  CAP)  or  cot.  CAP  =  cos.  AC  X  tang.  C;  and,  by  marking  as  in 
fig.  5,  the  rule  (sine  mid.  cc  tang.  adj.  or)  tang.  adj.  oc  sine  mid.  gives  tang.  (co.  AC)  :  sine 
(co.  CAP)  : :  tang.  (co.  AB)  :  sine  (co.  BAP) ;  then  A  ==  BAP  +  CAP.  By  marking  the 
segments  as  in  fig.  14,  the  rule  (sine  mid.  oc  cos.  opp.  or)  cos.  opp.  oc  sine  mid.  gives 
cos.  (co.  CAP)  :  sine  (co.  C)  :  :  cos.  (co.  BAP)  :  sine  (co.  B)  or  sine  CAP  :  cos.  C  :  :  sine  BAP  : 
COS.  B.  Having  A,  C,  and  AB,  BC  may  be  found  by  the  rule  sine  side  oc  sine  opp.  antrle, 
which  gives  sine  C  :  sine  AB  : :  sine  A  :  sine  BC. 

CASE  II.     Plate  XIII.,  fig.  3,  4. 
Given  AC,  BC,  and  the  included  angle  C,  to  find  AB,  and  the  angles  A,  B. 

The  rule  sine  mid.  =^ tang.  adj.  gives,  as  in  Case  I.,  tang.  CP  =  cos.  C  X  tang.  AC; 
then  BP  =  BC  +■  CP,  and  the  rule  cos.  opp.  oc  sine  mid.  gives,  by  marking  as  in  fig.  3, 
cos.  CP  :  sine  (co.  AC)  :  :  cos.  BP  :  sine  (co.  AB,)  and,  by  marking  as  in  fig.  4,  the  rule 
sine  mid.  oc  tang.  adj.  gives  sine  CP  :  tang.  (co.  C)  :  :  sine  BP  :  tang.  (co.  B).  Having 
found  AB,  we  may  find  A,  by  the  rule  siiie  side  cc  sine  opp.  angle,  which  gives  sine  AB  : 
sine  C  :  :  sine  BC  :  sine  A. 

If  the  angle  A  had  been  required,  and  not  B,  it  would  have  been  shorter  to  let  the  per- 
pendicular fall  from  the  point  B,  by  which  means  the  required  angle  A  would  not  be  divided 
mto  segments.  In  this  case,  the  side  AB  and  the  angle  A  might  be  found  in  a  similar 
manner  to  that  by  which  AB  and  B  are  found  above. 

CASE  III.     Plate  XIII.,  fig.  3,  4,  5,  14. 
Given  the  angles  C,  B,  and  the  opposite  side  AC,  to  find  BC,  AB,  and  the  angle  A. 

The  rule  sine  mid.  =  tang.  adj.  gives,  as  in  Case  I.,  tang.  CP  =  cos.  C  X  tang.  AC. 
Then  the  rule  tang.  adj.  cc  sine  mid.  gives,  by  marking  as  in  fig.  4,  tang.  (co.  C)  : 
sine  CP  :  :  tang.  (co.  B)  :  sine  BP ;  then  BC  =  CP  ^  BP.  Again,  the  rule  cos.  opp.  oc  sine  mid. 
gives,  by  marking  as  in  fig.  3,  cos.  CP  :  sine  (co.  AC)  :  :  cos.  BP  :  sine  (co.  AB).  Having 
found  BC,  the  rule  sine  side  cc  sine  opp.  angle  gives  sine  AC  :  sine  B  :  :  sine  BC  :  sine  A 

Othericise — The  rule  sine  mid.  =  tang.  adj.  gives,  as  in  Case  I.,  cot.  CAP  :=  cos.  AC  X 
tang.  C,  and  the  rule  sine  mid.  oc  cos.  opp.  gives,  by  marking  as  in  fig.  14,  sine  (co.  C)  : 
cos.  (co.  CAP)  :  :  sine  (co.  B)  :  cos.  (co.  BAP)  or  cos.  C  :  sine  CAP  :  :  cos.  B  :  sine  BAP, 
and  A  =  CAP  +  BAP.  Then  the  rule  sine  mid.  oc  tang.  adj.  gives,  by  marking  as  in 
fig.  5,  sine  (co.  CAP)  :  tang.  (co.  AC)  :  :  sine  (co.  BAP)  :  tang.  (co.  AB).  Having  found 
A,  the  rule  sine  side  oc  sine  opp.  angle  gives  sine  B  :  sine  AC  :  :  sine  A  :  sine  BC. 

CASE  IV.     Plate  XIII.,  fig.  5, 14. 
Given  the  angles  A,  C,  and  the  included  side  AC,  to  find  AB,  BC,  and  the  angle  B. 

The  rule  sine  mid.  =  tang.  adj.  gives,  as  in  Case  I.,  cot.  CAP  =  cos.  AC  X  tang.  C, 
and  BAP=A  +  CAP.  The  rule  sine  mid.  <x.  tang.  adj.  gives,  by  marking  as  in  fig.  5, 
sine  (co.  CAP)  :  tang.  (co.  AC)  :  :  sine  (co.  BAP)  :  tang.  co.  (AB).  The  rule  cos.  opp.  cc 
sine  mid.  gives,  by  marking  as  in  fig.  14,  cos.  (co.  CAP)  :  sine  (co.  C)  :  :  cos.  (co.  BAP)  • 
sine  (co.  B)  or  sine  CAP  :  cos.  C  :  :  sine  BAP  :  cos.  B.  Having  found  B,  the  rule 
sine  side  oc  sine  opp.  angle  gives  sine  B  :  sine  AC  :  :  sine  A  :  sine  BC. 

If  the  side  BC  had  been  required,  and  not  AB,  it  would  be  shorter  to  let  the  perpendicular 
fall  from  the  point  C,  by  which  means  the  required  side  BC  would  not  be  divided  into  seg- 
ments. In  tins  case,  the  side  BC  and  the  angle  B  might  be  found  in  a  similar  manner  tc 
that  by  which  AB  and  B  are  found  above. 

CASE  V.     Plate  XIII.,  fig.  3 

Given  AB,  AC,  and  BC,  to  find  either  of  the  angles,  as  A. 

Put  S  =  ^  (AB  -f  AC  4-  BC).  Then  the  angle  A  may  be  found  by  either  of  the  follow- 
ing  theorem.s,  in  which,  for  brevity,  the  words  sijie,  cosine,  &c.,  are  used  for  log.  sine 
log.  cosine,  &c. 

C3  'l  Sine  .^  A  =  sine  (S  —  AB)  -f-  sine  (S  —  AC)  -[-  cosec.  AB  -f-  cosec.  AC  — 20 

_  . 

(A  s  r'      1  A ^i"^^  S  4"  sine  (S  —  BC)  +  cosec.  AB  -{-  cosec.  AC  —  20 


SPHERIC  TRIGONOMETRy  441 

CASE  VI.     Plate  XIII.,  fig.  3. 

Given  the  angles  A,  B,  C,  to  find  either  of  the  sides,  as  BC. 

Put  S  =  ^  (A  -|-  B  4-  C).  Then  the  side  BC  may  be  found  by  either  of  the  following 
tlieorenis,  adapted  to  logarithms,  as  in  the  last  example. 

('  \  Sne  >■  BC cosine  S  -f-  cosine  (S  —  A)  -|-  cosec.  B  -|-  cosec.  C  —  20 

(C  )  Co  ■  le  P^  BC cosine  (S  —  B)  -{-  cosine  (S  —  C)  +  cosec.  B  -f-  cosec.  C  —  20 

^  .;        su     ,  - 

The  above  include  all  the  cases  of  Oblique  Trigonometry.  The  2d  and  4th  Cases  may 
be  solved  in  a  ditferent  manner  by  the  following  theorems,  which,  on  some  occasions,  may 
be  found  very  useful.  Thus,  both  the  angles  in  Case  II.  may  be  found  by  the  following 
theorems  : — 

(7.)  Sine  h  (AC  -f  BC)  :  sine  ^  (BC  M  AC)  :  :  cot.  h  C  :  tang,  i  (A  —  B). 

(S.)  Cosine  ^  (AC  -f-  BC)  :  cosine  i  (BC  M  AC)  : :  cot.  ^  C  :  tang.  ^  (A  -f  B). 
i  (A  —  B)  is  less  than  90°,  and  i  (A  +  B)  is  of  the  same  affection  as  ^  (AC  +  BC) 
The  sum  and  difference  of  the  terms  ^  (A  —  B)  and  J  (A  -|-  B)  will  give  A  and  B. 

Both  the  sides  in  Case  IV.  may  be  found  thus  : — 

(9.)  Sine  i  (A  +  C)  :  sine  ^  (A  M  C)  : :  tang,  i  AC  :  tang.  ^  (BC  02  AB). 
(10.)  Cosine  i  (A  +  C)  :  cosine  =^  (A  CQ  C)  :  :  tang.  ^  AC  :  tang.  ^  (BC  -f-  AB). 
4  (BCM  AB)  is  less  than  90°,  and  ^  (BC4-  AB)  is  of  the  same  affection  as  h  (A4-  C) 
Then  the  sum  and  difference  of  ^  (BC  ^  AB)  and  i  (BC  +  AB)  give  AB  and  BC. 


The  improved  rule  for  solving  the  cases  of  Obli/^ue  Spheric  Trigonometry  by  the  oirculai 
parts,  may  be  easily  deduced  from  those  given  by  Lord  Napier.  For  if  we  put  M  for  the 
middle  part,  A  for  the  adjacent  part,  and  B  for  the  opposite  part  of  the  triangle  APC, 
(fig.  3,  4,  5,  14,  Plate  XIII.,)  m,  a,  h,  for  the  corresponding  parts  of  the  triano-le  APB,  and 
P   for  the  perpendicular  AP  ;    then   if  P  is  an   adjacent  part,   the  rules  of  Napier    will 

„        sine  M  sine  to    ,  sine  M       sine  m 

give  tang,  r  =  ,  and  tang.  r  = ;  hence  = ;  consequently, 

tang.  A  tang,  a  tang.  A      tang,  a 

snie  HI  :  tang.  A   :  :   sine    m   :   tang.    a.      If   P    is   an   opposite   part,    the    same    rule 

...    .  „      sine  M         ,  ^        sine  to     ,  sine  M      sine  to 

will  give  COS.  r  = ,  and  cos.  r  = >   hence = j    consequentlv. 

cos.  B  COS.  h  cos.  B        cos.  h  '  ' 

sine  M  .  cos.  B  :  :  sine  m  :  cos.  &,  which  are  the  twfl  ruks  to  be  demonstrated. 
56 


442  TO  FIND  THE  LONGITUDE  OF  A  PLACE. 

PROBLEM  XXI. 

To  find  the  longitude  of  a  place  by  an  eclipse  of  ike  sun,  ivlien  the  beginning  or  end  is 
observed ;  the  apparent  time  being  estimated  from  noon  to  noon,  according  to  the  method 
of  astronomers ;  the  latitude  of  the  place  being  also  known. 

RULE. 

1.  With  the  longitude  by  account,  find  tlie  corresponding  Greenwich  incan  time  of  the 
observation.  For  tliis  time,  take  out  from  the  Nautical  Ahiianac  the  sun's  right  ascension, 
declination,  and  semi-diameter,  the  horizontal  parallaxes  of  the  sun  and  moon,  and  the 
moon's  declination  roughly  lu  the  minute. 

2.  Reduce  the  latitude,  and  the  moon's  horizontal  parallax,  by  subtracting  the  corrections 
found  in  Table  XXXVIII. ;  and  subtract  from  the  moon's  corrected  horizontal  parallax  the 
sun's  horizontal  parallax,  and  the  remainder  is  the  relative  parallax. 

3.  To  the  proportional  logarithm  of  the  relative  parallax  add  the  log.  secant  of  the 
reduced  latitude,  the  constant  logarithm  1.1761,  and  the  log.  cosecant  of  double  the  observed 
time  from  noon-;  (this  double  time  being  regarded  as  P.  M.  in  using  Table  XXVII.,  unless 
it  exceeds  twelve  hours,  in  which  case  the  excess  above  twelve  hours  is  to  be  regarded  as 
A.  M. ;)  the  sum,  rejecting  20  in  the  index,  is  (S). 

4.  To  the  sum  (S)  add  the  log.  cosine  of  the  moon's  declination,  and  the  constant  log. 
0.3010 ;  the  sum,  rejecting  10  in  the  index,  is  the  proportional  log.  of  an  arc  in  time,  which, 
subtracted  from  the  observed  time  from  noon,  gives  the  corrected  time  from  noon. 

5.  With  the  corrected  time,  the  reduced  latitude,  and  the  sun's  declination,  calculate  by 
Rule,  page  247,  the  sun's  true  altitude. 

6.  To  the  log.  secant  of  the  sun's  true  altitude  add  the  log.  sine  of  double  the  corrected 
time  from  noon,  (this  double  time  being  regarded  as  P.  M.  or  as  A.  M.,  in  the  same  way  as 
before,)  and  the  log.  cosine  of  the  reduced  latitude;  the  sum,  rejecting  20  in  the  index,  is 
the  log.  sine  of  the  parallactic  angle. 

7.  To  the  proportional  log.  of  the  relative  parallax  add  the  log.  secant  of  the  parallactic 
angle,  and  the  log.  secant  of  the  sun's  true  altitude  ,  the  sum,  rejecting  20  in  the  index,  is 
the  proportional  log.  of  the  correction  for  declination.  This  correction  is  of  the  same  name 
with  the  latitude,  when  the  observed  time  from  noon  is  less  than  six  hours,  and  of  the  dif- 
ferent name  when  this  time  is  greater  than  six  hours.  Correct  the  sun's  declination  by  adding 
to  it  the  correction  for  declination  if  of  the  same  name,  and  subtracting  if  of  the  different  name. 

8.  To  the  sum  (S)  add  the  log.  cosine  of  the  sun's  corrected  declination  ;  the  sum,  rejecting 
10  in  the  index,  is  the  proportional  log.  of  an  arc  in  time,  which  is  the  correction  for  right 
ascension,  and  is  additive  if  the  time  is  afternoon,  but  subtractive  if  the  time  is  forenoon. 
Correct  the  sun's  right  ascension  by  adding  the  correction  for  right  ascension  when  additive, 
and  subtracting  it  when  subtractive. 

9.  Multiply  the  nearest  numl)er  of  minutes  in  the  moon's  horizontal  parallax  by  the  nearest 
number  of  minutes  in  the  sun's  semi-diameter,  and  multiply  this  product 
by  the  factor  in  the  annexed  table  corresponding  to  the  sun's  true  altitude  ; 
the  product,  divided  by  100,  is  an  arc  expressed  in  seconds,  which,  subtracted 
from  the  sun's  semi-diameter,  gives  the  sun's  corrected  semi-diameter. 

10.  To  the  proportional  logarithm  of  tiie  moon's  horizontal  parallax  (not 
corrected)  add  the  constant  logarithm  0.5G46 ;  the  sura  is  the  proportional 
logarithm  of  the  moon's  semi-diameter. 

11.  When  the  observation  is  the  beginning  or  ending  of  an  eclipse,  the 
distance  of  the  centres  of  the  sun  and  moon  is  found  by  adding  the  sun's 
corrected  semi-diameter  to  the  moon's  semi-diameter.  But  when  the  obser- 
vation is  that  of  the  beginning  or  ending  of  total  darkness  in  a  total  eclipse, 
or  tliat  of  the  formation  or  of  the  breaking  up  of  the  ring  in  an  annular 
eclipse,  the  distance  of  the  centres  of  the  sun  and  moon  is  found  by  taking 
the  difference  between  the  sun's  corrected  semi-diameter  and  the  moon's  semi-diameter. 

12.  Assume,  from  inspection  of  the  Nautical  Almanac,  a  convenient  time  when  the 
moon's  right  ascension  differs  but  little  from  the  sun's  corrected  right  ascension,  and  for  this 
time  take  out  Jiew  right  ascensions,  and  neto  declinations  of  the  sun  and  moon,  and  their 
horary  motions  in  riglit  ascension  and  declination  by  Problems  I.  and  II. 

13.  From  the  hourly  motion  of  the  moon  in  right  ascension  subtract  that  of  the  sun; 
the  remainder  is  the  relative  motion  in  right  ascension. 

The  difference  between  the  hourly  motion  of  the  moon  in  declination  and  that  of  the  aun 
Is  the  relative  motion  in  declination. 

Correct  the  sun's  new  right  ascension,  by  adding  the  correction  for  right  ascension  when 
it  is  additive,  and  subtracting  when  it  is  subtractive. 

Correct  the  sun's  ncio  declination,  by  adding  the  correction  for  declination  when  it  is  of 
the  same,  and  subtracting  it  when  it  is  of  the  different  name. 

14.  Subtract  the  logarithm  of  the  difference  between  the  sun's  new  corrected  right 
ascension  and  the  moon's  right  ascension  from  the  logarithm  of  the  relative  motion  in  right 
ascension,  and  call  the  remainder  R. 

15.  To  the  remainder  R  add  the  constant  log.  0.4771  ;  the  sum  is  the  proportional  loga- 

-.,        r  •    »•        i    I     ( added  to  )  the  assumed  time  when  the  sun's  C  greater 

rithm  of  an  arc  m  tune,  to  be  <      , ,      .  j  /-        >  *  j    •   u*  •      •    •?  i„-. 

'  (  subtracted  from  3  neio  corrected  right  ascension  is  ^  less 

than  the  moon's  right  ascension,  to  get  the  nezc  corrected  time. 

16.  To  the  remainder  R  add  the  uroportional  logarithm  of  the  relative  motion  in  declina 


Sun's 

true  al- 

Factor. 

titude. 

0" 

0.01 

10 

0.31 

20 

0.61 

30 

0.89 

40 

1.15 

50 

1.37 

GO 

1.5-1 

70 

1.G7 

80 

1.75 

90 

1.77 

BY  AN  ECLIPSE  OF  THE  SUN. 


443 


tion;    the  sum  is  the  proportional  logarithm  of  a  correction  of  the  moon's  decimation 
Whether  this  correction  is  additive  or  subtractive  is  thus  determined  :  —  Find  three  numbers 

-fo"°-  =  -  ,^  increasing,  ^^,^,^^,„^^^,^^.j^; 


If  the  moon's  declination  is 


(^  decreasi 


Ul 


1  I 


If  the  moon's  motion  in  declination  is  <  ?  >  than  the  sun's,  the  second  number  is  <  o  ? 

Tf.,,  .  t  A    ■   ui  •      •    (  ereater  )  than  the  moon's  right  ascen-  C  ]  ) 

If  the  sun  s  new  corrected  right  ascension  ,s  ^  ^^^        ^      ^.^^^  ^1^^  ^,,j,.  ^  ^ -^^j^^^  ^3      J  2  5 

If  the  sum  of  these  numbers  is  J  °^^'^   I  the   correction    is  ?  gui^t/aTtive.  > 
The  result  gives  the  moon's  neio  corrected  declination. 

17.  To  the  logarithm  of  the  relative  motion  in  right  ascension  add  the  log.  cosine  of  tho 
moon's  new  declination,  (not  corrected,)  and  call  the  sum  (S/). 

18.  To  the  sum  (S,)  add  the  proportional  logarithm  of  the  relative  motion  in  declination, 
and  the  constant  logarithm  7.1427  ;  the  result  is  the  logarithm  cotangent  of  the  ^rsf  orbitical 
inclination,  which  is 

c'  ?  when  the  sun's  motion  in  declination  is  <  f'''^'^  ^^  i  than  the  moon's. 
S.  5  (  less        5 

19.  To  tlie  proportional  logarithm  of  the  difference  between  the  sun's  nac  declination 

corrected  and  the  moon's,  add  the  logarithm  secant  of  the  first  orbitical   inclination,  and 

from  the  sum  deduct  the  prop,  logarithm  of  the  distance  between  the  centres  of  the  sun 

and  moon  ;  the  remainder  is  tlie  log.  secant  of  the  second  orbitical  inclination,  which  has 

the  name  S.  )     ,        .,       ,  .■       ■         C  immersion, 

IV    >  wlien  tlie  observation  is  an  <  ■        ' 

JN.  3  ^  emersion. 

This  inclination  is  greater  than  90°  when  the  sun's   new  corrected  declination  is  greater 

than  tlie  moon's;  otherwise  less  than  90°. 

20.  Jldd  together  tlie  two  orbitical  inclinations  if  of  the  same  name,  and  suhtract  them  if 
of  different  names ;  and  call  the  result  the  relative  inclination,  which  must  have  the  same 
name  as  the  greater  of  the  two  orbitical  inclinations. 

To  the  log.  cosecant  of  the  relative  inclination  add  the  sum  (S,),  the  proportional  log.  of 
the  distance  of  tlie  centres  of  the  sun  and  moon,  and  the  constant  log.  7.G19S;  the  sum, 
rejecting  20  in  the  index,  is  the  prop.  log.  of  an  arc  in  time,  to  be  applied  to  the  new 
corrected  time  to  get  the  mean  time  at  Greenwich  ;    it  must  be 

suMrlcted  }  '^'^^'^  ^^'^  ''•^^^^'^^  inclination  is  |  ^; 

21.  By  applying  to  the  Greenwich  mean  time  the  equation  of  time  taken  from  page  II.  of 
the  Nautical  Almanac,  we  shall  have  the  apparent  time  at  Greenwich  ;  tlie  difference 
between  it  and  the  apparent  time  of  observation  will  show  the  longitude  of  the  place  from 
Greenwich. 

EXAMPLE. 

Suppose,  at  a  place  in  the  latitude  42°  31'  13''  N.,  and  estimated  longitude  4h.  43m.  38s.6 
the  end  of  a  solar  eclipse  was  observed,  November  30,  1834,  at  4h.  5in.  47s.5  apparent  time 
Bequired  the  longitude. 


ELEJIENTS   OF  THE  ECLIPSE. 


Apparent  time  of  observation November 

Estimated  longitude W. 

Apparent  time  at  Greenwich 

Equation  of  time stiUract 

Mean  time  at  Greenwich 

0's  rij^ht  ascension 

O's  declination S. 

0's  .semi-diameter 

0's  horizontal  parallax 

D  's  horizontal  paralla.x 

])  's  declination 

Latitude  of  the  place— Corr.  Table  XXX\l\l.  =  rcdiiced  latitude 

D' horizontal  parallax  — Corr.  Table  XXX Vin.=: CO' •20".14— 5".54  ) 

is  5  's  corrected  horizontal  parallar j 

B  's  corrected  horizontal  paralla,x  —  ©'s  hor.  par.  =  relative  parallax 

Elements  for  Nov.  30d.  8h. 

D  's  nciB  right  ascension 

D  's  iieiB  declination S. 

O's  new  right  ascension 

0's  neto  declination 

5  's  horary  motion  in  right  ascension 

D's  horary  motion  in  declination 

O's  horary  motion  in  right  ascension 

0's  horary  motion  in  declination 

J)  's  horary  motion  in  right  ascension  —  0's  horaiy  motion  in  right  ) 

ascension  :=retative  motion  in  right  ascension ( 

D  's  horary  motion  in  declination  —  0's  horary  motion  in  declination  | 

=  relative  motion  in  declination \ 

0's  new  right  ascension  -|-  corr.  for  right  ascen.  =  co7-r.  new  right  as. . 
O's  7ie«)  declination  —  corr.  declination  =  corr.  new  declination 


30d.  4h. 

5  m 

.  47s. 5 

4 

43 

38  .6 

30  8 

49 

26  .1 

11 

2  .4 

30  8 

38 

23  .7 

Ifi 

2.5 

53  .05 

21° 

41' 

52".6 

16 

14  .8 

8  .7 

no 

20  .14 

21 

7 

C  .8 

42 

19 

47  .4 

CO 

14  .6 

60 

5  .9 

16h. 

29m 

.  13S.35 

21" 

01' 

24".6 

]6h. 

2.'im 

.4HS.I5 

,21° 

41' 

39"  9 

2m 

.  333. 79 

& 

50"  U 

lOs.79 

2-l".05 

8     35  .25 


23 
5C' 


33  .3 
7".9 


444 


TO  FIND  THE  LONGITUDE. 


Relative  parallax        60    5'-.9 Prop.  Log.    0.47G4 

Reduced  latitude  42°  19' 47".4 Secant  10.1312 

Constant  Log.    1.1761 
Double  obs'd  time  fr.  noon  8h.  1  Im.  353.  Cosec.  10.0563 


Sum (S) 1.8400 

D's  declination  21°  7'  C".8 Cosine    9.9698 

Constant  Log.    0.3010 


Prop.  Log.  correction lm.23s.7 2.1108 

Observed  time  fr.  noon     41).  5     47  .5 


Corrected  time  from  noon  4h.4ni.23s.8  Log.ris.  4.71324 

Reduced  latitude  42°  19'  47".4 Cosine  9.86881 

0's  declination     21   4152.6 Cosine  9.96809 


Nat.  number 35493 4.55014 

Nat.  cosine 64  01  40         43794 

Nat.  sine  0's  true  altitade 08301  =  4°  45'  41" 

O's  true  altitude  4°'45'41" Secant  10.00150 

Double  corrected  time  fr.  noon  8h.  8m.  47s. 6  )  „  „  ,„_„ 

Sine  (P.M.)  j  9-9^223 

Reduced  latitude  42°  19' 47 ".4 Cosine    9.8G881 


Parallactic  angle sine  40°  30' 00" 9.81254 


Rel.  parallax  CO'  5".9 Prop.  Log.    0.4764 

Parallactic  angle  40°  30' Secant  10.1190 

0's  true  altitude  4°  45'  41' Secant  10.0015 


Correction  0's  declination  P.  L.  ■    45*  32"  N.    0.5969 
O's  declination 21°  41  52.6  S 

0'3  CORRECTED  DECLINATION.  .  .20    56  20.6  S 

Sum (S) 1.8400 

0's  corrected  declination  20°  56'  20".6..Cosine    9.9703 


Correction  ©'s  right  ascen.  P.  L.     2m.  47s. 15   1.8103 
O's  right  ascension 16h.  25m.  53s.05 

O's  CORRECTED  BIGHT  ASCEN.  16   28   40  .2 

60  X  16  X  0.143 


■  =  diminution  O's  semi-diam.      1".4 
O's  semi-diameter 16'  14". 8 


100 


O's  CORRECTED  SEMI-DIAMETER 10    13    .4 


D 's  horizontal  parallax  GO' 20".  14... Prop.  Log.   4747 
Constant  Log.  5646 


])  's  semi-diameter Prop.  Log.  16'  26".5    1 .0393 

O's  corrected  semi-diameter 16  13  .4 

Distance  OF  THE  CENTRES  OF  0<St  ])  32  39  .9 

The  time  when  the  moon's  right  ascension 
differs  but  little  from  the  sun's  corrected  right 
ascension,  is  November  30d.  8h.,  for  which 
Greenwich  time  we  take  the  following  by 
inspection :  — 

h.  m.        B. 

})'s  neiD  right  ascension 16  29  13.35 

J) 's  new  declination  south 21  01  24.6 

O's  right  ascension,  Nov.  30d 16  24  19.85 

Dec.    1  16  28  38.72 

Difference  for  24  hours 4  18.86 


for    Shours 1  26.29 

O's  NEW  RIGHT   ASCENSION 16  25  46.15 


0's  declination,  by  Problem  I 
Nov.  29d.+21°28'2a".8 

30    4-21  38  24  .8-1-  .0'02".0 
Dec.     1    --21   48    2.0  A   9  37  .9  —  24".8 
2    +21  57  14  .1-1-   9  12  ./  —25    1 


Second  declination -j-  21 

A  9'  37".2  Prop,  part 

B      25      Table  XLV 


-f-2]°38'24".8 
-f  3  12  .4 

-j-  2  .7 


D  's  horary  motion  in  rigid  ascension,  by  Problem  II. 

h.  m.       I. 
Nov.  30d.  7h.  D  's  right  ascension 16  26  39.67 

8  "      "  "         16  29  13.35 

9  "      "  "         .16  31  47.25 

Horary  motion  for  7h.  30m 2  33.68 

forS     30    2  33.90 

D's  HORARY  MOTION  AT  8h.  IN  R.  A3 2  33.79 

])  's  horary  motion  in  declination,  by  Problem  IT. 
Nov.  30d.  7h.  D  '3  declination 20°  52'  20'  .8 

8  "  "  21      1  24  .6 

9  "  "  21   10  19  .4 

Horary  motion  for  7h.  30ra 9    3  .8 

for  8     30    8  54  .8 


O's  NEW  DECLINATION 21    4139.9 


D's  HORARV    MOTION    AT  8h.  IN    DECLINATION    8    59    .3 

h.    m.       s. 

0's  right  ascension,  Nov.  30d 16  24  19.86 

Dec.    1  16  28  38.72 

Motion  in  right  ascension  in  24  hours 24  )  4  18.86 

O's  HOBART   MOTION    IN   RIGHT   ASCENSION  10.79 

O's  declination,  Nov.  30 21°  38'  24". 8 

Dec.     1 21   48    2  .0 


Motion  in  declination  in  24  hours 24  )  9  37  .2 

O's  HORARY  MOTION  IN    DECLINATION...  24   .05 


O's  7i«!c  corrected  right  ascension.... lOh.  28m.  33s.3 
D 's  right  ascension 16     29       13.35 

Difference 40  .05 

Log.  of  40.05 1.60360 

Relative  motion  in  right  ascens.  143" 2.15.534 

Remainder R 0.55274 

Constant  Log.  0.4771 

Arcintime Prop.  Log.      lGm.48s  4  1.0393 

Assumed  time 8h.  0       0 


New  CORRECTED  TIME 7    43     11  .6 


Remainder R 0.5527 

Relative  motion  in  declination  8'  35" .25.. P.  L.  1.3214 

Correction  D's  declination      P.  L.        2'  24".3    1.874i 

D's  declination 21    1  24  .6 

D's    NEW  CORRECTED    DECLIN 20  59    00    .3 

Relative  motion  in  right  ascen.  143"  2.1553 

D  's  new  declination 21"  1'  24  .6 Cos     9.9701 

Sum (S,) 2.1254 

Relative  motion  in  declination  6' 35".25...P.  L.  1.3214 
Constant  Log.  7.1427 

IsT  ORBITICAL  INCLINATION.  ..Cuif/n.  11°  2C'  S.  10.5895 

O's  new  declin.  corrected  20°  5Gi  7". 9 
D's  new  declin.  corrected  20  59   0  .3 

2.53  .4.. .P.  L.  1.7969 

Secant  1st  orbitical  inclination   14°  26'  10.0139 

11.8108 
Distance  of  the  centres  0  &  D  32'  39".9. .  .P.  L.  7413 
2d  ORBITICAL  iNCLiN.  Sec.  85°  7' N.  11.0696 
1st  orbitical  inclination.  ...14  26  S.  

Relative  inclination 70  41  N..  .Cosec   10.0252 

pum (s,) rrrrrr. 2.1254 

Distance  of  the  centres  0  &  D Prop.  Log.  0.7419 

Constant  Log.  7.6198 

Correction  in  time 55m.  95s.     Prop  Log.  0.5116 

JVeifl  corrected  time... 7h. 43      11  .0 


Grecnirichmea7itime..  .8   38  36  .6 

Equation  of  lime 11        2  .4 

Apparent  time 8   49  39.0    at  Oreenmch. 

Apparent  time ^4     5  47.5    observed. 

LONQITUDE 4    43  51    .5      REQUIRED 


TO  FIND   THE   LONGITUDE   BY  AN  OCCULTATION  445 

PROBLEM  XXU. 

To  find  the  longitude  of  a  place  by  an  occultation  of  a  star  by  the  moon  ;  the  apparent 
time  being  estimated  from  noon  to  noon,  according  to  the  method  of  astronomers,  and  the 
latitude  of  the  place  being  known. 

RULE. 

With  the  longitude  by  account  find  the  corresponding  Greenwicii  mean  time  of  obser- 
vation. For  this  time,  take  out  from  the  Nautical  Almanac  the  sun's  right  ascension,  the 
moon's  horizontal  parallax,  and  her  declination,  to  the  nearest  minute. 

Reduce  the  latitude  of  the  place,  and  the  moon's  horizontal  parallax,  by  subtracting  the 
corrections  found  in  Table  XXXVIII. 

To  the  sun's  right  ascension  add  the  apparent  time  of  observation;  the  sum  will  give  the 
right  ascension  of  the  meridian. 

Take  from  the  tables  "  for  facilitating  the  computation  of  occultations  of  certain  stars  by 
the  moon,"  in  the  Nautical  Almanac,  the  star's  right  ascension  and  declination.  The 
difference  between  the  right  ascension  of  the  meridian  and  the  star's  right  ascension  will 
give  the  hour-angle  of  the  star,  which  convert  into  degrees,  &c.,  naming  it  iccst  when  the 
right  ascension  of  the  meridian  is  greater  than  the  star's  right  ascension,  and  cast  when 
less. 

To  the  proportional  logarithm  of  the  moon's  corrected  horizontal  parallax  add  the  log. 
secant  of  the  reduced  latitude,  and  the  log.  cosecant  of  the  hour-angle,  rejecting  10  in  each 
index.  To  the  sum  (S)  add  the  log.  cosine  of  the  moon's  declination  and  the  constant  loga- 
rithm 0.3010  ;  the  result,  rejecting  10  in  the  index,  will  give  the  proportional  logarithm  of  an 
arc,  which,  subtracted  from  the  hour-angle,  will  give  the  hour-angle  corrected. 

Take  out  the  following  logarithms,  and  place  them  beneath  each  other,  in  two  columns  :  — 
the  proportional  log.  of  the  moon's  corrected  horizontal  parallax  in  both  columns  ;  the 
secant  of  the  star's  declination  in  col.  1,  and  its  cosecant  in  col.  2 ;  the  cosecant  of  the 
reduced  latitude  in  col.  1,  and  its  secant  in  col.  2;  and  the  secant  of  the  corrected  hour- 
angle  in  col.  2.  The  sum  of  the  logarithms  in  col.  1  will  give  the  prop.  log.  of  an  arc  of 
the  same  name  as  the  latitude,  and  the  sum  of  the  logarithms  in  col.  2  will  give  the  prop, 
log.  of  an  arc  of  a  different  name  from  the  star's  declination,  when  the  hour-angle  is  less 
than  90°,  but  of  the  same  name,  it' greater  than  1)0°.  The  sum  of  these  arcs,  having  regard 
to  their  names,  will,  being  applied  to  the  star's  declination,  give  the  declination  corrected. 

To  the  sum  (S)  add  the  constant  log.  L17G1,  and  the  log.  cosine  of  the  star's  corrected 
declination  ;  tlie  sum,  rejecting  10  in  the  index,  will  be  the  prop.  log.  of  an  arc  in  time, 
to  be  added  to  the  star's  right  ascension  when  it  is  loest  of  the  meridian,  but  subtracted  when 
east,  to  obtain  the  star's  right  ascension  corrected. 

Find  in  the  Nautical  Almanac  the  time  when  the  moon's  right  ascension  is  near  to  that 
of  the  star  corrected,  and  for  this  time  take  out  the  moon's  right  ascension  and  declination, 
and  their  hourly  variations. 

Subtract  the  common  log.  of  the  difference  between  the  corrected  right  ascension  of  the 
star  and  the  right  ascension  of  the  moon  from  the  common  log.  of  tlie  hourly  motion  in 
right  ascension  ;  to  the  remainder  add  the  constant  log.  0.4771  ;  to  the  same  remainder  add 
the  proportional  log.  of  the  hourly  motion  in  declination.  The  former  sum  will  be  the  pro- 
portional log.  of  a  time  to  be  added  to  the  assumed  time  when  the  star's  right  ascension  is 
greater  than  the  moon's,  otherwise  subtracted,  to  obtain  the  time  corrected.  The  latter  will 
be  the  proportional  log.  of  a  correction  of  the  moon's  declination,  to  be  applied  with  the  same 
name  as  the  hourly  variation  when  the  star's  right  ascension  is  greater  than  the  moon's 
right  ascension,  but  with  a  different  name  when  less. 

To  the  common  log.  of  the  hourly  motion  in  right  ascension  add  the  log.  cosine  of  the 
moon's  declination  ;  to  the  sum  (S,),  rejecting  10  in  the  index,  add  the  proportional  log.  of 
the  hourly  motion  in  declination,  and  the  constant  log.  7.1427.  The  result  will  be  the  log. 
cotangent  of  the  first  orbitical  inclination,  and  must  have  the  same  name  as  the  hourly 
motion  in  declination,  when  the  star  is  north  of  the  moon,  but  a  different  name  when  south 
of  the  moon. 

To  the  proportional  log.  of  the  difference  between  the  star's  declination  corrected  and  the 
moon's  declination  corrected  add  the  constant  log.  9.4354,  and  the  log.  secant  of  the  pre- 
ceding orbitical  inclination,  rejecting  10  in  tlie  index,  and  from  the  sum  subtract  the 
proportional  log.  of  the  horizontal  parallax  ;  the  remainder  will  be  the  log.  secant  of  the 
second  orbitical  inclination,  which  must  be  named  S.  when  the  observation  is  an  immersion, 
and  N.  when  an  emersion. 

Add  together  the  two  orbitical  inclinations,  having  proper  regard  to  their  names  ;  and  to 
the  log.  cosecant  of  this  sum  add  the  preceding  sum  (S,),the  proportional  log.  of  the  horizontal 
parallax,  and  the  constant  log.  8.1844.  The  sum  will  be  the  proportional  log.  of  a  correction 
to  be  applied  to  the  time  corrected  to  get  the  mean  time  at  Greenwich  ;  added  when  the  sum 
of  the  orbitical  inclinatio.is  is  N.,  subtracted  when  S. 

Apply  the  equation  of  time  to  this  Greenwich  mean  time,  and  we  shall  have  the  Green- 
wich apparent  time;  the  difference  between  it  and  the  apparent  time  at  the  place  of  obser- 
vation will  give  the  longitude  required,  icest  when  the  time  at  Greenwich  is  the  greatest 
east  when  less 


446 


TO  FIND   THE   LONGITUDE   BY   AN   OCCULTATION 


EXAMPLE. 

Suppose,  in  a  place  in  the  latitude  42°  19'  15''  north,  and  estimated  longitude  4h.44m.  west 
of  Greenwich,  the  immersion  of  y  Cancri  was  observed  April  20,  1839,  at  lOh.  45m.  35s.9 
apparent  time.     Required  the  longitude. 


P.  L.  D's  corrected  hor.  par. 

Secant  reduced  latitude 42' 

Cosecant  hour  angle 61 


56'    9".4 0.5059 

7  50      0.1298 

7  48      0.0576 


Sum (S) 

Cosine  D's  declination 22°  42' 06" 

Constant  Log. 


Proportional  Log.  correction..      19' 46"... 
Hour-angle 61°  7  48 


0.6933 
9.9650 
0.3010 


Hour-angle  corrected 60  46  02 

Col.  r.  _             Col.  2. 

P.  L.  D  's  c"T.  H.  P.  56'  9".4    0.5059|Sanie 

Secant*'-^'!  r.l.22°2'38"  0.0330  Cosecant.. . 

Cosec.  red.  lut.   42  7  50  0.1734  Secant 


Prop.  Log.     3i'  54".5  N.  0.7123N^-^°f;'V, 
°  )a.  oO  4o'2" 

7  37  .6  S Prop.  Log 


97  16  .9  N. 
•  '3declin....i>2°  2  33  .5N. 


Sum..{S)  , 
Const.  Log 


»'s  corr.  dec.  22  29  .55  .4  N Cosine 

Correction .' Oh.  2m.37s.93 P.  L. 

*'s  right  ascension.... 8    33     59  .03 


*'s  right  ascen.  corr...  8   36     36  .96 
]) 's  right  ascen 8   37       5.03 

Difference 28  .07 


ELEMENTS   OF   OCCULT.'iTION, 


Apparent  time  of  observation,...  April 

Estimated  longitude W. 

.Apparent  time,  at  Greenwich 

Equation  of  time subtract 

Mean  timeal  Greenwich April 

O's  right  ascension 

O's  right  ascension -|-appar.  time  of) 
obs.  =  right  ascension  meridian  j 

D  's  horizontal  parallax 

Correction  Table  XXXVIII 

5  's  corrected  horizontal  parallax 

D's  declination N. 

*'3  right  ascension 

*'s  hour-angle 

*'s  hour-angle,  in  degrees,  &c W. 

*'s  declination N. 

Lat.  place  —  corr.  Table  XXXVIII   ) 

=:  reduced  latitude j 

By  the  Nautical  Almanac  we  find  the 
moon's  right  ascension  to  be  near- 
est to  the  star's  corrected  right  as- 
cension on  20th  April,  at  16  hours, 
fcir  which  time  we  get  the 

D  's  right  ascension 

D's  horary  motion  in  right  ascension.. 

D's  declination N. 

D's  horarj'  motion  in  declination. ..S. 


20  10  45  35.9 
4  44    0 

20  15  29  35.9 
1   11.4 

20  15  28  24.5 
1  52  54.3 

12  38  30.2 

56'  14".5 

5  .] 

56    9  .4 

22°  42  06  .1 

8h.33<n.59s.03 

4      4     31  .17 

61°  7  48" 

22  2  38  .5 

42  7  49  .7 


h.  m. 

8  37    5.03 

2  11.38 
22°3G'55".0 

9  53.85 


Log 28".07 1.4482 

Log.  D  's  hor.  mo.  in  r.  as.  ]3I".38. . .  .2.1185 

Remainder 0.6703 Remainder 0.6703 

Constant  Log.   0.477J P.  L.  D  's  ho.  mo.  in  dec.   9'  53" .85 1 .2597 

Prop.Log 12m.49s.2 1 .1474 Prop.  Log 2'   6".9  N.   1.9300 

Assumed  time ....  .16h.O        0 j,  'g  declination 22°  36  55      N. 

Time  corrected 15  47       10  .8 T)^s  declination  corr..  .<i2  39  01  .9  N 


Log.  D's  hor.  mo.  in  right  ascension  131".38.... 2.1185 
Cosine  D  's  declination 22°  38'  55". . .  .9.9653 

Sum (S,) 2.0838 

P.  L.  D'slior.  mo.  in  declination.. 9'  53".85....  1.2597 
Constant  Log.  7.1427 


Cotangent  Ist  orbitical  inclination..  18° 5'  N..  .10.4862 


*'s  declination  corr.  22°  29'  55".4 
D's  declination  corr.  22  39  01  .9 

Difference 


Constant  Log. 
Secant  1st  orbitical  inclination 18°  5'... . 


9  06  .5... Prop.  Log.  1.2958 

9.43.54 
10.0220 


P.  L.  D's  horizontal  parallax... 56' 14". 5 

Secant  2d  orbitical  inclin 55°  36  S 

1st  orbitical  inclination 18  05  N. 

Difference 37  31  S..Cosec. 


10.7532 
.  0.5052 

,10.2480 


.  2.0838 
,.0.5052 
,  8.1844 

Prop.  Log.  of  correction 18m.  28s.3..  .0.9888 

Corrected  time 15h.  47      10.8 


Sum (S,) 

P.  L.  D 's  horizontal  parallax 56'  14i'..5 

Constant  lio 


Jl/etzn  Greenwich  time 15     28 

Equation  of  time 1 


42.5 
11  .4 


Apparent  time  at  Greenwich  15     29      53  .S 
Apparent  time  of  observ...  10     45      35  .9 

Longitude  in  time 4    44      18 


ON  WINDS  AND  STORMS. 

BY  TV.  C.  REDFIEL,!). 


Thk  earth  is  surrounded  by  a  fine,  invisible  and  elastic  fluid,  called  air;  which,  when 
spoken  of  in  its  general  relations  to  the  earth,  is  called  the  atmosphere.  Its  incumbent 
weight  or  pressure  upon  the  earth's  surface  is  determined  by  means  of  the  barometer, 
and  is  equal  to  a  column  of  mercury  of  about  thirty  inches  in  height,  at  the  sea  level. 
Wind,  is  air  in  motion.  It  is  found,  that  in  almost  every  country  and  in  every  sea, 
the  wind  is  more  or  less  predominant  in  a  particular  direction.  The  most  remarkable 
of  these  general  winds  are  distinguished  by  several  names,  as  trade  winds,  monsoons, 
variable  loinds,  cj-c. 

The  t?-ade  winds,  are  found  between  the  equator  and  the  30th  parallels  of  north  and 
south  latitude,  whdre  the  wind,  for  the  most  part,  blows  from  the  eastward :  but  near 
the  eastern  borders  of  any  ocean,  the  trade  winds  usually  blow  more  towards  the  equa- 
tor than  in  its  more  central  portions ;  while  on  the  western  borders,  the  wind  not  unfre- 
quently,  blows  in  a  direction  which  is  more  or  less  outward  from  the  equator. 

The  monsoons,  which  are  chiefly  found  in  the  Indian  seas,  are  regular  alternations 
of  the  trade  winds,  which  blow  for  six  or  eight  months  in  their  regular  course ;  but, 
during  the  other  portions  of  the  year,  are  replaced  by  a  westerly  wind :  which  is  pro- 
bably a  deflection  of  the  trade  wind  from  the  opposite  side  of  the  equator. 

The  variable  winds,  are  chiefly  found  extending  from  the  outward  borders  of  the 
trade  winds  to  the  polar  regions  ;  although  subject  to  frequent  changes,  both  of  velocity 
and  direction,  yet  their  predominating  direction  is  found  to  be  from  a  western  quarter 
being  opposite  to  the  general  course  of  the  trade  winds.  The  various  movements  of 
these  winds  are  often  exhibited  in  diflerent  strata,  superimposed  one  upon  another ;  and 
these  movements  viewed  in  their  extended  relations,  are  doubtless  connected  with  those 
of  the  trade  winds. 

The  land  and  sea  breezes,  are  daily  alternations  in  the  direction  of  the  general  winds, 
near  the  shores  of  an  island  or  continent.  They  appear  to  be  connected  with  the  daily 
changes  of  tem])erature  at  the  earth's  surface.  The  sea  breeze  generally  sets  in  about 
ten  in  the  forenoon  and  continues  till  about  five  or  six  in  the  evening :  at  seven  the  land 
breeze  begins  and  continues  till  about  eight  in  the  morning. 

A  whirlwind,  is  a  phenomenon  which  is  often  violent  and  dangerous.  The  identity 
of  waterspouts  and  whirlwinds  was  maintained  by  Franklin,  and  although  at  a  later 
period  this  has  sometimes  been  questioned,  it  appears  to  have  been  done  without  suf- 
ficient reason.  From  the  equal  distribution  of  the  atmosphere  as  the  envelop  of  our 
earth,  it  results,  that  no  violent  wind  can  take  place,  except  by  means  of  a  movement 
which  is  more  or  less  circuitous  in  its  character,  and  in  cases  of  great  violence,  the 
wind  is  exhibited  in  the  form  of  an  active  vortex  or  whirlwind;  which,  if  isolated  from 
other  violent  winds  and  of  small  extent,  is  often  called  a  tornado  or  loatcrspout. 

Waterspouts  and  whirlwinds  follow  the  course  either  of  the  surface  wind,  or  of  a 
higher  current  of  air  from  which  they  are  sometimes  depended ;  or  their  course  may 
be  modified  by  both  these  influences,  without  being  absolutely  determined  by  either. 
They  abound  most  in  those  calm  regions  which  are  found  near  the  external  limits  of 
the  trade  winds  and  in  like  regions  near  the  equator. 

Storms  and  hurricanes,  have  from  the  earliest  periods  been  considered  as  the  chief 
dangers  encountered  by  the  navigator.  It  was  discovered  by  Franklin,  that  northeast 
etorras,  in  tlie  United  States,  pursued  a  retrogressive  course,  commencing  sooner  in 
Philadelphia  than  in  Boston.  Careful  attention  having  been  given  to  the  phenomena  of 
ihe  Atlantic  storms,  in  later  years,  it  has  been  found  that  they  exhibit  certain  charac- 
teristics of  great  uniformity.  The  most  violent  of  these  storms,  often  known  by  the 
name  of  hurricanes,  appear  to  commence  in  the  intertropical  latitudes,  eastward  of  the 
West  Indies,  where  their  course  is  towards  the  northwest,  till  approaching  the  latitude 
of  30°,  their  westerly  progress  ceases  and  their  track  becomes  recurved  to  the  north- 
ward and  eastward ;  in  which  latter  direction  their  progress  usually  becomes  accelerated. 

On  the  annexed  chart,  the  routes  of  several  of  these  hurricanes  are  shown  by  dotted 
lines,  \yhich  indicate,  somewhat  nearly,  the  center  of  the  track  pursued  by  each  hur- 
ricane in  its  daily  progress,  on  such  part  of  its  route  as  has  become  known. 

The  rate  at  which  these  storms  advance  in  their  course,  is  from  eleven  to  thirty  miles 
an  hour;  their  average  progress  being  about  seventeen  miles.  This  cannot  explain  the 
velocity  of  the  wind,  which,  in  the  most  violent  part  of  the  gale,  sometimes  exceeds 
80  or  lOO  miles  an  hour:  but  the  observations  by  wluch  their  course  and  velocity  have 


448 


OP  WINDS. 


Cerogruphj 

been  determined  have  also  shown,  that  these  storms  act  in  the  manner  of  great  whirl- 
winds, turning  constantly  to  the  left  around  their  moving  axis  of  rotation :  the  most 
violent  portion  of  the  gale  being  found  toward  the  interior  or  heart  of  the  storm. 
Hence  it  is  also  found,  that  the  direction  of  the  wind  in  a  gale,  for  the  most  part,  does 
not  at  all  coincide  with  its  line  of  progress. 

When  a  storm  is  about  to  commence,  the  latitude  of  the  ship  will  indicate  its  proba- 
ble course,  as  is  seen  on  the  above  chart,  and  the  direction  of  wind  which  the  storm 
first  presents,  may  serve  to  determine  the  portion  of  the  gale  under  which  the  ship  is 
likely  to  fall,  either  by  preserving,  or  altering  her  course;  as  well  as  the  changes  and 
comparative  violence  of  the  wind,  to  which  she  will,  in  either  case,  be  exposed. 
In  the  hurricane  figures  which  are  found  on  the  chart  on  tracks  1,  5,  and  7,  tlie  curved 

arrows  show  the  various  directions  of  the 
wind  in  different  portions  of  the  advan- 
cing storm :  but  in  order  to  exhibit  this 
more  perfectly,  the  annexed  diagiam  may 
be  referred  to.  The  outward  circle  of 
this  diagram  represents  the  true  points  of 
the  compass,  and  the  curved  arrows  in  the 
figure  within,  represent  the  rotary  motion 
of  the  gale,  and  serve  to  show,  somewhat 
nearly,  the  direction  of  the  wind  in  all 
parts  of  the  storm.  The  indicator  C, 
shows  the  general  course  of  the  storm  in 
the  intertropical  latitudes,  whicli  as  the 
storm  moves  onward  to  the  higlicr  lati 
tudes,  changes  gradually  round  to  NE.  anc 
ultimately  to  nearly  east.  Thus,  on  the 
coast  of  the  United  Stales  northward  of 
Charleston  or  Cajie  Hatteras,  the  gale,  on 
its  central  section,  will  begin  from  E.  to 
SE.  and  its  close,  after  the  passage  of  its 
Diagram  for  north  latitude  center,  will  be  from  the  NW.  quarter ;  for 

the  wind  always  blows  across  the  path  of  the  storm  on  the  center  of  its  track. 


OP  WINDS. 


441) 


Toward  the  outward -maigin  of  the  storm  or  remote  from  the  center  of  its  path,  the 
ffale  becomes  less  severe  and  the  changes  of  wind  more  gradual  and  less  hazardous. 
The  navigator  should  so  lay  his  ship,  therefore,  as  to  avoid  the  heart  of  the  storm,  and 
at  the  same  time,  to  take  the  changes  of  wind  which  are  to  follow  in  the  moat  favor- 
able manner. 

It  will  be  found  more  difficult  to  form  a  just  estimate  of  the  direction  and  changes 
of  the  approaching  gale  in  the  latitudes  near  the  outward  borders  of  the  trade  winds 
and  thence  to  31°  or  32°,  than  in  other  latitudes;  for  here  the  storm  is  rapidly  chang- 
ing its  course,  and  the  changes  of  wind  cannot  therefore  be  so  accurately  estimated. 


True  6  s  jSTcFPth 


Diagram  for  south  latitude. 


In  south  latitudes  the  course  of  storms 
is  found  to  be  in  the  reverse  order  to  those 
which  are  traced  on  the  chart ;  their  pro- 
gress being  norlhioesterly  in  the  inter- 
tropical latitudes,  while  on  approaching 
the  latitude  of  30°  south,  they  recurve 
towards  the  south  and  southeast.  The 
rotary  motion  of  the  gale  is  also  in  the  op- 
posite direction  from  that  which  is  found 
in  the  northern  hemisphere,  being  to  the 
right,  around  the  moving  axis  of  rotation, 
as  is  shown  in  the  annexed  diagram  for 
south  latitude.  The  indicator  C,  here 
shows  the  storm  as  moving  southwesterly, 
and  a  gradual  change,  by  south,  to  south 
east,  wiU  show  the  various  directions  and 
changes  which  pertain  to  the  progress  of 
the  storm  in  different  south  latitudes. 

When  the  course  of  a  ship  at  the  com- 
mencement of  a  gale,  is  found  to  be  across 
the  track  of  the  storm,  the  direction  as 
well  as  force  of  the  gale  may  be  greatly  affected  by  this  gradual  change  of  position. 
These  changes  of  wind,  as  well  as  those  to  which  a  stationary  vessel  would  be  exposed, 
by  the  onward  movement  of  the  storm,  may  be  understood  by  consulting  the  chart  and 
diagrams :  the  latter,  for  the  sake  of  convenience,  may  be  drawn  on  a  card  and  used 
upon  the  common  charts. 

If  a  ship  is  hove  too  and  no  attempt  made  to  avoid  the  heart  of  the  storm,  it  is  ap- 
parent that  her  exposure  may  greatly  depend  upon  the  tack  on  which  she  is  laid.  This 
was  strikingly  exemplified  in  the  CuUoden's  storm  of  March,  1809,  in  the  Indian  ocean, 
lat.  23°  S.  The  Culloden,  with  a  convoy  of  Indiamen,  took  the  gale  at  SE.  and  the 
ships  which  were  hove  too  with  their  heads  to  the  southward  were  soon  out  of  the  gale, 
while  those  ships  which  stood  on  to  the  westward  were  either  lost,  or  continued  for  a 
long  time  exposed  to  the  full  severity  of  the  hurricane.* 

It  is  owing  probably,  to  the  centrifugal  action  of  these  rotative  storms,  that  the  ba- 
rometer always  sinks  under  the  first  portion  and  towards  the  center  of  the  storm,  in 
all  latitudes ;  and  this  fall  of  the  barometer  commonly  affords  the  earliest  and  surest 
indication  of  the  approaching  tempest.  When  the  center  or  axis  of  the  storm  has 
passed,  the  barometer  commences  rising,  whether  the  wind  has  already  been  violent  or 
not,  and  in  some  positions,  as  off  the  Cape  of  Good  Hope,  the  last  part  of  the  storm 
with  a  rising  barometer,  usually  exhibits  the  greatest  violence  of  wind.  The  state  of 
the  barometer  should  always  be  recorded  at  regular  and  frequent  intervals,  in  the  log 
book  or  journal. 

Observations  on  the  duration  and  strength  of  the  wind  and  the  movements  of  the 
barometer  on  opposite  sides  of  a  storm  have  shown  that  in  most  cases  the  rotary  action 
of  the  wind  is  not  entirely  uniform  in  its  development;  although  the  characteristic 
movements  of  rotation  are  clearly  distinguishable.  The  rotation  appears  to  be  mani- 
fested most  equally  in  storms  which  are  distinguished  for  their  activity  and  violence. 

Humboldt  estimates  the  extreme  velocity  of  the  tropical  tornadoes  at  two  or  three 
hundred  miles  an  hour.     The  velocity  of  a  heavy  gale  is  from  60  to  100  miles. 

Colonel  Beaufoy  states  that  wind  has  only  the  666th  part  of  the  effect  of  water,  when 
moving  with  equal  velocity.  He  observes  also,  that  it  frequently  happens  in  violent 
storms  of  wind  the  current  does  not  reach  any  considerable  altitude ;  for  often  at  the 
height  of  1,600  feet,  there  is  a  perfect  calm;  on  the  contrary,  it  is  not  uncommon  for 
the  wind  at  considerable  elevations  above  the  level  of  the  sea,  to  move  with  very  great 
celerity,  whilst  the  lower  parts  of  the  atmosphere  remain  in  a  state  of  tranquility. 


57 


♦  See  Reid  on  the  Law  of  Storraa  :  London,  1333,  p.  153-215 


Page  450] 


TABLE  LIV. 

(continued.) 

Latitudes  and  Longitudes. 


Islands,  Sfc,  in  the  SOUTH  .QJVD 
JVORTH  PACIFIC  OCEAjYS. 


Clermont  de  Tonnere, 
(S.  E.  pt.) 

Serle  Island,  (S.  E.  pt.)  . 

Henuake,  or  Honden  I.  . 

Disappointment  Islands, 

— Wytooliee(N.W.pt.). 

— Otooho,  (centre) 

Taiara,  or  King's  Island, 
(centre) 

Raraka  Isknd,  (entrance 
to  Lagoon)  

Kawahe,  or  Vincennes 
Island,  (S.  pt.) 

Aratica,  or  Carlshoff  Isl- 
and, (west  end) 

Manhii,  or  Wilson's  Isl- 
and, (west  end) 

Ahii,  or  Peacock  Island, 
(west  end) 

King  George's  group, 

—  Tiokea,  (S.  W.  pt.) 

—  Oura,  (S.pt.) 

Rurick,  or  Arutua  Island, 

(west  end) 

Nairsa,  or  Dean's  Island, 
(west  end) 

Tikehau,  or  Krusens- 
tern's  Island,  (N.  pt.). 

Mataiwa,  or  Lazareff  Isl- 
and, (N.  pt.) 

Metia  Island,  (N.  pt.)... 

Apataki,  (N.  pt.) 

Elizabeth,  or  Toau  Isl 
and,  (S.  E.pt.) , 

Sea-GuU  group, 

—  Tuinaka,  or  Ried, 
(south  island) 

—  Tipotu,  or  Bacon,  (S 
E.  pt.) 

—  Ohiti,  or  Clute,  (S 
W.  pt.) 

St.  Pablo,  (centre) 

Archangel,    or    Heretua, 

(centre) 

Margaret's,  or  Nukutipi- 

pi,  (centre) 

Four  Crowns,  or  Teku, 

(centre) 

Taweree,  or  St.  Simeon, 

or   Resolution    Isl.  (S. 

pt.)  (Isl.  near  Sandspit) 
Takurea,  or  VVtjlcousk}', 

(south  island)  . . 
Two   groups,  —  Dauhai- 

da,  RIanaka, 
— South  pt.  south    island 


D.   M. 

8  33  S 

&     21 

4  56 

4  lo 

4  o5 

5  42 

6  o6 


Lot. 


b  33 
4  26 
4  34 

4  3i 

4  44 

5  i5 
5  o5 
4  52 


5  58 

6  4o 
6  44 


Tahiti,    Har.  of    Papieti, 
(Motoutu  Island) . . . 


Rose  Island 

>    Manna.   (N.  W.  pt.)... 
-^  lOfoo,  (N.  W.  pt.) 


20  25 
20  42 
20  28 

17  22 
i5  48 

18  i3 

17  3i 

i4  32 
i4  i3 
i4  II 


Lous'- 


D.    M. 

i36  2i\V 

187  o4 
i38  48 

i4i  18 
i4i  3o 

i44  39 

i44  58 

i45  10 

145  39 

1 46  o4 

i46  25 

145  10 
145  20 

i46  5i 

1 47  59 
i48  i5 

1 48  42 
i48  i3 
i46  32 

1 45  49 

t44  10 

1 44  o3 

i44  16 
44  59 

i43  3i 

143  o4 

143  18 

i4i  3o 
142  i5 

142  10 

149  34 

168  07 

169  29 
169  36 


Pago-Pago  har.,  Island  of 
Tutuila 

Harbor  of  Ap'  i,  Itjland 
of  Upolu. ......    . . . . 

Harbor  of  Mataalu,  Isl- 
and of  Savaii 


Hoornlsland,  (N.  W.pt.) 
Uea,  or  Wallis  Island . . . 


Jarvis  Island,  (centre)  . . 
Penrhyn's  Island,  (N.pt.) 
Wostock,  or  Stavers  Isl- 

land,  (centre) 

Flint  Island,  (centre) .... 
Phoenix  Group, 

—  Birnie's  Isl'd,  (centre) 

—  Enderbury's    Island, 
(centre) 

—  Hull's    Island,    (west 
end) 

Gardner's  Isl'd,  (centre) 
McKean's  Isl'd,  (centre) 
Union  Group, 

—  Otafu,  or  Duke  of  York 
Island,  (N.  W.pt.)... 

—  Nukunono,  or  Duke  of 
Clarence  Isl'd,  (N.  pt.) 

—  Fakaafo,  or  Bowditch 
Island,  (village)  .... 

Swain's  Island,  (centre) 


Roaul,  or  Sunday  Island, 
(centre) 


Honolulu  har.,  Oahu  Isl 
Laliaina  har.,  Maui  Isl 
Waiakea  har.,  Hawaii, . . 


New  York,  or  Washing 
ton  Island,  (west  end) 

Necker  Island,  (centre)  . 

French  Frigate  School, 
or  Basse  de  Fregate 
Franqaise 


Maro  Reef. 


Smith  Island,  (centre) , 


Lat. 


D.  M 

i4  18  S 

3  49 
i3  28 

i4  i5 
i3  24 

o  22 

8  55 

10  o5 

11  26 

3  35 


4  3o 
4  38 
3  35 


8  36 

9  o5 

9  24 
II   10 

29  12 

21   19N 
20  5o 
19  44 


4  4i 
23  35 


23  45 
N.pt. 
25  19 
S.pt. 
16  48 


Taputeouca,  or  Drum- 
mond's  Island,  (Sands- 
pit  at  Utivoa) 

Nanouti,  or  Sydenham 
Island,  (N.pt.) 

Nanouki,  or  Hendervill 
Island,  (S.  pt.) 

Kuria,  or  Woodle  Island, 
(South  pt.) •. . 

Apamama,  or  Hopper  Isl- 
and, (N.  pt.) 

Maiana,  or  Hall's  Island, 
(N.pt.) 

Tarawa,  or  Knox  Island, 
(S.  W.  island) 

Apia,  or  Charlotte  Island, 
(entrance) 


I    i4S 
o  3o 
o  08N 
o  17 

0  3o 

1  02 
I  22 
I  48 


Long. 

D.   M. 

70  38  W 

71  4i 

72  iS 

78  02 
76  09 

59  5i 
58  07 

52  16 
5i  48 

71  39 

71  i4 

72  20 
l4  4i 
l4  17 

72  24 

71  38 

71  06 
70  53 

78  i5 

57  52 
56  4i 
55  o3 


60  1 3 
64.43 


65  59 
East  end 

70  32 
East  end 

69  46 


74  53  E 
l4  20 
73  4i 
73  26 
73  54 
73  04 
73  01 
73  02 


TABLE   LIV. 

(continued.) 

Latitudes  and  Longitudes. 


[Page  451. 


DCaraki,  or  Matthew's  Isl 
and,  (N.  pt.) 

Makin,  or  Pitt's  Island, 
(S.pt.)  


Funafuti,   or   Ellice    Isl 

and,  (N.  W.  island) .  . . 

Nukufetau,    or    Depeys 

ter's  Island,  (N.isl.)... 

Oditupu,  or  Tracy  Isl'nd, 

(centre) 

Hudson's  Island,  (N.  pt.) 
Speiden's  Isl'nd,  (centre) 
St.  Augustine,  (centre). 

Walpole  Island,  (centre) 
Elizabeth  Reef,  (N.  E.  pt.) 
Mathews'  Rock 


MacquarJe's  Isl'd,  (S.  pt.) 

Lord   Auckland    Group, 
(Sarah's  Bosom)  . . . 


Ovolau  Island,  (observa- 

^  tory)  

Lecumba  point,  (Sandal- 
wood Bay) 

Muthuata  har.  (cemetery 
on  Island) 

Unda  pt.  (east  entrance 
Vanu\a  Levu) 

Rewa  Roads,  (Nukalau 
Island) 


LmI. 


D.  M. 

2  o3N 

3  02 

8  26S 
7  56 
7  28 


Long. 


35 


Chesterfield  Group — 

N.  W.  pt.  Long  Island 
Kenn  Reef — 

Observatory, sand  cay 

S.  E.  pt.  of  reef. 

Frederick  Reef — 

South  Sand  Islet  . . . . 

North  Sand  Islet 

Saumarez  Reef — 

S.  W.  sand  cay 

S.  E.  elbow  of  reef. . . 
Lihon  Reef — 

N.  E.  point 

S.  \V.  point 


Percy  Group — 

Mid-Island,  "W.  bay  . 
N.  W.  bay,  S.  islet.. 
Pine  Peak  


Barrier  Reef — 

Inside,  No.  i  prong  . . 

"       No.  4  prong  . . 

Outside,  No.  i  prong. 

"         No.  3  prong. 


22  27 
29  34 
22  27 

54  AA 
5o  34 

17  4i 
16  52 
16  26 

16  08 

18  10 

19  52 

21  16 
21  i5 

21  02 

20  57 

21  5i 
21  55 

17  21 
17  39 


21  4o 
21  45 

21  3i 


22  09 

21  29 

20  o5 

21  00 


M 
•.16  E 


72  46 


i4 

28 
U 

23 

29 
06 

07 
24 
72  10 

59  49 


79 


66  27 


53 
35 
04 
55W 
32  E 

19 


5o  17 
5o  21 
5o  19 


52  12 

5i  10 

5o  55 

52  19 


Tokanova  pt.  (S.  E.  pt 
Vanua  Levu) 

Direction  Island,  or  Ne- 
niena 

Awakalo,  or  Round  I... 

Malolo,  (Avo  Town) 

Vomo  Island 

Kie  Island 

Ongea  Island 

Oneata  Island 

Nanuku  Island 


Turtle  Island,  (N.  pt.) 

Pescadores  Island,  (cast 
island) 

Korsakoff  Island,  (west 
island) 

Benham's  Island,  (south 
end) 

Hunter's  Island,  (centre) 

Bearing's  Island,  (centre) 

McKenzie's  Island,  (cen- 
tre)  

Wake's  Island,  (centre). 


Antique    Roads,    (Island 

of  Panay) 

Caldera    Roads,    (Island 

of  Mindanao) 

Soung  Roads,  (Island  of 

Sooloo) 

Manghee  Islands,  (Bala- 

bac  Straits) 


Lat.        Long. 


D.  M. 

16  46  S 

17  07 

16  4i 

17  46 

17  29 
16  4o 
19  04 

18  24 
16  42 

19  47 


II  23  N 

II  08 

5  47 
5  42 

5  35 

10  08 
19  17 

10  4o 

6  56 
6  01 


Bodegas  Poit  . .  . 
San  Franci-co .  .  . 
^lonterey ....    . . 

Santa  Biu-h.-u-a  . . 

San  Pedro 

Ju:in 

Diego 

Quentin 

Bartolomeo  .  , 

Magdalcna  Bay  ) 
North  Pt.  Entr.  J 
Observatory  . . . , , 
Cape  St.  Lucas . . , 


7  3o 

38  18  1 

37  47. 
36  37 
M  24. 
33  43. 
33  26. 
32  4i . 
3o  21.1 
27  39.1 

24  32. 

24  38.. 

22  52.. 


D.  M. 

179  56 E 

179  0-' 
177  43 
177  07 

177  i4 
179  o5 

178  3oW 

178  32 

179  26 

178  25 


167  37 E 

166  22 

169  36 
169  06 

168  26 

139  49 
166  32 


122 
122 
120 
117 


123 

123 
121 

"9 
1X8 
117 
117 

ii5 
ii4 


56 

19 
W 

0.7 
21 , 
5o.8 
38.8 
i3.S 
4i 
II. 3 
56.5 

5i.: 


112  1.2 


112 

9 


6.3 

52.  I 


452 


GREAT  CmCLE  SAILING. 


Tlie  shortest  distance  between  two  points  on  th.e  eartli's  surface  is  on  the  arc  of  a 
great  circle. 

The  following  rules  will  enable  the  navigator  to  calculate  the  courses  and  distance, 
and  to  project  the  great  circle  track  of  the  voyage,  on  the  chart.  By  projection,  the 
shortest  distance  between  the  two  given  points  will  be  readily  perceived. 

The  greatest  difference  between  the  tracks  is  found  in  the  higher  latitudes,  and 
between  points  in  about  the  same  parallels.  In  crossing  the  ei^uator,  this  difference, 
generally,  is  not  of  so  much  importance. 

In  Case  I.,  we  find  the  courses  on  the  great  circle  to  differ  from  the  course  on  the 
rhumb  line  about  two  points.  This  difference,  in  many  cases,  is  often  more ;  the  knowl- 
edge ,of  which  might  be  of  great  importance  to  the  mariner,  by  enabling  him  to  go  on 
his  course  with  a  wind  which  on  the  rhumb  line  would  be  adverse. 

The  course  on  the  rhumb  line  is  always  the  same ;  on  the  great  circle  it  is  continually 
changing.  It  would  be  well  to  calculate  the  course  two  or  three  times  during  the  day, 
working  out  the  position  of  the  ship  by  the  usual  methods. 

A  vessel  sailing  on  the  great  circle  track,  on  the  same  side  of  the  equator,  is  always 
in  a  higher  latitude  than  she  would  be  on  the  rhumb  line ;  consequently,  in  north  lat- 
itude, the  great  circle  line  will  be  north  of  the  rhumb 
line,  and  in  south  latitude,  will  be  south  of  the  rhumb 
line.  This  is  evident  on  inspectmg  the  annexed 
chart. 

"When  the  course  found  is  greater  than  90°,  its  sup- 
plement will  be  the  course  counted  from  the  opposite 
pole.  Thus,  in  Case  V.,  the  course  is  N.  131°  24'  E., 
or  S.  48°  36'  E. 

In  calculating  the  courses  PLL'  and  PL'L,  and  the 
distance  LL',  we  have  the  two  sides  PL  and  PL',  and 
their  included  angle  LPL'  given,  the  sides  being  the 
co-latitudes  of  the  given  places,  and  the  angle  P  being 
the  difference  of  longitude. 


CASE  I. 

Given  the  latitudes  and  longitudes  of  two  places  on  the  same  side  of  the  equator,  to  find  the 

courses. 
By  Theorems  (8),  (7),  page  441,  we  obtaua  the  following 

RULE. 

Make  two  columns,  and  write  down  the  following  logarithms.  The  log.  cosine  ol 
half  th3  difference  of  the  latitudes  in  Col.  1,  and  the  log.  sine  in  Col.  2,  the  log. 
cotangeat  of  half  the  difference  of  longitude  in  both  columns,  the  log.  cosecant  of  half 
the  sum  of  the  latitudes  in  Col.  1,  and  the  secant  in  Col.  2.  The  sum  of  the  logs,  in 
Col.  1  w^lU  give  log.  tangent  of  half  the  sum  of  the  courses,  and  the  sum  of  the  logs, 
in  Col.  2  will  give  the  log.  tangent  of  half  the  difference  of  the  courses.  The  sum  of 
these  results,  rejecting  20  in  the  index,  wUl  give  the  course  corresponding  to  the  greatest 
latitude,  and  their  difference  the  course  corresponding  to  the  least  latitude. 

EXAMPLE  I. 

Required  the  course  from  a  point  in  the  latitude  40°  N.  and  longitude  70°  W.  to  a 
place  in  the  latitude  50°  N.  and  10°  W. 


Half  the  sum  of  the  lats.  =  45°. 

Col.  1. 

i  diff.  lats.  5° COS. 

I  diff.  long.  30° cotang. 

i  sum  lats.  45° cosec. 

i  sum  courses, tang.       10.38741 

67°  43'  

12    03 


Half  their  difference  =  5°.     Half  the  difference  of 
long.  =  30°. 

Col.  2. 

i  diff.  lats.  5° sine        8.94030 

Same 10.23856 

i  sum  lats.  45° secant  10.15051 


9.99834 
10.23856 
10.15051 


i^  diff.  courses tang.       9.32937 

12°  03'  


Course  N.  79°  46'  W.,  from  latitude  50°  N. 
Course  N.  55°  40'  E.,  from  latitude  40°  N. 

By  Morcator's  sailing,  the  course  fi-om  50°  N.  is  S.  76°  42'  W.,  making  a  differenc9 
of  23°  32'. 

When  the  places  have  the  same  latitudes,  the  sum  of  the  logarithms  in  Col.  1  will  ba 
the  log.  tangent  of  the  course  from  either  place. 


90°    H0°     50°     60°     70"     80*= 


90= 


;60° 


60° 


50= 


50" 


40° 


40= 


SC^ 


30' 


20° 


20= 


10= 


10= 


10= 


10= 


20= 


20= 


30« 


30= 


'40= 


50° 


&0° 


90= 


-JK — ==^ 


40= 


50= 


60= 


40="     50°     60"     70°     80°     90  = 


p 

90° 

8 

o     7C 

o              GO"     SO" 

40° 

30°      io"     10°      0 

10 

20 

30° 

40° 

5 

°     G0° 

70" 

80°     90°  1 

60° 

1 

"    1        ! 

1 

-- 

A 

y 

^,-- 

^^ 

1 

y^' 

^'' 

^ 

--""^ 

-■'' 

-^ 

' 

40° 

'"^ 

40° 

B 

30° 

V^ 

K  - 

% 

X 

M° 

\  \ 

20° 
10° 

10° 

X 

\ 

\^ 

\ 

10° 

A 

V 

10° 
20° 
30° 

40° 

pn° 

V\ 

\ 

!40' 

1 

^^„^^ 

^ 

^^ 

^^— 

._ 

_^- 

fiO° 

L 

'^ 

50° 
60° 

co° 

1         1 

1 

1 

1  ' 

0° 

80 

"■     70 

"              60°     60° 

40° 

30°     20°     10° 

I)      1 

°     2 

°      20° 

40° 

J 

0°     00° 

70° 

80°     90°  Ij 

GREAT  CIRCLE   SAILING.  453 

EXAMPLE   II. 

Required  tlie  course  from  a  point  in  the  latitude  of  40°  S.  and  20°  E.,  to  anotiier 
point  in  latitude  40°  S.  and  longitude  80°  E. 

The  half  sum  latitudes  is  40°,  the  half  difference  is  0°,  and  half  the  difference  of 

longitude  30°. 

i  difference  lats.  0° cosine     10.00000 

J^  difference  long.  30° cotang.   10.23856 

I  sum  lats.  40° cosec.     10.19193 

Course,  69°  38' tang.       10.43049 

The  courses  are  S.  69°  38'  E.,  or  S.  69°  38'  W.,  and  the  difference  between  them  and 
the  Mercator  course  is  20°  22'. 

CASE  n. 

Given  the  latitudes  and  longitudes  of  the  two  places,  and  the  courses,  to  find  distance. 
Theorem  (2),  page  439,  gives  the  rule. 

RULE   I. 

Add  together  the  log.  sine  of  the  difference  of  the  longitude,  the  log.  cosine  of  the 
latitude,  the  log.  cosecant  of  the  course  ;  the  sum,  rejecting  20  in  the  index,  ■will  be  the 
log.  sine  of  the  distance. 

When  the  greatest  latitude  is  used,  the  least  course  must  be  taken,  and  when  the 
least  latitude  is  used,  the  greatest  course  must  be  taken. 

EXAMPLE   I. 

Let  the  latitudes  and  longitudes  of  the  places  be  as  in  Example  I.,  Case  I.,  and  the 
courses  as  therein  found,  55°  40'  and  79°  46'.     Required  the  distance. 

Difference  of  long.,  60° sine        9.93753 

Greatest  latitude,  50° cosine    9.80807 

Least  course,  55°  40' cosec.  10.08314 

Distance,  42°  23' sine        9.82874 

60 
2543  miles. 
By  Theorem  (10),  page  441,  we  obtain 

RULE   II. 

Add  together  the  log.  secant  of  half  the  difference  of  the  two  courses,  the  log.  cosine 
of  half  the  sum  of  the  two  courses,  and  the  log.  cotangent  of  half  the  sum  of  the  lat- 
itudes ;  the  sum,  rejecting  20  in  the  index,  will  be  the  log.  tangent  of  half  the 
distance. 

EXAMPLE  II, 
Given  the  parts  as  in  the  previous  example.    Required  the  distance. 

h.  difference  courses,  12°  03' ...secant         10.00968 

h.  sum  courses,  67°  43' cosine  9.57885 

h.  sum  latitudes,  45° cotangent  10.00000 

i  distance,  21°  llil' tangent         9.58853 

2_  

42°  23'    or  2543  miles. 

Distance  by  Mercator's  sailing, 2608 

Gain  on  the  great  circle, 65 

Use  Rule  II.  when  any  doubt  exists  as  to  the  result,  which  may  be  the  case  when  the 
difference  of  longitude  is  about  90°. 


GREAT  CIECLE  SAILING.  453 

EXAMPLE   II. 

Required  the  course  from  a  point  in  the  latitude  of  40°  S.  and  20°  E.,  to  another 
point  in  latitude  40°  S.  and  longitude  80°  E. 

The  half  sum  latitudes  is  40°,  the  half  difference  is  0°,  and  half  the  difference  of 

longitude  30°. 

i  difference  lats.  0° cosine     10.00000 

i  difference  long.  30° cotang.  10.23856 

I  sum  lats.  40° cosec.     10.19193 

Course,  69°  38' tang.       10.43049 

The  courses  are  S.  69°  38'  E.,  or  S.  69°  38'  W.,  and  the  difference  between  them  and 
the  Mercator  course  is  20°  22'. 

CASE   IL 
Given  tlie  latitudes  and  longitudes  of  the  two  places,  and  the  courses,  to  find  distance. 
Theorem  (2),  page  439,  gives  the  rule. 

RULE  I. 

Add  together  the  log.  sine  of  the  difference  of  the  longitude,  the  log.  cosine  of  the 
latitude,  the  log.  cosecant  of  the  course  ;  the  sum,  rejecting  20  in  the  index,  will  be  the 
log.  sine  of  the  distance. 

When  the  greatest  latitude  is  used,  the  least  course  must  be  taken,  and  when  the 
least  latitude  is  used,  the  greatest  course  must  be  taken. 

EXAMPLE   I. 

Let  the  latitudes  and  longitudes  of  the  places  be  as  in  Example  I.,  Case  I.,  and  the 
courses  as  therein  found,  55°  40'  and  79°  46'.     Required  the  distance. 

Difference  of  long.,  60° sine        9.93753 

Greatest  latitude,  50° cosine    9.80807 

Least  course,  55°  40' cosec.  10.08314 

Distance,  42°  23' sine        9.82874 

60 
2543  miles. 
By  Theorem  (10),  page  441,  we  obtain 

RULE   II. 

Add  together  the  log.  secant  of  half  the  difference  of  the  two  courses,  the  log.  cosine 
of  half  the  sum  of  the  two  courses,  and  the  log.  cotangent  of  half  the  sum  of  the  lat- 
itudes ;  the  sum,  rejecting  20  in  the  index,  will  be  the  log.  tar.gent  of  half  the 
distance. 

EXAMPLE  II. 
Given  the  parts  as  in  the  previous  example.     Required  the  distance. 

h  difference  courses,  12°  03' secant         10.00968 

I  sum  courses,  67°  43' cosine  9.57885 

I  sum  latitudes,  45° cotangent  10.00000 

i  distance,  21°  Hi' tangent         9.58853 

2_  

42°  23'    or  2543  miles. 

Distance  by  Mercator's  sailing, 2608 

Gain  on  the  great  circle, 65 

Use  Rule  II.  when  any  doubt  exists  as  to  the  result,  which  may  be  the  case  when  the 
difference  of  longitude  is  about  90°. 


454  GREAT  CIRCLE   SAILING. 

EXAMPLE   III. 

Required  tlie  distance  between  two  points,  one  in  40°  S.  and  20°  E.,  and  the  other 
in  40°  S.  and  80°  E. ;  tlie  course  from  eacli  point  being  69°  38'. 

Diiference  long.,    60"        sine        9.93753 

Latitude,  40°        cosine    9.88425 

Course,  69°  38' cosec.  10.02804 

Distance,  45°  03' '..sine       9.84982 

60  

2703 

Distance  by  Parallel  sailing, 2758 

'<         "     great  circle  '«        2703 

Gain, 55 

CASE   III. 

Give7i  the  latitude  and  lo7igitude  of  two  places,  to  find  the  maximum  separation  in  latitude. 

In  sailing  between  two  places  on  the  same  side  of  the  equator,  on  a  great  circle,  the 
vessel  always  keeps  in  a  higher  latitude  than  on  the  rhumb  line.  The  point  on  the 
great  circle  which  is  the  greatest  distance  from  the  rhumb  line,  measured  on  the 
meridian,  is  the  point  of  maximum  separation  in  latitude. 

RULE. 

For  the  latitude.  —  Find  the  course  between  the  places  by  Mercator's  sailing,  and 
take  the  supplement ;  also,  the  course  on  the  great  circle,  from  the  same  latitude ;  add 
together  the  log.  cosecant  of  the  Mercator  course,  the  log.  sine  of  the  great  circle  course, 
and  the  log.  cosine  of  the  latitude  of  the  place  of  the  great  circle  course  ;  the  sum  is 
the  log.  cosine  of  the  latitude  required. 

RULE. 

For  the  longitude.  —  Add  together  the  log.  secant  of  half  the  difference  of  the  given 
latitude  and  the  latitude  just  found,  the  log.  sine  of  half  the  sum  of  the  latitudes,  and 
the  log.  tangent  of  half  the  sum  of  the  great  circle  course,  and  the  supplement  of  the 
Mercator  course ;  the  sum  wUl  be  the  log.  cotangent  of  half  the  difference  of  longitude. 

EXAMPLE  I. 

Required  the  latitude  and  longitude  of  the  point  of  the  maximum  separation  between 
two  points,  one  in  the  latitude  40°  N.  and  longitude  70°  W.,  and  the  other  in  latitude 
50°  N.  and  longitude  10°  W. 

By  Case  I.,  Mercator's  sailing,  we  find  the  course. 

As  merid.  diff.  of  lat.,  851 2.92993 

Is  to  radius 10.00000 

So  is  the  diff.  of  long.  3600 .3.55630 

To  tang,  course,  76°  42' 10.62637 

Its  suppleme7it,    103°  18'. 
By  Case  L,  Example  I.,  we  find  the  great  circle  course,  from  40°  N.,  to  be  55°  40'. 

To  find  the  latitude. 

Mercator's  course,     76°  42' cosecant  10.01181 

Great  circle  course,  55°  40' sine  9.91686 

Latitude,  40° cosine        9.88425 

Latitude  required,  49°  28' cosine        9.81292 

To  find  the  longitude. 

Lowest  latitude,  40° 40°  Supplement  of  Mercator  course, ...103°  18 

Latitude  of  max.  separation, 49°  28'     Great  circle  course, 55°  40 

Sum, 89°  28'     Sum, 158°  58 

h  sum 44°  44'     h.  sum, 79°  29 


Difference, 9°  28 

i  difi"erence, 4°  44 


GEEAT   CmCLE   SAILING. 

i  diif.  of  latitudes,      4°  44 secant, 

i  sum  of  latitudes,    44°  44 sine, 

i  sum  of  com-ses,      79°  29' tangent, 

i  diff.  of  longitude,  14°  44' cotangent,  10.58026 

2  


10.00148 

9.84745 

10.73133 


Diff.  of  longitude,     29°  28' 
Longitude  left,  70° 


455 


40°  32'   longitude  of  the  required  point  of   max- 
imum separation. 


EXAMPLE   II. 

Required  the  latitude  and  longitude  of  the  point  of  maximum  separation  in  latitude, 
between  two  places,  one  in  lat.  40°  S.  and  20°  E.,  and  the  other  in  40°  S.  and  long. 
80°  E. 

The  courses  found  in  Case  I.,  Example  11.,  is  69°  38'.  The  Mercator  course  is  E.,  or 
"W.,  or  90°,  and  its  supplement,  therefore,  is  90°.     Their  half  sum  is  79°  49'. 


For  the  latitude. 

Merc,  course,  90° cosec.  10.00000 

Great  circle  co.,  69°  38' sine        9.97196 

Latitude,  40° cosine    9.88425 


Lat.  required,  44°  06' cosine    9.85621 


For  the  longitude. 
In  this  and  similar  cases  of  parallel  lat- 
itudes, the  difference  of  longitude  wiU  be 
equal  to  half  the  difference  of  longitude  be- 
tween the  given  places,  which,  in  this  ex- 
ample, is  30"^,  and  the  long,  required,  50.° 


CASE  IV. 

Given  the  latitudes  and  lo7igitudes  of  the  two  places,  and  the  great  circle  courses,  to  find  tJia 
maximimi  latitude  and  its  longitude. 

"When  both  courses,  counted  from  the  same  pole,  are  less  than  90°,  then  the  maximum 
latitude  of  the  arc  will  be  within  the  two  given  points. 
By  Theorem  (2),  page  439,  we  get  the 

RULE 

For  the  latitude.  —  Add  the  cosine  of  the  latitude  to  the  sine  of  the  great  circle 
course,  fi-om  the  same  latitude;  the  sum,  rejecting  10  in  the  index,  will  be  the  log. 
cosine  of  the  maximum  latitude. 

By  Theorem  (1),  page  439,  Ave  obtain  the 


RULE 

For  the  longitude.  —  To  the  log.  sine  of  the  latitude,  add  the  log.  tangent  of  the  cor- 
responding great  circle  course  ;  the  sum  will  be  the  log.  cotangent  of  the  difference  of 
longitude. 

EXAMPLE   I. 

Given  the  latitudes  and  longitudes  of  two  places,  as  in  Example  I.,  Case  I. ;  viz  :  40° 
N.  and  70°  W.,  and  50°  N.  and  10°  W.,  to  find  the  maximum  latitude.  The  great 
circle  courses  are  found,  in  Case  I.,  to  be  55°  40'  and  79°  46. 


To  find  the  latitude. 

Lat.  40° ^. cosine  9.88425 

Course,  55°  40' sine  9.91686 

Lat.  required,  50°  45^' cos.  9.80111 


To  find  the  longitude. 

Latitude,  40° sine        9.80807 

Course,  55°  40' tang.     10.16558 

Diff.  long.,  46°  44' cotang.  9.97305 

Long,  left,  70° 

Long.  req.  23°  16'  of  the  max.  lat. 

"^Tien  both  places  are  in  the  same  latitude,  the  maximum  latitude  and  the  point  of 
maximum  separation  will  be  the  same. 

EXAMPLE   II. 

Required  the  maximum  latitude  between  two  points  ;  one  in  lat.  40°  S.  and  20°  E. 
and  the  other  in  40°  S.  and  80°  E. 


456 


GREAT   CIRCLE   SAILING. 


In  Case  I.,  Example  11.,  tlie  great  circle  course  is  69°  38'. 

To  find  the  latitude. 

Latitude,  40° cosine  9.88425 

Course,  69°  38' sine      9.97196 

Latitude,  44°  06' cosine  9.85621 

Being  the  same  results  as  in  Case  III.,  Example  II. 

In  this  case,  the  longitude  -will  be  midway  between  the  two  given  longitudes. 

CASE   V. 

Given  tioo  places  on  the  opposite  sides  of  the  equator,  to  find  the  courses  and  distance  on 
an  arc  of  the  great  circle. 
Find  the  point  of  intersection  of  the  great  circle  with  the  equator,  by  the  following 
rule ;  then  with  this  point,  and  the  places  given,  proceed  as  in  Cases  I.  and  II.  for  the 
courses  and  distance. 

RULE. 

Add  together  the  sine  of  the  difference  between  the  two  latitudes,  (not  the  difference 
of  latitude,)  the  cosecant  of  the  sum  of  the  latitudes,  and  the  tangent  of  half  the 
difference  of  longitude  ;  the  sum,  rejecting  20  in  the  index,  will  be  the  tangent  of  an 
arc  X,  which,  added  to  half  the  difference  of  longitude,  will  give  the  difference  of  lon- 
gitude between  the  greatest  latitude  and  point  of  intersection. 

EXAMPLE. 

Given  two  points,  one  in  40°  N.  and  70°  AV.,  and  the  other  in  30°  S.  and  10°  W. 
Required  the  point  of  intersection  of  the  great  circle  with  the  equator,  and  the  courses 
and  distance  between  the  two  given  places. 

The  difference  between  40°  and  30°  =  10°.  The  sum  is  70°.  Half  the  difference  of 
longitude,  30°. 


Difference  between  the  lats. 
Sum  of  the  latitudes. 
Half  difference  of  long. 

ArcX, 

Half  diff.  of  long. 

Diff.  of  long,  from  40°  N. 
Long,  left, 


10° sine        9.23967 

70° cosec.  10.02701 

30° tang.      9.76144 


6°  05' tang 

30° 


9.02812 


36°  05' 
70° 


Long,  of  intersection,  33°  55'  "W. 

Having  the  latitudes  40°  N.  and  0°,  and  the  longitudes  70°  and  33°  55',  we  can 
calculate  the  courses  and  distance  by  rules  given  in  Cases  I.  and  H. 

To  calculate  the  courses. 
Half  sum  lats.,  20°,  and  half  difference  of  lats.,  20°.     Half  difference  long.,  18°  03  . 


CoL.  1. 

4  diff.  lats.  20° cos.  9.97299 

I  diff.  longs.  18°  03' cotang.    10.48694 

I  sum  lats.  20° cosec.      10.46595 

i  sum  courses,  83°  14' tang. 

48°  10' 


10.92588 


Col.  2. 

i  diff.  lats.  20° sine  9.53405 

Same 10.48694 

^  sum  lats.  20° secant  10.02701 

i  diff.  courses,  48°  10' tang.  10.04800 


Coxirse  N.  35°  04'   W.  from  the  equator. 

131°  24' 
180°         or 


S.  48°  36'  E.  from  latitude  40°  N. 

To  find  the  distance. 

Difference  of  long.  36°  05' sine 

Greatest  latitude,  40° cosine 

Least  course,  35°  04' cosec. 

Distance,  51°  45' sine 

_60 

3105  miles. 


9.77009 

9.88425 

10.24069 

9.89503 


GREAT   CmCLE   SAILING.  457 

To  find  the  courses  and  distance  fi-om  tlic  equator,  in  long.  33°  55'  "W.,  to  30°  S. 
and  10°  W. 
Half  sum  lats.,  15°.    Half  diff.  of  lats.,  15°.     Half  diff.  long.,  11°  57'. 

To  find  the  couirses. 


i  diff.  lats.  15° COS.  9.98494 

4  diff.  long.  11°  57' cotang.  10.67439 

I  sum  lats.  15° cosec.     10.58700 


i  diff.  lats.    15° sine  9.41300 

h.  diff.  long.  11°  57' cotang.  10.67439 

I  sum  lats.  15° secant    10.01506 

h  sum  courses,  86°  45'  . .  .tang.      11.24633 1  h.  diff-  course,  51°  42' tang.       10.10245 

51°  42' 

Course   S.    35°  03'  E.  from  the  equator. 

138°  27' 
180° 


N.  41°  33'  W.  from  30°  S. 

To  find  the  distance. 

Difference  long.  23°  55' sine        9.60789 

Greatest  lat.  30° cosine    9.93753 

Least  course,  35°  03' cosec.   10.24087 

Distance,  37°  41' sine        9.78629 

60  

2261  miles. 


CASE   VI. 
7b  project  the  track  on  a  great  circle. 

First,  (by  Eaper.)  \Vlicn  the  places  are  on  the  same  side  of  the  equator.  —  Dra-\v 
the  line  connecting  the  given  places,  find  the  position  of  the  point  of  the  maximum 
separation  of  latitude,  and  through  this  point  draw  a  line  parallel  to  the  line  connecting 
the  two  places.  Find  the  coui-ses  on  the  great  circle  from  the  two  places,  and  draAV 
them  on  the  chart.  We  can,  through  these  three  points,  roughly  trace  the  required 
cuiwe.     K  the  maximum  latitude  falls  on  the  curve,  we  shall  have  a  fourth  point. 

EXAMPLE  I. 

Given  the  latitudes  40°  N.  and  50°  N.,  and  their  corresponding  longitudes  70°  W. 
and  10°  W.,  to  project  the  great  circle  track  connecting  them. 

By  Case  I.,  the  courses  are  found  to  be  N.  55°  40'  E.,  and  N.  79°  46'  W.,  and  by 
Case  ni.,  the  position  of  the  point  of  maximum  separation  of  latitude  is  found  to  be  49° 
28'  N.  and  40°  32'  W. ;  and  by  Case  IV.,  the  maximum  latitude  is  in  50°  45^'  N.,  and 
23°  16'  W. 

Draw  on  the  chart  the  line  AB,  connecting  the  two  points  ;  from  A  and  B,  lay  off 
the  courses  N.  79°  46'  W.,  and  N.  55°  40'  E. ;  through  the  point  of  maximum  sep- 
aration draw  a  Kne  parallel  to  AB  ;  through  these  points,  and  the  point  of  maximiim 
latitude,  draw  the  dotted  Une,  which  will  be  the  track  required. 

Second.  — "When  the  places  are  on  opposite  sides  of  the  equator.  —  Find  the  course 
at  each  of  the  given  pomts,  and  the  points  of  maximum  separation  of  latitude,  for 
both  sides  of  the  equator ;  find  the  longitude  of  the  intersection  of  the  great  circle 
with  the  equator,  and  the  course  at  that  point ;  with  these  five  points,  construct  the 
track. 

EXAMPLE   II. 

Given  two  places,  one  in  the  latitude  40°  N.  and  longitude  70°  W.,  and  the  other  in 
30°  S.  and  10°  W.,  to  project  the  track  on  the  great  circle,  passing  through  them. 

The  courses  are,  by  Case  V.,  N.  41°  33'  W.,  from  30°  S.,  and  S.  48°  36'  E.,  from  40°  N. 
The  longitude  of  the  intersection  of  the  great  circle  on  the  equator,  by  Case  V.,  is  33° 
55'  TV.,  and  the  course  at  the  intersection  is  N.  35°  03'  W.,  and  S.  35°  03'  E. 

The  maximum  separation  of  latitude  north  of  the  equator,  is  in  25°  33'  N.  and  53° 
31'  W. ;  and  south  of  the  equator,  is  in  18°  21'  S.  and  20°  31'  W. 

With  these  points,  the  great  circle  track  can  be  constructed,  as  in  the  example  pre- 
ceding. 

58 


458  GREAT   CIRCLE   SAILING. 

EXAMPLE  III. 

Given  one  place  in  tlie  latitude  40°  S.  and  20'  E.,  and  another  in  40°  S.  and  80°  E., 
to  project  the  track. 

By  Case  I.,  Example  II.,  we  find  the  courses  to  be  S.  69°  38'  E.,  and  S.  69°  38'  "W. 

By  Case  III.,  Example  II.,  the  maximum  separation  of  latitude  is  in  44°  06'  S.  and 
longitude  50°  E. 

By  Case  IV.,  Example  II.,  the  maximum  latitude  in  this  case  is  the  same  as  the  max- 
imum separation  of  latitude. 

With  these  three  points,  the  track  can  be  easily  drawn. 


The  great-circle  track,  from  Cape  Clear  to  the  northern  portion  of  the  United  States, 
passes  so  near  Cape  Race,  that  mariners,  in  endeavoring  to  keep  on  this  track,  are  often 
placed  in  great  peril  when  approaching  tlie  vicinity  of  Newfoundland. 

The  following  track,  from  Cape  Clear,  passing  through  a  point  one  hundred  miles  south- 
east of  Cape  Race,  and  thence  to  Nantucket  South  Shoal,  by  Ifercator's  sailing,  is 
proposed : 

Lat.  Long.  Lat  Long. 

N.  W.  N.  W. 

1st.  From  Cape  Clear,  in    -     -     -     51°  26'  and    9°  29',  to  51°  16'  and  23°  27'. 
2d.    From  51°  16' and  23°  27',  to  49°  23' and  37°  24'. 

Sd.    From  49°  23'  and  37°  24',  to  45°  28'  and  51°  21'. 

4th.  From  100'  S.  E.  of  Cape  Race,  45°  28'  and  51°  21',  to  41°  04'  and  69°  51'. 

The  distance  on  these  four  courses,  by  Mercator's  sailing,  is        -        -         -         -         2532J 
The  distance  on  the  Great  Circle,  from  Cape  Clear  to  a  point  100  miles  S.  East  of 
Cape  Race,  and  thence  to  Nantucket  South  Shoal,  is      -         -        -        -        -         2528 

Making  a  saving  of  only  about 4^ 

The  distance  on  the  great  circle,  direct  from  Cape  Clear  to  Nantucket  South  Shoal,  is,  2505 
Being  only  27i  miles  less  than  the  route  proposed. 

In  sailing  easterly  beyond  the  Cape  of  Good  Hope,  we  have  (by  Example  III, 

page  454)  the  distance  by  parallel  sailing 2758  mile.s, 

And  by  great-circle  sailing 2703      " 

Now,  if  by  Mercator  sailing,  we  lay  off  the  track 
From  40°  S.  and  20°  E.  to  44°  06'  S.  and  50°  E.  and  thence 

From  44°  06'  S.  and  50°  E.  to  40°  S.  and  80°  E.,  we  shall  find  the  distance       2722      " 
Only  19  miles  more  than  the  great  circle. 

From  these  examples,  it  would  seem  that  the  advantages  in  most  cases  derived  from 
keeping  on  the  great-circle  track,  are  not  sufficient  to  authorize  the  mariner  to  run  the 
least  risk  in  pursuing  his  course ;  and  that  the  small  saving  of  distance  is  not  really  of  any 
comparative  importance. 


459 


ON  THE  COMPASS. 


Tlie  British  Admiralty  have  directed  that  Compasses  should  be  placed  at  least  4 
feet  6  inches  apart  on  board  of  the  ships  of  war.  This  is  to  avoid  the  disturbance 
knovpn  to  exist  when  two  needles  are  placed  near  each  other.  The  error  from 
this  source  has,  in  some  cases,  amounted  to  more  than  8°.  It  is  to  be  hoped  that 
the  mercantile  interest  of  the  country  will  adopt  this  rule.  If  the  steering  ap- 
paratus is  sufficiently  small  one  compass  is  strongly  recommended,  a  standard 
compass,  for  reference,  being  placed  on  the  centre  line  of  the  ship. 

No  Iron  should  be  allowed  within  seven  feet,  and  vertical  Iron  stancheons,  &c., 
should  be  at  least  fourteen  feet  from  the  compasses. 

Binnacles  should  be  made  without  doors,  to  prevent  improper  substances  from 
being  placed  therein. 

The  common  compasses  are  frequently  very  imperfectly  constructed,  and  the 
needles  poorly  magnetized.  Great  care  in  their  selection  cannot  be  too  strongly 
recommended ;  they  should,  like  the  chronometer,  be  carefully  handled,  and  not 
subjected  to  the  rough  usage  they  frequently  receive. 

Rules  for  ascertaining  the  deviation  of  the  compass  caused  by  the  Iron  in  the  ship. 

1st. — A  good  standard  compass  should  be  placed  on  the  centre  line  on  the  quarter 
deck,  as  far  as  possible  from  all  masses  of  Iron.  It  should  have  such  a  support  as 
will  render  bearings  and  amplitudes  easily  taken. 

2d. — Bearings  should  be  taken  only  on  that  part  of  the  ship  where  the  standard 
compass  is  placed,  or  where  the  observations  for  deviation  were  made. 

3d. — When  the  ship  is  fully  ready  for  sea,  with  every  thing  on  board,  allow  her 
head  to  come  up  successively  to  the  thirty-two  points  of  the  compass  ;  then  accu- 
rately observe  the  hearing  of  some  distant  hut  well  defined  object,  (the  real  mag- 
netic bearing  of  the  same  having  been  ascertained,)  and  record  the  same  as  in 
Table  I. 

4th. — The  real  magnetic  bearing  may  be  found  by  taking  the  standard  compass 
on  shore  and  placing  it  on  a  line  with  the  object  observed  and  that  part  of  the  ship 
where  the  compass  stood,  so  that  they  shall  be  in  a  line  with  the  observers  eye. 
The  difference  between  this  real  magnetic  bearing  and  the  bearing  in  col.  2  will 
give  the  deviation  which  is  found  in  col.  3. 

TABLE  I. 
Real  magnetic  bearing  of  the  distant  object  from  the  ship,  N.  80°  E. 


Ship's  Head 

Bearing  of  by 

Deviation  of 

Ship's  Head 

Bearing  of  by 

Deviation  of 

by  the  Stand- 

the Standard 

the  Standard 

by  the  Stand- 

the Standard 

the  Standard 

ard  Compass. 

Compass. 

Compass. 

ard  Compass. 

Compass. 

Compass. 

North. 

N.  81°  E. 

1°  w. 

South. 

N.  80°  E. 

Nothing. 

N.  by  E. 

N.  79    E. 

1°   E. 

S.  by  W. 

N. 81°  E. 

1°  W. 

N.  N.  E. 

N.  78    E. 

2°   E. 

s.  s.  w. 

N.  82°  E. 

2°  W. 

N.  E.  by  N. 

N.  76    E. 

4°   E. 

S.  W.  by  S. 

N. 83°  E. 

3°  w! 

N.  E. 

N. 75    E. 

5°   E, 

s.  w. 

N.  84°  E. 

4°  W. 

N.  E.  by  E. 

N.  74    E. 

6°   E. 

S.  W.byW. 

N.  85°  E. 

5°  W. 

E.  N.  E. 

N. 73    E. 

7°   E. 

W.  S.  W. 

N.  80°  E. 

6°  W. 

E.  by  N. 

N.  72    E. 

8°   E. 

W.  by  S. 

N. 87i  E. 

7^  W. 

East, 

N. 72^  E. 

7^  E. 

West. 

N. 87°  E. 

7°  W. 

E.  by  S. 

N.  73    E. 

7°   E. 

W.  by  N. 

N.  801-  E. 

6}  W. 

E.  S.  E. 

N.  74    E. 

6^   E. 

W.  N.  W. 

N. 80°  E. 

6°  W. 

S.  E.  byE. 

N.  75    E. 

5°   E. 

N.  W.  by  W. 

N. 85°  E. 

5°  W. 

S.  E. 

N.  76    E. 

4°   E. 

N.  W. 

N.  84°  E. 

4°  W. 

S.  E.  by  S. 

N.  77    E. 

3°   E. 

N.  W.  by  N. 

N. 83°  E. 

3°  W. 

S.  S.  E. 

N.  78    E. 

2°   E. 

N.  N.  W. 

N. 82°  E. 

2°  W. 

S.  by  E. 

N. 79i  E. 

0^  E. 

N.  by  W. 

N.  81°  E. 

1°  W. 

The  deviation  is  East  when  the  north  end  of  the  needle  is  drawn  to  the  eastward. 


460 


or  right  hand ;  and  West  when  the  north  end  of  the  needle  is  drawn  to  the  west- 
ward, or  left  hand. 

Example  : — When  the  ship's  head  is  E.  by  N.  the  bearing  by  the  standard  com- 
pass was  N.  72°  E.,  it  follows  that  the  north  end  of  the  needle  has  been  attracted 
8°  to  the  eastward. 

Should  there  be  no  proper  object  of  sufficient  distance  visible  from  the  ship,  then 
a  second  compass  must  be  taken  on  shore,  and  the  bearing  of  the  two  compasses 
from  each  other  observed  at  each  of  the  thirty- two  points,  and  the  results  registered 
as  in  Table  II.  The  standard  compass  should  be  compared  on  shore  with  the 
second  compass,  and  if  any  difference  is  found  it  should  be  noted. 

TABLE  II. 


Ship's  Head  by  the 
Standard  Com- 
pass. 

Bearing  of  the 

&hore  Compass 

from  the  Standard 

Compass. 

Bearing  of  the  Standard  Compass 

from  the  2d  Compass  on  shore  with 

the  correction  for  their  difference 

applied. 

Deviation  Stand 
ard  Ccmpass. 

Correct  Mag- 
netic Course. 

N. 
N.  by  E. 
N.  N.  E. 

&c. 

S.  31°  W. 

S.  28i  W. 
S.  27°  W. 

N.  30    E. 
N.  29^  E. 
N.  29°  E. 

1°  w. 

1°  E. 

2°  E. 

Nearly  N. 
N.  12°  E. 
N. 24°  E. 

Col.  5th  gives  the  correct  magnetic  course,  and  also  the  points,  when  the  iron 
causes  the  least  deviation,  which  generally  are  the  North  and  South  points ;  but 
as  this  is  not  always  so,  especially  for  steam  vessels,  we  should  depend  upon  obser- 
vations only.  ^ 

The  points  once  established  may  be  considered  permanent,  provided  every  thing 
remains  the  same  and  the  compass  used  in  the  same  place. 

An  azimuth  at  sea,  with  the  ship's  head  on  the  point  of  no  deviation,  will  give 
the  true  variation. 

The  amount  of  deviation  varies  with  the  latitude,  and  in  southern  latitudes  it  be- 
comes important  to  form  new  tables,  as  the  deviation  generally  changes  from  West 
to  East  and  from  East  to  West.  The  deviations  can  be  examined  at  sea,  by  ob- 
serving azimuths  with  the  ship  heading  on  different  points,  especially  on  the  point 
of  no  deviation.  If  the  results  conform  to  the  table  they  may  continue  to  be  used  ; 
if  not,  then  a  new  table  should  be  made. 

The  standard  compass  in  iron  vessels  should  be  raised  above  the  deck  much 
higher  than  in  sailing  vessels. 

In  steam  vessels  with  telescopic  funnels  the  deviation  is  sensibly  affected  when 
they  are  taken  down.     Observations  should  be  made  when  up  and  down. 

It  is  recommended  that  the  ship  should  be  directed  hy  the  standard  compass,  and 
the  binnacle  compass  should  be  used  by  the  helmsman  only  to  give  the  approximate 
course.     Direct  reference  should  be  frequently  had  to  the  standard  compass. 


E.   &   G.   W.   BLUNT. 


BOOISZS- 

Nautical  Almanac,  containing  the  Moon's  right  ascension  and  declination  for  eyery 

5  hours. 
Bowditch's  Navigator,  30th  edition. 
Blunt's  Coast  Pilot,  18th  edition. 
Erpeditious  Measurer,  for  measuring  cargo. 
Ward's  Lunar  Tables. 

Sheet  Anchor,  112  quarto  plates,  with  additions  by  G.  W.  Blunt. 
Commercial  Digest,  by  Joseph  Blunt,  9th  edition. 
TIDE  TABLES  for  tlie  Coast  of  the  United  States,  by  A.  D.  Bache,  Superintendent 

U.  S.  Coast  Survey. 

Chart  from  Cape  Cod  to  Labrador,  including  the  Grand  Bank  and  Gulf  of  New- 
foundland, &c. 
Eastern  Coast  of  the  United  States,  including  Nova  Scotia,  from  New- York  to  Capo 

Canso. 
Long  Island  Sound,  on  a  large  scale. 
Chart  from  Moutauk  Point  to  Cape  Antonio,  including  Bahama  Bank,  &c.,  on  a 

diagonal  scale. 
Chart  from  New-York  to  St.  Augustine,  in  three  sheets. 
Bahama  Banks,  including  the  Admiralty  Surveys  up  to  the  present  date. 
Bahama  Bank,  very  large  scale,  (Pilotage  Chart.) 
Bahama  Banks,  Island  of  Cuba  and  Passages,  on  a  large  scale. 
Florida  Reef,  on  a  large  scale. 

North  Coast  of  the  Gulf  of  Mexico,  from  St.  Mark's  to  New-Orleana. 
Chart  of  the  Coast  of  Texas. 
West  Indies  to  15*^  North,  including  Gulf  of  Mexico,  with  Admiralty  survey!  to 

present  date. 
West  Indies  to  9°  North,  including  Gulf  of  Mexico,  Spanish  Main,  Island  of  Trinidad, 

&c.,  two  sheets. 
Chart  of  Guyana,  from  recent  surveys. 
Coast  of  Brazil,  three  sheets. 
River  Plate. 
Cape  de  Verde  Islands. 
NORTH    ATLANTIC,  new  Chart,  on  a  large  scale,  with  a  Memoir,  PLANS  of 

AZORES,  MADEIRA,  and  TENERIFFE. 
North  Atlantic,  with  the  curves  of  magnetic  variation,  and  a  Memoir. 
South  Atlantic. 
Do.        do.       and  South  Pacific. 

North  Pacific,  including  China  Seas,  with  plans  of  Straits  Juan  de  Fuca,  &c. 
Behring's  Straits  and  Sea. 
Indian  and  Part  of  the  Pacific  Oceans. 
New  Charts  of  the  VINEYARD  and  NANTUCKET  SOUNDS,  on  a  very  largo  scal«. 

from  actual  surveys. 
New  Chart  of  Sagua  La  Grande. 

New  Chart  of  Tapo  Cod  and  Mnf5sachusctts  Bays  and  Coast. 
New  Chart  of  the  Wiiidward  Islands,  on  a  largo  scale, 
iJew  Chart  of  Magnetic  Variations  for  the  whole  world. 

The  subscribers  have  now  published  Charts  of  all  the   navigable  world,  from  the 
best  authorities,  and  hope-that  American  Ship-masters  will  use  American  Charts. 


IDX^^XJDXl^G-  Eisrc3-iisrE. 
They  have  just  completed  at  their  establishment,  after  a  labor  of  over  five  years,  a 
Dividing  Engine,  by  which  they  are  enabled  to  divide  Astronomical  and  Nautical 
Instruments  to  a  degree  of  precision  -which  they  vrill  guarantee  to  be  equal  to  the 
best  of  foreign  make.  The  subscribers,  therefore,  ask  that  American  ships  may  be 
navigated  by  American  made  instruments. 


Chronometers  of  the  best  makers,  for  sale  and  to  hire. 

Sextants,  Quadrants,  &c.,  of  American  manufacture. 

Spy  Glasses. 

Night  Glasses,  nevr  kind. 

Aneroid  Barometers. 

Compasses,  Dent's  Improved,  and  others. 

Binnacles. 

Globes,  Terrestrial  and  Celestial,  16  inch.     The  Terrestrial  with  Isothermal  Lines  of 

Temperature,  and  Deep  Sea  Soundings. 
Massg^,  o  Patent  Logs. 

Ogden's,  Ericsson's,  and  Massey's  Patent  Sounding  Instruments. 
Improved  Compasses  with  elastic  centres. 
ABBOTT'S  IIOROMETER,  a  new  and  simple  instrument  for  working  the  Longitude 

either  by  Lunar  Observations  or  the  Chronometer. 

E.  &  G.  W.  BLTJIJT,  179  Water-Street 


NOTICE     TO     SHIPMASTERS. 

Just  published,  Massachusetts  Bay. 


Office  of  the  Board  of  Underwriters. 
New-  York,  March  20th,  1858. 

There  is  reason  to  believe  that  disasters  to  vessels  have  recently  occurred  on  tiie 
Southern  Coast  of  the  United  States,  in  consequence  of  the  use  of  old  and  incorrect. 
Charts.  This  Board  would  earnestly  impress  upon  Shipmasters  the  great  importance 
of  being  provided  with  those  that  are  of  recent  date  and  from  a  reliable  source. 
Blunt's  Charts  of  the  Coast  of  the  United  States  are  corrected  in  conformity  with 
the  Government  Surveys,  and  have  accurately  laid  down  the  position  of  all  the  Lights 
now  in  use,  or  in  process  of  construction  on  our  coast,  and  these  Charts  should  be 
familiar  to  every  Shipmaster  in  the  trade. 

ELL  WOOD  WALTER,  Secretary  Board  of  Underwriters 


Eoctract   of  a  letter  from  Lieut.  John  Rodgers,  commanding  U.  S.  Ship  "  Hancock,^' 
attached  to  the  Surveying  Expedition  to  the  China  Seas,  North  Pacific. 

New-Bedford,  January  ith,  1852. 
I  had  a  long  discussion  on  Charts  of  the  extreme  North  Pacific,  Behring's  Straits, 
Sea  of  Okotsk,  &c.     All  the  Whalers  say  that  you  are  right. 

COMPASSES. — Attention  is  invited  to  the  new  Compasses  constructed  at  the 
establishment  of  the  subscribers.  It  is  a  fact  now  well  understood,  that  most  of  the 
losses  charged  to  Currents  are  due  to  the  imperfect  construction  of  Compasses,  and 
to  their  deviation  not  being  ascertained. 

Compasses  of  a  superior  quality,  are  manufactured  by  them,  and  are  constantly  on 
hand.     Also,  Dent's  Patent  and  other  approved  Compasses. 

E.  &  G.  W.  BLUNT. 

Agents  for  Rogers'  Signals.— Office  of  the  Marine  Register  or  American  Lloyds. 

Novtmber^  18G0. 


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